The Rethermalizing Bose-Einstein Condensate of Dark Matter Axions
Nilanjan Banik, Adam Christopherson, Pierre Sikivie, Elisa Maria Todarello
aa r X i v : . [ a s t r o - ph . C O ] S e p The Rethermalizing Bose-Einstein Condensate ofDark Matter Axions
Nilanjan Banik, Adam Christopherson, Pierre Sikivie, Elisa Maria Todarello
University of Florida, Gainesville, FL 32611, USA
DOI: w ill be assigned The axions produced during the QCD phase transition by vacuum realignment, stringdecay and domain wall decay thermalize as a result of their gravitational self-interactionswhen the photon temperature is approximately 500 eV. They then form a Bose-Einsteincondensate (BEC). Because the axion BEC rethermalizes on time scales shorter than theage of the universe, it has properties that distinguish it from other forms of cold darkmatter. The observational evidence for caustic rings of dark matter in galactic halos isexplained if the dark matter is axions, at least in part, but not if the dark matter is entirelyWIMPs or sterile neutrinos.
The story we tell applies to any scalar or pseudo-scalar dark matter produced in the earlyuniverse by vacuum realignment and/or the related processes of string and domain wall decay.However, the best motivated particle with those properties is the QCD axion since it is not onlya cold dark matter candidate but also solves the strong CP problem of the standard model ofelementary particles [1, 2]. So, for the sake of definiteness, we discuss the specific case of theQCD axion.The Lagrangian density for the axion field φ ( x ) may be written as L a = 12 ∂ µ φ∂ µ φ − m φ + λ φ + ... (1)where the dots represent interactions of the axion with the known particles. The propertiesof the axion are mainly determined by one parameter f with dimension of energy, called the‘axion decay constant’. In particular the axion mass is m ≃ f π m π f √ m u m d m u + m d ≃ · − eV 10 GeV f (2)in terms of the pion decay constant f π , the pion mass m π and the masses m u and m d of theup and down quarks, and the axion self-coupling is λ ≃ m f m d + m u ( m u + m d ) ≃ . m f . (3)All couplings of the axion are inversely proportional to f . When the axion was first proposed, f was thought to be of order the electroweak scale, but its value is in fact arbitrary [3]. However Patras 2015 f & · GeV [4].An upper limit f . GeV is obtained from the requirement that axions are notoverproduced in the early universe by the vacuum realignment mechanism [5], which maybe briefly described as follows. The non-perturbative QCD effects that give the axion itsmass turn on at a temperature of order 1 GeV. The critical time, defined by m ( t ) t = 1, is t ≃ · − sec( f / GeV) . Before t , the axion field φ has magnitude of order f . After t , φ oscillates with decreasing amplitude, consistent with axion number conservation. Thenumber density of axions produced by vacuum realignment is n ( t ) ∼ f t (cid:18) a ( t ) a ( t ) (cid:19) = 4 · cm (cid:18) f GeV (cid:19) (cid:18) a ( t ) a ( t ) (cid:19) (4)where a ( t ) is the cosmological scale factor. Their contribution to the energy density todayequals the observed density of cold dark matter when the axion mass is of order 10 − eV, withlarge uncertainties. The axions produced by vacuum realignment are a form of cold dark matterbecause they are non-relativistic soon after their production at time t . Indeed their typicalmomenta at time t are of order 1 /t , and vary as 1 /a ( t ), so that their velocity dispersion is δv ( t ) ∼ mt a ( t ) a ( t ) . (5)The average quantum state occupation number of the cold axions is therefore N ∼ (2 π ) n ( t ) π ( mδv ( t )) ∼ (cid:18) f GeV (cid:19) . (6) N is time-independent, in agreement with Liouville’s theorem. Considering that the axions arehighly degenerate, it is natural to ask whether they form a Bose-Einstein condensate [6, 7]. Wediscuss the process of Bose-Einstein condensation and its implications in the next section.The thermalization and Bose-Einstein condensation of cold dark matter axions is also dis-cussed in refs. [8, 9, 10, 11] with conclusions that do not necessarily coincide with ours in allrespects. Bose-Einstein condensation occurs in a fluid made up of a huge number of particles if four con-ditions are satisfied: 1) the particles are identical bosons, 2) their number is conserved, 3) theyare highly degenerate, i.e. N is much larger than one, and 4) they are in thermal equilibrium.Axion number is effectively conserved because all axion number changing processes, such asaxion decay to two photons, occur on time scales vastly longer than the age of the universe.So the axions produced by vacuum realignment clearly satisfy the first three conditions. Thefourth condition is not obviously satisfied since the axion is very weakly coupled. In contrast,for Bose-Einstein condensation in atoms, the fourth condition is readily satisfied whereas thethird is hard to achieve. The fourth condition is a matter of time scales. Consider a fluid thatsatisfies the first three conditions and has a finite, albeit perhaps very long, thermal relaxationtime scale τ . Then, on time scales short compared to τ and length scales large compared to a2 Patras 2015 ertain Jeans’ length (see below) the fluid behaves like cold dark matter (CDM), but on timescales large compared to τ , the fluid behaves differently from CDM.Indeed, on time scales short compared to τ , the fluid behaves as a classical scalar field sinceit is highly degenerate. In the non-relativistic limit, appropriate for axions, a classical scalarfield is mapped onto a wavefunction ψ by φ ( ~r, t ) = √ Re [ e − imt ψ ( ~r, t )] . (7)The field equation for φ ( x ) implies the Schr¨odinger-Gross- Pitaevskii equation for ψi∂ t ψ = − m ∇ ψ + V ( ~r, t ) ψ (8)where the potential energy is determined by the fluid itself: V ( ~r, t ) = m Φ( ~r, t ) − λ m | ψ ( ~r, t ) | . (9)The first term is due to the fluid’s gravitational self-interactions. The gravitational potentialΦ( ~r, t ) solves the Poisson equation: ∇ Φ = 4 πGmn , (10)where n = | ψ | . The fluid described by ψ has density n and velocity ~v = m ~ ∇ arg( ψ ). Eq. (8)implies that n and ~v satisfy the continuity equation and the Euler-like equation ∂ t ~v + ( ~v · ~ ∇ ) ~v = − m ~ ∇ V − ~ ∇ q (11)where q = − m ∇ √ n √ n . (12) q is commonly referred to as ‘quantum pressure’. The ~ ∇ q term in Eq. (11) is a consequence ofthe Heisenberg uncertainty principle and accounts, for example, for the intrinsic tendency of awavepacket to spread. It implies a Jeans length [12] ℓ J = (16 πGρm ) − = 1 . · cm (cid:18) − eV m (cid:19) − gr / cm ρ ! (13)where ρ = nm is the energy density. On distance scales large compared to ℓ J , quantum pressureis negligible. CDM satisfies the continuity equation, the Poisson equation, and Eq. (11) withoutthe quantum pressure term. So, on distance scales large compared to ℓ J and time scales shortcompared to τ , a degenerate non-relativistic fluid of bosons satisfies the same equations as CDMand hence behaves as CDM. The wavefunction describing density perturbations in the linearregime is given in ref. [13].On time scales large compared to τ , the fluid of degenerate bosons does not behave likeCDM since it thermalizes and forms a BEC. Most of the particles go to the lowest energy stateavailable to them through their thermalizing interactions. This behavior is not described byclassical field theory and is different from that of CDM. When thermalizing, classical fields Patras 2015 k B T where T is temperature. In contrast, for the quantum field,the average energy of each mode is given by the Bose-Einstein distribution, and the ultravioletcatastrophe is removed. To see whether Bose-Einstein condensation is relevant to axions onemust estimate the relaxation rate Γ ≡ τ of the axion fluid. We do this in the next section.When the mass is of order 10 − eV and smaller, the Jeans length is long enough to affectstructure formation in an observable way [14]. Because we are focussed on the properties ofQCD axions, we do not consider this interesting possibility here. It is convenient to introduce a cubic box of size L with periodic boundary conditions. In thenon-relativistic limit, the Hamiltonian for the axion fluid in such a box has the form H = X j ω j a † j a j + X j,k,l,m
14 Λ lmjk a † j a † k a l a m (14)with the oscillator label j being the allowed particle momenta in the box ~p = πL ( n x , n y , n z ),with n x , n y and n z integers, and the Λ lmjk given by [7]Λ ~p ,~p ~p ,~p = Λ ~p ,~p s ~p ,~p + Λ ~p ,~p g ~p ,~p (15)where the first term Λ ~p ,~p s ~p ,~p = − λ m L δ ~p + ~p ,~p + ~p (16)is due to the λφ self-interactions, and the second termΛ ~p ,~p g ~p ,~p = − πGm L δ ~p + ~p ,~p + ~p (cid:18) | ~p − ~p | + 1 | ~p − ~p | (cid:19) (17)is due to the gravitational self-interactions.In the particle kinetic regime, defined by the condition that the relaxation rate Γ ≡ τ issmall compared to the energy dispersion δω of the oscillators, the Hamiltonian of Eq. (14)implies the evolution equation˙ N l = X k,i,j =1 | Λ klij | [ N i N j ( N l + 1)( N k + 1) − N l N k ( N i + 1)( N j + 1)] 2 πδ ( ω i + ω j − ω k − ω l )(18)for the quantum state occupation number operators N l ( t ) ≡ a † l ( t ) a l ( t ). The thermalization ratein the particle kinetic regime, is obtained by carrying out the sums in Eq. (18) and estimatingthe time scale τ over which the N j change completely. This yields [6, 7]Γ ∼ n σ δv N (19)where σ is the scattering cross-section associated with the interaction, and N is the average stateoccupation number of those states that are highly occupied. The cross-section for scatteringby λφ self-interactions is σ λ = λ πm . For gravitational self-interactions, one must take the4 Patras 2015 ross-section for large angle scattering only, σ g ∼ G m ( δv ) , since forward scattering does notchange the momentum distribution.However, the axion fluid does not thermalize in the particle kinetic regime. It thermalizes inthe opposite “condensed regime” defined by Γ >> δω . In the condensed regime, the relaxationrate due to λφ self-interactions is [6, 7] Γ λ ∼ nλ m (20)and that due to gravitational self-interactions isΓ g ∼ πGnm ℓ (21)where ℓ = mδv is, as before, the correlation length of the particles. One can show that theexpressions for the relaxation rates in the condensed regime agree with those in the particlekinetic regime at the boundary δω = Γ.We apply Eqs. (20) and (21) to the fluid of cold dark matter axions described at the endof Section 1. One finds that Γ λ ( t ) becomes of order the Hubble rate, and therefore the axionsbriefly thermalize as a result of their λφ interactions, immediately after they are producedduring the QCD phase transition. This brief period of thermalization has no known impli-cations for observation. However, the axion fluid thermalizes again due to its gravitationalself-interactions when the photon temperature is approximately [6, 7] T BEC ∼
500 eV (cid:18) f GeV (cid:19) . (22)The axion fluid forms a BEC then. After BEC formation, the correlation length ℓ increases tillit is of order the horizon and thermalization occurs on ever shorter time scales relative to theage of the universe. As was emphasized in Section 3, the axion fluid behaves differently from CDM when it ther-malizes. Indeed when all four conditions for Bose-Einstein condensation the axions almost allgo to their lowest energy available state. CDM does not do that. One can readily show that infirst order of perturbation theory and within the horizon the axion fluid does not rethermalizeand hence behaves like CDM. This is important because the cosmic microwave backgroundobservations provide very strong constraints in this arena and the constraints are consistentwith CDM. In second order of perturbation theory and higher, axions generally behave differ-ently from CDM. The rethermalization of the axion BEC is sufficiently fast that axions thatare about to fall into a galactic gravitational potential well go to their lowest energy stateconsistent with the total angular momentum they acquired from nearby protogalaxies throughtidal torquing [7]. That state is a state of net overall rotation. In contrast, CDM falls intogalactic gravitational potential wells with an irrotational velocity field. The inner caustics aredifferent in the two cases. In the case of net overall rotation, the inner caustics are rings [15]whose cross-section is a section of the elliptic umbilic D − catastrophe [16], called caustic ringsfor short. If the velocity field of the infalling particles is irrotational, the inner caustics havea ‘tent-like’ structure which is described in detail in ref. [17] and which is quite distinct from Patras 2015
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