Constraints on dark matter particles from theory, galaxy observations and N-body simulations.

Constraints on dark matter particles from theory, galaxy observations and N-body simulations.

D. Boyanovsky boyan@pitt.edu Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA Observatoire de Paris, LERMA. Laboratoire Associé au CNRS UMR 8112.
61, Avenue de l’Observatoire, 75014 Paris, France.
LPTHE, Université Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), Laboratoire Associé au CNRS UMR 7589, Tour 24, 5ème. étage, 4, Place Jussieu, 75252 Paris, Cedex 05, France
   H. J. de Vega devega@lpthe.jussieu.fr LPTHE, Université Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), Laboratoire Associé au CNRS UMR 7589, Tour 24, 5ème. étage, 4, Place Jussieu, 75252 Paris, Cedex 05, France Observatoire de Paris, LERMA. Laboratoire Associé au CNRS UMR 8112.
61, Avenue de l’Observatoire, 75014 Paris, France.
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
   N. G. Sanchez Norma.Sanchez@obspm.fr Observatoire de Paris, LERMA. Laboratoire Associé au CNRS UMR 8112.
61, Avenue de l’Observatoire, 75014 Paris, France.
July 18, 2019
Abstract

Mass bounds on dark matter (DM) candidates are obtained for particles that decouple in or out of equilibrium while ultrarelativistic with arbitrary isotropic and homogeneous distribution functions. A coarse grained Liouville invariant primordial phase space density is introduced which depends solely on the distribution function at decoupling. The density is explicitly computed and combined with recent photometric and kinematic data on dwarf spheroidal satellite galaxies in the Milky Way (dShps) and the observed DM density today yielding upper and lower bounds on the mass, primordial phase space densities and velocity dispersion of the DM candidates. Combining these constraints with recent results from -body simulations yield estimates for the mass of the DM particles in the range of a few keV. We establish in this way a direct connection between the microphysics of decoupling in or out of equilibrium and the constraints that the particles must fulfill to be suitable DM candidates. If chemical freeze out occurs before thermal decoupling, light bosonic particles can Bose-condense. We study such Bose-Einstein condensate (BEC) as a dark matter candidate. It is shown that depending on the relation between the critical () and decoupling () temperatures, a BEC light relic could act as CDM but the decoupling scale must be higher than the electroweak scale. The condensate hastens the onset of the non-relativistic regime and tightens the upper bound on the particle’s mass. A non-equilibrium scenario which describes particle production and partial thermalization, sterile neutrinos produced out of equilibrium and other DM models is analyzed in detail and the respective bounds on mass, primordial phase space density and velocity dispersion are obtained. Thermal relics with that decouple when ultrarelativistic and sterile neutrinos produced resonantly or non-resonantly lead to a primordial phase space density compatible with cored dShps and disfavor cusped satellites. Light Bose-condensed DM candidates yield phase space densities consistent with cores and if also with cusps. Phase space density bounds on particles that decoupled non-relativistically combined with recent results from N-body simulations suggest a potential tension for WIMPs with .

pacs:
98.80.Cq,05.10.Cc,11.10.-z

I Introduction

Although the existence of dark matter (DM) was inferred several decades ago zwoo (), its nature still remains elusive. Candidate dark matter particles are broadly characterized as cold, hot or warm depending on their velocity dispersions. The clustering properties of collisionless DM candidates in the linear regime depend on the free streaming length, which roughly corresponds to the Jeans length with the particle’s velocity dispersion replacing the speed of sound in the gas. Cold DM (CDM) candidates feature a small free streaming length favoring a bottom-up hierarchical approach to structure formation, smaller structures form first and mergers lead to clustering on the larger scales.

Among the CDM candidates are weakly interacting massive particles (WIMPs) with . Hot DM (HDM) candidates feature large free streaming lengths and favor top down structure formation, where larger structures form first and fragment. HDM particle candidates are deemed to have masses in the few range, and warm DM (WDM) candidates are intermediate with a typical mass range .

The concordance standard cosmological model emerging from CMB, large scale structure observations and simulations favors the hypothesis that DM is composed of primordial particles which are cold and collisionless primack (). However, recent observations hint at possible discrepancies with the predictions of the concordance model: the satellite and cuspy halo problems.

The satellite problem, stems from the fact that CDM favors the presence of substructure: much of the CDM is not smoothly distributed but is concentrated in small lumps, in particular in dwarf galaxies for which there is scant observational evidence so far. A low number of satellites have been observed in Milky-Way sized galaxies kauff (); moore (); moore2 (); klyp (). This substructure is a consequence of the CDM power spectrum which favors small scales becoming non-linear first, collapsing in the bottom-up hierarchical manner and surviving the mergers as dense clumps moore (); klyp ().

The cuspy halo problem arises from the result of large scale -body simulations of CDM clustering which predict a monotonic increase of the density towards the center of the halos dubi (); frenk (); moore2 (); bullock (); cusps (), for example the universal Navarro-Frenk-White (NFW) profile frenk () which describes accurately clusters of galaxies, but indicates a divergent cusp at the center of the halo. Recent observations seem to indicate central cores in dwarf galaxies dalcanton1 (); van (); swat (); gilmore (), leading to the ’cusps vs cores’ controversy.

A recent compilation of observations of dwarf spheroidal galaxies dSphs gilmore (), which are considered to be prime candidates for DM subtructure spergel (), seem to favor a core with a smoother central density and a low mean mass density rather than a cusp gilmore (). The data cannot yet rule out cuspy density profiles which allow a maximum density and the interpretation and analysis of the observations is not yet conclusive dalcanton1 (); van2 (). These possible discrepancies have rekindled an interest in WDM particles, which feature a velocity dispersion larger than CDM particles, and consequently larger free-streaming lengths which smooth-out the inner cores and would be prime candidates to relieve the cuspy halo and satellite problems turok ().

A possible WDM candidate is a sterile neutrino dw (); este (); kusenko () with a mass in the range and produced via their mixing and oscillation with an active neutrino species either non-resonantly dw (), or through MSW (Mikheiev-Smirnov-Wolfenstein) resonances in the medium este (). Sterile neutrinos can decay into a photon and an active neutrino (more precisely the largest mass eigenstate decays into the lowest one and a photon) pal () yielding the possibility of direct constraints on the mass and mixing angle from the diffuse X-ray background Xray ().

Observations of cosmological structure formation via the Lyman- forest provide a complementary probe of primordial density fluctuations on small scales which yield an indirect constraint on the masses of WDM candidates. While constraints from the diffuse X-ray background yield an upper bound on the mass of a putative sterile neutrino in the range Xray (), the latest Lyman- analysis lyman () yields lower bounds in the range in tension with the X-ray constraints. More recent constraints from Lyman- yield a lower limit for the mass of a WDM candidate for an early decoupled thermal relic and for sterile neutrinos viel (). Strong upper limits on the mass and mixing angles of sterile neutrinos have been recently discussed beacom (), however, there are uncertainties as to whether WDM candidates can explain large cores in dSphs strigari (). It has been recently argued palazzo () that if sterile neutrinos are produced non-resonantly dw () the combined X-ray and Lyman- data suggest that these cannot be the only WDM component, with an upper limit for their fractional relic abundance . Recent boyarski2 () constraints on a radiatively decaying DM particle from the EPIC spectra of (M31) by XMM-Newton confirms this result and places a stronger lower mass limit .

All these results suggest that DM could be a mixture of several components with sterile neutrinos as viable candidates.

Motivation and goals: Although the paradigm describes large scale structure formation remarkably well, the possible small scale discrepancies mentioned above motivate us to study new constraints that different dark matter components must fulfill to be suitable candidates. Cosmological bounds on dark matter components primarily focused on standard model neutrinos bond (); TG (), heavy relics that decoupled in local thermodynamic equilibrium (LTE) when non-relativistic LW (); kt (); dominik () or thermal ultrarelativistic relics madsen (); madsenbec (); madsenQ (); salu (); hogan (). More recently, cosmological precision data were used to constrain the (HDM) abundance of low mass particles pastor (); steen (); raffelt (); mena () assuming these to be thermal relics.

The main results of this article are:

(a:) We consider particles that decouple in or out of LTE during the radiation dominated era with an arbitrary (but homogeneous and isotropic) distribution function. Particles which decouple being ultrarelativistic eventually become non-relativistic because of redshift of physical momentum. We establish a direct connection between the microphysics of decoupling in or out of LTE and the constraints that the particles must fulfill to be suitable DM candidates in terms of the distribution functions at decoupling.

(b:) We introduce a primordial coarse grained phase space density

where is the number of particles per unit physical volume and is the average of the physical momentum with the distribution function of the decoupled particle. is a Liouville invariant after decoupling and only depends on the distribution functions at decoupling. In the non-relativistic regime is simply related to the phase densities considered in refs. dalcanton1 (); TG (); hogan (); madsenQ () and can only decrease by collisionless phase mixing or self-gravity dynamics theo ().

In the non-relativistic regime we obtain

(1)

where is the primordial one-dimensional velocity dispersion and the dark matter density. Combining the result for the primordial phase space density determined by the mass and the distribution function of the decoupled particles, with the recent compilation of photometric and kinematic data on dSphs satellites in the Milky-Way gilmore () yields lower bounds on the DM particle mass whereas upper bounds on the DM mass are obtained using the value of the observed dark matter density today. Therefore the combined analysis of observational data from (dSphs), N-body simulations and the present DM density allows us to establish both upper and lower bounds on the mass of the DM candidates.

We thus provide a link between the microphysics of decoupling, the observational aspects of dark matter halos and the DM mass value.

(c:) Recent -body simulations numQ () indicate that the phase-space density decreases a factor during gravitational clustering. This result combined with eq.(1) and the observed values on dSphs satellites gilmore () yield

for the masses of thermal relics DM candidates, where ‘cored’ and ’cusp’ refer to the type of profile used in the dShps description and is the number of internal degrees of freedom of the DM particle. Wimps with masses decoupling in LTE at temperatures lead to primordial phase space densities many orders of magnitude larger than those observed in (dSphs). The results of -body simulations, which yield relaxation by orders of magnitudenumQ () suggest a potential tension for WIMPs as DM candidates. However, the -body simulations in ref.numQ () begin with initial conditions with values of the phase space density much lower than the primordial one. Hence it becomes an important question whether the enormous relaxation required from the primordial values to those of observed in dSphs can be inferred from numerical studies with suitable (much larger) initial values of the phase space density.

(d:) We study the possibility that the DM particle is a light Boson that undergoes Bose-Einstein Condensation (BEC) prior to decoupling while still ultrarelativistic. (This possibility was addressed in madsenbec ()). We analyze in detail the constraints on such BEC DM candidate from velocity dispersion and phase space arguments, and contrast the BEC DM properties to those of the hot or warm thermal relics.

(e:) Non-equilibrium scenarios that describe various possible WDM candidates are studied in detail. These scenarios describe particle production boydata () and incomplete thermalization dvd (), resonant dw () and non-resonant este () production of sterile neutrinos and a model recently proposed strigari () to describe cores in dSphs.

Our analysis of the DM candidates is based on their masses, statistics and properties at decoupling (being it in LTE or not). We combine observations on dSphs gilmore () and -body simulations numQ (), with theoretical analysis using the non-increasing property of the phase space density TG (); dalcanton1 (); hogan (); theo ().

The results from the combined analysis of the primordial phase space densities, the observational data on dSphs gilmore () and the -body simulations in ref.numQ () are the following:

  • (i): conventional thermal relics, and sterile neutrinos produced resonantly or non-resonantly with mass in the range that decouple when ultrarelativistic lead to a primordial phase space density of the same order of magnitude as in cored dShps and disfavor cusped satellites for which the data gilmore () yields a much larger phase space density.

  • (ii): CDM from wimps that decouple when non-relativistic with and kinetic decoupling at dominik () yield phase space densities at least eighteen to fifteen orders of magnitude [see eqs.(121), (122) and (128)] larger than the typical average in dSphs gilmore (). Results from -body simulations, albeit with initial conditions with much smaller values of the phase space density, yield a dynamical relaxation by a factor numQ (). If these results are confirmed by simulations with larger initial values there may be a potential tension between the primordial phase space density for thermal relics in the form of WIMPs with MeV and those observed in dShps.

  • (iii): Light bosonic particles decoupled while ultrarelativistic and which form a BEC lead to phase space densities consistent with cores and also consistent with cusps if . However if these thermal relics satisfy the observational bounds, they must decouple when , namely above the electroweak scale.

Section II analyzes the generic dynamics of decoupled particles for any distribution function, with or without LTE at decoupling, and for different species of particles. In section III we consider light thermal relics which decoupled in LTE as DM components: fermions and bosons, including the possibility of a Bose-Einstein condensate. Section IV deals with coarse grained phase space densities which are Liouville invariant and the new bounds obtained with them by using the observational dSphs data and recent results from -body simulations, bounds from velocity dispersion, and the generalized Gunn-Tremaine bound. In Section V we study the case of particles that decoupled out of equilibrium and the consequences on the dark matter constraints. Section VI summarizes our conclusions.

Ii Preliminaries: dynamics of decoupled particles

While the study of kinetics in the early Universe is available in the literature bernstein (); kt (); scott (), in this section we expand on the dynamics of decoupled particles emphasizing several aspects relevant to the analysis that follows in the next sections.

Consider a spatially flat FRW cosmology with length element

(2)

the non-vanishing Christoffel symbols are

(3)

The (contravariant) four momentum is defined as with an affine parameter, so that , where is the mass of the particle. This leads to the dispersion relation

(4)

The geodesic equations are

(5)
(6)
(7)

where and we used . The solution of eq.(7) is

(8)

where is the time independent comoving momentum. The local observables, energy and momentum as measured by an observer at rest in the expanding cosmology are given by

(9)

where form a local orthonormal tetrad (vierbein)

and the sign in eq.(9) corresponds to a space-like component. For the FRW metric

(10)

and we find,

(11)

is clearly the physical momentum, redshifting with the expansion. Combining the above with eq.(4) yields the local dispersion relation

(12)

A frozen distribution describing a particle that has been decoupled from the plasma is constant along geodesics, therefore, taking the distribution to be a function of the physical momentum and time, it obeys the Liouville equation or collisionless Boltzmann equation

(13)

Taking as an independent variable this equation leads to the familiar form

(14)

Obviously a solution of this equation is

(15)

where is the time independent comoving momentum. The physical phase space volume element is invariant, , where refer to physical and comoving volumes respectively.

The scale factor is normalized so that

(16)

and , where is the cosmic time at decoupling and is the redshift.

If a particle of mass has been in LTE but it decoupled from the plasma with decoupling temperature its distribution function is

(17)

for fermions or bosons respectively allowing for a chemical potential at decoupling.

In what follows we consider general distributions as in eq.(15) unless specifically stated.

The kinetic energy momentum tensor associated with this frozen distribution is given by

(18)

where is the number of internal degrees of freedom, typically . Taking the distribution function to be isotropic it follows that

(19)
(20)

where is the energy density and is the pressure. In summary,

(21)

The pressure can be written in a manner more familiar from kinetic theory as

(22)

where is the physical (group) velocity of the particles measured by an observer at rest in the expanding cosmology.

To confirm covariant energy conservation recall that , furthermore from eq.(5) it follows that , leading to

(23)

the first term results from the measure and the last term from ; from the expression of the pressure eq.(20) the covariant conservation equation

(24)

follows. The number of particles per unit physical volume is

(25)

and obeys

(26)

namely, the number of particles per unit comoving volume is conserved.

These are generic results for the kinetic energy momentum tensor and the particle density for any distribution function that obeys the collisionless Boltzmann equation (13).

The entropy density for an arbitrary distribution function for particles that decoupled in or out of LTE is

(27)

where the upper and lower signs refer to Fermions and Bosons respectively. Since it follows that

(28)

therefore the entropy per comoving volume is constant. In particular the ratio

(29)

is a constant for any distribution function that obeys the collisionless Liouville equation kt ().

In the case of LTE, using the distribution eq.(17) in the entropy density eq.(27) yields the result

(30)

for either statistics, where are evaluated at the decoupling time . The entropy of the gas of decoupled particles does not affect the relationship between the photon temperature and the temperature of ultrarelativistic particles that decouple later which can be seen as follows.

Consider several species of particles, one of which decouples at an earlier time in or out of equilibrium with the distribution function and entropy given by eq.(27) while the others remain in LTE with entropy density , until some of them decouple later while ultrarelativistic. Here is the temperature at time and is the effective number of ultrarelativistic degrees of freedom. Entropy conservation leads to the relation,

(31)

however, because , the usual relation , relating the temperature of a gas of ultrarelativistic decoupled particles to the photon temperature follows.

For light particles that decouple in LTE at temperature we can approximate

(32)

where

(33)

are the decoupling temperature and chemical potential red-shifted by the expansion, therefore for particles that decouple in LTE with we can approximate

(34)

This distribution function is the same as that of a massless particle in LTE which is also a solution of the Liouville equation, or collisionless Boltzmann equation.

Since the distribution function is dimensionless, without loss of generality we can always write for a particle that decoupled in or out of LTE

(35)

where are dimensionless constants determined by the microphysics, for example dimensionless couplings or ratios between and particle physics scales or in equilibrium etc. To simplify notation in what follows we will not include explicitly the set of dimensionless constants , etc, in the argument of , but these are implicit in generic distribution functions. If the particle decouples when it is ultrarelativistic, .

It is convenient to introduce the dimensionless ratios

(36)

and

(37)

For example, for a particle that decouples in equilibrium while being non-relativistic, is the Maxwell-Boltzmann distribution function kt ()

(38)

where is the effective number of ultrarelativistic degrees of freedom at decoupling, and is the solution of the Boltzmann equation, whose dependence on and the annihilation cross section is given in chapter 5.2 in ref. kt ().

Changing the integration variable in eqs.(21)-(25) to we find

(39)
(40)
(41)
(42)
(43)

leading to the equation of state:

(44)

In the ultrarelativistic and non-relativistic limits, and , respectively, we find

(45)
(46)
(47)

In the ultrarelativistic limit the energy density and pressure become,

(48)

describing radiation behaviour. In the non-relativistic limit

(49)

and the equation of state becomes

(50)

corresponding to cold matter behaviour. In the non-relativistic limit, it is convenient to write

(51)

where is the number of ultrarelativistic degrees of freedom at decoupling, and is the photon number.

The average squared velocity of the particle is given in the non-relativistic limit by

(52)

Therefore, the equation of state in thermal equilibrium is given by

(53)

where is the one dimensional velocity dispersion given at redshift by

(54)

and we used that

(55)

is the photon temperature today pdg ().

The results above, eqs.(39)-(54) are general for any distribution of decoupled particles whether or not the particles decoupled in equilibrium.

Using the relation (51) for a given species of particles with degrees of freedom, their relic abundance today is given by

(56)

where we used that today eV pdg ().

If this decoupled species contributes a fraction to dark matter, with and using that pdg () for non-baryonic dark matter, then:

(57)

Since we find the constraint

(58)

where in general depends on the mass of the particle as in eq.(35). For a particle that decouples while non-relativistic with the distribution function eq.(38) this is recognized as the generalization of the Lee-Weinberg lower bound LW (); kt (), whereas if the particle decouples in or out of LTE when it is ultrarelativistic, in which case does not depend on the mass, eq.(58) provides and upper bound which is a generalization of the Cowsik-McClelland cow (); kt () bound.

The constraint eq.(58) suggests two ways to allow for more massive particles: by increasing , namely the particle decouples earlier, at higher temperatures when the effective number of ultrarelativistic species is larger, and/or decoupling out of LTE with a distribution function that favors smaller momenta, thereby making the denominator in eq.(58) smaller, the smaller number of particles allows a larger mass to saturate the DM abundance.

For the particle to be a suitable dark matter candidate, the free streaming length must be much smaller than the Hubble radius. Although we postpone to a companion article free () a more detailed study of the free streaming lengths in terms of the generalized distribution functions, here we adopt the simple requirement that the velocity dispersion be small, namely the particle must be non-relativistic

(59)

From eq.(52) this constraint yields

(60)

where is given by eq.(36). From eqs.(16), (37), (55) and (60) we obtain the following condition for the particle to be non-relativistic at redshift

(61)

Taking the relevant value of the redshift for large scale structure to be the redshift at which reionization occurs wmap3 (), we find the following generalized constraint on the mass of the particle of species which is a dark matter component

(62)

The left side of the inequality corresponds to the requirement that the particle be non-relativistic at reionization (taking ), namely a small velocity dispersion , corresponding to a free streaming length much smaller than the Hubble radius (), while the right hand side is the constraint from the requirement that the decoupled particle is a dark matter component, namely eq. (58) is fulfilled.

Iii Light Thermal relics as Dark Matter Components.

In this section we consider particles that decouple in LTE.

iii.1 Fermi-Dirac and non-condensed Bose-Einstein gases of light particles as DM components.

The functions in the density and pressure denoted by respectively for Fermions () and Bosons and the equation of state eq.(44) for each case are depicted in figs. 1-2 for vanishing chemical potential in both cases. We have also numerically studied these functions for values of the chemical potential in the range but the difference with the case of vanishing chemical potentials is less than even for the largest value studied which is about the maximum consistent with constraints on lepton asymmetries allowed by BBN and CMB kneller ().

Figure 1: Fermions without chemical potential. Left panel: and vs . Right panel: vs. .
Figure 2: Bosons without chemical potential. Left panel: and vs . Right panel: vs. .

These figures make clear that the onset of the non-relativistic behavior occurs for in both cases. It is useful to compare this result, with the generalized constraint eq.(60) for the case of thermal relics. Replacing the LTE distribution functions (Fermi-Dirac or Bose-Einstein, without chemical potentials) in eq.(60) we obtain

(63)

The detailed analysis of the corresponding functions yields the more precise estimate in both cases for the transition to the non-relativistic regime.

Therefore, the decoupled particle of mass becomes non-relativistic at a time when . At the time of Big Bang Nucleosynthesis (BBN) when kt () MeV and , the decoupled particle is non-relativistic if

(64)

in which case it does not contribute to the effective number of ultrarelativistic degrees of freedom during BBN and would not affect the primordial abundances of light elements. If the particle remains ultrarelativistic during BBN the total energy density in radiation is kt ()

(65)

where is the (LTE) temperature of the fluid, for Bosons (Fermions), is the effective number of ultrarelativistic degrees of freedom at time from particles that remain in LTE at this time, and is the effective number of degrees of freedom at decoupling. The second term in eq.(65) is an extra contribution to the effective number of ultrarelativistic degrees of freedom.

At the time of BBN, kt () and early decoupling of the light particle, , leads to a negligible contribution to the effective number of ultrarelativistic degrees of freedom well within the current bounds verde (). Therefore, provided that the decoupled particle is stable, either for light particles that remain relativistic during (BBN) but that decouple very early on when or when the particle’s mass MeV, there is no influence on the primordial abundance of light elements and BBN does not provide any tight constraints on the particle’s mass.

iii.2 A Bose condensed light particle as a Dark Matter component

Consider the case of a light bosonic particle, for example an axion-like-particle. Typical interactions involve two types of processes, inelastic reactions are number-changing processes and contribute to chemical equilibration, while elastic ones distribute energy and momenta of the intervening particles, these do not change the particle number but lead to kinetic equilibration. Consider the case in which chemical freeze out occurs before kinetic freeze-out, such is the case for a real scalar field with quartic self-interactions. In this theory, number-conserving processes such as establish kinetic (thermal) equilibrium, but conserve particle number, a cross section for such process is where is the quartic coupling. The lowest order number-changing processes that contribute to chemical equilibrium are , with cross sections . Hence this is an example of a theory in which chemical freeze out occurs well before kinetic freeze out for small coupling.

Another relevant example is the case of WIMPs studied in ref. dominik () where it was found that , while where are the chemical and kinetic (thermal) decoupling temperatures respectively. Although this study focused on a fermionic particle, it is certainly possible that a similar situation, namely chemical freeze-out much earlier than kinetic freeze out, may arise for bosonic DM candidates.

Under this circumstance, the number of particles is conserved if the particle is stable, but the temperature continues to redshift by the cosmological expansion, therefore the gas of Bosonic particles cools at constant comoving particle number. This situation must eventually lead to Bose Einstein condensation (BEC) since the thermal distribution function can no longer accomodate the particles with non-vanishing momentum within a thermal distribution. Once thermal freeze out occurs, the frozen distribution must feature a homogeneous condensate and the number of particles for zero momentum becomes macroscopically large. Although some aspects of Bose Einstein condensates were studied in ref. madsenQ (); madsenbec (), we study new aspects such as the impact of the BEC upon the bound for the mass and the velocity dispersion of DM candidates.

The bosonic distribution function for a fixed number of particles includes a chemical potential and is given by eq.(17) where for the distribution function to be manifestly positive for all . Separating explicitly the contribution from the mode the number of particles per comoving volume is

(66)

where

(67)

is the comoving condensate density. In the infinite volume limit the condensate term vanishes unless . For we find

(68)

where

(69)

The maximum value that can achieve is , therefore, neglecting we replace by . If the comoving particle density

(70)

then, there must be a zero momentum condensate with and in the infinite (comoving) volume limit. In this limit we find,

(71)

where the critical temperature is given by

(72)

The solution of the equation (66) that determines the condensate fraction shows that for

(73)

In the infinite volume limit the distribution function for particles that decouple while ultrarelativistic , for becomes

(74)

From eq.(25) the total number of particles for is given by

(75)

where

(76)

For eq.(71) implies that

(77)

hence for the total density is given by

(78)

The enhancement factor over the thermal result reflects the population of particles in the condensed, zero momentum state. The energy density and pressure are given by

(79)
(80)

where

(81)