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HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS IN MILKY
WAY HALO SUBSTRUCTURE
LEANNE DELMA DUFFY
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
This work is based on research performed by the Axion Dark Matter eX-
periment (ADMX). I am grateful to my ADMX collaborators for their efforts,
particularly in running the experiment and providing the high resolution data.
Without these efforts, this work would not have been possible.
I thank my advisor, Pierre Sikivie, for his support and guidance throughout
graduate school. It has been a priviege to collaborate with him on this and other
projects. I also thank Dave Tanner for his assistance and advice on this work.
I would like to thank the other members of my advisory committee, Jim Fry,
Guenakh Mitselmakher, Pierre Ramond, Richard Woodard and Fred Hamann,
for their roles in my progress. I am also grateful to the other members of the
University of Florida Physics Department who have contributed to my graduate
I am especially grateful to my family and friends, both near and far, who have
supported me through this long endeavor. Special thanks go to Lisa Everett and
TABLE OF CONTENTS
ACKNOW LEDGMENTS ............................. ii
LIST OF TABLES ................................. v
LIST OF FIGURES ..................... ......... vi
ABSTRACT ................... .............. viii
1 INTRODUCTION .................... ....... 1
2 AXIONS. .......... ................... ..... 7
2.1 Introduction . . . . . . . 7
2.2 The Strong CP Problem .......... ............. 7
2.3 The Axion ...................... ........ 11
2.3.1 Introduction ........................... 11
2.3.2 The Peccei-Quinn Solution to the Strong CP Problem .. 11
2.3.3 The Axion Mass ............... .... .. 15
2.3.4 The Axion Electromagnetic Coupling . . ..... 19
2.4 Axions in Cosmology .................. ....... .. 21
3 DISCRETE FLOWS AND CAUSTICS IN THE GALACTIC HALO 27
3.1 Introduction .................. ............ .. 27
3.2 Existence .................. ............. .. 27
3.3 Densities .................. .............. .. 37
3.4 Discussion .. ... .. .. .. .. .. .. .. .. .. .... .. .. 39
4 HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS .... 41
4.1 Introduction .................. ............ .. 41
4.2 Axion Dark Matter eXperiment ............. ... .. .. 42
4.3 Axion Signal Properties .................. .... .. 45
4.4 Noise Properties .................. .......... .. 49
4.5 Removal of Systematic Effects ...... ......... .. 53
4.6 Axion Signal Search ............... . 61
4.7 R results . . ... . . . ..... . 64
4.8 Discussion ............... ............. .. 69
5 SUMMARY AND CONCLUSION ............ ....... .... 72
REFERENCES ................................... 74
BIOGRAPHICAL SKETCH ........ .................... 79
LIST OF TABLES
4-1 Effective power thresholds for all n-bin searches, with the frequency res-
olutions, b, and corresponding maximum flow velocity dispersions, 6v',
for a flow velocity of 600 km/s. ............. .... 67
4-2 Numerically calculated values of the form factor, C, and amplifier noise
temperatures, T1, from NRAO specifications. .. . ..... 67
LIST OF FIGURES
3-1 A 2-D slice of 6-D phase-space. The line is a cross-section of the sheet
of width 6v on which the dark matter particles lie prior to galaxy for-
mation. The wiggles are the peculiar velocities due to density perturba-
tions. When overdensities become non-linear, the sheet begins to wind
up clockwise in phase-space, as shown. .................... .. 30
3-2 The phase-space distribution of dark matter particles in a galactic halo
at a particular time, t. The horizontal axis is the galactocentric distance,
r, in units of the halo radius, R, and v is the radial velocity. Spherical
symmetry has been assumed for simplicity. Particles lie on the solid line. 31
3-3 The cross-section of the tricusp ring. Each line represents a particle tra-
jectory. The caustic surface is the envelope of the triangular feature, in-
side which four flows are contained. Everywhere outside the caustic sur-
face, there are only two flows. Illustration courtesy of A. Natarajan. 33
3-4 The tricusp ring caustic. Axial symmetry has been used for illustrative
purposes. Illustration courtesy of A. Natarajan. ........... .. 34
4-1 Schematic diagram of the receiver chain. ................ 45
4-2 Sketch of the ADMX detector. .................. ..... 46
4-3 Power distribution for a large sample of 1-bin data. .......... ..51
4-4 Power distribution for a large sample of 2-bin data. .......... ..52
4-5 Power distribution for a large sample of 4-bin data. .......... ..53
4-6 Power distribution for a large sample of 8-bin data. .......... ..54
4-7 Power distribution for a large sample of 64-bin data. ......... ..55
4-8 Power distribution for a large sample of 512-bin data ........... ..56
4-9 Power distribution for a large sample of 4096-bin data. . .... 57
4-10 An environmental peak as it appears in the MR search (top) and the 64-
bin HR search. The unit for the vertical axis is the rms power fluctua-
tion in each case. . . . . .. . .. .. .58
4-11 HR filter response calibration data (512 bin average). The power has
been normalized to the maximum power output. ............. ..59
4-12 Sample 4096-bin spectrum before correction for the cavity v- ii1.1 ifr v cou-
pling. The line is the fit obtained using the equivalent circuit model. 60
4-13 The same 4096-bin spectrum of Fig. 4-12 after correction for the cavity-
amplifier coupling. .................. .. ...... 61
4-14 Illustration of the addition scheme for the 2, 4 and 8-bin searches. The
numbers correspond to the data points of the 1-bin search. Numbers within
the same box are bins added together to form a single datum in the n-
bin searches with n > 1. .................. .... 62
4-15 97.7'. confidence level limits for the HR 2-bin search on the density of
any local axion dark matter flow as a function of axion mass, for the DFSZ
and KSVZ a77 coupling strengths. Also shown is the previous ADMX
limit using the MR channel. The HR limits assume that the flow veloc-
ity dispersion is less than 6v2 given by Eq. (4-13). ............ ..68
4-16 97.7'. confidence level limits for the HR 4-bin search on the density of
any local axion dark matter flow as a function of axion mass, for DFSZ
and KSVZ a77 coupling strengths. Densities above the lines are excluded.
For comparison, the predicted density of the Big Flow is also shown. The
HR limits assume that the flow velocity dispersion is less than 6v2 given
by Eq. (4-13). . . . . . .. . . 70
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
HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS IN MILKY
WAY HALO SUBSTRUCTURE
Leanne Delma Duffy
C('! r: Pierre Sikivie
M, 1 ri Department: Physics
The axion is one of the leading particle candidates for the universe's dark
matter component. Despite possessing very small couplings, the axion's interaction
with photons can be utilized to search for it using a microwave cavity detector. The
Axion Dark Matter eXperiment (ADMX) uses such a detector to search for axions
in our galactic halo.
ADMX has recently added a new, high resolution channel to search for axions
in discrete flows. ADMX's medium resolution channel searches for axions in the
thermalized component of the halo.
We review the motivation for the axion and its properties which make it
a good dark matter candidate. We also review the arguments for the existence
of discrete flows in galactic halos. A flow of discrete axions with small velocity
dispersion will appear as a very narrow peak in the output of a microwave cavity
detector. A high resolution search can detect such a peak with large signal to noise.
We have performed such a search.
The details of the high resolution axion search and analysis procedure are
presented. In this search, no axion signal was found in the mass range 1.98-2.17
peV. We place upper limits on the density of axions in local discrete flows based on
This work is on a new search for axion dark matter. The Axion Dark Matter
eXperiment (ADMX) has achieved improved sensitivity by implementing a high
resolution channel to search for axions in halo substructure. In this chapter, we
give background information on dark matter and outline the contents of this work.
The i i i i ily of the total energy density of the universe is contributed by com-
ponents that are not understood. Only approximately !.' of the energy budget is
contributed by baryonic matter, that is, particles which interact electromagnetically
and can thus be observed by radiation. The remaining contributions to the energy
budget come from components that are called dark matter and dark energy. The
dark matter component acts as matter, but interacts only gravitationally with the
observable baryonic matter. This component contributes approximately 2"'-. to
the total energy density. Dark energy acts as a fluid with negative pressure and
contributes the remaining 7!' The dark energy component is causing the recent
epoch of accelerated expansion of the universe [1, 2]. The dark matter component
is the concern of this dissertation.
Dark matter was first postulated by Fritz Zwicky in 1933 . While observing
the Coma cluster of galaxies, he noted that the amount of visible matter was too
small for the system to be gravitationally bound. Given the observed galactic
velocities, the system should fly apart. Zwicky proposed additional matter that
was not visible to provide the necessary gravitational potential energy to bind the
The strongest evidence for dark matter tod i- is provided by the rotation
curves of spiral galaxies. Plots of observed circular velocity against radial distance
are flat to large distances. The contributions to this curve from the disk are not
enough to support this rotation curve. It is thus believed that spiral galaxies
consist of a visible disk embedded in a much larger elliptical dark matter halo. For
a review of evidence for dark matter, see Bertone et al. .
Dark matter particles must have the following two properties: (1) They are
effectively collisionless as far as structure formation is concerned; i.e., the only
significant long-range interactions are gravitational, and (2) The dark matter must
be cold; i.e., it must be non-relativistic well before the onset of galaxy formation.
The first property means that dark matter can interact only weakly with
baryonic matter. The second property is necessary to form the structure in the
universe that we observe tod i-. If dark matter was more energetic, it would be
able to freely stream out of the initial density perturbations that have formed into
What the dark matter consists of is still unknown, despite knowledge of the
properties it must possess. The standard model of particle physics does not contain
a particle that can provide the dark matter of the universe. Extensions to the
standard model do, however, provide viable particle candidates. The leading dark
matter particle candidates are axions and weakly-interacting massive particles
The axion is the pseudo-Nambu-Goldstone boson from the Peccei-Quinn
solution to the strong CP problem [5, 6, 7, 8]. The axion mass, ma, is constrained
to lie in the range 10-6 to 10-2 eV [9, 10, 11]. There are two benchmark axion
models that are minimal extensions of the standard model: the Kim-Shifman-
Vainshtein-Zakharov (KSVZ) [12, 13] model and the Dine-Fischler-Srednicki-
Zhitnitsy (DFSZ) [14, 15] model. In the early universe, one population of axions
is produced by thermal processes and has temperature of order 1 K tod iv. In
addition, cold axion populations arise from vacuum realignment [16, 17, 18] and
string and wall decay [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Which
mechanisms contribute depends on whether the Peccei-Quinn symmetry breaks
before or after inflation. The cold axions were never in thermal equilibrium with
the rest of the universe.
WIMPs are a class of dark matter candidates: heavy particles that interact
via forces of weak-scale strength. WIMPs occur in many models, particularly in
extensions of the standard model which include a new parity symmetry to prevent
proton decay. The most popular WIMP candidate is arguably the lightest super-
symmetric particle (LSP), i.e., the lightest, ..--iet-undetected particle provided by
the minimal supersymmetric extension to the standard model ( LSSM). In contrast
to axions, WIMPs are thermal relics. They began in thermal equibrium with the
primordial heat bath. When the interaction rate of WIMPs with the rest of the
heat bath falls below the expansion rate of the universe, these particles decouple
or I. out." Their evolution is then governed by the universe's expansion and
Much work is currently underway to detect dark matter and deduce its particle
properties. ADMX is a direct detection experiment searching for dark matter
axions [32, 33, 34, 35]. This experiment uses a tunable Sikivie microwave cavity
 to search for axions. When the resonant frequency of the cavity, v, corresponds
to the energy, Ea, of axions passing through the cavity (i.e., v = Ea/h), resonant
conversion of axions to photons will occur. The signal is a peak in the energy
spectrum of the output from the cavity. Many direct detection experiments are
also searching for WIMPs. WIMP direct detection is based on looking for nuclear
recoils from the elastic scattering of passing WIMPs. Additionally, attempts are
being made to detect dark matter indirectly using -1 i .i i cal signatures. These
primarily focus on detecting the products of WIMP annihilations: neutrinos,
positrons, anti-protons and gamma-rays.
These signals are dependent on the dark matter density. The power output
from resonant axion conversion to photons is proportional to the local axion
density. For WIMPs, the rate of nuclear recoil events from WIMP scattering is
proportional to the local WIMP density. Also, the flux of annihilation products
searched for in indirect detection is proportional to the square of the WIMP
density at the site of annihilation. Thus it is necessary to make assumptions about
the distribution of dark matter in our galactic halo. In particular, galactic halo
substructure is of interest for dark matter detection. The presence of substructure
in a galactic halo means that there will be regions of enhanced dark matter density,
improving detection prospects due to the signal dependence on density.
While the power in an axion signal observed by a microwave cavity detector is
proportional to the local axion density, the signal width is caused by the velocity
dispersion of dark matter axions. In searching for axions, it is thus also necessary
to make assumptions about their velocity distribution in the Milky Way halo. A
number of models are used to guide ADMX's search. These are the isothermal
model, the results from N-body simulations [37, 38] and a description of galactic
halos in terms of discrete flows from late infall of dark matter onto the galaxy. A
specific model which considers late infall is the caustic ring model [39, 40].
In the isothermal model, it is assumed that the dark matter halo has ther-
malized via virialization and thus has a Maxwell-Boltzmann velocity distribution.
ADMX's medium resolution (i lR) channel  searches for such axions, assuming
that the velocity dispersion is 0(10-3c) or less, where c is the velocity of light.
(The escape velocity from our galaxy is approximately 2 x 10-3c.)
Of particular interest to this work is the existence of cold flows of dark matter
axions within the halo. Such flows are associated with the tidal disruption of
subhalos predicted by N-body simulations and with late infall of dark matter onto
the galactic halo.
Numerical simulations indicate that hundreds of smaller clumps, or subhalos,
exist within the larger halo [37, 38]. Tidal disruption of these subhalos leads to
flows in the form of "tidal tails" or -I. ,I i- The Earth may currently be in a
stream of dark matter from the Sagittarius A dwarf galaxy [41, 42].
Non-thermalized flows from late infall of dark matter onto the halo are also
expected [43, 44]. Dark matter that has only recently fallen into the gravitational
potential of the galaxy will have had insufficient time to thermalize with the rest
of the halo. This dark matter will be present in the halo in the form of discrete
flows. There will be one flow of particles falling onto the galaxy for the first time,
one due to particles falling out of the galaxy's gravitational potential for the
first time, one from particles falling into the potential for the second time, etc.
Furthermore, where the gradient of the particle velocity diverges, particles "pile
up" and form caustics. In the limit of zero flow velocity dispersion, caustics have
infinite particle density. The velocity dispersion of cold axions at a time, t, prior to
galaxy formation is approximately 6va ~ 3 x 10-17(10-5eV/mT)(to/t)2/3 , where
to is the present age of the universe. Thus, a flow of dark matter axions will have a
small velocity dispersion, leading to a large, but finite density at the location of a
The caustic ring model predicts that the Earth is located near a caustic feature
. This model, fitted to bumps in the Milky Way rotation curve and a triangular
feature seen in the IRAS maps, predicts that the flows falling in and out of the
halo for the fifth time contain a significant fraction of the local halo density. The
predicted densities are 1.7 x 10-24 g/cm3 and 1.5 x 10-25 g/cm3 , comparable
to the local dark matter density of 9.2 x 10-25 g/cm3 predicted by Gates et al. .
The flow of the greatest density is referred to as the "Big Flow."
The possible existence of discrete flows provides an opportunity to increase the
discovery potential of ADMX. A discrete axion flow produces a narrow peak in the
spectrum of microwave photons in the experiment and such a peak can be searched
for with higher signal-to-noise than the signal from axions in an isothermal halo. A
high resolution (HR) channel has been built to take advantage of this opportunity.
If a signal is found, the HR channel will also provide detailed information on the
structure of the Milky Way halo.
The HR channel is the most recent addition to ADMX, implemented as a
simple addition to the receiver chain, running in parallel with the MR channel.
This channel and the possible existence of discrete flows can improve ADMX's
sensitivity by a factor of thi [;,], significantly enhancing its discovery potential.
This work is arranged as follows. Background information on the axion is given
in C'! lpter 2. The strong CP problem of the standard model of particle physics,
the motivation for the axion, is described and the Peccei-Quinn solution, resulting
in the axion, is discussed. Properties important to axion detection, such as its mass
and coupling, are also reviewed. Some .,-I i. .1li, -ii J1 and cosmological consequences
of the axion are outlined, particularly the production of cold axion populations. A
review of discrete flows and caustics in the galactic halo is presented in Chapter 3.
C'! lpter 4 describes ADMX and provides details of the HR analysis. The new
result, improving ADMX's search sensitivity by a factor of three, is also shown. A
summary and conclusions are presented in C'! Ipter 5.
The axion is the pseudo-Nambu-Goldstone boson implied by the Peccei-Quinn
solution to the strong CP problem [5, 6, 7, 8]. It is also a good candidate for the
dark matter of the universe. This chapter provides background information on
axions. The strong CP problem is described in Section 2.2. In Section 2.3, the
Peccei-Quinn solution to this problem is outlined, using the original Peccei-Quinn-
Weinberg-Wilczek axion model. Axions are shown to be a natural consequence
of this solution. Axion properties important to detection are also reviewed.
Section 2.4 discusses cosmological aspects of axions, specifically how axion dark
matter arises and the limits that cosmology and .i-1 |i1, ,!-ii place on the axion
2.2 The Strong CP Problem
Quantum chromodynamics (QCD) is the theory of the strong nuclear forces.
Its gauge symmetry is SUc(3), the color symmetry group. In nature, QCD is not a
stand-alone theory. It is embedded within the standard model of particle physics.
The full gauge symmetry of the standard model is SUc(3) xSUL(2)x Uy(1), i.e., the
direct product of the color, left-handed and hypercharge symmetries, respectively.
The unified SUL(2) x Uy(l) forms the electroweak symmetry. Breaking of the
electroweak symmetry down to UEM(1) via the Higgs mechanism results in the W'
and Z boson masses and quark and lepton masses. In this section, we outline the
strong CP problem and explain that both QCD and the electroweak effects that
give the quarks mass combine to create this problem.
The Lagrangian of QCD is
S(a a + N g ca -a ( )
LQCD = GG + [iij7Y D/,q (mjqIJtQJ + h.c.)] + 1G""" (2-1)
The qi are the quark fields, the subscript i indicating each of the N = 6 quark
flavors and the additional subscript L or R denoting a left- or right-handed field.
The mi are the quark masses, g is the color coupling and the 7, are the gamma
matrices. The notation "h.c." stands for hermitian conjugate. The gluon field
strength tensor is
CG, PA -P A ga fabcAb A (22)
where Aa is the gluon vector potential, the superscripts referring to the eight
possible gluon color assignments, and fabc are the structure constants of SUc(3).
Acting on a spinor field, i, the covariant derivative, D., is
D,, = (, +gA- a ) (2-3)
where the A4 are the Gell-Mann matrices. The final term of Eq. (2-1) is the "0-
term." The angle, 0, is a parameter and G0a" is the dual tensor to the gluon field
strength, defined by
Gap" = -e""G a (2-4)
with d"P", the Levi-Civita tensor.
The parameters of QCD are thus g, mj and 0. The color coupling, g is energy
dependent and in defining the theory, it is normally exchanged for the QCD
confinement scale, AQCD, of order 200 MeV. The parameter, 0, is the QCD vacuum
angle. This parameter is necessary to fully describe QCD because the SUc(3)
gauge symmetry is non-Abelian. Non-Abelian gauge potentials have disjoint
sectors, labelled by an integer topological winding number. These sectors are
disjoint as they cannot be transformed continuously into each other. There exists a
vacuum configuration corresponding to each n, between which quantum tunnelling
can occur. The gauge invariant QCD vacuum state is thus a superposition of vacua
of different n, i.e.,
10) e nOI)n. (2-5)
This is the origin of 0.
In the limit of massless quarks, QCD has a classical chiral symmetry, UA(1).
However, this symmetry is anomalous. The existence of the Adler-Bell-Jackiw
anomaly [47, 48] means that this symmetry is not present in the quantum theory.
In the full quantum theory, including quark masses, the physics of QCD remains
unchanged under the transformations,
qi eiiy/2qi (2-6)
mi C e-iim (2-7)
0 0 aj. (28)
While the physics remains the same, this is not a symmetry because the
parameter 0 has changed. The transformations of Eq. (2-6) through Eq. (2-8) can
be used to move phases between the quark masses and 0. However, the quantity,
S0 arg(mlm2...mvN) (2-9)
is invariant and therefore observable, unlike 0. This is commonly written as
0 0 arg det M (2-10)
where M is quark mass matrix.
The 0-term violates the discrete parity symmetry (P) and the combined
operation of a parity transformation followed by charge conjugation (CP). If CP
was a good symmetry of the standard model, the 0-term would not be permitted.
However, this is not the case; CP violation has been observed in the electroweak
sector. Consequently, there is no apparent reason why the 0-term would not be
present in the standard model.
While CP violation is present in the electroweak sector of the standard model,
it has not been observed in QCD. An electric dipole moment for the neutron is the
most easily observed consequence of strong CP violation. The 0-term results in a
neutron electric dipole moment of [9, 49, 10, 11],
|dn ~ 10-16 ecm, (2-11)
where e is the electric charge. The current experimental limit is 
Sdn, < 6.3 x 10-26 cm (212)
thus |81 < 10-9. However, there is no reason to expect that 0 should be so close
to zero. Since CP violation is introduced in the standard model by allowing the
quark mass matrices to have arbitrary complex entries, 0 is naturally expected to
be of order one. This is the strong CP problem, i.e. the question of why the angle 0
should be nearly zero, when CP violation is present in the standard model.
A number of solutions to the strong CP problem have been proposed. The
Peccei-Quinn (PQ) solution [5, 6] results in the presence of an axion [7, 8], which
has the additional motivation of being a good candidate for the dark matter of the
universe. This solution is outlined in detail in the following section. Other solutions
include the up quark mass being zero and that CP is spontaneously broken. If the
bare up quark mass is zero, the 0 dependence of the QCD Lagrangian disappears
and the strong CP problem is solved. This solution is, however, disfavored by
lattice calculations and by the success of first order chiral perturbation theory in
reproducing the pattern of pseudo-scalar meson masses. The Nelson-Barr model
is an example of a theory where the strong CP problem is solved by properly
engineered spontaneous CP violation [51, 52]. We focus on only the PQ solution in
the following section.
2.3 The Axion
This section provides important background information for axion detection.
In Section 2.3.2, we discuss the PQ solution to the strong CP problem. The
original Peccei-Quinn-Weinberg-Wilczek axion model is used for illustration, but
other axion models are also discussed. A derivation of the axion mass is given in
Section 2.3.3, using the methods of low energy effective theory. In Section 2.3.4,
the axion-electromagnetic coupling is reviewed. This coupling is the basis for
axion detection experiments. The resulting power developed in a microwave cavity
detector, using this coupling, is also given.
2.3.2 The Peccei-Quinn Solution to the Strong CP Problem
The Peccei-Quinn solution to the strong CP problem promotes 0 from a
parameter to a dynamical variable. To implement this mechanism, a global
symmetry, U(1)pQ, is introduced. This symmetry has a color anomaly and is
spontaneously broken. The resulting pseudo-Nambu-Goldstone boson is the
axion. The axion field, a, can be redefined to absorb the parameter 0. The non-
perturbative effects which make QCD 0 dependent result in a potential for the
axion field, causing it to relax to the CP conserving minimum and solving the
strong CP problem.
To realize the PQ solution, it is necessary to add new fields to the standard
model, otherwise there are no degrees of freedom available to accommodate the
axion. In the original, Peccei-Quinn-Weinberg-Wilczek (PQWW) axion model an
additional Higgs doublet was introduced. We review this model to demonstrate the
Peccei-Quinn mechanism in this section.
The simplest way to introduce additional degrees of freedom is via an extra
Higgs doublet. We assume that one of the Higgs doublets, 0,, couples to the up-
type quarks and the other, Qd, couples to the down-type quarks. We distinguish
between the up- and down-type quarks by labelling them ui and di, respectively
(rather than qj, as in the previous section). As there are N quarks, there are N/2
up-type quarks and down-type quarks. The leptons can acquire mass through
Yukawa couplings to either of the Higgs doublets or to a third Higgs doublet. We
ignore this complication here and simply examine the couplings to quarks. The
quarks acquire their masses from the expectation values of the neutral components
of the Higgs, Q% and j0. The mass generating couplings are
S y"u. ",UR + yi d + h.c. (2-13)
Peccei and Quinn chose the Higgs potential to be
V (,1 ,-d) 2 t 2, 7i, t, > (2-14)
where the matrices (aij) and (bi) are real and symmetric and the sum is over
the two types of Higgs fields. With this choice of potential, the full Lagrangian,
including the kinetic term and 0-term, has the following global invariance, UpQ(1):
SC i2au (2-15)
Se d (216)
ui e-i"Y5Ui (2-17)
di e-id sd (2-18)
0 0-N(a+ ad) (2-19)
Note that it is possible to write down lepton couplings which also observe the PQ
symmetry. It is necessary that these couplings do so, otherwise a potential term for
a will result, d, -1i. i-; the PQ mechanism.
When the electroweak symmetry breaks, the neutral Higgs components acquire
One linear combination of the Nambu-Goldstone fields, P, and Pd, is the longi-
tudinal component of the Z-boson, as per electroweak symmetry breaking in the
standard model. This combination is
h = cos 3P, sin 3Pd .
The orthogonal combination is the axion field,
a = sin 3, P, + cos OPd .
sin pOa + cos po h
cos Oa sin 3h .
Using Eqs. (2-20), (2-21), (2-24) and (2-25) in Eq. (2-13), the axion couplings to
quarks arise from
i t sin 4a
-m =mu e u aURi + m die
i Li i Li
'dRi + h.c. ,
where m" = y"'iv and m = 11' The axion field dependence can be removed
from the mass terms using the transformations of Eqs. (2-17), (2-18) and (2-19).
Direct couplings between the axion and quarks will still remain in the Lagrangian,
through the associated change in the quark kinetic term. The resulting change in 0
S- N(v,/vd + vd/v,)a/v (2-27)
where v = v + v. The axion field can be redefined to absorb 0 on the right-
hand side of Eq. (2-27). Defining
VPQ = (2-28)
Vu/Vd + dVu,
the 0-term of Eq. (2-1) is replaced by
La G G""" (2-29)
167 2VpQ P
Non-perturbative QCD effects explicitly break the Peccei-Quinn symmetry, but
do not become important until the universe cools to the quark-hadron transition.
These effects give the axion field a potential and when they become important, the
field relaxes to the minimum, which conserves CP. Hence the PQ mechanism, which
replaces 0 with the dynamical axion field, solves the strong CP problem.
However, the PQWW axion has been ruled out by observation. Under the
PQWW scheme, the axion mass is inherently tied to the electroweak symmetry
breaking scale, v. As VpQ ~ v and v = 247 GeV, the axion mass is of the order
of 100 keV. Such a heavy axion would have been observed at particle colliders and
has thus been ruled out. The calculation of the axion mass is reviewed in the next
While the PQWW axion model is not viable, this does not, however, eliminate
the possibility of an axion solving the strong CP problem. "Invi- il!. axion
models, named such for their extremely weak couplings, are still possible. In an
invisible axion model, the PQ symmetry is decoupled from the electroweak scale
and instead is spontaneously broken at a much higher temperature, decreasing
the axion mass and coupling strength. Two benchmark, invisible axion models
exist: the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) and Kim-Shifman-Vainshtein-
Zhakharov (KSVZ) models. In both the KSVZ and DFSZ models, an axion with
permissable mass and couplings results.
In the KSVZ model, only the Higgs doublet of the standard model occurs.
The axion is introduced as the phase of an additional electroweak singlet scalar
field. The known quarks cannot be directly coupled to such a field, as the Yukawa
couplings would lead to unreasonably large quark masses. Instead, this scalar
is coupled to an additional heavy quark, also an electroweak singlet. The axion
couplings are then induced by the interactions of the heavy quark with the other
The DFSZ model has two Higgs doublets, as in the PQWW model, and an
additional electroweak singlet scalar. It is the electroweak singlet scalar which
acquires a vev at the PQ symmetry breaking scale. The scalar does not couple di-
rectly to quarks and leptons, but through it's couplings to the two Higgs doublets.
Thus, it is possible for the existence of an axion to solve the strong CP
problem. While significant for that alone, the axion also provides an interesting
candidate for the cold dark matter of the universe.
2.3.3 The Axion Mass
We review how the axion mass can be obtained from the low-energy effective
field theory, using the chiral Lagrangian. For this purpose, we consider only
the two lightest quarks, up and down. The chiral Lagrangian is invariant under
SUL(2) x SUR(2) x Uv(1). We will introduce an extra UA(1) symmetry, but break
it explicitly by giving a large mass to the eta particle. Indeed, the group UA(1) is
not actually a symmetry of QCD, as it is broken at the quantum level by instanton
effects. The symmetries are spontaneously broken down to SUL+R(2) x Uv(1) by
the quark condensation at the quark-hadron transition. At the scale, A, the quark
condensate acquires expectation value
(q<9 j)o -A3U() (2-30)
where 7r(x) is the pion field. The scale, A, is of the order of AQCD, but not equal to
it. The matrix U is given by
U() = exp ,i (2 31)
where r is the Pauli matrices and f, is the pion decay constant, equal to 93 MeV.
For the SUL+R(2) triplet, the pions, the effective Lagrangian is
L, = Tr(0,UtaPU) + A3Tr(mU + h.c.) (2-32)
where mq is the diagonal quark mass matrix,
m, = (7 0 (2-33)
Expansion of the Lagrangian shows the pion mass to be
2 3 Md+ (3d
m = 2A3 (2-34)
To find the axion mass, we also need to introduce the would-be Nambu-
Goldstone boson associated with the spontaneous breaking of the UA(1) symmetry.
We denote this particle as Tl (eta) in the following (this state is actually some linear
combination of the rl and rl' pseudo-scalar mesons). The expectation value of the
quark condensate becomes
(qqLu)o= --A3U(r) exp f) (2-35)
L \J 1 A
where f, is the T] decay constant. The effective Lagrangian is
L7,n ,T989" + Tr(, Ut8WU)
+A3Tr(mqUexp ,) + h.c.) + -m, cs (2-36)
where the final term is the potential for rotations in the UA(1) direction.
When the 0-term is included, the expectation value of the condensate remains
that given in Eq. (2-35), except that is replaced by + 2. Indeed, under a
qj eg q, (2-37)
rriq e 2Tmq (2 38)
f + (2-39)
fA 2 f,
In an axion model, 0 is replaced by 0 + N, where Va is the scale at which the PQ
symmetry breaks. The constant N is defined by the anomaly,
N = 2 ptf, (2-40)
where pf is the appropriate charge and tf is the second casimir operator of the
algebra. The axion decay constant, f, is defined by
a = (2-41)
The effective Lagrangian, including the 0-term and the axion field, is
1 1 f2
7na 2 0,aaa + 2989a + f-Tr(aUtaPU)
+A3Tr(,qU exp (f) + h.c.) + mi os 2 cos 0 + (2-42)
The variable 0 defines the origin of the axion field, so we may choose this to be
zero. The quark mass matrix can be written as a real matrix times a phase and
we may rotate to move 0 onto the quark mass term. This illustratres that the 0
dependence is alv--i a dependence on 0.
The eta, neutral pion and axion fields mix, as they all have the same quantum
numbers. Firstly, consider rf-a mixing. The physical rl field is
lphys = r + f (2-43)
and we use the redefinition
a' = a (244)
As the minimum of the potential occurs when the cosine term is zero, we may set
rlphys to zero. The new Lagrangian is
1 f2 _iTa' + (
aa,a Oa'a' + Tr( ,UtaPU) + A3Tr(mUexp + h.c.) (245)
2 4 k 2Va
We find that the minimum of the potential occurs at r = 0 and a' = 0. The
physical neutral pion and axion fields are
md mf a'+ O(f (2-46)
md + m,2f f( )
Smd m f, o, (f
aphys =a + 0 (2-47)
md+m, 2fa f j
with corresponding masses
mo A md+ o (2-48)
m mumd o+ Of (2 49)
a f2i(m, +md)
expressed as [9, 49, 10, 11]
(10 12GeV (251)
ma 6 x 10-6 eV (012Ge (2-5 1)
2.3.4 The Axion Electromagnetic Coupling
Axion detection is based on its electromagnetic coupling . We discuss how
this coupling arises in effective field theory and review the power developed in a
microwave cavity experiment in this section.
The axion electromagnetic coupling is due to mixing between the axion,
neutral pion and eta. The couplings of the Lagrangian for any of these particles to
decay to two photons is
a (r 5 rTI N, a) F\ P
Cw/o/o--,,_ 4 j + y + F JF^Mt' (2-52)
47 f7 3 f, 2 Va
The coefficients in the above equation arise from the trace over the anomaly loop.
The constant N, is given by
N, 2 pf(ef)2 (2-53)
where pf is the PQ charge of a right-handed quark field. Using the definition of the
physical axion field given in Eqs. (2-44) and (2-47), the resulting axion coupling to
two photons is
r,7 = g7 Faa FL"V p (2-54)
12 NI 3 md + n(
and we have relabelled the physical axion field as a. In grand unified theories, N,
and N are related, with N/N = 8/3. In this case, g, 0.36. Both the PQWW
and DFSZ axion models are grand-unifiable. In the KSVZ axion model, this is not
the case. The introduction of an additional heavy neutral quark means that the
KSVZ axion model cannot fit within a grand unified theory. In this case, N, = 0,
as the up and down quarks carry no PQ charge, and g, = -0.97.
The full Lagrangian for the interaction of axions with photons in free space is
1 ac 1 1 a2
L = F, + g- FgF + -,a8l"a -m+a2 2 + O (2-56)
4 4fa 2 2 v
In terms of the electric and magnetic fields, E and B, and introducing a medium
with dielectric constant, c, Eq. (2-56) can be written as
1 1 1 22 a
= (cE2 B- 2) + -aaaa a- _m -g -E B (2-57)
2 2 2 47Tfa
In a cavity permeated by a -1I i.- inhomogeneous magnetic field, resonant
conversion of axions to photons can be induced if the cavity frequency corresponds
to that of the axion energy. The resulting power developed in a microwave cavity
= ( VB min(Q,Qa) (2-58)
where V is the cavity volume, Bo is the magnetic field strength, pa is the local
density of axions with energy corresponding to the cavity frequency, Q is the
quality factor of the cavity and Qa is the ratio of the energy of halo axions to
their energy spread, equivalent to a "quality factor" for the halo axion signal. The
mode-dependent form factor, C, is given by
|I d3xE, Bo 2
C B2V/f d3 E2 1 (2 59)
BV fv xe E,| '
in which E,(x) is the time-dependent electric field of the mode under consideration
and c is the dielectric constant of the medium inside the cavity. This is more
conveniently expressed as
P = 0.5 x 10-21W (V BO) x 1024g.cm
500L 7T 0.36 0.5 x -24g.-3
( va min [Q, Qa] (260)
where Va is the axion energy frequency.
Thus, when such a cavity is tuned to the correct frequency, resonant con-
version of axions to photons results. This conversion is observed as a peak in the
frequency spectrum of the detector output.
2.4 Axions in Cosmology
Axions may p1 iv an important role in cosmology. We focus on two aspects of
this here. Firstly, for a mass in the range 10-6 10-4 eV, the axion is an interesting
dark matter candidate. Secondly, we outline the restrictions that cosmology and
.1- i' r,-lics place on the axion mass and coupling.
Axions satisfy the two criteria necessary for cold dark matter: (1) a non-
relativistic population of axions could be present in our universe in sufficient
quantities to provide the required dark matter energy density and (2) they are
effectively collisionless, i.e., the only significant long-range interactions are gravita-
tional. There are three mechanisms via which cold axions are produced: vacuum
realignment [16, 17, 18], string decay [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]
and domain wall decay [29, 30, 31]. We discuss the history of the axion field as
the universe expands and cools to see how and when these mechanisms occur. We
also review the process of vacuum realignment in detail, as there will alv--b i be a
contribution to the cold axion populations from that mechanism and, as discussed
below, it is possible that this provides the only contribution.
There are two important scales in the problem of axions as dark matter. The
first is the temperature at which the PQ symmetry breaks, TpQ. Which of the
above mechanisms contribute significantly to the cold axion population depends
on whether this temperature is greater or less than the inflationary reheating
temperature, TR. The second is the temperature at which the axion mass, arising
from non-perturbative QCD effects, becomes significant. At high temperatures, the
latter effects are not significant and the axion mass is negligible . The axion
mass becomes significant at a critical time, ti, when majt ~ 1 [16, 17, 18]. The
corresponding temperature is T1 1 GeV.
At initial early times, the PQ symmetry is unbroken. At TpQ, it breaks spon-
taneously and the axion field, which is proportional to the phase of the complex
scalar field acquiring a vev, may have any value. The phase varies continuously,
changing by order one from one horizon to the next. Axion strings appear as topo-
logical defects. If TpQ > TR, the axion field is homogenized over huge distances and
the string density is diluted by inflation, to the point where it is extremely unlikely
that any axion strings remain in our visible universe. In the case TpQ < TR, the
axion field is not homogenized and strings radiates cold, massless axions until
non-perturbative QCD effects become significant. At this time, the axion strings
become the boundaries of N domain walls. If N 1= the walls bounded by string
rapidly radiate cold axions and decay (domain wall decay). For N > 1, the domain
wall problem occurs  because the vacuum is multiply degenerate and there is at
least one domain wall per horizon. The domain walls end up dominating the energy
density and cause the universe to expand as S o t2, where S is the scale factor.
Although other solutions to the domain wall problem have been proposed , we
assume here that N = 1 or TpQ > TR. Thus, if TpQ < TR, string and wall decay
contribute to the axion energy density. If TR < TpQ, and the axion string density is
diluted by inflation, these mechanisms do not contribute significantly to the density
of cold axions. Then, only vacuum realignment will contribute a significant amount.
Vacuum realignment will result in a population of cold axions, independent
of TR. This is discussed in more detail below, however, an overview is as follows.
At TpQ, the axion field amplitude may have any value. If TpQ > TR, the homoge-
nization from inflation will result in a single value of the axion field over our visible
universe. Non-perturbative QCD effects cause a potential for the axion field. When
these effects become significant, the axion field will begin to oscillate in the po-
tential. These oscillations do not decay and contribute to the local energy density
as non-relativistic matter. Thus, a cold axion population results from vacuum
realignment, regardless of the inflationary reheating temperature.
To understand the contribution from vacuum realignment, consider a toy axion
model with one complex scalar field, O(x), in addition to the standard model fields.
Let the potential for O(x) be
V(0) = (2 4 v)2, (2-61)
When the universe cools to a temperature TpQ ~ Va, Q acquires a vacuum
() = a exp(ia(x)) (2-62)
The axion field is related to a(x), the phase of the scalar field, O(x), by
a(x) vaa(x) (2-63)
At T ~ A, non-perturbative QCD effects give the axion a mass. They produce
an effective potential
V() = m (T) (1 cos) (2-64)
ma(T) O .lma A .1D (2-65)
The minimum of the potential occurs at
(x) = Na(x)= 0. (2-66)
The axion acquires mass, ma, due to the curvature of the potential at this mini-
mum. Given the definition of the axion field in Eq. (2-63), the effective Lagrangian
2 '2 \ a ) (267)
In a Friedmann-Robertson-Walker universe, the equation of motion is
a + 3H(t)a V2 + (T(t)) sin(Na) 0 (2-68)
Near the potential minima,
V(a) -mava2 (2-69)
sin(Na) Na (2-70)
We now restrict the discussion to the zero momentum mode. For TPQ > TR,
this will be the only mode with significant occupation, so the final energy density
calculated will be for this case. In the case, TR > TpQ, higher modes will also be
occupied. For the zero momentum mode, the equation of motion reduces to
S+ 3H(t) + mi(t)a 0 (2-71)
i.e., the field satisfies the equation for a damped harmonic oscillator with time-
dependent parameters. As no initial value of a is preferred, the most general
a = al + a2t (2-72)
where ac and a2 are constants. Thus, at T >> TQCD, a is approximately constant.
The field will, however, begin to oscillate in its potential when the universe cools to
the critical temperature, T1, defined by 
ma(Ti(ti)) 3H(TI(ti)) -= (2-73)
As the axion field can realign only as fast as causality permits, the correspond-
ing momentum of a quantum of the axion field is
pa(ti) ~ 10-9eV (2-74)
for t1 ~ 2 x 10-' s, i.e. the age of the universe at which the quark-hadron transition
occurs. As discussed below, the axion mass is restricted to the range 10-6 10-2 eV
and thus this population is non-relativistic or cold.
This mechanism can produce a sufficient quantity of cold axions to provide
the dark matter of the universe. We show this by reviewing the energy density
of axions produced by the realignment mechanism. The energy density for a
homogeneous scalar field around its potential minimum is
p T +mt (2-75)
By the Virial theorem,
(2) =m2(a2) = (2-76)
As axions are non-relativistic and decoupled,
p oc M (2-77)
Thus, the number of axions per comoving volume is conserved, provided the axion
mass varies adiabatically.
The initial energy density in the coherent oscillations is
P1 = m(ti)a /2 (2-78)
12 / ) ( )2 (2-79)
-f2 jr na(tI) ( -JN (279)
The energy density in axions t.--l-v is
po = (2S80)
m (ti) S3(to)
-1 f2 1 fS31(t> 1\2
2 tJ S3 (to)
where S(t) is the scale factor at time, t. Eq. (2-81) implies the axion energy
~0.15 ( v) (2-82)
As the axion couplings are very small, these coherent oscillations do not decay and
make a good candidate for the dark matter of the universe.
The mass is related to the Peccei-Quinn decay constant, fa, by Eq. (2-51) and
the couplings of the axion mass are inversely proportional to fa. Thus limits on any
of the axion mass, axion couplings or PQ decay constant is also a restriction on the
other two. Since Q, < QCDM = 0.22, fa < 1012 GeV and thus, m, > 10-6 eV. This
is the lower bound on the axion mass range. If the axion mass were any greater,
too much dark matter would be produced via the realignment mechanism.
The upper limit on the axion mass is 10-2 eV, from observations of SN1987a.
The number of neutrinos observed on Earth due to this supernovae and its duration
are in good agreement with models of supernovae. Light particles, such as axions,
present novel cooling mechanisms that can alter the duration of supernovae. If the
axion mass is less than 10-2 eV, axions are not produced in significant numbers
to affect supernovae. However, for a range of axion masses above this, axion
production and escape from supernovae will significantly shorten the supernova
duration by efficiently transporting energy away. Above approximately 0.5 eV, the
mean free path of an axion will be too short for significant numbers of axions to
escape from supernovae. At this point, other .,-I i .!r,~ i-i I1 processes, such as the
lifetime of red giants forbid axion masses in higher ranges .
DISCRETE FLOWS AND CAUSTICS IN THE GALACTIC HALO
ADMX's high resolution channel searches for discrete flows of cold axions
passing the detector. As discussed in the introduction, such flows occur due to tidal
stripping of dwarf galaxies and late infall of dark matter onto the galactic halo.
In this chapter, we review the arguments why such flows are expected to occur in
cold dark matter cosmology, thus providing an interesting possibility to search for
In Section 3.2, we review literature demonstrating that discrete flows are a
natural consequence of a cold dark matter cosmology. Section 3.3 discusses the
densities of such flows. A significant fraction of the local halo density should be
contained in discrete flows, which is important when searching for them, as the
signal observed is proportional to the density. A brief discussion of evidence for
flows and detection of axions in these flows concludes this chapter, in Section 3.4.
This work demonstrates that searching for discrete flows of cold axions in the
galactic halo improves the sensitivity of a microwave cavity detector. However, it
is necessary that such halo substructure exists for us to benefit from this improved
detector sensitivity. Natarajan and Sikivie have shown that discrete flows and
caustics are a necessary consequence of cold dark matter cosmology. In this section,
we review the arguments for the presence of discrete flows and caustics in the
galactic halo. First, we describe why it is expected that such halo substructure
forms and then we review the mathematical proof for the existence of both inner
and outer caustics in galactic halos.
While we are interested in this substructure from the point of view of axion
detection, it should be noted that the existence of discrete flows and caustics is
independent of the type of cold dark matter. The only requirement for flows and
caustics to form is the assumption of cold dark matter itself. Cold dark matter
particles are assumed to possess the following properties:
(1) The particles must be collisionless, i.e., the only significant interactions of
these particles are gravitational. This property explains why the particles are dark
(2) The particles have negligible initial velocity dispersion, where the initial
conditions are those when the dark matter first falls into a galaxy's gravitational
potential. This is discussed further in the following.
The primordial velocity dispersion of both axions and WIMPs is negligible 
as far as large scale structure formation is concerned. For WIMPs, the primordial
velocity dispersion is determined by the temperature, TD, at which they decouple
from the primordial heat bath. Considering Hubble expansion to be the only
significant effect to alter the WIMP velocity dispersion, the velocity dispersion,
6vw, of a WIMP of mass mw falling into a galaxy tod i- is
(6- 2TD ( S(tD)\ (3-t
\mw ) S(to) I
where S is the scale factor, given at the time of decoupling, tD, and todi-, to. For a
WIMP with mass of 1 GeV that decoupled when the temperature was 10 MeV, the
velocity dispersion todci- is 6vw ~ 10-12
For axions, the primordial velocity dispersion is due to the inhomogeneities in
the axion field when the axion mass, m,, becomes significant, i.e., when m, ~ H
at temperature Ta, 1 GeV and time ta ~ 2 x 10-' s. The magnitude of the field
inhomogeneity depends on whether the Peccei-Quinn (PQ) symmetry breaks before
or after inflationary reheating. If the PQ symmetry is broken after 1,. 1 iir the
axion field is inhomogeneous on the scale of the horizon size (~ ti) when the mass
becomes significant and hence,
Va 1 (S(ta) 017 -5eV (32)
6v, -- ^ 10-17 x (3-2)
mat, S(to)) ma
If the PQ symmetry is broken before reheating, inflation homogenizes the axion
field over enormous distances and the velocity dispersion, 6va, due to quantum
mechanical fluctuations in the axion field, is even smaller than in Eq. (3-2).
The primordial velocity dispersion of dark matter particles falling onto a
galaxy at any time, t, can be obtained by substitution of S(to) for the scale factor,
S(t). For both axions and WIMPs, we see that the initial velocity dispersion is so
small as to be negligible.
The formation of discrete flows and caustics can be understood by considering
the phase-space distribution of dark matter particles falling into a gravitational
potential. At early times, prior to the onset of galaxy formation, these particles
will lie on a thin 3-dimensional (3D) sheet in 6D phase-space, as illustrated in
Fig. 3-1. The thickness of the sheet is proportional to the local velocity dispersion
of the dark matter particles, 6v, and thus the sheet is thin. This sheet will also be
continuous, as the number density of particles is very large over the scale at which
the sheet is bent in phase space.
As dark matter particles are collisionless, the evolution of the sheet is de-
termined by the influence of gravity only. Where density perturbations become
non-linear, the 3D sheet will begin to 'iiLd up" clockwise in phase-space. Whereas
previously, in the linear regime, the sheet covered physical space only once, it will
now begin to cover physical space multiple times. After much time, the phase-space
particle distribution will look as shown in Fig. 3-2. As particles fall into a grav-
itational potential, there will be a number of discrete flows present at each point
at any time . There will be one flow of particles falling in for for the first time,
Figure 3-1. A 2-D slice of 6-D phase-space. The line is a cross-section of the sheet
of width 6v on which the dark matter particles lie prior to galaxy
formation. The v-i.- .--~ are the peculiar velocities due to density per-
turbations. When overdensities become non-linear, the sheet begins to
wind up clockwise in phase-space, as shown.
10 10-2 10- 100
Figure 3-2. The phase-space distribution of dark matter particles in a galactic
halo at a particular time, t. The horizontal axis is the galactocentric
distance, r, in units of the halo radius, R, and v is the radial velocity.
Spherical symmetry has been assumed for simplicity. Particles lie on
the solid line.
one flow of particles falling out for the first time, one of particles falling out for
the second time, etc. Also, at the locations where the sheet folds, caustics form.
There are two types of caustics that occur within a galactic halo: "outer" and
"inner." Outer caustics form near where a flow of particles falling out of the halo's
gravitational potential turnaround and fall back in. These caustics are topologically
spheres. Inner caustics form where particles falling into the potential reach their
distance of closest approach to the center of the galaxy. When the initial velocity
of infalling particles is dominated by a rotational component, inner caustics are a
"tricusp imi; , whose cross-section is a D_4 catastrophe. The cross-section is
illustrated in Fig. 3-3 and the ring shown in Fig. 3-4. Axial symmetry has been
used in these figures, but is not a neccesary condition for the formation of caustics.
We proceed to review the mathematical arguments for the existence of outer
and inner caustics . Parametrize the particles on the phase-space sheet using
a = (ca, a2, a3). This parametrization may be chosen as convenient. Let x(a; t)
be the physical position of the particle labelled a at time t. At early times, before
galactic evolution becomes non-linear, the mapping a -- x will be one-to-one.
At late times, when the sheet covers physical space multiple times, for any given
physical location r there will be, in general, multiple solutions aj(r, t) with
j = 1, 2, n(r, t), to r = x(a; t). That is, there will be particles with different a
at the same physical location, r. The number of flows at r at time t is n(r, t). The
number density of particles on the sheet is d- It follows that the mass density in
physical space is 
p(r, t) da3(a) dD(at) (3-3)
j =1 = (r,t)
Figure 3-3. The cross-section of the tricusp ring. Each line represents a particle
trajectory. The caustic surface is the envelope of the triangular feature,
inside which four flows are contained. Everywhere outside the caustic
surface, there are only two flows. Illustration courtesy of A. Natarajan.
Figure 3-4. The tricusp ring caustic. Axial symmetry has been used for illustrative
purposes. Illustration courtesy of A. Natarajan.
1, i't l -7.-11 f 1 4 -,
where m is the particle mass and
D(a, t) detO (a ) (3-4)
The magnitude of D is the Jacobian of the map a x. Eq. (3-3) is the sum over
the mass density in each discrete flow at r.
Caustics occur where D = 0 and the map is singular . At these points,
the mapping from phase-space to physical space changes from n-to-one to (n 2)-
to-one. The physical density at the location of caustics becomes very large, as the
phase-space sheet is tangent to velocity space. In the limit of zero initial velocity
dispersion, the dark matter particle density diverges at the location of a caustic. In
reality, these flows will have a small velocity dispersion and thus the caustics will
have a large, but finite, density.
The presence of outer caustics is easily seen from Fig. 3-2. Natarajan and
Sikivie  demonstrated that inner caustics must also be present in the galactic
halo. Consider a continuous flow of cold dark matter particles falling in and out
of a gravitational potential and a spherical surface of radius R surrounding the
potential well. Using the parametrization, a = (0, ), where 0 and Q are the
polar coordinates where a particle falling into the potential crosses the sphere at
time, 7. Then x(0, Q, T; t) gives the particle's position at time, t. Natarajan and
Sikivie demonstrated that
9(x,y, z) Ox Ox x\
D det 9(0, ) x x) (3-5)
vanishes at at least one point inside the sphere at any t. Thus, a caustic is present
within the sphere. Such a caustic is an inner caustic. We review their proof in the
following. The variable t will be suppressed.
For each (0, 0), the time at which a particle within the sphere crossed its
surface lies in the range -Tout(O, 4) < T < T-r(O, ) where nr,(Torut) is the initial
crossing time of particles currently crossing the sphere on the way in(out). The
sphere's center is chosen to lie at the origin, x = 0. The distance from the sphere's
center to a particle's position is
r(0, Q, r) = x(O, ) x(O, 0, r)
r < 0 and
Thus, fr all (0, ) there exists a ( ) such that
Thus, for all (0, 0) there exists a Too(0, 0) such that
r(0, To(0, )) -minr(0,,r) -- rm(0, ) (3-8)
where the minimum is over r for fixed (0, 0). The distance rmi,(0, Q) is the smallest
distance to the origin among all particles labelled (0, 0).
There are two cases to be considered: rm,i(O, Q) / 0 for some (0, 0) and
rmin(0, 4) = 0 for all (0, 0). In the first case,
Or x xo
or -x. -0 (3-9)
OT ,o(eO) r OT O,6,ro(0,6)
for all (0, 0) such that rmi,(0, Q) / 0. The distance r,in(O, Q) has a maximum value
over the sphere S2(0, ). C'! (0, 00) be such that rmin(0o, 0o) = max rmin(O, ).
Or x OX
0 o r o00
Or x Ox
where ao (0o, o, ro(0o, Qo)) and x(ao) / 0. Eqs.
that ao' Ox and o are all perpendicular
are linearly dependent and D(ao) = 0. Thus, xo is
(3-9), (3-10) and (3-11) imply
to xo, i.e. these three vectors
the location of a caustic. As
xo depends on the choice of origin, such a caustic is spatially extended, which is as
expected; caustics are generically surfaces.
In the special case, rni(0, ) = 0 for all (0, ), x(O, To((0, 0)) =0 for all (0, )
and thus, for T near To(O, 0):
x(0, Q, T) (, )(T To(0, 4)) + 0((T To(O, 0))2) (3-12)
v(O,) aX (3-13)
O ,T e,(to(Oe6)
Using the reparametrization, 0' = 0, = and T' = T To(0, Q) and relabelling,
(," (0) 0,T),
x(0, Q, 7) v(8, )r + 0(r2) (314)
D(0,0,T) -v(0,). x2 (3-15)
As D = 0 at T7 0, the origin is the location of a caustic in this special case. In
this case, the caustic has collapsed to a point.
Thus, both inner and outer caustics must be present in a galactic halo.
Discrete flows and caustics are a natural consequence of a cold dark matter
In order to be detectable by a microwave cavity experiment, flows in the
galactic halo must have sufficient density. In this section, we review arguments that
demonstrate that discrete flows are expected to contain a significant fraction of
the local dark matter density. In particular, the flow density is enhanced near the
location of a caustic. Evidence sl.-.-, -1 that the Earth is located near a caustic
The local density of the first few flows was first estimated by Sikivie and
Ipser , for cases both without and with angular momentum. We review their
The initial estimate was calculated for the first flow, i.e., the flow of particles
passing Earth for only the first time. These particles had a maximum galactocen-
tric distance of rm ~ 1 Mpc, which was reached 5 x 109 years ago. The density at
this location is estimated to be the average cosmological dark matter density .I-iv,
pCDM(t0). In the case of no angular momentum, the local density of the first flow
will be the density at rm multiplied by the appropriate geometrical focussing factor,
i.e., (rF/re)2, thus,
pi(r., to) ) CDM (t ~ l0-25g/cm3 (3-16)
When angular momentum is included, not all particles falling into the galaxies will
pass through the center. Defining d as the average distance of closest approach for
particles falling in for the first time, the estimated density is
pi (r, to) ~ PCDM (t) ( )2 ( 2 -g/cm3 (3 17)
The more detailed calculations of Sikivie, Tkachev and Wang  confirm this
estimate and provide estimates of densities of the same order of magnitude for the
other flows. Their calculations show that each of the first eight in and out flows
have densities of the order of ''- of the local halo density (assuming a local dark
matter density of 9.2 x 10-25 g/cm3 ). Thus, these estimates lead us to expect
that flows contain a significant fraction of the local dark matter density.
At the location of a caustic, the dark matter density will be greatly enhanced.
This will be reflected by rising bumps in the galactic rotation curve at these
locations. Fitting the caustic ring model to rises in the Milky Way rotation curve
and to a triangular feature in the IRAS map predicts that the flows falling in
and out for the fifth time contain a significant fraction of the halo density at the
location of our solar system. The predicted densities are 1.7 x 10-24 g/cm3 and
1.5 x 10-25 g/cm3 . The flow of the greatest density is called the "Big Flow."
This flow is predicted to have a velocity dispersion of 53 m/s and velocity of
approximately 300 km/s relative to the Sun. Thus this flow is of particular interest
for axion dark matter detection.
In this section, we discuss evidence for discrete flows and caustics and the
consequences for microwave cavity detection of axion dark matter.
As demonstrated in Section 3.2, discrete flows and caustics are a necessary
consequence of cold dark matter cosmology. It is significant in this regard that
caustics of luminous matter are also believed to exist and have been observed in
bright elliptical galaxies. Malin and Carter first observed ripples in the distribution
of light in these galaxies . Computer simulations demonstrate that when a small
galaxy falls into the fixed gravitational potential of a large elliptical galaxy, the
small galaxy is tidally disrupted and its stars end up on a thin ribbon in phase-
space. These phase-space ribbons are like the phase-space sheets of dark matter
discussed earlier, except for being limited in spatial extent. The folding of these
phase-space ribbons will lead to the observed ripples in the light distribution of
an elliptical galaxy which has swallowed a smaller galaxy [59, 60, 61]. There is no
explanation other than the existence of caustics for the presence of these ripples in
elliptical galaxies. The existence of caustics of visible matter further supports the
expectation that dark matter caustics are present in galactic halos.
While virialization will thermalize the halo and destroy the oldest flows, flows
will be present today from particles which have only lately fallen onto the halo.
These particles will not have had sufficient time to thermalize with the rest of the
Discrete flows are expected to contain a significant fraction of the local halo
density, as discussed in Section 3.3. Discrete flows produce a distinct signal in
an axion detector. A series of narrow peaks, one per flow, will appear in the
spectra output. The width of each peak is proportional to the velocity dispersion
of the corresponding flow. The power in each peak is directly proportional to the
density of axions in the flow. Such narrow peaks have higher signal-to-noise ratio
in a high resolution axion search. Thus, if a significant fraction of the local halo
density consists of axions in such flows, a high resolution axion search increases the
experiment sensitivity to axions. Furthermore, if a signal is found, it will provide
detailed information on the structure of axion dark matter within our galaxy.
HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS
ADMX uses a microwave cavity detector to search for axions in our galactic
halo [33, 62, 63, 64, 65, 66]. In its present search mode, the ADMX detector
spends approximately 50 seconds at each cavity setting. As a result it can look
for features in the axion frequency spectrum with a resolution of order 20 mHz.
This potential has recently been realized by building the HR channel, which
became fully operational in August 2002. It offers the opportunity to improve the
sensitivity of the experiment by searching for the spectral features expected from
the presence of discrete flows of dark matter axions. It has been demonstrated
that the HR channel increases ADMX's sensitivity to an axion signal by a factor of
ADMX can operate its two channels simultaneously. The MR channel searches
for broad signals, with width of order 1 kHz and a Maxwell-Boltzmann energy dis-
tribution. The HR channel searches for narrow signals arising from discrete axion
flows. Each discrete flow produces a peak in the axion signal. The frequency at
which a peak occurs is indicative of the square of the velocity of the corresponding
flow in the laboratory frame. In searching for cold flows of axions, it is assumed
that the flows are steady, i.e., the rates of change of velocity, velocity dispersion
and flow density are slow compared to the time scale of the experiment. The as-
sumption of a steady flow implies that the signal we are searching for is alv-- i
present. Even so, the signal frequency will change over time due to the Earth's
rotation and orbital motion [67, 68]. In addition to a signal frequency shift in
data taken at different times, apparent broadening of the signal occurs because its
frequency shifts while the data are being taken. The HR channel has a frequency
resolution of 0.019 Hz. To conduct a search without making assumptions about
flow velocity dispersions, searches are conducted for peak power spread across
several bins. We refer to the associated sum of power across n single bins as n-bin
searches. These searches are performed for n =1, 2, 4, 8, 64, 512 and 4096.
This chapter is on ADMX's HR channel search . The experiment is
described in Section 4.2. In Section 4.3, the signal expected from a microwave
cavity detector observing a cold flow of axions is discussed. The detector noise
characteristics are analyzed in Section 4.4. Section 4.5 contains details of the
systematic corrections performed on the data. The complete analysis and axion
signal search procedure are in Section 4.6. The HR search has covered the axion
mass range 1.98-2.17 peV. No axion signal was found in this range. Exclusion
limits on the density of axions in local discrete flows, based on this result, are
presented in Section 4.7. A discussion of the results is in Section 4.8.
4.2 Axion Dark Matter eXperiment
The microwave cavity detector utilizes the axion-electromagnetic coupling to
induce resonant conversion of axions to photons. The relevant interaction is
,,ay 9 ,y,a E B (4-1)
where a is the axion field, E and B are the electric and magnetic fields, respec-
tively, and ga,, the axion-electromagnetic field coupling. The coupling depends on
the fine structure constant, c, the axion decay constant, f,, and a model dependent
iafa (4 2)
In the KSVZ model, g, -0.97, whereas in the DFSZ model, g, 0.36. The axion
decay constant is related to its mass by
m e 6x 10-6( 12Gef eV. (4-3)
This coupling allows resonant conversion of axions to photons to be induced in a
microwave cavity permeated by a strong magnetic field .
As axions in the galactic halo are non-relativistic, the energy of any single
axion with velocity, v, is
E = mac2 + mav2 .(4 4)
The axion-to-photon conversion process conserves energy, i.e., an axion of energy,
Ea, converts to a photon of frequency, v = Ea/h. When v falls within the
bandwidth of a cavity mode, the conversion process is resonantly enhanced. The
signal is a peak in the frequency spectrum of the voltage output of the detector.
The power developed in the cavity due to resonant axion-photon conversion is
P ga V amin(Q, a), (4-5)
where V is the cavity volume, Bo is the magnetic field strength, pa is the density
of galactic halo axions at the location of the detector, Qa is the ratio of the energy
of the halo axions to their energy spread, equivalent to a "quality factor" for the
halo axion signal, and C is a mode dependent form factor which is largest in the
fundamental transverse magnetic mode, T.I,,,, C is given by
S Jdx E, Bo 2
C B0Vf d3xE, 2 (4-6)
B VfV Ud3x |E,|2
in which E,(x'. is the time dependent electric field of the mode under considera-
tion, Bo(x) is the static magnetic field in the cavity and c is the dielectric constant
of the medium inside the cavity. The frequ'-ii- -dependent form factor is evaluated
numerically. Eq. (4-5) can be recast in the convenient form,
P -0.5 x 10-21W (V) (B 2 C 2 )
500 L 7 T 0.36 0.5x10-24 g.cm3
x (- ) mini(Q') (4-7)
A schematic of ADMX, showing both the MR and HR channels, is given
in Fig. 4-1. A more detailed illustration of the magnet, cavity and cryogenic
components is shown in Fig. 4-2. The microwave cavity has an inner volume, V,
of 189 L. The frequency of the T.I,,,, mode can be tuned by moving a pair of
rods inside. The rods may be metal or dielectric and can be replaced as necessary
to reach the desired frequency range. The cavity is located in the bore of a
superconducting solenoid, which generates a magnetic field, Bo, of 7.8 T. The
voltage developed across a probe coupled to the electromagnetic field inside the
cavity is passed to the receiver chain. As the experiment operates with the cavity
at critical coupling, half the power developed in the cavity is lost to its walls and
only half is passed to the receiver chain. During operation, the quality factor of
the cavity, Q, is approximately 7 x 104 and the total noise temperature for the
experiment, T,, is conservatively estimated to be 3.7 K, including contributions
from both the cavity and the receiver chain.
The first segment of the receiver chain is common to both the MR and HR
channels. It consists of a cryogenic GaAs HFET amplifier built by NRAO, a crystal
bandpass filter and mixers. At the end of this segment, the signal is centered at 35
kHz, with a 50 kHz span. The MR signal is sampled directly after this part of the
receiver chain. The HR channel contains an additional bandpass filter and mixer,
resulting in a spectrum centered at 5 kHz with a 6 kHz span.
Time traces of the voltage output from the receiver, consisting of 220 data
points, are taken with sampling frequency 20 kHz in the HR channel. This results
in a data stream of 52.4 s in length, corresponding to 0.019 Hz resolution in the
REJECT 10.7MHz MIXER
MIXER #1 IF #2
RFI t 35kHz AF 125Hz BIN
r------ --- ------
1.3K HFET MIXER
5kHz AF 0.02FHF BIN
r u DISK
Figure 4-1. Schematic diagram of the receiver chain.
frequency spectrum. The data were primarily taken in parallel with the operations
of the MR channel over a period beginning in November, 2002 and ending VT ic,
2004. Continuous HR coverage has been obtained and candidate peak elimination
performed for the frequency range 478-525 MHz, corresponding to the axion mass
range 1.98-2.17 peV. Data with Q less than 40 000 and/or cavity temperature
above 5 K were discarded. When this was the case, additional data were taken to
ensure coverage of the full range.
4.3 Axion Signal Properties
The HR channel is used to search for narrow peaks caused by flows of cold
axions through the detector. It is assumed that the flows are steady, i.e., the
rates of change of velocity, velocity dispersion and density of these flows are slow
compared to the time scale of the experiment. The assumption of a steady flow
implies that the signal we are searching for is ahv-w present. Even so, the kinetic
energy term in Eq. (4-4) and the corresponding frequency change over time due to
the Earth's rotational and orbital motions. In addition to a signal frequency shift
TTT Stepping motors
" Cavity LHe reservoir
Magnet LHe reservoir
1.3K J-T refrigerator
Cavity vacuum chamber
Diplpotrir tnnino rnd
Figure 4-2. Sketch of the ADMX detector.
in data taken at different times, apparent broadening of the signal occurs because
its frequency shifts while the data are being taken.
Using Eq. 4-4, one sees that ratio of the shift in frequency, Af, to the base
frequency, f, due to a change in velocity, Av, is
f mavc2 + m(4 8)
f mi2 +!n#,,V2
The velocity of a dark matter flow relative to the Earth will be in the range
100 1000 km/s. We chose v = 600 km/s as a representative value for the purpose
of estimation. A frequency of f = 500 MHz is chosen as typical for the data under
The magnitude of the velocity on the surface of Earth at the equator due to
the Earth's rotation is vR = 0.4 km/s. It is less than this at the location of the
axion detector, but this value is used for the purpose of illustration. Assuming the
extreme case of alignment of the Earth's rotational velocity with the flow velocity,
Av = 2vR. The resulting daily signal modulation is 3 Hz. Approximating the
Earth's orbit as circular, the magnitude of it's velocity with respect to the Sun
is VT = 30 km/s. Again, considering the extreme case of velocity alignment, the
frequency modulation due to the orbit of Earth around the Sun is at most 200 Hz.
The bandwidth of the HR channel is 6 kHz. After identifying candidate
frequencies, they are reexamined to see if they satisfy the criterion of a constantly
present signal. Thus, if the spectrum is centered on the candidate frequency when
it is reexamined, the signal will still be within the detector bandwidth as it will
move at most 200 Hz from its original frequency.
In addition, both the rotation of the Earth and its motion around the Sun
will result in a small change in the flow velocity relative to the detector while
each spectrum is taken and a subsequent increase in the signal line-width relative
to what would be expected in the static case. Similarly to Eq. (4-9), the signal
bN .I.. 1,i,,- 6f, due to a change in the flow velocity, 6v, is
6f f (4-10)
Taking the time of integration to be At 50 s, the change in relative velocity is at
6v 27,, ( ) (4 11)
where T is the period of the motion (diurnal or annual) and v, is the respective
velocity (vO or vT). The line-width is increased by 4 x 10-3 Hz due to the Earth's
rotation. The Earth's orbital motion increases the line-width by 10-3 Hz. The
spectral resolution of the HR channel is 0.019 Hz, large enough to make these
For flows of negligible velocity dispersion, the sensitivity of the experiment
is proportional to the frequency, f, and the time of integration, At, provided
the resolution, B = 1/At, is less than the shift of the signal frequency during
measurement. This requirement allows a measurement integration time as long as
t < 160s 50 (4-12)
This -i,-.- -r that for the data this note is based on, a more sensitive limit could
have been achieved with a longer integration time than the actual 52 s.
The velocity dipersion of the flow may, however, be a limiting factor. While
no value for velocity dispersion is assumed in performing the HR analysis, for
illustrative purposes, let us consider a particular case: the "Big Flow," discussed
by Sikivie . The upper bound on the velocity dispersion of this flow is 6v < 50
m/s. This leads to a maximum line broadening of 6fBF < 8 x 10-2 Hz, i.e.,
a signal from axions in the Big Flow is spread over four frequency bins in the
detector spectrum if the limit 6v < 50 m/s is saturated. Let us emphasize, however,
that there is no reason to believe this bound is saturated.
In general, we do not know the velocity dispersion of the cold axion flows for
which we search. Subsequently, we do not know the signal width. To compensate,
searches are performed at multiple resolutions by combining 0.019 Hz wide bins.
These searches are referred to as n-bin searches, where n = 1, 2, 4, 8, 64, 512
and 4096. For f = 500 MHz and v =600 km/s, the corresponding flow velocity
6nm/s (600 km/s (4 13)
Further details of the n-bin searches are given in Section 4.6.
4.4 Noise Properties
The power output from the HR channel is expressed in units of a, the rms
noise power. This noise power is related to the noise temperature, T,, via
S- kBT, A (414)
where kB is Boltzmann's constant and b is the frequency resolution. The total noise
temperature, T, = Tc + Tel, where Tc is the physical cavity temperature and
Tei is the electronic noise contribution from the receiver chain. As no averaging is
performed in HR sampling, b = I/At. Thus, the rms noise power is
a = kBbT (4-15)
Output power is normalized to a and T, is used to determine this power. Eq. (4
15) has been verified experimentally by allowing the cavity to warm and observing
that a is proportional to Tc.
The noise in the HR channel has an exponential distribution. The noise in
a 1-bin is the sum of independent sine and cosine components, as no averaging is
performed in HR sampling. The energy distribution should be proportional to a
Boltzmann factor, exp(-E/kT), and non-relativistic and classical energies, such
as E = (1/2)mv2 or E = (1/2)kx2, are proportional to squares of the amplitude.
Thus, the noise amplitude, a, for a single component (i.e., sine or cosine) has a
Gaussian probability distribution,
dP 1 / a2 (41)
da = a exp (416
where Ja is the standard deviation.
As there are two components per bin, the addition of n bins is that of 2n
independent contributions. The probability distribution, dP/dpn, of observing noise
power p, in an n-bin is
dP f1 o exp( a) a (1
(- 2n 2 2ndai) W6n k (4-17)
dPn i 1 a (-co a 2n7 k21 2
Evaluating the above expression,
dP n-P l n2) (418)
dn p- p,
dp ( -. exp 418
For n = 1,
dp exp (4-19)
which is indeed a simple exponential, as expected.
Using this noise distribution, we can easily see that the average (rms) noise
power in the one bin search is a = ac2. Substituting this in (4-19), the noise power
distribution function becomes
dP 1 pl
d exp (4-20)
For each HR spectrum, a is determined by plotting the number of frequency
bins, Np, with power between p and p + Ap against p. According to Eq. (4-20),
N Ap (4-21)
Np a C (421)
108 .-- .........08
10 6 104 ........... ... ......... --------........................................................
1 0 2 .. ...................................... ... ..................................................
102J-----'-------'------------ -----i-----i--- ----I------
0 10 20 30 40
Figure 4-3. Power distribution for a large sample of 1-bin data.
where N is the total number of frequencies. As
In N+ln = NA (4-22)
a is the inverse of the slope of the In Np versus p plot. Figure 4-3 demonstrates
that the data is in good agreement with this relation for p less than 20J. The
deviation of the data from Eq. (4-22) for p greater than 20a is due to the fact
that the background is not pure noise, but also contains environmental signals of a
As we combine an increasing number of bins, the noise power probability
distribution approaches a Gaussian, in accordance with the central limit theorem.
The right-hand side of Eq. (4-18) approaches a Gaussian in the limit of large
n. We have examined a large sample of noise in each n-bin search and verified
that it is distributed according to Eq. (4-18). Figures 4-4 through 4-9 show the
0 2 4 6 8 10.
Figure 4-4. Power distribution for a large sample of 2-bin data.
progression from the exponential distribution of Fig. 4-3 to a near Gaussian curve
for the 4096-bin search.
In addition to examining the behavior of the noise statistics, we have per-
formed a cross-calibration between the HR and MR channels. The signal power
of an environmental peak, observed at 480 MHz and shown in Fig. 4-10, was
examined in both the HR and MR channels. The observed HR signal power was
(1.8 0.1) x 10-22 W, where the error quoted is the statistical uncertainty. The MR
channel observed signal power 1.7 x 10-22 W, in agreement with the HR channel.
Note that the MR signal was acquired with a much longer integration time than
that of the HR signal (2000 for MR versus 52 s for HR).
The combination of the calibration of the noise power with cavity temperature,
the consistency between expected and observed noise statistics and the agreement
6 . . . . . . . . . .
5 , , , ,
the consistency between expected and observed noise statistics and the agreement
0 5 10 15
Figure 4-5. Power distribution for a large sample of 4-bin data.
of signal power observed in both the HR and MR channels, makes us confident that
the signal power is accurately determined in the HR channel.
4.5 Removal of Systematic Effects
There are two systematic effects introduced in the receiver chain shown in
Fig. 4 1. Two passband filters are present on the HR receiver chain: one with
bandwidth 35 kHz on the shared MR-HR section and a passive LC filter of
bandwidth 6 kHz, seen by the HR channel only. The combined response of both
these filters has been analyzed and removed from the data. The second systematic
effect is due to the frequency-dependent response of the coupling between the
cavity and the first cryogenic amplifier. This effect is removed using the equivalent
circuit model described later.
The combined passband filter response was determined by taking data with a
white noise source at the rf input of the receiver chain. A total of 872 time traces
5 ..,.. ....
0 5 10 15 20 25
Figure 4-6. Power distribution for a large sample of 8-bin data.
were recorded over a two dv period. In order to achieve a reasonably smooth
calibration curve, 512 bins in the frequency spectrum for each time trace were
averaged giving 9.77 Hz resolution. The combined average of all data is shown
in Fig. 4-11. This measured response was removed from all data used in the HR
search, as follows. The raw power spectra have frequency 0-10 kHz, where the
center frequency of 5 kHz has been mixed down from the cavity frequency. Each
raw power spectrum is cropped to the region 2-8 kHz to remove the frequencies
not within the LC filter bandwidth. Each remaining frequency bin is then weighted
by a factor equal to the receiver chain response at the given frequency divided
by the maximum receiver chain response. Interpolation for frequency points not
specifically included in the calibration curve is performed by assuming that each
point on the calibration curve was representative of 512 bins centered on that
frequency, so all power corresponding to frequencies within that range is normalized
5 .. . . . . . . .. . . .. . .
30 40 50 60 70 80 90 100
Figure 4-7. Power distribution for a large sample of 64-bin data.
by the same factor. As the calibration curve varies slowly with frequency within
the window to which each spectra is cropped, this is an adequate treatment of the
In the MR channel, the effect of the cavi' i i-,plnif. v coupling is described
using an equivalent-circuit model . This model has been adapted for use in the
HR channel. The frequency dependent response of the cavity amplifier coupling
is most evident in the 4096-bin search, thus this is the data used to apply the
equivalent circuit model. A sample spectrum before correction is shown in Fig. 4-
In the equivalent-circuit model, each frequency is given by A, the number
of bins it is offset from the bin of the center frequency, measured in units of the
4096-bin resolution, i.e. b4096 = 78.1 Hz. The equivalent-circuit model predicts
that the power (in units of the rms noise) at the NRAO amplifier output (the point
2 6 .. ....... ....... .................................
... i .. . . . . ... . . . .... .... .
00 450 500 550 600 650
Figure 4-8. Power distribution for a large sample of 512-bin data.
labelled "RF" in Fig. 4-1) in the 4096-bin search at the frequency offset A is
S 8a3+ S A st 2 4
P(A) + 8a3 + 4a4 ( Aa2 (4-23)
1+ 4 ( )a5
where the parameters al through a5 are
a, = (b4096/b)(T + T, + Tv)/T, (4-24)
a2 f /(b4096Q) (4-25)
a3 (b4096/b)(T + Tv + (T T) cos(2kL))/T,, (4-26)
a4 (b4096/b)((T Tv) sin(2kL))/T, and (4-27)
a5 (fo fcen)/b4096 (4-28)
In the above expressions, Tc is the physical temperature of the microwave cavity,
T1 and Tv are the current and voltage noise, respectively, contributed by the
6 0 0 0 .............................. ........ .....
S... . . . . . . . . . . . . . . . . .
1000 ....... .
3%00 3900 4000 4100 4200 4300 4400
Figure 4-9. Power distribution for a large sample of 4096-bin data.
amplifier, T, is the noise temperature contributed from all components, b is the
frequency resolution of the HR channel, i.e. 0.019 Hz, L is the electrical (cable)
length from the cavity to the HFET amplifier, fo is the cavity resonant frequency,
fcen is the center frequency of the spectrum and k is the wavenumber corresponding
to frequency fen + bA. The factor b4096/b appears in the parameters a1, a3 and a4
as it is an overall factor which results from normalizing the power to the single bin
noise baseline. In practice, the parameters al through a5 are established by fitting.
The line in Fig. 4-12 shows the fit obtained using the equivalent circuit model.
Large peaks in the data, e.g. an axion signal or environmental peak, are re-
moved before fitting to prevent bias. The 4096-bin spectrum is used to perform
the fit and then the original 1-bin spectrum is corrected to remove the systematic
effect. The weighting factors are calculated using Eq. (4-23) and the fitted pa-
rameters, al through as, at the center of each bin of width b4096. These factors are
BW= 1.21 Hz
t = 52 sec
479.9966 479.9970 479.9974
An environmental peak as it appears in the MR search (top) and the
64-bin HR search. The unit for the vertical axis is the rms power
fluctuation in each case.
BW= 125 Hz
t =2000 sec
1 --- -
0 2000 4000 6000 8000 10000
FFT frequency (Hz)
Figure 4-11. HR filter response calibration data (512 bin average). The power has
been normalized to the maximum power output.
S. . . . . . . . . . . . . . . .
4 2 0 0 .............................. ..... ................... .................. .O.o.. ..
4 2 0 0 .................... ... ... ..... ...... ..... ...... ..... .... ..
4 0 0 0 .. ...... 0- .. ...
499.805 499.807 499.809 499.811
Figure 4-12. Sample 4096-bin spectrum before correction for the cavity-amplifier
coupling. The line is the fit obtained using the equivalent circuit
4250 .. .
4 1 5 0 ............................ *...................... .. ..... ....
4 ** *. 0*
4 0* _* o
S 6 *
4100 ** *
4 O .805 499.807 499.809 499.811
Figure 4-13. The same 4096-bin spectrum of Fig. 4-12 after correction for the
cavi i,- ,lin l.ift. r coupling.
the ratio of the fit at a given point to the maximum value of the fit. Each 1-bin is
multiplied by the factor calculated for the bin of width b4096 within which it falls.
Figure 4-13 shows the spectrum of Fig. 4-12 after removal of systematic
effects. The removal of the cavi i i:plifi. r coupling and the passband filter
response using the techniques described above results in flat HR spectra.
4.6 Axion Signal Search
We now describe the search for an axion signal and summarize the analysis
performed on each time trace.
The width of an axion signal is determined by the signal frequency, axion
velocity and flow velocity dispersion (Eq. (4-10)), the latter being the most
uncertain variable. n-bin searches, where n is the number of .,lIi ient 1-bins added
together (n = 1, 2, 4, 8, 64, 512 and 4096), are conducted to allow for various
1-bin search: 2 4 5 6 7 8 9 10 11 12 13 14 1 16
2-bin search: 12 3 4 5 6 7 8 910 11 12 13 14 15 16
23 345 67 S89 10 11 1213 1415
4bin search: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
34 56 78910 11121314
8 bin search: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
S5 6 78 9 10 1112
Figure 4-14. Illustration of the addition scheme for the 2, 4 and 8-bin searches.
The numbers correspond to the data points of the 1-bin search. Num-
bers within the same box are bins added together to form a single
datum in the n-bin searches with n > 1.
velocity dispersions. For searches with n > 1, there is an overlap between successive
n-bins such that each n-bin overlaps with the last half of the previous and first half
of the following n-bin. This scheme is illustrated for the 2, 4 and 8-bin searches in
The search for an axion signal is performed by scanning each spectrum for
peaks above a certain threshold. All such peaks are considered candidate axion sig-
nals. The thresholds are set at a level where there is only a small probability that
a pure noise peak will occur and such that the number of frequencies considered as
candidate axion peaks is manageable. The candidate thresholds used were 20, 25,
30, 40, 120, 650 and 4500 a, in increasing order of n.
All time traces are analyzed in the same manner. A fast Fourier transform
is performed and an initial estimate of a is obtained by fitting the 1-bin noise
distribution to Eq. (4-22). Systematic effects are then removed, i.e. the corrections
described in Section 4.5 for the filter passband response and cavity-amplifier
coupling are performed. "L ,;, peaks not included in the equivalent circuit model
fit for the cavity-amplifier response are defined to be those greater than 12 1' of
the search threshold for each n-bin search. After the removal of systematic effects,
the 1-bin noise distribution is again fitted to Eq. (4-22) to obtain the true value of
a and the search for peaks above the thresholds takes place.
The axion mass is not known, requiring that a range of frequencies must
be examined. Full HR coverage has been obtained for the region 478-525 MHz,
corresponding to axion masses between 1.98 and 2.17 peV. The selected frequency
range is examined in three stages for axion peaks, as follows:
Stage 1: Data for the entire selected frequency range is taken. The frequency
step between successive spectra is approximately 1 kHz, i.e. the center frequency of
each spectrum differs from the previous spectrum by 1 kHz. Frequencies at which
candidate axion peaks occur are recorded for further examination during stage 2.
Stage 2: Multiple time traces are taken at each candidate frequency from stage
1. The steady flow assumption means that a peak will appear in spectra taken with
center frequency equal to the candidate frequency from stage 1 if such a peak is
an axion signal. The frequencies of persistent peaks, i.e. peaks that appear during
both stage 1 and 2 are examined further in stage 3.
Stage 3: Frequencies of persistent peaks undergo a three-part examination.
The first step is to repeat stage 2, to ensure the peaks still persist. Secondly, the
warm port attenuator is removed from the cavity and multiple time traces taken.
If the peak is due to external radio signals entering the cavity (an environmental
peak), the signal power will increase dramatically. If the signal originates in the
cavity due to axion-photon conversion, the power developed in the cavity will
remain the same as that for the normal configuration. The third step is to use an
external antenna probe as a further confirmation that the signal is environmental.
Some difficulties were encountered with the antenna probe, due to polarization of
environmental signals. However, the second step is adequate to confirm that peaks
are environmental. If a persistent peak is determined to not be environmental, a
final test will confirm that it is an axion signal. The power in such a signal must
grow proportionally with the square of the magnetic field (Bo in Eq. (4-7)) and
disappear when the magnetic field is switched off.
No axion peaks were found in the range 478-525 MHz using this approach.
The exclusion limit calculated from this data is discussed in the following section.
Over the frequency range 478-525 MHz, we derive an upper limit on the
density of individual flows of axion dark matter as a function of the velocity
dispersion of the flow. The corresponding axion mass range is 1.97-2.17 /eV. Each
n-bin search places an upper limit on the density of a flow with maximum velocity
dispersion, 6v,, as given by Eq. (4-13).
Several factors reduce the power developed in an axion peak from that given
in Eq. (4-7). The experiment is operated near critical coupling of the cavity to
the preamplifier, so that half this power is observed when the cavity resonance
frequency, fo, is precisely tuned to the axion energy. If fo is not at the center of a
1-bin, the power is spread into .,1i ,i:ent bins, as discussed below. When the axion
energy is off-resonance, but still within the cavity bandwidth at a frequency f, the
Lorentzian cavity response reduces the power developed by an additional factor of
To be conservative, we calculate the limits at points where successive spectra
overlap, i.e. at the frequency offset from fo that minimizes h(f).
If a narrow axion peak falls at the center of a 1-bin, all power is deposited
in that 1-bin. However, if such a peak does not fall at the center of a 1-bin, the
power will be spread over several 1-bins. We now calculate the minimum power in a
single n-bin caused by a randomly situated, infinitely narrow axion line. The data
recorded is the voltage output from the cavity as a function of time. The voltage
as a function of frequency is obtained by Fourier transformation and then squared
to obtain a raw "pc.v, i spectrum. The actual power is obtained by comparison to
the rms noise power. The data are sampled for a finite amount of time and thus,
the Fourier transformation of the output, F(f), will be of the voltage multiplied by
a windowing function, i.e.,
F(f) v(t)w(t) exp(i27rft)dt (4-30)
where v(t) is the measured output voltage and w(t) is the windowing function for a
sampling period T,
w(t) 1 if T/2 < t < T/2 ,
w t) = (4-31)
Eq. (4-30) is equivalent to
F(f) V(k)W(f k)dk, (4-32)
where V(f) and W(f) are the Fourier transforms of the output voltage, v(t), and
the windowing function, w(t), i.e., F(f) is the convolution of V(f) and W(f),
W(f) sin(fT) (433)
W(f)= (4 33)
Discretizing Eq. (4-32) and inserting Eq. (4-33), we have
1 sin((f (m + 1+
F(f) V((m + -)b) b s( ( ( 434)
mO 2 'r(- (m +))
where b is the frequency resolution of the HR channel, 2N points are taken in the
original time trace, and the center frequency of the jth 1-bin is (j + 1/2)b. Thus,
for an axion signal of frequency f falling in 1-bin j, a fraction of the power,
g(m) sin(mr + 5 ) (4-35)
Sm7 +6 )
is lost to the mth 1-bin from 1-bin j, where 6 = 7(m + 1/2 f/b). If 6 = 0, i.e, the
axion signal frequency is exactly equal to a 1-bin center frequency, all the power is
deposited in a single 1-bin. However, if this is not the case, power is lost to other
1-bins. In setting limits, we assume that the power loss is maximal.
The maximum power loss occurs when a signal in the 1-bin search falls exactly
between the center frequency of two .,-li i:ent 1-bins. In this case, when 6 = 7/2,
Eq. (4-35) shows that 40.5'. of the power will be deposited in each of two 1-bins.
In n-bin seaches with n > 2, not as much power is lost to other n-bins, due to the
overlap between successive n-bins. The minimum power deposited in an n-bin is
81 for n = 2, ,7', for n = 4 and 9 ;', for n = 8. For n = 64, 512 and 4096, the
amount of power not deposited in a single n-bin is negligible.
For the n-bin searches with n = 64, 512 and 4096, a background noise sub-
traction was performed which will lead to exclusion limits at the 97.7'. confidence
level. These limits are derived using the power at which the sum of the signal
power and background noise power have a 97.7'. probability to exceed the candi-
date thresholds. We call this power the "effectli, threshold for each search. The
effective thresholds are obtained by integrating the noise probability distribution,
Eq. (4-18), numerically solving for the background noise power corresponding to
the 97.7'. confidence level for each n and subtracting these values from the original
candidate thresholds. For n = 64, 512 and 4096, the effective thresholds are 71,
182 and 531 a, respectively. For smaller values of n, background noise subtraction
does not significantly improve the limits and the effective threshold was taken to
be the candidate threshold. Table 4-1 summarizes this information and shows the
frequency resolution of each search with the corresponding maximum flow velocity
dispersion from Eq. (4-13) for v = 600 km/s.
Our exclusion limits were calculated for an axion signal with power above the
effective threshold reduced by the appropriate factors. These factors arise from the
critical coupling, the Lorentzian cavity response and the maximum power loss due
to the peak not falling in the center of an n-bin, as outlined above. Equations (4-7)
and (4-15) were used, for both KSVZ and DFSZ axion couplings. The cavity
volume, V, is 189 L. Measured values of the quality factor, Q, the magnetic field,
Table 4-1. Effective power thresholds for all n-bin searches, with the frequency
resolutions, bn and corresponding maximum flow velocity dispersions,
6v,, for a flow velocity of 600 km/s.
n Effective b, 6v,
threshold (a) (Hz) (m/s)
1 20 0.019 6
2 25 0.038 10
4 30 0.076 20
8 40 0.15 50
64 71 1.2 400
512 182 9.8 3000
4096 531 78 20000
Table 4-2. Numerically calculated values of the form factor,
temperatures, Te1, from NRAO specifications.
Frequency (\!I.:) C Tei (K)
450 0.43 1.9
475 0.42 1.9
500 0.41 1.9
520 0.38 1.9
550 0.36 2.0
C, and amplifier noise
Bo, and the cavity temperature, Tc, are recorded in each data file. Numerically
determined values of the form factor, C are given in Table 4-2. The electronic
noise temperature, Te1, was conservatively taken from the specifications of the
NRAO amplifier, the dominant source of noise in the receiver chain, although our
measurements indicate that Te1 is less than specified. These values are also given in
Table 4-2. Linear interpolation between values at the frequencies specified was used
to obtain values of C and Te1 at all frequencies.
The 2-bin search density exclusion limit obtained using these values is shown
in Fig. 4-15. For values of n other than n = 2, the exclusion limits differ by only
constant factors. The constant factors are 1.60, 1.00, 1.12, 1.39, 2.53, 5.90 and 17.2
for n = 1, 2, 4, 8, 64, 512 and 4096, respectively.
2.00 2.05 2.10 2.15
axion mass (geV)
97.7'. confidence level limits for the HR 2-bin search on the density of
any local axion dark matter flow as a function of axion mass, for the
DFSZ and KSVZ a77 coupling strengths. Also shown is the previous
ADMX limit using the MR channel. The HR limits assume that the
flow velocity dispersion is less than 6v2 given by Eq. (4-13).
We have obtained exclusion limits on the density in local flows of cold axions
over a wide range of velocity dispersions. The most stringent limit, shown in
Fig. 4-15, is from the 2-bin search. For a flow velocity of 600 km/s relative to the
detector, the 2-bin search corresponds to a maximum flow velocity dispersion of 10
m/s. The 1-bin search limit is less general, in that the corresponding flow velocity
dispersion is half that of the 2-bin limit. It is also less stringent; much more power
may be lost due to a signal occurring .1 -li,- from the center of a bin than in the
n = 2 case. For n > 2, the limits are more general, but the larger power threshold
of the searches make them less stringent.
The largest flow predicted by the caustic ring model has density 1.7 x
10-24 g/cm3 (0.95 GeV/cm3), velocity of approximately 300 km/s relative to the
detector, and velocity dispersion less than 53 m/s . Using Eq. (4-13) with
Table 4-1 and the information di-1 i'-, '1 in Fig. 4-15 multiplied by the appropriate
factors of 1.12 to obtain the 4-bin limit, it can be seen that the 4-bin search,
corresponding to maximum velocity 50 m/s for v = 300 km/s, would detect this
flow if it consisted of KSVZ axions. For DFSZ axions, this flow would be detected
for approximately half the search range. These limits and the Big Flow density are
illustrated in Fig. 4-16.
Figure 4-15 demonstrates that the high resolution analysis improves the
detection capabilities of ADMX when a significant fraction of the local dark matter
density is due to flows from the incomplete thermalization of matter that has only
recently fallen onto the halo. The addition of this channel to ADMX provides an
improvement of a factor of three over our previous medium resolution analysis.
It is possible that an even more sensitive limit could have been achieved with a
longer integration time, as discussed in Section 4.3. This issue should be considered
--- Big Flow
-r --------- --
97.7'. confidence level limits for the HR 4-bin search on the density
of any local axion dark matter flow as a function of axion mass, for
DFSZ and KSVZ a77 coupling strengths. Densities above the lines
are excluded. For comparison, the predicted density of the Big Flow is
also shown. The HR limits assume that the flow velocity dispersion is
less than 6v2 given by Eq. (4-13).
Axion Mass (peV)
at the beginning of future data runs in order to maximize the discovery potential of
the HR channel.
SUMMARY AND CONCLUSION
This work demonstrates that the new, high resolution channel of the Axion
Dark Matter eXperiment improves its sensitivity for axion detection by a factor of
three, provided a large fraction of the local density is in a single cold flow.
Axions present an interesting candidate for the cold dark matter component
of the universe's energy density. The original motivation for the axion was to solve
the strong CP problem of the standard model of particle physics. The axion is
the pseudo-Nambu-Goldstone boson associated with breaking the Peccei-Quinn
symmetry, implemented to solve the strong CP problem. It was later realized that
the axion was also a good particle candidate for dark matter. The Peccei-Quinn
symmetry breaking scale is the parameter which governs the properties of the
axion and is inversely proportional to the axion mass and couplings. The axion
mass is constrained to lie between 10-6 10-2 eV, by cosmological and astrophysical
processes. Thus, the axion parameter space is bounded and we know in which
range to search for the axion.
While the axion has very small couplings, it is possible to search for them by
utilizing the axion-electromagnetic coupling. The Axion Dark Matter eXperiment
(ADMX) uses a tunable microwave cavity detector to search for axions. When the
magnetic field inside the cavity is tuned to the axion energy, resonant conversion
of axions to photons will occur, which can be observed as a voltage peak in the
output of the detector.
A new, high resolution channel has recently been added to the ADMX
detector. This channel was designed to improve detector sensitivity by teaching for
axions in a specific form of halo substructure: discrete flows. The original, medium
resolution channel searches for axions in a thermalized component of the Milky
Way halo. These axions have a Maxwellian velocity distribution. Axions in discrete
flows have a small velocity dispersion, resulting in a narrow peak in the spectrum
output by the cavity detector. The high resolution channel can search for these
peaks with a high signal-to-noise ratio, improving detector sensitivity.
Discrete flows are expected to be present in the halo from tidal disruption of
dwarf galaxies and from late infall of dark matter into the gravitational potential.
Dark matter which has only recently fallen into the potential will not have had
sufficient time to thermalize with the rest of the halo. Examining the phase-space
structure of such particles shows that discrete flows will occur due to this late
The first analysis for this channel has been successfully completed. After
analysis of the noise background and removal of systematic effects, no axion
signal was found in the mass range 1.97-2.17 peV. A broad range of flow velocity
dispersions was considered by searching for signals across multiple bins by adding
.,ili .ient bins together. The new exclusion limits obtained from the high resolution
channel increase the sensitivity of the ADMX detector by up to a factor of three
over the previous medium resolution result. The high resolution channel thus
enhances ADMX's detection ability. Should an axion signal be found, the high
resolution channel will also yield valuable information about the phase-space
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Leanne Delma Duffy was born on August 3, 1975, in Sydney, Australia. In
1980, her family moved to the Gold Coast, Australia, where she attended Miami
State Primary School and Merrimac State High School. After graduating from
high school, she moved to Brisbane, Australia, where she obtained a Bachelor
of Science with honours in physics from The University of Queensland in 1997.
She spent two years working for an environmental consulting firm, Pacific Air &
Environment, Pty Ltd, and then moved to Gainesville, Florida, to study for a
Doctor of Philosophy in physics.