<%BANNER%>

Quantum Gravitational Correction to Scalar Field Equations during Inflation

Permanent Link: http://ufdc.ufl.edu/UFE0022050/00001

Material Information

Title: Quantum Gravitational Correction to Scalar Field Equations during Inflation
Physical Description: 1 online resource (73 p.)
Language: english
Creator: Kahya, Emre
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: astrophysics, cosmology
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We computed the one loop corrections from quantum gravity to the self-mass-squared of a massless, minimally coupled scalar on a locally de Sitter background. The calculation was done using dimensional regularization and renormalized by subtracting fourth order BPHZ counterterms. We used this computation of the self-mass-squared from quantum gravity to include quantum corrections to the scalar evolution equation. The plane wave mode functions are shown to receive no significant one loop corrections at late times. This result probably applies as well to the inflaton of scalar-driven inflation. If so, there is no significant correction to the phi phi correlator that plays a crucial role in computations of the power spectrum.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Emre Kahya.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Woodard, Richard P.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022050:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022050/00001

Material Information

Title: Quantum Gravitational Correction to Scalar Field Equations during Inflation
Physical Description: 1 online resource (73 p.)
Language: english
Creator: Kahya, Emre
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: astrophysics, cosmology
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We computed the one loop corrections from quantum gravity to the self-mass-squared of a massless, minimally coupled scalar on a locally de Sitter background. The calculation was done using dimensional regularization and renormalized by subtracting fourth order BPHZ counterterms. We used this computation of the self-mass-squared from quantum gravity to include quantum corrections to the scalar evolution equation. The plane wave mode functions are shown to receive no significant one loop corrections at late times. This result probably applies as well to the inflaton of scalar-driven inflation. If so, there is no significant correction to the phi phi correlator that plays a crucial role in computations of the power spectrum.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Emre Kahya.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Woodard, Richard P.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022050:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101106_AAAAFW INGEST_TIME 2010-11-07T00:50:57Z PACKAGE UFE0022050_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 6901 DFID F20101106_AABTGN ORIGIN DEPOSITOR PATH kahya_e_Page_39thm.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
01f3bdd03cfddc6341f9661b3c413ea9
SHA-1
ec6d98e635819d28a9b10bb961a600a40a36ba25
97115 F20101106_AABSPV kahya_e_Page_48.jpg
bfbf61b3400745433a564ad10dfa649f
764fb954ffe0d5d65cf59f3ba677df4501df3383
25271604 F20101106_AABSZQ kahya_e_Page_29.tif
81dd1c365f8753cdaebd04ce3c24fdf1
a9a82779d6e56e3285c97a3699b518dd3ae6534d
44403 F20101106_AABTBQ kahya_e_Page_38.pro
523c2c3f077ebfeab739e2f501faae2e
61ab28a472a3f0287c9eac95bf21b7100387f972
6534 F20101106_AABSUS kahya_e_Page_10thm.jpg
bbd384a5528f282bf31dec364fb623c4
b70780070c5f2e52e7c83a54a2795daa8ab1c7d3
29272 F20101106_AABTGO kahya_e_Page_40.QC.jpg
b0f9115067269cc6274b9185708f52f1
da5df6a7f012cabcee537726ab83a90ffb5c2b98
22914 F20101106_AABSPW kahya_e_Page_62.QC.jpg
f2742deb21176f27d839433b29eda344
aed6c2666bf131120aeaed66e2de019cee1daf53
F20101106_AABSZR kahya_e_Page_30.tif
57b59d7f1b8c29be969ed748c7e4c1ea
23a8a67c6e9ea064c78b1ba4408b73ade12b0703
36051 F20101106_AABTBR kahya_e_Page_39.pro
0f24e54c0c9d098d7c54f6bd0f0ef257
3ae2a4e6f2dbe8a5f44b0a4f78976ba5c76e8c36
689377 F20101106_AABSUT kahya_e_Page_65.jp2
5988841dde84e80bd68fcfdf8f586c69
fef29cab74f8520b37e61431cb4c243aea36c2fe
7263 F20101106_AABTGP kahya_e_Page_40thm.jpg
fc6344dbd03410adba8dee84d7bc9389
9bab752b65f0432c0e2d35e15e67b48b6ef74593
4797 F20101106_AABSPX kahya_e_Page_63thm.jpg
15e70bd6a1f30eb41fc92c1dca9d532c
8a49f08f9ce6d482732e8ac0a959c71b5f35f3c2
40631 F20101106_AABTBS kahya_e_Page_40.pro
4db7b042e0cd8811fde202078678eaf3
b02fb0bb9ec1ef695fc287c71e665d26781fa06c
24270 F20101106_AABSUU kahya_e_Page_42.pro
862e240c49c7a09fea1b4a456d03e2fd
d6bfb415cca4e03bc0f0f2606e073ecc8d76e299
23749 F20101106_AABTGQ kahya_e_Page_41.QC.jpg
e07a90dbd217b74344872ebd2c9e5fbd
e1bd235798a898d541ce26b65d419d664ca1bdf4
1632 F20101106_AABSPY kahya_e_Page_51.txt
42d06db527e3a522e2430b2c8f624f28
50cf73bce2f177c9f4ce04db5f8d89c8ea6cff4b
F20101106_AABSZS kahya_e_Page_32.tif
b0d55c40ce780021b873080b97094e7d
350d0a1792c8021aa02c25c22e6e261691095c18
19859 F20101106_AABTBT kahya_e_Page_43.pro
97a5487adcacd79453f2fbab4872a301
57a9b60e257cfabe3697c400b8c59017157abc63
50194 F20101106_AABSUV kahya_e_Page_69.jp2
b6cfb1142f2aa98d59937d64da6d0dbd
64a2bea0b364d6fffc1957f830b675428408cfb4
6636 F20101106_AABTGR kahya_e_Page_41thm.jpg
2cecbeecc23fc06a0aa4126ede46db88
ae55c048176264474f129a84a1900f2efa8bd88b
5311 F20101106_AABSPZ kahya_e_Page_68thm.jpg
c08656fbf45ee55412b19a9345394bb7
c2537463cff4a9e198ea37bbaeba5eb472d74cec
F20101106_AABSZT kahya_e_Page_35.tif
7c6c5412827bb2babfe87ef9284fc915
9c7c9c9b65e1b2073041fc2732887945ae1248b9
46343 F20101106_AABTBU kahya_e_Page_44.pro
f0e29e70a36bf1d5da040e0d2a7e7ec1
5256dd5cc36a01c93e3575e67ef7f463fca2a5f4
691455 F20101106_AABSUW kahya_e_Page_26.jp2
4e8967698a3dc0fef7bff1a0d9e43bc5
fd7faad854732bf4f6e2fee5bac4ee59f05dd70c
15091 F20101106_AABTGS kahya_e_Page_42.QC.jpg
bb7333164397ddfe4c59e0eca2656405
2775297ae4ca0567e09a3b626e2daf22cbf61770
F20101106_AABSZU kahya_e_Page_36.tif
fe0f8afd93e97abe4613ed47353ed356
14884271dcce97eec3381976768c42d24137cf79
38900 F20101106_AABTBV kahya_e_Page_45.pro
ec482f01c2dc7ef8df2323f042ae3fd7
727e70515732e34048f728de081d2c787b5a469c
27778 F20101106_AABSUX kahya_e_Page_08.pro
ac3c38071994f4cf195b32e3c1bcd563
dbab16a066096dfed2a48babb377b473ff307657
4151 F20101106_AABTGT kahya_e_Page_42thm.jpg
6675ba3d6f57c8876bccc6ff400a298d
25542c9f05dd0d4a9f777f20f44fb5da376e5aae
F20101106_AABSZV kahya_e_Page_37.tif
a3889da06515d2abf268e7771c1ec553
48c5b065259b042877df639ee4723dd321313476
44550 F20101106_AABTBW kahya_e_Page_47.pro
ee971f33ce868687978f5937656c26cc
0397f38359f25c827305477d8a2a7549ba03bd3d
71118 F20101106_AABSSA kahya_e_Page_19.jpg
f569b1661e37a93cb377ae12613ed4db
9cd57b09e01cc9b553f8c5f9822a28ef75f1d0e9
74000 F20101106_AABSUY kahya_e_Page_57.jpg
5b73ce2cea1ea1b43c48f1e75dc51eb9
47498d345455ded62dd1afcb73f649ce01132511
7339 F20101106_AABTGU kahya_e_Page_44thm.jpg
62a0b0ba34f880093c8ec033a7e65808
8de3e61da9b8d3bfbd162bcf16ae486e12b64177
F20101106_AABSZW kahya_e_Page_41.tif
cacee4ab2642d0ae3ba07efa892ec91f
e23d9d49eeeb4e99b4b1f4a48249b93d12a40ac6
50079 F20101106_AABTBX kahya_e_Page_49.pro
abfb1ad6787d4d0b265e95e316cc53bd
c5d0dba6edd465225f67227fdf9c4c81a21b6f34
2171 F20101106_AABSSB kahya_e_Page_70.txt
408160417b29b581bd58a632c771d138
171e285af59e268332f149e354db9a275b085f62
4096 F20101106_AABSUZ kahya_e_Page_02.jpg
35531bb7cf619fd77a3739e7b98a7e24
448b18a6dff24a359b5bc17bcbca2f7991ac5a70
28352 F20101106_AABTGV kahya_e_Page_46.QC.jpg
9bf625a5f2d74fc92a1f9c3fd358ba93
95ee588dacdeb1fe57c41c0a0d2b8180b0aea2fb
F20101106_AABSZX kahya_e_Page_42.tif
2bbbfc69fb4b0581ec5fa0a72121827e
9109b6628ef143b3eda6cd3787a32c2fa76b83bd
43284 F20101106_AABTBY kahya_e_Page_50.pro
556bb7e8562ad053c7c49169916e5c78
d4210e67d4696ea521d4c4b1f948409a238ea868
38540 F20101106_AABSSC kahya_e_Page_41.pro
881a989788a485b79d87047acfdf2778
8e7b18e9f233a7b4935ab9c9c505f8aa579bdfe4
7233 F20101106_AABTGW kahya_e_Page_46thm.jpg
e5e1573ea148e7716fa9764c457d65e7
086c1c44a2b3991ac7b5885c403da7fc7f21b3e2
72828 F20101106_AABSXA kahya_e_Page_66.jpg
927a7ed58a3ea3423f2ce119fce53655
5329ff1dded6646dbdf331f33968f8dbaca92702
1053954 F20101106_AABSZY kahya_e_Page_43.tif
5b281670ccf250d7e7a4a015afb14cfa
ff34d702fdcdb074024672c69e6696e1f32e5ab5
19498 F20101106_AABSSD kahya_e_Page_72.jpg
8d59db88a7dbc6932cfeb11d8754be95
df54bf67d8c7567ba93d8945a8a4e1b8ab028bae
8030 F20101106_AABTGX kahya_e_Page_48thm.jpg
d661c7aefaad541571c26fe6fefa0346
616224fa6fec28883820410084cdee97e4e2c444
54790 F20101106_AABSXB kahya_e_Page_67.jpg
a6e5072807692fdf1828b182f08815d8
a4c7cfd439cc071c4e659b9b2e055e16b75eab68
F20101106_AABSZZ kahya_e_Page_44.tif
e4277a50376351e9e5e444ed18733213
544cc4f7d7259d31c7fc92456be897d4ec95b7e1
31815 F20101106_AABTBZ kahya_e_Page_51.pro
14be95d03cc1150072782bed402ebaad
b37517731e6ea4f5443c9b954f7c2927d7173eb9
488 F20101106_AABSSE kahya_e_Page_02thm.jpg
8c03b3acca2fe4ad1b0caf7cc92b3db3
6c3181581df3d0bfc9e12252cb570756a7eb0ac1
29561 F20101106_AABTGY kahya_e_Page_49.QC.jpg
b905df7ec577ff2f555461f52ebf5414
930cb929efa80bb1412125aadcef6a95a0ae425c
F20101106_AABSSF kahya_e_Page_08.tif
dca7795e365769c54aa4252bcb9d6698
cec51cfc569478280ac0c0c671c3733baf7ed0a9
109422 F20101106_AABSXC kahya_e_Page_70.jpg
cbafcf1dc719d1d5b9d98311cd6a997d
5c7e6089a8bbd5d1ac048ee36c2c30dffc7398aa
7191 F20101106_AABTGZ kahya_e_Page_49thm.jpg
1395511e47d29405de3dae73a2f686d3
6a9a27c55c1c788e1033263502bd69c34cef93ac
1741 F20101106_AABTEA kahya_e_Page_52.txt
386967271616cc2ad9bf61c56e573088
208647f43aa4e5f07cac4c7c0fe812557db79017
F20101106_AABSSG kahya_e_Page_58.tif
725cecf262841595e389e33de452e0f2
4c2750a211c1b62e730573484667c221047e7f34
105646 F20101106_AABSXD kahya_e_Page_71.jpg
009d0bd1c9012b9db99549d368c91ab7
cb0a3c825d791c51ac589df986de094ce79976fb
1738 F20101106_AABTEB kahya_e_Page_54.txt
0a3ae098ba9d178341e244a4ae7d0473
7864957bebd0bd6a89538a409338473fe3065e9b
674351 F20101106_AABSSH kahya_e_Page_52.jp2
e53ca08a80c9de09f926c9e43848c58c
8d8058d55d5d33d3eac8e9e104d51c025fb08d6f
24350 F20101106_AABSXE kahya_e_Page_01.jp2
233b6a5efab5cdc45a80629ca7af9989
10d403e3320dacd3c74257e4060e4f43f686cb48
1265 F20101106_AABTEC kahya_e_Page_55.txt
c4580934d4e0100e06d402a18a486c71
40161be938960a5a68237e91c7c0a8e897780b95
68010 F20101106_AABSSI kahya_e_Page_13.jpg
d44c05186c182f07350f90adcf763a18
ed61a09fdb891899414ee0149db6426df91c161b
5481 F20101106_AABSXF kahya_e_Page_02.jp2
e1115083a489fd8fb9eeb0095a9a6246
e5a77829f294b2cd9001430f5159b1dbe2699938
2188 F20101106_AABTED kahya_e_Page_56.txt
ccdb4ccdcd92abd7fc58136f39b882b7
087442668e4da3dbc95ebaec7909e545f90c496b
F20101106_AABSSJ kahya_e_Page_40.tif
1d80146709d2bd4d0cb1d91a2a41ce3a
15ea4541d356689131df4f16af25b7be2f730aa1
1051966 F20101106_AABSXG kahya_e_Page_05.jp2
b56dd565287c262ac7daceadd1134ad2
35544b7f7edf7a075a872f4a5575c34bb1abf86a
1901 F20101106_AABTEE kahya_e_Page_57.txt
0c7270456063420e37a892ed4a9bc0ef
cf33f0dbe511a1da864b23ce7f8f139659d51885
7258 F20101106_AABSSK kahya_e_Page_47thm.jpg
2f9f03160361a821a97c954697801039
823d184ded73ffc7d30fc2447d5abca9d765a419
62214 F20101106_AABSXH kahya_e_Page_08.jp2
ae9de27ad601f2f3aea2f92917e7702a
998d4aa36bef0c33623563826176536acd6857c0
1509 F20101106_AABTEF kahya_e_Page_58.txt
85967df148c8ed68c892ca6046d46daf
4acbbc386913efea455a62eac039d5f84bd8fe2a
1051979 F20101106_AABSXI kahya_e_Page_09.jp2
503fec9f2e3bc451d7a6a8d6c6001c18
d1d18632b6454e031521f953a25ece065b497129
2117 F20101106_AABTEG kahya_e_Page_59.txt
67f2d0e31f596230871bd3a0f8d8fed5
ab1c1a1bd31950daa3d053372fdf6806860b1b4a
64045 F20101106_AABSSL kahya_e_Page_58.jpg
12dbf7de907f33f6901b43f14203e3e6
1c8b79214e0757f70ec3761006f0917faac8c52b
989074 F20101106_AABSXJ kahya_e_Page_10.jp2
90fe82d984b13b707451b82740f6fe9e
3b677057e89bad406090b507ed0e791e23502e2e
581 F20101106_AABTEH kahya_e_Page_61.txt
68c6abe6c09cb3deef51d23a8cc5f8bd
7b29f518ca23fbda4e07021e277bb3fab2ed381b
1594 F20101106_AABSSM kahya_e_Page_45.txt
145d91dca3c352213c10b944383fa941
f9eaaa3645677b416c885f090499a63f0f267d2b
74504 F20101106_AABSXK kahya_e_Page_11.jp2
53c942c82ee20e318daf84db44ae8a58
52212e7b38e5d95d8610bb91989708bfada9255f
1482 F20101106_AABTEI kahya_e_Page_62.txt
5f9107efa51253a7d835c18a3375d2f2
14b1f5f215286dd0c6bac415ce015bfd88d4c9dd
22178 F20101106_AABSSN kahya_e_Page_29.QC.jpg
9475d0a02753904aa4d2c695ab764c4c
385cba2fab589c3c2f749ed65bf19818b64c4cb8
774302 F20101106_AABSXL kahya_e_Page_12.jp2
73f866c920fde093642fc2008f6e9b00
623777760e6aed48fdf68001d0288b658d731974
1139 F20101106_AABTEJ kahya_e_Page_63.txt
936b20ce488b4fe9af9cb1711cbfad87
3244e737c0116c15ff863661eb3801a52716b9a2
81030 F20101106_AABSSO kahya_e_Page_36.jpg
223ca5f76a7a372f601cc36297dd6cfd
ea21b151f79e7861d4b6409201175fcfa19cac19
619200 F20101106_AABSXM kahya_e_Page_16.jp2
d8af9360423e507153f077d5d62799d4
3769ed79b541d79ed9d35f7aeb9737038a254b28
1537 F20101106_AABTEK kahya_e_Page_64.txt
f9345475e986bc9e8853cbbc2b553dcc
772d17250986303309fdd1a6306f0bb15c9ed0f3
43746 F20101106_AABSSP kahya_e_Page_46.pro
9d7fe90bfbc8e02fca217b478d755015
f8929f2b8af72537c898007d4793eb9ce138dd2d
715201 F20101106_AABSXN kahya_e_Page_17.jp2
a52e456940b6f20c3402926197b2422b
f25abd1c9b0e3da196e77beaaf0f733da78c58bb
1591 F20101106_AABTEL kahya_e_Page_66.txt
aed113dfdaa70c944a3e29d26c998e29
3d095fc9e25d320b58c811f02441038d188837cb
103090 F20101106_AABSSQ kahya_e_Page_53.jpg
327791ca3fa44cf7230b1c77b4d94cd6
d5b3e836a367cc30ad5831178873fac3a64ea4e9
712949 F20101106_AABSXO kahya_e_Page_18.jp2
de731a7eefe5fb4ffba29049be0c9af6
8f5e2905fd136796cb3ff00a7f43dd340eb4a09f
1160 F20101106_AABTEM kahya_e_Page_67.txt
bd0b9ae3eed198e1094876c8c2304602
84e659ac28713654be2ded8baa25fbd8b1f6966f
54354 F20101106_AABSSR kahya_e_Page_63.jpg
5b1e7dca9466b629ef9a4a7322f208e7
597c0642d02d25bd91106eb7ff12ef351bb0ad92
762388 F20101106_AABSXP kahya_e_Page_19.jp2
617565b937e589576d28cac73942fae6
76bd908b7b802e426059c73e1637f2739947fe5f
1220 F20101106_AABTEN kahya_e_Page_69.txt
f41232d363869be4c6f8cd39e8b3e5e3
4f6163064cda1b580ad3f6f235223c6b69c9e639
1028313 F20101106_AABSSS kahya_e_Page_44.jp2
e52467f56511834958a751e2a64e73ce
e4bfeafde702989f178ebc7fdff88a1dc5d216e1
2051 F20101106_AABTEO kahya_e_Page_71.txt
e0aeaa673412ec4930ca84b0bf19787c
b1080d932db55ca0064d84151e92f2af7a78a4bb
1051950 F20101106_AABSST kahya_e_Page_06.jp2
571fc7cdad423a95966e5490826e7d9f
136b786fff6012ef54b58f2375f5daeec7ec71e5
928731 F20101106_AABSXQ kahya_e_Page_20.jp2
b7d838bbd6c9023f94b81346b63052e0
9da736f1857b1c6821b1387500bd9d19daee5255
339 F20101106_AABTEP kahya_e_Page_72.txt
719542415a904348d07659c89ec26d74
d65ce8ec413fbc5ea0634b9c51ea393ba77bd3ec
1924 F20101106_AABSSU kahya_e_Page_40.txt
bed00272e129e3b60e969f1b0f8e46ec
02ffa4700b2fd2a28aa19bbf4f91bae6818a28b0
610628 F20101106_AABSXR kahya_e_Page_21.jp2
a6ced71d4a7a8f58d5dc85ed194b562e
b2be9b95ba1bb59ec1f1ad192f4ec43bfb83f5bd
1943 F20101106_AABTEQ kahya_e_Page_01thm.jpg
b18f3c150c1eadc9bc125531e7e2bc3f
1ce3c6dbe79bbd57f8441de269fcbb476d735261
1051944 F20101106_AABSSV kahya_e_Page_23.jp2
2b3ddf1c5e3fc56ec07797a5bd1e1ae0
78b910db3ebd8ae3145aecd343a3dd05f56b7c3a
776833 F20101106_AABSXS kahya_e_Page_24.jp2
08af54c1e6bdf7884b824827f97b714c
2ed68f12224431f912e68713b6e6c0609cf7799c
3649 F20101106_AABTER kahya_e_Page_04thm.jpg
37dc7586126ad5c1acbca292fb1a873e
73dc544c1855e870b1411d80d3185c7514265309
29119 F20101106_AABSSW kahya_e_Page_33.jpg
0bd79f8cd4794353df3785ffc66a8e52
4c1d69471df9fe660391e05daaeb9f252ae5be1c
545948 F20101106_AABSXT kahya_e_Page_27.jp2
0dd1b33ca6f1274c84919a11f0fee39d
0f579953279efd800ef80a530f93d5eb46f986cd
24327 F20101106_AABTES kahya_e_Page_05.QC.jpg
3fd8c7bf7e373e6c44f12b8ecd2b153e
43c7c43d47449b0bd300b38424352194329e6079
5017 F20101106_AABSSX kahya_e_Page_16thm.jpg
c56801657917fcffc0a4a6a622231b0d
9265e62267725a6c6bb2539cbb07f636bea31eb9
67075 F20101106_AABSXU kahya_e_Page_28.jp2
0275bbc7db85207052824a47a7f36088
c7589f0b3bf14d5908d03e06f7178a38edbacd2f
5149 F20101106_AABTET kahya_e_Page_05thm.jpg
721b17d49834968d86f96a755d65b5f4
0037dceb4425b5b77a30f5ae60619358889f08ba
51338 F20101106_AABSSY kahya_e_Page_48.pro
c974f123a339c4b8a5cfafc13949eb79
bfc963c1c97a87ede1347d772f46a9e6865ce0c0
696454 F20101106_AABSXV kahya_e_Page_29.jp2
83222f2679bc9af67f70ab29d80be916
1125947d4a0a2dc303ccdb2d270595082d437aeb
700837 F20101106_AABSQA kahya_e_Page_13.jp2
f53e6aeac714398f8e657448eb707df6
e36de9b7b4e4d42f9c0281a462ce923af9739b2f
34244 F20101106_AABTEU kahya_e_Page_06.QC.jpg
c7a88f4f4fbf6ff4ef7ddf5b0343e396
ab8fe59ecce4537c471a4ca5d63e2ab40012a4a8
52780 F20101106_AABSSZ kahya_e_Page_56.pro
4379aded312e492153bde426a89bd44a
1a3e4b8a00bd24a4e83510eb8b9bd1f3851aa2ff
975320 F20101106_AABSXW kahya_e_Page_30.jp2
a90e1e2158c09b19145b7266f59f89d9
cb8eea92f5a765555d438ef6f51c0f5b8114ae45
18149 F20101106_AABSQB kahya_e_Page_55.QC.jpg
9ad66f0ac08fd9996b2dfb894a41e34c
f9e216950e86b0e8bf2faa17f3435e8fae82b8cb
7480 F20101106_AABTEV kahya_e_Page_06thm.jpg
94ebcf8b3c6ed41af2f8f58c3cb32bba
967cfe5abf26584f1da7437dad3321ec6e4196ea
662975 F20101106_AABSXX kahya_e_Page_31.jp2
29de735cbb382d5e6dc8469453a47b44
9cd917cae2ffb5ccadf44cb41cf7dc19ad06b854
95505 F20101106_AABSQC kahya_e_Page_23.jpg
b81c0d06a9fbb66d352fda730fa23160
1197a6bfb046be5cbe31e30ecb53c8bba7e93522
4294 F20101106_AABTEW kahya_e_Page_08thm.jpg
7d35bbf967e3d1d646d8efbf844a9d9d
665401b185876e4ed47cad2bdd39c9ac524f8855
F20101106_AABSVA kahya_e_Page_19.tif
1e5fb8def8c726826681aba2d38cd461
c60ca800e0aa2583e4c0a41bb8fc316665e8769f
605467 F20101106_AABSXY kahya_e_Page_32.jp2
5b62755001a5f2a78a9029425349f437
be69dd36dba8084ab0071ed016297b59b8d649e4
1897 F20101106_AABSQD kahya_e_Page_35.txt
05ed0c5499ff53bfe02cd48222b7fd86
ca69885220e8433f184f40e7a1c1e5124b02881d
34304 F20101106_AABTEX kahya_e_Page_09.QC.jpg
35acf3ff3e9cdedb624f11bddcac979a
0cde1cd54fc4082be123e79237868201ee9deb7f
89542 F20101106_AABSVB kahya_e_Page_47.jpg
915e089e3a0e1530d07ef0ae982ad202
b8ff892103886a68ca6af8ca2bf1e54f8f292a43
23272 F20101106_AABSXZ kahya_e_Page_33.jp2
263da533e6617a9bcf130f16a975226f
82c2efe88a9e75208342f0bea42d51fe66239ca8
6459 F20101106_AABSQE kahya_e_Page_72.QC.jpg
050a22d210b8e5608017768881850b15
f0b8c4ba7685b0818e95c789b25e137b22a8750a
6172 F20101106_AABTEY kahya_e_Page_11thm.jpg
6384df2f6841bf2189f8128616bfab20
42cf3b1df3b94c03147861d3db10b6d39b15f239
59738 F20101106_AABSVC kahya_e_Page_25.jpg
2979dbb029a8adbcf9baf5720fac0774
d31ae6046649cec670dc498ed8225e7f3300803c
110912 F20101106_AABSQF kahya_e_Page_09.jpg
0308718063e735aaba9aca3b8a9b7b6a
4bb6a1dffc64ad37f4559c12b9e62785ba6ffd46
25500 F20101106_AABTEZ kahya_e_Page_12.QC.jpg
c76169ca8203e10526c14abc825c8142
f09941442a6bc9fce848afe8ef36ad0809d78e6d
50042 F20101106_AABTCA kahya_e_Page_53.pro
60fa1a8216c824b992a1c56531e53794
a2d8fdc120d6b3d05c54e24a3d791a83f52071d6
47385 F20101106_AABSVD kahya_e_Page_34.pro
e4fdc0f394ead1e7e74c167267c361ef
3a1053c2deb028d6830ed075ba14d7702006c7d5
1915 F20101106_AABSQG kahya_e_Page_11.txt
8523d29d36b784be0a938542c245f0fd
e4fc28f3d0c10cd36a6f925da8927d7fd805f96c
23952 F20101106_AABTCB kahya_e_Page_55.pro
76ec53174e8634b660b3d55708dd7b1c
2efe0f3f6e4923e3cdc8ac5e4d598224bc31bd44
1051887 F20101106_AABSVE kahya_e_Page_34.jp2
3430602b377193ae4f3ec7ccda13fcfb
d2ed0991664cb29533fdd982dadd027482fe0a06
F20101106_AABSQH kahya_e_Page_71.tif
00fd976e724ca667934174e40c378c3a
b835ca71756722b47140bec05fb725f9c946e9d8
38638 F20101106_AABTCC kahya_e_Page_57.pro
f11c7ce743b4eb051903ab8fe3248cb9
fbd256a86ba726403c371c653cc88e6ded300f3a
68715 F20101106_AABSVF kahya_e_Page_17.jpg
8f8c2b4ade4a668d83773cd05d004702
d619ef85cf038b6eb340830151c2c9daed23c7f6
60710 F20101106_AABSQI kahya_e_Page_69.jpg
beb13fb70dbe18d30d64d53dddd6ec80
20b25d34875183e768596808767b98b2534cfa85
6961 F20101106_AABTHA kahya_e_Page_50thm.jpg
bd713b4c5a684dd73047e54cb78a913a
6c29b48c57e6b25ebebedd2ca5f30dc6e8941b43
27734 F20101106_AABTCD kahya_e_Page_58.pro
2d0c240184c7b00c4ec23cb6819801ac
49a3b4ef36ec43c321b21d0c3272f050ec744c84
2042 F20101106_AABSVG kahya_e_Page_30.txt
eff3bf693052d8b8eb1e2a8a50b142a1
af74503a881bc97c314d4bca5c2826ce930329c2
21325 F20101106_AABTHB kahya_e_Page_51.QC.jpg
55cb72cff6968c4e1021205b5f5cd726
d350bcae181d9c4abd170c02831faba46410dcb5
49109 F20101106_AABTCE kahya_e_Page_59.pro
e3dedb4df787d233e2ee131ac57dfed9
9e9a441e67f1fbca77bfb0872e924f282cabc884
4840 F20101106_AABSVH kahya_e_Page_03.jp2
be6e3b174a1070f11c2c688e6e0189c3
f9f782b4a7d2752e5008454bc435ce1efe96aa27
46322 F20101106_AABSQJ kahya_e_Page_67.jp2
e8d313771167ace4122bffc75ab2ab9d
5b6d6a84261009cb4c1af05bbc495ba4697a5187
5860 F20101106_AABTHC kahya_e_Page_51thm.jpg
4cb7b5903f4327bd741ca6d260b7b78a
0488cd16e0456e397e4d1adc9e342ba8c36fb639
7399 F20101106_AABSVI kahya_e_Page_30thm.jpg
bd08121e9adac15c63bd007ad449bc43
9de23a64dc24558d4d7cee2a14c2dbca8a412db8
F20101106_AABSQK kahya_e_Page_04.tif
e21989fa9be5f9280aa7b66f04a6451d
5607fe39aa8685fe08ef60905247571fb0af9f10
13933 F20101106_AABTCF kahya_e_Page_61.pro
d2a942d80118c07f5b78679d1def3d78
412c7afdf3b0120f8189d7da3de1eeeb6fde20ab
22170 F20101106_AABTHD kahya_e_Page_52.QC.jpg
a78b3455297ce6091d709d3939efd0f8
b4655943298fe10a005296184c113496ced7d7bc
5811 F20101106_AABSVJ kahya_e_Page_64thm.jpg
6605dad20e5c2a386fff19982c62e593
3b4fec8ff36a08311b02caebb066c9d7956d836f
27765 F20101106_AABSQL kahya_e_Page_45.QC.jpg
461d18f79e9b78dc91c8ec40dfae55fe
888e61f260a09a2b9fd48c2c9819ab66f95dc942
33500 F20101106_AABTCG kahya_e_Page_62.pro
67c75341cb82c7182e6c3d9573ea7b2f
5f7535219c8d90a0039bd0c1981e140608f6998e
6605 F20101106_AABTHE kahya_e_Page_52thm.jpg
6e3f1eee4d029f1bf4652d5cb0484b82
7e052500d28d115f120dcb6a91db833e9a045343
27070 F20101106_AABSVK kahya_e_Page_37.QC.jpg
938ee3acf239265318a8a7cea95eaff6
2892e939fb604f4756b46da5abcae9d99eaa7ddb
71867 F20101106_AABSQM kahya_e_Page_65.jpg
e9d95ff9462c9a2446f195106010027d
175dc04a61296d0b0b42bde6f80df713b7f32605
21971 F20101106_AABTCH kahya_e_Page_63.pro
915dbf1d6736396d4ce868071870a863
6bd7cb0014a4ea176e6e4f94d258c061094c0798
7702 F20101106_AABTHF kahya_e_Page_53thm.jpg
8ca3593b18c8e4d3a303f7b17e8c4a33
81aa3fa7d940db1bbce946cbd8f836c5b7db730d
31242 F20101106_AABSVL kahya_e_Page_48.QC.jpg
d0c493077240e619d7cc05d48a66c9fd
2176e25ee989e3094dd22cc2eb73e6dd25d3a69c
23739 F20101106_AABSQN kahya_e_Page_57.QC.jpg
c643d75fdca8989ad340b108c0b41cee
9ee6d4af3305e3593a86e5e8af324475046a3c63
30763 F20101106_AABTCI kahya_e_Page_66.pro
6ade013049c89c15462294c1e11bfcf9
306a37e2e40229d9eae46f399acfac1080545c96
21791 F20101106_AABTHG kahya_e_Page_54.QC.jpg
f23ac75832ed9f5dde061e3ab8467d47
f6e3f3565343d5033c6cc2fb079bf8d8c150f9c8
1004173 F20101106_AABSVM kahya_e_Page_49.jp2
75e317ed09fd1eae049c6285edcef1a8
cca27cd70bb9cdf8bedb2cebc0b3e8e21e0f584c
50032 F20101106_AABSQO kahya_e_Page_04.jp2
0750e8070776e64196d425683c8b813a
be54ee69e3d3122585d0cf05fc19f098d0f2bb6f
23116 F20101106_AABTCJ kahya_e_Page_67.pro
59afa69d385787c3f88ac69202604656
ee227ec049c885f94ea09da8a373d7c9a54299fe
5793 F20101106_AABTHH kahya_e_Page_54thm.jpg
a91c9e0a6fda22299d9e69417e8102ca
0977ae1bdebbe07369495c0050592d2adde65bae
798889 F20101106_AABSVN kahya_e_Page_62.jp2
2dc9c57c761e8551d0ffadf1e3c9676b
55b84cd52532bee17d2c2b9467150f42a90f317c
F20101106_AABSQP kahya_e_Page_26.tif
3b8cb0ae5aed939200c9072502fa76eb
4ea9d84604a06a746ffab8af048352b4c4da8c35
30501 F20101106_AABTCK kahya_e_Page_68.pro
f562c1ba588f47b5f873583f7641e313
3d6e4ffab3a089d8e618693c8d75a2288e81be84
4677 F20101106_AABTHI kahya_e_Page_55thm.jpg
6f4289849e645d2ff7abbfaf1ecdcd94
ef9f3554817d03893263cb9d1336d9c405a5c6fa
F20101106_AABSQQ kahya_e_Page_59.tif
7ab415cf905876b069ce95ec948efb29
cdc8b86c06bbbd39eb86be8068354ada7d7a271b
25125 F20101106_AABTCL kahya_e_Page_69.pro
cd4bae3a53f7940805356e6240ed8f4c
05b3ffddcb9e1e2eaf12daa74ec43d16a4c3c35e
32531 F20101106_AABTHJ kahya_e_Page_56.QC.jpg
5095a368087bd5cd5b7d5c3f6082768c
2d178c60277d4ee927f0476f3186fef6b16339a1
110470 F20101106_AABSVO UFE0022050_00001.xml FULL
0f7d583cf4340440f75bac04f2d41e07
328db38ed5ac0f9fd444f9fac987cd8e162e27fe
7660 F20101106_AABSQR kahya_e_Page_71thm.jpg
116e7abdc95a41abe928b0d62ec55b75
eb4e30d26d8f35f57dbe720c1f009e6075c60544
53886 F20101106_AABTCM kahya_e_Page_70.pro
805b567d5c156fd8eccea5df4ef59c58
ff9a80b5ee03ca3c8dfd2615015c9b2e9c3f6d8c
7750 F20101106_AABTHK kahya_e_Page_56thm.jpg
37fdd22db65059704180756c74d91a3a
d56e542ece7bf68b801d393bf919afa5b1afe451
1446 F20101106_AABSQS kahya_e_Page_02.QC.jpg
8402f480d1b15c695257a97e27e54d80
e04667ebb2e1878b6ca3358db44708e50fdba801
51548 F20101106_AABTCN kahya_e_Page_71.pro
e93d403efeae582f872f553639b354ad
ed12450a6a23981fb9afdabb63a112a864e65d9f
6475 F20101106_AABTHL kahya_e_Page_57thm.jpg
4d268e7da2a79a21c78086ed51ee8759
e133f3fa1b11c6ca85ee365dacf0242021c93bcb
20273 F20101106_AABSQT kahya_e_Page_15.QC.jpg
2a9ae664740bb2d95df2e42ce5b5e6ab
2980387926c3166c0699484cfe4ed1d163ff5318
10864 F20101106_AABTCO kahya_e_Page_73.pro
5244136e97cb9aabfd345675b3e160c6
ce3bff9aa770d80861eba1b38ce6491b5331292d
21410 F20101106_AABTHM kahya_e_Page_58.QC.jpg
275e1c11b1ac0d27d086132007d8d5dd
c7d641c4a28885184091a95904cac02801e5bda0
3302 F20101106_AABSVR kahya_e_Page_03.jpg
efb739544bec5933ba4dd7037169c3e2
d1d432d109ebcae0f9d8b77df8a5c83a7510c7bc
85752 F20101106_AABSQU kahya_e_Page_38.jpg
dc14ebcc0e8b8ade27d91f21851f7f0c
3f52fe43fe938c542d060cc1ac2fe97cfe0e222e
475 F20101106_AABTCP kahya_e_Page_01.txt
f958d6c20eb785cecaabc56ab6d08941
b76c0087d0c58fa343e9f8b2cad3ccbc4120424f
31184 F20101106_AABTHN kahya_e_Page_59.QC.jpg
09ccbd6d54d07cdaf80115f4dadb50fa
0b8c8b7339d4923d326dec4d116e60ca4b5d21a3
47999 F20101106_AABSVS kahya_e_Page_04.jpg
334c7c959fa005b8aec8d153f0f4bf22
3c1266ad2c1a25335ecb4875f981d105fa756e01
632134 F20101106_AABSQV kahya_e_Page_64.jp2
87db37f33d08a097b7c731f1b8813ae0
3ae04add0f590dbdab85c7e8049ee83bb54c110e
91 F20101106_AABTCQ kahya_e_Page_02.txt
d8cda674a230d0fabdab413df348d9b7
5c6993866e56994aafa60688d7322fb775c90023
7319 F20101106_AABTHO kahya_e_Page_59thm.jpg
814660e8c4b8e585e14f6b584cf9efd1
5fce199bbd4d691666f1efc7b665e26ec8b2858a
77977 F20101106_AABSVT kahya_e_Page_05.jpg
41fa2c5c3968389f1654d5a2b82f06de
237457d38abda13c437834383207b5aace371e6e
13413 F20101106_AABSQW kahya_e_Page_43.QC.jpg
6108991b62d01036f07a881e02ffaea3
5fa0681c0b27f4400dec1493b3b2844242494447
89 F20101106_AABTCR kahya_e_Page_03.txt
5cc71c325e31781eefdc228d7c698961
2e8fe3f2c2fa5ce41c8d9473673b7ca9e88c56db
37529 F20101106_AABTHP kahya_e_Page_60.QC.jpg
dc4abdd3204846e2da20ed63f3777536
59afefa125c5e2eeee151df6a48815e92b83e9f3
110419 F20101106_AABSVU kahya_e_Page_06.jpg
9aa100b84eaa3bcfec8c81242b96f264
30d012e8819cfb8f737f3f3144386d971a9c1115
616854 F20101106_AABSQX kahya_e_Page_25.jp2
c54c3767778163c03be7b405fa477dd1
efa5f0058e1f6cf7d4945e29ca46c5c07d9db774
936 F20101106_AABTCS kahya_e_Page_04.txt
d198f69c65575f8fe61bbfc793edd56b
fdc4ddd512d4f05d8d949a08b218b7675b339257
9086 F20101106_AABTHQ kahya_e_Page_61.QC.jpg
bbebc79e5446e916588c896a783d7f8e
4b8dbb15d6257537a8d96649d64ffb9ba42db7d7
18436 F20101106_AABSVV kahya_e_Page_07.jpg
24f5403de09551c18767ff0bfe7b0d43
5f48b66095de71705b3988e906be75ae394ba5ad
22536 F20101106_AABSQY kahya_e_Page_68.QC.jpg
46377de0598936d3a80c3be200dc949c
7ba69c2b546061c27fe547bc1b64cabb3d5f9eda
1717 F20101106_AABTCT kahya_e_Page_05.txt
89f699633b1a06cab52569d43aee0958
48d5da71a2705cf2b41e136951b9e6d3eec6a89b
6498 F20101106_AABTHR kahya_e_Page_62thm.jpg
3b1fb3308a78f99330a70e01d8d77bb4
d8e8a10826598709c27b93b73b5995fd6b40e277
57277 F20101106_AABSVW kahya_e_Page_08.jpg
e7ffef0322114cd6843f70cf47e2e23c
0494579897ff4400f9b7f7133cf7e61b4d6689c5
94153 F20101106_AABSQZ kahya_e_Page_44.jpg
9c315216b10541755f9ded8950f55a20
6a700c61d17ccee951160303de44c4a5eef49194
1950 F20101106_AABTCU kahya_e_Page_06.txt
12213e30bc55963b33d85199660d7232
bfa29d8fd108612db700aeb9e3998d5559ae2f26
17289 F20101106_AABTHS kahya_e_Page_63.QC.jpg
ff3a1e11e56e6709c8d701041c1136c9
dff427338c21ddd1c011fa97c399755bbfc69d5f
89009 F20101106_AABSVX kahya_e_Page_10.jpg
cbf6849ad376ae6afdb1fe8931af23a0
ce957052b32d814452d9e33acc2f2e1a95d133d7
390 F20101106_AABTCV kahya_e_Page_07.txt
730a14437e9f3bf39a818b5ca09dedf0
2801a0809c90d606b618f986182069605b28ff42
20573 F20101106_AABTHT kahya_e_Page_64.QC.jpg
8258313c00553b2b097ed4b92b1f6bff
1d9df6cbc5f0c98df09456bb05e5253fb2756ba3
67680 F20101106_AABSVY kahya_e_Page_11.jpg
d2731a59db84f1319296ce0afa36a7d7
f2ac5bea10d1ff0104c6774769e674684c7f75ad
1303 F20101106_AABTCW kahya_e_Page_08.txt
cc87fd01e88679a2f25991ff132fbee4
da9cbde00d1018a1f2d64fd33b8b0abb95261559
84988 F20101106_AABSTA kahya_e_Page_37.jpg
7d4cf589fe895f13f21df39d40a4a7cb
4ec409810eed2eec5b387810828235eacbfeccaf
22819 F20101106_AABTHU kahya_e_Page_66.QC.jpg
48d4f2de8084c40696953e6460da56dc
c93d37498750ddf0b3ad43d4f383e4fa5e64d219
76490 F20101106_AABSVZ kahya_e_Page_12.jpg
70ec4b0bac0cbc105f0506758644d905
94106709f6b87472c765d00e60d9fe10489f99b7
2311 F20101106_AABTCX kahya_e_Page_09.txt
79d55ef3d728835e9e78a680eb027dd4
20e74c904ee43ced7185eac1654fbbd2cbf30f55
F20101106_AABSTB kahya_e_Page_09.tif
aa8dba4c235e16d635dd43c5707545d0
02d60b85ed44feb1f3a7963ad85499a62530e6e1
5733 F20101106_AABTHV kahya_e_Page_66thm.jpg
dfcf41ea0c2d93cdf8bda4ec87de8624
12c3003028e8c37f514e835f6a34edc588d3ecb1
1792 F20101106_AABTCY kahya_e_Page_10.txt
40ddb6f0951106fb30be8a8f647e74d7
6907beeb77be22ab97405346157712cc18b804ed
F20101106_AABSTC kahya_e_Page_70.tif
439c00e374ec1d0d5a32145632696f10
3b39aba93780e0ec5d42cac857ef0481493767a2
16797 F20101106_AABTHW kahya_e_Page_67.QC.jpg
baa3d727e6424baac056b383c68ca21e
2f246384da90ad6825e373431540f0b8c5d31747
908743 F20101106_AABSYA kahya_e_Page_35.jp2
bdc20e4721efd947112cff9ac20c9fd7
e7ceec4b17f6e1905b60bd1371ba6a412c79c864
F20101106_AABTAA kahya_e_Page_46.tif
6d168f319cfbd4d79af64195a3bce8cc
748fb01b03457e6ea6abd1cb90ef982214ba7d13
1700 F20101106_AABTCZ kahya_e_Page_14.txt
029ed83298109530fc0e891c1629e901
4a5743e9f43c4aa1d90837a8b3bb89b0bdcc549b
F20101106_AABSTD kahya_e_Page_39.tif
dc5950e998e413d7ea88e84f86b49863
ead26f397ba7ba36fe6fb5e2aabce33e70c6d14e
4032 F20101106_AABTHX kahya_e_Page_67thm.jpg
e46acba64df97c4455ce8a503183e185
d392e3ca16a025972e67b9d884b7551eed1712f1
897363 F20101106_AABSYB kahya_e_Page_37.jp2
cbd75e6788ead69f9d14fa29ae74e0ad
f7de88700bef2947f1bccb1d0faa7d9f3bb0054f
F20101106_AABTAB kahya_e_Page_47.tif
1c392b7c9e275dddef5fc137d660bb26
fb7139c392cb71cd99a8084df6037c734f54758d
F20101106_AABSTE kahya_e_Page_45.tif
944652d110290f09883f5b537f9ad08d
a9ed025a7265041de3faf5e933e75eb2bf65aa02
18611 F20101106_AABTHY kahya_e_Page_69.QC.jpg
d18ba3029eb5a04045eab34500b6838a
1275ecbd5bf6a0a97007a3ad5029e65b39b9dc1b
909852 F20101106_AABSYC kahya_e_Page_38.jp2
65d8867da0bdebf81046716f1ea2d1cd
7ade94b93476588dc0fa7b414d49b31e69e8fe59
F20101106_AABTAC kahya_e_Page_48.tif
688191306ea425ebd3950414be97307b
d7e11d7a9a3a5f4c57a71422abc54916bb870f75
8336 F20101106_AABSTF kahya_e_Page_01.QC.jpg
b6da17c1fc54289974d0336371ddbc48
ab32c21c0e8bbe1f9e1d7f0b2abb6a6eb547faf1
4754 F20101106_AABTHZ kahya_e_Page_69thm.jpg
8206f8a1cda085d0a2dc995da0667499
1e89cbcca4a6154dfec3405b1233f8f040821c7a
6506 F20101106_AABTFA kahya_e_Page_12thm.jpg
432701656edf45869c982d024350d987
06184139775cee95c9d9d362c677d15236887d35
817067 F20101106_AABSYD kahya_e_Page_39.jp2
d69a56d813f92920b00fb8d768f81a97
e31c5ef7fb4c2a7e3f6c9a37090f50d1708969f2
F20101106_AABTAD kahya_e_Page_49.tif
61f4f6a731487491a40566331caf470e
dff1318036cf224bbaf80ffbd4be6519fa5cd643
1492 F20101106_AABSTG kahya_e_Page_68.txt
2fa6b474ece31b309cf7d0c2025275e8
2e8277551d6391e4e7a0dffb88b635e620f34860
21468 F20101106_AABTFB kahya_e_Page_13.QC.jpg
3898c6ca6d7839fb948ae88d1ef9b2dc
0584b6e669efb6d9a486e2f752b07b05818e03cf
98688 F20101106_AABSOJ kahya_e_Page_59.jpg
343822da47c3c92a1a18c4ecf814407a
2ed1fb9850fdc628f8849f1ac6e849144a5ffd0f
920233 F20101106_AABSYE kahya_e_Page_40.jp2
6f37f06379e7c4368701dd772cb59277
54fafa6f6d12f2f95a3aa7cef9babb0251d35e2e
F20101106_AABTAE kahya_e_Page_51.tif
518e102c7738ce0b193caf7b6661fafa
08d2cab201676189039d004eaaaae87b845a8b2f
56938 F20101106_AABSTH kahya_e_Page_09.pro
eea0e85f6311996aeca6c08c557ad498
c6770a5d216ea98660c5a9e0fd5d3518ee8d5c80
20228 F20101106_AABTFC kahya_e_Page_14.QC.jpg
9cddaecf10e18a7fa129dde03371aa52
fd28bc5d38c90f0686536bcd36c084d1aacac09d
1658 F20101106_AABSOK kahya_e_Page_07thm.jpg
af07440f7369e5372fe2a37df2650dfa
320db15c6a757a1e5b24c50a57455b9fe536bacf
777124 F20101106_AABSYF kahya_e_Page_41.jp2
fd702303616371f6261580c0e7a69a37
060603af51ced8afc5e6a3e03a0a6fe69955255b
F20101106_AABTAF kahya_e_Page_56.tif
f9069a1e79b55e31fe5eae15026cab14
9af893750c2c48818bc802019280d80b9e667425
6828 F20101106_AABSTI kahya_e_Page_45thm.jpg
5a2dab021da89ca51c9da7cf33ea4342
8642c158b5e707246683ea195c16c91cea43ab47
5966 F20101106_AABTFD kahya_e_Page_14thm.jpg
57979430c4beb87b50270f025b8b643e
7677423218a96582f82a9e6513f1854e1536d4ac
15106 F20101106_AABSOL kahya_e_Page_04.QC.jpg
061d0fe8e3df914daee42c43fb06d781
a1087e3e3d81f3185a54e6333bdf1c2b82afe5b3
442768 F20101106_AABSYG kahya_e_Page_42.jp2
025c3abd95ff36e4b80a78a1a1cd41ca
c5c8351b236cc2ae7fb6cda18c77737bc3be75dd
F20101106_AABTAG kahya_e_Page_57.tif
8591c9fbb768d5dd226dcb95e9ecd564
4d7547bda02cb3a355ba47b47a46b490b51d1e34
32646 F20101106_AABSTJ kahya_e_Page_19.pro
92e05b1a736a4e1f77423ece10593984
d62cd969c27b4529805fe15e54bd47f55f573a94
5460 F20101106_AABTFE kahya_e_Page_15thm.jpg
e5087167625b9281b204c90171db9cfe
64aae2c63f88d55f484fa75b6914e996cc26df86
F20101106_AABSOM kahya_e_Page_31.tif
d6ca5a7551f4dd986eb45a7cbbabce71
3e3492fe067548f2354bc3e501f40ac933729d82
36737 F20101106_AABSYH kahya_e_Page_43.jp2
7cedc04e01b619e8efcf25f729e3faea
f6fd4d5bd81d24744eb29d6f74c8f40b9bc6a2b7
F20101106_AABTAH kahya_e_Page_60.tif
3b0d28092f3abcd8da0c8537a68ed3ab
163d80467f954157a04eaedc804faf3f00e75f7c
F20101106_AABSTK kahya_e_Page_53.tif
dd5ba2f8693f702e216dea147877b0df
81290554fc6c58674678c45a4ea0af3d2558ac54
18655 F20101106_AABTFF kahya_e_Page_16.QC.jpg
7d726fea431db38044c729e47ace10fe
3c138c927e7227ae2ac170fce4ab454d36ff7c72
940375 F20101106_AABSYI kahya_e_Page_45.jp2
2ce40d38e61a47469ec8ccd089c0641d
e049789315c3ad9e8de9c2371b7963a7e176edea
F20101106_AABTAI kahya_e_Page_61.tif
df7c0239ae79ca7b1c65d6750693f0ac
d2d3658c2fa244cac498408a2ae7c79c0fa68e33
32273 F20101106_AABSTL kahya_e_Page_54.pro
f2a57b3fc616271e92bc34d93419a919
dedba47da2f974a59f52b48c29f70fbc05c7a1b2
F20101106_AABSON kahya_e_Page_54.tif
19c48d06caa0117ed13213c249ba3f82
af198a9b595a161966b8a287f683dcf27bf54bd6
22316 F20101106_AABTFG kahya_e_Page_17.QC.jpg
60ed5d3e501eaab8571b19bbbe5ad22c
644ad0622d0135b23a4030a73cb1cba7659ebdf3
974425 F20101106_AABSYJ kahya_e_Page_47.jp2
7acdb73ac756de43777f3fcbc6f3b76f
c9f7e3a94834bed3203af70954565561309caa72
F20101106_AABTAJ kahya_e_Page_63.tif
465e716434d7e3c306ecfe608586cff4
43387eb6020160c58c03832a2e86f48ceef03b7e
550 F20101106_AABSOO kahya_e_Page_03thm.jpg
569e04b82df4956b406ecf4f8e0dc93a
aafe1df6953d14859c5b3ac846e0bb8ce613a30c
22599 F20101106_AABTFH kahya_e_Page_18.QC.jpg
ba66fbf771edad6493a8eb9fd36d51c1
b73d7892b682fda7009ecfd9deeb66692808d14f
1051976 F20101106_AABSYK kahya_e_Page_48.jp2
1845130915c38a51fb4c37f436138a46
db2b39c04992d087a3d0650e76f3d1c76984fec8
F20101106_AABTAK kahya_e_Page_64.tif
e4ae0aa3c8df03b9e6a0d1417af82607
f966d69ab6bd75867992010d16b3c17fe932de2e
668907 F20101106_AABSTM kahya_e_Page_15.jp2
09176fbb77f09305d9141728de9cc527
14ad56c4e3d523148708f263014ad0d3afd4a235
5289 F20101106_AABSOP kahya_e_Page_25thm.jpg
1f2b11fd22984aa38ddbaa38fc0fe0a3
c5fd92b9e471c45f26247d7291ed3dc6430b12e8
5983 F20101106_AABTFI kahya_e_Page_18thm.jpg
9c039f0c9509d6e30b68838072d6973a
92e3443ad6b61517b4755ed84bac597c357c8a20
904822 F20101106_AABSYL kahya_e_Page_50.jp2
23634a6498aadeaf023eba59030e6041
2c75d47e371cb3e3f09a8fad3e964bcbe45472eb
F20101106_AABTAL kahya_e_Page_67.tif
a8839eefdb84b81f947eb397e640b7b9
35f8cdce7217c695c16099b285f0b2818cf350cc
27802 F20101106_AABSTN kahya_e_Page_22.pro
f8eb139022262c343456d34aeead98c7
6b0716d33507ccb52915e7419dd5177a4d55db55
67441 F20101106_AABSOQ kahya_e_Page_51.jpg
0f719a7e7737effb6f9611299a695b49
f3f99cdc9ae846db389598683e60cc2a7755ca09
22578 F20101106_AABTFJ kahya_e_Page_19.QC.jpg
302b90e9d3c579cd237b76b428e0f312
f74d642c5a09ac59f8392e18d1fa36bc17ea3beb
642120 F20101106_AABSYM kahya_e_Page_54.jp2
15ea2b4c140e504c5933b463b560dc7c
a4d136c8b36104a2f1c927edf35a9e1778bd6b02
F20101106_AABTAM kahya_e_Page_68.tif
4a96a17dbbf0ea14d5aad70540b9f8a1
f11808cceae776a55f9f86036f8732fbea5325fc
3471 F20101106_AABSTO kahya_e_Page_43thm.jpg
77e98c58ed2649e99108420bbe7397f9
5f09f35ad137f31436f6a131e689cfa542f29f36
F20101106_AABSOR kahya_e_Page_25.tif
04abc584bfa202d0aec2f72e59660124
06898034243aa07d80f7a368a8997ec00c34b191
6251 F20101106_AABTFK kahya_e_Page_19thm.jpg
e1424fdaf5e18c0efc5468ebd9ace608
94c2c6734e0d285c01b5b191d30ddafb7ff1db0b
50445 F20101106_AABSYN kahya_e_Page_55.jp2
2504d60e7b52008f9bcddac5e0103aad
35078005a356b3167ed2c9249033f156e97142ab
F20101106_AABTAN kahya_e_Page_69.tif
6b49c09fa5464092744a1843646f4641
12909b4bafdf0b0b838146a1121e6286e1e6e4d6
29606 F20101106_AABSTP kahya_e_Page_44.QC.jpg
dd9df5a3dd30af26cf6252ec25c4645a
9e65859587ec14ea854d70c31b58c3e14cb01059
32517 F20101106_AABSOS kahya_e_Page_53.QC.jpg
fc9d2e4199f88daef91086f49916967b
7d253f53740831f7f83f31ecc623c0b35a61f83e
26336 F20101106_AABTFL kahya_e_Page_20.QC.jpg
38e4c7a37f21925af90519ba191ce097
cab6b9d073788a0fd35f41c57c89533547ca1483
1051978 F20101106_AABSYO kahya_e_Page_56.jp2
fc55cb714f84c5101ffdc8fa3685bd56
a539d6a94ce97c1397898f999dc687eabd39f25a
F20101106_AABTAO kahya_e_Page_73.tif
d24d19a464b22fa88aa400bdc16db228
504d011e51911e1ffbcde295e2cf285c38b7ca9a
211449 F20101106_AABSTQ kahya_e_Page_07.jp2
d64bcb69e46f0f7224a3ec4f180e2d23
afe435ccccffc69310b22de7f51de52aabed968d
7896 F20101106_AABSOT kahya_e_Page_09thm.jpg
fdbe8f7dbff8133e0dc5b1c1d7e8f1f5
504df69087b2bb4607cfe4d037395bced1f6538e
7095 F20101106_AABTFM kahya_e_Page_20thm.jpg
2b1399ab7b9c319f548883341bf8885a
cd35f44d24eee1c42e616e0a3d006d0300c0be20
834248 F20101106_AABSYP kahya_e_Page_57.jp2
e136a5169426d84f037b85f9f68a04dc
b2ac00688f3449de6fa4f0b76261852e654aebf6
8145 F20101106_AABTAP kahya_e_Page_01.pro
16e353322c2f2f46d827ef2ee2561fb6
1c08253b46e68e1f5ff1b9837305f61d066c156a
7570 F20101106_AABSTR kahya_e_Page_37thm.jpg
b48edf94a0dc379fa9deffe0b1a05d9d
00ec9aa54a434bc1e04a0f1f181ff1fcfe6d3868
6328 F20101106_AABSOU kahya_e_Page_65thm.jpg
ba3e29f34327cad7f7bbd2d02c421cf3
e2ac6bef23db17f144e74df9853c6c12345393d3
20597 F20101106_AABTFN kahya_e_Page_21.QC.jpg
39fb03b3585e5c5b681b1ea54b90e444
58e4adeeaa2d3836a4d3275d0fd00e6aa068a0b8
651127 F20101106_AABSYQ kahya_e_Page_58.jp2
6b927c1331294f793d5abecbe605d123
251d5da1d7e58ebe4d6e8b3a35d7efc4615cbdc7
805 F20101106_AABTAQ kahya_e_Page_03.pro
0de2606bbc2d5c1a4a353a404a9a1a71
48f406c0189d8a1cee6dc584cce5227c897be1e8
25183 F20101106_AABSTS kahya_e_Page_36.QC.jpg
09a6778ed1ea62fa434f07907dc66ac7
0108127a743e93c985d72cf48e7ab8e169681287
5942 F20101106_AABSOV kahya_e_Page_13thm.jpg
e4bca568674857fdf31f5304fc131074
a59c9f8b34d28cc1fff1cc754ba74c49aed341f1
5466 F20101106_AABTFO kahya_e_Page_21thm.jpg
313a65c89b46c2c899fa19f9875a4fc3
595229034efab940f1c46704a67e5a7d92743ab0
39182 F20101106_AABTAR kahya_e_Page_05.pro
189bebcbe57e26170ba93fcd4cf946c4
07cd2a5a54f431d1885e4eb35762c3fdb5290624
F20101106_AABSTT kahya_e_Page_07.tif
f194c83cf6a6a581166da6f20d582238
2b120e69e11d4851a651fd23513150389a755f23
23680 F20101106_AABSOW kahya_e_Page_65.QC.jpg
fbd2cafbc007e6ee4e25180bb4e7e5c6
daac5aa1f3b1a805ba317d9d2856829b3f11faec
6260 F20101106_AABTFP kahya_e_Page_22thm.jpg
034a9683c6f659de38097dde826ae596
faecfadf928c0192867e954b97137c94b82c0972
1051981 F20101106_AABSYR kahya_e_Page_59.jp2
2aaa823d02466cbf9c61a2ff26e2da7a
6abe35ebcc978fc27e0b610e5a856e3582ab964f
7827 F20101106_AABTAS kahya_e_Page_07.pro
6723f8be50f5654b137079030058dbfa
cb0269f79fdcdb988be02756f7c96bb9e9e866ab
1602 F20101106_AABSTU kahya_e_Page_65.txt
3d1ef9e0a68513356be46ae11f5334b2
0be9f3945609e80c637730c7326ec73e214063e5
F20101106_AABSOX kahya_e_Page_02.pro
fef9ba708ff3322ced975b821d1101e5
b88028ae2400885e014538e019d97e5fe457f92e
29756 F20101106_AABTFQ kahya_e_Page_23.QC.jpg
690c3bfb1bcc2b272de402aaad1c87f2
4021c8c6ee83eec4b762ca46a68815a86a489c83
1051984 F20101106_AABSYS kahya_e_Page_60.jp2
129a60bfa30b076d3d59697d901225c1
da037ef601ab15c311174a8249e40848cb9ddba9
44981 F20101106_AABTAT kahya_e_Page_10.pro
86920f4c9098bd0c17c6094cd22262e4
f46c7299ed6eef482f858e7da2bb0636f9ea4cdb
8486 F20101106_AABSTV kahya_e_Page_60thm.jpg
8124eb592df0c8c6eb6db12f47e7bea2
2fb4c6430a2ca692f741ff3e427c8d5df6c64ad3
468 F20101106_AABSOY kahya_e_Page_73.txt
22784c698a6456505515e3128c227518
b68dde6fcfa5ec6aa434b99fbec03211a6a312d5
7447 F20101106_AABTFR kahya_e_Page_23thm.jpg
d13f1e41511e832cde9f2d6fdc156809
2ea8246ab52ba067edb15a1396020b7abd443548
535754 F20101106_AABSYT kahya_e_Page_63.jp2
960555e306cb1d44fb07666e718efe80
91a918c94c51cb6fe3c5242e443618ae5dd523a7
36965 F20101106_AABTAU kahya_e_Page_11.pro
c3b6a1c42e957e4c566bca39d68c246a
c56e228514e7111fa477215503f2b3b564f13b9e
1051983 F20101106_AABSTW kahya_e_Page_53.jp2
6cd4b1883cc52ae931774bacdf6b9477
8483774c3197b5f29175680fda14fcb02a5c430d
2139 F20101106_AABSOZ kahya_e_Page_53.txt
fb463e03750b27f01d6160b1851dfa7d
c1b67a4fac48a948ee8ccc545b56ea32d873b661
22855 F20101106_AABTFS kahya_e_Page_24.QC.jpg
86107d93458124b16cae658dfbf899b6
60c979c5a4bd0392f031926843b59d5aedfac7b7
58697 F20101106_AABSYU kahya_e_Page_68.jp2
31a4a0f7e1608430d98df7c707fba5fe
19f12d8d657d7989938b75bf041cdcb07b9929fb
36224 F20101106_AABTAV kahya_e_Page_12.pro
d858a08deffb06a4b816751820a2d5df
382312f274e2611efd4e1cacabf28c47082f37b8
19663 F20101106_AABSTX kahya_e_Page_72.jp2
e0f689d08dec1ec3d11be1ed58d7ec57
4a9bba7128a89d88ec60c02641f491a8601a8113
6369 F20101106_AABTFT kahya_e_Page_24thm.jpg
cf04a66f2aeeefd4ebb51132f3326ce5
6ea4c8813c963d65f5a9c8e6aa8855fa52b59cbd
117797 F20101106_AABSYV kahya_e_Page_70.jp2
0634bd75f51facadaab6301b903ed8b9
09f867dd22e85f50fe701fcd8c7f46af79fc8ec9
30475 F20101106_AABTAW kahya_e_Page_13.pro
bfce54b0d15f58235a6e58dc5a6c8d77
1f3ebf8b94aaaafdf70dd56f9731a2916156b1f7
2143 F20101106_AABSRA kahya_e_Page_38.txt
e2a8a253169d456d8d5e7ba5a7b5764c
aa9432180b61026ba37182345d07cd0610b13bac
31493 F20101106_AABSTY kahya_e_Page_61.jp2
95a39100370fe122d05e6a9830f76347
41c21bfb0579b010d2b8195791e26120397cd84c
19410 F20101106_AABTFU kahya_e_Page_25.QC.jpg
08398fa90a70d48cd2469ad639ba229e
a529527136ac8c7c4d382e0ce6ade4e85e6f0c61
31121 F20101106_AABTAX kahya_e_Page_14.pro
1f8076abb6ada48b2466d40c9d516110
ae04c72776a148fe8b81885eb821d41b67c5c743
25517 F20101106_AABSRB kahya_e_Page_73.jpg
5e32aaf8ee0f19f2ab0d8b3be83315a0
bbcf832036da255ae152ba2158a9529592f51f8e
59621 F20101106_AABSTZ kahya_e_Page_21.jpg
108ce4da0f10aefcd3035c7d160bd10b
ecfe8ccfd2bbe66496394d99970ec79c729eb4ea
110926 F20101106_AABSYW kahya_e_Page_71.jp2
ef14d5615d19926fcc1a8acbcda85a40
847b86a525e4f0459b1a037ebb7346fca5d93516
20748 F20101106_AABTFV kahya_e_Page_26.QC.jpg
cb9132c42aa4abe7097f3d4eb4784192
2ea05c44769afea5fc62ddb34e7f7589c173473e
F20101106_AABSRC kahya_e_Page_52.tif
8a16c54373a93a9fe929a0347de82bb1
715ae03800d3cb44e61ca0d4d8cd4e65e14266a3
27048 F20101106_AABSYX kahya_e_Page_73.jp2
a131b82cc04474438dd22178408efc19
8f40df23250cb8f0f6f811cd21291936cbbf506e
5779 F20101106_AABTFW kahya_e_Page_26thm.jpg
08e4b0e727e58ffec75f2f6240911790
8c02b8eca4e4131d4b3c2078e1d6653e73b6317e
29232 F20101106_AABTAY kahya_e_Page_15.pro
6afa112a38801d1fc74acf880623efcc
b31bb2633440e0716c0763fc3d5114ae835d18a4
F20101106_AABSRD kahya_e_Page_38.tif
7fd0d963aaa117f3ed9a2eafc11b0229
56321d633e154b78aa9957e8c8fd866ac920372e
62633 F20101106_AABSWA kahya_e_Page_14.jpg
d790d8a60fa635e94fc269ed8126966a
87bce5413b0adf3f016b2b26bd3999873114f6f9
F20101106_AABSYY kahya_e_Page_01.tif
cd1769af66248b9176d5310b690ff845
d09f91189bceb4fdb534ca38f2fdde962a22411b
18141 F20101106_AABTFX kahya_e_Page_27.QC.jpg
848c0d59d9f123afe4a0a936fc0ee095
3bb7ae9fb3c2e6b115f2b14729168d4b50352b32
25471 F20101106_AABTAZ kahya_e_Page_16.pro
4cc55086a97d4d7213cfbeefbae22974
e2a86be536ec05d2ebe29615e2bb292cb22ee1ef
72212 F20101106_AABSRE kahya_e_Page_68.jpg
16b725d6647f9e0dfec146825296c7c5
5e4fdbfb8993a6b0c94a92d7b65d416d91940d97
61889 F20101106_AABSWB kahya_e_Page_15.jpg
94580ad084a5002bb6d0fcd2d7d1a0cd
01b7bbf28c4c824626aa57bb8d55bce8a89af33f
F20101106_AABSYZ kahya_e_Page_02.tif
96cd670761151634c167fe4851a93c1d
66eb230cd8b2afd263845ab7c65283b004b618e0
4983 F20101106_AABTFY kahya_e_Page_27thm.jpg
1e45a133361eedc14d57b9fd6abd2e9d
af1189d1737d7811d8db4ec425f7d9a1d5ff47a5
881063 F20101106_AABSRF kahya_e_Page_36.jp2
e08e9e09e0ef55223afcecb71975e3cd
685d614f1505d5ae46b6167546e1b359acc62786
57221 F20101106_AABSWC kahya_e_Page_16.jpg
1c2a59922128ec7594b139a260a184db
e3727bec78411b082e6f1e8e1abe21da96bdf0f0
20074 F20101106_AABTFZ kahya_e_Page_28.QC.jpg
57f3bb78f82a68ad6f5a27b94b83f925
66297ad3e8d084d3f9a8623c40f9e613dcd19159
62066 F20101106_AABSRG kahya_e_Page_22.jpg
977f1946790c752c66946d72e1c9b433
524aff34486fabab009e7868afd2182a7aab7ab0
1737 F20101106_AABTDA kahya_e_Page_15.txt
33d25adda26c1a68a8a9cd57c60f13ec
b15c0327f02ebdf4e1317f571488c943364a5c37
68576 F20101106_AABSWD kahya_e_Page_18.jpg
501cb397c76ffb372d83c3b3eb01cc40
5d221bdf2db08c639dc43bf858be6a02aa6f6645
31028 F20101106_AABSRH kahya_e_Page_52.pro
848aaef36d1d7b6cf6b8ee9202500e82
e52febc24b091b320fdcd1903b06a76181d02a29
1278 F20101106_AABTDB kahya_e_Page_16.txt
3b89b599804f519fe27d14c351990271
65c639f820c324151e1a2b9a09cd8a99def73fe2
80634 F20101106_AABSWE kahya_e_Page_20.jpg
62c44fcdcc5cd6c8bf08c4cdb9427022
937f2743fe24592e0bfe68de92020884d4a48f76
40348 F20101106_AABSRI kahya_e_Page_20.pro
c311bbcc91333ca632163a32031e0e82
67cc83d246fabaffb1452d730651258684e93fbc
1514 F20101106_AABTDC kahya_e_Page_17.txt
49e5cb1338ebd376532800a8ec5fd22c
e0cd4da5cf81359fdd9a8b74ce660077d9c9f3d1
71190 F20101106_AABSWF kahya_e_Page_24.jpg
014a9d128b78b2d6794cd09d8e5f305d
b1a9c50539436166da649e568b940e62b8c7eefa
7762 F20101106_AABTIA kahya_e_Page_70thm.jpg
1953d05a9ff2aae44945d9e7d83fac84
af0106f9c388cd3c77e0e287d9de1d442b07bf46
22397 F20101106_AABSRJ kahya_e_Page_04.pro
dc9bcfaa05309d7e615d88deff1dd53b
fe58c8e02ca739fe355566dd8aae8c24cc5a0d39
1620 F20101106_AABTDD kahya_e_Page_19.txt
864166a470835f050a43bbc5a19330fe
8c5cff6bccce2512f86a20e1927ff638d6e2dd5f
61745 F20101106_AABSWG kahya_e_Page_26.jpg
9b7d4827e1c3bd3c498b1b3fdae9b419
cd14e540cf7d616d5f56c0a9bc5f60c667c095ff
32717 F20101106_AABTIB kahya_e_Page_71.QC.jpg
650993e12240bd2013860df342a0f0a8
62f9ce96d10aedaec3236ce95c5c432c017ddda9
1810 F20101106_AABTDE kahya_e_Page_20.txt
70e7effe50b9b86b935b726424a30d78
245d9c4dc6e2ac473d7fbfe40cb8d6fdd88cda29
55096 F20101106_AABSWH kahya_e_Page_27.jpg
3f48c6df1b7d5780f70c258427c05071
13ea40c202ecad9bb0079255dc751e1137d51f7b
8257 F20101106_AABTIC kahya_e_Page_73.QC.jpg
9891171c1be46bd6b6aeb2d15aa12791
fb2e1321cc66bf746fe5eeef2cfb07b92d58eecf
1345 F20101106_AABTDF kahya_e_Page_21.txt
2991eb4798c92dde734bd2d8669da44b
c9a3cdf91dc10ef38bd0909ed4ba3e1b9d292e2d
63292 F20101106_AABSWI kahya_e_Page_28.jpg
4c3bcc9b563ef37eab284b185d959fa1
1149557f72ca6ce649d336e5b2f46b8213a51d35
2304 F20101106_AABSRK kahya_e_Page_61thm.jpg
c00c5dfd2f163154487da6a174c981ae
f80009c744a6268738faa5651ba03e65894c7c3c
2024 F20101106_AABTID kahya_e_Page_73thm.jpg
72b63c310c7dd22d7c6d2175185df207
3b0986cba6468c4021c1f2626ebd23d163d94085
1621 F20101106_AABTDG kahya_e_Page_22.txt
b5aaa0ee64d29b79fccd1f56d503b8ff
cd85adb15978332215c057ecab0ebf44826a6682
66673 F20101106_AABSWJ kahya_e_Page_29.jpg
2f3e338adb4778f9a4b6aa702ad43fe4
96d5b2a14eddb2a604eeb1e990c35126e0100c28
F20101106_AABSRL kahya_e_Page_50.tif
4da75ef2513634424d39f77eede16306
88822d289b350fb8896b48862bebfcc22f9e5f5c
85637 F20101106_AABTIE UFE0022050_00001.mets
05f3ece746e0bb58dbe08f0e3a84bedf
bc64d6b7da36fc7ccd19a8642e70ed1117686b01
2110 F20101106_AABTDH kahya_e_Page_23.txt
27c24eec739667b6c2552648381b715f
cc247d5a809efd72e3439cee37c80f4456ad34d5
67876 F20101106_AABSWK kahya_e_Page_31.jpg
b48c7418f8edb18f98c4548c3e64b945
8a6c48e519cb81a50c1490b62c7b30c0d960dcff
29510 F20101106_AABSRM kahya_e_Page_64.pro
b6bcf9bacc9b5bc5df9c0ea82308ee8d
48051324693a002fc9849dd90c1ede0d5d5315b4
1817 F20101106_AABTDI kahya_e_Page_24.txt
0c6c6372c11880407b7698ce9aadb952
278cb958b3205628dbe4cfeac46ac30b85df1452
96942 F20101106_AABSWL kahya_e_Page_34.jpg
9d856a211c448e89d8dc4ad771ecf397
828a0e84807256763915404f26d340e630eac944
17603 F20101106_AABSRN kahya_e_Page_08.QC.jpg
b0b2b8f685b6ff57a30ac705f30606b2
465b51f62fa2cb889b6b5ff56e2c9b9f21c7617d
1107 F20101106_AABTDJ kahya_e_Page_25.txt
961d20cbf05b66c13dd29e1e6e72f072
35ed9f2cec69274acc7a27ee3a0b624d8a157717
84710 F20101106_AABSWM kahya_e_Page_35.jpg
2201e7aca48188d2f179da0aa22255e1
10d82b1a11685691523f40a031b1785863f846ab
F20101106_AABSRO kahya_e_Page_33.tif
52d8ce38b172915919868d5008e35b67
4dc6f67ffbe98048b203269c582f53b7ba333a53
1336 F20101106_AABTDK kahya_e_Page_26.txt
83ce591141845416605ef58b421f8963
d12cc2fd2f514c880b2f89caa9def7f84871781d
83388 F20101106_AABSWN kahya_e_Page_39.jpg
37ce6c485203d2750b2018793046c3e1
b06ce4fd89c332ca5752974d392b99c28f578a43
F20101106_AABSRP kahya_e_Page_72.tif
2233f88cb2dca315872c24b2163d8d01
62f1975687f314cde020496afbd8810d833fab1b
1275 F20101106_AABTDL kahya_e_Page_27.txt
6d033d2b9646f985884ad62cad0e134b
c8481a31e024dd254b70c1b48eef37dda664ae2c
72338 F20101106_AABSWO kahya_e_Page_41.jpg
a024d4f1b846c811a31ed5c722a96d26
4e7c96eb033414c1085007aa2bcaff8b1d8bdb50
F20101106_AABSRQ kahya_e_Page_34.tif
308befc23796d2e0f7b130fa89de6616
788550aba15015ebedde65616ab9cd98cfa7fd52
1522 F20101106_AABTDM kahya_e_Page_28.txt
004afb0a96ee994f341aaa4b52011fca
33f6aedb18870f3976bcae9526cd99911794871a
2462 F20101106_AABSRR kahya_e_Page_60.txt
986103ef1e825a2ecdcd8df938e70262
a53eb954b87a3ff70b35ac0153f4b011779f1d34
1682 F20101106_AABTDN kahya_e_Page_29.txt
5a1ba554659189369e0b4a9707ca18d2
82da3a1cb3963e39152390063d7a88ccbc0e915d
48985 F20101106_AABSWP kahya_e_Page_42.jpg
c3e5abf593899ab117eebff0fa674dd3
d5be74a247790925295ebbc39aac0fb37c2a611d
44871 F20101106_AABSRS kahya_e_Page_06.pro
33ed48cf5ad69525ce743c791fd6b4d1
2d579e26afcbb09a9b28c378788226797408173a
1686 F20101106_AABTDO kahya_e_Page_31.txt
55f1f10f525dbbcc443c506844fadcc6
2ead795ff0830c89632c03378b1cf2cc82edeba7
89557 F20101106_AABSWQ kahya_e_Page_45.jpg
c1306d482eaf04997552b31040090296
1f1ba98145b9dc7383f0f415069764196a57bf62
6257 F20101106_AABSRT kahya_e_Page_17thm.jpg
27d42820def3abbdfd156c6144d0a53e
c0464ed1b1ae2c07c53592ea2a7c8ba9f7da5bac
1891 F20101106_AABTDP kahya_e_Page_32.txt
4b3f275854e4b0934e3ec4c7832931c6
5d2fb5bad51d26a881a3adcaf6b9403c4a2601b2
85167 F20101106_AABSWR kahya_e_Page_46.jpg
b565b4a0f2d621aba3dd3002bcde935e
005878371091aed5e0d60c383690b04c18540ae4
60438 F20101106_AABSRU kahya_e_Page_66.jp2
e549890d2e734710cb0eaae23b9a3828
e6a1be18559f143a1bb9c29586059c54be873c79
2012 F20101106_AABTDQ kahya_e_Page_36.txt
665109fac88f631b9b009db339f426e4
e9d6806eac09a8d2bead02153e2a90701defa9b7
93291 F20101106_AABSWS kahya_e_Page_49.jpg
e4bc85ba8de00c6d593d6e51ccd0683d
c349ee25c9e4dc65b0edbad55c828db001c6d7d4
26465 F20101106_AABSRV kahya_e_Page_50.QC.jpg
cc76275b4de0d396b5418ccf4b1103d1
928cdb1bccb891002388721569bff9175802f690
2063 F20101106_AABTDR kahya_e_Page_37.txt
70ee7b7f83a143d2439e4e0dad4457e9
b13fd5853611e586566c6c93008891f72bbe02ac
83340 F20101106_AABSWT kahya_e_Page_50.jpg
23792aef09ff9d94eb8b8ba3a2f001d6
bffdba9b5385f0206327b9c65bbcdf07103177a9
22152 F20101106_AABSRW kahya_e_Page_11.QC.jpg
71d0e5216e85ecb1d5a4c93a56f11060
79b945d59c3a4c2ed4bd3900e496e78e1bedda1e
1568 F20101106_AABTDS kahya_e_Page_42.txt
6ade0a92428fdb1a148c858c71024a1a
48ca3a1ea98b3b9630d2f5cf76667b37413d48f5
67204 F20101106_AABSWU kahya_e_Page_54.jpg
f62851afaf0dc8ca52f1d57b83bf4cb2
180c3a6e6a5775037a9e959d9701acdcb7cc11bf
5727 F20101106_AABSRX kahya_e_Page_31thm.jpg
b9a47c694b3cb8c19dc28e8b054fc1f3
4a4bc864f10236cbe0eb460c1bbfb3d0dcd1c835
1232 F20101106_AABTDT kahya_e_Page_43.txt
ef80ae03f003cb0cb4e359932b54a88f
73330b79dba2d189d6e081bf11be33c8d5f7fcfa
59304 F20101106_AABSWV kahya_e_Page_55.jpg
8d3bcf7ac26a9233f4c2e578653e0d1c
dfeeaa71e7e8e2080a87e6466efe899450d9fdde
1523 F20101106_AABSPA kahya_e_Page_13.txt
854a85d02df3bb64ad2e8d99b3d0a917
ababc838bab5c4ef47c3319b0999675f9820a6bb
1473 F20101106_AABSRY kahya_e_Page_18.txt
9d734677d61bd19e6f7c2162f249598c
8f0ef36763e284d879e3eec1bbfd3556289d5216
2003 F20101106_AABTDU kahya_e_Page_44.txt
84e1cb9b83f43ce8f8c2bcd5d4c59921
07cb6e5710d6bd4210b221d62c1128ca0389a017
104010 F20101106_AABSWW kahya_e_Page_56.jpg
7f3b2610f7f3b18011f7efacd2f0228f
bbeb0c74cc2bf6581cd02c513945a72ba9079585
F20101106_AABSPB kahya_e_Page_65.tif
652c60928b141afb240445cc372cb481
fc028536bb83e716e4fd44b792d62d486ead517f
68345 F20101106_AABSRZ kahya_e_Page_52.jpg
48f494fb0b0a9f57bca56ab6c9ea83fb
72b76fe963fba2c0d0593678f8c37c5517e4f53f
1866 F20101106_AABTDV kahya_e_Page_46.txt
a642db1d6db1653752c7fbb1a8f7910a
234270513244d5b467c8efd009532c76ba13ad0f
27964 F20101106_AABSWX kahya_e_Page_61.jpg
dda55d6a7f2889a5c4b57c854f5c4997
50d1bb6a27e0acb4cbffb1ae1d3e7480683c9c12
2004 F20101106_AABSPC kahya_e_Page_41.txt
1ae39bb8b187a408bf3c2bd19db52f94
92ef71ec79fcec6c1cf8fa422ee98d20206402eb
2053 F20101106_AABTDW kahya_e_Page_47.txt
9f29cf7d4333a2b9b95c7b7599f4006d
fd3644f9fbd517de363e94c2f2852c6518c89cb5
120390 F20101106_AABSUA kahya_e_Page_60.jpg
f0c2be9f6119d45ccbfdabdbeb948f18
0e86f23d51ca7d685587dbeeb01d9431becbc031
71229 F20101106_AABSWY kahya_e_Page_62.jpg
de0ca25e908269f5ddf4ea9d1be24fef
47a1bbd6963f0f8f495b2fccd76679d457548e9d
906 F20101106_AABSPD kahya_e_Page_33.txt
82f367073c30b24ef8cc8653a3f6f6dd
723d8581bcaeee0a1bb1b1a12bc8301123540091
2259 F20101106_AABTDX kahya_e_Page_48.txt
955536a362e80cb064fc7617a8cf4e54
499a38c48e902bee936c655c8946e8d06708d068
20086 F20101106_AABSUB kahya_e_Page_22.QC.jpg
79061b11008e4623c5543052f0a77121
a1b86f22bdb96951fd1f09bc2e1c58ecfa3ef2c1
62427 F20101106_AABSWZ kahya_e_Page_64.jpg
e8ae2d8e138924fce6374c6bd779573d
8ecc1811f5959d7b2125369f7e2a0d92c80e982b
29381 F20101106_AABSPE kahya_e_Page_28.pro
bbcda3eaaa18161bfb87b3f5f9dae9c4
a84f235961411cf2f0cb33cd3ccb1d91c8ec5e22
2483 F20101106_AABTDY kahya_e_Page_49.txt
fa47204d35548c52fb13a38db64c55ac
8609ac86e492df80d0dfa2a5dffffc879592a7d5
2141 F20101106_AABSUC kahya_e_Page_39.txt
45edf951db2bab90e1f0628a5268fc37
add26c033e4a4e2498549680857f514b02eb4638
21593 F20101106_AABSPF kahya_e_Page_31.QC.jpg
3ed15c3d6948b968a72d95836a0f010a
f59f0025b3e1daad81ff91f1508335e6fd3f065f
2168 F20101106_AABTDZ kahya_e_Page_50.txt
26eddc625bbc4f92dd3bd6222ab59556
74571265bc677f7477f3dc84c33e0006689edfbe
5823 F20101106_AABSUD kahya_e_Page_07.QC.jpg
6be8b6bc8eb65c9ebcdcaa6b95f47f4e
bdb5d8fdb6b5f0acf1b8cca50524366a3b79d83e
89904 F20101106_AABSPG kahya_e_Page_40.jpg
8fb80c8be6eb9cac471dc97106d9726c
9f647c1a4f49edc8eb839bec5b5c15c9f2183856
F20101106_AABSZA kahya_e_Page_03.tif
9578c62c9139b3666de4ee197d41ceee
38a6e51ce87c756425eee9d480e9e83b54024a74
32673 F20101106_AABTBA kahya_e_Page_17.pro
0d6c3987ff59df49a5dc38f396bc7b15
8065fe137303f5177429eaad85938d874e8c83a7
43721 F20101106_AABSUE kahya_e_Page_43.jpg
e3496a08474f6bbd02bd403e6331e694
3217badf66a3b8d60a91bf14d0033295cbfd3b7a
667784 F20101106_AABSPH kahya_e_Page_22.jp2
bfa27ab7836bf35a3796f719d3f2b843
a2b674acf154be20c2e3b0a42b5fab1d6b981b7e
F20101106_AABSZB kahya_e_Page_05.tif
f7ac4c3ab5b22377e8fa00a3d7f4e823
a8793db05b53da59202c80ae501788dbcda06cf1
30012 F20101106_AABTBB kahya_e_Page_18.pro
8c0b2e931980092eb92c4f1e1aaf263d
c679851604da046042fbd3897079686c8726fb2f
F20101106_AABSUF kahya_e_Page_55.tif
8643fa2e7cf338caadcf6da61534741b
7f0cf82b9f18102eca4c344f4c7472863c17b1f8
F20101106_AABSZC kahya_e_Page_06.tif
a2a0bb470ed573566a261a64de377153
cb8dbec186f428e79328b83958c66c9b0fb55fff
25653 F20101106_AABTBC kahya_e_Page_21.pro
5bb699b82239f6ba8cb1329f882c27de
e5ee9d0265f6a84e3020b531e016e4fbaf5ea5d1
5367 F20101106_AABTGA kahya_e_Page_28thm.jpg
3b95d2ed95550fee2c86b23d16d8558c
2a6c926d8b961a69abd23636d9e5c846a7863704
1666 F20101106_AABSPI kahya_e_Page_12.txt
7a4e48d416363f031d715eaba81ec685
e3f8e0433243adf2cbd8a7c7f1b4368275fcbaca
F20101106_AABSZD kahya_e_Page_10.tif
1434924994f9a97e4446ae53330d83b9
bbf15fc0102fbb926ebdde86a2392506a584bc39
47245 F20101106_AABTBD kahya_e_Page_23.pro
eb380f0da40560638ebb57f32fa9c686
e13ba0b2949d2206d7b35e3e9ff8a5f81d1d028e
1149 F20101106_AABSUG kahya_e_Page_03.QC.jpg
1fbc91eb5a7fd7d48d43f4864cb8d5db
4b667f3be94fed79b1c260f7129c957545f97db1
5684 F20101106_AABTGB kahya_e_Page_29thm.jpg
9fe4c6a6165caeb64973a3d3a70b4e8a
07d7050a187538d451393f9aa8061d7da7dcac31
62703 F20101106_AABSPJ kahya_e_Page_60.pro
f0962f2f25730f499232a78d61ee337b
002d0d8974acd5f508264b6d03fe71b185fd9add
F20101106_AABSZE kahya_e_Page_12.tif
8c588c05eab0abec632306e1f7c14eb3
24e0acb0625e664a0bcb3d946a2cc088dd3542c5
35144 F20101106_AABTBE kahya_e_Page_24.pro
ce590f7fc11171e162a1b7df07c74107
fbf3a98728bc970581fbd20a39a01c894c67c8ee
1942 F20101106_AABSUH kahya_e_Page_34.txt
b6768976f14b91139bb0e17778d3bfa0
a373340685d8799cadeb2804f34e48023e52eef5
21386 F20101106_AABTGC kahya_e_Page_32.QC.jpg
6c4ca3414556e5cfad6ae332a8d16ab3
b8fbc2e354ab5b10a5f71d164083c8f8e9cede24
91157 F20101106_AABSPK kahya_e_Page_30.jpg
f14e5d07f0272cea0d1f41979bd391ed
2b1f097d019760508edf5ab0df703ba533b80511
F20101106_AABSZF kahya_e_Page_14.tif
d1e4a2538637ccfd9408e3fe1611b4ff
6f1d41ca2bce0a534d5be8f89c301a5902f37b64
26320 F20101106_AABTBF kahya_e_Page_25.pro
7c74135103a79f2983e9d64366bfdf0d
151bb376b0f46d43ca258ae3ad36aa60c27f2373
7722 F20101106_AABSUI kahya_e_Page_72.pro
20bb5b9de228c63b5dd1e6fd8ac7c65a
f0b0f5e326cad44774da9a8589702ec87550bbf4
5708 F20101106_AABTGD kahya_e_Page_32thm.jpg
8ef233ee7fbc8d695afec24b59ddc1c6
bd3777a233257e9181a09e28efcce18a43f6cb8e
964235 F20101106_AABSPL kahya_e_Page_46.jp2
68b4d9185fbd102c4722946df89eaf1f
18af3716268f51e72a1ed4bfb4aa0b41374570a1
F20101106_AABSZG kahya_e_Page_15.tif
cd5551294eb656d2916ea41d3c88b5cd
09595abc2a0f1a7306454c0ada545d5f8abb7ad7
28511 F20101106_AABTBG kahya_e_Page_26.pro
2ba333c6a3ed64c1566168559d09c150
1c14a62ab6f3a9998557d839e74c7b9c080affab
27773 F20101106_AABSUJ kahya_e_Page_10.QC.jpg
74289a645d82ec8d12f85e6179c0bcb4
6d1a084f85acd9969c1693bda9a44d6295fe044b
2518 F20101106_AABTGE kahya_e_Page_33thm.jpg
0d6b94382bbd280f8ff709f547827006
ec06318f6aee7305f3ae6475ea2d5c190e7e081d
9476 F20101106_AABSPM kahya_e_Page_33.QC.jpg
5b5d9ea5028481fb747f360f31e96433
412f32cd8410d3a58980ec68efd72cc633dcbd1c
F20101106_AABSZH kahya_e_Page_16.tif
a9af9eb030ab7e99241e0efdf79dcae4
23e6b7b86bccd8e1ffb2e972dfb19f48c9629057
24048 F20101106_AABTBH kahya_e_Page_27.pro
305b529b5c1d8e89a80300f5f334fa27
95ec222b4cc2f49f5ee68f96506b7917bc33b224
26833 F20101106_AABSUK kahya_e_Page_01.jpg
20566f80d7fab35f163eb0d4098f8afb
a127bdbe07d5a4154c886aea8e2af0085a5934aa
30086 F20101106_AABTGF kahya_e_Page_34.QC.jpg
4deb7cc2db9c2963a0cf9f2d4dee5e12
5e28171796b6423820a7dda63a722d772badca20
F20101106_AABSPN kahya_e_Page_20.tif
2cd8bdd2bd7e990d5c47b618f5863a97
3278e1821c4b2ad0233e58b1b3e15fe2f2558d91
F20101106_AABSZI kahya_e_Page_17.tif
5cf099546fef2c38757cff48b68d6d7e
3b8cdaa21a4b61b461848eff74b2904548ca8ccb
32842 F20101106_AABTBI kahya_e_Page_29.pro
0bd7ccf129b80b77fb331c930f8f2e37
1516fe139c16c714cabf34abf09a17a231872c66
717523 F20101106_AABSUL kahya_e_Page_51.jp2
0c73ec9b3bd257353de4708d80b2ae36
eb7c962af177d95be850a1ae794ab0ec83770cce
7161 F20101106_AABTGG kahya_e_Page_34thm.jpg
8012b7f4668bb82430551f35c1cdcdb2
09304f4b886e107d984bac3471a252fecc66cb45
27637 F20101106_AABSPO kahya_e_Page_47.QC.jpg
ec26a4e7afb2380bda863cc2fc705efd
e33847d0af0325a6529fc3d8037618ea247326d5
F20101106_AABSZJ kahya_e_Page_18.tif
7fd2cf9135627c6082d78f6fcb3bd7f0
a59bcef55664d719dabfe09153adbb8404d5a350
45105 F20101106_AABTBJ kahya_e_Page_30.pro
c363104164a5926d0dea765854327d61
7635bdd7ac6bebd02ad103453eb08a9b2e211656
F20101106_AABSUM kahya_e_Page_13.tif
e421f4f0a92ee5a21ca1a0a8d03f3b3d
b5cef70f1628219bdcf1a7e3e3b91bfce02c67b6
27031 F20101106_AABTGH kahya_e_Page_35.QC.jpg
850ac5fcc092a17f70cdaffd56cea1ef
9e2f0b1c0f98b394a8a59235220c4bca3180a2ca
449641 F20101106_AABSPP kahya_e.pdf
4303922a5436a254244fadf56ddcf3e9
bf14d616ba261bbf538b4cf9824d68d84e59bf8a
F20101106_AABSZK kahya_e_Page_21.tif
41fe8ac6f1945d5759025c4d2e782db8
fd27b7855353f554d4bab471e28f7b7a561d4b48
32648 F20101106_AABTBK kahya_e_Page_31.pro
406ac1773ce9cf2cf82160cbd69dc8ff
c57f588565dbc7139f425322188731751507e755
7028 F20101106_AABTGI kahya_e_Page_35thm.jpg
1ed6d0725cc5514ee9c34d39dc7a64b9
44d8df3d7b60c6f33cfe25cb585a67f1a2076580
F20101106_AABSPQ kahya_e_Page_11.tif
1115b8287758f529684edcb4efc19bb8
a5a3879ed45db24c5ed7c92aeab3892ab556e997
F20101106_AABSZL kahya_e_Page_22.tif
7d244f8095225fdd5b02e8909f0e9696
304e5b53ed102911b2525a0d2e121bf43dc608a9
30178 F20101106_AABTBL kahya_e_Page_32.pro
f7252f974cd2b2cc2747d55afd453562
b81fefe926fd2348158064b8eb88da46cb094a4e
29102 F20101106_AABSUN kahya_e_Page_30.QC.jpg
fd1cb1d70b66b30acdf54adaa692eae6
51303355f63cf8ee5aa630d0d879aad30a7823bc
6888 F20101106_AABTGJ kahya_e_Page_36thm.jpg
5d60c0e327db198201d95ea3d961f02d
ac1bbbe24da00579b703cb70346cf812af86041b
30094 F20101106_AABSPR kahya_e_Page_65.pro
cc2b4a2f854a4f94dfede3e078921f74
4dab1ea7d768d03aa32dcdb8a283f07492ae956d
F20101106_AABSZM kahya_e_Page_23.tif
5b950c9c63cd731c73ff50cb19d255e8
2f368c9d4b3e3b9b957535fc74e9f5169aa4ea56
11576 F20101106_AABTBM kahya_e_Page_33.pro
da003e23b2a926ca13106c89a1b84525
f0e0c4a10fadbe8278a3773a4d04dd89ee13b07f
32857 F20101106_AABSUO kahya_e_Page_70.QC.jpg
16853809e9d8c4ede8b18c0ae618a095
490f452e5cbf98810b9562064e98ac6f97d2dd37
27897 F20101106_AABTGK kahya_e_Page_38.QC.jpg
65dcfa959bc762af3d0d233cf730c448
8abbcd251da7902305ca54ff370298827033cf9c
6153 F20101106_AABSPS kahya_e_Page_58thm.jpg
e446f5303068410b52bbb9055d00443d
493e27cfd2a04f3a1092e4713667108730e11d8a
F20101106_AABSZN kahya_e_Page_24.tif
0cd16dd552392166f24d2a6fc5f3ca18
4d819785e0ff242d1d4cc6230caf9ac61ae3e461
41286 F20101106_AABTBN kahya_e_Page_35.pro
34f81291442d839978e8525cac8ff64e
bc393b748498f618e26c3619a724efcf260103de
F20101106_AABSUP kahya_e_Page_62.tif
40a80e298c502378f9e25723f3105e68
aa8d426dff17bc43a69898733c95a867ed41ad72
7403 F20101106_AABTGL kahya_e_Page_38thm.jpg
7ffa77006c5d37a657d3aaa4c738db2f
bce2d0f3ce6b57e275aeeef8464655d5897a10c7
674872 F20101106_AABSPT kahya_e_Page_14.jp2
616f889f3b86c0aa510bb2e90a448708
47da50bad1a8bfb4e51b91cb28c07c769d83d49b
F20101106_AABSZO kahya_e_Page_27.tif
c1955f96a34610e99b6bb45db3cd5b1a
fae7c9b6e8cb5f8f0466139cf879921e20b79c4b
43079 F20101106_AABTBO kahya_e_Page_36.pro
80a4c4c18a840f74dde7b2920d4720ba
6c226f96a25ea6902bac6e9870362cc5d29eadf2
1485 F20101106_AABSUQ kahya_e_Page_72thm.jpg
b688ce702e65db73c27f8fdf621b3d30
b160c71764150cb5dff75cbe00a4fa964a06f549
26597 F20101106_AABTGM kahya_e_Page_39.QC.jpg
831e546e41468a2a1c29e772ddb1aeba
c1158534179f6dd091d75ae85614a620e501bc40
62944 F20101106_AABSPU kahya_e_Page_32.jpg
2d4b870cbbcdc265b404e5318e5986f6
8060cc6ae800d5af6875ec3f9187b8ab02a982c5
F20101106_AABSZP kahya_e_Page_28.tif
e18bce1d6abc34f7f3116dc5441280f3
5565fcee7a3f569ae326d5101e7dd874e20be440
42210 F20101106_AABTBP kahya_e_Page_37.pro
210ca897084e71782bd1093645731be2
c217a049daa5223bb052e147aa2111c692a6c1a3
F20101106_AABSUR kahya_e_Page_66.tif
e6110197c9b3e43741be2b0cc56d4039
76a2b45e73c3bd0907a4f3d65e744e8f92dea0b8







QUANTUM GRAVITATIONAL CORRECTION TO SCALAR FIELD EQUATIONS
DURING INFLATION



















By

EMRE ONUR KAHYA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008




































2008 Emre Onur K i,-



































To my dearest mother









ACKNOWLEDGMENTS

I am indebted to my advisor, Professor Richard Woodard. He is a very hardworking,

meticulous and a dedicated person as probably many know. But he also is a teacher who

really cares about his students. He would not hesitate to sacrifice his time and effort for

the benefit of his students. Generosity and self-discipline are two words that come into my

mind when I think of him. I hope that one di- his contributions to the Bessel Functions

will be acknowledged.

I would like to thank my mother, Hiirmet K !ri i. my father, Or'-i- K !.i- and my

sister, Hilal Kahya, for their constant support. I would like to thank my wife, Selva; and

my daughter, Hafsa Asude, for making my life enjovi- 1b-

I also would like to thank Professor Pierre Sikivie and Professor A' i, c,- Karasu for

writing letters of recommendation on my behalf.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 6

LIST OF FIGURES .................................... 7

A BSTR A CT . . . . . . . . .. . 8

CHAPTER

1 INTRODUCTION .................................. 9

2 FEYNMAN RULES ........ ........................ 11

3 ONE LOOP SELF-MASS-SQUARED ......... .............. 16

3.1 Contributions from the 4-Point Vertices ......... .......... 16
3.2 Contributions from the 3-Point Vertices ......... .......... 19
3.2.1 Local Contributions ........................... 20
3.2.2 Logarithm Contributions ......... .............. 22
3.2.3 Normal Contributions ........... .............. 23

4 RENORMALIZATION. .................. ............ .. 34

5 EFFECTIVE MODE EQUATION .................. ..... .. 44

5.1 The Schwinger-Keldysh Formalism .................. ... .. 44
5.2 Restrictions on Our Solution .................. ..... .. 47
5.3 Local Corrections .................. ............. .. 50
5.4 Nonlocal Corrections .................. ........... .. 51

6 CONCLUSION .................. ................. .. 56

APPENDIX

A EXTRACTING DERIVATIVES ........... .... . .. 62

REFEREN CES . . . . . . . ... ... 70

BIOGRAPHICAL SKETCH ............. ... . ... .. 73









LIST OF TABLES


Tabl

2-1

2-2

3-1

3-2

3-3

4-1

4-2

4-3



4-4

4-5

5-1

5-2

5-3

5-4


(47r) 2 -.V^-^
Other Local Normal Contributions from Table 4-1.

All Finite Nonlocal Contributions with x .....

Integrals with a'3 . ................

Integrals with a'4 ..................

Integrals with a'5 ... ............... .

The ) (4i )4 d' (Ext. Operator) x {f( ) f(
a (4 4


e

Three-Point Vertex Operators V~a3 contracted into 0102 h . ...

Four-Point Vertex Operators Uap3P contracted into 01' .

Nonlocal Logarithm Contributions from relation (3-37) with x .

Divergent Normal Contributions . .................

Finite Normal Contributions in terms of x .- ......

Local Normal Contributions from Table 3-2 . ...........

Finite Normal Contributions from Table 3-2 with x

Normal Contributions to Counterterms from Table 4-1. All terms are mu
b i2HD-4 r()
b y D7 . . . . . . . .


terms .


A-i Contributions acted upon by a


Contributions

Contributions

Contributions

Contributions

Contributions

Contributions

Contributions

Contributions


acted upon

acted upon

acted upon

acted upon

acted upon

acted upon

acted upon

acted upon


by -

by 71

by 72

by 73

by 6

by e,

by 62

by 63


A-10 Contributions acted upon by (


(aa')DD 2. . . .

(aa )D-2( + a'2)H2. ..

S(aa')DH 4 .. ......

S(a/)D-1(a2 + a'2)H4 .

(aa/)D-1(a + a')2H4.....

(aa ')D-2( + a'2)V .2.

(aa/)D-1H2V2 .......

(aa /)D-2 (a2 + a/2)H2V2.

(aa)D-2(a + a')2H2V2.

(aa/)D-2 4 .. . .


[tiplied


page

12

12

27

32

33

39

40


42


' ' '










LIST OF FIGURES
Figure page

3-1 Contribution from 4-point vertices. .................. ..... .. 16

3-2 Contribution from two 3-point vertices. .................. .... 19

4-1 Contribution from counterterms. .................. ....... .. 34









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

QUANTUM GRAVITATIONAL CORRECTION TO SCALAR FIELD EQUATIONS
DURING INFLATION

By

Emre Onur K !r,-

August 2008

('C! i: Richard P. Woodard
Major: Physics

We computed the one loop corrections from quantum gravity to the self-mass-squared

of a massless, minimally coupled scalar on a locally de Sitter background. The calculation

was done using dimensional regularization and renormalized by subtracting fourth order

BPHZ counterterms. We used this computation of the self-mass-squared from quantum

gravity to include quantum corrections to the scalar evolution equation. The plane wave

mode functions are shown to receive no significant one loop corrections at late times. This

result probably applies as well to the inflation of scalar-driven inflation. If so, there is no

significant correction to the pc correlator that p1l i- a crucial role in computations of the

power spectrum.









CHAPTER 1
INTRODUCTION

My study was understanding quantum loop effects to massless minimally coupled

scalars during inflation. To do that I computed the self-mass-squared of massless

minimally coupled scalars coupled with gravitons at one-loop order. Then I used that

to solve the quantum corrected equations of motion in order to see whether this one-loop

effect alters the dynamics of a scalar inflation.

Quantum loop effects can be understood as the reaction of classical field theory to

virtual particles. The energy-time uncertainty principle tells us that the number density

of these virtual particles, which persist forever during primordial inflation, is enhanced[l]

for the ones which are massless and not conformally invariant. Massless, minimally

coupled scalars and gravitons are the only two particles which have zero mass and are

not conformally invariant. As a result of inflation quantum effects for these particles are

vastly enhanced which is the origin of the primordial scalar [2] and tensor [3] perturbations

predicted by inflation [4, 5]. Weinberg has recently shown that loop corrections to these

perturbations are also enhanced by logarithms of the ratio of the scale factor to its value

at first horizon crossing, although not enough to make them observable [6, 7]. However,

much larger loop corrections (to other things) can be obtained either from interactions

with larger loop counting parameters, or by studying things we would perceive as spatially

constant such as the vacuum energy or particle masses.

Explicit computations have been made on de Sitter background in different models

which involve either scalars or gravitons. A massless, minimally coupled scalar with a

quartic self-interaction is pushed up its potential by inflationary particle production,

thereby inducing a violation of the weak energy condition [8, 9] and a nonzero scalar

mass [10, 11]. The vacuum polarization from a charged, massless, minimally coupled

scalar induces a nonzero photon mass [12, 13] and a small negative shift in the vacuum

energy [14]. The inflationary creation of massless, minimally coupled scalars which are









Yukawa-coupled to a massless fermion gives the fermion mass [15, 16] and induces a

negative vacuum energy that grows without bound [17, 18].

It is natural to extend these studies by combining general relativity with a massless,

minimally coupled scalar to probe the effects of gravitons on scalars. Such an investigation

has great phenomenological interest because the inflation potential of scalar-driven

inflation is so flat that inflation mode functions are effectively those of a massless,

minimally coupled scalar. Indeed, it is standard to compute the power spectrum of scalar

perturbations by setting the scalar-scalar correlator to its value for a massless, minimally

coupled scalar on de Sitter background [4, 5]. The subject of quantum gravitational

corrections to the scalar effective potential has a long history [19, 20] but we will here

look at all one loop corrections to the linearized, effective scalar field equation, including

corrections to the derivative terms and also the fully nonlocal corrections. This is what

one must do to fix the scalar field strength in addition to its mass.

Our result is that the scalar mode functions experience no significant corrections at

one loop. We prove this by solving the linearized, effective scalar field equation. We derive

the Feynman Rules of MMCS+GR in chapter 2. ('! lpter 3 gives a review of one loop

computation of the scalar self-mass-squared [21], which represents the quantum correction

to the linearized effective field equation. In chapter 4 we extract finite terms from the

one-loop results using BPHZ renormalization scheme. In chapter 5 we solve the equation

in the relevant regime of late times. Our conclusions comprise chapter 6.









CHAPTER 2
FEYNMAN RULES

To facilitate dimensional regularization we work in D spacetime dimensions. Our

Lagrangian is,

1 1
--t, ., "' + (D-2)A) g. (2 1)
2 167G

Here G is Newton's constant and A (D 1)H2 is the cosmological constant. Because our

scalar is a spectator to A-driven inflation, its background value is zero. Our background

geometry is the conformal coordinate patch of D-dimensional de Sitter space,

ds2 2 d2+ d&. d) where a(') (22)


Perturbation theory is expressed using the graviton field h,,(x),


9v,(x) = a2 (T + Khl(x)) where K2 167rG (2-3)

The inverse metric and the volume element have the following expansions,


9g p 2 tKhP + hK2 p -...), (2-4)

a aD (+ Kh + 2h2 2 hPhp + ... (2-5)
S8 4

This computation requires the 02h and 02h2 interactions which derive from expanding the

scalar kinetic term,

1 1 11
-t 1t'g g 0= 0 -D-2P-aj,aD -2hPv + IpKh + 2 h2
2 2 2 8

4 2
1r 22hPu ph- l2hh2 + 2&hpphv + O(s3)}. (2 6)

We represent the 3-point and 4-point interaction terms as vertex operators acting on

the fields. For example, the first of the 3-point vertices is,

aD-2 aa hap VP = i=aD-20a02 (2-7)
2 K 03hV3-M 27









I va3
1 i aDD-2 aa2
2 D-2 a
Table 2-1. Three-Point Vertex Operators V, 3 contracted into 0102 ha /

I UjaO3pa
1 K2aD-2 a P1'02
2 K2aD-29ap 01 02
3 IKt2 nD-2 aaPa
4 -i2 aD-2 a I9
Table 2-2. Four-Point Vertex Operators UPI" contracted into 1 li _, .


We number the fields "1", "2", "3", etc, starting with the two scalars and proceeding

to the gravitons. Although we extract a factor of 1 for the two identical scalars, it is

more efficient, for our computation, not to extract a similar factor of 1 for the identical

gravitons of the 4-point vertices. Then we can dispense with the symmetry factor. So our

first 4-point vertex is,


ta2
1 aD2a a[ h
16


tUa3pp 2 D-2 poa_1.02 .
8


The 3-point vertices are listed in Table 2-1; Table 2-2 gives the 4-point vertices.

Three notational conventions will simplify our discussion of propagators. The first is

to denote the background geometry with a hat,


1"
a2


, = aD and R =D(D-1)H2.


(2-9)


Second, because time and space are treated differently in the gauge we shall employ, it

is useful to have expressions for the purely spatial parts of the Lorentz metric and the

Kronecker delta,


and =6& 1 0)o
by~ v/ "OV


(2-8)


, = +

(2-10)








Finally, the various propagators have simple expressions in terms of y(x; x'), a function of
the de Sitter invariant length (x; xl) from x" to x'P,

y(x; x') 4 sin2( Hl(x; x')) = aa'H2 2 ( r/ )2 (2-11)

where a = a(Qr) and a' = a((').
The massless minimally coupled scalar propagator obeys,

a( A,) iAx;x') i 6D(x x'). (2-12)

It has long been known that there is no de Sitter invariant solution [22]. The de-Sitter
breaking solution which is relevant for cosmology is the one which preserves homogeneity
and isotropy. This is known as the "E(3)" vacuum [23], and the minimal solution takes the
form [8, 9],
HD-2 F(D-1)
iAA(x; x') A(y) + kln(aa') where k H- 2D (2-13)
(47) 72 Fr()
The de Sitter invariant function A(y) is [9],

HD-2 f(D1) 4D-1 r(-+D ) 4D-2 D F(D-1)
A(y) z 21 C yOt (D-1)
(4x) 2 D '.) y 2 )(D
SF(n+D-1) y t 1 r(n++ 1) pYn-+2
Sn r(n+D 1) n--+2 F(n+2) 4 1

To get the graviton propagator, we add the following gauge fixing term to the
invariant Lagrangian [24],

GF = + (D-2)Hahpp). (2-15)








We can partially integrate the quadratic part of the gauge fixed Lagrangian to put it in
the form hPvD fhp,o where the kinetic operator is,


+ 0 (Po) 0 7 1 ( -
PV 2" 4 pa 2(Dl-3) Pi tt 66 DA

+p) 6 Di + (D_ ) o60606 Dc (2-16)
(P ") 0 2 \D-3 /' v

The three scalar differential operators are,

DA p (V Pv) (2-17)

DB ( Op t-()-2) R g, (2-18)

DC a (p 2a)(- (2 )VR g. (2-19)

The graviton propagator in this gauge has the form of a sum of constant tensor
factors times scalar propagators,

i [PA,] (x; x') /= T id(x; x'). (2-20)
I=A,B,C
We can get the scalar propagators by inverting the scalar kinetic operators,

D, x i(x; x') i6D(x x') for I A,B,C. (2-21)

The tensor factors are,

[,T, ] D- (2 22)
[p ] = -4 )(6 (2-23)
[PTp] (D 2) [(D 3) (- 3) + 6 + ] [(D-3)6 6+ (2-24)
(D 2)(D 3)

With these definitions and equation (2-21) we can see that the graviton propagator
satisfies the following equation,

D p x i [P,A (x; x') -= 6(b6)ibD(X x) (2-25)









The most singular part of the scalar propagator is the propagator for a massless,
conformally coupled scalar [25],

HD-2 D \ D
iAf (x; x') r 1)( (2-26)
(47r) 2 y/
The A-type propagator obeys the same equation as that of a massless, minimally coupled
scalar. The de Sitter invariant B-type and C-type propagators are,

H -2 F(n+D-_2) y\
iAB(x; x')= iAcf(x; x') 2)
(4T) 2o F(n+D) 4)
F)(n+ D) -+2
2) ((2-27)
F(n+2) \4) '

iAc(x;x') iAcf(x;x') + HD 2 F(n+D3) (yD "


2 F(n+2) 4

They can also be expressed as hypergeometric functions [26, 27],

H 2(D-2)*(' 2 1-1)' (229)
iAB(x;.x') H"D F(D 2)F(1) (D 2,1; ; 1-) (2 29)
(4) 2 F( 2) 2 4

iAc(x; ') D -2F(D 3)F(2) 2F( -3,2; ;1-). (2 30)
(47) 2 F( 2) 2 4
These propagators might look complicated but they are actually simple to use since the
sums vanish in D = 4, and every term in these sums goes like a positive power of y(x; x').
Therefore only a small number of terms in the sums can contribute when multiplied by a
fixed divergence.









CHAPTER 3
ONE LOOP SELF-MASS-SQUARED

This is the heart of the paper. We first evaluate the contribution from the 4-point

vertices of Table 2-2. Then we compute the vastly more difficult contributions from

products of two 3-point vertices from Table 2-1. We do not renormalize at this stage,

although we do take D = 4 in finite terms. Renormalization is postponed until the next

section.





6T



Figure 3-1. Contribution from 4-point vertices.


3.1 Contributions from the 4-Point Vertices

The generic diagram topology is depicted in Fig. 3-1. The analytic form is,
4
im (r; r') = YUi"" x I ["Ai] (r; X) 6D XXI) (_-)
I=
In reading off the various contributions from Table 2-2 one should note that, whereas "02"

acts upon x's, the derivative operator "i1" must be partially integrated back onto the

entire contribution. For example, the contribution from U PP is,

,--2s --, 9a1 a2 x [i/2] (x; x) 6D(Xx')


+i 2a D-2 {aD2i-a AP](X;X)a/J6D(X_ X')} (3-2)









Reading off the other terms from Table 2-2 gives,


K20p aD-2 [aAp (X;X)D(XXl + iK22p aD-2


ax A1L 3 (x; x) 6D (x x') + i 20 D-2 L [ a D _


-I2aa aD-2 apAP] (X; X)a D(X )


(3 3)


It is apparent from expression (3-3) that we require the coincidence limits of each of
the three scalar propagators [28],

imAAX; X) D (D {- cot D) + 21 n(a) (3-4)
x' x (47r) 2 F2) 2

lim iAB(x; x') D x -- -1 -- 2 (3-5)
X'-z (47r) F () D-2 162
HD-2 F(D 1) 1 H2
lim Ac (x; x') x ) -) (3-6)
X'-z (4)) F D) (D -2)(D -3) 1672

Note that the B-type and C-type propagators are finite for D = 4. The four contractions
of the coincident graviton propagator we require are [28],


[i AP (x; x)
i aPUp (x; x)


S[a a] (x; x) -

[a (A]x; x) X


(D D- 3 t7H 2
-4 31 A XI 161r2,
H2
(D-1)(D2-3D-2) A( H2
iA(x; x) 2 X 2
D-3 163r2
4 KXPXU + 2r6p] H2
D- iAA(x; x) + [2i + 2;o x 2 ,
DD-2 3D 2 A6 H2
K D 3A(X; X) + 26 6' X 1
D-3 167


To save space we have taken D = 4 in the finite contributions from the B-type and C-type

propagators.


(3 7)

(3 8)

(3-9)

(3-10)


-iM42pt(X; x')








Substituting these relations into expression (3-3) and performing some trivial algebra
gives the final result,

i2HD-2 F(D -1) ctl
-iM4t (x; x') (4) ( X cot D 4D(D-1)

xa (aD-2a 1D X) DaDx -2I26)D _x)
(47r)2 4 /) \4 /



iKjH {30( 2ln(a) 04(x2_x)) 4ln(a)a2V2t(xx'/)
472

(a264(x-x')) + a2V2764xx')} + O(D-4). (3-11)

Note that each of these terms vanishes in the flat space limit of H -+ 0 with the comoving
time t ln(a)/H held fixed. The reason for this is that the coincidence limit of the flat
space graviton propagator vanishes in dimensional regularization.
In order to combine -iMIpt with the 3-point contributions it is useful to introduce
notation for the scalar d'Alembertian in de Sitter background,

S1 a( g ) 0(aD-2) (3-12)


We also extract the logarithm from inside the d'Alembertian,

Sa2 ln(() aa')a 44(xx') + H2a 464_') (3-13)

With these conventions the final result takes the form,

-iM_,t(x;x) H= (4{ ) [ 1 )D(D 1)- cot( D) -2

+31n(aa') a DO + D7 cot( D) +2-41n(aa') aD-2V2

+9H2aD + O(D-4) 6D(xx) (3-14)















Figure 3-2. Contribution from two 3-point vertices.

3.2 Contributions from the 3-Point Vertices
In this section we calculate the contributions from two 3-point vertex operators. It
is diagrammatically represented in Fig. 3-2. Consulting Table 2-1 and remembering to
partially integrate any derivative that acts upon an outer leg gives,
2 2
-im2t(; x' = V1a(x) V"(x') x I La, (x; x') iAA(x; x') (3-15)
I=1 J= 1
= 2 aa /)D-2. a/ 'Apa 20,0 A /)D-2
= i- { (aa/)^D 2 1 0,0,A'a0-2 ddA +
[a,. /)D-2. a,p 0 ,)--
KaAPUI aaAA} + ^2 { (aa) 2 VA1AA}
2
{ (aa'i-2 a iA8/ } (3-16)

Upon substituting the graviton propagator, performing the contractions and segregating
terms with the same scalar propagators, one finds three generic sorts of terms. The first
are those which involve two A-type propagators,

27V.'/ /)D-2 AA V A ( 000, /D-2 A 00
D-3 aa AA 00 iAA
+n2a 2 + a'/D-2 AA 9 iA-] + ~ [(aa') / 2 AA 0, iAA] (3-17)

The second kind of term involves one A-type and one B-type propagator,

2o00 Laa/D-2 iABVV A -AA 2 ia0 L Daa/D-2 LAB 9O90 iAA]
-20o' (aa/)D-2 LAB 9oi0 iAAI 2VV/ (aa/)D-2 LAB a0o0o iAA] (3-18)








Finally, there is the case of one propagator of A-type and the other of C-type,

2K2 20) 0o [(a/)D-2 LAC 909 iAA (3- 9)
D-3 I A

Each of the nine terms in expressions (3-17-3-19) has the form,

K~2"a [(aa/)D-2i,(; x')a, i1AA(X; x')] (3-20)

where "I" might be A, B or C. Note that the three propagators can be written almost
entirely as functions of y(x; x') defined in (2-11),

iAA(x;x')= A(y) + kln(aa') iA(x; x') = B(y) and iAc(x; x') = C(y) (3-21)

The functions A(y), B(y) and C(y) can be read off from expressions (2-14), (2-27) and
(2-28), respectively. Note also that the inner derivatives eliminate the de Sitter breaking
term of iAA,

ai y ay a2Y
pjAA(X;X') Ojo' ~ D 2(-x') + A"(y + A'(y) xx (3-22)

It follows that the analysis breaks up into three parts:
Local contributions from the delta function in (3-22);
I. ',.n:thm contributions from the factor of kln(aa') in the A-type propagator when
I = A in expression (3-20); and
Normal contributions to expression (3-20) of the form,

2a ai(aalI)D 2(y) AiQy ax'y aa2
K2 (aa A//-2 O A 5 + A/ 2 (3-23)

We shall devote a separate part of this subsection to each.
3.2.1 Local Contributions
These are the simplest contributions. They only come from the 2nd term of (3-17),
the 4th term of (3-18) and from (3-19). To avoid overlap with the logarithm contributions
of the next part we define the local contribution from the 4th term of (3-17) without the








logarithm,


2(D-1\ /2A(y) x 2 D l)] i2HD-2 F(D)
t 3 x aD (47) (D-3)F(D)

x rcot( D -a D D( -) +a-2v2D(x _')}. (3-24)

Note that we have chosen to convert primed derivatives into unprimed, and to absorb the
temporal derivatives into a covariant d'Alembertian D,

0o(aD-2/6D(xxl')) ( _p aD-2,,D(x )) +aD-2V26D(x x') (3-25)
S-aD D (-x') + aD-2 V2 D(-') (3-26)

This will facilitate renormalization.
The other two local contributions are finite. The 4th term of (3-18) gives,

-2 v (aa )D-2B(y) x _(xX)]
ix2H2
-- 2 x aV24_(x x') + O(D-4) (327)
1672
And (3-19) gives,

K2 D -2) I D-2C( ) X -^ 6D( -l
2 D-320 / J a D -
i22 a4D4(X -) a2264(x- _) + O(D-4) (3-28)


Summing the three local contributions gives,

-iM 3 ) X; x') = D [()D cot (D) ] aDD
loc (47) 2 F(D) ID-3 2

+ -( )7 cot D) ]aD-2v2 + O(D-4) 6n(D_-). (3-29)








3.2.2 Logarithm Contributions
These all come from expression (3-17). They can be simplified by using the
propagator equation (2-12),

Oo(aDV-2 oAA(x;x')) -6D x') + aD- 2A(y) (3 30)

S(a/'D -20iAA(x') -i6Dx- x') + d 'D- 2V2A(y) .(3-31)

One can also take the limit D = 4 because all the logarithm contributions are finite. For
example, the function A(y) is,

A(y) 1- 2 n 1 + O(D-4) (3-32)
167 2 4

The first term of (3-17) gives,

K2 / [(aa/)D-2 x kln(aa') x V-V'iAA(;x')]
K2H2
8K2 ln(aa')(aa)2V4A(y) + O(D-4) (3-33)

The second term of (3-17) has the most complicated reduction,

-K2(D- 00o jDaa)D-2 x k ln(aa') x a0oiAAx; x')
D-3
i3K2H2 In(a')- _aa + 22V2}64(x-x')
87w2
3 2 H2 i9 2 H4
2 In(aa')(aa')2V4A(y) 8 a4 4 (x -x')
3,2H3
872 (aa)2(a0o+a' )V2A(y) + O(D-4) (334)

The third term of (3-17) gives,

Ki0, [(aa/)D-2 x klIn(aa') x oLAA ('XA;x')
i2H2 2 K 2 H2
8 2 in(aa)a"V24- ) + 2 n(aa')(aa')4A(y)
K2 H3
+ 2 -(aa')2a' V2A(y) + O(D-4) (3 35)
87 22









A very similar contribution derives from the final term of (3-17),


K2oa0 (aa')D-2 x kln(aa') x i0AA(; x')]
iK2 H 2 H2
8 2 ln( ()a2V2 4 ( ') + ln(aa')(aa' )24A(y)
8wr2 8r2
S2H3
+ 82 (aa')a0oV2A(y) + O(D-4) (3-36)

Combining all four terms results in some significant cancellations,


p (; ) = n(aa') 3a + 4a22x
log 87;,

-9H2a4 x x) 2H(aa')2(a0o+a'0)V A(y) + O(D-4)}. (3-37)


Each of the local terms in (3-37) cancels a similar finite, local 4-point contribution in

(3-14), leaving only the nonlocal contribution involving derivatives of A(y). It is possible

to eliminate the temporal derivatives in this expression. However, the procedure is best

explained in the final part of this section.

3.2.3 Normal Contributions

These contributions are the most challenging. Our strategy for reducing them is to

first extract the ap and 0' derivatives from (3-23) gene, t.:. ll. without exploiting the

functional forms of A(y), B(y) and C(y). We also convert all primed derivatives into

unprimed ones and express the final result in terms of ten "External Operators". This not

only makes it possible to perceive general relations, it also reduces the superficial degree

of divergence of the terms we must eventually expand. And it leaves functions of the de

Sitter invariant variable y(x; x') for which an improved expansion procedure is possible

[29].

This step of extracting derivatives is still quite involved so we shall describe only the

essentials in the body of the thesis and consign the details to an appendix. The appendix

also gives tabulated results for each of the ten External Operators. The final reduction of

these generic tabulated results is straightforward. This subsection closes with a description









of the technique and a pair of tables giving the final potentially divergent and manifestly
finite contributions, respectively.
Our generic method for extracting derivatives requires one to carry out many
indefinite integration of functions of y. We define this operation by the symbol I[f](y),


I[f](y) dy'f ('). (3-38)

If the function F(y) is the product of two propagator functions, then acting two
derivatives on it can never produce a delta function,

)')F(y) = F"(y) + F'(y) (3-39)
axP ax', ( xPax',

It follows that we can express the inner part of the basic normal contribution (3-23) in
terms of integrals of such products,


(y) A ) +A'(y) 2 a 2 [fA"] (y) + a02 I[f'A'](y) (3-40)
A OXP Ox'" OxPOx',7 OxPx'

We must still deal with the final term of (3-40). In conformal coordinates the mixed
second derivative of y(x; x') is [30],


X Xy H2 aa/y606 2a6oHAx, + 2a'HAxspo 2, }. (3-41)
axPax'a

Breaking this up into spatial and temporal components gives,

S=H2a'[2-y+2aa'H2 -] =H2aa' x -2aHAxj (3-42)
axoax'o I oax'
H LH2aa' x -22ij H (343)
-y02 = H2aa' x 2a'HAx, H2 aa'x -2 (3-43)
axiax'0 a xiax'j

One consequence is,

aa'H2Al f(y) -= (D- )I[f](y) [ ) (3-44)
2 4aa'H2








Another consequence is the relations,1


f (y)aOoOaA(y)


f (y)O aoOA(y)

f (y) a0a'A(y)

f (y)88 A(iy)


2
aa2 [f A" (y) + Ha .I2[f'A'](y)
+Haa'{ 2-Ay)[+'A']) (D-a)l [I'A']([I) ,
3, f2 HaQi[f',A,](y),

o812 [f A"](y) + Ha'd J2[f'A'](y) ,

I2 [fA'"](y) 2H2aa'/i [f'A'](y) .


(3-45)

(3-46)

(3-47)

(3-48)


Using these identities it is possible to extract the derivatives from the first of the
A-terms,


VV [(aa)D-2A(y)V-V'A(y)]

= (aa')D-2(V./)212 [AA"](y) 2(D-1)H2(aa')D-1V.V'I[A'2]()

= (aa')D- 2412[AA"](y) + 2(D- 1)H2(aa)D- 12I[A'2]() .


(3 49)

(3-50)


Only the first term in the expansion of 12 [AA"](y) contributes a divergence; we can set
D= 4 in the higher terms. Similarly, only the first two terms in the expansion of I[A'2](y)
can diverge.
It is very simple to convert the primed spatial derivatives to unprimed ones,


(3-51)


1 On the left hand side of relation (3-45)
the delta function.


we mean the naive second derivative, without


Of (y) = -Of (y) .









We already used this relation in reducing the first of the A-terms. For time derivatives it
is useful to note,


Ha (y 2a'HA)

Ha' (y + 2aHA


a/

a
2 + 2)
2 2-a- .


(3-52)

(3-53)


From this follow three important identities. The simple one is,


(0 + a')f(y) H(a+a')yf'()


(3-54)


Another result is,


(a'o + a$) f (/)


2Haa' x aa'H2 11A ; f'(y ) ,
V2
-(D- 1)Haa' f(y) + -I[f() .
2 H


The final identity results from combining (3-54) and (3-56),


(a9o + a'o) f (y)


(a+a')(ao,+)f(y) (a'aO+a o)f(y) ,

H(a+a')2yf'(y) +(D- 1)Haa'f (y) I-[f]() .
2H


We can now reduce the nonlocal logarithm contribution from equation (3-37).

Applying (3-58) gives,

82 H2 62H2
x -2H(aa')24(aa + )V2A(y) 1 '2 12(aa')3H2V2A

-4(aa')2(a+a')2V2(yA') + 2(aa')24I[A] .


(3-57)

(3-58)


(3-59)


The derivative and the integral are straightforward using the D = 4 expansion for A(y)

given in (3-32). The final result is reported in Table 3-1. Of course we have neglected
terms which eventually vanish such as V4y.


(3 55)

(3-56)









External Operator Coefficient of (4)
(aa') H2V2 12 + 24 in x
(aa')2a + a/ ')2H2V2
(aa')2V4 8 Inx 16x lnx
Table 3-1. Nonlocal Logarithm Contributions from relation (3-37) with x y
4.

We eventually want to absorb all double time derivatives into covariant d'Alembertian's,

H 1 (D-2)H 1(
S 2 2)H0 + 2 60(360)
a a a

This is most effectively done with the internal factors of (aa')D-2. For example, consider
reducing one of the mixed A-terms,

a0a [(aa ')D-2A(y)aoa0A(y)

v'2o0 (aa't)D-2ao2[AA/](y) + HV'2 o0 [aD- laD-212 [A'2](y)] (3 61)

S-aD a'D-2V212[AA"](y) + (aa')D-2V412[AA"](y)

+HaD-la'D-2V29oI2[A'2](y) + (D-1)H2aDa'D-2V212[A'2](y) (3 62)

Note also that we can convert a primed covariant d'Alembertian to an unprimed one if it
acts on a function of just y(x; x'),

Of (y) H2 [(y-y2)f"(y) + D(2-y)f'(y)] ['f(y) (3 63)








This is used in reducing the other mixed A-term,


a9, [(aa')D-2A(y)OaQ'A(y)]
V2 0 [(aa/)D-21o2[AA"/]()] I + HV20, [aD-2a'D-112[A'2]()] (3 64)

-aD-2a'DV2I2[AA"](y) + (aa')D-2V412[AA"](y)

+HaD-2a'D-1V20'I2[A'2](y) + (D-1)H2aD-2a'DV212[A'2]( ) (365)

aD- V2 DI2[AA"](y) + (aa')D-2V412[AA"](y)

-HaD-3 a'DV2oI2[A'2]() + a'D -1413[A'2]() (366)
2

The previous point can be summarized by the relations,

[0 [(aa')D-2o0f(y)] a= -DNaD-f() + (aa')D-2V2f(y), (3 67)

O' [(aa')D-2a0f()] -a D-2a'DDf() + (aa')D-2V2f() (3 68)

Another important point is that it is almost .i. ,- best to write any single factor of the
mixed product 0o0' as follows,

1 1 (3169)
aoa (~o + a)2 -2a -a2 (369)
2 2 2

So we find the ubiquitous reduction,

0o [(aa')D-2f(y)] ((aa/)D -20 f(y)
+(D-2)H(aa')D-2(a'Oo+aO')f (y) + (D- 2)2H2(aa')D- f(' ) (370)

(aa/)D- (a2 + a2) [ODf()+H2 ?y)] aa/)D-2V2f()
1 1
+(D- 2)(aa')D-2V2JI[](y) + (D-2)(D-3)H2 (aa/)D- f(/)
+H24)2 ) .2(
+ tH2(a+a')2(aa')D-2[(D- )yfl(y)y2f/(y)] (3 71)
2








Another example is the two B-terms,


i, [(aa/r)D-2 B(y)iA(y)] + 0o [( (aa')D-2B(y)aiA(y)]
a (a0 o+a) [(aa )D-2 (y) (ao + 0)aA(y)]
-ao0 [(aa)D-2 B(y)iA(y)] aa [(aa')D-2B(y)a 8 A(y) (3-72)
S-(a + '2) (aa/')D-27V2 {D[A"B](y) + H2I[A'B+ yA"B](y) }

+(aa/)D-24 {2J2[A"B](y) 2[A'B'] (y) H2(a+a/)2(aa/)D-2V2
x { (D- 2)I[A'B+yA"B] (y) +yA'(y)B(y) + y2A(y)B(y)
-yI[A'B'](y)} + (D-1)H2(a2 aa'+a'2)(aa )D-2V212[A'B'](y) (3 73)

Extracting derivatives in this way from the various normal contributions results in
functions of y which are acted upon by ten external operators,

a (aa')D2 (3 74)

/ = (aa')D- (a2 + a'2)H2 (3-75)
71 (aa)DH4 (376)
72 = (aa)D-(a2 + a'2)H4, (3 77)
73 (aa/)D-l(a + a')2H4 271 + 72, (3 78)
6 (aa')D-2(a2 + a'2)V2 (3 79)

c1 (aa')D-1H2V2 (3-80)

62 (aa/)D-2(a2 +a'2)H22, (381)
C3 (aa/)D-2(a + a')2H2V2 2c1 + 2 (3-82)
( (aa')D-24. (383)

Tables A-i-A-10 of the Appendix give explicit results for each of these ten operators. Note
that in addition to the three propagator functions A(y), B(y) and C(y), we also employ








the following less singular differences:


AB B A and AC 2( (C A) (3-84)

The next step is substituting the explicit forms (2-14), (2-27), (2-28) for the
propagator functions into the results of Tables A-1-A-10 and expanding to the required
order. To understand what this is, note that we will be integrating the result with respect
to x against a smooth function (the zeroth order mode solution) with the derivatives
of the "External Operators" acted outside the integrals. Because y(x; x') vanishes like

(x x')2 at coincidence, it is only necessary to retain the dimensional regularization for
terms which would go like 1/y2 and higher for D = 4.
Although these tables involve a bewildering v ii' I v of different integrals and
derivatives, careful examination of the results shows that they derive from just eight
products of the propagator functions,

A'2 AA" A'B' A"B A'AB' A B, AA'AC' and A"AC (3-85)

The most singular products of A'2 and AA" ahl--iv appear either doubly integrated
e.g., 12[AA"] in Table A- or else integrated once and then multiplied by y e.g.,
- I[A'2] in Table A-2. Hence we need only retain the dimensional regularization for the

1/yD terms of these expansions,

F 2(D) H2D-4 4)D (43 (4)2 D-4)
A 2y +4 +4 + O 3 (3-86)
16 (4w)D y/ y/3 Y Y
A" 2(D)H2D-4 D 4 D 4)3
AA/" 2 In_
16 (47 )D D-2 y y 4

-4(4) In -2(4) o(D2 + (3-87)
y 4/ D y

The product A'B' can appear with only a single integration -e.g., DI[A'B'] in Table A-2
-or multiplied by a single factor of y e.g., DyA'B' in Table A-4. We must therefore








retain the dimensional regularization for the 1/yD-1 term,
F2( D) H2D-4 (4D 4D-1 4
A'B' 1 )D- + (D -2)( +O(D-4 (3 88)
16 (47r)"D (Y// (// \/2
However, the product A"B is .i. li,- shielded by two or more powers of y, so the
expansion we require for it is,
2 D) H2D-4 4 D 4 3 D-4 (389)
A"B = (7) D 4 O (3-89)
16 (4)D -2 y 3
The products involving AB and AC are less singular,
PF2) 2D4 D1 H4 42 (D_4~1 (390)
A'AB' 2T -2 -4 +- ( + O (3-90)
16 (47)D /2 j
2D 2) H2D-4 ( Y3 Q (4)3
A//AB ) 2 4- In l + 2 -
16 (4x)D L 4 4

+4(4)2n() + 2(4) + O( (3-91)

p2( ) H2D-4 (4)D-1 (4)2 D-_4
A'AC' ( 2 -8 1( +o (D-4 (3-92)
16 (4x)D y2 ) Y
P2 D) H2D-4 i4,3 ,4 3
A"AC=- 2 6 In + 8 -
16 (4x)D y 47
16(4)2( +(4)2 o(D 4)} (3 93)

One next substititues these expansions into the totals of Tables A-1-A-10 and
performs the necessary integration, differentiations, multiplications and summations.
We must also multiply by the overall factor of K2. For example, the result for "External









Ext. Op. Coef. of D 4- (D)( -1 Coef. of (7 p2D Y-2
D
a0 0 (D-1)(D-2)2
o D D3-3D2-4D+8
4(D-1) 4(D-1)(D-2)
D(D-2) D3-3D2-4D+8
71 4 4
D D3-3D2-4D+8
72 4 4(D-1)
D 0
73 4
0 0
(D2-6D+4)
1C 0 2(D-1)(D-2)
(1-2D)
C2 0 (D-1)(D-2)
C3 0 0
i0 0
Table 3-2. Divergent Normal Contributions.


Operator" a is,


2I2[AA//"] + I2[A"AC]

F 2D) 2H2D-42 D (4 D 4 3 4
16 (4) t 2 + 12 In + 8

(4)2 4 )42 D-4\
+12(- ln ) +6(- +0( -, (3-94)

K2H2D-4F2 (D) D 4 D-2 4) t (4)
(47r)D 2 (D 1)(D 2)2y +6() (y +13

-6 I2( 18 n( + 0(-. (395)


We have tabulated the results for each of the ten "External Operators". Table 3-2 gives

the quadratically and logarithmically divergent terms; Table 3-3 gives the terms which are

manifestly finite. In all cases the expressions were worked out by hand and then checked

with Mathematica [31].






























Ext. Op. Coefficient of 4

a 61n+13- 61n2x -8nx

/ 5 -18n x
x
1 3 108 n x 36

72 18

73 -5 36
6 4 In 49 +1n2 x +10 nx + 12x lnx
x 6x
i + 601nx 120x In x + 72x
12 -12 Inx + 12x lnx
2 6x
3 1+ 36xInx + 12x
10 24xln x + 242 n x 36X2
3
Table 3-3. Finite Normal Contributions in terms of x y
4"









CHAPTER 4
RENORMALIZATION

In this section we obtain a completely finite result for the self-mass-squared

by subtracting 4th-order BPHZ counterterms [32]. We first identify two invariant

counterterms which can contribute to this 1PI (One Particle Irreducible) function at

one loop. Because our gauge fixing functional (2-15) breaks de Sitter invariance [24], we

must also consider noninvariant counterterms. We identify the only possible candidate

based on a careful discussion of the residual symmetries of our gauge fixing functional. It

remains to collect and compute the actual divergences. Contributions from the 4-point

vertices are already local, as are the "local contributions" from the 3-point vertices. Using

a now standard technique of partial integration [8] we segregate the divergences from the

ii .. in i, contributions" of Table 3-2. In the end we identify the divergent parts of the

three counterterms and report a completely finite result.






Figure 4-1. Contribution from counterterms.


One renormalizes the scalar self-mass-squared by subtracting diagrams of the form

depicted in Fig. 4-1. Because our scalar-graviton interactions have the form ~"h"Q00,

compared to the K"h"Ohah interactions of pure gravity, the superficial degree of divergence

at one loop order is four, the same as that of pure quantum gravity. Of course the

corresponding counterterms must contain two scalar fields, each of which has the

dimension of a mass. Because we are dealing with one loop corrections from quantum

gravity, all these counterterms must also carry a factor of the loop counting parameter

K2 = 167G, which has the dimension of an inverse mass-squared. Each counterterm must

therefore have an additional mass dimension of four, either in the form of explicit masses









or else as derivatives. The term with no derivatives is,


2 4 2 g (4 1 )

There is no way to obtain an invariant with one derivative. Two derivatives can act either

on the scalars or on the metric to produce a curvature. We can take the distinct terms to

be,

K 2gd.,. 20 0--g and K2m2 2R g (4-2)

There are no invariants with three derivatives. By judicious partial integration and use of

the Bianchi identity we can take the distinct terms with four derivatives to be,


g 1911'1. PgVP g K2tQaRg^V Og k t4 vpR" g

2 l2R 2 -g and K22RvR1g (4-3)


Because our scalar is massless and mass is multiplicatively renormalized in dimensional

regularization, we can dispense with (4-1) and (4-2). The last two counterterms of

(4-3) cannot occur because the unrenormalized Lagragnian (2-1) is invariant under

0 0 + const. The second and third terms of (4-3) become degenerate when one uses

the background equation, R, = (D-1)H2',. In the end just two independent invariant

counterterms survive, each with its own coefficient,

1 1
Sai1200 0 D and 2- a2K2H20/Q00aD2 (4 4)

The associated vertices are,


c1t K2 DOOaD ia1 2 aDD26D(x-x) (45)

2K2 H2 aD-2 a2K2H2aD6D(Xx') (4 6)


Had our gauge condition respected de Sitter invariance, all the divergences in

-iM22(; ') could have been absorbed using (4-5) and (4-6) with appropriate choices

for the divergent parts of the coefficients ac and a2. Although the reasons for it are not









completely understood, there seems to be an obstacle to adding a de Sitter invariant gauge

fixing functional [24, 33, 34]. This is why we employ, ,1 the noninvariant functional (2-15).

We must therefore describe how de Sitter transformations act in our conformal coordinate

system and which subgroup of them is respected by our gauge condition. The D (D+1)

de Sitter transformations can be decomposed as follows:

Spatial translations -(D-1) distinct transformations.

r]7 r] x'i = xi + Ci (4-7)


Rotations (D- 1)(D- 2) distinct transformations.


]' = ] x' = Rjx (4-8)

Dilatation 1 distinct transformation.


]' = k x'i kx (4-9)

Spatial special conformal transformations -(D-1) distinct transformations.


1'/ = x1i -. (4-10)
1- 2 -. + 1i,11- ,-x 1- 20- + 1i,11- ,-x

It turns out that our gauge choice breaks only spatial special conformal transformations

(4-10) [18]. Hence we can use the other symmetries to restrict possible noninvariant

counterterms. Spatial translational invariance means that there can be no dependence

upon x' except through the fields. Rotational invariance implies that spatial indices

on derivatives must be contracted into one another. Dilatation invariance implies that

derivatives and the conformal time T] can only occur in the form a-1,.

We can alv--i- use the invariant counterterms (4-5-4-6) to absorb a 02 in favor of V2

and a single 0o,

1 1 H- V2
] [-0 (D-2)Hao + V2] 02 = (D-2) -0 + (411)
a2L J (a a









We can also avoid (80c)2,


,(00 2 1 0 2 V0 ,. + Vp (4 12)
a2 a2 a2 a2

One might think we need HaD-10ocpOlp, but a partial integration allows it to be written in

terms of an invariant counterterm and one with purely spatial derivatives,

HaD-1pp HaaD10D-30cp H D 0, r" H2a D-2(000)2 (4 13)


1
= HaD-3do ({9^9),F + HUdaa. ,,'"* /-9 VaD-.V^ V (4-14)

S(D 1)H2, ,.,i" g H2aD-2V V (4 15)
2

Another term one might consider is HaD-30o~V2cp, but it can be partially integrated

(twice) to give purely spatial derivatives,

HaD-3oa0V2 -HaD-30oVy Vp (4-16)

--HaD-300(V. V.) (4 17)
2
(D 3)H2aD-2V V (4-18)
2

Based on these considerations we conclude that only three noninvariant counterterms

might be needed in addition to the two invariant ones,

K2 aD-2ngV2 2 a-4V2V2 and 2H2aD 2V* V2. (4-19)
2 2 2

Because our gauge fixing term (2-15) becomes Poincar6 invariant in the flat space limit of

H -- 0 with the comoving time held fixed, any noninvariant counterterm must vanish in

this limit. Hence we require only the final term of (4-19). The vertex it gives is,

1 V V
2 ,.2H2D a2H2aD-272VD(X-') (4-20)
2 a a

The structure of the three possible counterterms serves to guide our further reduction

of -iM2(X; x'). First, we must convert all the factors of a' into a on the local terms.

Second, we see that factors of HaD-3V2 o6D(x x') are not possible. Finally, it is not









possible to get a divergence proportional to H3aD- M0D(x x) after using the delta

function to convert all the factors of a' into factors of a.

It is now time to collect the divergent terms from the previous two sections. Those

from the 4-point contributions, and from the "local" 3-point contributions are already in a

form which can be absorbed into the three counterterms. However, we must still bring the

Sii. ii ii" 3-point contributions of Table 3-2 to this form. Recall that these terms involve

powers of y that are not integrable for D = 4 dimensions,

(4)- and (4)-2 (4-21)
y Y

Our procedure is to extract d'Alembertians from these terms until they become integrable

using the identity,

Of(y) = H2 [(4y -y2)f"(y) + D(2-y)f'(y)]
7zD4 H 2-D
+Res 4y 2 H2 D (Dx-x') (4-22)
e Y 'i r ( f 1 ) ^ g
Here Res[F] stands for the residue of F(y); that is, the coefficient of 1/y in the Laurent

expansion of the function F(y) around y = 0.

The key identity (4-22) allows us to extract a covariant d'Alembertian from each of

the nonintegrable terms,

(4\D-1 2 4 D-2 4 D-2 (423)
S(D- )2 H2y D-2 y
(4)D-2 2 O 4D-3 4 4 D-3 (4-24)
y) -(D-3)(D-4) H2y D-4 y)

We could use (4-24) on (4-23) to reduce them both to the power l/yD-3. The power

l/yD-3 is integrable, so we could take D = 4 at this point were it not for the explicit

factors of 1/(D-4).

To segregate the divergence on the local term we add zero in the form,

O 4 -1 DD 4 --1 (4) H-D "
0 4) -1 D(D ~ 1 (bD ) ( '). (4 25)
H2k y2 2K 2 /y/ F(D ) aD










External Operator Coef. of i2 (D 4 (D-) r
(47w)D/2 (D-3)(D-4) aD
D
a (D-1)(D-2)
D D o (D+2)(D-4)
2(D-1)(D-2) H2 4
D2 O (D4-5D3+16D-16)
71 2(D-2) H2 4
(D-2)(D3-3D2-4D+8)
72 4(D-1)
6 0
(D2 -6D+4)
C1 2(D-1)
(1-2D)
2 (D-1)
Table 4-1. Local Normal Contributions from Table 3-2.
Table 4-1. Local Normal Contributions from Table 3-2.


Using (4-25) in (4-24) gives,

(4)D-2 2
Y (D-3)(D-


H-D D)
(D-3)(D-4)F


(4)2H- DjD X') 4D-3
4) Fr(D ) D + H2[y
4 4 -3 D(D-2) 4\
D-4(y) y 8(D-3) Y)

SD(-) f4D Y
(D) 2) D H2

+2() In() + O(D
Y 4 Y-


1}






-4).


The analogous result for the quadratically divergent term is,

(4D-1 iH-(47) 2 D (x -x) 12 4In
(4) (D-3)(D-4)F() D-2H)2 D 2 H4 (y
H]2 4 2 H4 4


(4-26)





(4-27)







(4-28)


The divergent local terms that result from applying (4-27) and (4-28) to Table 3-2 are

reported in Table 4-1. Table 4-2 gives the corresponding finite terms. In each case we have

eliminated the redundant External Operators 73 = 271 + 72 and C3 = 2ei + C2.

The next step is to reexpress the local terms of Table 4-1 as local counterterms.

This is done by using the delta function to convert all factors of a' from the External









External Operator Coefficient of (47,4

SH2 I 3x 3x 3x
in2 [11] [ + I




6 0
di+ H2[1 n x1 2ln 1
2 HV2 6 3 63
J_________________________H2 [ 7111x 7lnx
0
D [1nx] _71na + 7
2___ __ 6x U_______ 3x6x
0
Table 4-2. Finite Normal Contributions from Table 3-2 with x .


Operators into factors of a, and then passing all factors of a to the left. In most cases this

is straightforward but /3- and 3 require the following identities:


(aa')D- (a2 + a'2) 2[a-D6D(x ') = 2aDD2 -12H2 aD

+8aD-2H2V2 + 2(D2-2D+2)H4aD] D(x-') ,

(aa')D-(a 2+a/)- 2)H2 [a-DD(x -_') 2aD(H20 H4)D(x ')


(4-29)

(4 30)


Our results for the three possible counterterms (4-5), (4-6) and (4-20) are reported

in Table 4-3. Note that the contribution to (4-5) vanishes, as it must because this

counterterm happens to be zero in flat space.

Another important consistency check comes from the local terms proportional to

it2H4aD6D(x-x'), which are reported in Table 4-4. Recall that a counterterm of this form

is forbidden by the symmetry Q -- 0 + const of the bare Lagrangian (2-1). Although three

of the four contributions to Table 4-4 diverge, their sum is finite for D = 4. It doesn't

vanish because the A-type propagator equation implies,


iaD6D(x-x') ( aa)D {A(y) D-)kH2


(4 31)


Because the total for Table 4-4 is finite one can take D = 4 and then use (4-31) to

subsume the result into finite, nonlocal terms of the same form as have already been








reported in Table 3-3,


Table 4 4 2HD-4 (-D4+5D3 -16D+16)F(D) aDH4hD(x-x') (4 32)
Table 4 4 =2 XDH4 D /) (432)
(4-) 4(D -2)(D- 3)
t2H4
82 x ia4(x-x') (4 33)
862

4 (aa')4H2D[2 x 41n -(aa)4 12 (4-34)
(47)4 x n 4

Table 4-5 includes this with the similarly finite results of Tables 3-1, 3-3 and 4-2.
Our final result for the regulated but unrenormalized, one loop self-mass-squared
derives from combining expressions (3-14), (3-29), and the local parts of (3-37), with
Tables 4-3 and 4-5. It takes the form,

,2 (x; x') i= 2aD (1D2 + 32D +3 a -D(x -')+ Table 4- 5 +O(D-4) (4 35)

The coefficients fi are

1= 0, (4-36)
HD-4 (-D3+ D-4)F(D 1) (D+1)(D- 4)F(D) cot(ID)
02 =- 2 (D (4-37)
2 (4D7 4(D-1)(D-3) 4(D-3)F() 3)

D-4 + O(D-4) (438)
(47) 3
HD-4z 4 58 }
(4 D-4 + + 27 + O(D-4) (4-39)
( 4) 7 D -4 3

(Here 7 ~ .577215 is Euler's constant.) The obvious renormalization convention is to
choose each of the three ai's to absorb the corresponding ,i, leaving an arbitrary finite
term Aai,
at = -,i + Aoa (4-40)

We can now take the unregulated limit (D = 4) to obtain the final renormalized result,

iM (x; x') = i2U4(Aa12 + A AoD + 4A) (x x) + Table 4 5. (441)
Va 2J
































Normal Contributions to Counterterms from
b i2HD-4 (D)
y (47) (D-3)(D-4)


Table 4-1. All terms are multiplied


Contrib. from Coef. of iKH (D-3) ) x aDH44D(x -')
(47-), (D-3)(D-4)
0 _D(D2-2D+2)
'7 f2 (D-1)(D-2)
(D+2)(D-4)

(D4-5D3+16D-16)
1 4
(D-2)(D3-3D2-4D+8)
I2 2(D-1)
Total (D-4)(-D4+5D3-16D+16)
4(D-2)
Table 4-4. Other Local Normal Contributions from Table 4-1.


From a 2 (x x') aD H2 D (-x') aD-2H2X2D( x')
D 0 0
a (D-1)(D-2)
O0 D 6D 4D
H2 (D-1)(D-2) (D-1)(D-2) (D-1)(D-2)
/3 0 (D+2)(D-4) 0
2
71 2 0 D2 0

e 0 0 (D2-6D+4)
1 2(D-1)
C2 0 0 (2-4D)
(D-1)
Total 0 (D-4)(-D3+D-4) (D3-16D2+28D-16)
2(D-1)(D-2) 2(D-1)(D-2)


Table 4-3.






















External Operator Coefficient of 2H4
(47)4
(aa')43 /H2 nx
(aa')4D2 261nx + 3- 61n2x-181nx
(aa')4H2D 61n + 4 41nx
(aa')4H4 4 + 18 120 1081nx
(aa'3( + a123/H2 6
(aa')3(a2 + a'2)D2 -2 x + 1
3x 67
(aa')3 (a2 + a2)H2E 2I llx 5 lnx
(aa')3(a2 + a'2)H4 41x 32 54
(aa')3H2V2 -21 16 + 841nx 48xlnx + 96x
(aa')3V2 Inn
3x
(aa)2(a2 a'2)H2V2 7 + 12lnx + 48xlnx + 12x
( a a ( + a ) 3 T X
(aa')(a2 + 2)v 2D 1x + 41n2 x + 10ln x + 12xlnx
(aa)2V4 In x 24x In x + 242 In x 36x2
Table 4-5. All Finite Nonlocal Contributions with x
Table 4-5. All Finite Nonlocal Contributions with x -
4.









CHAPTER 5
EFFECTIVE MODE EQUATION

This is the heart of the paper. We begin by clarifying what is meant by the effective

mode equation, then we explain the restricted sense in which we solve it. Finally we work

out the contributions from the local counterterms in (4-41) and from the nonlocal terms of

Table 4-5.

5.1 The Schwinger-Keldysh Formalism

We seek to find plane wave mode solutions to "the effective field equations."

Although the quoted phrase is common parlance, it is nonetheless ambiguous because

there are different sorts of effective field equations whose solutions mean different things

in terms of the unique canonical operator formalism. Introductory courses in quantum

field theory typically concern the in-out effective field equations. The plane wave mode

solutions of these equations give in-out matrix elements of commutators of the full field

with a tree order creation operator of the in vacuum,


) io( r; ) Oout [(r), ain()] i (5-1)

This quantity is of great interest for flat space scattering problems but it has little

relevance to cosmology where there may be an initial singularity and where particle

production precludes the in vacuum from evolving to the out vacuum.

The more interesting cosmological experiment is to release the universe from a

prepared state at finite time and let it evolve as it will. The mode solutions of interest to

this experiment are the expectation values of commutators of the full field with the tree

order creation operator of the initial vacuum,


Nx;k) K0 [4(x),at(k)] 2). (5-2)

The effective field equation that 4((x; k) obeys is given by the Schwinger-Keldysh

formalism [18, 30]. This is a covariant covariant extension of Feynman diagrams which









produces true expectation values instead of in-out matrix elements [35-38]. Because there

are excellent reviews on this subject [39-42], we will confine ourselves to explaining how to

use the formalism.

The chief difference between the Schwinger-Keldysh and in-out formalisms is that

the endpoints of particle lines have a polarity. Therefore, every propagator iA(x; x')

of the in-out formalism gives rise to four Schwinger-Keldysh propagators: iA,+(x; x'),

iA+ (x; x'), iA (x; x') and iA (x; x'). Each of these propagators can be obtained

by making simple changes to the Feynman propagator. For our model, the Feynman

propagators of the scalar and graviton happen to depend upon the length function y(x; x')
defined in expression (2-11), and also upon the two scale factors. The four polarities

derive from making the following substitutions for y(x; x'):





iA (x; x') : y y (x; x') a(Q)a(Q')[- '2 (- _T-/ i )2] (5-6)
iA (x;x') y y- i (x; x') a(ll)a'') [il 2 (I-'+iM)2] (5 6)


Vertices in the Schwinger-Keldysh formalism either have all + lines or all lines. The +

vertex is identical to that of the in-out formalism, whereas the vertex is its conjugate.

Because any external line can be either + or -, each N-point one particle irreducible

(1PI) function of the in-out formalism gives rise to 2N 1PI functions in the Schwinger-Keldysh

formalism. The Schwinger-Keldysh effective action is the generating functional of these

1PI functions.








We can express the effective function in terms of fields o+, to access the + lines, and
o_, to access the lines,

[+, ] S[ S[y_] Jd4xJ d4x'

S +(x)M (x; x'I c+(x') + +(x)M2 (x; x')- (x) +,(5 7)
< +0(91), (5-7)
+ (x)M2 (x;x')c+(x') + p_(x)M2 (x; x' x')

where S[K] is the classical scalar action.
At the order we are working, -iM2 (x; x') is the same as the in-out self-mass-squared.
We can therefore read it off from (4-41). We get -iM2 (x; x') by dropping the delta
function terms, reversing the sign and replacing y(x; x') by y (x; x') in Table 4-5. The
other two 1PI 2-point functions derive from conjugating these two,

-iM2 x;x') -M2 x;x) -iM2 (- IM2 x;x'). (5-8)

To get the Schwinger-Keldysh effective field equations one varies the action with
respect to the field of either polarity, then sets the two polarities equal to 4(x). At
linearized level this gives,

a4D x) j d'fdx' M/ 3 2 (X x') + M+2 (x;x')} I x') 0. (5-9)

Here i = -1/H is the initial conformall) time at which the universe is released in free
Bunch-Davies vacuum. One can see from relations (5-3) and (5-4), and from the extra
conjugated vertex in M (x; x'), that the bracketed term vanishes for rl' > rl. The fact that
y (x; x') is the complex conjugate of y++(x; x') for Tl' < Tr means that the bracketed term
is real. One also sees that it must involve the imaginary part of at least one propagator,
which means the only net effect comes from points x'" on or inside the past light-cone of
x". Hence the Schwinger-Keldysh effective field equations are real and causal, unlike those
of the in-out formalism.









5.2 Restrictions on Our Solution

Two limitations on our knowledge impose important restrictions on the sense in which

we can solve (5-9):

1. We only know the scalar self-mass-squared at one loop order; and

2. We took the initial state to be free, Bunch-Davies vacuum.

The first limitation means we must solve (5-9) perturbatively. The full scalar self-mass-squared

can be expanded in powers of the loop-counting parameter K2 = 167G,


M2 (x;x') + M2 (x') J (x; x) (5-10)
= 1

A similar expansion applies for plane wave solutions to (5-9),
00
i(x; k) = Z (TI, k) x ik (5-11)
= 0

To make K(x; k) agree with (5 2) we must normalize the tree order solution appropriately,


o(, k) u(, k) H (1 ) exp [ (5-12)
\2k 3 aH/ aH

The > 1 solutions obey,


a2 Lo + 2Ha0 + k2 t,(T, k) = jd'J d3x'M x x')-k( Q', k)ekik)('-x) (5-13)


We know only M2\(x; x') so the sole correction we can compute is Ki(I, k).

The second limitation means it only makes sense to solve for +i(q, k) at late times,

i.e., as Tr 0-.1 Interactions result in important corrections to free vacuum on a

flat background and it is unthinkable that this does not happen as well for de Sitter



1 This raises the issue of how applicable our results are to using the mode functions for
the power spectrum of a realistic model in which inflation ceases before the conformal time
reaches zero. We shall comment further at the end of this sub-section. For now, note that
a mode which experiences horizon crossing 30 e-foldings before the end of inflation would
still experience the scale factor growing by an additional factor of about 1013.









background. In the Schwinger-Keldysh formalism these corrections would correspond

to vertices on the initial value surface [30, 43]. In the in-out formalism the free vacuum

is automatically corrected by time evolution. One can follow the progress of this in the

Schwinger-Keldysh formalism by isolating terms that decay with increasing time after the

release of the initial state. For example, the two loop expectation value of the stress tensor

of a massless, minimally coupled scalar with a quartic self-interaction gives the following

energy density and pressure [8, 9],

AH4 1 1 1 (+2)a-"-1
p 2 Iln2(a) + (++2 + O(A2) (514)
S (2)4 (t8a 3 8 (n+8 )2
AH4 1 1 1a-
p n2(a) In(a)- (n2 a + O(A 2) (5-15)
(27)4 8 t2 T4 (n+1)2

We suspect that the (separately conserved) terms which fall like powers of 1/a can be

absorbed into an order A correction of the initial state. On the other hand, the terms

which grow like powers of In(a) represent the effect of inflationary particle production (in

this case, of scalars) pushing the field up its quartic potential.
Because we have not worked out the order K and K2 corrections to Bunch-Davies

vacuum, we cannot trust corrections to )l(T, k) that fall off at late times relative to the

tree order solution u(qr, k),

H k2 ik3 k4
u_2k3 ^2H2a2+2 3H3 a 3 + H4 a4)1

To understand what this means, it is best to convert equation (5-13) for 4t from

conformal time r to comoving time t In(-Hr/)/H,

+ 3H + 1 =i -4 T' u(l', k)fd3x'Mi/ x X 'x '- ). (5-17)
at2 Ot a2 a 4 1

Because the factor of k2/a2 on the left hand side redshifts to zero at late times, its effect

on the late time limit of )i can be at most a constant (about which more shortly). Given

a putative form for the late time behavior of the right hand side it is easy to infer the









leading late time behavior of 41, for example,


r.h.s- C ln(a) 4= In2(a) (5-18)
C
r.h.s- C I(H ln(a), (5-19)
3H2
ln(a) C ln(a)
r.h.s C = K1+Constant (520)
a 2H2 a
1 C(7 1
r.h.s C- 4 )1 Constant (5-21)
a 2H2 a

The only effects which can be distinguished from (the currently unknown) corrections to

the initial state derive from contributions to the right hand side that fall off no faster than

1/ n(a). It turns out that inverse powers of In(a) cannot occur, so the practical dividing

line is between contributions to the right hand side which grow or approach a nonzero

constant and those which fall off.

Of course ,;, correction to the left hand side of (5-17) is liable to induce a

time independent shift in the late time limit of )1. Shifts of this form which are also

independent of the wave number k could be absorbed into a field strength renormalization

and are unobservable. However, significant k-dependent shifts could induce observable

tilts into the power spectrum and are an important potential outcome from a study of this

sort. The possibility for such a k-dependent shift provides an excellent justification for the

laborious task of working out the order K and K2 corrections to Bunch-Davies vacuum.

It is easy to see that only a small range of k-dependent, constant shifts in t) can be

significant. First, note that all one loop corrections are suppressed by the loop counting

parameter, whose largest value consistent with the current limit on the tensor-to-scalar

ratio is GH2 < 10-12 [44]. Next, observe that any constant shift in tI) can be expressed as

the late time limit of the tree order mode function times a dimensionless function of the

ratio k/H,

Constant H x f (5-22)
,V2k3 /2H









Observable modes experienced first horizon crossing (which means k ~ Ha) within the last

60 e-foldings of inflation, hence they must have huge values for k/H under our convention

that inflation begins with a = 1. One cannot get any more that (k/H)2 because then

the flat space limit would diverge. In fact, even (k/H)2 is ruled out because it would give

rise to a Gk2 correction in the flat space limit, which is not seen. So the only chance for a

significant effect seems to be a single power of k/H, possibly times logarithms of k/H. We

will see an example of such a correction in the next sub-section, although it seems likely

that this particular correction can be nulled by an appropriate correction to the initial

state.

5.3 Local Corrections

Because the scalar d'Alembertian annihilates the tree order solution, only the third,

noncovariant counterterm makes any contribution to (5-17),

t J d x' a4 (Aaiw2 + Aa2D + Aa3_) 6(x -x')
a 4 a 2 1 2

XU(, k/,~i( -A' u(_, k) (5 23)
a2

Because this term rapidly redshifts to zero, we see from the preceding discussion that it

can only shift Ki by a (possibly k-dependent) constant at late times. It is easy to work

this out explicitly.

Substituting (5-23) into (5-17) gives,
aa k2 u(qk)
+3H + A Aa3 k2 x (5-24)
at2 Ot a2 a2

The solution can be written as an integral over co-moving time,

A(q, k) A 1.2 dt' G(t, t'; k)( k(5-25)
Jo a2 )

where the Green's function is,

G(t, t'; k) = -t) [(u(, k)u*(q', k) u*(q, k)u(q', k) (5-26)
WG (, k)I
w kl J>ii["" ~i~ii U(ii(1ii 56









Here the Wronskian is,


H2 -2ik3
W(t', k) (Tr', k)u*(', k) u(r', k)u*(', k) 2k3 H2a3( ,' (5-27)


and a dot denotes differentiation with respect to co-moving time.

The integrals in (5-25) are straightforward and the result is,

A 4,(,, k)
Aca3


2 k H2a(T/') I 2Ha(T/') 0
SH 3 i3k k2 1 ikHa
/2k[4 4Ha 2H2a2

+ )u(,k) + + W)e'2k/Hu* *(,k) (5 29)
2H 2k 4 2k

Taking the late time limit reveals a correction of order k/H,

lim A4i (, k) A= 3 {H ik 3 iH l 2ik/H iH 2ik/H } (5 30)
lira ,(/,,k) -Aa3 + -+-C + .5
t0oo \2k3 2H 4 2k 4 k

The difficult but crucial question is, can this correction be subsumed into a redefinition of

the initial state? Although we have no proof, the strong suspicion is that it can be. The

order k/H term in (5-30) derives entirely from the homogeneous, lower limit terms on the

last line of (5-29), and this is precisely the property that an initial state correction would

have.

5.4 Nonlocal Corrections

Although Table 4-5 might seem to present a bewildering variety of nonlocal

contributions to (5-17), a series of seven straightforward steps suffices to evaluate each

one:








1. Eliminate any factors of 1/y using the identities,

SH2 In(l +3, (5-31)

In( ) H2 In2 ( -In()} + 31n ) 2 (5-32)
y 4 HO 2 4 4 4

2. Extract the factors of D and V2 from the integration over x'" using the identities,

D 2 92 k2
H2 a2 + 4a + a2 ] (5-33)
H2 a2 Oa a2H2
V2 k2. (5-34)

3. Combine the ++ and +- terms to extract a factor of i and make causality manifest,

In Y++- In ) 2 IO(A- Ax) (5-35)

In2(Y++) 1n 2(Y+) 4 ( (AT/- Ax) In(aa/H2 (A2 2) (5-36)
4 4 (4 \ / \4

Here we define a a(]), a' a(T'), AT r] r and Ax Y |l i- r'|. Note that any
positive powers of y become,

Y+ -aa'H2(Ar2-Ax2) (5-37)
4

4. Make the change of variables r' x x, perform the angular integration and make
the further change of variable r = Ar z,

Sd3x' (A, -Ax) F (aa/ 'H2(2 2_ A k2) ('

S470(A,) Odr r2F (/aa/H2( 2 sin(kAx (5-38)

4T0(A,)AT3 dz Z2 aa (1 1)2(1 2)) (5-39)
o a a 4 kAz
5. Reduce the z integration to a combination of elementary functions and sine and
cosine integrals [45].









6. Make the change of variables a' = -1/Hq', expand the integrand and perform the

integration over a'.
7. Act any derivatives with respect to a.

Much of the labor involved in implementing these steps derives from the spacetime

dependence of the zeroth order solution, u(q', k)eikM'. For example, one can see from

(5-39) that only elementary functions would result from the z integration if the zeroth

order solution were constant. Because we do not yet possess corrections to the initial state

there is no reason to avoid making the simplification,

__ j!dfdx'A4(x; x')u(qI', k)(a' x-)

u(0, k)0fA4
a4- dfdxM x'); (5-40)

To see why, note from expressions (5-16) and (5-39) that the deviation of the zeroth

order solution from u(0, k) introduces at least two factors of 1/a' or 1/a. Because the

constant mode function can at best result in powers of In(a), and because we cannot trust

contributions to (5-17) which fall off, we may as well discard any terms which acquire an

extra factor of 1/a. Extra factors of 1/a' effectively restrict the integration to early times,

which again causes the net result to fall off at late times.

One consequence of the simplification (5-40) is that we can neglect any contribution

from Table 4-5 which contains a factor of V2. We therefore need only compute terms of

the form,

u(0,k) ( 4)4 da/KdxIf(-)-f(Y- (5 41)
-H4(0,k)i 4K ) NJ0 / f (5-41)

where the constant K takes the values of 3, 4 and 5, and the functions f(x) are 1/x,

in x/x, In x and In2 x. The action of the d'Alembertian derives from setting k = 0 in

(5-33). The integration for K = 3 (given in Table 5-1) and K = 4 (given in Table 5-2)

were worked out in a previous paper [29]. Applying the same technique -which is just

the 7-step procedure given above for k = 0 -gives the results for K = 5 in Table 5-3.










f(x) -i x fd4 'a'3{f( )( )
1 In(a) 1 + (
a +1a+ T2}
In(x) n2 (a) 2 1n(a) 3 + 2 ln(a)
S2a a a '3a ( a2
ln(x)1 ln(a) 1 + 0(1)
6 2a 4a+
in2 x) 1 11 In(a) 21n(a) 49 i 72 I n(a)
In () )- 2a a 4a 3a' 'O( )
Table 5-1. Integrals with a'3.


f(x) -H4 x j d4 x'a'4f( ) f( _)}

ln(x) 3 )
+ 0()________
ln(x) In(a) (6 -(
In2 () lln2(a) In(a) + ( )

Table 5-2. Integrals with a'4.


It remains just to act the d'Alembertians and sum the results for each of the

contributions from Table 4-5. That is done in Table 5-4. Although individual contributions

can grow as fast as n2 (a), all growing or even finite terms cancel. It follows that the right

hand side of (5-17) falls off at least as fast as 1/a times powers of In(a). Hence there are

no significant corrections to the mode function at one loop order.


f(x) W x fd4x'a'5{f(Y-) f(-)}
1- + 0(1)
ln ) 10+(1)
x T6 a + 0(a
In(x) a -+ O( )
In2 (X 13 1 5 ln()
ln2 (-) a a- In (a) +9 (a)
Table 5-3. Integrals with a'5
















External Operator x f(x) Coefficient of u(0, k) x 16
(aa')4D3/H2 x 0
(aa')4D2 X 261nx 0
(aa')4D2 x 3 0
(aa')4 2 x -6 n2 -18
(aa')4 2 x -18 In x 0
(aa')4H2[ x -61nx 0
(aa')4H2D x 0
(aa')4H20 x -41nx 2
(aa')4H4 x 4ln 3
(aa')4H4 x 1 -9
(aa')4H4 x -08 In x 33 18 ln(a)
(aa')(a2 + a'2) H /H2 x I (-325+12"2) + 1 ln(a) In2(a)
(aa')3 2 x -'x (85-122) 41n(a) 21n2(a)
1 8 2ln(a)
(aa')(a2 + 2 x Iln(a)
(aa')32 + a2H2 x 21 (160-12r2) ln(a) + 2 ln2(a)
(aa')3(a2 + a'2)2x H 10l n(a)
(aa')(a2 + a'2)H2 x -18lnx -15 + 18ln(a)
(aa3a2 + 2)H4 41 (-91+12"2 + n(a)- 1n2(a)
(aa')32 '2)H4 32 + n(a)
Total 0


(40 diH 4 (Ext. Operator) x f{(Y)-) f (Y)} terms.


Table 5-4. The









CHAPTER 6
CONCLUSION

We have computed one loop quantum gravitational corrections to the scalar

self-mass-squared on a locally de Sitter background. We used that to solve the one

loop-corrected, linearized effective field equation for a massless, minimally coupled

scalar. The computation was done using dimensional regularization and renormalized by

subtracting the three possible BPHZ counterterms. Because our gauge condition (2-15)

breaks de Sitter invariance, one of these counterterms is noninvariant.

Unlike previous analysis of the scalar effective potential [20], our technique uses

the full self-mass-squared, including corrections to the derivative terms and nonlocal

corrections. It should therefore be sensitive not only to the scalar's mass but also to

its field strength. We do not yet possess the computational tools needed to search for

k-dependent but time independent shifts of the field strength but we find no growing

corrections at one loop order.

The point of this exercise is to discover whether or not the inflationary production

of gravitons has a significant effect upon minimally coupled scalars as it does on fermions

[18]. In order to check this we computed one loop corrections to the scalar mode functions

using the effective field equation,


( (x)) d4 r(x; x')(x') = 0 (6-1)

Similar studies have already probed the effects of scalar self-interactions [10, 11], fermions

[45] and photons [21], but none has so far considered the effects of gravitons. Although our

scalar is a spectator to A-driven inflation, the near flatness of inflation potentials -i-i-. --

that the result we shall obtain may apply as well to the inflation of scalar-driven inflation.

A significant difference between this and previous scalar studies [10, 11, 21, 45] is

that quantum gravity is not renormalizable. Although we could absorb divergences with

quartic, BPHZ counterterms, no physical principle fixes the finite coefficients Aca of these









counterterms. That ambiguity is one way of expressing the problem of quantum gravity.
However, a little thought reveals that we will be able to get unambiguous results for
late time corrections to the mode functions. The reason is that the scalar d'Alembertian

annihilates the tree order mode solution,

H [t ik ik
o(x; k) u(, k)eu ) where u(Tq, k) = -] exp[ (62)
/2k Ha

Hence only the third counterterm makes a nonzero contribution, and its effect rapidly
redshifts away,

JdV fa (Aajiw + Aa2D + Aa3a2)4(x x-') x K)o(x';
j 4 / 2 4 1 2 2 83 4 / )a2
k2
= -a x Aa3 2 0o(x; k) (6-3)

It is instructive to compare our de Sitter background result (4-41) with its flat space
analogue. In the flat space limit of H -+ 0 with fixed comoving time, the scalar and

graviton propagators become,

P( 1) 1

2iLa 42') 2 (F64)
[Aflat] (x; x') 2= [2(9-D-2I 42 PrD-2 (6-5)

Here Ax2 is the Poincar4 length function analogous to y(x; x'),

Ax2 (x;x) -= _'- I i (6-6)

Two features of the flat space propagators deserve comment:
1. Both propagators are manifestly Poincar6 invariant; and

2. The coincidence limits of both propagators vanish in dimensional regularization.

We exploited the first property to drop all but the final noninvariant counterterm on
the list (4-19). And the second property explains why our 4-point contribution (3-14)









vanishes in the flat space limit,


2 (X; '1)
-iM ( A.(; ;x) ) 6 i D(xX; ) +j -uD _'

+* 2 A ](x; ) L axD } p a fla ]x; Xx)


-i2a { Xap ( ) flatD (x x') = 0. (6-7)

An only slightly less trivial computation reveals that the flat space limit of our total
3-point contribution should also vanish,

-ziML (X; x'I)= -,A2fa aX;p XI 1 'flat X X
3pt

2 aI ( ]c 0,0 1 A + 2 1 r f T1 I A

K OP in -[a iAlat 0'0" iA flati 8itl
a-da '" P [A ] (x; ') x ,,ii (x; ') (6-8)


)2 )a{p ,a90, 2} ,, -,, ] (6-9)
16rD AXD-2 AxD-2

-2 a2 a D2 (6 10)
167 D A D-2 AD -2
=0. (6-11)

This result has a number of consequences:
1. It explains why the highest dimension counterterm (4-5) fails to appear;
2. It explains why all the entries of Table 4-5 vanish in the flat space limit except the
first line,

H (aa)4 x 4 n()
(47)4 y) ( Y

1 K
Sx 6 In (H2A; 2 (6 12)
3 (4)4 A2 42









and the fifth line,




1 2 u
(4) (aa') (a 2 + a'2)D3 t n-


1 6x 4 l6 n H2A x2 1 (6-13)
3 (47)4 AX2 4

3. It explains why (6-12) and (6-13) cancel in the flat space limit; and

4. It means that any physical effect we find must derive entirely from the nonzero

Hubble constant.

This is the right point to comment on accuracy. This has been a long and tedious

computation, involving the combination of many distinct pieces. It is significant when

these pieces join together to produce results that can be checked independently, such

as the vanishing of the flat space limit. One sees that in the way the a and 37-

contributions to the ac counterterm cancel in Table 4-3. Another example is the way

three individually divergent terms combine in Table 4-4 to produce a finite result for a

counterterm that is forbidden by the shift symmetry of the bare Lagrangian (2-1).

Although the ac counterterm had to vanish by the flat space limit, we do not yet

understand why the coefficient the a2 counterterm is finite. The contribution of this

term to the scalar self-mass-squared vanishes in flat space, but it would seem to affect

the Q + h -- 0 + h scattering amplitude. The divergences on this were explored in the

classic paper of 't Hooft and Veltman [46]. Unfortunately, their on-shell analysis makes no

distinction between R(00)2 which we have -and (a0)4 which we do not have.

We should also comment on what subset of the full de Sitter group is respected by

our result (4-41). Recall that our gauge fixing term breaks spatial special conformal

transformation (4-10). This is why the noninvariant counterterm (4-20) occurs. It is also

responsible for the noninvariant factors of V2 and a2 + a'2 in Table 4-5. Because these

breaking derive entirely from the gauge condition, we expect them to have no physical

consequence.









The graviton and scalar propagators also break the dilatation symmetry (4-9). Unlike

the violation of spatial special conformal transformations, the breaking of dilatation

invariance is a physical manifestation of inflationary particle production and can have

important consequences. Dilatation breaking comes in the ln(aa') term of the A-type

propagator (2-14). These logarithms were responsible for the secular growth that was

found in the fermion field strength [18], so one might expect them to drive any effect

on scalars as well. However, it turns out that the factors of ln(aa') all drop out. For

the scalar propagator this is a trivial consequence of the fact that it alv--iv carries one

primed and one un-primed derivative. Logarithms from the graviton propagator do appear

in the 4-point contributions (3-14), and in the 3-point logarithm contributions (3-37).

But all factors of ln(aa') cancel in the final result (4-41), which turns out to respect

dilatation invariance. Because of this we suspect that there will be no significant late time

corrections to the mode functions at one loop order.

Physically our result means that the sea of infrared gravitons produced by inflation

has little effect on the scalar. Although our scalar is a spectator to A-driven inflation,

typical scalar inflation potentials are so flat that the inflation is also unlikely to suffer

significant one loop corrections from quantum gravity. If so, it is consistent to use the tree

order scalar mode functions to compute the power spectrum of cosmological perturbations.

There are good reasons why the scalar should not acquire a mass from quantum

gravity [20] but there seems to be no reason why its field strength cannot suffer a time

dependent renormalization of the type experienced by massless fermions [18]. Weinberg's

result for the power spectrum allows for ln(a) corrections [6], and they do arise from

individual terms in Table 5-4. However, there is no net correction of this form at one loop.

In the end, our null result may not be as surprising as it seems. It is very simple to

show that the scalar self-mass-squared vanishes at one loop order on a flat background

[21]. It is also wrong to think of the scalar as weaker than the fermion; what is weaker is

the fractional one loop correction. Although massless fermions experience a time-dependent










field strength renormalization, this represents a ln(a) enhancement of tree order mode
3
functions which fall like 1/av. By contrast, the tree order scalar mode functions approach

a nonzero constant. Indeed, it is the fact that derivatives of this constant vanish which

makes the fractional correction of the Kh p coupling so small. The sea of infrared

gravitons is present, but it can only couple, at this order, to the scalar's stress-energy, and

the stress-energy of a single scalar redshifts to zero at late times.








APPENDIX A
EXTRACTING DERIVATIVES
We group the various normal contributions into seven parts:


V- [(aa')D-2AV-V'A] ,

M [(Kaa /)D-2AaO/A] + o0/ [(aa /)D-2A0oaAl ,I

008 [(aa/')tD-2 AoA] ,

-800[(aa )D-2BV-V'A ,

-A0 [(aa ')D-2/oDB A 0;0 [(aa')D-2 Ba'4A1 ,I
-V-V (7'aa/)D-2 0o8Al ,I

00v [(aa')D-2 a'a oA .]


(A-l)

(A-2)

(A-3)

(A-4)

(A-5)

(A-6)

(A-7)


In these definitions the expression "OoOa'A(y)" means the naive derivative, without the delta
function. Also note that we have suppressed the unbiquitous factors of K2.
An important simplification in reducing P2 is to achieve a symmetric form which has
no 00. This can be done by adding equations (3-62) and (3-65) and then using equation
(3-58),


P2 (-6 + 2C)12[AA"] + (D-l)e212[A'2]

+H(aa/)D-2(a0o +a')V22 [A2] ,

(-6 + 2)12 [AA"] + (D-l )c12[A'2]

+e3yI[A'2] + (D- 1)112 [A/2] 3[A2] .
2


(A-8)


(A-9)


Another organizational point concerns removing factors of y from inside integral.
This is desirable because it reduces the number of distinct integrals which appear. It can
alv-i-, be accomplished by partial integration. We will illustrate using the function acted
upon by -63 in equation (3-73),


F(y) (D-2)I[A'B+yA"B] + yA'B + y2A"B yI[A'B'] .


(A-10)








Note the relations,


A'B + yA"B A'B + y I[A"B] (A-ll)

iy[A"B] + A'B- I[A"B] (A-12)

0 yI[A"B] + I[A'B'] (A-13)

We can therefore write,

F(y) = y2A"B + (D-l)y[A"B] + (D-2)I2[A'B'] (A-14)

With (A-13) and (A-14) we can read off the following result for P5 from equation
(3 73),

P5 = 6I2[BA"] (D- 1)112[B'A'] (D-1)212[B'A']

+( -212 [BA//] + I [B'A'] + eC2 yI[A"B] + 12 [A'B'}

+e y2A"B + (D 1)yl[A"B] + (D 2)12[A'B'] (A-15)

Many terms involving A in P2 combine with cognate terms involving B in P5 to produce
the less singular propagator function AB = B A. Summing expressions (A-9) and
(A-15) gives,

P2+5 -62[ABA"] (D-1)C12[AB'A'] (D-1)C12[AB'A'] + C3yI[A'2]

+~(-232 [AA"/ + I'- ['A']B + (D-2) A"B + I'A'B']
+esy2A"B+ D-1)y[A"B + (D1-IA I ( 2)12[A4'B'] (A-16)








In contradistinction to P2 and Ps, the reduction of the other parts is facilitated by
further sub-division immediately after employing identities (3-45) and (3-48),

f(y)OoO'oA(y) = 0ooI2[fA"] + 2aa'H21[f'A']
-aa'H2(D-1) + y I2 [fA] .V'f'A] (A-17)
f(y)VV'A(y) = VV'I2[fA"] 2(D-1)aa'H21[f'A] (A-18)

One employs (A-17) on P3 (from which we can read off the result for P7) and P6 to give
the sub-parts,

P3 00 (aa'/)D-2a0a/i2[AA//"] (A-19)
P3b 2 2oOo [(aa/)D-1H21[A/2]] (A-20)
P3, -00o [(aa')D-1H2{(D-) + y P}2[A/2] (A-21)

P3d (2^ )D- VV3[A/2] (A-22)
P7a 00 [(aa/)D'-2a0/2[ACA//"] (A-23)
P7b 2000o [(aa/)D- 1H2 /[A/AC/]] (A-24)
P7Tc -00o'[(aa)D-1H2{(D- 1)+y 2[[A/AC'] (A 25)

P7d a -00[ (aa'/)D-2V. 3 [AC'] (A-26)
P6a V -.V (aa')D-200012 [B4"]] (A-27)
P6b E 2V.V/[(aa/)D-1H21[A/ '/] (A-28)
P6, VV /[(aa/)D-1H2{(D-1) + y }2[A/'B/] (A-29)

P V V[(aa/)D-2V. V 3 [AB'] (A-30)
2 1








Part Contribution Acted upon by a
Psa 12 [AA"]
P7a 12[ACA"]
Total 2 [AA" + 2 [ACA"
Table A-i. Contributions acted upon by a = (aa')D2.

Applying the second identity (A-18) to P1 and P6 gives their sub-parts,

Pi, V-V [(aa/')D-2. /2[AA/"] (A 31)
Plb -2(D- )V.V/(aa/)D-1H21[A2] (A 32)
P4I -o0 aa[('/)D-2V l [BA"l] (A 33)
P4b 2(D-1)o [(aaI)D- H2 [A/B/] (A-34)

Of course there is no problem further reducing the spatial derivatives. The following
generic reductions serve to reduce terms involving the operator 0oa',

a0oa [(aar)D- 20 ()] = 6 + (}f(y) (A-35)

(aa')D- H2 f(y) 3 + (D 1)(D 2)71 + 72
+ 2a2 2_s
73 a 2 82 D
+ [(D -)y + 21 +C-1+ f(y) (A-36)

000[(aa')D-2 v f. y) =f (D-2)(D-3)i 2
12 2 Oy

[(D-1)y + 2] + [t 2(D-2)I] }f(Y) (A-37)

V.V/[(aa')D-2aO f(y)] { (D- )(D-2)>1 -
3 a 2
22 O-O

2b [(D-1)y 2 ] gv (D-r2)] }f(y) (A-38)
Tables A-1-A-10 give the results for each of the ten External Operators.










Part Contribution Acted upon by 3
P36 I [A'2]
P1 2 ][A'2] (D2l)12[A'/2
P3c -2l[A [A
P4b (D-1)I[A'B']
P7b I[A'AC']
P7c yl[A'AC'] ( )I2[A'AC']
Total DI[A'B']- I[A'AB']- 'yI[A2- (D1)D2[A/2]
+I[A'AC'] ylI[A'AC'] (-1)I2[A'AC']
Table A-2. Contributions acted upon by 3 = (aa')D-2(a2 + a'2)H2D.



Part Contribution Acted upon by 71
P3b (D-1)(D-2)I[A'2]
P3c (D-1)(D-2)ylI[A'2] (D-I)(D-2)I2[A/2]
P4b (D-1)2(D-2)I[A'B']
P7b (D-1)(D-2)I[A'AC']
P7c (D-1)(D-2)y[A'AC'] (D-1)2 (D-2) 2[A/AC/']
D(D-1)(D-2)I[A'B'] (D-1)(D-2)I[A'AB'] I(D-1)(D-2)JYI[A/2]
Total (D-1)2(D-2)2[A'2 + (D-1)(D-2)I[A'AC']
(D-1)(D-2)YI[A'AC'] (D-1) (D-2) 2[A'AC']
Table A-3. Contributions acted upon by 71 (aa')DH4.



Part Contribution Acted upon by 72
P3b yA'2
P3c~ 1 2 j2A2 D 1y[A2]
P4b (D-1)yA'B'
P7b yA'AC'
Pe -I 2A'AC'- DyI[A'AC']
Total DyA'B' yA'AB' yA'2 DyI[A'2
+yA'AC' DyI[A'AC' y2A'AC
Table A-4. Contributions acted upon by 2 = (aa')D-1(a2 + a'2)H4.













Part Contribution Acted upon by 73
P3b (D-1)yA'2 + 12(A 2)/
c --Dy2 1A2 Y3 (A12)/ (D-1)Dy 2
P4b (D-1)2A'B' + (D-)D1)y 2(A/B)/(
P7b (D-1)yA'AC' + y2(A'AC')'
P7, -D~2 A'A C 3(A'AC/)' (D-)DyI[A'AC']
(D-1)DyA'B' + D2(A'B')' (D-1)yA'B' y2 (A'B')'
Total -Dy2A'2 3(A12 (D-I)DyI[A'2 + (D-i)yA'C'
+y/2(A/'C')' Dy2 A'AC'- 1 3 (AAC ')- l(D-I)DyI[A'AC']
Table A-5. Contributions acted upon by s73 (aa')D-l(a + a')2H4.










Part Contribution Acted upon by 6
P2+5 2 [ABA"]
P3a 12[AA"]
P3d lI3[A'/2]
P4a 112 [BA]
P6a 2 [BA/"
P7a 12 [ACA"]
P7d I3[A'AC']
Total 22 + [A'2 C" +
Total 2 [ABA"] + 4 [A ] [ACA//] + [A/AC/


Table A-6. Contributions acted upon by 6 :


(aa')D-2 (a2 + a'2)V2.











Part Contribution Acted upon by e1
Pib 2(D-1)I[A'2]
P2+5 -(D-1)12[A'AB']
P3b -21[A'2]+ 12 [A/2]
Pc yI[A'21 (D-1)2l[A'2 D 12[A/2 1D(D-2)I3[A/'2
P3d (D_2)(D-3) J[A/2
P4a (D-2)(D-3)2 [BA"]
P4b -2(D-1)I[A'B'] + I(D-1)D2 [A'B']
P6a 1(D-)(D-2)2 [BA"]
P6b 21[A'B']
P6c -yl[A'B'] (D-1)2[A'B']
P7b -21[A'AC'] + D2 [A'AC']
P7r yI[A'AC'] + (D-1)j2[A'AC'] DyI2[A'AC'] D(D-2)I3[AlAC']
P7d (D-2)(D-3) [A'AC']
-2(D-2)[A'AB']-yI[A'AB']- (5 4)12 [AA'I] + 2 [A'B']
Total -o2 [A'2] + (D-2)2[A"B] (D-2) [A2 21[A'AC/]
+yI [A/AC']+( 3"- I2[A'AC'/] D T2[A'AC'] (D-2)13[A'AC']
Table A-7. Contributions acted upon by c1 = (aa/)D-1H2V2.





Part Contribution Acted upon by C2
P2+5 -(D-1)2[A/AB/] + yI[BA"] + 12[A'B']
P3d gyl2 A/2
P4a_ yI [ BA"\]
P4. lI[BA"]
P6a I [BA"'t
P7d ,y12 [A'AC']
Total 2y1[BA"] + 12[A/B']
-(D-1)2[A'AB's] + yI2[A 2]+ yI [A'AC']


Table A-8. Contributions acted upon by C2


(aa/)D-2(a2 + a'2)H2.












Part Contribution Acted upon by C3
P2+5 y2A"B + (D-1)yI[A"B] + (D-2)I2[A'B'] + yl[A'2]
P3d (D- )y12[A/2 + 1 2 [A/2]
P4a (D- )yI[A"B] + y2A"B
P6a (DI )yI[A/"B] + jy2A"B
P7d (D )yl2[A'AC'] + 2I [A'AC']
Total 2y2A"B + 2(D-1)yI[A"B] + (D-2)2 [A'B'] + yI[A'2]
+(Dn )y2[A/2] + y2[A'/2 + (D- 1)y [A'AC'] + y2I[A'AC']
Table A-9. Contributions acted upon by C3 = (aa')D-2(a + a)2H2V2.


Part Contribution Acted upon by
Pla, 2 [AA"]
P2+5 -22 [ABA"] + 13 [A'AB']
P3a J2 [AA"]
P3d 113[A,2] + (D 2)J4[A/2]
P4a -12[A"B] + (D )I3[A"B]
P6a [-IA"B]- (D )3[A"B]
P6d I 3 [AB'
P7a 12 [ACA"]
Pra -I[A'AcC'] + (D2)14 [A'AC']
Total -42 [A"AB] + 13[A'AB'] + (D 2)4 [A'2]
+J2[ACA"] 13 [A'AC'] + ( D 2)J4[A'AC']
Table A-10. Contributions acted upon by ( (aa)D-2V4.









REFERENCES

[1] R. P. Woodard, "Quantum Effects during Inflation," in Quantum Field Theory
under the Inrla, ..i of External Conditions (Rinton Press, Princeton, 2004) ed. K. A.
Milton, pp. 325-330, astro-ph/0310757.

[2] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33 (1981) 532.

[3] A. A. Starobinskif, JETP Lett. 30 (1979) 682.

[4] V. Mukhanov, H. Feldman and R. Brandenberger, Phys. Rep. 215 (1992) 203.

[5] A. R. Liddle and D. H. Lyth, Phys. Rep. 231 (1993) 1, astro-ph/9303019.
[6] S. Weinberg, Phys. Rev. D72 (2005) 043514, hep-th/0506236; Phys. Rev. D74 (2006)
023508, hep-th/0605244.

[7] K. C'!I I. rdsakul, Phys. Rev. D75 (2007) 063522, hep-th/0611352.
[8] V. K. Onemli and R. P. Woodard, Class. Quant. Gray. 19 (2002) 4607,
gr-qc/0204065.

[9] V. K. Onemli and R. P. Woodard, Phys. Rev. D70 (2004) 107301, gr-qc/0406098.
[10] T. Brunier, V. K. Onemli and R. P. Woodard, Class. Quant. Gray. 22 (2005) 59,
gr-qc/0408080.
[11] E. O Kahya and V. K. Onemli, Phys. Rev. D76 (2007) 043512, gr-qc/0612026.

[12] T. Prokopec, O. Tornkvist and R. P. Woodard, Phys. Rev. Lett. 89 (2002) 101301,
astro-ph/0205331; Ann. Phys. 303 (2003) 251, gr-qc/0205130.

[13] T. Prokopec and R. P. Woodard, Ann. Phys. 312 (2004) 1, gr-qc/0310056;

[14] T. Prokopec, N. C. Tsamis and R. P. Woodard, "Stochastic Inflationary Scalar
Electrodynamics," arXiv:0707.0847.

[15] T. Prokopec and R. P. Woodard, JHEP 0310 (2003) 059, astro-ph/0309593.

[16] B. Garbrecht and T. Prokopec, Phys. Rev. D73 (2006) 01. 111 ;, gr-qc/0602011i.

[17] S. P. Miao and R. P. Woodard, Phys. Rev. D74 (2006) 044019, gr-qc/0602110.

[18] S. P. Miao and R. P. Woodard, Class. Quant. Gray. 23 (2006) 1721, gr-qc/0511140;
Phys. Rev. D74 (2006) 024021, gr-qc/0603135.

[19] L. Smolin, Phys. Lett. 93B (1980) 95.

[20] A. D. Linde, Particle Ph,;-. and I, nl.,i:.I ,, C ..-I,.,...,;i (Harwood, Chur,
Switzerland, 1990), hep-th/0503202; "Inflationary Comsology," arXiv: 0705.0164.

[21] E. O. K ri- and R. P. Woodard, Phys. Rev. D76 (2007) 124005, arXiv:0709.0536.









[22] B. Allen and A. Folacci, Phys. Rev. D35 (1987) 3771.

[23] B. Allen, Phys. Rev. D32 (1985) 3136.

[24] N. C. Tsamis and R. P. Woodard, Commun. Math. Phys. 162 (1994) 217.

[25] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge
University Press, 1982).

[26] P. Candelas and D. J. Raine, Phys. Rev. D12 (1975) 965.

[27] J. S. Dowker and R. Critchley, Phys. Rev. D13 (1976) 3224.

[28] N. C. Tsamis and R. P. Woodard, Ann. Phys. 321 (2006) 875, gr-qc/05-6056.

[29] T. Prokopec, N. C. Tsamis and R. P. Woodard, Class. Quant. Gray. 24 (2007) 201,
gr-qc/0607094.
[30] E. O. K .i, i and R. P. Woodard, Phys. Rev. D72 (2005) 104001, gr-qc/0508015;
Phys. Rev. D74 (2006) 084012, gr-qc/0608049.

[31] S. Wolfram, The Mathematica Book, Third Edition (Cambridge University Press,
1996).

[32] N. N. Bogoliubov and 0. Parasiuk, Acta Math. 97 (1957) 227; K. Hepp, Commun.
Math. Phys. 2 (1966) 301; W. Zimmermann, Commun. Math. Phys. 11 (1968) 1; 15
(1969) 208; in Lectures on Elementary Particles and Quantum Field Ti ..-',;/ ed. S.
Deser, M. Grisaru and H. Pendleton (MIT Press, Cambridge, 1971), Vol. I.

[33] I. Antoniadis and E. Mottola, J. Math. Phys. 32 (1991) 1037.

[34] R. P. Woodard, "De Sitter Breaking in Field Theory," in Deserfest: A Celebration of
the Life and Works of S'.,l;,,. Deser (World Scientific, Hackensack, 2006) eds J. T.
Liu, M. J. Duff. K. S. Stelle and R. P. Woodard, pp. 339-351.

[35] J. Schwinger, J. Math. Phys. 2 (1961) 407.
[36] K. T. Mahanthappa, Phys. Rev. 126 (1962) 329.

[37] P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4 (1963) 1; J. Math. Phys. 4
(1963) 12.

[38] L. V. Keldysh, Sov. Phys. JETP 20 (1965) 1018.

[39] R. D. Jordan, Phys. Rev. D33 (1986) 444.
[40] K. C. Chou, Z. B. Su, B. L. Hao and L. Yu, Phys. Rept. 118 (1985) 1.

[41] E. Calzetta and B. L. Hu, Phys. Rev. D35 (1987) 495.

[42] L. H. Ford and R. P. Woodard, Class. Quant. Gray. 22 (2005) 1637, gr-qc/0411003.





72


[43] L. H. Ford, Phys. Rev. D31 (1985) 710.
[44] D. N. Spergel et al., Astrophys. J. Suppl. 170 (2007) 377, astro-ph/0603449.
[45] L. D. Duffy and R. P. Woodard, Phys. Rev. D72 (2005) 024023, hep-ph/0505156.
[46] G. 't Hooft and M. Veltman, Ann. Inst. Henri Poincar6, XX (1974) 69.









BIOGRAPHICAL SKETCH

Emre Kahya was born in Turkey and got his B.S. and M.S. at Middle East Technical

University. He decided to pursue a doctoral study in physics at USA and admitted by the

University of Florida. Been interested in theoretical physics, after taking a quantum field

theory course of Prof. Richard Woodard, he decided study Quantum Gravity with him.

He was awarded his Ph.D. in summer 2008.





PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

Iamindebtedtomyadvisor,ProfessorRichardWoodard.Heisaveryhardworking,meticulousandadedicatedpersonasprobablymanyknow.Buthealsoisateacherwhoreallycaresabouthisstudents.Hewouldnothesitatetosacricehistimeandeortforthebenetofhisstudents.Generosityandself-disciplinearetwowordsthatcomeintomymindwhenIthinkofhim.IhopethatonedayhiscontributionstotheBesselFunctionswillbeacknowledged.Iwouldliketothankmymother,HurmetKahya;myfather,OrbayKahya;andmysister,HilalKahya,fortheirconstantsupport.Iwouldliketothankmywife,Selva;andmydaughter,HafsaAsude,formakingmylifeenjoyable.IalsowouldliketothankProfessorPierreSikivieandProfessorAtalayKarasuforwritinglettersofrecommendationonmybehalf. 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2FEYNMANRULES ................................. 11 3ONELOOPSELF-MASS-SQUARED ....................... 16 3.1Contributionsfromthe4-PointVertices .................... 16 3.2Contributionsfromthe3-PointVertices .................... 19 3.2.1LocalContributions ........................... 20 3.2.2LogarithmContributions ........................ 22 3.2.3NormalContributions .......................... 23 4RENORMALIZATION ................................ 34 5EFFECTIVEMODEEQUATION ......................... 44 5.1TheSchwinger-KeldyshFormalism ...................... 44 5.2RestrictionsonOurSolution .......................... 47 5.3LocalCorrections ................................ 50 5.4NonlocalCorrections .............................. 51 6CONCLUSION .................................... 56 APPENDIX AEXTRACTINGDERIVATIVES ........................... 62 REFERENCES ....................................... 70 BIOGRAPHICALSKETCH ................................ 73 5

PAGE 6

Table page 2-1Three-PointVertexOperatorsVIcontractedinto12h. ........... 12 2-2Four-PointVertexOperatorsUIcontractedinto12hh. ......... 12 3-1NonlocalLogarithmContributionsfromrelation( 3{37 )withxy ....... 27 3-2DivergentNormalContributions. .......................... 32 3-3FiniteNormalContributionsintermsofxy .................. 33 4-1LocalNormalContributionsfromTable 3-2 .................... 39 4-2FiniteNormalContributionsfromTable 3-2 withx=y ............. 40 4-3NormalContributionstoCountertermsfromTable 4-1 .Alltermsaremultipliedbyi2HD4 (D3)(D4). ................................. 42 4-4OtherLocalNormalContributionsfromTable 4-1 ................ 42 4-5AllFiniteNonlocalContributionswithxy ................... 43 5-1Integralswitha03. ................................... 54 5-2Integralswitha04. ................................... 54 5-3Integralswitha05. ................................... 54 5-4Theu(0;k) ....... 55 A-1Contributionsacteduponby=(aa0)D2. .................... 65 A-2Contributionsacteduponby=(aa0)D2(a2+a02)H2. ............. 66 A-3Contributionsacteduponby1=(aa0)DH4. .................... 66 A-4Contributionsacteduponby2=(aa0)D1(a2+a02)H4. ............. 66 A-5Contributionsacteduponby3=(aa0)D1(a+a0)2H4. .............. 67 A-6Contributionsacteduponby=(aa0)D2(a2+a02)r2. ............. 67 A-7Contributionsacteduponby1=(aa0)D1H2r2. ................. 68 A-8Contributionsacteduponby2=(aa0)D2(a2+a02)H2r2. ............ 68 A-9Contributionsacteduponby3=(aa0)D2(a+a0)2H2r2. ............ 69 A-10Contributionsacteduponby=(aa0)D2r4. ................... 69 6

PAGE 7

Figure page 3-1Contributionfrom4-pointvertices. ......................... 16 3-2Contributionfromtwo3-pointvertices. ....................... 19 4-1Contributionfromcounterterms. .......................... 34 7

PAGE 8

Wecomputedtheoneloopcorrectionsfromquantumgravitytotheself-mass-squaredofamassless,minimallycoupledscalaronalocallydeSitterbackground.ThecalculationwasdoneusingdimensionalregularizationandrenormalizedbysubtractingfourthorderBPHZcounterterms.Weusedthiscomputationoftheself-mass-squaredfromquantumgravitytoincludequantumcorrectionstothescalarevolutionequation.Theplanewavemodefunctionsareshowntoreceivenosignicantoneloopcorrectionsatlatetimes.Thisresultprobablyappliesaswelltotheinatonofscalar-drivenination.Ifso,thereisnosignicantcorrectiontothe''correlatorthatplaysacrucialroleincomputationsofthepowerspectrum. 8

PAGE 9

Mystudywasunderstandingquantumloopeectstomasslessminimallycoupledscalarsduringination.TodothatIcomputedtheself-mass-squaredofmasslessminimallycoupledscalarscoupledwithgravitonsatone-looporder.ThenIusedthattosolvethequantumcorrectedequationsofmotioninordertoseewhetherthisone-loopeectaltersthedynamicsofascalarinaton. Quantumloopeectscanbeunderstoodasthereactionofclassicaleldtheorytovirtualparticles.Theenergy-timeuncertaintyprincipletellsusthatthenumberdensityofthesevirtualparticles,whichpersistforeverduringprimordialination,isenhanced[ 1 ]fortheoneswhicharemasslessandnotconformallyinvariant.Massless,minimallycoupledscalarsandgravitonsaretheonlytwoparticleswhichhavezeromassandarenotconformallyinvariant.Asaresultofinationquantumeectsfortheseparticlesarevastlyenhancedwhichistheoriginoftheprimordialscalar[ 2 ]andtensor[ 3 ]perturbationspredictedbyination[ 4 5 ].Weinberghasrecentlyshownthatloopcorrectionstotheseperturbationsarealsoenhancedbylogarithmsoftheratioofthescalefactortoitsvalueatrsthorizoncrossing,althoughnotenoughtomakethemobservable[ 6 7 ].However,muchlargerloopcorrections(tootherthings)canbeobtainedeitherfrominteractionswithlargerloopcountingparameters,orbystudyingthingswewouldperceiveasspatiallyconstantsuchasthevacuumenergyorparticlemasses. ExplicitcomputationshavebeenmadeondeSitterbackgroundindierentmodelswhichinvolveeitherscalarsorgravitons.Amassless,minimallycoupledscalarwithaquarticself-interactionispushedupitspotentialbyinationaryparticleproduction,therebyinducingaviolationoftheweakenergycondition[ 8 9 ]andanonzeroscalarmass[ 10 11 ].Thevacuumpolarizationfromacharged,massless,minimallycoupledscalarinducesanonzerophotonmass[ 12 13 ]andasmallnegativeshiftinthevacuumenergy[ 14 ].Theinationarycreationofmassless,minimallycoupledscalarswhichare 9

PAGE 10

15 16 ]andinducesanegativevacuumenergythatgrowswithoutbound[ 17 18 ]. Itisnaturaltoextendthesestudiesbycombininggeneralrelativitywithamassless,minimallycoupledscalartoprobetheeectsofgravitonsonscalars.Suchaninvestigationhasgreatphenomenologicalinterestbecausetheinatonpotentialofscalar-driveninationissoatthatinatonmodefunctionsareeectivelythoseofamassless,minimallycoupledscalar.Indeed,itisstandardtocomputethepowerspectrumofscalarperturbationsbysettingthescalar-scalarcorrelatortoitsvalueforamassless,minimallycoupledscalarondeSitterbackground[ 4 5 ].Thesubjectofquantumgravitationalcorrectionstothescalareectivepotentialhasalonghistory[ 19 20 ]butwewillherelookatalloneloopcorrectionstothelinearized,eectivescalareldequation,includingcorrectionstothederivativetermsandalsothefullynonlocalcorrections.Thisiswhatonemustdotoxthescalareldstrengthinadditiontoitsmass. Ourresultisthatthescalarmodefunctionsexperiencenosignicantcorrectionsatoneloop.Weprovethisbysolvingthelinearized,eectivescalareldequation.WederivetheFeynmanRulesofMMCS+GRinchapter2.Chapter3givesareviewofoneloopcomputationofthescalarself-mass-squared[ 21 ],whichrepresentsthequantumcorrectiontothelinearizedeectiveeldequation.Inchapter4weextractnitetermsfromtheone-loopresultsusingBPHZrenormalizationscheme.Inchapter5wesolvetheequationintherelevantregimeoflatetimes.Ourconclusionscomprisechapter6. 10

PAGE 11

TofacilitatedimensionalregularizationweworkinDspacetimedimensions.OurLagrangianis, 2@@gp g+1 16GR(D2)p g:(2{1) HereGisNewton'sconstantand(D1)H2isthecosmologicalconstant.Becauseourscalarisaspectatorto-drivenination,itsbackgroundvalueiszero.OurbackgroundgeometryistheconformalcoordinatepatchofD-dimensionaldeSitterspace, Perturbationtheoryisexpressedusingthegravitoneldh(x), Theinversemetricandthevolumeelementhavethefollowingexpansions, g=aD1+1 2h+1 82h21 42hh+:::: Thiscomputationrequiresthe2hand2h2interactionswhichderivefromexpandingthescalarkineticterm, 2@@gp g=1 2@@aD2(h+1 2h+1 82h21 42hh1 22hh+2hh+O(3)): Werepresentthe3-pointand4-pointinteractiontermsasvertexoperatorsactingontheelds.Forexample,therstofthe3-pointverticesis, 2aD2@@h=)V1=iaD2@1@2:(2{7) 11

PAGE 12

Three-PointVertexOperatorsVIcontractedinto12h. I Four-PointVertexOperatorsUIcontractedinto12hh. Wenumbertheelds\1",\2",\3",etc,startingwiththetwoscalarsandproceedingtothegravitons.Althoughweextractafactorof1 2forthetwoidenticalscalars,itismoreecient,forourcomputation,nottoextractasimilarfactorof1 2fortheidenticalgravitonsofthe4-pointvertices.Thenwecandispensewiththesymmetryfactor.Soourrst4-pointvertexis, The3-pointverticesarelistedinTable 2-1 ;Table 2-2 givesthe4-pointvertices. Threenotationalconventionswillsimplifyourdiscussionofpropagators.Therstistodenotethebackgroundgeometrywithahat, Second,becausetimeandspacearetreateddierentlyinthegaugeweshallemploy,itisusefultohaveexpressionsforthepurelyspatialpartsoftheLorentzmetricandtheKroneckerdelta, 12

PAGE 13

2H`(x;x0)=aa0H2nk~x~x0k2j0ji2o;(2{11) whereaa()anda0a(0). Themasslessminimallycoupledscalarpropagatorobeys, IthaslongbeenknownthatthereisnodeSitterinvariantsolution[ 22 ].Thede-Sitterbreakingsolutionwhichisrelevantforcosmologyistheonewhichpreserveshomogeneityandisotropy.Thisisknownasthe\E(3)"vacuum[ 23 ],andtheminimalsolutiontakestheform[ 8 9 ], (D ThedeSitterinvariantfunctionA(y)is[ 9 ], (D (n+D (n+2)y Togetthegravitonpropagator,weaddthefollowinggaugexingtermtotheinvariantLagrangian[ 24 ], 2aD2FF;Fh;1 2h;+(D2)Hah0:(2{15) 13

PAGE 14

2hDh,wherethekineticoperatoris, 2 41 2(D3)0000DA+0( 2D2 Thethreescalardierentialoperatorsare, Thegravitonpropagatorinthisgaugehastheformofasumofconstanttensorfactorstimesscalarpropagators, Wecangetthescalarpropagatorsbyinvertingthescalarkineticoperators, Thetensorfactorsare, (D2)(D3)h(D3)00+ Withthesedenitionsandequation( 2{21 )wecanseethatthegravitonpropagatorsatisesthefollowingequation, 14

PAGE 15

25 ], TheA-typepropagatorobeysthesameequationasthatofamassless,minimallycoupledscalar.ThedeSitterinvariantB-typeandC-typepropagatorsare, (n+D (n+2)y (n+D (n+2)y Theycanalsobeexpressedashypergeometricfunctions[ 26 27 ], (D (D ThesepropagatorsmightlookcomplicatedbuttheyareactuallysimpletousesincethesumsvanishinD=4,andeveryterminthesesumsgoeslikeapositivepowerofy(x;x0).Thereforeonlyasmallnumberoftermsinthesumscancontributewhenmultipliedbyaxeddivergence. 15

PAGE 16

Thisistheheartofthepaper.Werstevaluatethecontributionfromthe4-pointverticesofTable 2-2 .Thenwecomputethevastlymoredicultcontributionsfromproductsoftwo3-pointverticesfromTable 2-1 .Wedonotrenormalizeatthisstage,althoughwedotakeD=4inniteterms.Renormalizationispostponeduntilthenextsection. Figure3-1. Contributionfrom4-pointvertices. 3-1 .Theanalyticformis, InreadingothevariouscontributionsfromTable 2-2 oneshouldnotethat,whereas\@2"actsuponx0,thederivativeoperator\@1"mustbepartiallyintegratedbackontotheentirecontribution.Forexample,thecontributionfromU1is, 16

PAGE 17

2-2 gives, Itisapparentfromexpression( 3{3 )thatwerequirethecoincidencelimitsofeachofthethreescalarpropagators[ 28 ], limx0!xiA(x;x0)=HD2 (D limx0!xiB(x;x0)=HD2 (D limx0!xiC(x;x0)=HD2 (D (D2)(D3)!H2 NotethattheB-typeandC-typepropagatorsareniteforD=4.Thefourcontractionsofthecoincidentgravitonpropagatorwerequireare[ 28 ], TosavespacewehavetakenD=4inthenitecontributionsfromtheB-typeandC-typepropagators. 17

PAGE 18

3{3 )andperformingsometrivialalgebragivesthenalresult, (D 4D(D1)@aD2@D(xx0)DaD2r2D(xx0))+i2H2 NotethateachofthesetermsvanishesintheatspacelimitofH!0withthecomovingtimetln(a)=Hheldxed.Thereasonforthisisthatthecoincidencelimitoftheatspacegravitonpropagatorvanishesindimensionalregularization. InordertocombineiM24ptwiththe3-pointcontributionsitisusefultointroducenotationforthescalard'AlembertianindeSitterbackground, Wealsoextractthelogarithmfrominsidethed'Alembertian, 2ln(aa0)a44(xx0)+3 2H2a44(xx0):(3{13) Withtheseconventionsthenalresulttakestheform, (D 4D(D1)cot 18

PAGE 19

Contributionfromtwo3-pointvertices. 3-2 .ConsultingTable 2-1 andrememberingtopartiallyintegrateanyderivativethatactsuponanouterleggives, =2@@0n(aa0)D2ihi@@0iAo+2 Uponsubstitutingthegravitonpropagator,performingthecontractionsandsegregatingtermswiththesamescalarpropagators,onendsthreegenericsortsofterms.TherstarethosewhichinvolvetwoA-typepropagators, ThesecondkindofterminvolvesoneA-typeandoneB-typepropagator, 19

PAGE 20

22D2 Eachoftheninetermsinexpressions( 3{17 3{19 )hastheform, where\I"mightbeA,BorC.Notethatthethreepropagatorscanbewrittenalmostentirelyasfunctionsofy(x;x0)denedin( 2{11 ), ThefunctionsA(y),B(y)andC(y)canbereadofromexpressions( 2{14 ),( 2{27 )and( 2{28 ),respectively.NotealsothattheinnerderivativeseliminatethedeSitterbreakingtermofiA, aD2D(xx0)+A00(y)@y @x@y @x0+A0(y)@2y @x@x0:(3{22) Itfollowsthattheanalysisbreaksupintothreeparts: 3{22 ); 3{20 );and 3{20 )oftheform, @x@y @x0+A0@2y @x@x0#):(3{23) Weshalldevoteaseparatepartofthissubsectiontoeach. 3{17 ),the4thtermof( 3{18 )andfrom( 3{19 ).Toavoidoverlapwiththelogarithmcontributionsofthenextpartwedenethelocalcontributionfromthe4thtermof( 3{17 )withoutthe 20

PAGE 21

aD2D(xx0)i=i2HD2 (D3)(D Notethatwehavechosentoconvertprimedderivativesintounprimed,andtoabsorbthetemporalderivativesintoacovariantd'Alembertian, Thiswillfacilitaterenormalization. Theothertwolocalcontributionsarenite.The4thtermof( 3{18 )gives, aD2D(xx0)i=i2H2 And( 3{19 )gives, 22D2 aD2D(xx0)i=i2H2 Summingthethreelocalcontributionsgives, (D 2iaD2r2+O(D4))D(xx0): 21

PAGE 22

3{17 ).Theycanbesimpliedbyusingthepropagatorequation( 2{12 ), OnecanalsotakethelimitD=4becauseallthelogarithmcontributionsarenite.Forexample,thefunctionA(y)is, Thersttermof( 3{17 )gives, Thesecondtermof( 3{17 )hasthemostcomplicatedreduction, Thethirdtermof( 3{17 )gives, 22

PAGE 23

3{17 ), Combiningallfourtermsresultsinsomesignicantcancellations, Eachofthelocaltermsin( 3{37 )cancelsasimilarnite,local4-pointcontributionin( 3{14 ),leavingonlythenonlocalcontributioninvolvingderivativesofA(y).Itispossibletoeliminatethetemporalderivativesinthisexpression.However,theprocedureisbestexplainedinthenalpartofthissection. 3{23 )generically,withoutexploitingthefunctionalformsofA(y),B(y)andC(y).Wealsoconvertallprimedderivativesintounprimedonesandexpressthenalresultintermsoften\ExternalOperators".Thisnotonlymakesitpossibletoperceivegeneralrelations,italsoreducesthesupercialdegreeofdivergenceofthetermswemusteventuallyexpand.AnditleavesfunctionsofthedeSitterinvariantvariabley(x;x0)forwhichanimprovedexpansionprocedureispossible[ 29 ]. Thisstepofextractingderivativesisstillquiteinvolvedsoweshalldescribeonlytheessentialsinthebodyofthethesisandconsignthedetailstoanappendix.TheappendixalsogivestabulatedresultsforeachofthetenExternalOperators.Thenalreductionofthesegenerictabulatedresultsisstraightforward.Thissubsectioncloseswithadescription 23

PAGE 24

Ourgenericmethodforextractingderivativesrequiresonetocarryoutmanyindeniteintegrationsoffunctionsofy.WedenethisoperationbythesymbolI[f](y), IfthefunctionF(y)istheproductoftwopropagatorfunctions,thenactingtwoderivativesonitcanneverproduceadeltafunction, @x@y @x0+F0(y)@2y @x@x0:(3{39) Itfollowsthatwecanexpresstheinnerpartofthebasicnormalcontribution( 3{23 )intermsofintegralsofsuchproducts, @x@y @x0+A0(y)@2y @x@x0)=@@0I2[fA00](y)+@2y @x@x0I[f0A0](y):(3{40) Wemuststilldealwiththenaltermof( 3{40 ).Inconformalcoordinatesthemixedsecondderivativeofy(x;x0)is[ 30 ], @x@x0=H2aa0ny002a0Hx+2a0Hx02o:(3{41) Breakingthisupintospatialandtemporalcomponentsgives, @x0@x00=H2aa0h2y+2aa0H2k~xk2i;@2y @x0@x0j=H2aa02aHxj; @xi@x00=H2aa02a0Hxi;@2y @xi@x0j=H2aa02ij: Oneconsequenceis, 2(D1)I[f](y)rr0 24

PAGE 25

2rr0I3[f0A0](y)+H2aa0n(2y)I[f0A0](y)(D1)I2[f0A0](y)o; UsingtheseidentitiesitispossibletoextractthederivativesfromtherstoftheA-terms, =(aa0)D2r4I2[AA00](y)+2(D1)H2(aa0)D1r2I[A02](y): OnlythersttermintheexpansionofI2[AA00](y)contributesadivergence;wecansetD=4inthehigherterms.Similarly,onlythersttwotermsintheexpansionofI[A02](y)candiverge. Itisverysimpletoconverttheprimedspatialderivativestounprimedones, 3{45 )wemeanthenaivesecondderivative,withoutthedeltafunction. 25

PAGE 26

@x0=Hay2a0H=Hay2+2a0 @x00=Ha0y+2aH=Ha0y2+2a a0: Fromthisfollowthreeimportantidentities.Thesimpleoneis, Anotherresultis, =(D1)Haa0f(y)+r2 Thenalidentityresultsfromcombining( 3{54 )and( 3{56 ), =H(a+a0)2yf0(y)+(D1)Haa0f(y)r2 Wecannowreducethenonlocallogarithmcontributionfromequation( 3{37 ).Applying( 3{58 )gives, ThederivativeandtheintegralarestraightforwardusingtheD=4expansionforA(y)givenin( 3{32 ).ThenalresultisreportedinTable 3-1 .Ofcoursewehaveneglectedtermswhicheventuallyvanishsuchasr4y. 26

PAGE 27

Coecientof2H4 4 NonlocalLogarithmContributionsfromrelation( 3{37 )withxy Weeventuallywanttoabsorballdoubletimederivativesintocovariantd'Alembertian's, a@0+1 Thisismosteectivelydonewiththeinternalfactorsof(aa0)D2.Forexample,considerreducingoneofthemixedA-terms, =aDa0D2r2I2[AA00](y)+(aa0)D2r4I2[AA00](y)+HaD1a0D2r2@0I2[A02](y)+(D1)H2aDa0D2r2I2[A02](y): Notealsothatwecanconvertaprimedcovariantd'Alembertiantoanunprimedoneifitactsonafunctionofjusty(x;x0), 27

PAGE 28

=aD2a0Dr20I2[AA00](y)+(aa0)D2r4I2[AA00](y)+HaD2a0D1r2@00I2[A02](y)+(D1)H2aD2a0Dr2I2[A02](y); =aD2a0Dr2I2[AA00](y)+(aa0)D2r4I2[AA00](y)HaD3a0Dr2@0I2[A02](y)+1 2aD3a0D1r4I3[A02](y): Thepreviouspointcanbesummarizedbytherelations, Anotherimportantpointisthatitisalmostalwaysbesttowriteanysinglefactorofthemixedproduct@0@00asfollows, 2(@0+@00)21 2@201 2@020:(3{69) Sowendtheubiquitousreduction, =1 2(aa0)D2(a2+a02)hf(y)+H2yf0(y)i(aa0)D2r2f(y)+1 4(D2)(aa0)D2r2I[f](y)+1 2(D2)(D3)H2(aa0)D1f(y)+1 2H2(a+a0)2(aa0)D2h(D1)yf0(y)+y2f00(y)i: 28

PAGE 29

=(a2+a02)(aa0)D2r2nI2[A00B](y)+H2I[A0B+yA00B](y)o+(aa0)D2r4n2I2[A00B](y)1 2I3[A0B0](y)oH2(a+a0)2(aa0)D2r2n(D2)I[A0B+yA00B](y)+yA0(y)B(y)+y2A00(y)B(y)yI[A0B0](y)o+(D1)H2(a2+aa0+a02)(aa0)D2r2I2[A0B0](y): Extractingderivativesinthiswayfromthevariousnormalcontributionsresultsinfunctionsofywhichareacteduponbytenexternaloperators, Tables A-1 A-10 oftheAppendixgiveexplicitresultsforeachofthesetenoperators.NotethatinadditiontothethreepropagatorfunctionsA(y),B(y)andC(y),wealsoemploy 29

PAGE 30

BBAandC2D2 Thenextstepissubstitutingtheexplicitforms( 2{14 ),( 2{27 ),( 2{28 )forthepropagatorfunctionsintotheresultsofTables A-1 A-10 andexpandingtotherequiredorder.Tounderstandwhatthisis,notethatwewillbeintegratingtheresultwithrespecttox0againstasmoothfunction(thezerothordermodesolution)withthederivativesofthe\ExternalOperators"actedoutsidetheintegrals.Becausey(x;x0)vanisheslike(xx0)2atcoincidence,itisonlynecessarytoretainthedimensionalregularizationfortermswhichwouldgolike1=y2andhigherforD=4. Althoughthesetablesinvolveabewilderingvarietyofdierentintegralsandderivatives,carefulexaminationoftheresultsshowsthattheyderivefromjusteightproductsofthepropagatorfunctions, ThemostsingularproductsofA02andAA00alwaysappeareitherdoublyintegrated|e.g.,I2[AA00]inTable A-1 |orelseintegratedonceandthenmultipliedbyy|e.g.,1 2yI[A02]inTable A-2 .Henceweneedonlyretainthedimensionalregularizationforthe1=yDtermsoftheseexpansions, 16H2D4 16H2D4 D24 TheproductA0B0canappearwithonlyasingleintegration|e.g.,DI[A0B0]inTable A-2 |ormultipliedbyasinglefactorofy|e.g.,DyA0B0inTable A-4 .Wemusttherefore 30

PAGE 31

16H2D4 However,theproductA00Bisalwaysshieldedbytwoormorepowersofy,sotheexpansionwerequireforitis, 16H2D4 D24 TheproductsinvolvingBandCarelesssingular, 16H2D4 16H2D4 16H2D4 16H2D4 OnenextsubstitituestheseexpansionsintothetotalsofTables A-1 A-10 andperformsthenecessaryintegrations,dierentiations,multiplicationsandsummations.Wemustalsomultiplybytheoverallfactorof2.Forexample,theresultfor\External 31

PAGE 32

4(D1)(D2) 4 4 4(D1) 0 2(D1)(D2) (D1)(D2) 0 0 Table3-2. DivergentNormalContributions. Operator"is, 162H2D4 D24 =2H2D4 Wehavetabulatedtheresultsforeachoftheten\ExternalOperators".Table 3-2 givesthequadraticallyandlogarithmicallydivergentterms;Table 3-3 givesthetermswhicharemanifestlynite.InallcasestheexpressionswereworkedoutbyhandandthencheckedwithMathematica[ 31 ]. 32

PAGE 33

x+13 1 30 x49 6x+4ln2x+10lnx+12xlnx 1 3x+60lnx120xlnx+72x 2 13 6x12lnx+12xlnx 3 6x+36xlnx+12x FiniteNormalContributionsintermsofxy 33

PAGE 34

Inthissectionweobtainacompletelyniteresultfortheself-mass-squaredbysubtracting4th-orderBPHZcounterterms[ 32 ].Werstidentifytwoinvariantcountertermswhichcancontributetothis1PI(OneParticleIrreducible)functionatoneloop.Becauseourgaugexingfunctional( 2{15 )breaksdeSitterinvariance[ 24 ],wemustalsoconsidernoninvariantcounterterms.Weidentifytheonlypossiblecandidatebasedonacarefuldiscussionoftheresidualsymmetriesofourgaugexingfunctional.Itremainstocollectandcomputetheactualdivergences.Contributionsfromthe4-pointverticesarealreadylocal,asarethe\localcontributions"fromthe3-pointvertices.Usinganowstandardtechniqueofpartialintegration[ 8 ]wesegregatethedivergencesfromthe\normalcontributions"ofTable 3-2 .Intheendweidentifythedivergentpartsofthethreecountertermsandreportacompletelyniteresult. Figure4-1. Contributionfromcounterterms. Onerenormalizesthescalarself-mass-squaredbysubtractingdiagramsoftheformdepictedinFig. 4-1 .Becauseourscalar-gravitoninteractionshavetheformnhn@@,comparedtothenhn@h@hinteractionsofpuregravity,thesupercialdegreeofdivergenceatonelooporderisfour,thesameasthatofpurequantumgravity.Ofcoursethecorrespondingcountertermsmustcontaintwoscalarelds,eachofwhichhasthedimensionofamass.Becausewearedealingwithoneloopcorrectionsfromquantumgravity,allthesecountertermsmustalsocarryafactoroftheloopcountingparameter2=16G,whichhasthedimensionofaninversemass-squared.Eachcountertermmustthereforehaveanadditionalmassdimensionoffour,eitherintheformofexplicitmasses 34

PAGE 35

g:(4{1) Thereisnowaytoobtainaninvariantwithonederivative.Twoderivativescanacteitheronthescalarsoronthemetrictoproduceacurvature.Wecantakethedistincttermstobe, gand2m22Rp g:(4{2) Therearenoinvariantswiththreederivatives.ByjudiciouspartialintegrationanduseoftheBianchiidentitywecantakethedistincttermswithfourderivativestobe, g;2@@Rgp g;2@@Rp g;22R2p gand22RRp g: Becauseourscalarismasslessandmassismultiplicativelyrenormalizedindimensionalregularization,wecandispensewith( 4{1 )and( 4{2 ).Thelasttwocountertermsof( 4{3 )cannotoccurbecausetheunrenormalizedLagragnian( 2{1 )isinvariantunder!+const.Thesecondandthirdtermsof( 4{3 )becomedegeneratewhenoneusesthebackgroundequation,bR=(D1)H2bg.Intheendjusttwoindependentinvariantcountertermssurvive,eachwithitsowncoecient, 1 212aDand1 222H2@@aD2:(4{4) Theassociatedverticesare, 1 212aD!i12aD2D(xx0); 222H2@@aD2!i22H2aDD(xx0): HadourgaugeconditionrespecteddeSitterinvariance,allthedivergencesiniM2(x;x0)couldhavebeenabsorbedusing( 4{5 )and( 4{6 )withappropriatechoicesforthedivergentpartsofthecoecients1and2.Althoughthereasonsforitarenot 35

PAGE 36

24 33 34 ].Thisiswhyweemployedthenoninvariantfunctional( 2{15 ).WemustthereforedescribehowdeSittertransformationsactinourconformalcoordinatesystemandwhichsubgroupofthemisrespectedbyourgaugecondition.The1 2D(D+1)deSittertransformationscanbedecomposedasfollows: 2(D1)(D2)distincttransformations. Itturnsoutthatourgaugechoicebreaksonlyspatialspecialconformaltransformations( 4{10 )[ 18 ].Hencewecanusetheothersymmetriestorestrictpossiblenoninvariantcounterterms.Spatialtranslationalinvariancemeansthattherecanbenodependenceuponxiexceptthroughtheelds.Rotationalinvarianceimpliesthatspatialindicesonderivativesmustbecontractedintooneanother.Dilatationinvarianceimpliesthatderivativesandtheconformaltimecanonlyoccurintheforma1@. Wecanalwaysusetheinvariantcounterterms( 4{5 4{6 )toabsorba@20infavorofr2andasingle@0, a@0+r2 36

PAGE 37

OnemightthinkweneedHaD1@0'',butapartialintegrationallowsittobewrittenintermsofaninvariantcountertermandonewithpurelyspatialderivatives, =1 2HaD3@0(@'@')+H2@'@'gp gH2aD2r'r'; 2(D1)H2@'@'gp gH2aD2r'r': AnothertermonemightconsiderisHaD3@0'r2',butitcanbepartiallyintegrated(twice)togivepurelyspatialderivatives, =1 2HaD3@0(r'r'); 2(D3)H2aD2r'r': Basedontheseconsiderationsweconcludethatonlythreenoninvariantcountertermsmightbeneededinadditiontothetwoinvariantones, 1 22aD2'r2';1 22aD4r2'r2'and1 22H2aD2r'r':(4{19) Becauseourgaugexingterm( 2{15 )becomesPoincareinvariantintheatspacelimitofH!0withthecomovingtimeheldxed,anynoninvariantcountertermmustvanishinthislimit.Hencewerequireonlythenaltermof( 4{19 ).Thevertexitgivesis, 232H2aDr ThestructureofthethreepossiblecountertermsservestoguideourfurtherreductionofiM2(x;x0).First,wemustconvertallthefactorsofa0intoaonthelocalterms.Second,weseethatfactorsofHaD3r2@0D(xx0)arenotpossible.Finally,itisnot 37

PAGE 38

Itisnowtimetocollectthedivergenttermsfromtheprevioustwosections.Thosefromthe4-pointcontributions,andfromthe\local"3-pointcontributionsarealreadyinaformwhichcanbeabsorbedintothethreecounterterms.However,wemuststillbringthe\normal"3-pointcontributionsofTable 3-2 tothisform.RecallthatthesetermsinvolvepowersofythatarenotintegrableforD=4dimensions, Ourprocedureistoextractd'Alembertiansfromthesetermsuntiltheybecomeintegrableusingtheidentity, gD(xx0): HereRes[F]standsfortheresidueofF(y);thatis,thecoecientof1=yintheLaurentexpansionofthefunctionF(y)aroundy=0. Thekeyidentity( 4{22 )allowsustoextractacovariantd`Alembertianfromeachofthenonintegrableterms, (D2)2 (D3)(D4) Wecoulduse( 4{24 )on( 4{23 )toreducethembothtothepower1=yD3.Thepower1=yD3isintegrable,sowecouldtakeD=4atthispointwereitnotfortheexplicitfactorsof1=(D4). Tosegregatethedivergenceonthelocaltermweaddzerointheform, 0= aDD(xx0):(4{25) 38

PAGE 39

Coef:ofi2HD4 (D3)(D4)D(xx0) 4 4 (D2)(D33D24D+8) 4(D1) (D26D+4) 2(D1) (12D) (D1) Table4-1. LocalNormalContributionsfromTable 3-2 Using( 4{25 )in( 4{24 )gives, (D3)(D4)((4)D 8(D3)4 =iHD(4)D Theanalogousresultforthequadraticallydivergenttermis, 22 24 Thedivergentlocaltermsthatresultfromapplying( 4{27 )and( 4{28 )toTable 3-2 arereportedinTable 4-1 .Table 4-2 givesthecorrespondingniteterms.IneachcasewehaveeliminatedtheredundantExternalOperators3=21+2and3=21+2. ThenextstepistoreexpressthelocaltermsofTable 4-1 aslocalcounterterms.Thisisdonebyusingthedeltafunctiontoconvertallfactorsofa0fromtheExternal 39

PAGE 40

Coecientofi2H4 3x 6x] x]+ x+2 x2 3x 3x 6x Table4-2. FiniteNormalContributionsfromTable 3-2 withx=y Operatorsintofactorsofa,andthenpassingallfactorsofatotheleft.Inmostcasesthisisstraightforwardbut (aa0)D1(a2+a02)2haDD(xx0)i=h2aD212H2aD+8aD2H2r2+2(D22D+2)H4aDiD(xx0); (aa0)D1(a2+a02)H2haDD(xx0)i=2aD(H2H4)D(xx0): Ourresultsforthethreepossiblecounterterms( 4{5 ),( 4{6 )and( 4{20 )arereportedinTable 4-3 .Notethatthecontributionto( 4{5 )vanishes,asitmustbecausethiscountertermhappenstobezeroinatspace. Anotherimportantconsistencycheckcomesfromthelocaltermsproportionaltoi2H4aDD(xx0),whicharereportedinTable 4-4 .Recallthatacountertermofthisformisforbiddenbythesymmetry!+constofthebareLagrangian( 2{1 ).AlthoughthreeofthefourcontributionstoTable 4-4 diverge,theirsumisniteforD=4.Itdoesn'tvanishbecausetheA-typepropagatorequationimplies, BecausethetotalforTable 4-4 isniteonecantakeD=4andthenuse( 4{31 )tosubsumetheresultintonite,nonlocaltermsofthesameformashavealreadybeen 40

PAGE 41

3-3 Table 44 =i2HD4 4(D2)(D3)aDH4D(xx0); =2H4 Table 4-5 includesthiswiththesimilarlyniteresultsofTables 3-1 3-3 and 4-2 Ournalresultfortheregulatedbutunrenormalized,oneloopself-mass-squaredderivesfromcombiningexpressions( 3{14 ),( 3{29 ),andthelocalpartsof( 3{37 ),withTables 4-3 and 4-5 .Ittakestheform, 45 +O(D4):(4{35) Thecoecientsiare 4(D1)(D3)(D+1)(D4)(D)cot( 4(D3)(D =HD4 3+O(D4)); 3+2+O(D4)): (Here:577215isEuler'sconstant.)Theobviousrenormalizationconventionistochooseeachofthethreei'stoabsorbthecorrespondingi,leavinganarbitrarynitetermi, Wecannowtaketheunregulatedlimit(D=4)toobtainthenalrenormalizedresult, 45 41

PAGE 42

0 6D 2 0 2(D1) 0 (D1) 0 2(D1)(D2) (D316D2+28D16) 2(D1)(D2) NormalContributionstoCountertermsfromTable 4-1 .Alltermsaremultipliedbyi2HD4 (D3)(D4). Contrib:from Coef:ofi2HD4 (D3)(D4)aDH4D(xx0) (D1)(D2) 2 4 (D2)(D33D24D+8) 2(D1) 4(D2) OtherLocalNormalContributionsfromTable 4-1 42

PAGE 43

Coecientof2H4 26lnx 3x6ln2x18lnx x+4 4lnx x+18 lnx 6x 4lnx 3x54 (aa0)3H2r2 2x12lnx+48xlnx+12x 6x+4ln2x+10lnx+12xlnx 10 3lnx24xlnx+24x2lnx36x2 AllFiniteNonlocalContributionswithxy 43

PAGE 44

Thisistheheartofthepaper.Webeginbyclarifyingwhatismeantbytheeectivemodeequation,thenweexplaintherestrictedsenseinwhichwesolveit.Finallyweworkoutthecontributionsfromthelocalcountertermsin( 4{41 )andfromthenonlocaltermsofTable 4-5 io(x;~k)=Douth'(x);yin(~k)iinE:(5{1) Thisquantityisofgreatinterestforatspacescatteringproblemsbutithaslittlerelevancetocosmologywheretheremaybeaninitialsingularityandwhereparticleproductionprecludestheinvacuumfromevolvingtotheoutvacuum. Themoreinterestingcosmologicalexperimentistoreleasetheuniversefromapreparedstateatnitetimeandletitevolveasitwill.Themodesolutionsofinteresttothisexperimentaretheexpectationvaluesofcommutatorsofthefulleldwiththetreeordercreationoperatoroftheinitialvacuum, (x;~k)=Dh'(x);y(~k)iE:(5{2) Theeectiveeldequationthat(x;~k)obeysisgivenbytheSchwinger-Keldyshformalism[ 18 30 ].ThisisacovariantcovariantextensionofFeynmandiagramswhich 44

PAGE 45

35 { 38 ].Becausethereareexcellentreviewsonthissubject[ 39 { 42 ],wewillconneourselvestoexplaininghowtousetheformalism. ThechiefdierencebetweentheSchwinger-Keldyshandin-outformalismsisthattheendpointsofparticlelineshaveapolarity.Therefore,everypropagatori(x;x0)ofthein-outformalismgivesrisetofourSchwinger-Keldyshpropagators:i++(x;x0),i+(x;x0),i+(x;x0)andi(x;x0).EachofthesepropagatorscanbeobtainedbymakingsimplechangestotheFeynmanpropagator.Forourmodel,theFeynmanpropagatorsofthescalarandgravitonhappentodependuponthelengthfunctiony(x;x0)denedinexpression( 2{11 ),andalsouponthetwoscalefactors.Thefourpolaritiesderivefrommakingthefollowingsubstitutionsfory(x;x0): VerticesintheSchwinger-Keldyshformalismeitherhaveall+linesoralllines.The+vertexisidenticaltothatofthein-outformalism,whereasthevertexisitsconjugate. Becauseanyexternallinecanbeeither+or,eachN-pointoneparticleirreducible(1PI)functionofthein-outformalismgivesriseto2N1PIfunctionsintheSchwinger-Keldyshformalism.TheSchwinger-Keldysheectiveactionisthegeneratingfunctionalofthese1PIfunctions. 45

PAGE 46

['+;']=S['+]S[']1 2Zd4xZd4x08><>:'+(x)M2++(x;x0)'+(x0)+'+(x)M2+(x;x0)'(x0)+'(x)M2+(x;x0)'+(x0)+'(x)M2(x;x0)'(x0)9>=>;+O('3); whereS[']istheclassicalscalaraction. Attheorderweareworking,iM2++(x;x0)isthesameasthein-outself-mass-squared.Wecanthereforereaditofrom( 4{41 ).WegetiM2+(x;x0)bydroppingthedeltafunctionterms,reversingthesignandreplacingy(x;x0)byy+(x;x0)inTable 4-5 .Theothertwo1PI2-pointfunctionsderivefromconjugatingthesetwo, TogettheSchwinger-Keldysheectiveeldequationsonevariestheactionwithrespecttotheeldofeitherpolarity,thensetsthetwopolaritiesequalto(x).Atlinearizedlevelthisgives, Herei=1=Histheinitial(conformal)timeatwhichtheuniverseisreleasedinfreeBunch-Daviesvacuum.Onecanseefromrelations( 5{3 )and( 5{4 ),andfromtheextraconjugatedvertexinM+(x;x0),thatthebracketedtermvanishesfor0>.Thefactthaty+(x;x0)isthecomplexconjugateofy++(x;x0)for0
PAGE 47

5{9 ): 1. Weonlyknowthescalarself-mass-squaredatonelooporder;and 2. Wetooktheinitialstatetobefree,Bunch-Daviesvacuum. Therstlimitationmeanswemustsolve( 5{9 )perturbatively.Thefullscalarself-mass-squaredcanbeexpandedinpowersoftheloop-countingparameter2=16G, Asimilarexpansionappliesforplanewavesolutionsto( 5{9 ), (x;~k)=1X`=02``(;k)ei~k~x:(5{11) Tomake(x;~k)agreewith( 5{2 )wemustnormalizethetreeordersolutionappropriately, 0(;k)=u(;k)H aHexphik aHi:(5{12) The`1solutionsobey, WeknowonlyM21(x;x0)sothesolecorrectionwecancomputeis1(;k). Thesecondlimitationmeansitonlymakessensetosolvefor1(;k)atlatetimes,i.e.,as!0. 47

PAGE 48

30 43 ].Inthein-outformalismthefreevacuumisautomaticallycorrectedbytimeevolution.OnecanfollowtheprogressofthisintheSchwinger-Keldyshformalismbyisolatingtermsthatdecaywithincreasingtimeafterthereleaseoftheinitialstate.Forexample,thetwoloopexpectationvalueofthestresstensorofamassless,minimallycoupledscalarwithaquarticself-interactiongivesthefollowingenergydensityandpressure[ 8 9 ], 8ln2(a)+1 18a31 81Xn=1(n+2)an1 8ln2(a)1 12ln(a)1 241Xn=1(n24)an1 Wesuspectthatthe(separatelyconserved)termswhichfalllikepowersof1=acanbeabsorbedintoanordercorrectionoftheinitialstate.Ontheotherhand,thetermswhichgrowlikepowersofln(a)representtheeectofinationaryparticleproduction(inthiscase,ofscalars)pushingtheeldupitsquarticpotential. Becausewehavenotworkedouttheorderand2correctionstoBunch-Daviesvacuum,wecannottrustcorrectionsto1(;k)thatfalloatlatetimesrelativetothetreeordersolutionu(;k), Tounderstandwhatthismeans,itisbesttoconvertequation( 5{13 )for1fromconformaltimetocomovingtimetln(H)=H, @t+k2 Becausethefactorofk2=a2onthelefthandsideredshiftstozeroatlatetimes,itseectonthelatetimelimitof1canbeatmostaconstant(aboutwhichmoreshortly).Givenaputativeformforthelatetimebehavioroftherighthandsideitiseasytoinferthe 48

PAGE 49

r:h:s!Cln(a)=)1!C r:h:s!C=)1!C r:h:s!Cln(a) r:h:s!C1 Theonlyeectswhichcanbedistinguishedfrom(thecurrentlyunknown)correctionstotheinitialstatederivefromcontributionstotherighthandsidethatfallonofasterthan1=ln(a).Itturnsoutthatinversepowersofln(a)cannotoccur,sothepracticaldividinglineisbetweencontributionstotherighthandsidewhichgroworapproachanonzeroconstantandthosewhichfallo. Ofcourseanycorrectiontothelefthandsideof( 5{17 )isliabletoinduceatimeindependentshiftinthelatetimelimitof1.Shiftsofthisformwhicharealsoindependentofthewavenumberkcouldbeabsorbedintoaeldstrengthrenormalizationandareunobservable.However,signicantk-dependentshiftscouldinduceobservabletiltsintothepowerspectrumandareanimportantpotentialoutcomefromastudyofthissort.Thepossibilityforsuchak-dependentshiftprovidesanexcellentjusticationforthelaborioustaskofworkingouttheorderand2correctionstoBunch-Daviesvacuum. Itiseasytoseethatonlyasmallrangeofk-dependent,constantshiftsin1canbesignicant.First,notethatalloneloopcorrectionsaresuppressedbytheloopcountingparameter,whoselargestvalueconsistentwiththecurrentlimitonthetensor-to-scalarratioisGH21012[ 44 ].Next,observethatanyconstantshiftin1canbeexpressedasthelatetimelimitofthetreeordermodefunctiontimesadimensionlessfunctionoftheratiok=H, Constant=H H:(5{22) 49

PAGE 50

5{17 ), 1 Becausethistermrapidlyredshiftstozero,weseefromtheprecedingdiscussionthatitcanonlyshift1bya(possiblyk-dependent)constantatlatetimes.Itiseasytoworkthisoutexplicitly. Substituting( 5{23 )into( 5{17 )gives, @t+k2 Thesolutioncanbewrittenasanintegraloverco-movingtime, (;k)=3k2Zt0dt0G(t;t0;k)u(0;k) wheretheGreen'sfunctionis, 50

PAGE 51

andadotdenotesdierentiationwithrespecttoco-movingtime. Theintegralsin( 5{25 )arestraightforwardandtheresultis, 1(;k) 3=i k(u(;k)ha(0)k2 =H 4i3k 4+iH Takingthelatetimelimitrevealsacorrectionoforderk=H, limt!11(;k)=3H 4iH 4e2ik=H+iH ke2ik=H):(5{30) Thedicultbutcrucialquestionis,canthiscorrectionbesubsumedintoaredenitionoftheinitialstate?Althoughwehavenoproof,thestrongsuspicionisthatitcanbe.Theorderk=Htermin( 5{30 )derivesentirelyfromthehomogeneous,lowerlimittermsonthelastlineof( 5{29 ),andthisispreciselythepropertythataninitialstatecorrectionwouldhave. 4-5 mightseemtopresentabewilderingvarietyofnonlocalcontributionsto( 5{17 ),aseriesofsevenstraightforwardstepssucestoevaluateeachone: 51

PAGE 52

Eliminateanyfactorsof1=yusingtheidentities, 4 4 2ln2y 2. Extractthefactorsofandr2fromtheintegrationoverx0usingtheidentities, @a+k2 3. Combinethe++and+termstoextractafactorofiandmakecausalitymanifest, lny++ ln2y++ 4aa0H2(2x2): Herewedeneaa(),a0a(0),0andxk~x~x0k.Notethatanypositivepowersofybecome, 4aa0H2(2x2):(5{37) 4. Makethechangeofvariables~r=~x0~x,performtheangularintegrationsandmakethefurtherchangeofvariabler=z, 4aa0H2(2x2)ei~k(~x0~x)=4()Z0drr2F1 4aa0H2(2r2)sin(kx) =4()3Z10dzz2Faa01 5. Reducethezintegrationtoacombinationofelementaryfunctionsandsineandcosineintegrals[ 45 ]. 52

PAGE 53

Makethechangeofvariablesa0=1=H0,expandtheintegrandandperformtheintegrationovera0. 7. Actanyderivativeswithrespecttoa. Muchofthelaborinvolvedinimplementingthesestepsderivesfromthespacetimedependenceofthezerothordersolution,u(0;k)ei~k~x0.Forexample,onecanseefrom( 5{39 )thatonlyelementaryfunctionswouldresultfromthezintegrationifthezerothordersolutionwereconstant.Becausewedonotyetpossesscorrectionstotheinitialstatethereisnoreasontoavoidmakingthesimplication, (5{40) Toseewhy,notefromexpressions( 5{16 )and( 5{39 )thatthedeviationofthezerothordersolutionfromu(0;k)introducesatleasttwofactorsof1=a0or1=a.Becausetheconstantmodefunctioncanatbestresultinpowersofln(a),andbecausewecannottrustcontributionsto( 5{17 )whichfallo,wemayaswelldiscardanytermswhichacquireanextrafactorof1=a.Extrafactorsof1=a0eectivelyrestricttheintegrationtoearlytimes,whichagaincausesthenetresulttofalloatlatetimes. Oneconsequenceofthesimplication( 5{40 )isthatwecanneglectanycontributionfromTable 4-5 whichcontainsafactorofr2.Wethereforeneedonlycomputetermsoftheform, wheretheconstantKtakesthevaluesof3,4and5,andthefunctionsf(x)are1=x,lnx=x,lnxandln2x.Theactionofthed'Alembertianderivesfromsettingk=0in( 5{33 ).TheintegrationsforK=3(giveninTable 5-1 )andK=4(giveninTable 5-2 )wereworkedoutinapreviouspaper[ 29 ].Applyingthesametechnique|whichisjustthe7-stepproceduregivenabovefork=0|givestheresultsforK=5inTable 5-3 53

PAGE 54

2a+2ln(a) 1 6ln(a) 2a+1 4a+O(1 1 3ln(a)11 9ln2(a) 2a+2ln(a) 4a+2 Table5-1. Integralswitha03. 2+O(1 4+O(1 1 6ln(a)11 36+O(1 1 6ln2(a)8 9ln(a)+7 42 Table5-2. Integralswitha04. Itremainsjusttoactthed'AlembertiansandsumtheresultsforeachofthecontributionsfromTable 4-5 .ThatisdoneinTable 5-4 .Althoughindividualcontributionscangrowasfastasln2(a),allgrowingorevennitetermscancel.Itfollowsthattherighthandsideof( 5{17 )fallsoatleastasfastas1=atimespowersofln(a).Hencetherearenosignicantcorrectionstothemodefunctionatonelooporder. 6a+O(1 36a+O(1 1 24a1 6+O(1 144a1 3ln(a)+5 9+O(ln(a) Table5-3. Integralswitha05. 54

PAGE 55

Coecientofu(0;k)H4 (aa0)4226lnx (aa0)4238 3x (aa0)426ln2x (aa0)4218lnx (aa0)4H26lnx x (aa0)4H24 (aa0)4H24lnx (aa0)4H44lnx x (aa0)4H418 (aa0)4H4108lnx (aa0)3(a2+a02)3=H2lnx 3ln(a)2 3ln2(a) (aa0)3(a2+a02)2lnx 3ln2(a) (aa0)3(a2+a02)21 6x 92 3ln(a) (aa0)3(a2+a02)H22lnx 3ln(a)+2 3ln2(a) (aa0)3(a2+a02)H25 310ln(a) (aa0)3(a2+a02)H218lnx (aa0)3(a2+a02)H44lnx 3ln(a)2 3ln2(a) (aa0)3(a2+a02)H432 3x 9+32 3ln(a) Total 0 Table5-4. Theu(0;k) 55

PAGE 56

Wehavecomputedoneloopquantumgravitationalcorrectionstothescalarself-mass-squaredonalocallydeSitterbackground.Weusedthattosolvetheoneloop-corrected,linearizedeectiveeldequationforamassless,minimallycoupledscalar.ThecomputationwasdoneusingdimensionalregularizationandrenormalizedbysubtractingthethreepossibleBPHZcounterterms.Becauseourgaugecondition( 2{15 )breaksdeSitterinvariance,oneofthesecountertermsisnoninvariant. Unlikepreviousanalysisofthescalareectivepotential[ 20 ],ourtechniqueusesthefullself-mass-squared,includingcorrectionstothederivativetermsandnonlocalcorrections.Itshouldthereforebesensitivenotonlytothescalar'smassbutalsotoitseldstrength.Wedonotyetpossessthecomputationaltoolsneededtosearchfork-dependentbuttimeindependentshiftsoftheeldstrengthbutwendnogrowingcorrectionsatonelooporder. Thepointofthisexerciseistodiscoverwhetherornottheinationaryproductionofgravitonshasasignicanteectuponminimallycoupledscalarsasitdoesonfermions[ 18 ].Inordertocheckthiswecomputedoneloopcorrectionstothescalarmodefunctionsusingtheeectiveeldequation, Similarstudieshavealreadyprobedtheeectsofscalarself-interactions[ 10 11 ],fermions[ 45 ]andphotons[ 21 ],butnonehassofarconsideredtheeectsofgravitons.Althoughourscalarisaspectatorto-drivenination,thenearatnessofinatonpotentialssuggeststhattheresultweshallobtainmayapplyaswelltotheinatonofscalar-drivenination. Asignicantdierencebetweenthisandpreviousscalarstudies[ 10 11 21 45 ]isthatquantumgravityisnotrenormalizable.Althoughwecouldabsorbdivergenceswithquartic,BPHZcounterterms,nophysicalprinciplexesthenitecoecientsiofthese 56

PAGE 57

0(x;~k)=u(;k)ei~k~xwhereu(;k)=H Haiexphik Hai:(6{2) Henceonlythethirdcountertermmakesanonzerocontribution,anditseectrapidlyredshiftsaway, ItisinstructivetocompareourdeSitterbackgroundresult( 4{41 )withitsatspaceanalogue.IntheatspacelimitofH!0withxedcomovingtime,thescalarandgravitonpropagatorsbecome, 4D xD2; 4D xD2: Herex2isthePoincarelengthfunctionanalogoustoy(x;x0), x2(x;x0)k~x~x0k2j0ji2:(6{6) Twofeaturesoftheatspacepropagatorsdeservecomment: 1. BothpropagatorsaremanifestlyPoincareinvariant;and 2. Thecoincidencelimitsofbothpropagatorsvanishindimensionalregularization. Weexploitedtherstpropertytodropallbutthenalnoninvariantcountertermonthelist( 4{19 ).Andthesecondpropertyexplainswhyour4-pointcontribution( 3{14 ) 57

PAGE 58

Anonlyslightlylesstrivialcomputationrevealsthattheatspacelimitofourtotal3-pointcontributionshouldalsovanish, =22(D 16D@@(1 xD2@@1 xD2)h2()i; =22(D 16D@2(1 xD2@21 xD2); =0: Thisresulthasanumberofconsequences: 1. Itexplainswhythehighestdimensioncounterterm( 4{5 )failstoappear; 2. ItexplainswhyalltheentriesofTable 4-5 vanishintheatspacelimitexcepttherstline, 34 32 x2ln1 4H2x2); 58

PAGE 59

64 32 x2ln1 4H2x2); (6{13) 3. Itexplainswhy( 6{12 )and( 6{13 )cancelintheatspacelimit;and 4. ItmeansthatanyphysicaleectwendmustderiveentirelyfromthenonzeroHubbleconstant. Thisistherightpointtocommentonaccuracy.Thishasbeenalongandtediouscomputation,involvingthecombinationofmanydistinctpieces.Itissignicantwhenthesepiecesjointogethertoproduceresultsthatcanbecheckedindependently,suchasthevanishingoftheatspacelimit.Oneseesthatinthewaytheand 4-3 .AnotherexampleisthewaythreeindividuallydivergenttermscombineinTable 4-4 toproduceaniteresultforacountertermthatisforbiddenbytheshiftsymmetryofthebareLagrangian( 2{1 ). Althoughthe1countertermhadtovanishbytheatspacelimit,wedonotyetunderstandwhythecoecientthe2countertermisnite.Thecontributionofthistermtothescalarself-mass-squaredvanishesinatspace,butitwouldseemtoaectthe+h!+hscatteringamplitude.Thedivergencesonthiswereexploredintheclassicpaperof'tHooftandVeltman[ 46 ].Unfortunately,theiron-shellanalysismakesnodistinctionbetweenR(@)2|whichwehave|and(@)4|whichwedonothave. WeshouldalsocommentonwhatsubsetofthefulldeSittergroupisrespectedbyourresult( 4{41 ).Recallthatourgaugexingtermbreaksspatialspecialconformaltransformation( 4{10 ).Thisiswhythenoninvariantcounterterm( 4{20 )occurs.Itisalsoresponsibleforthenoninvariantfactorsofr2anda2+a02inTable 4-5 .Becausethesebreakingsderiveentirelyfromthegaugecondition,weexpectthemtohavenophysicalconsequence. 59

PAGE 60

4{9 ).Unliketheviolationofspatialspecialconformaltransformations,thebreakingofdilatationinvarianceisaphysicalmanifestationofinationaryparticleproductionandcanhaveimportantconsequences.Dilatationbreakingcomesintheln(aa0)termoftheA-typepropagator( 2{14 ).Theselogarithmswereresponsiblefortheseculargrowththatwasfoundinthefermioneldstrength[ 18 ],soonemightexpectthemtodriveanyeectonscalarsaswell.However,itturnsoutthatthefactorsofln(aa0)alldropout.Forthescalarpropagatorthisisatrivialconsequenceofthefactthatitalwayscarriesoneprimedandoneun-primedderivative.Logarithmsfromthegravitonpropagatordoappearinthe4-pointcontributions( 3{14 ),andinthe3-pointlogarithmcontributions( 3{37 ).Butallfactorsofln(aa0)cancelinthenalresult( 4{41 ),whichturnsouttorespectdilatationinvariance.Becauseofthiswesuspectthattherewillbenosignicantlatetimecorrectionstothemodefunctionsatonelooporder. Physicallyourresultmeansthattheseaofinfraredgravitonsproducedbyinationhaslittleeectonthescalar.Althoughourscalarisaspectatorto-drivenination,typicalscalarinatonpotentialsaresoatthattheinatonisalsounlikelytosuersignicantoneloopcorrectionsfromquantumgravity.Ifso,itisconsistenttousethetreeorderscalarmodefunctionstocomputethepowerspectrumofcosmologicalperturbations. Therearegoodreasonswhythescalarshouldnotacquireamassfromquantumgravity[ 20 ]butthereseemstobenoreasonwhyitseldstrengthcannotsueratimedependentrenormalizationofthetypeexperiencedbymasslessfermions[ 18 ].Weinberg'sresultforthepowerspectrumallowsforln(a)corrections[ 6 ],andtheydoarisefromindividualtermsinTable 5-4 .However,thereisnonetcorrectionofthisformatoneloop. Intheend,ournullresultmaynotbeassurprisingasitseems.Itisverysimpletoshowthatthescalarself-mass-squaredvanishesatonelooporderonaatbackground[ 21 ].Itisalsowrongtothinkofthescalarasweakerthanthefermion;whatisweakeristhefractionaloneloopcorrection.Althoughmasslessfermionsexperienceatime-dependent 60

PAGE 61

2.Bycontrast,thetreeorderscalarmodefunctionsapproachanonzeroconstant.Indeed,itisthefactthatderivativesofthisconstantvanishwhichmakesthefractionalcorrectionoftheh@'@'couplingsosmall.Theseaofinfraredgravitonsispresent,butitcanonlycouple,atthisorder,tothescalar'sstress-energy,andthestress-energyofasinglescalarredshiftstozeroatlatetimes. 61

PAGE 62

Wegroupthevariousnormalcontributionsintosevenparts: Inthesedenitionstheexprssion\@0@00A(y)"meansthenaivederivative,withoutthedeltafunction.Alsonotethatwehavesuppressedtheunbiquitousfactorsof2. AnimportantsimplicationinreducingP2istoachieveasymmetricformwhichhasno@0.Thiscanbedonebyaddingequations( 3{62 )and( 3{65 )andthenusingequation( 3{58 ), =(+2)I2[AA00]+(D1)2I2[A02]+3yI[A02]+(D1)1I2[A02] Anotherorganizationalpointconcernsremovingfactorsofyfrominsideintegrals.Thisisdesirablebecauseitreducesthenumberofdistinctintegralswhichappear.Itcanalwaysbeaccomplishedbypartialintegration.Wewillillustrateusingthefunctionacteduponby3inequation( 3{73 ), 62

PAGE 63

@yI[A00B]; =@ @ynyI[A00B]o+A0BI[A00B]; =@ @ynyI[A00B]o+I[A0B0]: Wecanthereforewrite, With( A{13 )and( A{14 )wecanreadothefollowingresultforP5fromequation( 3{73 ), 2I3[B0A0]o+2nyI[A00B]+I2[A0B0]o+3ny2A00B+(D1)yI[A00B]+(D2)I2[A0B0]o: ManytermsinvolvingAinP2combinewithcognatetermsinvolvingBinP5toproducethelesssingularpropagatorfunctionB=BA.Summingexpressions( A{9 )and( A{15 )gives, 2I3[B0A0]o+2nyI[A00B]+I2[A0B0]o+3ny2A00B+(D1)yI[A00B]+(D2)I2[A0B0]o: 63

PAGE 64

3{45 )and( 3{48 ), @yoI2[f0A0]1 2rr0I3[f0A0]; Oneemploys( A{17 )onP3(fromwhichwecanreadotheresultforP7)andP6togivethesub-parts, @yoI2[A02]i; 2@0@00h(aa0)D2rr0I3[A02]i; @yoI2[A0C0]i; 2@0@00h(aa0)D2rr0I3[A0C0]i; @yoI2[A0B0]i; 2rr0h(aa0)D2rr0I3[A0B0]i: 64

PAGE 65

ContributionActeduponby P3a Total TableA-1. Contributionsacteduponby=(aa0)D2. Applyingthesecondidentity( A{18 )toP1andP6givestheirsub-parts, Ofcoursethereisnoproblemfurtherreducingthespatialderivatives.Thefollowinggenericreductionsservetoreducetermsinvolvingtheoperator@0@00, 2(D1)(D2)1+2 @y+3 @y+y2@2 2(D2)(D3)12 @y3 @y+y2@2 4(D2)Ii)f(y); 2(D1)(D2)12 @y3 @y+y2@2 4(D2)Ii)f(y): Tables A-1 A-10 givetheresultsforeachofthetenExternalOperators. 65

PAGE 66

ContributionActeduponby P3b 2yI[A02](D1 2)I2[A02] 2yI[A0C0](D1 2)I2[A0C0] Total 2yI[A02](D1 2)I2[A02] +I[A0C0]1 2yI[A0C0](D1 2)I2[A0C0] TableA-2. Contributionsacteduponby=(aa0)D2(a2+a02)H2. Part ContributionActeduponby1 2(D1)(D2)yI[A02]1 2(D1)2(D2)I2[A02] 2(D1)(D2)yI[A0C0]1 2(D1)2(D2)I2[A0C0] 2(D1)(D2)yI[A02] Total 2(D1)2(D2)I2[A02]+(D1)(D2)I[A0C0] 2(D1)(D2)yI[A0C0]1 2(D1)2(D2)I2[A0C0] TableA-3. Contributionsacteduponby1=(aa0)DH4. Part ContributionActeduponby2 2y2A02D 2y2A0C0D Total 2y2A02D +yA0C0D 2y2A0C0 Contributionsacteduponby2=(aa0)D1(a2+a02)H4. 66

PAGE 67

ContributionActeduponby3 2y3(A02)01 2(D1)DyI[A02] 2y3(A0C0)01 2(D1)DyI[A0C0] 2y3(A02)01 2(D1)DyI[A02]+(D1)yA0C0 2y3(A0C0)01 2(D1)DyI[A0C0] TableA-5. Contributionsacteduponby3=(aa0)D1(a+a0)2H4. Part ContributionActeduponby P2+5 4I3[A02] 2I2[BA00] 2I2[BA00] 4I3[A0C0] Total 2I2[BA00]+1 4I3[A02]I2[CA00]+1 4I3[A0C0] TableA-6. Contributionsacteduponby=(aa0)D2(a2+a02)r2. 67

PAGE 68

ContributionActeduponby1 4D(D2)I3[A02] 4(D2)(D3)I3[A02] 2(D2)(D3)I2[BA00] 2(D1)DI2[A0B0] 2(D1)(D2)I2[BA00] 4D(D2)I3[A0C0] 4(D2)(D3)I3[A0C0] 2)I2[A0B0]+D2 Total 4(D2)I3[A02]2I[A0C0] +yI[A0C0]+(3D2 2)I2[A0C0]D 4(D2)I3[A0C0] TableA-7. Contributionsacteduponby1=(aa0)D1H2r2. Part ContributionActeduponby2 4yI2[A02] 2yI[BA00] 2yI[BA00] 4yI2[A0C0] Total 2yI[BA00]+I2[A0B0] 4yI2[A02]+1 4yI2[A0C0] TableA-8. Contributionsacteduponby2=(aa0)D2(a2+a02)H2r2. 68

PAGE 69

ContributionActeduponby3 4)yI2[A02]+1 4y2I[A02] 2)yI[A00B]+1 2y2A00B P6a 2)yI[A00B]+1 2y2A00B P7d 4)yI2[A0C0]+1 4y2I[A0C0] Total 2y2A00B+2(D1)yI[A00B]+(D2)I2[A0B0]+yI[A02] +(D1 4)yI2[A02]+1 4y2I[A02]+(D1 4)yI2[A0C0]+1 4y2I[A0C0] TableA-9. Contributionsacteduponby3=(aa0)D2(a+a0)2H2r2. Part ContributionActeduponby P1a 2I3[A0B0] 2I3[A02]+(D2 8)I4[A02] 4)I3[A00B] 4)I3[A00B] 2I3[A0B0] 2I3[A0C0]+(D2 8)I4[A0C0] Total 8)I4[A02] +I2[CA00]1 2I3[A0C0]+(D2 8)I4[A0C0] TableA-10. Contributionsacteduponby=(aa0)D2r4. 69

PAGE 70

[1] R.P.Woodard,\QuantumEectsduringInation,"inQuantumFieldTheoryundertheInuenceofExternalConditions(RintonPress,Princeton,2004)ed.K.A.Milton,pp.325-330,astro-ph/0310757. [2] V.F.MukhanovandG.V.Chibisov,JETPLett.33(1981)532. [3] A.A.Starobinski,JETPLett.30(1979)682. [4] V.Mukhanov,H.FeldmanandR.Brandenberger,Phys.Rep.215(1992)203. [5] A.R.LiddleandD.H.Lyth,Phys.Rep.231(1993)1,astro-ph/9303019. [6] S.Weinberg,Phys.Rev.D72(2005)043514,hep-th/0506236;Phys.Rev.D74(2006)023508,hep-th/0605244. [7] K.Chaicherdsakul,Phys.Rev.D75(2007)063522,hep-th/0611352. [8] V.K.OnemliandR.P.Woodard,Class.Quant.Grav.19(2002)4607,gr-qc/0204065. [9] V.K.OnemliandR.P.Woodard,Phys.Rev.D70(2004)107301,gr-qc/0406098. [10] T.Brunier,V.K.OnemliandR.P.Woodard,Class.Quant.Grav.22(2005)59,gr-qc/0408080. [11] E.OKahyaandV.K.Onemli,Phys.Rev.D76(2007)043512,gr-qc/0612026. [12] T.Prokopec,O.TornkvistandR.P.Woodard,Phys.Rev.Lett.89(2002)101301,astro-ph/0205331;Ann.Phys.303(2003)251,gr-qc/0205130. [13] T.ProkopecandR.P.Woodard,Ann.Phys.312(2004)1,gr-qc/0310056; [14] T.Prokopec,N.C.TsamisandR.P.Woodard,\StochasticInationaryScalarElectrodynamics,"arXiv:0707.0847. [15] T.ProkopecandR.P.Woodard,JHEP0310(2003)059,astro-ph/0309593. [16] B.GarbrechtandT.Prokopec,Phys.Rev.D73(2006)064036,gr-qc/0602011i. [17] S.P.MiaoandR.P.Woodard,Phys.Rev.D74(2006)044019,gr-qc/0602110. [18] S.P.MiaoandR.P.Woodard,Class.Quant.Grav.23(2006)1721,gr-qc/0511140;Phys.Rev.D74(2006)024021,gr-qc/0603135. [19] L.Smolin,Phys.Lett.93B(1980)95. [20] A.D.Linde,ParticlePhysicsandInationaryCosmology(Harwood,Chur,Switzerland,1990),hep-th/0503202;\InationaryComsology,"arXiv:0705.0164. [21] E.O.KahyaandR.P.Woodard,Phys.Rev.D76(2007)124005,arXiv:0709.0536. 70

PAGE 71

[22] B.AllenandA.Folacci,Phys.Rev.D35(1987)3771. [23] B.Allen,Phys.Rev.D32(1985)3136. [24] N.C.TsamisandR.P.Woodard,Commun.Math.Phys.162(1994)217. [25] N.D.BirrellandP.C.W.Davies,QuantumFieldsinCurvedSpace(CambridgeUniversityPress,1982). [26] P.CandelasandD.J.Raine,Phys.Rev.D12(1975)965. [27] J.S.DowkerandR.Critchley,Phys.Rev.D13(1976)3224. [28] N.C.TsamisandR.P.Woodard,Ann.Phys.321(2006)875,gr-qc/05-6056. [29] T.Prokopec,N.C.TsamisandR.P.Woodard,Class.Quant.Grav.24(2007)201,gr-qc/0607094. [30] E.O.KahyaandR.P.Woodard,Phys.Rev.D72(2005)104001,gr-qc/0508015;Phys.Rev.D74(2006)084012,gr-qc/0608049. [31] S.Wolfram,TheMathematicaBook,ThirdEdition(CambridgeUniversityPress,1996). [32] N.N.BogoliubovandO.Parasiuk,ActaMath.97(1957)227;K.Hepp,Commun.Math.Phys.2(1966)301;W.Zimmermann,Commun.Math.Phys.11(1968)1;15(1969)208;inLecturesonElementaryParticlesandQuantumFieldTheory,ed.S.Deser,M.GrisaruandH.Pendleton(MITPress,Cambridge,1971),Vol.I. [33] I.AntoniadisandE.Mottola,J.Math.Phys.32(1991)1037. [34] R.P.Woodard,\DeSitterBreakinginFieldTheory,"inDeserfest:ACelebrationoftheLifeandWorksofStanleyDeser(WorldScientic,Hackensack,2006)edsJ.T.Liu,M.J.Du.K.S.StelleandR.P.Woodard,pp.339-351. [35] J.Schwinger,J.Math.Phys.2(1961)407. [36] K.T.Mahanthappa,Phys.Rev.126(1962)329. [37] P.M.BakshiandK.T.Mahanthappa,J.Math.Phys.4(1963)1;J.Math.Phys.4(1963)12. [38] L.V.Keldysh,Sov.Phys.JETP20(1965)1018. [39] R.D.Jordan,Phys.Rev.D33(1986)444. [40] K.C.Chou,Z.B.Su,B.L.HaoandL.Yu,Phys.Rept.118(1985)1. [41] E.CalzettaandB.L.Hu,Phys.Rev.D35(1987)495. [42] L.H.FordandR.P.Woodard,Class.Quant.Grav.22(2005)1637,gr-qc/0411003.

PAGE 72

[43] L.H.Ford,Phys.Rev.D31(1985)710. [44] D.N.Spergeletal.,Astrophys.J.Suppl.170(2007)377,astro-ph/0603449. [45] L.D.DuyandR.P.Woodard,Phys.Rev.D72(2005)024023,hep-ph/0505156. [46] G.'tHooftandM.Veltman,Ann.Inst.HenriPoincare,XX(1974)69.

PAGE 73

EmreKahyawasborninTurkeyandgothisB.S.andM.S.atMiddleEastTechnicalUniversity.HedecidedtopursueadoctoralstudyinphysicsatUSAandadmittedbytheUniversityofFlorida.Beeninterestedintheoreticalphysics,aftertakingaquantumeldtheorycourseofProf.RichardWoodard,hedecidedstudyQuantumGravitywithhim.HewasawardedhisPh.D.insummer2008. 73