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HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS IN MILKY WAY HALO SUBSTRUCTURE By 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 2006 ACKNOWLEDGMENTS 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 school experience. 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 Ethan Siegel. TABLE OF CONTENTS page ACKNOW LEDGMENTS ............................. ii LIST OF TABLES ................................. v LIST OF FIGURES ..................... ......... vi ABSTRACT ................... .............. viii CHAPTER 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 PecceiQuinn 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 Table page 41 Effective power thresholds for all nbin searches, with the frequency res olutions, b, and corresponding maximum flow velocity dispersions, 6v', for a flow velocity of 600 km/s. ............. .... 67 42 Numerically calculated values of the form factor, C, and amplifier noise temperatures, T1, from NRAO specifications. .. . ..... 67 LIST OF FIGURES Figure page 31 A 2D slice of 6D phasespace. The line is a crosssection 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 nonlinear, the sheet begins to wind up clockwise in phasespace, as shown. .................... .. 30 32 The phasespace 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 33 The crosssection 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 34 The tricusp ring caustic. Axial symmetry has been used for illustrative purposes. Illustration courtesy of A. Natarajan. ........... .. 34 41 Schematic diagram of the receiver chain. ................ 45 42 Sketch of the ADMX detector. .................. ..... 46 43 Power distribution for a large sample of 1bin data. .......... ..51 44 Power distribution for a large sample of 2bin data. .......... ..52 45 Power distribution for a large sample of 4bin data. .......... ..53 46 Power distribution for a large sample of 8bin data. .......... ..54 47 Power distribution for a large sample of 64bin data. ......... ..55 48 Power distribution for a large sample of 512bin data ........... ..56 49 Power distribution for a large sample of 4096bin data. . .... 57 410 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. . . . . .. . .. .. .58 411 HR filter response calibration data (512 bin average). The power has been normalized to the maximum power output. ............. ..59 412 Sample 4096bin spectrum before correction for the cavity v ii1.1 ifr v cou pling. The line is the fit obtained using the equivalent circuit model. 60 413 The same 4096bin spectrum of Fig. 412 after correction for the cavity amplifier coupling. .................. .. ...... 61 414 Illustration of the addition scheme for the 2, 4 and 8bin searches. The numbers correspond to the data points of the 1bin search. Numbers within the same box are bins added together to form a single datum in the n bin searches with n > 1. .................. .... 62 415 97.7'. confidence level limits for the HR 2bin 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. (413). ............ ..68 416 97.7'. confidence level limits for the HR 4bin 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. (413). . . . . . .. . . 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 By Leanne Delma Duffy August 2006 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 1.982.17 peV. We place upper limits on the density of axions in local discrete flows based on this result. CHAPTER 1 INTRODUCTION 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 [3]. 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 system. 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. [4]. 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 longrange interactions are gravitational, and (2) The dark matter must be cold; i.e., it must be nonrelativistic 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 galaxies. 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 weaklyinteracting massive particles (WIMPs). The axion is the pseudoNambuGoldstone boson from the PecceiQuinn solution to the strong CP problem [5, 6, 7, 8]. The axion mass, ma, is constrained to lie in the range 106 to 102 eV [9, 10, 11]. There are two benchmark axion models that are minimal extensions of the standard model: the KimShifman VainshteinZakharov (KSVZ) [12, 13] model and the DineFischlerSrednicki 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 PecceiQuinn 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 weakscale 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, ..ietundetected 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 gravitational interactions. 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 [36] 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, antiprotons and gammarays. 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 Nbody 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 MaxwellBoltzmann velocity distribution. ADMX's medium resolution (i lR) channel [34] searches for such axions, assuming that the velocity dispersion is 0(103c) or less, where c is the velocity of light. (The escape velocity from our galaxy is approximately 2 x 103c.) 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 Nbody simulations and with late infall of dark matter onto the galactic halo. 5 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]. Nonthermalized 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 1017(105eV/mT)(to/t)2/3 [40], 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 caustic. The caustic ring model predicts that the Earth is located near a caustic feature [45]. 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 1024 g/cm3 and 1.5 x 1025 g/cm3 [45], comparable to the local dark matter density of 9.2 x 1025 g/cm3 predicted by Gates et al. [46]. 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 signaltonoise 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 PecceiQuinn 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. CHAPTER 2 AXIONS 2.1 Introduction The axion is the pseudoNambuGoldstone boson implied by the PecceiQuinn 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 PecceiQuinn solution to this problem is outlined, using the original PecceiQuinn WeinbergWilczek 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 .i1 i1, ,!ii place on the axion mass. 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 standalone 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, lefthanded 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""" (21) j=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 righthanded 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 Da D,, = (, +gA a ) (23) where the A4 are the GellMann matrices. The final term of Eq. (21) 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 (24) 2 (24) with d"P", the LeviCivita 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 nonAbelian. NonAbelian 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. (25) 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 AdlerBellJackiw 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 (26) mi C eiim (27) N 0 0 aj. (28) i= 1 While the physics remains the same, this is not a symmetry because the parameter 0 has changed. The transformations of Eq. (26) through Eq. (28) can be used to move phases between the quark masses and 0. However, the quantity, S0 arg(mlm2...mvN) (29) is invariant and therefore observable, unlike 0. This is commonly written as 0 0 arg det M (210) where M is quark mass matrix. The 0term 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 0term 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 0term 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 0term results in a neutron electric dipole moment of [9, 49, 10, 11], dn ~ 1016 ecm, (211) where e is the electric charge. The current experimental limit is [50] Sdn, < 6.3 x 1026 cm (212) thus 81 < 109. 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 PecceiQuinn (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 pseudoscalar meson masses. The NelsonBarr 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 2.3.1 Introduction 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 PecceiQuinnWeinbergWilczek 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 axionelectromagnetic 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 PecceiQuinn Solution to the Strong CP Problem The PecceiQuinn 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 pseudoNambuGoldstone 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, PecceiQuinnWeinbergWilczek (PQWW) axion model an additional Higgs doublet was introduced. We review this model to demonstrate the PecceiQuinn 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 downtype quarks. We distinguish between the up and downtype 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 uptype quarks and downtype 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. (213) Peccei and Quinn chose the Higgs potential to be V (,1 ,d) 2 t 2, 7i, t, > (214) ij i,3 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 0term, has the following global invariance, UpQ(1): SC i2au (215) Se d (216) ui ei"Y5Ui (217) di eid sd (218) 0 0N(a+ ad) (219) 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 vevs: ( 0) SvueiPu/v diPdv . VdCe (220) (221) One linear combination of the NambuGoldstone fields, P, and Pd, is the longi tudinal component of the Zboson, 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. (220), (221), (224) and (225) in Eq. (213), 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. (217), (218) and (219). Direct couplings between the axion and quarks will still remain in the Lagrangian, (222) Thus, (223) (224) (225) (226) through the associated change in the quark kinetic term. The resulting change in 0 is S N(v,/vd + vd/v,)a/v (227) where v = v + v. The axion field can be redefined to absorb 0 on the right hand side of Eq. (227). Defining 2v VPQ = (228) Vu/Vd + dVu, the 0term of Eq. (21) is replaced by 2 La G G""" (229) 167 2VpQ P Nonperturbative QCD effects explicitly break the PecceiQuinn symmetry, but do not become important until the universe cools to the quarkhadron 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 subsection. 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 DineFischlerSrednickiZhitnitsky (DFSZ) and KimShifmanVainshtein 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 fields. 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 lowenergy 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 quarkhadron transition. At the scale, A, the quark condensate acquires expectation value (q<9 j)o A3U() (230) 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 f2 L, = Tr(0,UtaPU) + A3Tr(mU + h.c.) (232) where mq is the diagonal quark mass matrix, m, = (7 0 (233) 0 rrmd Expansion of the Lagrangian shows the pion mass to be 2 3 Md+ (3d m = 2A3 (234) To find the axion mass, we also need to introduce the wouldbe 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' pseudoscalar mesons). The expectation value of the quark condensate becomes (qqLu)o= A3U(r) exp f) (235) L \J 1 A 17 where f, is the T] decay constant. The effective Lagrangian is 1 f,2 L7,n ,T989" + Tr(, Ut8WU) +A3Tr(mqUexp ,) + h.c.) + m, cs (236) where the final term is the potential for rotations in the UA(1) direction. When the 0term is included, the expectation value of the condensate remains that given in Eq. (235), except that is replaced by + 2. Indeed, under a UA(1) transformation, qj eg q, (237) to rriq e 2Tmq (2 38) f + (239) 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, (240) f 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 = (241) N The effective Lagrangian, including the 0term and the axion field, is 1 1 f2 7na 2 0,aaa + 2989a + fTr(aUtaPU) +A3Tr(,qU exp (f) + h.c.) + mi os 2 cos 0 + (242) 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 alvi a dependence on 0. The eta, neutral pion and axion fields mix, as they all have the same quantum numbers. Firstly, consider rfa mixing. The physical rl field is Naf, lphys = r + f (243) 2va and we use the redefinition Nf a' = a (244) 2va 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 (246) md + m,2f f( ) Smd m f, o, (f aphys =a + 0 (247) md+m, 2fa f j with corresponding masses mo A md+ o (248) m mumd o+ Of (2 49) a f2i(m, +md) S( (2o50) expressed as [9, 49, 10, 11] (10 12GeV (251) ma 6 x 106 eV (012Ge (25 1) \ Ja 2.3.4 The Axion Electromagnetic Coupling Axion detection is based on its electromagnetic coupling [36]. 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' (252) 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 (253) f where pf is the PQ charge of a righthanded quark field. Using the definition of the physical axion field given in Eqs. (244) and (247), the resulting axion coupling to two photons is r,7 = g7 Faa FL"V p (254) 47 fa where 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 grandunifiable. 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 [36] 1 ac 1 1 a2 L = F, + g FgF + ,a8l"a m+a2 2 + O (256) 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. (256) can be written as 1 1 1 22 a = (cE2 B 2) + aaaa a _m g E B (257) 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 detector is = ( VB min(Q,Qa) (258) \7jfa/ ma 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 modedependent 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 timedependent 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 1021W (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 106 104 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 longrange 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 alvb 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 nonperturbative QCD effects, becomes significant. At high temperatures, the latter effects are not significant and the axion mass is negligible [53]. 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 nonperturbative 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 [54] 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 [29], 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. Nonperturbative 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 nonrelativistic 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, (261) 4 When the universe cools to a temperature TpQ ~ Va, Q acquires a vacuum expectation value, () = a exp(ia(x)) (262) The axion field is related to a(x), the phase of the scalar field, O(x), by a(x) vaa(x) (263) At T ~ A, nonperturbative QCD effects give the axion a mass. They produce an effective potential 2 V() = m (T) (1 cos) (264) where [53] ma(T) O .lma A .1D (265) The minimum of the potential occurs at (x) = Na(x)= 0. (266) 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. (263), the effective Lagrangian is 2 '2 \ a ) (267) In a FriedmannRobertsonWalker universe, the equation of motion is a + 3H(t)a V2 + (T(t)) sin(Na) 0 (268) R2(t) Near the potential minima, 1 222 V(a) mava2 (269) 2 and thus, sin(Na) Na (270) 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 (271) 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 solution is a = al + a2t (272) 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 [55] 3 ma(Ti(ti)) 3H(TI(ti)) = (273) 2t1 As the axion field can realign only as fast as causality permits, the correspond ing momentum of a quantum of the axion field is 1 pa(ti) ~ 109eV (274) for t1 ~ 2 x 10' s, i.e. the age of the universe at which the quarkhadron transition occurs. As discussed below, the axion mass is restricted to the range 106 102 eV and thus this population is nonrelativistic 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 2 p T +mt (275) By the Virial theorem, (2) =m2(a2) = (276) As axions are nonrelativistic and decoupled, mra(t) p oc M (277) R3(t) 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 (278) 12 / ) ( )2 (279) f2 jr na(tI) ( JN (279) The energy density in axions t.lv is m,(to) S31(tl) po = (2S80) m (ti) S3(to) 1 f2 1 fS31(t> 1\2 f (281) 2 tJ S3 (to) where S(t) is the scale factor at time, t. Eq. (281) implies the axion energy density, ~0.15 ( v) (282) 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 PecceiQuinn decay constant, fa, by Eq. (251) 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, > 106 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 102 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 102 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,~ ii I1 processes, such as the lifetime of red giants forbid axion masses in higher ranges [11]. CHAPTER 3 DISCRETE FLOWS AND CAUSTICS IN THE GALACTIC HALO 3.1 Introduction 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 axions. 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. 3.2 Existence 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 matter. (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 [56] 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)\ (3t \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 ~ 1012 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 PecceiQuinn (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,  ^ 1017 x (32) 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. (32). 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 phasespace 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 3dimensional (3D) sheet in 6D phasespace, as illustrated in Fig. 31. 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 nonlinear, the 3D sheet will begin to 'iiLd up" clockwise in phasespace. 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 phasespace particle distribution will look as shown in Fig. 32. As particles fall into a grav itational potential, there will be a number of discrete flows present at each point at any time [43]. There will be one flow of particles falling in for for the first time, 30 A * Figure 31. A 2D slice of 6D phasespace. The line is a crosssection of the sheet of width 6v on which the dark matter particles lie prior to galaxy formation. The vi. .~ are the peculiar velocities due to density per turbations. When overdensities become nonlinear, the sheet begins to wind up clockwise in phasespace, as shown. 31 4 S2 10 3 4 10 102 10 100 r/R Figure 32. The phasespace 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; [40], whose crosssection is a D_4 catastrophe. The crosssection is illustrated in Fig. 33 and the ring shown in Fig. 34. 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 [44]. Parametrize the particles on the phasespace 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 nonlinear, the mapping a  x will be onetoone. 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 [40] n(r,t) 3 p(r, t) da3(a) dD(at) (33) 1 (r,t) j =1 = (r,t) 33 S*.1 I. Figure 33. The crosssection 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. 0.002  z 0 0.002  tr .04 094 Figure 34. 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 ) (34) The magnitude of D is the Jacobian of the map a x. Eq. (33) is the sum over the mass density in each discrete flow at r. Caustics occur where D = 0 and the map is singular [39]. At these points, the mapping from phasespace to physical space changes from ntoone to (n 2) toone. The physical density at the location of caustics becomes very large, as the phasespace 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. 32. Natarajan and Sikivie [44] 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) (35) 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 < Tr(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) and ar 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 (36) >0. (37) r(0, To(0, )) minr(0,,r)  rm(0, ) (38) 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 (39) 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, ). Then Or x OX 0 o r o00 and 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 (310) (311) (39), (310) and (311) 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) (312) where 89x v(O,) aX (313) 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) Hence, D(0,0,T) v(0,). x2 (315) 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 cosmology. 3.3 Densities 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 feature. The local density of the first few flows was first estimated by Sikivie and Ipser [43], for cases both without and with angular momentum. We review their estimates below. 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 .Iiv, 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 ~ l025g/cm3 (316) 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 [57] 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 1025 g/cm3 [46]). 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 1024 g/cm3 and 1.5 x 1025 g/cm3 [45]. 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. 3.4 Discussion 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 [58]. 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 phasespace ribbons are like the phasespace sheets of dark matter discussed earlier, except for being limited in spatial extent. The folding of these phasespace 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 halo. 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 signaltonoise 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. CHAPTER 4 HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS 4.1 Introduction 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 three [35]. ADMX can operate its two channels simultaneously. The MR channel searches for broad signals, with width of order 1 kHz and a MaxwellBoltzmann 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 nbin searches. These searches are performed for n =1, 2, 4, 8, 64, 512 and 4096. This chapter is on ADMX's HR channel search [69]. 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 1.982.17 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 axionelectromagnetic coupling to induce resonant conversion of axions to photons. The relevant interaction is ,,ay 9 ,y,a E B (41) where a is the axion field, E and B are the electric and magnetic fields, respec tively, and ga,, the axionelectromagnetic field coupling. The coupling depends on the fine structure constant, c, the axion decay constant, f,, and a model dependent factor, g,: iafa (4 2) 9 /fo 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 106( 12Gef eV. (43) This coupling allows resonant conversion of axions to photons to be induced in a microwave cavity permeated by a strong magnetic field [36]. As axions in the galactic halo are nonrelativistic, the energy of any single axion with velocity, v, is E = mac2 + mav2 .(4 4) 2 The axiontophoton 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 axionphoton conversion is [36] 2 VBopaC P ga V amin(Q, a), (45) ma 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 (46) 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. (45) can be recast in the convenient form, P 0.5 x 1021W (V) (B 2 C 2 ) 500 L 7 T 0.36 0.5x1024 g.cm3 x ( ) mini(Q') (47) A schematic of ADMX, showing both the MR and HR channels, is given in Fig. 41. A more detailed illustration of the magnet, cavity and cryogenic components is shown in Fig. 42. 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 IMAGE REJECT 10.7MHz MIXER MIXER #1 IF #2 RFI t 35kHz AF 125Hz BIN FFT r   1.3K HFET MIXER AMPLIFIER #3 5kHz AF 0.02FHF BIN r u DISK Bo Sf MAGNET CAVITY & TUNING RODS Figure 41. 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 478525 MHz, corresponding to the axion mass range 1.982.17 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 ahvw present. Even so, the kinetic energy term in Eq. (44) and the corresponding frequency change over time due to the Earth's rotational and orbital motions. In addition to a signal frequency shift Stepping motors TTT Stepping motors Cryostat vessel " Cavity LHe reservoir Magnet LHe reservoir 1.3K JT refrigerator Cavity vacuum chamber Amplifiers Tuning mechanism Microwave Cavity Diplpotrir tnnino rnd Figure 42. 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. 44, 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 consideration. 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 linewidth relative to what would be expected in the static case. Similarly to Eq. (49), the signal bN .I.. 1,i,, 6f, due to a change in the flow velocity, 6v, is 6f f (410) c2 Taking the time of integration to be At 50 s, the change in relative velocity is at most 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 linewidth is increased by 4 x 103 Hz due to the Earth's rotation. The Earth's orbital motion increases the linewidth by 103 Hz. The spectral resolution of the HR channel is 0.019 Hz, large enough to make these effects negligible. 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 (412) 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 [45]. 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 102 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 nbin searches, where n = 1, 2, 4, 8, 64, 512 and 4096. For f = 500 MHz and v =600 km/s, the corresponding flow velocity dispersions are 6nm/s (600 km/s (4 13) Further details of the nbin 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 (415) 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 1bin 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 50 Boltzmann factor, exp(E/kT), and nonrelativistic 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 nbin is dP f1 o exp( a) a (1 ( 2n 2 2ndai) W6n k (417) dPn i 1 a (co a 2n7 k21 2 Evaluating the above expression, dP nP l n2) (418) dn p p, dp ( . exp 418 For n = 1, dP 1 dp exp (419) 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 (419), the noise power distribution function becomes dP 1 pl d exp (420) dpi a 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. (420), N Ap (421) Np a C (421) 51 108 . .........08 C% 10 6 104 ........... ... ......... ........................................................ 102 1 0 2 .. ...................................... ... .................................................. 102J'' ii I 100 , 0 10 20 30 40 Power (c) Figure 43. Power distribution for a large sample of 1bin data. where N is the total number of frequencies. As In N+ln = NA (422) a is the inverse of the slope of the In Np versus p plot. Figure 43 demonstrates that the data is in good agreement with this relation for p less than 20J. The deviation of the data from Eq. (422) for p greater than 20a is due to the fact that the background is not pure noise, but also contains environmental signals of a nonstatistical nature. As we combine an increasing number of bins, the noise power probability distribution approaches a Gaussian, in accordance with the central limit theorem. The righthand side of Eq. (418) approaches a Gaussian in the limit of large n. We have examined a large sample of noise in each nbin search and verified that it is distributed according to Eq. (418). Figures 44 through 49 show the 52 x 108 * 6 5 0 2 4 6 8 10. Power (T) Figure 44. Power distribution for a large sample of 2bin data. progression from the exponential distribution of Fig. 43 to a near Gaussian curve for the 4096bin search. In addition to examining the behavior of the noise statistics, we have per formed a crosscalibration between the HR and MR channels. The signal power of an environmental peak, observed at 480 MHz and shown in Fig. 410, was examined in both the HR and MR channels. The observed HR signal power was (1.8 0.1) x 1022 W, where the error quoted is the statistical uncertainty. The MR channel observed signal power 1.7 x 1022 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 , , , , 2 I QS the consistency between expected and observed noise statistics and the agreement 53 x 108 2.5 * S1.5 C1 0.5 S;* 0 5 10 15 Power (o) Figure 45. Power distribution for a large sample of 4bin 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 MRHR 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 frequencydependent 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 54 x 107 6 5 ..,.. .... C) 6 *o4 S :114 3  .................................................. 2i 0 5 10 15 20 25 Power (o) Figure 46. Power distribution for a large sample of 8bin 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. 411. This measured response was removed from all data used in the HR search, as follows. The raw power spectra have frequency 010 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 28 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 x 106 3.5 2.5 0 5 .. . . . . . . .. . . .. . . .5 ............ a*** 30 40 50 60 70 80 90 100 Power (a) Figure 47. Power distribution for a large sample of 64bin 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 normalization. In the MR channel, the effect of the cavi' i i,plnif. v coupling is described using an equivalentcircuit model [70]. 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 4096bin search, thus this is the data used to apply the equivalent circuit model. A sample spectrum before correction is shown in Fig. 4 12. In the equivalentcircuit 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 4096bin resolution, i.e. b4096 = 78.1 Hz. The equivalentcircuit model predicts that the power (in units of the rms noise) at the NRAO amplifier output (the point 56 x 104 14 10 . * . 8 O * o_ 2 6 .. ....... ....... ................................. ... i .. . . . . ... . . . .... .... . 2^^^^^^^^ 00 450 500 550 600 650 Power (o) Figure 48. Power distribution for a large sample of 512bin data. labelled "RF" in Fig. 41) in the 4096bin search at the frequency offset A is S 8a3+ S A st 2 4 P(A) + 8a3 + 4a4 ( Aa2 (423) 1+ 4 ( )a5 ( a2) where the parameters al through a5 are a, = (b4096/b)(T + T, + Tv)/T, (424) a2 f /(b4096Q) (425) a3 (b4096/b)(T + Tv + (T T) cos(2kL))/T,, (426) a4 (b4096/b)((T Tv) sin(2kL))/T, and (427) a5 (fo fcen)/b4096 (428) 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 7000 6 0 0 0 .............................. ........ ..... 3000 000  S... . . . . . . . . . . . . . . . . . 1000 ....... . 3%00 3900 4000 4100 4200 4300 4400 Power (G) Figure 49. Power distribution for a large sample of 4096bin 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. 412 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 4096bin spectrum is used to perform the fit and then the original 1bin spectrum is corrected to remove the systematic effect. The weighting factors are calculated using Eq. (423) and the fitted pa rameters, al through as, at the center of each bin of width b4096. These factors are ' 30 0 t20 3 10 o 20 a Q  479.994 479.997 480.000 BW= 1.21 Hz t = 52 sec 479.9966 479.9970 479.9974 Frequency (MHz) Figure 410. An environmental peak as it appears in the MR search (top) and the 64bin HR search. The unit for the vertical axis is the rms power fluctuation in each case. BW= 125 Hz t =2000 sec 1   1.0 0.8 0.6 0 0.4 0.2 0.0 0 2000 4000 6000 8000 10000 FFT frequency (Hz) Figure 411. HR filter response calibration data (512 bin average). The power has been normalized to the maximum power output. 60 4500  0 S. . . . . . . . . . . . . . . . . 4400 i S2, *0 6 4 2 0 0 .............................. ..... ................... .................. .O.o.. .. 4300  4 * \ * 4 2 0 0 .................... ... ... ..... ...... ..... ...... ..... .... .. 4100 * 4 0 0 0 .. ...... 0 .. ... *. ' 499.805 499.807 499.809 499.811 Frequency (MHz) Figure 412. Sample 4096bin spectrum before correction for the cavityamplifier coupling. The line is the fit obtained using the equivalent circuit model. 61 4350 4300 0* 4250 .. . o O 4 1 5 0 ............................ *...................... .. ..... .... 4 ** *. 0* 4 0* _* o S 6 * 4100 ** * O0* 4050 . 4 O .805 499.807 499.809 499.811 Frequency (MHz) Figure 413. The same 4096bin spectrum of Fig. 412 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 1bin is multiplied by the factor calculated for the bin of width b4096 within which it falls. Figure 413 shows the spectrum of Fig. 412 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. (410)), the latter being the most uncertain variable. nbin searches, where n is the number of .,lIi ient 1bins added together (n = 1, 2, 4, 8, 64, 512 and 4096), are conducted to allow for various 1bin search: 2 4 5 6 7 8 9 10 11 12 13 14 1 16 2bin 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 414. Illustration of the addition scheme for the 2, 4 and 8bin searches. The numbers correspond to the data points of the 1bin search. Num bers within the same box are bins added together to form a single datum in the nbin searches with n > 1. velocity dispersions. For searches with n > 1, there is an overlap between successive nbins such that each nbin overlaps with the last half of the previous and first half of the following nbin. This scheme is illustrated for the 2, 4 and 8bin searches in Fig. 414. 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 1bin noise distribution to Eq. (422). Systematic effects are then removed, i.e. the corrections described in Section 4.5 for the filter passband response and cavityamplifier coupling are performed. "L ,;, peaks not included in the equivalent circuit model fit for the cavityamplifier response are defined to be those greater than 12 1' of the search threshold for each nbin search. After the removal of systematic effects, the 1bin noise distribution is again fitted to Eq. (422) 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 478525 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 threepart 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 axionphoton 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. (47)) and disappear when the magnetic field is switched off. No axion peaks were found in the range 478525 MHz using this approach. The exclusion limit calculated from this data is discussed in the following section. 4.7 Results Over the frequency range 478525 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 1.972.17 /eV. Each nbin search places an upper limit on the density of a flow with maximum velocity dispersion, 6v,, as given by Eq. (413). Several factors reduce the power developed in an axion peak from that given in Eq. (47). 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 1bin, the power is spread into .,1i ,i:ent bins, as discussed below. When the axion energy is offresonance, but still within the cavity bandwidth at a frequency f, the Lorentzian cavity response reduces the power developed by an additional factor of 1 h(f) (429) 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 1bin, all power is deposited in that 1bin. However, if such a peak does not fall at the center of a 1bin, the power will be spread over several 1bins. We now calculate the minimum power in a single nbin 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 (430) o0 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) = (431) 0 otherwise. Eq. (430) is equivalent to F(f) V(k)W(f k)dk, (432) 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), given by W(f) sin(fT) (433) W(f)= (4 33) 7f Discretizing Eq. (432) and inserting Eq. (433), 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 1bin is (j + 1/2)b. Thus, for an axion signal of frequency f falling in 1bin j, a fraction of the power, g(m) sin(mr + 5 ) (435) Sm7 +6 ) is lost to the mth 1bin from 1bin j, where 6 = 7(m + 1/2 f/b). If 6 = 0, i.e, the axion signal frequency is exactly equal to a 1bin center frequency, all the power is deposited in a single 1bin. However, if this is not the case, power is lost to other 1bins. In setting limits, we assume that the power loss is maximal. The maximum power loss occurs when a signal in the 1bin search falls exactly between the center frequency of two .,li i:ent 1bins. In this case, when 6 = 7/2, Eq. (435) shows that 40.5'. of the power will be deposited in each of two 1bins. In nbin seaches with n > 2, not as much power is lost to other nbins, due to the overlap between successive nbins. The minimum power deposited in an nbin 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 nbin is negligible. For the nbin 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. (418), 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 41 summarizes this information and shows the frequency resolution of each search with the corresponding maximum flow velocity dispersion from Eq. (413) 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 nbin, as outlined above. Equations (47) and (415) 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 41. Effective power thresholds for all nbin 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 42. 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 42. 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 42. Linear interpolation between values at the frequencies specified was used to obtain values of C and Te1 at all frequencies. The 2bin search density exclusion limit obtained using these values is shown in Fig. 415. 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) Figure 415. 97.7'. confidence level limits for the HR 2bin 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. (413). en E U 0 C a N in LL r) 1 M E U1 U . *0" 4.8 Discussion 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. 415, is from the 2bin search. For a flow velocity of 600 km/s relative to the detector, the 2bin search corresponds to a maximum flow velocity dispersion of 10 m/s. The 1bin search limit is less general, in that the corresponding flow velocity dispersion is half that of the 2bin 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 1024 g/cm3 (0.95 GeV/cm3), velocity of approximately 300 km/s relative to the detector, and velocity dispersion less than 53 m/s [45]. Using Eq. (413) with Table 41 and the information di1 i', '1 in Fig. 415 multiplied by the appropriate factors of 1.12 to obtain the 4bin limit, it can be seen that the 4bin 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. 416. Figure 415 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 70 DFSZ KSVZ  Big Flow r   101 c 0 0 1 10 C 101 102 Figure 416. 97.7'. confidence level limits for the HR 4bin 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. (413). 2.15 2.05 2.1 Axion Mass (peV) 71 at the beginning of future data runs in order to maximize the discovery potential of the HR channel. CHAPTER 5 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 pseudoNambuGoldstone boson associated with breaking the PecceiQuinn 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 PecceiQuinn 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 106 102 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 axionelectromagnetic 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 signaltonoise 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 phasespace structure of such particles shows that discrete flows will occur due to this late infall. 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 1.972.17 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. 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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. 