# Axions and the Galactic Angular Momentum Distribution

###### Abstract

We analyze the behavior of axion dark matter before it falls into a galactic gravitational potential well. The axions thermalize sufficiently fast by gravitational self-interactions that almost all go to their lowest energy state consistent with the total angular momentum acquired from tidal torquing. That state is a state of rigid rotation on the turnaround sphere. It predicts the occurrence and detailed properties of the caustic rings of dark matter for which observational evidence had been found earlier. We show that the vortices in the axion Bose-Einstein condensate (BEC) are attractive, unlike those in superfluid He and dilute gases. We expect that a large fraction of the vortices in the axion BEC join into a single big vortex along the rotation axis of the galaxy. The resulting enhancement of caustic rings explains the typical size of the rises in the Milky Way rotation curve attributed to caustic rings. We show that baryons and ordinary cold dark matter particles are entrained by the axion BEC and acquire the same velocity distribution. The resulting baryonic angular momentum distribution gives a good qualitative fit to the distributions observed in dwarf galaxies. We give estimates of the minimum fraction of dark matter that is axions.

###### pacs:

95.35.+d## I Introduction

One of the outstanding problems in science today is the identity of the dark matter of the universe PDM . The existence of dark matter is implied by a large number of observations, including the dynamics of galaxy clusters, the rotation curves of individual galaxies, the abundances of light elements, gravitational lensing, and the anisotropies of the cosmic microwave background radiation. The energy density fraction of the universe in dark matter is observed to be 26.7% Planck . The dark matter must be non-baryonic, cold and collisionless. Non-baryonic means that the dark matter is not made of ordinary atoms and molecules. Cold means that the primordial velocity dispersion of the dark matter particles is sufficiently small, less than about today, so that it may be set equal to zero as far as the formation of large scale structure and galactic halos is concerned. Collisionless means that the dark matter particles have, in first approximation, only gravitational interactions. Particles with the required properties are referred to as ‘cold dark matter’ (CDM). The leading CDM candidates are weakly interacting massive particles (WIMPs) with mass in the 100 GeV range, axions with mass in the eV range, and sterile neutrinos with mass in the keV range. To try and tell these candidates apart on the basis of observation is a tantalizing quest.

Recently it has been argued that the dark matter is axions CABEC ; case ; therm because this assumption provides a natural explanation and detailed account of the existence and properties of caustic rings of dark matter in galactic halos. Axions are different from the other cold dark matter candidates, such as WIMPs and sterile neutrinos, because axions form a Bose-Einstein condensate (BEC). They do so as a result of their gravitational self-interactions. On time scales long compared to their thermalization time scale , almost all axions go to the lowest energy state available to them. The other dark matter candidates do not do this. The (re)thermalization of the axion BEC is sufficiently fast that axions that are about to fall into a galactic gravitational potential well go to their lowest energy available state consistent with the total angular momentum they acquired from nearby protogalaxies through tidal torquing therm . That state is a state of net overall rotation. In contrast, ordinary cold dark matter falls into galactic gravitational potential wells with an irrotational velocity field inner . The inner caustics are different in the two cases. In the case of net overall rotation, the inner caustics are rings crdm whose cross-section is a section of the elliptic umbilic catastrophe sing , called caustic rings for short. 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. inner and which is quite distinct from caustic rings. Evidence was found for caustic rings. A summary of the evidence is given in ref. MWhalo . Furthermore, it was shown in ref. case that the assumption that the dark matter is axions explains not only the existence of caustic rings but also their detailed properties, in particular the pattern of caustic ring radii and their overall size.

The main purpose of the present paper is to give a more detailed description of the behavior of axion dark matter before it falls into the gravitational potential well of a galaxy. In particular we want to investigate the appearance and evolution of vortices in the rotating axion BEC, and ask whether they have implications for observation. We obtain two results that may be somewhat surprising. The first is that, unlike the vortices in superfluid He and in BECs of dilute gases, the vortices in axion BEC attract each other. The reason for the difference in behavior is that atoms have short range repulsive interactions whereas axions do not. The vortices in the axion BEC join each other producing vortices of ever increasing size. When two vortices join, their radii (not their cross-sectional areas) are added. We expect a huge vortex to form along the rotation axis of the galaxy as the outcome of the joining of numerous smaller vortices. We call it the ‘big vortex’. The presence of a big vortex implies that the infall is not isotropic as has been assumed in the past STW ; crdm ; sing ; MWhalo . The axions fall in preferentially along the equatorial plane. Caustic rings are enhanced as a result because the density of the flows that produce the caustic rings is larger. We propose this as explanation of the fact that the rises in the Milky Way rotation curve attributed to caustic rings MWcr are typically a factor five larger than predicted assuming that the infall is isotropic, a puzzle for which no compelling explanation had been given in the past.

The second perhaps surprising finding is that baryons are entrained by the axion BEC and acquire the same velocity distribution as the axion BEC. The underlying reason for this is that the interactions through which axions thermalize are gravitational and gravity is universal. The condition for the baryons to acquire net overall rotation by thermal contact with the axion BEC is the same as the condition for the axions to acquire net overall rotation by thermalizing among themselves. When baryons and axions are in thermal equilibrium, their velocity fields are the same since otherwise entropy can be generated by energy-momentum exchanging interactions between them. We expect that a big vortex forms in the baryon fluid as well, although one of lesser size than the big vortex in the axion fluid. The resulting angular momentum distribution of baryons agrees qualitatively with that observed by van den Bosch et al. in dwarf galaxies Bosch . In contrast, if the dark matter is WIMPs or sterile neutrinos, the predicted angular momentum distribution of baryons in galaxies differs markedly from the observed distribution, a discrepancy known as the ‘galactic angular momentum problem’ Nava ; Burk .

We consider the possibility that the dark matter is a mixture of axions and another form of cold dark matter. For the purposes of our discussion WIMPs and sterile neutrinos behave in the same way. So we call the other form of cold dark matter WIMPs for the sake of brevity. There is a minimum fraction of dark matter that must be axions for the axions to rethermalize by gravitational interactions before they fall into galactic gravitational potential wells. That fraction is of order 3%. If the axion fraction of dark matter is larger than of order 3%, the axions acquire net overall rotation through their thermalizing gravitational self-interactions and entrain the baryons and WIMPs with them. The baryons and WIMPs acquire the same velocity distribution as the axions before falling onto galactic halos. The WIMPs therefore produce the same caustic rings, and at the same locations, as the axions. However, to account for the typical size of the rises in the Milky Way rotation curve attributed to caustic rings, we find that the fraction of dark matter in axions must be of order 37% or more.

To investigate the issues of interest here we generalize the statistical mechanics of many body systems in thermal equilibrium to the case when the system is rotating and total angular momentum is conserved. When total angular momentum is conserved, a system of identical particles in thermal equilibrium is characterized by an angular frequency in addition to its temperature and its chemical potential . Broadly speaking, the state of thermal equilibrium of such systems is one of rigid rotation with angular frequency . Incidentally, we find that there is no satisfactory generalization of the isothermal sphere model of galactic halos with . This is a serious flaw of that model since galactic halos acquire angular momentum from tidal torquing. For a Bose-Einstein condensate, we derive the state that most particles condense into when . In superfluid He this state is one of rigid rotation except for a regular array of vortices embedded in the fluid. For a BEC of collisionless particles contained in a cylindrical volume the state is one of quasi-rigid rotation with all the particles as far removed from the axis of rotation as allowed by the Heisenberg uncertainty principle. For an axion BEC about to fall into a galactic gravitational well the state is one in which each spherical shell rotates rigidly but the rotational frequency varies with the shell’s radius as . (The motion is similar to that of water draining through a hole in a sink). If the axions were to equilibrate fully, they would all move very close to the equatorial plane. Because their thermalization rate is not much larger than the Hubble rate, we expect that the axions start to move towards the equator but there is not enough time for the axions to get all localized there. The motion of the axions toward the equatorial plane is another way to see why the big vortex forms.

There is a large and growing literature exploring the hypothesis that the dark matter is a Bose-Einstein condensate of spin zero particles with mass of order eV, or so dmBEC ; RS . When the mass is that small, the de Broglie wavelength of the BEC in galactic halos is large enough (of order kpc) that the wave nature of the BEC has observable effects. In contrast, our proposal is that the dark matter is composed of ordinary QCD axions, or of axion-like particles with properties similar to QCD axions. That QCD axions form a BEC is not an assumption on our part but a consequence of their standard properties. The axion BEC forms because the axions thermalize as a result of their gravitational interactions CABEC ; therm . The axion BEC behaves differently from WIMPs because it rethermalizes on time scales less than the age of the universe. The process of thermalization is key to understanding the properties of any BEC. It is only by thermalizing that a macroscopically large fraction of degenerate identical bosons go to their lowest energy available state. The process of thermalization is not described by the Gross-Pitaevskii equation. That equation describes the properties of the state that the particles condense into, but not the process by which the particles condense into that state. We emphasize in Section II that vortices appear in a BEC only as part of the process of rethermalization. The Gross-Pitaevskii equation describes the properties of vortices and their motions, but not their appearance.

The outline of this paper is as follows. In Section II, we generalize the rules of statistical mechanics to the case where the many body system conserves angular momentum. We derive the rule that determines the state that a rotating BEC condenses into. We verify that the rule is consistent with the known behaviour of superfluid He and derive the expected behavior of a rotating BEC of quasi-collisionless particles. In Section III, we obtain the expected behavior of an axion BEC about to fall into a galactic gravitational potential well, and the response of baryons and WIMPS to the presence of the axion BEC. In Section IV, we show that the axion BEC provides a solution to the galactic angular problem and derive a minimum fraction of dark matter in axions (37%) from the typical size of rises in the Milky Way rotation curve attributed to caustic rings. Section V provides a summary.

## Ii Statistical mechanics of rotating systems

In this section we discuss theoretical issues related to the statistical mechanics of rotating many-body systems. First we generalize the well-known equilibrium Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann distributions to the case where angular momentum is conserved. Systems which conserve angular momentum are characterized by an angular velocity, in addition to their temperature and chemical potential. Next we discuss the self-gravitating isothermal sphere Chandra as a model for galactic halos. We show that the model is a reasonably good description only when the angular momentum is zero. Next we discuss rotating Bose-Einstein condensates (BEC) and obtain the rule that determines the state which particles condense into. We analyze the properties of the vortices that must be present in any rotating Bose-Einstein condensate, and discuss the contrasting behaviours of vortices in superfluid He and in a fluid of quasi-collisionless particles. Finally, we identify rethermalisation as the mechanism by which vortices appear in a BEC after it has been given angular momentum.

### ii.1 Temperature, chemical potential and angular velocity

A standard textbook result gives the average occupation number of particle state in a system composed of a huge number of identical particles at temperature and chemical potential :

(1) |

where is the energy of particle state , and = 0, +1 or -1. If the particles are distinguishable, one must take and the distribution is called Maxwell-Boltzmann. If the particles are bosons, = + 1 and the distribution is called Bose-Einstein. If the particles are fermions, and the distribution is called Fermi-Dirac.

To obtain Eq. (1), one considers a system with given total energy and given total number of particles . The are the values of the which maximize the entropy Huang ; Pathria . One may repeat this exercise in the case of a system that conserves total angular momentum about some axis, say . Maximizing the entropy for given total energy , total number of particles and total angular momentum , one finds:

(2) |

where is an angular velocity. The system at equilibrium is characterized by , and . If the total number of particles is not conserved, one must set . Likewise if total angular momentum is not conserved one must set .

### ii.2 The self-gravitating isothermal sphere revisited

Consider a huge number of self-gravitating identical classical particles. (Although identical, they are disinguishable by arbitrarily unobtrusive labels.) A particle state is given by its location in phase-space. According to Eq. (2), the particle density in phase-space is given at thermal equilibrium by

(3) |

where is the particle mass, , and is the gravitational potential. Newtonian gravity is assumed. The gravitational potential satisfies the Poisson equation:

(4) |

where

(5) |

is the physical space density.

(6) |

where is the velocity dispersion of the particles and 4) and (6) one obtains . Combining Eqs. (

(7) |

This equation permits a spherically symmetric ansatz, . It can then be readily solved by numerical integration. The solutions have the form

(8) |

where

(9) |

and is a unique function with the limiting behaviours: as and as . A plot of the function is shown, for example, in Fig. 1 of ref. galcen or Fig. 4.7 of ref. BT . The function is often approximated by for convenience. The phase-space distribution

(10) |

is called an ‘isothermal sphere’ Chandra .

The isothermal sphere is often used as a model for galactic halos SL ; JKG . As such it has many attractive properties. First, the isothermal sphere model is very predictive since it gives the full phase-space distribution in terms of just two parameters, and . Second, these two parameters are directly related to observable properties of a galaxy: is related to the galactic rotation velocity at large radii by and is related to the galactic halo core radius by . Third, since for large , the isothermal model predicts galactic rotation curves to be flat at large . This is consistent with observation. Fourth, since for small , galactic halos have inner cores where the density is constant. This is also consistent with observation. Fifth, the model is based on a simple physical principle, namely thermalization.

For all its virtues, we do not believe the isothermal model to be a good description of galactic halos. The reason is that present day galactic halos, such as that of the Milky Way, are unlikely to be in thermal equilibrium. If for some unexplained reason the Milky Way halo were in thermal equilibrium today, it would soon leave thermal equilibrium because it accretes surrounding dark matter. The infalling dark matter particles only thermalize on time scales that are much longer than the age of the universe velpeak ; rob . The flows of infalling dark matter produce peaks in the velocity distribution. A large fraction of the halo, over 90% in the model of ref. MWhalo , is in cold flows. This disagrees with the smooth Maxwell-Boltzmann distribution, Eq. (10), of the isothermal model. The presence of infall flows, with high density contrast in phase-space, has been confirmed by cosmological N-body simulations DDT .

Here we point to another flaw of the isothermal sphere as a model of galactic halos. Galactic halos acquire angular momentum from tidal torquing. If they are in thermal equilibrium, as the isothermal model supposes, the phase-space distribution must be given by Eq. (3) with . However, Eq. (3) is an unacceptably poor description of galactic halos as soon as . Indeed, Eq. (3) may be rewritten

(11) |

where . Compared to the case, the velocity distribution is locally boosted by the rigid rotation velocity . The physical space density is

(12) |

Substituting this into Eq. (4), one obtains

(13) |

where are cylindrical coordinates. Eq. (13) does not have any solutions for which the density goes to zero for large . Indeed at large unless there. But this implies, through Eq. (4), that the density goes to the constant value at large . The particles at large have huge bulk motion with average velocity . Thus the rotating isothermal sphere is an object of infinite extent in a state of rigid rotation. This is certainly inconsistent with the properties of galactic halos.

As was discussed by Lynden-Bell and Wood LB , isothermal spheres and self-gravitating systems in general are unstable because their specific heat is negative, i.e. they get hotter when energy is extracted from them. The instability implies a gravo-thermal catastrophe on a time scale which may be inconsistent with the age of galactic halos and thus cause further difficulties in using isothermal spheres as models for galactic halos. The instability of rotating isothermal spheres is discussed in ref. Chav .

### ii.3 The rotating Bose-Einstein condensate

#### ii.3.1 Bose-Einstein condensation

Bose-Einstein condensation occurs when the following four conditions are satisfied: 1) the system is composed of a huge number of identical bosons, 2) the bosons are highly degenerate, i.e. their average quantum state occupation number is larger than some critical value of order one, 3) the number of bosons is conserved, and 4) the system is in thermal equilibrium. When the four conditions are satisfied a fraction of order one of all the bosons are in the same state.

Let us recall a simple argument Pethik why Bose-Einstein condensation occurs. We first set . For identical bosons, Eq. (2) states

(14) |

Let be the ground state, i.e. the particle state with lowest energy. It is necessary that the chemical potential remain smaller than the ground state energy at all times since Eq. (14) does not make sense for . The total number of particles is an increasing function of for fixed since each has that property. Let us imagine that the total number of particles is increased while is held fixed. The chemical potential increases till it reaches . At that point the total number of particles in excited () states has its maximum value

(15) |

In three spatial dimensions, is finite Pethik . Consider what happens when, at a fixed temperature , is made larger than . The only possible system response is for the extra particles to go to the ground state. Indeed the occupation number of that state becomes arbitrarily large as approaches from below. This is the phenomenon of Bose-Einstein condensation.

We may repeat the above argument for . In this case the chemical potential must remain smaller than the smallest . The existence of a minimum is guaranteed because the particle energy always contains a piece that is quadratic in , namely the kinetic energy associated with motion in the direction. Let

(16) |

and the particle state that minimizes . (We are relabeling the states compared to the case.) The largest possible number of particles in states at a given temperature is

(17) |

If the total number of particles is larger than , the extra particles go to the state.

#### ii.3.2 Fluid description of the condensed state

Each particle state is described by a wavefunction which satisfies the Schrödinger equation (we suppress for the time being the index that labels particle states)

(18) |

where is the gravitational potential and is the non-gravitational potential energy of the particle in its background. We assume throughout that the particles are non-relativistic. As is well-known, the probability density and the probability flux density satisfy the continuity equation because the total probability to find the particle someplace is conserved.

When a state is occupied by a huge number of particles, that state’s wavefunction describes the properties of a macroscopic fluid. The fluid density is

(19) |

and the fluid flux density is

(20) |

They satisfy the continuity equation

(21) |

for the reason mentioned at the end of the previous paragraph. The fluid velocity is defined by . If we write the wavefunction as

(22) |

then , and hence Pethik

(23) |

When a state is occupied by a huge number of particles, its wavefunction satisfies Eq. (18) as does the wavefunction of any other state. However, the gravitational potential and the potential energy may have important contributions from the particles themselves, i.e.

(24) |

where the dots indicate other contributions to the gravitational potential, and likewise

(25) |

if the particles interact pairwise with forces derived from a potential . Upon substituting Eqs. (24) and/or (25), Eq. (18) takes on a non-linear form. This non-linear form of the Schrödinger equation is called the Gross-Pitaevskii equation.

Eq. (18) implies an Euler-type equation for the fluid velocity Pethik

(26) |

where

(27) |

Except for the term, Eq. (26) is the Euler equation for a fluid of classical particles moving in the potentials and . The term is a consequence of the Heisenberg uncertainty principle and accounts, for example, for the intrinsic tendency of a wavepacket to spread. However, the term is irrelevant on scales large compared to the de Broglie wavelength. On such large length scales the fluid described by the Schrödinger equation is indistinguishable from a fluid of classical particles moving in the potentials and .

#### ii.3.3 Vortices Ons ; Feyn

It may appear at first that the requirement , implied by Eq. (23), disagrees with the principle just stated, since a fluid of particles may have a rotational velocity field whereas the wave description allows apparently only irrotational flow. However, that appearance is deceiving because Eq. (23) is valid only where . Indeed is not well defined where . The following example is instructive

(28) |

where is the Bessel function of index . This wavefunction solves the Schrödinger equation with . It describes an axially symmetric flow with energy and -component of angular momentum per particle. The fluid is clearly rotating since Eq.(23) implies the velocity field

(29) |

The curl of that velocity field vanishes everywhere except on the -axis: where and are Cartesian coordinates in the plane perpendicular to . vanishes on the -axis if . Furthermore, since the Bessel function for Abram , the density implied by Eq. (28) is tiny for much less than the classical turnaround radius . Thus, on length scales large compared to , the fluid motion described by the wavefunction (28) is the same as the motion of a fluid composed of classical particles, in agreement with the principle stated at the end of the previous paragraph.

That the motion of a fluid of cold dark matter particles can be described by a wavefunction was emphasised by Widrow and Kaiser some twenty years ago Widrow . Ordinary (non BEC) cold dark matter particles have irrotational flow because their rotational modes have been suppressed by the expansion of the universe and, when density perturbations start to grow, Eq.(26) implies at all times if initially inner . The previous paragraph emphasizes that a fluid of cold dark matter particles can be described by a wavefunction whether or not the velocity field is irrotational. The only requirement is that the wavefunction vanish on a set of lines, called vortices, if the velocity field is rotational.

Each vortex carries an integer number of units of angular momentum, units in the example of Eq. (28). The following rule applies. Let be the circulation of the velocity field along a closed path

(30) |

If the fluid is represented by a wave, Eq. (23) implies

(31) |

where is the change of the phase when going around . Since the wavefunction is single valued, is an integer. The surface enclosed by a closed path with non-zero circulation must be traversed by vortices whose units of angular momentum add up to . Another rule is that as long as the fluid is described by a wavefunction , vortices cannot appear spontaneously in the fluid. They can only move about. Indeed, consider an arbitrary closed path . The total number of vortices encircled by , counting a vortex with angular momentum as vortices, is determined by the constraint of Eq. (31). The only way the RHS of that equation can change is by having the wavefunction vanish somewhere on and letting a vortex cross that curve. Finally, a third rule: vortices must follow the motion of the fluid. This is a corollary of Kelvin’s theorem which states that, in gradient flow - i.e. if the RHS of Euler’s equation is a gradient, as is the case for Eq. (26) - the circulation of the velocity field along a closed path that moves with the fluid is constant in time.

#### ii.3.4 Superfluid He

Let us see how the above considerations apply first to the case of superfluid He, and next to the case of a collisionless fluid. Of course, we have nothing new to say about superfluid He, which we discuss merely to build confidence in the general approach described so far. We found that, when Bose-Einstein condensation occurs, a macroscopically large number of particles, say , condense into the state with lowest . We have, in terms of the quantities defined earlier,

(32) | |||||

and

(33) | |||||

He atoms have an interparticle potential that describes forces that are strongly repulsive at short range and weakly attractive at long range. In the liquid state, the average interatomic distance is of order the atom size. Thus the density of superfluid He has an approximately constant value . In that case

(34) |

and therefore

(35) |

is minimized when . So, when superfluid He carries angular momentum, it is in a state of rigid rotation when viewed on length scales large compared to the de Broglie wavelength. Because rigid rotation implies that the velocity field has non-zero curl, vortices are present. The vortices are parallel to the -axis and their number density per unit area is

(36) |

according to Eqs. (30) and (31). The transverse size of a vortex is determined by balancing the competing effects of and in Eq. (26). tends to increase the transverse size of the vortex whereas tends to decrease it assuming, as is the case in superfluid He, that the interparticle interactions are repulsive at short distances. The outcome determines the transverse size of a vortex to be a characteristic length , called the ‘healing length’ Pethik . For the interparticle potential

(37) |

the healing length is

(38) |

The transverse size of a -vortex, i.e. a vortex that carries units of angular momentum, is . Indeed the behaviour at short distances to the vortex center is the same as in Eq. (28) with replaced by . The cross-sectional area of a -vortex is therefore of order . Also its energy per unit length Pethik is proportional to . The vortices repel each other because a -vortex has more energy per unit length than 1-vortices. The lowest energy configuration for given angular momentum per unit area is a triangular lattice of parallel 1-vortices Tkach . Such triangular vortex arrays were observed in superfluid He Pack and in Bose-Einstein condensed gases Kett .

#### ii.3.5 Quasi-collisionless particles

We finally arrive at the object of our interest, a Bose-Einstein condensate of quasi-collisionless particles. The particles cannot be exacly collisionless since they must thermalize to form a BEC and they can only thermalize if they interact. However, the interaction by which the particles thermalize can be arbitrarily weak since the thermalization may, in principle, occur on an arbitrarily long time scale. In that limit we may set in Eq. (18). For the sake of definiteness we set the gravitational field as well. We have then

(39) | |||||

If one approximates the Bose-Einstein condensate as a fluid of classical particles (setting and taking and to be independent variables), the state of lowest is one of rigid rotation with angular velocity and all particles placed as far from the -axis as possible. For the reasons stated earlier, this is a good approximation only on length scales large compared to the BEC de Broglie wavelength. To obtain the exact BEC state, one must solve the eigenvalue problem

(40) |

The BEC state is then such that . Since by assumption the system conserves angular momentum, the operators and are simultaneously diagonalizable. Thus

(41) |

and

(42) |

Let us consider the particular example of a BEC contained in a cylinder of radius and height . In this case, the operators , are simultaneously diagonalized by and

(43) |

where , , , and is the root of with . Since

(44) |

is minimized by setting and where minimizes , the first zero of Abram . For large

(45) |

Hence

(46) |

When the BEC is approximated as a fluid of classical particles, the BEC state is rigid rotation with all the particles located at , not necessarily in a uniform way. In the actual BEC state the particles are, for large , uniformly located just inside the surface, in a film of thickness .

Unlike the case of superfluid He, vortices in a collisionless BEC attract each other. Indeed the lowest energy state for given total angular momentum is a single -vortex with transverse size as large as possible. We may imagine turning off the interparticle repulsion in superfluid He placed in a cylindrical container. Starting with a triangular array of parallel 1-vortices but progressively decreasing , the vortices grow in transverse size till they join into a single -vortex and all matter is uniformly concentrated near the surface.

### ii.4 Thermalization and vortex formation

We emphasized that vortices cannot appear spontaneously in a fluid that is described by a (single) wavefunction . The Gross-Pitaevskii equation can only describe the motion of vortices, not their appearance. How then do the vortices appear? The vortices appear when the bosons move between different particle states, some of which have vortices and some of which don’t. When angular momentum is given to a BEC that is free of vortices, it will at first remain free of vortices even though it carries angular momentum. The vortices only appear when the BEC rethermalizes and the particles go to the new lowest energy state consistent with the angular momentum the BEC received.

Consider, for example, a BEC of spin zero particles in a cylindrical volume. The wavefunctions of the particle states are given by Eq. (43). The Hamiltonian is the sum of free and interacting parts: . The free Hamiltonian is:

(47) |

where and are annihilation and creation operators satisfying canonical commutation relations and generating a Fock space in the usual fashion. We assume that the interaction has the general form

(48) |

where , and so forth, so that the total number of particles

(49) |

is conserved. In addition we require that

(50) |

so that the total angular momentum

(51) |

is conserved as well. The interaction causes the system to thermalize on some time scale . Ref. therm estimates the thermalization rate of cold dark matter axions through their and gravitational self-interactions. The relevant thing for our discussion here is only that there is a finite time scale over which the system thermalizes.

Let us suppose that particles are in thermal equilibrium in the cylinder with and temperature well below the critical temperature for Bose-Einstein condensation. A macroscopically large number of particles are in the ground state , which we label for short. The remaining particles are in excited () states. The vorticity of each state equals its quantum number. The particles in the ground state form a fluid with zero vorticity. Many excited states carry vorticity but their occupation numbers are small compared to . The particles in excited states merely constitute a gas at temperature . Let us suppose that the fluid is then given some angular momentum. This can be done, for example, by having a large mass which gravitationally attracts the particles in the cylinder go by, producing a time-dependent potential energy . We assume for the sake of definiteness that the mass passes by the cylinder on a time scale which is much shorter than the thermal relaxation time scale . While the mass passes by, each particle stays in whatever state it was in to start with since the interaction that allows particles to jump between states is, by assumption, too feeble to have any effect on the time scale. The wavefunction of each state satisfies the time-dependent Schrödinger equation:

(52) |

with the initial condition . Although each changes in time, for the reasons given earlier, its vorticity does not. Therefore, just after the mass has passed, the macroscopic fluid described by has no vorticity although it generally has angular momentum. After a time of order , the particles acquire a thermal distribution, Eq. (2) with , consistent with the total number of particles , the angular momentum acquired from the passing mass and total energy including some energy acquired from the passing mass. Assuming the temperature is still below the critical temperature for Bose-Einstein condensation, a macroscopically large number of particles are in the state with given by Eq. (46). That state describes a fluid which carries a single vortex with units of angular momentum.

## Iii Axions, baryons and WIMPs

In this section we apply the considerations of Section II to dark matter axions when they are about to fall into a galactic gravitational potential well. We also discuss the behavior, in the presence of dark matter axions, of baryons and of a possible ordinary cold dark matter component made of weakly interacting massive particles (WIMPs) and/or sterile neutrinos. First we discuss the axions by themselves, ignoring the other particles.

### iii.1 Axions

Axions behave differently from ordinary cold dark matter particles, such as WIMPs or sterile neutrinos, on time scales long compared to their thermalization time scale because on time scales long compared to the axions form a BEC and almost all axions go to their lowest energy available state CABEC ; therm . Ordinary cold dark matter particles do not do this.

It may be useful to clarify the notion of lowest energy available state. Thermalization involves interactions. By lowest energy available state we mean the lowest energy state that can be reached by the thermalizing interactions. In general the system has states of yet lower energy. For example, and at the risk of stating the obvious, when a beaker of superfluid He is sitting on a table, the condensed atoms are in their lowest energy available state. This is not their absolute lowest energy state since the energy of the condensed atoms can be lowered by placing the beaker on the floor.

Axions behave in the same way as ordinary cold dark matter on time scales short compared to their thermalization time scale CABEC . So, to make a distinction between axions and ordinary cold dark matter it is necessary to observe the dark matter on time scales long compared to . The critical question is then: what is the thermalization time scale ?

#### iii.1.1 Axion thermalization

The relaxation rate of axions through gravitational self-interactions is of order CABEC ; therm ; Saik

(53) |

where and are their density and mass, and their correlation length. is their momentum dispersion. A heuristic derivation of Eq. (53) is as follows. If the axions have density and correlation length , they produce gravitational fields of order . Those fields completely change the typical momentum of axions in a time . is the inverse of that time. To estimate the axion relaxation time today, let us substitute gr/cc (the average dark matter density today), eV (a typical mass for dark matter axions) and (the horizon during the QCD phase transition, stretched by the universe’s expansion until today). This yields a relaxation time of order years, much shorter than the present age of the universe. So dark matter axions formed a BEC a long time ago already. It is found in refs. CABEC ; therm that the axions first thermalize and form a BEC when the photon temperature is approximately 500 eV

It may seem surprising that axions thermalize as a result of their gravitational self-interactions since gravitational interactions among particles are usually negligible. 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 ) and those states have enormous correlation lengths, as was just discussed.

It has been claimed CABEC ; case ; therm that the dark matter is axions, at least in part, because axions explain the occurrence of caustic rings of dark matter in galactic halos. For the explanation to succeed it is necessary that the axions that are about to fall onto a galaxy thermalize sufficiently fast that they almost all go to the lowest energy available state consistent with the angular momentum they acquired from neighboring protogalaxies by tidal torquing TTT . Heuristically, the condition is therm

(54) |

where is the acceleration necessary for the axions to remain in the lowest energy state as the tidal torque is applied. Here must be taken to be of order the size of the system, i.e. some fraction of the distance between neighboring protogalaxies. It was found in ref. therm that the inequality (54) is satisfied by a factor of order 30 - i.e. that its LHS is of order 30 times larger than its RHS - independently of the system size.

#### iii.1.2 Caustic rings

The evidence for caustic rings is summarized in ref. MWhalo . It is accounted for if the angular momentum distribution of the dark matter particles on the turnaround sphere of a galaxy is given by

(55) |

where is time since the Big Bang, is the radius of the turnaround sphere, the galactic rotation axis, the unit vector pointing to an arbitrary point on the turnaround sphere, and a dimensionless parameter that characterizes the amount of angular momentum the particular galaxy has. 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. Eq. (55) states that the particles on the turnaround sphere rotate rigidly with angular velocity vector . The time-dependence of the turnaround radius is predicted by the self-similar infall model FGB to be is related to the slope of the evolved power spectrum of density perturbations on galaxy scales Dor . This implies that is in the range 0.25 to 0.35 STW . The evidence for caustic rings is consistent with that particular range of values of . . The parameter

Each property of the angular momentum distribution given in Eq. (55) maps onto an observable property of the inner caustics of galactic halos: the rigid rotation implied by the factor causes the inner caustics to be rings of the type described in refs. crdm ; sing ; inner , the value of determines their overall size, and the time dependence causes, in the stated range, the caustic radii to be proportional to (). The prediction for the caustic radii is

(56) |

where is the galactic rotation velocity. To account for the evidence for caustic rings, axions must explain Eq. (55) in all its aspects. We now show, elaborating the arguments originally given in ref.case , that axions do in fact account for each factor on the RHS of Eq. (55).

#### iii.1.3

Consider a comoving spherical volume of radius centered on a protogalaxy. At early times where is the cosmological scale factor. At later times deviates from Hubble flow as a result of the gravitational pull of the protogalactic overdensity. At some point it reaches its maximum value. At that moment it equals the galactic turnaround radius. is taken to be of order but smaller than the distance to the nearest protogalaxy of comparable size, say one third of that distance. In the absence of angular momentum, the axions have a purely radial motion described by a wavefuntion where is the radial coordinate relative to the center of the sphere. When angular momentum is included the radial motion is modified at small radii by the introduction of an angular momentum barrier. This modification of the radial motion is relatively unimportant and we neglect it. The wavefunctions of the states that the axions occupy are thus taken to be

(57) |

where and are the usual spherical angular coordinates (, ), and and are quantum numbers. is as before the eigenvalue of the -component of angular momentum. The -direction is the direction of the total angular momentum acquired inside the sphere as a result of tidal torquing. We will see below that that direction is time independent. is an additional quantum number, associated with motion in . We normalize and the various such that

(58) |

We suppress the quantum numbers and henceforth.

According to Eq. (54) the axions thermalize on a time scale that is short compared to the age of the universe. Hence we expect most axions to keep moving to the state of lowest . The angular frequency is time dependent since the angular momentum is growing by tidal torquing and the moment of inertia is increasing due to the expansion of the volume under consideration. We have

(59) | |||||

where

(60) |

is similar to a moment of inertia but differs from the usual definition because the volume to which it refers is not rotating like a rigid body in three dimensions. Eq. (57) implies instead that each spherical shell of that volume rotates with the same angular momentum distribution. The associated angular velocities vary with shell radius as . The inner shells rotate faster than the outer shells because all shells have the same angular momentum distribution.

We take the gravitational potential to be spherically symmetric. The first term on the RHS of Eq. (59) is then independent of the angular variables and irrelevant to what follows. We will ignore it henceforth. Since

(61) |

we have

(62) | |||||

The dependence of that minimizes is

(63) |

However that exact dependence is not allowed because the wavefunction must be single-valued. Instead we have

(64) |

by which we mean that is as given in Eq. (63) except for the insertion of small defects (vortices) that allow to be single-valued. The vortices are discussed below.

After Eq. (64) is satisfied, we have

(65) |

is further minimized by having peaked at , i.e. at the equator. The width of the peak is of order . Eq. (64) shows that is the angular momentum per particle in the galactic plane. A typical value is

(66) |

Therefore in their state of lowest the axions are almost all within a very small angular distance, of order radians, from the galactic plane.

The state just described is the state most axions would be in after a sufficiently long period of thermalization. Because the thermalization criterion of Eq. (54) is only satisfied by a factor of order 30 we expect that, although the axions start to move towards the equator, there is not enough time for all the axions to get localized there. We expect the system to behave as follows. As the axions acquire angular momentum they go to a state, described by Eqs. (57) and (64), in which each spherical shell rotates rigidly with angular velocity proportional to where is the shell radius. The axion velocity field is

(67) |

The sign indicates that the LHS and RHS equal each other except for the presence of vortices. The vortices have direction and density per unit surface given by [see Eqs. (30) and (31)]