Citation
Role of Axions in Structure Formation in the Universe

Material Information

Title:
Role of Axions in Structure Formation in the Universe
Creator:
Banik, Nilanjan
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (143 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
SIKIVIE,PIERRE
Committee Co-Chair:
WOODARD,RICHARD P
Committee Members:
TAN,JONATHAN CHARLES
GONZALEZ,ANTHONY HERNAN

Subjects

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

Notes

Abstract:
One of the most profound puzzles of modern day science is to understand the nature of dark matter. Axions that arise as a result of the solution to the \textit{CP} problem in the theory of strong interactions are one of the leading dark matter candidates. A remarkable property that sets axions apart from other dark matter candidates is that they thermalize via gravitational self-interactions to form a Bose-Einstein Condensate (BEC). Axion BEC behaves differently from other dark matter candidates over time scales longer than their thermalization time. This dissertation is a compilation of my published works, in which we have explored novel ways axion dark matter can affect structure formation in the Universe. After a brief historical review on dark matter we review in detail the Bose-Einstein condensation of axion dark matter. We show that density perturbations in the early Universe can be given a wavefunction description. We then show that the evolution of axion dark matter can only be described by quantum field theory on time scales longer than their thermalization time. Next, we show that imparting angular momentum to a system of axion BEC gives rise to vortices and these vortices can combine to form bigger vortices. Axion BEC vorticity will give rise to vorticity in baryons due to thermalization. We show that during recombination, the baryon vorticity will seed primordial magnetic fields. We then present a technique for obtaining the leading behavior of the velocity dispersion near caustics. The results are used to derive an upper limit on the energy dispersion of the local flow of dark matter from the sharpness of the nearby caustic, and a prediction for the dispersions in its velocity components is obtained. Ultralight axionlike particles (ULALPs) have been predicted in many String Theory based extensions of Standard Model and is a popular dark matter candidate. We show that like QCD axions, ULALPs form a BEC. We then derive lower mass bounds on ULALPs based on heating effect of infalling ULALPs on galactic disk stars and the thickness of the nearby caustic ring. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: SIKIVIE,PIERRE.
Local:
Co-adviser: WOODARD,RICHARD P.
Statement of Responsibility:
by Nilanjan Banik.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
LD1780 2017 ( lcc )

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Full Text

PAGE 5

4

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^n(^z^n) R(t)2 t jmax

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(80;0)m 12m

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5%26%

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10)]TJ /F3 7.9701 Tf 6.586 0 Td[(8c10)]TJ /F3 7.9701 Tf 6.586 0 Td[(8c

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10)]TJ /F3 7.9701 Tf 6.586 0 Td[(12c10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5c210)]TJ /F3 7.9701 Tf 6.586 0 Td[(17cc2

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10)]TJ /F3 7.9701 Tf 6.586 0 Td[(8c

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L=g2s 322~GaGa2gsGaL6=0.0:710)]TJ /F3 7.9701 Tf 6.586 0 Td[(11 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(11 U(1)PQ

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(x) fafaU(1)PQ(x)(x)=0 (x) ma'10)]TJ /F3 7.9701 Tf 6.586 0 Td[(6eV1012GeV fa (x) DD(x)+V0a((x))=0V0aD Va=ma(t)2f2a1)]TJ /F4 11.9552 Tf 11.955 0 Td[(cos(x) fa: ma t1 m(t1)t1=1:~=c=1t1210)]TJ /F3 7.9701 Tf 6.587 0 Td[(7s)]TJ /F7 7.9701 Tf 20.362 -4.428 Td[(fa 1012GeV1=3T1'1GeV1012GeV fa1=6(x)

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t)]TJ /F3 7.9701 Tf 6.586 0 Td[(11t1t1 p(t)1 t1R(t1) R(t)R(t)t1U(1)PQ n(t)'41047 cm3Xfa 1012GeV5=3R(t1) R(t)3XXX1 2sin211=(t1)=fa

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Nn(2)3 4 3(mv)31061Xfa 1012GeV8=3:N Tc(t)=2n(t) (3)1=3'300GeVX1=3fa 1012GeV5=9R(t1) R(t): 4 V=L3(~x;t) (~x;t)=X~n(a~n(t)~n(~x)+ay~n(t)~n(~x)) (~x;t)=X~n()]TJ /F5 11.9552 Tf 9.299 0 Td[(i!~n)(a~n(t)~n(~x))]TJ /F5 11.9552 Tf 11.955 0 Td[(ay~n(t)~n(~x))

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~n(~x)=ei~p~n~x p 2!~nV;~p~n=2~n=L~n=(n1;n2;n3)n1n2n3!~n=p p2~n+m2 [a~n(t);ay~n0(t)]=~n;~n0;[a~n(t);a~n0(t)]=0:4 H=X~n!~nay~na~n+X~n1;~n2;~n3;~n41 4~n3~n4s~n1;~n2ay~n1ay~n2a~n3a~n4 ~n3~n4s~n1;~n2=)]TJ /F5 11.9552 Tf 9.299 0 Td[( 4m2V~n1+~n2;~n3+~n4: aaaaayayayayayaaaayayaya Hg=)]TJ /F5 11.9552 Tf 10.494 8.087 Td[(G 2Zd3xd3x0(~x;t)(~x0;t) j~x)]TJ /F5 11.9552 Tf 11.76 0 Td[(~x0j(~x;t)=1 2(2+m22) Hg=X~n1;~n2;~n3;~n41 4~n3~n4g~n1;~n2ay~n1ay~n2a~n3a~n4 ~n3~n4g~n1;~n2=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(4Gm2 V~n1+~n2;~n3+~n41 j~p~n1)]TJ /F5 11.9552 Tf 12.022 0 Td[(~p~n3j2+1 j~p~n1)]TJ /F5 11.9552 Tf 12.022 0 Td[(~p~n4j2:Hg

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)]TJ /F1 11.9552 Tf 11.211 0 Td[<001c004d002f>-326<0069003f0032>-326<0032004d00320060003b0076>]TJ -383.818 -23.908 Td[<002f004200620054>-27<00320060006200420051004d>]TJ /F5 11.9552 Tf 51.73 0 Td[(!)]TJ /F5 11.9552 Tf 10.798 0 Td[(<>! )]TJ /F5 11.9552 Tf 11.397 0 Td[(<>1 )]TJ /F6 11.9552 Tf 10.635 0 Td[(nvN:

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N1061a(~p1)+a(~p2)$a(~p3)+a(~p4)N1N2N3NN4<>! )]TJ /F5 11.9552 Tf 12.557 0 Td[(<
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klij!k+!l)]TJ /F5 11.9552 Tf 12.375 0 Td[(!i)]TJ /F5 11.9552 Tf 12.374 0 Td[(!jclCldl cl(t)=Cl(t)+dl(t):Clp Nl _Cl(t)=)]TJ /F5 11.9552 Tf 9.299 0 Td[(iMXi;j;k=11 2ijklCkCiCjeiklijt:dldylCl(t)Cl(t) Kklijt<<1 _Cl(t))]TJ /F5 11.9552 Tf 21.918 0 Td[(iKXi;j;k=11 2ijklCkCiCj: 4 _C~p1(t)i 4m2VX~p2;~p31 2C~p2C~p3C~p4~p4=~p1+~p2)]TJ /F5 11.9552 Tf 12.381 0 Td[(~p3N3=2K2 _C~p 4m2VKN3=2 4m2VNN3=2

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KN=NClp N4 )]TJ /F7 7.9701 Tf 7.314 -1.793 Td[(1 4nm)]TJ /F3 7.9701 Tf 6.587 0 Td[(2n=N=V )]TJ /F7 7.9701 Tf 7.314 -1.793 Td[(g4Gnm2`2`=1=p !)]TJ /F1 11.9552 Tf 7.314 0 Td[<002d>-325<0069003f0032>-326<0069>26<0072>27<0051>-325<0032006200690042004b001c006900320062>-326<001c003b0060003200320058>-435<001b>26<0069>-325<0069003f001c0069>-326<0023>-27<0051006d004d002f001c>26<00600076>82<002d>]TJ -245.902 -23.908 Td[<006d0054>-326<00690051>-326<0037001c002b00690051>26<00600062>-325<00510037>-326<0051>26<0060002f00320060>-325<006b>-326<0051>26<0060>-325<00620051002d>]TJ ET BT /F5 11.9552 Tf 143.172 -365.098 Td[(vNvn (p)3n m2!n m2)]TJ /F5 11.9552 Tf 20.202 8.201 Td[(: N

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44 t)]TJ /F6 11.9552 Tf 11.851 0 Td[(1=H1=t !(t)=(p(t))2 m(t): !(t1)1=t1t1\(t1)>1=t1t1!(t)<1=tm(t)t1(p(t))2/R(t))]TJ /F3 7.9701 Tf 6.586 0 Td[(2/1=tR(t)/p tt14t1 )]TJ /F7 7.9701 Tf 7.314 -1.793 Td[((t1)H(t1)t14t1 )]TJ /F7 7.9701 Tf 7.314 -1.793 Td[((t)=H(t)/R(t))]TJ /F3 7.9701 Tf 6.587 0 Td[(3t/t)]TJ /F8 5.9776 Tf 7.782 3.259 Td[(1 2

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t1`(t)1=p(t)t1R(t)=R(t1) )]TJ /F7 7.9701 Tf 7.314 -1.794 Td[(g(t)=H(t)8Gnm2`2t510)]TJ /F3 7.9701 Tf 6.586 0 Td[(7R(t1) R(t)t t1Xfa 1012GeV2 3t1)]TJ /F7 7.9701 Tf 7.315 -1.793 Td[(g=H/R)]TJ /F3 7.9701 Tf 6.587 0 Td[(1(t)t/R(t) tBEC TBEC500eVXfa 1012GeV1 2:4 H(t)tBEC

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~p=0~p=0~p=0~p=0

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D)]TJ /F3 7.9701 Tf 6.587 0 Td[(4 4

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N N (~r;t) i@t (~r;t)=)]TJ /F4 11.9552 Tf 18.27 8.088 Td[(1 2mr2+m(~r;t) (~r;t);m(~r;t)~=c=1 n(~r;t)=N (~r;t) (~r;t):

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r2=4Gmn(~r;t): @tn+~r~j=0; ~j=N 2m( ~r )]TJ /F5 11.9552 Tf 11.955 0 Td[( ~r ):~v(~r;t)~j(~r;t)n(~r;t)~v(~r;t) (~r;t)=p n(~r;t)ei(~r;t) ~v=1 m~r: @t~v+(~v~r)~v=)]TJ /F5 11.9552 Tf 10.434 3.022 Td[(~r)]TJ /F5 11.9552 Tf 13.09 3.022 Td[(~rq q=)]TJ /F4 11.9552 Tf 17.979 8.087 Td[(1 2m2r2p n p n:q 0(~r;t)=p n0(t)ei1 2mH(t)r2;

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H(t) ~v=H~r: @tn0+3Hn0=0 0=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(1 2(@tH+H2)r2: @tH+H2=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(4G 3mn0(t): H(t)2+K a(t)2=8G 3mn0(t)K=)]TJ /F4 11.9552 Tf 9.299 0 Td[(1;0;+1a(t)H(t)=_a a (~r;t)= 0(~r;t)+ 1(~r;t):~k 1(~r;t)= 0(~r;t)Zd3k 1(~k;t)ei~k~r a(t): 1(~r;t)=Zd3k1(~k;t)ei~k~r a(t):

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i@t 1=)]TJ /F4 11.9552 Tf 15.613 8.088 Td[(1 2mr2 1+m(0 1+1 0); r21=4Gm( 0 1+ 0 1): (~k;t) 1(~k;t)+ 1()]TJ /F5 11.9552 Tf 8.697 3.155 Td[(~k;t); (~k;t) 1(~k;t))]TJ /F5 11.9552 Tf 11.956 0 Td[( 1()]TJ /F5 11.9552 Tf 8.698 3.155 Td[(~k;t); i@t(~k;t))]TJ /F5 11.9552 Tf 34.996 8.088 Td[(k2 2ma2(t)(~k;t)=0; i@t(~k;t)+ 8Gm2n0(t) k2a2(t))]TJ /F5 11.9552 Tf 27.69 8.087 Td[(k2 2ma2(t)!(~k;t)=0: (~k;t) @2t(~k;t)+4 3t@t(~k;t))]TJ /F4 11.9552 Tf 11.955 0 Td[(4G(~k;t)+k4 4m2a4(t)(~k;t)=0;=mn0 1(~k;t)=1 2(~k;t)+ima(t)2 k2@t(~k;t) n1(~r;t)=j 0(~r;t)j2Zd3k 1(~k;t)+ 1()]TJ /F5 11.9552 Tf 8.697 3.155 Td[(~k;t)ei~k~r a(t); (~r;t)=n1(~r;t) n0(~r;t)=Zd3k(~k;t)ei~k~r a(t):

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(~r;t)=p n0(t)+n1(~r;t)ei0(~r;t)+1(~r;t); 1(~r;t)=1 2iZd3k(~k;t)ei~k~r a(t): ~v(k;t)=ia(t)~k ~k~k@t(~k;t); `J=(16Gm2))]TJ /F8 5.9776 Tf 7.782 3.259 Td[(1 4=1:021014cm10)]TJ /F3 7.9701 Tf 6.586 0 Td[(5eV m1 2 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(29g=cm3 !1 4:k>a(t) `Jk<
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=2 3Gr2b E(b)1 2mb2+2 3Gmb2:E(b)b(4 3Gm2))]TJ /F8 5.9776 Tf 7.782 3.258 Td[(1 4`J (~r;t)=Aei~k~r+e)]TJ /F7 7.9701 Tf 6.586 0 Td[(i~k~re)]TJ /F7 7.9701 Tf 6.586 0 Td[(i!t:A!=~k~k 2mN (~x;t)n1=n2=NjAj2~v1=~k m~v2=)]TJ /F7 7.9701 Tf 11.509 6.81 Td[(~k m (~r;t)n(~r)=4jAj2cos2(~k~r)~v=0n(~r;t)~v(~r;t) (~r;t)

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2 kv=k mk!1k m!0m!1~v(~r;t)m100n010)]TJ /F3 7.9701 Tf 6.586 0 Td[(6=cm3v010)]TJ /F3 7.9701 Tf 6.586 0 Td[(1210)]TJ /F3 7.9701 Tf 6.586 0 Td[(3~v10)]TJ /F3 7.9701 Tf 6.586 0 Td[(3m10)]TJ /F3 7.9701 Tf 6.586 0 Td[(5n0109=cm3

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v010)]TJ /F3 7.9701 Tf 6.587 0 Td[(1710181061'(~r;t) )]TJ /F5 11.9552 Tf 11.955 0 Td[(c2D@'+!20'+ 3!'3=0DL'4=)]TJ /F7 7.9701 Tf 11.317 4.707 Td[( 4!'4 (~r;t) mc2~c!0 g00=)]TJ /F5 11.9552 Tf 9.298 0 Td[(c2)]TJ /F4 11.9552 Tf 12.811 0 Td[(2g0i=0gij=ij (@2t)]TJ /F5 11.9552 Tf 11.955 0 Td[(c2r2+!20)'+ 3!'3)]TJ /F4 11.9552 Tf 11.956 0 Td[((2 c2@2t+~r~r+1 c2@t@t)'=0: '(~r;t)=p 2Re[e)]TJ /F7 7.9701 Tf 6.586 0 Td[(i!0t(~r;t)]c2@t!0@t!0e2i!0te)]TJ /F3 7.9701 Tf 6.586 0 Td[(2i!0t i@t=)]TJ /F5 11.9552 Tf 14.546 8.088 Td[(c2 2!0r2+ 8!0jj2+!0 c2:

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i~@t =)]TJ /F14 11.9552 Tf 12.944 8.088 Td[(~2 2mr2 +V(~r;t) ; (~r;t)=r ~ !0 (~r;t): m=~!0 c2 V(~r;t)=m(~r;t)+~4 8m2c4j (~r;t)j2: V=L3 (~r;t)=X~nr ~ 2!~nV[a~n(t)ei ~~p~n~r+ay~n(t)e)]TJ /F12 5.9776 Tf 8.342 3.259 Td[(i ~~p~n~r]~n=(n1;n2;n3)nk(k=1;2;3)~p~n=2~ L~n!=c ~p ~p~p+c2m2a~nay~n [a~n(t);ay~n0(t)]=~n;~n0;[a~n(t);a~n0(t)]=0:

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~!0~NN(~r;t) H=X~n~!~nay~na~n+X~n1;~n2;~n3;~n41 4~~n3;~n4~n1;~n2ay~n1ay~n2a~n3a~n4~n3;~n4~n1;~n2 ~n3;~n4~n1;~n2=~n3;~n4s~n1;~n2+~n3;~n4g~n1;~n2: ~n3;~n4s~n1;~n2=+~3 4m2c4V~n1+~n2;~n3+~n44 ~n3;~n4g~n1;~n2=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(4Gm2~ V~n1+~n2;~n3+~n41 j~p~n1)]TJ /F5 11.9552 Tf 12.022 0 Td[(~p~n3j2+1 j~p~n1)]TJ /F5 11.9552 Tf 12.022 0 Td[(~p~n4j2 i_a~n1=)]TJ /F4 11.9552 Tf 10.797 8.088 Td[(1 ~[H;a~n1]=!~n1a~n1+1 2X~n2;~n3;~n4~n3;~n4~n1;~n2ay~n2a~n3a~n4: (~r;t)=X~nr ~ 2!~nV[A~n(t)ei ~~p~n~r+Ay~n(t)e)]TJ /F12 5.9776 Tf 8.342 3.259 Td[(i ~~p~n~r]:A~n(t) i_A~n1=!~n1A~n1+1 2X~n2;~n3;~n4~n3;~n4~n1;~n2A~n2A~n3A~n4: a~n(t)A~n(t)

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N~n(t)=ay~n(t)a~n(t);t~n _Ni=Xk;i;j=11 2jklijj2[NiNj(Nl+1)(Nk+1))-222(NlNk(Ni+1)(Nj+1)]2(!i+!j)]TJ /F5 11.9552 Tf 10.194 0 Td[(!k)]TJ /F5 11.9552 Tf 10.193 0 Td[(!l):~njj !)-471(=1 fNjg!>>)]TJ /F1 11.9552 Tf 7.314 0 Td[<002d>-325<0069003f>-1<0032>-326<00620076006200690032004b>-326<00420062>-326<0062001c0042002f>-326<00690051>-326<0023>-27<0032>-326<0042004d>-326<0069003f0032>]TJ -273.52 -23.908 Td[<01f30054001c>26<006000690042002b00480032>-325<00460042004d003200690042002b01f4>-326<00600032003b0042004b00320058>-434<0068003f0032>-326<0062001c004b0032>-326<002f0032006000420070001c006900420051004d>-326<0069003f001c0069>-326<0076004200320048002f0062>-326<0031005b0058>-326<0055>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 308.989 -272.367 Td[<006a011c0038006b>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 332.421 -272.367 Td[<0056>-326<0023006d0069>-326<001c00540054004800420032002f>-326<00690051>-326<0069003f0032>]TJ -332.421 -23.908 Td[<002b0048001c006200620042002b001c0048>-326<002b0051006d004d0069003200600054001c>26<006000690062>]TJ ET BT /F5 11.9552 Tf 183.836 -320.183 Td[(N~n(t)=A~n(t)A~n(t) _Ni=Xk;i;j=11 2jklijj2[NiNjNl+NiNjNk)]TJ /F5 11.9552 Tf 11.955 0 Td[(NlNkNi)]TJ /F5 11.9552 Tf 11.955 0 Td[(NkNlNj]2(!i+!j)]TJ /F5 11.9552 Tf 11.955 0 Td[(!k)]TJ /F5 11.9552 Tf 11.955 0 Td[(!l); i+j!k+lijklijkl6=0

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fNjg NiNj(Nl+1)(Nk+1))-222(NlNk(Ni+1)(Nj+1)=0!i+!j=!k+!l=~! (1+1 Ni)(1+1 Nj)=(1+1 Nk)(1+1 Nl)i+j=k+l i=Cln1+1 NiCC=kBT Ni=1 ei kBT)]TJ /F4 11.9552 Tf 11.955 0 Td[(1: fNjg (Ni+Nj)NkNl=NiNj(Nk+Nl)i+j=k+l 1 Ni+1 Nj=1 Nk+1 Nl; 1 Ni=C0i:

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C0=1 kBT Nii=kBTTkBT ~(~r;t)~(~r;t)fN~g~(~r;t)~(~r;t) t t

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4

PAGE 47

t4

PAGE 48

4

PAGE 49

!T

PAGE 50

!!6=0!6=04rr)]TJ /F3 7.9701 Tf 6.587 0 Td[(24

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4 NiiT Ni=1 e1 T(i)]TJ /F7 7.9701 Tf 6.586 0 Td[())]TJ /F5 11.9552 Tf 11.955 0 Td[(ii=0=)]TJ /F4 11.9552 Tf 9.298 0 Td[(1 E=PiniiN=PiniNini L=Pinili^zENL Ni=1 e1 T(i)]TJ /F7 7.9701 Tf 6.587 0 Td[()]TJ /F7 7.9701 Tf 6.587 0 Td[(!li))]TJ /F5 11.9552 Tf 11.955 0 Td[(

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!T!=0!=0 (~r;~v) N(~r;~v)=N0e)]TJ /F12 5.9776 Tf 7.782 3.259 Td[(m T[1 2~v~v+(~r))]TJ /F7 7.9701 Tf 6.586 0 Td[(!^z(~r~v)]mN0e T(~r) r2(~r)=4Gmn(~r) n(~r)=Zd3vN(~r;~v)!=0 n(~r)=n0e)]TJ /F8 5.9776 Tf 15.762 3.259 Td[(3 (~r)=3T mn0=N0(2 3)3 2 r2(~r)=4Gmn0e)]TJ /F8 5.9776 Tf 15.763 3.258 Td[(3 (~r):

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(~r)=(r) n(r)=n0d(r=s) s= 12Gmn01 2d(x)d(x)!1x!0d(x)!2=x2x!1d(x) d(x)2 2+x2 N(~r;~v)=N0e)]TJ /F8 5.9776 Tf 15.762 3.258 Td[(3 ~v~vd(r=s) svrotvrot=q 2 3saa=p 2sn(r)/1=r2rrn(r)'n0r

PAGE 54

!6=0 !6=0 N(~r;~v)=N0e)]TJ /F12 5.9776 Tf 7.782 3.258 Td[(m T[1 2(~v)]TJ /F7 7.9701 Tf 6.632 0 Td[(~!~r)2+(~r))]TJ /F8 5.9776 Tf 7.782 3.258 Td[(1 2(~!~r)2]~!=!^z!=0~!~r n(~r)=n0e)]TJ /F12 5.9776 Tf 7.782 3.258 Td[(m T[(~r))]TJ /F8 5.9776 Tf 7.782 3.258 Td[(1 2(~!~r)2]: r2(~r)=4Gmn0em T[)]TJ /F3 7.9701 Tf 6.586 0 Td[((~r)+1 2!22]:(;z;) n(~r)d(~r)/em 2T!22!1 2!22 !2 2Gm~!~r

PAGE 55

!=0 Ni=1 e1 T(i)]TJ /F7 7.9701 Tf 6.586 0 Td[())]TJ /F4 11.9552 Tf 11.955 0 Td[(1:i=00 i0 Nex(T;=0)=Xi>01 e1 T(i)]TJ /F7 7.9701 Tf 6.587 0 Td[(0))]TJ /F4 11.9552 Tf 11.955 0 Td[(1:Nex(T;=0) TNNex(T;=0)N)]TJ /F5 11.9552 Tf 12.083 0 Td[(Nex(T;=0)

PAGE 56

N00!6=0ii)]TJ /F5 11.9552 Tf 12.278 0 Td[(!liiili^ 0mini[i)]TJ /F5 11.9552 Tf 11.956 0 Td[(!li]i=0!=0i6=0T Nex(T;=0)=Xi6=01 e1 T(i)]TJ /F7 7.9701 Tf 6.586 0 Td[(0))]TJ /F4 11.9552 Tf 11.955 0 Td[(1:NNex(T;=0)N)]TJ /F5 11.9552 Tf 11.713 0 Td[(Nex(T;=0)i=0 ~r~v=0 6=0=0 (;;t)=AJl(k)eile)]TJ /F7 7.9701 Tf 6.586 0 Td[(ik2 2mtJll=V=0k2 2mzl

PAGE 57

~v=l m^:z~r~v=2l m^z2(x;y)xy^zzl6=0Jl(s)'0s<-434<00610042004d002b0032>-326<0069003f0032>-326<0072>26<001c007000320037006d004d002b006900420051004d>-325<00420062>-326<00620042004d003b00480032>]TJ -300.041 -23.908 Td[<0070001c0048006d0032002f002d>]TJ /F5 11.9552 Tf 38.065 0 Td[(l[]1 2l[]

PAGE 58

(~r;t))]TJ /F1 11.9552 Tf 7.314 0 Td[<0058>-434<0068003f0032>-326<006900510069001c0048>-326<004d006d004b0023>-27<00320060>-326<00510037>-326<00700051>26<006000690042002b00320062>-325<0032004d002b00420060002b00480032002f>-326<0023>26<0076>]TJ /F4 11.9552 Tf 213.851 0 Td[()]TJ /F1 11.9552 Tf 7.314 0 Td[<002d>-325<002b0051006d004d00690042004d003b>-326<001c>-326<00700051>26<0060006900320074>-325<007200420069003f>-326<001c>-1<004d003b006d0048001c>26<0060>]TJ -287.609 -23.908 Td[<004b0051004b0032004d0069006d004b>]TJ /F5 11.9552 Tf 58.951 0 Td[(ll )]TJ /F1 11.9552 Tf -435.534 -23.908 Td[<001c004d002f>-326<00480032006900690042004d003b>-326<001c>-326<00700051>26<0060006900320074>-325<002b0060005100620062>-326<0069003f001c0069>-326<002b006d0060007000320058>-434<00360042004d001c004800480076>81<002d>-325<001c>-326<0069003f00420060002f>-326<0060006d00480032002c>-435<00700051>26<006000690042002b00320062>-325<004b006d00620069>-326<00370051004800480051>26<0072>-325<0069003f0032>-326<004b0051006900420051004d>-326<00510037>]TJ 0 -23.908 Td[<0069003f0032>-326<007e006d0042002f0058>-434<0068003f00420062>-326<00420062>-326<001c>-326<002b0051>26<0060005100480048001c>27<00600076>-325<00510037>-326<00450032004800700042004d01f60062>-326<0069003f00320051>26<00600032004b>-325<0072003f0042002b003f>-326<00620069001c006900320062>-326<0069003f001c0069002d>-326<0042004d>-326<003b0060001c002f00420032004d0069>-326<007e0051>26<0072>-325<0040>-326<0042005800320058>-435<00420037>]TJ 0 -23.908 Td[<0069003f0032>-326<005f003e0061>-326<00510037>-326<0031006d00480032006001f60062>-326<0032005b006d001c006900420051004d>-326<00420062>-326<001c>-326<003b0060001c002f00420032004d0069002d>-326<001c0062>-326<00420062>-326<0069003f0032>-326<002b001c00620032>-326<00370051>26<0060>-325<0031005b0058>-326<0055>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 318.797 -179.31 Td[<006a011c0064>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 336.371 -179.31 Td[<0056>-325<0040>-326<0069003f0032>-326<002b00420060002b006d0048001c006900420051004d>-326<00510037>-326<0069003f0032>]TJ -336.371 -23.908 Td[<0070003200480051>-27<002b00420069>26<0076>-325<007d00320048002f>-326<001c00480051004d003b>-326<001c>-326<002b0048005100620032002f>-326<0054001c0069003f>-326<0069003f001c0069>-326<004b0051007000320062>-326<007200420069003f>-326<0069003f0032>-326<007e006d0042002f>-326<00420062>-326<002b0051004d00620069001c004d0069>-326<0042004d>-326<00690042004b00320058>]TJ ET BT /F9 11.9552 Tf 0 -227.126 Td[<0039005800520058006a0058006a>-1000<0061006d0054>-31<00320060007e006d0042002f>]TJ /F3 7.9701 Tf 108.744 4.339 Td[(444N=)]TJ /F5 11.9552 Tf 12.526 0 Td[(!l =Zd3r()]TJ /F4 11.9552 Tf 15.613 8.087 Td[(1 2mr2+m+V)=1 NZd3rn[m 2~v~v+m(q+)+V] l=Zd3r^z(~r1 i~r)=Zd3r1 i@ @=1 NZd3rnm^z(~r~v): Vp4

PAGE 59

n0 'n0 NZd3rm 2~v~v;l'n0 NZd3rm^z(~r~v) 'n0m 2NZd3r[(vz)2+(v)2+(v)]TJ /F5 11.9552 Tf 11.955 0 Td[(!)2)]TJ /F5 11.9552 Tf 11.955 0 Td[(!22]:~v=!^z~r4z m 2^z(~r~v)=m! )]TJ /F5 11.9552 Tf 10.434 3.022 Td[(~rq)]TJ /F3 7.9701 Tf 12.122 4.707 Td[(1 m~rV )]TJ /F5 11.9552 Tf 10.434 3.022 Td[(~rq)]TJ /F3 7.9701 Tf 12.122 4.707 Td[(1 m~rV4 Vp(~x)]TJ /F5 11.9552 Tf 11.76 0 Td[(~x0)=U03(~x)]TJ /F5 11.9552 Tf 11.76 0 Td[(~x0) =1 p 2mn0U0:lll k)]TJ /F3 7.9701 Tf 6.586 0 Td[(1l2l2 l2ll

PAGE 60

4 V=0 =0 =Zd3r()]TJ /F4 11.9552 Tf 15.613 8.088 Td[(1 2mr2)]TJ /F5 11.9552 Tf 13.151 8.088 Td[(! i@ @)=1 NZd3rn[m 2~v~v+mq)]TJ /F5 11.9552 Tf 11.955 0 Td[(m!^z(~r~v)]: q=0n(~r)~v(~r)!^zz ()]TJ /F4 11.9552 Tf 15.614 8.087 Td[(1 2mr2)]TJ /F5 11.9552 Tf 11.956 0 Td[(!1 i@ @)i=ii:00=minii)]TJ /F3 7.9701 Tf 14.24 4.707 Td[(1 2mr21 i@ @(i=kl) kl(z;;)=Akl(z;)eil;kl=kl)]TJ /F5 11.9552 Tf 11.956 0 Td[(!l )]TJ /F4 11.9552 Tf 18.27 8.088 Td[(1 2m(@2 @z2+1 @ @@ @)]TJ /F5 11.9552 Tf 18.192 8.088 Td[(l2 2)Akl=klAkl:

PAGE 61

Rh)]TJ /F3 7.9701 Tf 14.239 4.707 Td[(1 2mr21 i@ @)]TJ /F3 7.9701 Tf 6.675 -4.977 Td[(1 i@ @z2 lpn=eilsin(p hz)Jl(xln R)l=0;1;2;:::p=1;2;3;:::n=1;2;3;:::xlnnthJl(x)xl1
PAGE 62

H=H0+H1 H0=Xlpnlpnaylpnalpnalpnaylpn H1=Xi;i0;i00;i0001 4ii0i00i000ayi000ayi00ai0aiilpni0l0p0n0 N=Xlpnaylpnalpn ii0i00i000=0unlessl+l0=l00+l000 L=XlpnlaylpnalpnH1=1 )]TJ /F1 11.9552 Tf 6.484 4.123 Td[<0058>-434<005f003200370058>-435<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 67.601 -604.592 Td[<006b0038>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 79.317 -604.592 Td[<0029>-325<0032006200690042004b001c006900320062>-326<0069003f0032>-326<0069003f00320060004b001c004800420078001c006900420051004d>-326<0060>-1<001c00690032>]TJ /F4 11.9552 Tf 172.07 0 Td[()]TJ /F1 11.9552 Tf 11.212 0 Td[<00510037>-326<002b00510048002f>-326<002f001c>26<00600046>-325<004b001c0069006900320060>-326<001c007400420051004d0062>-326<0069003f00600051006d003b003f>-326<0069003f003200420060>]TJ ET BT /F1 11.9552 Tf 227.766 -687.422 Td[<0065006b>]TJ ET Q

PAGE 63

4=1 )]TJ /F1 11.9552 Tf 10.382 4.123 Td[<0051007000320060>-326<0072003f0042002b003f>-326<0069003f0032>-326<00620076006200690032004b>-326<0069003f00320060004b001c004800420078003200620058>]TJ -140.311 -23.908 Td[<004700320069>-326<006d0062>-326<0062006d00540054>-27<005100620032>-326<0069003f001c0069>]TJ /F5 11.9552 Tf 100.256 0 Td[(N!=0TN0(l;p;n)0=(0;1;1)i=0N)]TJ /F5 11.9552 Tf 12.374 0 Td[(N0i6=0lN0N0TMVext(~r;t)MMMH1M i@ti(~r;t)=[)]TJ /F4 11.9552 Tf 15.613 8.087 Td[(1 2mr2+Vext(~r;t)]i(~r;t)i(~r;t=)=i(~r)i(~r;t)M0(~r;t)N =+1NLE

PAGE 64

(l;p;n)00=(l00;1;1)l00 l00 1 )]TJ /F1 11.9552 Tf -426.01 -19.785 Td[<0023>-27<0032002b001c006d00620032>-326<0051004d>-326<00690042004b0032>-326<0062002b001c004800320062>-326<00480051004d003b>-326<002b0051004b0054001c>26<00600032002f>-325<00690051>]TJ /F5 11.9552 Tf 203.656 0 Td[( )]TJ /F6 11.9552 Tf 10.635 0 Td[(4Gnm2`2nm`1 pp n`g4Gnm`pp gm)]TJ /F1 11.9552 Tf 11.212 0 Td[<00420062>]TJ -413.262 -23.908 Td[<0069003f0032>-326<0042004d00700032006000620032>-326<00510037>-326<0069003f001c0069>-326<00690042004b00320058>-434<0068>81<0051>-325<0032006200690042004b001c00690032>-326<0069003f0032>-326<001c007400420051004d>-326<006000320048001c0074001c006900420051004d>-326<00690042004b0032>-326<00690051>-27<002f001c>26<0076>82<002d>-325<004800320069>-326<006d0062>-326<0062006d00230062006900420069006d00690032>]TJ /F5 11.9552 Tf 0 -23.908 Td[(nm'0:2310)]TJ /F3 7.9701 Tf 6.587 0 Td[(29m'10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5

PAGE 65

`'210)]TJ /F3 7.9701 Tf 6.587 0 Td[(7secGeV 10)]TJ /F8 5.9776 Tf 5.756 0 Td[(4eV=0:61017cm105 )]TJ /F7 7.9701 Tf 20.893 -4.428 Td[(fa 1012GeV1 2fa`1061 4Gnm2`>_p=m_v_v`

PAGE 66

~l(^n;t)=mjmax^n(^z^n)R(t)2 ttR(t)^z^njmax ~!=jmax t^z R(t)/t2 3+2 9 ^n(^z^n) jmaxR(t)2 tan1=nn=1;2;3::: an'40kpc nvrot 220km=sjmax 0:18vrot

PAGE 67

^n(^z^n)S(t)S(t)=a(t)Sa(t)S(t)S(r;t)=U(r;t)r l;p(r;;;t)=U(r;t)Al;p(;)00<2lplzzpU(r;t)A(;) ZS(t)0r2drjU(r;t)j2=Z0sindZ20djA(;)j2=1:lp =)]TJ /F5 11.9552 Tf 12.362 0 Td[(!(t)l!

PAGE 68

=Zr
PAGE 69

A A(;;t)=(;t)ei!(t)I(t)sin2: A(;;t)'(;t)ei!(t)I(t)sin2A(;;t) A 'Zd[1 2I(t)jd dj2)]TJ /F4 11.9552 Tf 13.151 8.088 Td[(1 2!2(t)I(t)sin2jj2]:= 21 p !I L!I L(500km s)(10kpc)m'2:61019m 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5eV:10)]TJ /F3 7.9701 Tf 6.586 0 Td[(10

PAGE 70

r)]TJ /F3 7.9701 Tf 6.586 0 Td[(2r ~v'vr^r+v^+1 mrL(t)sin^:' m 2~r~v'~rL(t) mrsin^=L(t) r2cos^r: Z20dZ 20sindr2m 2~r~v=L: ^za0'L0 kkL0a=L kaL0
PAGE 71

dM ddt^z=0Mt2 3 dM ddt=M 6t dM ddt(;t)=N(sin)M 6tv N=\(+2) 2(\( 2+1))2 R(t)2 t R(t)2 t S(t)

PAGE 72

V(t) ~(t)=ZV(t)d3r(~r;t)~r()]TJ /F5 11.9552 Tf 10.434 3.022 Td[(~r(~r;t))(~r;t)=(~r;t))]TJ /F5 11.9552 Tf 12.335 0 Td[(0(t)0(t)(~r=a(t)~x;t)=(~x)(~r;t)(~r;t) 0(t)(~r=a(t)~x;t)=a(t)(~x) ~(t)=0(t)a(t)4ZVd3x(~x)~x()]TJ /F5 11.9552 Tf 10.434 3.022 Td[(~rx(~x)): 0(t)/a(t))]TJ /F3 7.9701 Tf 6.586 0 Td[(3(t)/a(t)/t2 3t5 3R(t)/t2 3+2 9 =0:330:25<<0:35 jmaxjmax LjEj1 2 GM5 2;LME

PAGE 73

jmaxjmaxjmax'=0jmax =r 6 5)]TJ /F4 11.9552 Tf 11.956 0 Td[(38 10+31 jmax: =jmaxjmax'0:180:05=0jmaxLEM jmax6=0jmax 1+=2 1+=3 jmax'0:4260:05jmax'0:18

PAGE 74

4Gnmm0`>m0_v;m0_vm0 nm&1 30DMDMT! v10)]TJ /F3 7.9701 Tf 6.587 0 Td[(3c T.1 2mv2'610)]TJ /F3 7.9701 Tf 6.586 0 Td[(8Km 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5eV:m0m<10)]TJ /F3 7.9701 Tf 6.587 0 Td[(3=)]TJ /F5 11.9552 Tf 13.005 0 Td[(!l!

PAGE 75

r)]TJ /F3 7.9701 Tf 6.586 0 Td[(2r t dM0 ddt=N0(sin)0M0 6t0 mm0 ~l0(^n;t)=m0jmax^n(^z^n)R(t)2 t:

PAGE 77

dM dl=Mvl0 (l0+l)2for0llmax=l0 )]TJ /F4 11.9552 Tf 11.955 0 Td[(1>1Mvlmaxlog10()]TJ /F4 11.9552 Tf 9.427 0 Td[(1)0:06<)]TJ /F4 11.9552 Tf 12.102 0 Td[(1<1:0lav

PAGE 78

lav lmax=()]TJ /F4 11.9552 Tf 11.955 0 Td[(1)ln )]TJ /F4 11.9552 Tf 11.955 0 Td[(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(1:lav lmax =1:25 dM dll=0l=0l=lav lav lmax lmax'2:6lavlav lmax lmax lav t l(;t)=^z~l(~n;t)=lmax(sin)2t t05 3

PAGE 79

s=l lavp(s)/dM dl(l)

PAGE 80

t0 lmax=mjmaxR20 t0R0~l0m0M0 l/t5 3 dM dl(l)=ZdZt00dtdM ddt(;t)(l)]TJ /F5 11.9552 Tf 11.955 0 Td[(l(;t)): t=1=3 dM dl(l)=6 5M0 lmaxl lmax1 5Il lmax I(r)=NZp 1)]TJ /F7 7.9701 Tf 6.586 0 Td[(r0dx (1)]TJ /F5 11.9552 Tf 11.956 -.001 Td[(x2)6 5)]TJ /F12 5.9776 Tf 7.782 3.258 Td[( 2: =0dM dl(l)l=0dM dl(0)=3M0 lmax>0dM dl/l1 5l=0dM dl(l) lav=1 M0Zlmax0dlldM dl(l)=lmax3 11p 2( 2+1)\(+2) \( 2+1)\( 2+5 2):lmax=lav =0=10:5<<2:0lmax=lavlmax=lav

PAGE 81

n

PAGE 82

N N 2)]TJ /F5 11.9552 Tf 12.081 0 Td[(m=1 2p u s0'p p 2ap us0n=5 p=0:018am=5:5(cosm)>0:5Nn=5 =0XaXW=1)]TJ /F5 11.9552 Tf 12.122 0 Td[(Xa

PAGE 83

XaN+XWN0'5N.100.5 N0.2 Xa&3 8; 4

PAGE 84

r)]TJ /F3 7.9701 Tf 6.587 0 Td[(2r 0

PAGE 85

0:5.0.2lmax=lavlmax=lav'2:6 lmax=lav0<5

PAGE 87

(80;0)m

PAGE 88

10kpcB10)]TJ /F3 7.9701 Tf 6.587 0 Td[(23G BIGMF10)]TJ /F3 7.9701 Tf 6.587 0 Td[(18)]TJ /F4 11.9552 Tf 12.627 0 Td[(10)]TJ /F3 7.9701 Tf 6.586 0 Td[(15G 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(30)]TJ /F4 11.9552 Tf 12.733 0 Td[(10)]TJ /F3 7.9701 Tf 6.586 0 Td[(20G

PAGE 89

B10)]TJ /F3 7.9701 Tf 6.586 0 Td[(19G 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(17Gz100 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(26)]TJ /F4 11.9552 Tf 12.016 0 Td[(10)]TJ /F3 7.9701 Tf 6.586 0 Td[(23G

PAGE 90

)]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(G4Gnmmb` pb;nmmb`1 Hpbp 3mbTTT<1keV )]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(e910)]TJ /F3 7.9701 Tf 6.586 0 Td[(21s)]TJ /F3 7.9701 Tf 6.587 0 Td[(1bh2T T04 T0bh2)]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(G=H)]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(G=)]TJ /F3 7.9701 Tf 7.315 -1.793 Td[(e i=i)]TJ /F5 11.9552 Tf 12.614 0 Td[((t)liiliith(t)

PAGE 91

zr1500 ~FT=4T 3c~vT~v~ve~vpd~ve dt=)]TJ /F5 11.9552 Tf 15.032 8.088 Td[(e me~E+~ve c~B)]TJ /F4 11.9552 Tf 13.151 8.088 Td[(4T~ve 3cme+~vp)]TJ /F5 11.9552 Tf 11.479 0 Td[(~ve ep+~agrav; d~vp dt=e mp~E+~vp c~B)]TJ /F4 11.9552 Tf 13.151 8.088 Td[(4T~vp 3cmp)]TJ /F5 11.9552 Tf 12.674 8.088 Td[(~vp)]TJ /F5 11.9552 Tf 11.479 0 Td[(~ve ep+~agrav; epmempe~agrav

PAGE 92

~J=ene(~vp)]TJ /F5 11.9552 Tf 12.265 0 Td[(~ve)ne'npz900 ~!e d dt~r ~J ene!=e me~r~E+4T 3mec~!e)]TJ /F4 11.9552 Tf 11.955 0 Td[(2~r ~Je me!; e me~r~E=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(4T 3mec~!e: 1 c@~B @t=)]TJ /F5 11.9552 Tf 10.434 3.022 Td[(~r~E; @~B @t=4T 3e~!e: 1 a2@(a2~B) @t=4T0a)]TJ /F3 7.9701 Tf 6.586 0 Td[(4 3e~!e(t);a(t)

PAGE 93

!L=MR2LMR t5=3t1=3z10zt)]TJ /F3 7.9701 Tf 6.586 0 Td[(4=3 !(z)=8>><>>:!0(1+z)5=2(1+z))]TJ /F3 7.9701 Tf 6.586 0 Td[(1=2;zz>900 B(z) z210)]TJ /F3 7.9701 Tf 6.587 0 Td[(22Gz 105=2!0 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(15s)]TJ /F3 7.9701 Tf 6.586 0 Td[(1lnzr z:z900B10)]TJ /F3 7.9701 Tf 6.586 0 Td[(17GB010)]TJ /F3 7.9701 Tf 6.587 0 Td[(23G

PAGE 94

B10)]TJ /F3 7.9701 Tf 6.586 0 Td[(17Gz=90010kpc10)]TJ /F3 7.9701 Tf 6.587 0 Td[(23G

PAGE 95

h=Ea=mac2+1 2mav2Eama~v10)]TJ /F3 7.9701 Tf 6.586 .001 Td[(3c10)]TJ /F3 7.9701 Tf 6.586 0 Td[(6

PAGE 96

105D=c2'0:310)]TJ /F3 7.9701 Tf 6.586 0 Td[(11

PAGE 97

1:510)]TJ /F3 7.9701 Tf 6.586 0 Td[(2430:510)]TJ /F3 7.9701 Tf 6.587 0 Td[(243

PAGE 98

V(x) E=m 2v2+V(x):x v(x)=r 2 m(E)]TJ /F5 11.9552 Tf 11.955 0 Td[(V(x)):V(x)x1x2E=V(x1)=V(x2)x10v(x)<0xx2@n @t=0 @n @t+@ @x(nv)=0;

PAGE 99

n(x)=J q 2 m(E)]TJ /F5 11.9552 Tf 11.956 0 Td[(V(x));JJx1x2A2xx1 V(x)=E+(x)]TJ /F5 11.9552 Tf 11.955 0 Td[(x1)dV dx(x1)+O(x)]TJ /F5 11.9552 Tf 11.956 0 Td[(x1)2dV dx(x1)<0 n(x)=J q )]TJ /F3 7.9701 Tf 12.123 4.707 Td[(2 mdV dx(x1)1 p x)]TJ /F5 11.9552 Tf 11.955 0 Td[(x1xx1 n(x)=J q +2 mdV dx(x2)1 p x2)]TJ /F5 11.9552 Tf 11.955 0 Td[(xxx2 EdN dE(E)EE E=<(E)]TJ /F4 11.9552 Tf 14.688 3.022 Td[(E)2>1 2:x v(x)'v(x;E)=r 1 2m(E)]TJ /F5 11.9552 Tf 11.956 0 Td[(V(x)) v(x)=<(v(x;E))]TJ /F4 11.9552 Tf 12.398 0 Td[(v(x))2>1 2'<(E)]TJ /F4 11.9552 Tf 14.688 3.022 Td[(E)@v @E(x;E)2>1 2=E mjv(x;E)j:

PAGE 100

dN dE(E)jE)]TJ /F4 11.9552 Tf 15.209 3.022 Td[(Ej E!0 @E @vjx=mv(x;E) EE N=n(x) mv(x)'J E:xE6=0j=1;2 xj=E dV dx(xj): nmax;js m 2dV dx(xj)J p xj=r m 2EJ: vmax;jr E 2m:nmaxvmax v=0Em 2(vmax)2 V(r)

PAGE 101

V(r)(r;)EL ~v(r;)=vr(r)^r+v(r)^ v(r)=L mrandvr(r)=r 2 m(E)]TJ /F5 11.9552 Tf 11.955 0 Td[(Ve(r)) Ve=V(r)+L2 2mr2:L6=0aE=Ve(a)RE=Ve(R)R>aa0rR n(r)=J rjvr(r)jJJr=ar=Rra n(r)=J aq )]TJ /F3 7.9701 Tf 12.123 4.707 Td[(2 mdVe dr(a)1 p r)]TJ /F5 11.9552 Tf 11.955 0 Td[(a; n(r)=J Rq +2 mdVe dr(R)1 p R)]TJ /F5 11.9552 Tf 11.955 0 Td[(rrR

PAGE 102

d2N dEdL=N 2ELe)]TJ /F8 5.9776 Tf 7.782 3.258 Td[(1 2(E)]TJ /F8 5.9776 Tf 7.36 1.511 Td[(E E)2)]TJ /F8 5.9776 Tf 7.782 3.258 Td[(1 2(L)]TJ /F8 5.9776 Tf 6.653 1.511 Td[(L L)2:E=EL=L <(E)]TJ /F4 11.9552 Tf 14.687 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 13.336 3.022 Td[(L)>=0:<(E)]TJ /F4 11.9552 Tf 14.869 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 13.518 3.022 Td[(L)>6=0 vpvp10)]TJ /F3 7.9701 Tf 6.587 0 Td[(17vp10)]TJ /F3 7.9701 Tf 6.587 0 Td[(12vp10)]TJ /F3 7.9701 Tf 6.587 0 Td[(8vEmvvvLmDvDEL L=EDav vav

PAGE 103

vavDavDav vL=mDavvE=mvavvL=0<(E)]TJ /F4 11.9552 Tf 13.397 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 12.047 3.022 Td[(L)>L!)]TJ /F5 11.9552 Tf 24.575 0 Td[(L<(E)]TJ /F4 11.9552 Tf 13.397 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 12.047 3.022 Td[(L)>L =LjEj1 2 GM5 2LEMG0:05a=RaRa=R0:1 jmaxjmax<(E)]TJ /F4 11.9552 Tf 15.311 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 13.959 3.022 Td[(L)><(E)]TJ /F4 11.9552 Tf 14.074 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 12.723 3.022 Td[(L)>EL<(E)]TJ /F4 11.9552 Tf 12.563 3.022 Td[(E)(L)]TJ /F4 11.9552 Tf 11.212 3.022 Td[(L)>ELd2N dEdLdE=E)]TJ /F4 11.9552 Tf 14.885 3.022 Td[(EdL=L)]TJ /F4 11.9552 Tf 13.535 3.022 Td[(Ldvr=vr)]TJ /F4 11.9552 Tf 12.321 0 Td[(vrdv=v)]TJ /F4 11.9552 Tf 12.842 0 Td[(v

PAGE 104

(r;)m=1 dE=vrdvr+vdvanddL=rdv: (dEdL)0B@1 2E001 2L1CA0B@dEdL1CA=(dvrdv)0B@v2r 2Evrv 2Evrv 2Ev2 2E+r2 2L1CA0B@dvrdv1CA: 0B@dvrdv1CA=0B@cossin)]TJ /F4 11.9552 Tf 11.291 0 Td[(sincos1CA0B@dv1dv21CA tan2=2vrv v2+E Lr2)]TJ /F5 11.9552 Tf 11.955 0 Td[(v2r dE E2+dL L2=dv1 12+dv2 22 1 (12)2=1 2v2r+v2 2E+r2 2Ls 1 4v2r+v2 2E+r2 2L2)]TJ /F5 11.9552 Tf 15.932 8.087 Td[(r2v2r 2E2L: 12=EL rjvr(r)j: d2N dvrdv=d2N dEdLjdet@(E;L) @(vr;v)j=rjvr(r)jd2N dEdL: (vr)2=cos2(1)2+sin2(2)2(v)2=sin2(1)2+cos2(2)2

PAGE 105

=(22)]TJ /F5 11.9552 Tf 11.955 0 Td[(21)sincos: N=n(r) 12=n(r)rjvr(r)j EL=J ELr!0vr!02v 2!1 r L rcE2+rc L2v(rc)rc=aR1vr vrn(r) vr! vrvr vr;maxjrcr EL rcv(rc) nmaxjrcJs v(rc) rcEL: v(a)=1 r L aE2+a L2=1 r v(a) vavv2+a Davv2'vvav v(a)a<
PAGE 106

nmaxjaJ p LL: v(R)=1 r L RE2+R L2=1 r av(a) Rvavv2+R Davv2'vDav Ra<
PAGE 107

x()=x0)]TJ /F4 11.9552 Tf 13.151 8.088 Td[(1 2g2x v()=@x @=)]TJ /F5 11.9552 Tf 9.298 0 Td[(g:yyx0y=s =1 gjjx0:dN dx0dv=)]TJ /F5 11.9552 Tf 9.299 0 Td[(gdy v(y)=g=x0 jj=x0r g 2(x0)]TJ /F5 11.9552 Tf 11.956 0 Td[(y):

PAGE 108

N=n=v A3 z=z0+bx=x0)]TJ /F5 11.9552 Tf 11.956 0 Td[(c)]TJ /F4 11.9552 Tf 13.151 8.088 Td[(1 2s2 (x;z) (;)z=z0 vz=@z @=bvx=@x @=)]TJ /F5 11.9552 Tf 9.298 -.001 Td[(c: d(x;z)d2N dxdz=d2N dd1 jD(;)j

PAGE 109

D(;)=det@(x;z) @(;)=b()]TJ /F5 11.9552 Tf 9.298 0 Td[(s2+c):D(;)=0D(;)=0 x=x0)]TJ /F4 11.9552 Tf 13.151 8.088 Td[(3 2sc2 b21 3jz)]TJ /F5 11.9552 Tf 11.955 0 Td[(z0j2 3;(x;z)(;) (x0;z0)(x0;z0)1 p hh1 p h

PAGE 110

(x;z)(x;z) z=z0x>x0=0=)]TJ /F4 11.9552 Tf 9.299 0 Td[((x)]TJ /F5 11.9552 Tf 12.016 0 Td[(x0)=c d(x;z0)=d2N dd1 bjx)]TJ /F5 11.9552 Tf 11.955 0 Td[(x0j:x=x0=)]TJ /F7 7.9701 Tf 12.487 4.707 Td[(s 2c2z)]TJ /F5 11.9552 Tf 11.955 0 Td[(z0=)]TJ /F7 7.9701 Tf 10.676 4.707 Td[(bs 2c3 d(x0;z)=d2N dd1 32 bsc21 31 jz)]TJ /F5 11.9552 Tf 11.955 0 Td[(z0j2 3:z=z0x
PAGE 111

0B@bbsc1CA0B@dd1CA=0B@)]TJ /F5 11.9552 Tf 9.299 0 Td[(dz0)]TJ /F5 11.9552 Tf 11.956 0 Td[(dbdx0)]TJ /F5 11.9552 Tf 11.955 0 Td[(dc)]TJ /F3 7.9701 Tf 13.151 4.707 Td[(1 22ds1CA:!0!0 )]TJ /F7 7.9701 Tf 6.587 .001 Td[(dz0dx0 0B@dd1CA=1 b(c)]TJ /F5 11.9552 Tf 11.955 0 Td[(s2)0B@c)]TJ /F5 11.9552 Tf 11.955 0 Td[(b)]TJ /F5 11.9552 Tf 9.298 0 Td[(sb1CA0B@)]TJ /F5 11.9552 Tf 9.298 0 Td[(dz0dx01CA: dvz=bd=)]TJ /F5 11.9552 Tf 21.767 8.088 Td[(cdz0 c)]TJ /F5 11.9552 Tf 11.955 0 Td[(s2dvx=)]TJ /F5 11.9552 Tf 9.299 0 Td[(dc: vz=p <(dvz)2>=cz0 jc)]TJ /F5 11.9552 Tf 11.955 0 Td[(s2j;vx=c:dz0dc =c c)]TJ /F5 11.9552 Tf 11.955 0 Td[(s2:^x^z

PAGE 112

~v5'(505^120^r)km=s^^r1:510)]TJ /F3 7.9701 Tf 6.587 0 Td[(2430:1510)]TJ /F3 7.9701 Tf 6.586 0 Td[(243~v5)]TJ /F1 11.9552 Tf 10.982 1.794 Td[<001c004d002f>-325<0069003f0032>-326<004700420069006900480032>-326<003600480051>25<0072>-324<003f>-1<001c0062>-326<0070003200480051>-27<002b00420069>26<0076>]TJ /F5 11.9552 Tf 157.128 0 Td[(~v5+

PAGE 113

EL E=1 2vr(r)2+L2 2r2+V(r)vr(r)rV(r) V(r)=v2rotln(r)vrotvrot'a E=L2 2a2+V(a):dEdLdaELa dE=LdL a2)]TJ /F5 11.9552 Tf 13.15 8.088 Td[(L2 a3da+dV dr(a)da: da=a v2rot)]TJ /F5 11.9552 Tf 11.956 0 Td[(v(a)2(dE)]TJ /F5 11.9552 Tf 13.151 8.088 Td[(LdL a2)v(r)=L=ra (a)2<(da)2>=a2 (v2rot)]TJ /F5 11.9552 Tf 11.955 0 Td[(v(a)2)2[(E)2+L2 a4(L)2)]TJ /F4 11.9552 Tf 11.955 0 Td[(2L a2]

PAGE 114

d2N dEdLEp <(dE)2>Lp <(dL)2> a2E LL=avav v(a)Davvavv(a)Dava ELa R<
PAGE 115

E=mvavvvav E m.2:410)]TJ /F3 7.9701 Tf 6.586 0 Td[(10: aa+p

PAGE 116

x

PAGE 118

10)]TJ /F3 7.9701 Tf 6.586 0 Td[(22eV10)]TJ /F3 7.9701 Tf 6.586 0 Td[(2310)]TJ /F3 7.9701 Tf 6.586 0 Td[(20eV 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(33eVm10)]TJ /F3 7.9701 Tf 6.586 0 Td[(18eV `J=(16Gm2))]TJ /F8 5.9776 Tf 7.782 3.259 Td[(1 4=1:011014cm10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5eV m1 2 10)]TJ /F3 7.9701 Tf 6.586 0 Td[(29g=cm3 !1 4

PAGE 119

0:310)]TJ /F3 7.9701 Tf 6.586 0 Td[(2210)]TJ /F3 7.9701 Tf 6.586 0 Td[(24eV m10)]TJ /F3 7.9701 Tf 6.586 0 Td[(22 m10)]TJ /F3 7.9701 Tf 6.586 0 Td[(21 m3:7)]TJ /F4 11.9552 Tf 11.999 0 Td[(5:610)]TJ /F3 7.9701 Tf 6.586 0 Td[(22eV m<0:410)]TJ /F3 7.9701 Tf 6.586 0 Td[(22 m&10)]TJ /F3 7.9701 Tf 6.586 0 Td[(22eV

PAGE 120

mfmft11=m n(t1)m2111ft1 n(t)m21a(t1) a(t)3;a(t) n(t1)c;0 mt0 teq2teq t13=2;t0c;0t1m>210)]TJ /F3 7.9701 Tf 6.587 0 Td[(28

PAGE 121

1p c;0t0 (mteq)1 431017GeV10)]TJ /F3 7.9701 Tf 6.586 0 Td[(22eV m1 4: p(t)1 t1a(t1) a(t); N(2)3 4=3n(t) (p)35109810)]TJ /F3 7.9701 Tf 6.586 0 Td[(22eV m5 2: )]TJ /F7 7.9701 Tf 7.315 -1.794 Td[(a1 64 2m3 f2=1 2:510110secm 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(22eV31017GeV f21=f)]TJ /F6 11.9552 Tf 11.907 0 Td[(1=4!)]TJ /F1 11.9552 Tf 7.315 0 Td[<0058>-434<0041004d002f00320032002f002d>-326<00620042004d002b0032>-326<0031005b0058>-326<0055>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 292.235 -617.192 Td[<0064011c0038>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 309.809 -617.192 Td[<0056>-326<003b0042007000320062>-326<001c004d>-326<006d00540054>-27<00320060>-326<00480042004b00420069>-326<0051004d>-326<0069003f0032>]TJ ET BT /F1 11.9552 Tf 224.837 -687.422 Td[<0052006b0052>]TJ ET Q

PAGE 122

!(t)(p(t))2 2m<1 2mt21a(t1) a(t)2=Hfortteq H<)]TJ /F1 11.9552 Tf 11.212 0 Td[<00420062>-325<004d0032002b003200620062001c>26<00600076>-326<00370051>26<0060>-325<0069003f00320060004b001c004800420078001c006900420051004d>-326<00690051>-326<0051>-27<002b002b006d00600058>-434<0068003f0032>-326<002b0051004d002f0042006900420051004d>]TJ /F5 11.9552 Tf 271.044 0 Td[(!<)]TJ /F1 11.9552 Tf 11.212 0 Td[<002f0032007d004d00320062>-325<0069003f0032>]TJ -380.079 -23.908 Td[<01f3002b0051004d002f0032004d00620032002f>-326<00600032003b0042004b003201f4>135<0058>]TJ 23.384 -23.907 Td[<0041004d>-326<0069003f0032>-326<002b0051004d002f0032004d00620032002f>-326<00600032003b0042004b0032002d>-326<0069003f0032>-326<006000320048001c0074001c006900420051004d>-326<0060001c00690032>-326<002f006d0032>-326<00690051>]TJ /F5 11.9552 Tf 254.227 0 Td[(4 )]TJ /F7 7.9701 Tf 7.315 -1.793 Td[(jjn 4m2:jjm2 f21f )]TJ /F7 7.9701 Tf 7.315 -1.794 Td[((t)=H(t)n(t1) 2f2a(t1) a(t)3tt1 t1 2fortteq: 4 )]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(g4Gnm2`2`1=p )]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(g(t)=H(t)3H 2n(t)m2 tot(t)1 (p(t))2&3H 2m(t) tot(t)a(t) a(t1)2

PAGE 123

&s t teqfortteq; =nmtot

PAGE 124

)]TJ /F6 11.9552 Tf 10.635 0 Td[(nvNnvN g4G2m2 (v)4: N(2)3n 4 3(mv)3: )]TJ /F7 7.9701 Tf 7.314 -1.794 Td[(g242(G)2 m3(v)6:tcrq 3 4Grvtcr= g=1 )]TJ /F7 7.9701 Tf 7.314 -1.793 Td[(g2 27r4m3(v)2:

PAGE 125

A (~r;t)=A(~r;t)ei(~r;t): n(~r;t)=(A(~r;t))2 ~v(~r;t)=1 m~r(~r;t):n~v (~r;t)=p nei(~p~r)]TJ /F7 7.9701 Tf 6.586 0 Td[(!t);~p=m~v!=p2=2m

PAGE 126

ni~vi(i=1;2) (~r;t)=p n1ei(~p1~r)]TJ /F7 7.9701 Tf 6.586 0 Td[(!1t)+p n2ei(~p2~r)]TJ /F7 7.9701 Tf 6.587 0 Td[(!2t)]TJ /F7 7.9701 Tf 6.587 0 Td[(): n(~r;t)=j(~r;t)j2=n1+n2+2p n1n2cos(~p~r)]TJ /F4 11.9552 Tf 11.955 0 Td[(!t)]TJ /F5 11.9552 Tf 11.955 0 Td[();~p=~p2)]TJ /F5 11.9552 Tf 12.748 0 Td[(~p1!=!2)]TJ /F5 11.9552 Tf 12.68 0 Td[(!1`=1 j~pjm~v ~g=)]TJ /F4 11.9552 Tf 9.298 0 Td[(8Gp n1n2m`sin(~p~r)]TJ /F4 11.9552 Tf 11.955 0 Td[(!t)]TJ /F5 11.9552 Tf 11.956 0 Td[()^n^n=`~p nnn1010yr=108yr=100n.10

PAGE 127

n.10 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(26310)]TJ /F3 7.9701 Tf 6.586 0 Td[(3c `1 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(3m64pc10)]TJ /F3 7.9701 Tf 6.587 0 Td[(22eV m T1 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(6m2105year10)]TJ /F3 7.9701 Tf 6.587 0 Td[(22eV m: g4G`0:5km secGyear10)]TJ /F3 7.9701 Tf 6.586 0 Td[(22eV m:gTT vgTr t0 Tr 1 2N(N)]TJ /F4 11.9552 Tf 11.955 0 Td[(1)0:5km sec10)]TJ /F3 7.9701 Tf 6.587 0 Td[(22eV m3 2t01010N20vgNN2z20)]TJ /F3 7.9701 Tf 6.587 0 Td[(1 m&10)]TJ /F3 7.9701 Tf 6.587 0 Td[(23eVv
PAGE 128

n~r~v6=0D)]TJ /F3 7.9701 Tf 6.587 0 Td[(4

PAGE 129

n n=5(l;b) 12m n=5 md2r dt2=)]TJ /F5 11.9552 Tf 10.494 8.088 Td[(dVe(r) dr

PAGE 130

rVe(r) vr(r)=r )]TJ /F4 11.9552 Tf 12.687 8.088 Td[(2 mdVe dr(a)(r)]TJ /F5 11.9552 Tf 11.955 0 Td[(a)a n(r)/1 r2jvr(r)j:r=a1=p r)]TJ /F5 11.9552 Tf 11.956 0 Td[(a (~r) )]TJ /F14 11.9552 Tf 15.602 8.088 Td[(~2 2m1 r2d drr2d(r) dr+Ve(r)(r)=E(r)E=Ve(a)r=a (r)=C1 rAi((a)]TJ /F5 11.9552 Tf 11.955 0 Td[(r)) =)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(2m ~2dVe dr(a)1 3:)]TJ /F3 7.9701 Tf 6.586 0 Td[(1)]TJ /F3 7.9701 Tf 6.587 0 Td[(1)]TJ /F7 7.9701 Tf 10.495 5.112 Td[(dVe dr(a)mv(a)2=av(a)'500

PAGE 131

m&10)]TJ /F3 7.9701 Tf 6.587 0 Td[(20eV)]TJ /F3 7.9701 Tf 6.587 0 Td[(1.r=a)]TJ /F3 7.9701 Tf 6.587 0 Td[(1/m)]TJ /F3 7.9701 Tf 6.586 0 Td[(2=3 12m ma&10)]TJ /F3 7.9701 Tf 6.587 0 Td[(20eV

PAGE 132

m10)]TJ /F3 7.9701 Tf 6.587 0 Td[(22eV

PAGE 133

10)]TJ /F3 7.9701 Tf 6.587 0 Td[(2310)]TJ /F3 7.9701 Tf 6.586 0 Td[(20eV

PAGE 142

st