aa r X i v : . [ a s t r o - ph . GA ] M a r The Case for Axion Dark Matter
P. Sikivie
Department of Physics, University of Florida, Gainesville, FL 32611, USA (Dated: March 10, 2010)Dark matter axions form a rethermalizing Bose-Einstein condensate. This provides an opportunityto distinguish axions from other forms of dark matter on observational grounds. I show that if thedark matter is axions, tidal torque theory predicts a specific structure for the phase space distributionof the halos of isolated disk galaxies, such as the Milky Way. This phase space structure is preciselythat of the caustic ring model, for which observational support had been found earlier. The otherdark matter candidates predict a different phase space structure for galactic halos.
PACS numbers: 95.35.+d
One of the outstanding problems in science today isthe identity of the dark matter of the universe [1]. Theexistence of dark matter is implied by a large numberof observations, including the dynamics of galaxy clus-ters, the rotation curves of individual galaxies, the abun-dances of light elements, gravitational lensing, and theanisotropies of the cosmic microwave background radia-tion. The energy density fraction of the universe in darkmatter is 23%. The dark matter must be non-baryonic,cold and collisionless.
Cold means that the primordialvelocity dispersion of the dark matter particles is suffi-ciently small, less than about 10 − c today, so that itmay be set equal to zero as far as the formation of largescale structure and galactic halos is concerned. Colli-sionless means that the dark matter particles have, infirst approximation, only gravitational interactions. Par-ticles with the required properties are referred to as ‘colddark matter’ (CDM). The leading CDM candidates areweakly interacting massive particles (WIMPs) with massin the 100 GeV range, axions with mass in the 10 − eVrange, and sterile neutrinos with mass in the keV range.To try and tell these candidates apart on the basis ofobservation is a tantalizing quest.In this regard, the study of the inner caustics of galac-tic halos may provide a useful tool [2, 3]. An isolatedgalaxy like our own accretes the dark matter particlessurrounding it. Cold collisionless particles falling in andout of a gravitational potential well necessarily form aninner caustic, i.e. a surface of high density, which may bethought of as the envelope of the particle trajectories neartheir closest approach to the center. The density divergesat caustics in the limit where the velocity dispersion ofthe dark matter particles vanishes. Because the accreteddark matter falls in and out of the galactic gravitationalpotential well many times, there is a set of inner caustics.In addition, there is a set of outer caustics, one for eachoutflow as it reaches its maximum radius before fallingback in. We will be concerned here with the catastrophestructure and spatial distribution of the inner caustics ofisolated disk galaxies.The catastrophe structure of inner caustics dependsmainly on the angular momentum distribution of the in- falling particles [3]. There are two contrasting cases toconsider. In the first case, the angular momentum dis-tribution is characterized by ‘net overall rotation’; in thesecond case, by irrotational flow. The archetypical exam-ple of net overall rotation is instantaneous rigid rotationon the turnaround sphere. The turnaround sphere is de-fined as the locus of particles which have zero radial ve-locity with respect to the galactic center for the first time,their outward Hubble flow having just been arrested bythe gravitational pull of the galaxy. Net overall rotationimplies that the velocity field has a curl, ~ ∇ × ~v = 0. Thecorresponding inner caustic is a closed tube whose cross-section is a section of the elliptic umbilic ( D − ) catastro-phe [2, 3]. It is often referred to as a ‘caustic ring’, or‘tricusp ring’ in reference to its shape. In the case of irro-tational flow, ~ ∇ × ~v = 0, the inner caustic has a tent-likestructure quite distinct from a caustic ring. Both typesof inner caustic are described in detail in ref.[3].If a galactic halo has net overall rotation and its timeevolution is self-similar, the radii of its caustic rings arepredicted in terms of a single parameter, called j max .Self-similarity means that the entire phase space struc-ture of the halo is time independent except for a rescalingof all distances by R ( t ), all velocities by R ( t ) /t and alldensities by 1 /t [4–7]. t is time since the big bang. Fordefiniteness, R ( t ) will be taken to be the turnaround ra-dius at time t . If the initial overdensity around which thehalo forms has a power law profile δM i M i ∝ ( 1 M i ) ǫ , (1)where M i and δM i are respectively the mass and excessmass within an initial radius r i , then R ( t ) ∝ t + ǫ [4]. Inan average sense, ǫ is related to the slope of the evolvedpower spectrum of density perturbations on galaxy scales[8]. The observed power spectrum implies that ǫ is in therange 0.25 to 0.35 [6]. The prediction for the caustic ringradii is ( n = 1, 2, 3, .. ) [2, 7] a n ≃
40 kpc n (cid:18) v rot
220 km / s (cid:19) (cid:18) j max . (cid:19) (2)where v rot is the galactic rotation velocity. Eq.( 2) is for ǫ = 0 .
3. The a n have a small ǫ dependence. However,the a n ∝ /n approximate behavior holds for all ǫ in therange 0.25 and 0.35, so that a change in ǫ is equivalentto a change in j max . ( ǫ, j max ) = (0.30, 0.180) impliesvery nearly the same radii as ( ǫ, j max ) = (0.25, 0.185)and (0.35, 0.177).Observational evidence for caustic rings with the radiipredicted by Eq. (2) was found in the statistical distribu-tion of bumps in a set of 32 extended and well-measuredgalactic rotation curves [9], the distribution of bumps inthe rotation curve of the Milky Way [10], the appearanceof a triangular feature in the IRAS map of the MilkyWay in the precise direction tangent to the nearest caus-tic ring [10], and the existence of a ring of stars at thelocation of the second ( n = 2) caustic ring in the MilkyWay [11]. Each galaxy may have its own value of j max .However, the j max distribution over the galaxies involvedin the aforementioned evidence is found to be peaked at0.18. There is evidence also for a caustic ring of darkmatter in a galaxy cluster [12]. The caustic ring modelof galactic halos [7] is the phase space structure that fol-lows from self-similarity, axial symmetry, and net overallrotation.Self-similarity requires that the time-dependence ofthe specific angular momentum distribution on theturnaround sphere be given by [6, 7] ~ℓ (ˆ n, t ) = ~j (ˆ n ) R ( t ) t (3)where ˆ n is the unit vector pointing to a position on theturnaround sphere, and ~j (ˆ n ) is a dimensionless time-independent angular momentum distribution. In case ofinstantaneous rigid rotation, which is the simplest formof net overall rotation, ~j (ˆ n ) = j max ˆ n × (ˆ z × ˆ n ) (4)where ˆ z is the axis of rotation and j max is the param-eter that appears in Eq. (2). The angular velocity is ~ω = j max t ˆ z . Each property of the assumed angular mo-mentum distribution maps onto an observable propertyof the inner caustics: net overall rotation causes the innercaustics to be rings, the value of j max determines theiroverall size, and the time dependence given in Eq. (3)causes a n ∝ /n .The angular momentum distribution assumed by thecaustic ring halo model may seem implausible becauseit is highly organized in both time and space. Galactichalo formation is commonly thought to be a far morechaotic process. However, since the model is motivatedby observation, it is appropriate to ask whether it is con-sistent with the expected behaviour of some or any ofthe dark matter candidates. In addressing this questionwe make the usual assumption, commonly referred to as‘tidal torque theory’, that the angular momentum of agalaxy is due to the tidal torque applied to it by nearbyprotogalaxies early on when density perturbations are still small and protogalaxies close to one another [13, 14].We divide the question in three parts: 1. is the value of j max consistent with the magnitude of angular momen-tum expected from tidal torque theory? 2. is it possiblefor tidal torque theory to produce net overall rotation?3. does the axis of rotation remain fixed in time, and isEq. (3) expected as an outcome of tidal torque theory? MAGNITUDE OF ANGULAR MOMENTUM
The amount of angular momentum acquired by agalaxy through tidal torquing can be reliably estimatedby numerical simulation because it does not depend onany small feature of the initial mass configuration, so thatthe resolution of present simulations is not an issue in thiscase. The dimensionless angular momentum parameter λ ≡ L | E | GM , (5)where G is Newton’s gravitational constant, L is the an-gular momentum of the galaxy, M its mass and E its netmechanical (kinetic plus gravitational potential) energy,was found to have median value 0.05 [15]. In the causticring model the magnitude of angular momentum is givenby j max . As mentioned, the evidence for caustic rings im-plies that the j max -distribution is peaked at j max ≃ j max implied by the evidence for causticrings compatible with the value of λ predicted by tidaltorque theory?The relationship between j max and λ may be easilyderived. Self-similarity implies that the halo mass M ( t )within the turnaround radius R ( t ) grows as t ǫ [4]. Hencethe total angular momentum grows according to d~Ldt = Z d Ω dMd Ω dt ~ℓ = 49 ǫ M ( t ) R ( t ) t j max ˆ z (6)where we assumed, for the sake of definiteness, that theinfall is isotropic and that ~j (ˆ n ) is given by Eq. (4). In-tegrating Eq. (6), we find ~L ( t ) = 410 + 3 ǫ M ( t ) R ( t ) t j max ˆ z . (7)Similarly, the total mechanical energy is E ( t ) = − Z GM ( t ) R ( t ) dMdt dt = − − ǫ GM ( t ) R ( t ) . (8)Here we use the fact that each particle on the turnaroundsphere has potential energy − GM ( t ) /R ( t ) and approxi-mately zero kinetic energy. Combining Eqs. (5), (7) and(8) and using the relation R ( t ) = π t GM ( t ) [4], wefind λ = r − ǫ
810 + 3 ǫ π j max . (9)For ǫ = 0.25, 0.30 and 0.35, Eq. (9) implies λ/j max =0.281, 0.283 and 0.284 respectively. Hence there is excel-lent agreement between j max ≃ .
18 and λ ∼ . j max and λ gives further cre-dence to the caustic ring model. Indeed if the evidence forcaustic rings were incorrectly interpreted, there would beno reason for it to produce a value of j max consistent with λ . Note that the agreement is excellent only in Concor-dance Cosmology. In a flat matter dominated universe,the value of j max implied by the evidence for caustic ringsis 0.27 [2, 7]. NET OVERALL ROTATION
Next we ask whether net overall rotation is an expectedoutcome of tidal torquing. The answer is clearly no if thedark matter is collisionless. Indeed, the velocity field ofcollisionless dark matter satisfies d~vdt ( ~r, t ) = ∂~v∂t ( ~r, t )+( ~v ( ~r, t ) · ~ ∇ ) ~v ( ~r, t ) = − ~ ∇ φ ( ~r, t ) (10)where φ ( ~r, t ) is the gravitational potential. The initialvelocity field is irrotational because the expansion of theuniverse caused all rotational modes to decay away [16].Furthermore, it is easy to show [3] that if ~ ∇ × ~v = 0 ini-tially, then Eq. (10) implies ~ ∇ × ~v = 0 at all later times.Since net overall rotation requires ~ ∇ × ~v = 0, it is incon-sistent with collisionless dark matter, such as WIMPs orsterile neutrinos. If WIMPs or sterile neutrinos are thedark matter, the evidence for caustic rings, including theagreement between j max and λ obtained above, is purelyfortuitous.Axions [17–20] differ from WIMPs and sterile neu-trinos in this respect. Axions are not collisionless be-cause they form a rethermalizing Bose-Einstein conden-sate [21]. Bose-Einstein condensation (BEC) may bebriefly described as follows: if identical bosonic particlesare highly condensed in phase space, if their total num-ber is conserved and if they thermalize, most of themgo to the lowest energy available state. The condensingparticles do so because, by yielding their energy to theremaining non-condensed particles, the total entropy isincreased. In the case of cold dark matter axions, ther-malization occurs because of gravitational interactionsbetween the low momentum modes in the axion fluid.This process is quantum mechanical in an essential wayand not described by Eq. (10).Axions form a rethermalizing Bose-Einstein conden-sate when the photon temperature reaches of order 100eV( f / GeV) [21] where f is the axion decay constant.By rethermalizing we mean that thermalization rate re-mains larger than the Hubble rate so that the axion statetracks the lowest energy available state. The compres-sional (scalar) modes of the axion field are unstable andgrow as for ordinary CDM, except on length scales too small to be of observational interest [21]. Unlike ordinaryCDM, however, the rotational (vector) modes of the ax-ion field exchange angular momentum by gravitationalinteraction. Most axions condense into the state of low-est energy consistent with the total angular momentum,say ~L = L ˆ z , acquired by tidal torquing at a given time.To find this state we may use the WKB approximationbecause the angular momentum quantum numbers arevery large, of order 10 for a typical galaxy. The WKBapproximation maps the axion wavefunction onto a flowof classical particles with the same energy and momen-tum densities. It is easy to show that for given totalangular momentum the lowest energy is achieved whenthe angular motion is rigid rotation. So we find Eq. (4) tobe a prediction of tidal torque theory if the dark matteris axions.Thermalization by gravitational interactions is only ef-fective between modes of very low relative momentum[21]. After the axions fall into the gravitational poten-tial well of the galaxy, they form multiple streams andcaustics like ordinary CDM [22]. The momenta of parti-cles in different streams are too different from each otherfor thermalization by gravitational interactions to occuracross streams. The wavefunction of the axions insidethe turnaround sphere is mapped by the WKB approxi-mation onto the flow of classical particles with the sameinitial conditions on that sphere. The phase space struc-ture thus formed has caustic rings since the axions reachthe turnaround sphere with net overall rotation. The ax-ion wavefunction vanishes on an array of l lines, whichmay be thought of as the vortices characteristic of a BECwith angular momentum. However, the transverse size ofthe axion vortices is of order the inverse momentum asso-ciated with the radial motion in the halo, ( mv r ) − ∼ − eV) of the axion mass.In a BEC without radial motion the size of vortices is oforder the healing length [23], which is much larger than( mv r ) − .One might ask whether there is a way in which netoverall rotation may be obtained other than by BEC ofthe dark matter particles. I could not find any. Gen-eral relativistic effects may produce a curl in the velocityfield but are only of order ( v/c ) ∼ − which is far toosmall for the purposes described here. One may proposethat the dark matter particles be collisionfull in the senseof having a sizable cross-section for elastic scattering offeach other. The particles then share angular momentumby particle collisions after they have fallen into the galac-tic gravitational potential well. However, the collisionsfuzz up the phase space structure that we are trying toaccount for. The angular momentum is only fully sharedamong the halo particles after the flows and caustics ofthe model are fully destroyed. Axions appear singled outin their ability to produce the net overall rotation impliedby the evidence for caustic rings of dark matter. SELF-SIMILARITY
The third question provides a test of the conclusionsreached so far. If galaxies acquire their angular momen-tum by tidal torquing and if the dark matter particlesare axions in a rethermalizing Bose-Einstein condensate,then the time dependence of the specific angular momen-tum distribution on the turnaround sphere is predicted.Is it consistent with Eq. (3)? In particular, is the axis ofrotation constant in time?Consider a comoving sphere of radius S ( t ) = Sa ( t ) cen-tered on the protogalaxy. a ( t ) is the cosmological scalefactor. S is taken to be of order but smaller than half thedistance to the nearest protogalaxy of comparable size,say one third of that distance. The total torque appliedto the volume V of the sphere is ~τ ( t ) = Z V ( t ) d r δρ ( ~r, t ) ~r × ( − ~ ∇ φ ( ~r, t )) (11)where δρ ( ~r, t ) = ρ ( ~r, t ) − ρ ( t ) is the density perturbation. ρ ( t ) is the unperturbed density. In the linear regime ofevolution of density perturbations, the gravitational po-tential does not depend on time when expressed in termsof comoving coordinates, i.e. φ ( ~r = a ( t ) ~x, t ) = φ ( ~x ).Moreover δ ( ~r, t ) ≡ δρ ( ~r,t ) ρ ( t ) has the form δ ( ~r = a ( t ) ~x, t ) = a ( t ) δ ( ~x ). Hence ~τ ( t ) = ρ ( t ) a ( t ) Z V d x δ ( ~x ) ~x × ( − ~ ∇ x φ ( ~x )) . (12)Eq. (12) shows that the direction of the torque is timeindependent. Hence the rotation axis is time indepen-dent, as in the caustic ring model. Furthermore, since ρ ( t ) ∝ a ( t ) − , τ ( t ) ∝ a ( t ) ∝ t and hence ℓ ( t ) ∝ L ( t ) ∝ t . Since R ( t ) ∝ t + ǫ , tidal torque theory predictsthe time dependence of Eq. (3) provided ǫ = 0 .
33. Thisvalue of ǫ is in the range, 0 . < ǫ < .
35, predictedby the evolved spectrum of density perturbatuions andsupported by the evidence for caustic rings. So the timedependence of the angular momentum distribution on theturnaround sphere is also consistent with the caustic ringmodel.In conclusion, if the dark matter is axions, the phasespace structure of galactic halos predicted by tidal torquetheory is precisely, and in all respects, that of the caus-tic ring model proposed earlier on the basis of observa-tions. The other dark matter candidates predict a differ-ent phase space structure for galactic halos. Although theQCD axion is best motivated, a broader class of axion-like particles behaves in the manner described here.I am grateful to Ozgur Erken, James Fry, Qiaoli Yang and the members of the ADMX collaboration for usefuldiscussions. This work was supported in part by theU.S. Department of Energy under contract DE-FG02-97ER41029. [1] For a recent review, see
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