It was found in ref. [1] that cold dark matter axions thermalize as a result of their gravitational self-interactions. When they thermalize, they form a Bose-Einstein condensate. It may seem surprising that axions thermalize as a result of their gravitational self-interactions since gravitational interactions among particles are usually thought to be negligible. Gravitational interactions among particles are in fact almost always negligible but cold dark matter axions are an exception because the axions occupy in huge numbers a small number of states (the typical quantum state occupation number is 1061) and those states have enormous correlation lengths (of order parsec to Gpc, today).

Let us call Γ = 1/τ the axion thermalization rate. On time scales short compared to τ, cold dark matter axions form a degenerate Bose gas described by a classical field equation. Their behavior is then indistinguishable from that of ordinary CDM except on length scales that are too short (1014 cm or so) to be of observational interest. On times scales large compared to τ, cold dark matter axions thermalize. The thermalization of a degenerate Bose gas is a quantum-mechanical entropy generating process, not described by classical field equations. On time scales larger than τ the axion state, i.e. the state that most axions are in, tracks the lowest energy state available to them. The behaviour of such a rethermalizing axion BEC is different from that of ordinary CDM and the differences are observable.

The thermalization of cold dark matter axions is discussed in detail in ref. [2]. It is found there that rethermalization of the axion BEC by gravitational self-interactions is sufficiently fast that the axions that are about to fall into a galactic potential well almost all go to the lowest energy state consistent with the angular momentum they have acquired from tidal torquing. That state is one of net overall rotation, implying \( \nabla \times v\ne 0\) where v(x, t) is the velocity field of the infalling dark matter. In contrast, ordinary cold dark matter (e.g. WIMPs and sterile neutrinos) falls in with an irrotational velocity field, \( \nabla \times v=0\). The inner caustics of galactic halos are different in the two cases. If the dark matter falls in with net overall rotation, the inner caustics are rings whose cross-section is a section of the elliptic umbilic (D −4) catastrophe, called caustic rings for short [3]. If the velocity field of the infalling particles is irrotational, the inner caustics have a ‘tent-like’ structure which is described in detail in ref. [4] and which is quite distinct from that of caustic rings. Evidence was found for caustic rings of dark matter. The evidence is summarized in ref. [5]. The evidence for caustic rings is reproduced if the specific angular momentum distribution on the turnaround sphere is given by [5, 6]

$ l(\stackrel{\wedge }{n},t)={j}_{\mathrm{max}}\stackrel{\wedge }{n}\times \left(\stackrel{\wedge }{z\text{\hspace{0.05em}}}\times \stackrel{\wedge }{n}\right)\frac{R{(t)}^{2}}{t}$
(3.1)

where \( \widehat{n}\) is the unit vector pointing to a position on the turnaround sphere, \( \widehat{z}\) is the axis of rotation and j max is a parameter which takes a specific value for each galaxy. The evidence for caustic rings implies that the j max distribution is peaked near 0.18. R(t) is the radius of the turnaround sphere. The turnaround sphere is defined as the locus of particles which have zero radial velocity with respect to the galactic center for the first time, their outward Hubble flow having just been arrested by the gravitational pull of the galaxy. The present turnaround radius of the Milky Way is of order 2 Mpc. In the self-similar infall model [7], \( R(t)\alpha t{\scriptscriptstyle \frac{2}{3}}+{\scriptscriptstyle \frac{2}{9\epsilon }}\) where ε is related to the slope of the power spectrum of density perturbations on galactic scales. The observed power spectrum implies that ε is in the range 0.25–0.35 [6]. This range is also consistent with the evidence for caustic rings. Equation (3.1) states that the turnaround sphere at time t rotates with angular velocity \( \omega =\frac{{j}_{\mathrm{max}}}{t}\widehat{Z}\). Each property of the angular momentum distribution (3.1) maps onto an observable property of the inner caustics: net overall rotation causes the inner caustics to be rings, the value of j max determines their overall size, and the time dependence given in Eq. (3.1) is responsible for the observed pattern \( {a}_{n}\propto 1/n(n=1,2,3\dots )\) of the caustic ring radii a n . Each of these properties follows from the assumption that the infalling dark matter is a rethermalizing axion BEC [8].

First, the parameter j max is related to the dimensionless angular momentum parameter

$ \lambda \equiv \frac{L{\left|E\right|}^{\frac{1}{2}}}{G{M}^{\frac{5}{2}}},$
(3.2)

where G is Newton’s gravitational constant, L is the angular momentum of the galaxy, M its mass and E its net mechanical (kinetic plus gravitational potential) energy. λ was found in numerical simulations [9] to have median value 0.05. The relationship between j max and λ is [8]

$ \lambda =\sqrt{\frac{6}{5-3e}}\frac{8}{10+3e}\frac{1}{\pi }{j}_{\mathrm{max}}.$
(3.3)

For j max = 0.18 and ε in the range 0.25–0.35, Eq. (3.3) implies λ = 0.051. The excellent agreement between j max and λ gives further credence to the caustic ring model. Indeed if the evidence for caustic rings were incorrectly interpreted, there would be no reason for it to produce a value of j max consistent with λ.

Second, rigid rotation on the turnaround sphere is explained by the fact that most axions go to the lowest energy state available to them and that, for given total angular momentum, the lowest energy is achieved when the angular motion is rigid rotation.

Third, one can show [8] that, during the linear regime of evolution of density perturbations, the total torque applied to a halo grows as the scale factor a(t) \( \alpha t{\scriptscriptstyle \frac{2}{3}}\) and hence ℓ(t) \( \alpha t{\scriptscriptstyle \frac{5}{3}}\). Since \( R(t)\alpha t{\scriptscriptstyle \frac{2}{3}}+{\scriptscriptstyle \frac{2}{9\epsilon }}\), tidal torque theory predicts the time dependence of Eq. (3.1) provided ε = 0.33. This value of ε is in the range, 0.25 < ε < 0.35, predicted by the evolved spectrum of density perturbations and supported by the evidence for caustic rings. So the time dependence of the angular momentum distribution on the turnaround sphere is also consistent with the caustic ring model.

The above is the gist of the argument. It is elaborated in greater detail in [10]. A few comments may be in order. One question is: what fraction of the dark matter must be axions to justify the evidence for caustic rings. We hope to comment on this soon. Another question is: to what extent does the evidence for caustic rings require the dark matter to be QCD axions [11], as opposed to some other kind of axion-like particle(s). The evidence requires that a sizable fraction of the dark matter be identical bosons, whose number is conserved on cosmological time scales, and which are sufficiently cold and thermalize sufficiently fast that they form a BEC. Furthermore the BEC must rethermalize sufficiently fast that the particles go to a state of net overall rotation as they are about to fall into galactic potential wells. It happens that the QCD axion with mass of order 10−5 eV has all these properties and since it solves in addition the strong CP problem of the Standard Model of elementary particles, it is reasonable to assume that the dark matter is in fact QCD axions. However, there are many axion-like particles [12] that can equally well reproduce the evidence for caustics rings. Furthermore, whether or not the particle in question is the QCD axion, the prediction of Bose-Einstein condensation and subsequent caustic ring formation is rather insensitive to the particle mass and therefore does not provide a good guide to it. The axion is being searched for as a constituent of the Milky Way halo [13], as a particle radiated by the Sun [14], and in experiments that convert photons to axions and axions back to photons behind a wall [15].

Finally, many authors have proposed [16, 17] that the dark matter is a Bose-Einstein condensate of particles with mass of order 10−21 eV or less. When the mass is that small, the dark matter BEC behaves differently from CDM on scales of observational interest as a result of the tendency of the BEC to delocalize. Due to the Heisenberg uncertainty principle, a BEC has Jeans’ length [1, 18, 19]

$ {l}_{\text{J}}={\left(16\pi Gp{m}^{2}\right)}^{-\frac{1}{4}}={\mathrm{1.02.10}}^{14}\text{cm}{\left(\frac{{10}^{-5}eV}{m}\right)}^{\frac{1}{2}}{\left(\frac{{10}^{-29}g/c{m}^{3}}{r}\right)}^{\frac{1}{4}},$
(3.4)

where ρ is the BEC density and m the constituent particle mass. As mentioned earlier, this length scale is unobservably small in the QCD axion case. In contrast, when \( m~{10}^{-21}\text{eV}\), the Jeans’ length is of order 3 kpc and has implications for observation. It leads to a suppression of the dark matter density near the galactic center. This is proposed as a remedy for the excessive concentration of dark matter near galactic centers seen in numerical simulations of structure formation with ordinary CDM [20].