Galaxy Phase-Space Density Data Preclude That Bose–Einstein Condensate Be the Total Dark Matter

Abstract

Ultralight scalars with a typical mass of the order $m\sim {10}^{-22}$ eV and light scalars forming a Bose–Einstein condensate (BEC) exhibit a Jeans length in the kpc scale and were therefore proposed as dark matter (DM) candidates. Our treatment here is generic, independent of the particle physics model and applies to all DM BEC, in both in or out of equilibrium situations. Two observed quantities crucially constrain DM in an inescapable way: the average DM density ${\rho }_{DM}$ and the phase-space density Q. The observed values of ${\rho }_{DM}$ and Q in galaxies today constrain both the possibility to form a BEC, and the DM mass m. These two constraints robustly exclude the axion DM that decouples after inflation. Moreover, the value $m\sim {10}^{-22}$ eV can only be obtained with a number of ultrarelativistic degrees of freedom at decoupling in the trillions, which is impossible for decoupling in the radiation dominated era. In addition, we find for the axion vacuum misalignment scenario that axions are produced strongly out of thermal equilibrium and that the axion mass in such a scenario turns to be 17 orders of magnitude too large to reproduce the observed galactic structures. Moreover, we also consider inhomogenous gravitationally bounded BEC’s supported by the bosonic quantum pressure independently of any particular particle physics scenario. For a typical size $R\sim$ kpc and compact object masses $M\sim {10}^7{M}_{\odot }$ they remarkably lead to the same particle mass $m\sim {10}^{-22}$ eV as the BEC free-streaming length. However, the phase-space density for the gravitationally bounded BEC’s turns out to be more than sixty orders of magnitude smaller than the galaxy-observed values. We conclude that the BEC cannot be the total DM. The axion can be candidates to be only part of the DM of the universe. Besides, an axion in the mili-eV scale may be a relevant source of dark energy through the zero point cosmological quantum fluctuations.

1. Introduction

Deciphering the nature of dark matter (DM) is nowadays one of the most active domains in astrophysics, cosmology and particle physics. Cold dark matter (CDM) particles heavier than a GeV succeed to reproduce the observations for large scales beyond the Mpc. DM particles with mass m below the eV (HDM-hot DM) are ruled out because their too large Jeans lengths exclude the formation of the observed galaxies. There is a way out for scalar particles if they form Bose–Einstein condensates (BEC), where the Jeans length can be estimated as [1,2,3,4,5]

$${\lambda }_J\sim 4\sqrt{\frac{{10}^{-22}\mathrm{eV}}{m}}\mathrm{kpc}\simeq 1.2\times {10}^{17}\sqrt{\frac{{10}^{-22}\mathrm{eV}}{m}}\mathrm{km}.$$

(1)

In BEC dark matter, in order to reproduce the observed galactic structures, one should have typically [1,2,3,4,5]:

$$m\sim {10}^{-22}\mathrm{eV}.$$

(2)

In the linear approximation, the mass value is constrained up to ${10}^{-26}$ eV approximately; the value ${10}^{-22}$ eV arises from a non-perturbative analysis of small scales.

The same requirement but for non-BEC dark matter gives m in the keV scale, that is warm dark matter (WDM) [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. CDM and WDM yield identical results for large scales beyond the Mpc; WDM also provides the correct medium, galactic and small scales in agreement with observations Refs. [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and references therein.

BEC of alkali atoms, BEC of molecules and BEC of magnons have been observed experimentally in the laboratory [22,23].

The galactic phase space density is an important physical quantity and its analysis is crucial to constrain the nature of dark matter from the by now robust observational data for it, as described in Section 3 and Section 4 here below and references therein.

In this paper, we study the density in physical space $\rho$ and the density in phase-space Q in order to constrain for the first time with these two observables, the Bose–Einstein condensates (BECs) as dark matter candidates, mainly the mass of generic light scalars forming a DM BEC and other relevant properties of such BECs.

Axion cosmological scenarii and BECs as dark matter have a wide literature from many years, see for example [24,25,26,27,28,29], and is not our aim here to review all them; this is not a review paper, and our aim here is to provide constraints never considered before for DM BECs. We stress that previous DM BEC literature has not introduced the modern DM galactic phase density and the constraints imposed by the real galaxy data for the DM phase density, and none of previous DM BEC papers relate to the aspects of the BEC DM constraints we are treating for the first time in BEC DM here.

Building particle physics models of axion scenarii, although interesting in its own right, is not the aim of this paper. The DM BEC phase space density constraints and the galaxy data for them are not treated in the previous DM BEC literature.

Moreover, the phase space density and its galactic data to constraint DM is a treatment rather generic and universal, independent of the details of such models. This applies to any kind of DM and is manifest in the astrophysical and dark matter galactic literature.

In this paper, we consider and constraint with galaxy data, in different and consistent ways, different observables and physical magnitudes as:

• the density in real space,
• the phase space density,
• the surface density,
• the free streaming length,
• the number of effective degrees of freedom,
• the mass range of the different Ultralight mass particles in the BECs,
• in different situations, in thermal and out of thermal equilibrium, for homogeneous as well as for gravitational non homogeneous BECs.
• Cross-correlation and self-consistency of the obtained constraints for all of the above relevant physical magnitudes, to make the results of this paper strongly robust and far beyond the literature in the field.

The two observables: the average DM density ${\rho }_{DM}$and the galactic phase space density Q robustly constrain in an inescapable way both the possibility to form a BEC, e.g., $(T_d/T_c)$, and the DM particle mass, m, ruling out BEC DM in general, and the BEC axion DM in particular. Moreover, the typical value $m\sim {10}^{-22}$ eV can only be obtained with a number of ultrarelativistic degrees of freedom at decoupling in the trillions, which is impossible for decoupling in the radiation dominated era. The situation for lighter DM particles is even worse and makes the exclusion result even stronger.

This paper is organized as follows:

In Section 2, we analyze the Bose–Einstein condensate as a dark matter candidate. Our treatment applies to any shape of the distribution function and is valid for any particle physics model. A detailed and updated analysis of the BEC phase space density and BEC coarse-grained phase space density is provided in Section 3 and Section 4, both in theory and observations.

Section 3 provides an updated synthesis and clarification on the phase density in DM and a useful state-of-the art. From such analysis, robust DM BECs constraints are derived in Section 4.

DM BEC decoupling at Thermal Equilibrium is treated in Section 5, including its implications for the DM axion. DM BEC in decoupling out of Thermal Equilibrium is treated in Section 6, including its implications for the typical Jeans Lengths and for the BEC number of ultra-relativistic degrees of freedom.

In Section 7, we discuss inhomogeneous, gravitationally bounded BE condensates of finite size. The constraints from the galactic phase space density remarkably provide similar values to the homogeneous BEC constraints found in Section 4 and Section 5, and confirm the generic character and robustness of the results.

In Section 8, we analyze DM axions in the canonical axion vacuum misalignment scenario [30,31,32,33,34] and we constraint it with the DM phase space density and galactic data. The exclusion constraints obtained for the axion as a DM candidate confirm the universal constraints obtained in this paper for the DM BECs in general.

Section 9 summarizes our results, conclusions and remarks. We notice that axions with masses in the meV = ${10}^{-3}$ eV range can play an important role in astrophysics and cosmology [35,36,37,38,39,40,41], not for dark matter but for dark energy as we proposed and studied in Ref. [35], and the misalignment scenario could produce axions with mass in such an meV scale.

2. The Bose–Einstein Condensate (BEC) as a Dark Matter Candidate

• After decoupling, the DM distribution function freezes out and is a function of the covariant momentum p. We consider generic distribution functions $f_d$ out of thermal equilibrium or thermal. The specific form of $f_d$ in the non-thermal cases depends on the details of the interactions before decoupling.
• Our treatment applies to any shape of $f_d$ and is valid for any particle physics model. For convenience and without loss of generality, we choose $f_d$ as a function of $(p/T_d):f_d(p/T_d)$, where $T_d$ is the covariant decoupling temperature.

In a BEC, a sizeable fraction of the particles is in the zero momentum state, while the rest is on excited states. We call ${\rho }_0$ the zero-momentum comoving contribution to the mass density. The contribution from the excited states $\rho -{\rho }_0$ follows as usual by integrating the distribution function. When the particles became nonrelativistic, we thus have

$$\rho -{\rho }_0=m{\int }_0^{\infty }\frac{p^{2}dp}{2{\pi }^2}{f}_d\left(\frac{p}{T_d}\right)=\frac{m}{2{\pi }^2}{T}_d^{3}U,$$

(3)

$$U\equiv {\int }_0^{\infty }{y}^2{f}_d\left(y\right)dy,$$

(4)

where we consider neutral scalars. The case where the particles remain ultra-relativistes (UR) is considered in Equation (29) below.

The BEC density ${\rho }_0$ vanishes at the BEC covariant critical temperature $T_c$. Therefore, the BEC can be present if $T_d<T_c$ and we have from Equation (3) [42],

$$\rho =\frac{mU}{2{\pi }^2}{T}_c^3,{\rho }_0=\frac{mU}{2{\pi }^2}\left(T_c^3-T_d^3\right),T_d<T_c.$$

(5)

[$T_c$ is defined by the above equation even in the out of thermal equilibrium case]. $T_d$ and $T_c$ are related to the respective effective number of UR degrees of freedom $g_d$ and $g_c$, and to the photon temperature today $T_{\gamma }$ by entropy conservation [30]:

$$T_c={\left(\frac{2}{g_c}\right)}^{1/3}{T}_{\gamma },T_d={\left(\frac{2}{g_d}\right)}^{1/3}{T}_{\gamma },T_{\gamma }=0.2348\times {10}^{-3}\mathrm{eV}.$$

(6)

The DM density $\rho$ must reproduce the observed average DM in the universe ${\Omega }_{DM}{\rho }_{\mathrm{crit}}$. Hence,

$${\rho }_{DM}\equiv {\Omega }_{DM}{\rho }_{\mathrm{crit}}=\frac{mU}{{\pi }^2{g}_d}{\left(\frac{T_c}{T_d}\right)}^3{T}_{\gamma }^3,$$

(7)

$${\rho }_{DM}=0.9259\times {10}^{-23}{\mathrm{keV}}^4$$

Therefore, the DM particle mass m can be related to $(T_d/T_c)$ as

$$m={\pi }^2\frac{{\rho }_{DM}{g}_d}{T_{\gamma }^{3}U}{\left(\frac{T_d}{T_c}\right)}^3=7.059\mathrm{eV}\frac{g_d}{U}{\left(\frac{T_d}{T_c}\right)}^3.$$

(8)

The value of $g_d$ depends on the detailed particle physics of the light scalar particle. For QCD axions decoupling soon after the QCD phase transition, one has $g_d\sim 25$. The covariant critical temperature $T_c$ as well as $g_c$ are parameters that depend on the BEC state.

$$\mathrm{It} \mathrm{must} \mathrm{be} g_c<g_d(\mathrm{and} \mathrm{hence} T_d<T_c) \mathrm{in} \mathrm{order} \mathrm{to} \mathrm{have} \mathrm{a} \mathrm{BEC}.$$

The continuous and bounded function $f_d\left(p/T_d\right)$ stands for the excited states of the distribution function. The total distribution function can be written as a Dirac delta function representing the zero momentum BEC plus the excited states piece $f_d\left(p/T_d\right)$, as follows

$$f_d^{total}\left(p\right)={\left(2\pi \right)}^3\frac{{\rho }_0}{m}\delta \left(\overrightarrow{p}\right)+f_d\left(\frac{p}{T_d}\right)=2{\pi }^2\frac{{\rho }_0}{m}\frac{\delta \left(p^2\right)}{p^2}+f_d\left(\frac{p}{T_d}\right)$$

(9)

The continuum Dirac delta notation is convenient for calculations but in reality the BEC is in a finite comoving volume $V_c$ and the wavenumbers $\overrightarrow{p}$ are discretized:

$$\overrightarrow{p}=\frac{2\pi }{V_c^{1/3}}\overrightarrow{n},\overrightarrow{n}ϵ{\mathcal{Z}}^3,\delta \left(\overrightarrow{p}\right)=\frac{V_c}{{\left(2\pi \right)}^3}{\delta }_{\overrightarrow{p},\overrightarrow{0}}.$$

Therefore, ${\displaystyle \delta (\overrightarrow{0})=\frac{V_c}{{\left(2\pi \right)}^3}}$ is finite as well as $f_d^{total}\left(0\right)$:

$$f_d^{total}\left(0\right)=\frac{{\rho }_0}{m}{V}_c+f_d\left(0\right).$$

The comoving square velocity $<v^2>$ can be then expressed as

$$<v^2>=\frac{<p^2>}{m^2}=\frac{1}{m^2}\frac{{\int }_0^{\infty }{p}^4{f}_d\left(p/T_d\right)dp/\left(2{\pi }^2\right)}{({\rho }_0/m)+{\int }_0^{\infty }{p}^2{f}_d\left(p/T_d\right)dp/\left(2{\pi }^2\right)}$$

(10)

$$<v^2>=\frac{T_d^5}{m^2{T}_c^3}\frac{V}{U},V\equiv {\int }_0^{\infty }{y}^4{f}_d\left(y\right)dy$$

(11)

and U is given by Equation (4). The BEC does not contribute to the integral in the numerator of $<v^2>$ in Equation (10) because the integral of the Dirac delta function in $f_d^{total}$ Equation (9) vanishes upon integration over $\overrightarrow{p}$ due to the extra $p^2$ factor in the numerator of Equation (10) with respect to the denominator.

As noticed in Ref. [42] the Bose–Einstein distribution, for massless particles is of the order of the comoving volume $V_c$, as discussed above. The BEC distribution function is well defined. This is also the case for massive particles because the chemical potential at decoupling ${\mu }_d$ must be equal to the particle mass m in order to form a BEC [42]. Even the part of the distribution function that describes the particles outside the condensate is of the order of $V_c$.

Therefore, the two pieces of the total distribution function: the BEC part and the excited states part $f_d\left(p/T_d\right)$ are of the order of the comoving volume $V_c$, which in this case is very small. This is so because for the QCD phase transition $g_d\sim 25$ and $(z_d+1)\sim 1.7\times {10}^{12}$, giving for $V_c$,

$$V_c=\frac{V_{today}}{{(z_d+1)}^3}\sim 2\times {10}^{-37}{V}_{today}.$$

(12)

Taking for example $V_{today}^{1/3}=1$ kpc, yields $V_c^{1/3}\sim \mathrm{18,200}$ km.

3. The BEC Phase Space Density

Let us discuss now the phase space density, which is defined by [43]:

$${\displaystyle Q\equiv \frac{\rho }{<{\sigma }^2{>}^{3/2}}=\sqrt{27}\frac{\rho }{<v^2{>}^{3/2}}=\sqrt{27}\frac{m^3\rho }{{〈{\overrightarrow{P}}_f^2〉}^{3/2}},}$$

(13)

where ${\sigma }^2=<v^2>/3$ is the velocity dispersion, and ${\overrightarrow{P}}_f$ is the physical momentum.

Including explicitly the BEC, using Equations (5) and (10), the phase space density Q Equation (13) is given by

$${\displaystyle Q=\sqrt{27}{m}^3\frac{{\left[({\rho }_0/m)+{\int }_0^{\infty }{p}^2{f}_d\left(p/T_d\right)dp/\left(2{\pi }^2\right)\right]}^{5/2}}{{\left[{\int }_0^{\infty }{p}^4{f}_d\left(p/T_d)\right)dp/\left(2{\pi }^2\right)\right]}^{3/2}}}$$

(14)

$$Q=\frac{3\sqrt{3}}{2{\pi }^2}{m}^4\frac{U^{5/2}}{V^{3/2}}{\left(\frac{T_c}{T_d}\right)}^{15/2}$$

• In the absence of self-gravity Q is the Liouville invariant because both $\rho$ and ${〈{\overrightarrow{P}}_f^2〉}^{3/2}$ redshift as ${(z+1)}^3$.
• Because the distribution function is frozen and is a solution of the collisionless Boltzmann (Liouville) equation, it is clear that Q is a constant, namely a Liouville invariant, in the absence of self-gravity [42].
• The value of Q given by Equation (14) is valid after decoupling and before structure formation when Q is invariant under the universe expansion.

4. The BEC Coarse-Grained Phase Density Constraint

The expression of Q Equation (13) provides a good approximation to the coarse-grained phase-space density. Given the central role played by the phase-space density in the galaxy DM context, we derive here the connection between the coarse-grained phase-space distribution function $F(\mathbf{r},\mathbf{v})$ and $Q\left(\mathbf{r}\right)$.

The matter density express in terms of $F(\mathbf{r},\mathbf{v})$ as

$$\rho \left(\mathbf{r}\right)=\int d^{3}vF(\mathbf{r},\mathbf{v})=<v^2{>}^{3/2}\int d^{3}wF(\mathbf{r},\mathbf{w}),$$

(15)

where we change the integration variable to $\mathbf{w}\equiv \mathbf{v}/\sqrt{<v^2>}$.

We find from Equation (15) the inequality

$$\frac{\rho \left(\mathbf{r}\right)}{<v^2{>}^{3/2}}=\int d^{3}wF(\mathbf{r},\mathbf{w})\ge {\int }_{\left|\mathbf{w}\right|\le 1}{d}^{3}wF(\mathbf{r},\mathbf{w})$$

where we used the positiveness of the distribution function $F(\mathbf{r},\mathbf{w})\ge 0$.

Now, we can apply the first mean value theorem [44] to the last integral over $\mathbf{w}$,

$$Q\left(\mathbf{r}\right)\equiv \frac{\rho \left(\mathbf{r}\right)}{<v^2{>}^{3/2}}\ge {\int }_{\left|\mathbf{w}\right|\le 1}{d}^{3}wF(\mathbf{r},{\mathbf{w}}_1)=\frac{4\pi }{3}F(\mathbf{r},{\mathbf{w}}_1),$$

(16)

where ${\mathbf{w}}_1$ is a point inside the unit sphere.

• We find that Q approximates the coarse-grained phase-space distribution function by excess by a factor of order one. Previous estimations [45] of Q yielded values similar to the rigorous derivation presented here.
• Therefore, Equation (13) provides an appropriate coarse-grained phase-space density of Q. The phase-space density expressions Equation (13) are always valid: in the primordial universe where it is constant as well as afterwards in the presence of self-gravity when structure formation occurs.

Tremaine and Gunn [46] argued that the value of the coarse grained phase space density is always smaller than, or equal to, the maximum value of the (fine grained) microscopic phase space density, which is the distribution function.

Such argument relies on the theorem that states that the coarse-grained phase space density can only diminish by collision-less phase mixing or violent relaxation by gravitational dynamics [47].

A similar argument was presented by Dalcanton and Hogan [43,48], and confirmed by numerical studies.

In Ref. [6], Q was implemented in a model independent manner to evaluate the DM particle mass scale in non-BEC DM.

The phase-space density Q is a constant in the absence of self-gravity, and Q can only decrease by collision-less phase mixing or self-gravity dynamics [43,46,47,48,49,50,51,52]. For these reasons, $Q^{-1}$ behaves as an entropy that can only increase or stay constant during the universe expansion.

Therefore, necessarily:

$$Q_{\mathrm{today}}\le Q,\mathrm{where}{Q}_{\mathrm{today}}\equiv \frac{Q}{Z}$$

(17)

$Z\ge 1$ being a numerical constant, namely the decreasing factor Z introduced in Ref. [6]. The value of $Q_{\mathrm{today}}$ can be computed with galaxy data today for $\rho$ and $<{\sigma }^2>$, namely

$${\displaystyle Q_{\mathrm{today}}=\frac{{\rho }_{\mathrm{today}}}{<{\sigma }_{\mathrm{today}}^2{>}^{3/2}}}$$

(18)

Normally, $\rho$ and ${\sigma }^2$ are averaged over the galaxy core.

$Q_{\mathrm{today}}$ has been well measured by different galaxy observations and it is galaxy dependent. $Q_{\mathrm{today}}$ is the largest for ultracompact dwarf galaxies and the smallest for large and dilute spiral galaxies [53,54,55,56,57,58,59,60,61,62]. From the compilation of different and well established sets of galaxy data, e.g., in Table 1 of Ref. [15] we have

$$5\times {10}^{-6}<{\left(\frac{Q_{\mathrm{today}}}{{\mathrm{keV}}^4}\right)}^{2/3}<1.4.$$

(19)

Now, from Equations (8) and (14), we express m and $(T_d/T_c)$ in terms of $Q,U$ and V with the result:

$$\begin{array}{c}m=\frac{2^{2/3}}{3{\pi }^2}V{Q}^{2/3}\frac{T_{\gamma }^5}{{\left({\rho }_{DM}{g}_d\right)}^{5/3}}=44.62\mathrm{keV}{\left(\frac{Q}{{\mathrm{keV}}^4}\right)}^{2/3}{\left(\frac{25}{g_d}\right)}^{5/3}V,\\ (\frac{T_d}{T_c}{)}^3=\frac{2^{2/3}}{3{\pi }^4}\mathit{UV}\frac{T_{\gamma }^8}{({\rho }_{DM}{g}_d{)}^{8/3}}{Q}^{2/3}=248.43(\frac{Q}{{\mathrm{keV}}^4}{)}^{2/3}\mathit{UV}(\frac{25}{g_d}{)}^{8/3}.\end{array}$$

(20)

$g_d\sim 25$ corresponds to DM decoupling just after the QCD phase transition as it is the case for axions.

5. Decoupling at Thermal Equilibrium

For decoupling at thermal equilibrium (TE), from Equations (3) and (10), we have

$$U^{TE}=2\zeta \left(3\right)=2.404114,V^{TE}=24\zeta \left(5\right)=24.88627$$

From these values, Equation (20) yields

$$\begin{array}{c}\frac{T_d}{T_c}=24.58709(\frac{Q}{{\mathrm{keV}}^4}{)}^{2/9} (\frac{25}{g_d}{)}^{8/9}\\ m=1.110\mathrm{MeV}\left(\frac{Q}{{\mathrm{keV}}^4}{)}^{2/3}\right(\frac{25}{g_d}{)}^{5/3} :\mathrm{TE}.\end{array}$$

(21)

Therefore, from Equations (21) and (19) the condition $Q\ge Q_{\mathrm{today}}$ implies

$$\frac{T_d}{T_c}\ge 27.5(\frac{25}{g_d}{)}^{8/9} , m\ge 1.55\mathrm{MeV}(\frac{25}{g_d}{)}^{5/3} :\mathrm{TE}.$$

(22)

We see that for $g_d\sim 25$, is always $T_d>T_c$ and hence no BEC forms for TE decoupling.

In order to form a BEC, it is necessary that the decoupling temperature $T_d$ be below the critical temperature $T_c$. The condition

$$T_d\le T_c$$

yields from Equation (22)

$$g_d\ge 1040:\mathrm{BEC},\mathrm{TE}.$$

(23)

• This requires particle models possessing a huge number of particle states and where DM decouples presumably in the Grand Unification scale (GUT), where the number of ultrarelativistic degrees of freedom is in the hundreds (well above the electroweak scale).
• Recall that at the TeV scale in the standard model of particle physics, $g_d\sim 100$ [30]. In addition, for $g_d=1040$ Equation (22) yields

  $$m>3.10\mathrm{keV},g_d=1040:\mathrm{BEC},\mathrm{TE}.$$

  (24)
  This particle mass value is much larger than the DM particle mass appropriate for BEC DM Equation (2) $m\sim {10}^{-22}$ eV. This is a huge difference of the orders of magnitude.

Implications for the Axions

•

QCD axions can decouple well before the QCD phase transition, at temperatures $T_d\sim {10}^{11}$ GeV.

For $T_d\sim {10}^{11}$ GeV we can have $g_d$ in the hundreds and from Equation (22) the axion mass m turns to be in the keV scale, a huge difference of orders of magnitude above the mass range values for the axion mass given by the present experimental limits [2]:

$$6\times {10}^{-6}\mathrm{eV}<m_a<2\times {10}^{-3}\mathrm{eV},$$

and even more, a huge difference with respect to the typically ultra-light BEC mass value Equation (2) $m\sim {10}^{-22}$ eV.

Besides, it must be noticed that an isolated system which is not integrable and can thermalize because is an ergodic system both classically and quantum mechanically [63]. Namely, the particle trajectories explore ergodically the constant energy manifold in phase-space, covering it uniformly according to precisely the microcanonical measure and yielding to a thermal situation [63]. This is the case for axions and more generally for QCD systems. Moreover, it is generally the case of self-gravitating DM particles in galaxies [16,17,18,64,65].

In order to determine how the thermalization happens, as well as the thermalization time scale and further physical features, specific calculations must be performed in the corresponding model, see Refs. [2,66,67,68,69,70]. The methods used in these calculations are appropriately chosen for the system considered: Boltzmann equations, classical evolution equations, Schwinger–Keldysh approach for quantum fields, expansions in 1/N in field theory and other methods.

6. Decoupling out of Thermal Equilibrium

For decoupling out of TE we recall that typically, thermalization is reached by the mixing of the particle modes and the scattering between particles, which redistribute the particles in the phase space as following:

The higher momentum modes are populated by a cascade, whose wave front moves towards the ultraviolet region akin to a direct cascade in turbulence, leaving in its wake a state of nearly local TE but with a temperature lower than that of equilibrium [66,71].

Hence, when the dark matter particles at decoupling are not at thermodynamical equilibrium, their momentum distribution is expected to be peaked at smaller momenta than in the TE case because the ultraviolet cascade is not yet completed [66,71]. Therefore, the distribution function at decoupling out of TE can be written as

$$f_d^{outTE}\left(p\right)=\frac{f_0}{e^{\frac{p}{\xi T_d}}-1}\theta (p^0-p),$$

(25)

where $\xi =1$ at TE and $\xi \lesssim 1$ before the thermodynamical equilibrium is attained; $f_0\sim 1$ is a normalization factor and $p^0$ cuts the spectrum in the ultraviolet region not yet reached by the cascade.

• The above features and the distribution function out of TE Equation (25) are generic and universal, the result is unique irrespective of the different ways the massive bosons forming a BEC can be out of TE, because the formation of a BEC is a unique process requiring one universal condition $T_d\le T_c$.
• Equation (25) describes out of equilibrium massive scalar particles [66,71]. Out of equilibrium massless scalar particles were studied in Refs. [67,68,69,72].

The distribution function Equation (25) yields for U and V through Equations (3) and (10):

$$U^{outTE}=f_0{\xi }^{3}U\left(s\right),V^{outTE}=f_0{\xi }^{5}V\left(s\right),s\equiv \frac{p^0}{\xi T_d},$$

$$U\left(s\right)\equiv {\int }_0^s\frac{y^{2}dy}{e^y-1},V\left(s\right)\equiv {\int }_0^s\frac{y^{4}dy}{e^y-1}.$$

Then, from Equation (20), the condition $Q\ge Q_{\mathrm{today}}$ implies for decoupling out of TE:

$$\begin{array}{ccc}\hfill & & m\ge 44.62\mathrm{keV}{\left(\frac{Q_{\mathrm{today}}}{{\mathrm{keV}}^4}\right)}^{2/3}{\left(\frac{25}{g_d}\right)}^{5/3}{V}^{outTE}\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill & & m\ge 1.110\mathrm{MeV}{\left(\frac{Q_{\mathrm{today}}}{{\mathrm{keV}}^4}\right)}^{2/3}{\left(\frac{25}{g_d}\right)}^{5/3}{f}_0{\xi }^5\frac{V\left(s\right)}{V(\infty )},\hfill \\ & & {\left(\frac{T_d}{T_c}\right)}^3\ge 1.48635\times {10}^4{\left(\frac{Q_{\mathrm{today}}}{{\mathrm{keV}}^4}\right)}^{2/3}{f}_0^2{\xi }^8\frac{U\left(s\right)V\left(s\right)}{U(\infty )V(\infty )}:\mathrm{out}\mathrm{of}\mathrm{TE}.\hfill \end{array}$$

(27)

Typically, out of TE we have

$$s=\mathcal{O}\left(1\right),f_0=\mathcal{O}\left(1\right),U\left(1\right)/U(\infty )=0.147,$$

$$V\left(1\right)/V(\infty )=0.00658$$

From the bound Equation (26), the limiting condition $T_d\sim T_c$ for the presence of a BEC is satisfied for $s\sim 1,\xi \sim 0.7,f_0\sim 1$. As a consequence, in the BEC limiting case of decoupling out of TE, we find:

$$T_d\sim T_c,f_0{\xi }^5\frac{V\left(s\right)}{V(\infty )}\sim {10}^{-3},m\ge 14\mathrm{eV}{\left(\frac{25}{g_d}\right)}^{\frac{5}{3}}:\mathrm{BEC}\mathrm{out}\mathrm{of}\mathrm{TE}.$$

(28)

We conclude that:

• BEC DM decoupling at thermal equilibrium requires a particle model with a huge number $g_d\ge 1040$ of particle states ultrarelativistic at DM decoupling [Equation (23)]. For $g_d=1040$ the particle mass must be $m>3$ keV [Equation (24)], that is, twenty-five orders of magnitude larger than the appropriate BEC mass value Equation (2).
• BEC DM decoupling out of thermal equilibrium requires for $g_d\sim 25$ a particle mass m of at least 0.03 eV [Equation (28)]. For $T_d\sim {10}^{11}$ GeV, $g_d$ is in the hundreds and we obtain from Equation (28) m at least twenty orders of magnitude larger than the appropriate BEC mass value Equation (2).

6.1. Implications for the BEC Jeans Lengths

From Equation (1) the Jeans lengths corresponding to the above cases Equations (24) and (28) are:

$${\lambda }_J\left(\mathrm{keV}\right)=3.8\times {10}^4\mathrm{km}\mathrm{for}\mathrm{TE},$$

$${\lambda }_J\left(10\mathrm{eV}\right)=6.9\times {10}^6\mathrm{km}\mathrm{for}\mathrm{out}\mathrm{of}\mathrm{TE}.$$

• These BEC Jeans length values are unrealistically small by eleven to thirteen orders of magnitude [see Equation (1)] in order to form the observed galaxy structures. Namely, DM structures of all sizes above these minuscule Jeans lengths will be formed in contradiction with astronomical observations. These Jeans length values are even worse than the cold DM Jeans length which is $\sim 3\times {10}^{12}$ km.
• Therefore, the BEC particle masses compatible with the DM average density and the DM galaxy phase-space density constraints, namely:

  $$m>3\mathrm{keV}(\mathrm{in} \mathrm{TE}) \mathrm{and} m>0.03 \mathrm{eV} (\mathrm{out} \mathrm{of} \mathrm{TE} ),$$

  have exceedingly small Jeans lengths, results which strongly disfavour BEC DM.

6.2. Implications for the BEC Number of Ultra-Relativistic Degrees of Freedom

It is interesting to see which value of $g_d$ corresponds to the particle mass value $m\sim {10}^{-22}$ eV appropriate for galaxy structure formation. We find from Equations (2), (22) and (28) that $g_d$ must take the values

$$g_d\sim 2\times {10}^{11}\mathrm{TE};g_d\sim 2\times {10}^{14}\mathrm{out}\mathrm{of}\mathrm{TE}.$$

• These gigantic values of $g_d$ are totally impossible for decoupling in the radiation dominated era. Namely, these values of degrees of freedom are absolutely unrealistic for whatever particle physical model one considers. Hence, there is no way to realize a tiny DM mass $m\sim {10}^{-22}$ eV.

In the case DM stays ultra-relativistic until today, Equation (5) for the DM density becomes

$$\rho =\frac{T_c^4}{2{\pi }^2}W,W\equiv {\int }_0^{\infty }{y}^3{f}_d\left(y\right)dy$$

(29)

This equation is valid both in the BEC case and in the absence of a BEC.

For out of thermal equilibrium decoupling we have from Equation (25) that

$$W^{outTE}<W^{TE}={\pi }^4/15$$

We thus obtain from Equations (6) and (29),

$$g_d^{TE}=0.4443>g_d^{outTE}$$

(30)

• Equation (30) cannot be satisfied because it must always be $g_d\ge 2$ due to the existence of the photon. Therefore, scalar particles, which are ultra-relativistic today cannot describe the DM.
• The treatment we presented here is independent of the particle physics model describing the DM particle and applies to all DM BEC. All the results found here only follow from the gravitational interaction of the particles, their bosonic nature and the robust DM observational constraints from the average DM density ${\rho }_{DM}$ and the DM phase-space density Q.

7. Gravitationally Bounded Bose–Einstein Condensates of Finite Size

So far, we have here considered homogeneous Bose–Einstein condensate (BEC) i.e., a BEC in all the space. Gravitationally bounded BEC’s with a finite size R can also exist. That is to say, a two phase situation in which the BEC is inside the radius R and the normal phase is outside. This gravitationally bounded BEC can be considered as the final stationary state, dynamically produced by a gravitational BEC phase transition.

The results provided by this gravitational BEC study are robust irrespective of any particular particle physics model. A gravitationally bounded object formed by a BEC can be obtained by equating the bosonic quantum pressure and the gravitational pressure.

The BEC quantum pressure is the flux of the quantum momentum, $P_Q=nvp$, p being the minimum momentum from the Heinsenberg principle $p\sim \hslash /R$ and $n=\rho /m$ is the number density of particles. Therefore,

$$P_Q=\rho {\left(\frac{\hslash }{Rm}\right)}^2=\frac{3}{4\pi }\frac{{\hslash }^{2}M}{R^5{m}^2},$$

(31)

where $\rho =\left(3M\right)/\left(4\pi R^3\right)$ and M is the mass of the BEC.

For an object of radius R and mass M the gravitational pressure is

$$P_G=\frac{G{M}^2}{4\pi R^4}.$$

(32)

Thus, $P_Q=P_G$ implies for the BEC size R,

$$R=\frac{3}{GM}{\left(\frac{\hslash }{m}\right)}^2=2.861\times {10}^{-36}\frac{M_{\odot }}{M}{\left(\frac{\mathrm{eV}}{m}\right)}^2\hslash \mathrm{kpc}.$$

(33)

The DM particle mass becomes from Equation (33)

$$m=5.349\times {10}^{-22}\mathrm{eV}\sqrt{\frac{{10}^7{M}_{\odot }}{M}\frac{\mathrm{kpc}}{R}}.$$

(34)

That is, for a typical compact BEC object of kpc size and mass $M\sim {10}^7{M}_{\odot }$ we obtain the BEC DM particle mass in agreement with Equation (2). This remarkable result shows the consistency of the self-gravitating BEC quantum estimate Equations (31)–(33) with the BEC free-streaming length Equation (2).

The phase space density$Q=\rho /{\sigma }^3$ can be estimated following similar lines as above, namely

$$Q=\sqrt{27}\rho {\left(\frac{mR}{\hslash }\right)}^3=\frac{\sqrt{27}}{4\pi }{\left(\frac{m}{\hslash }\right)}^{3}M,$$

(35)

with the result

$$\frac{Q}{{\mathrm{keV}}^4}=0.461\times {10}^{-68}{\left(\frac{m}{{10}^{-22}\mathrm{eV}}\right)}^3\left(\frac{M}{{10}^7{M}_{\odot }}\right).$$

(36)

• BEC objects would correspond to compact halos, i.e., typically M about ${10}^7{M}_{\odot }$, thus, $Q\sim {10}^{-68}$ for the typical $m\sim {10}^{-22}$ eV. That is, Q turns out more than sixty orders of magnitude smaller than the observed values Equation (19).
• Although $m\sim {10}^{-22}$ eV provides reasonable BEC free-streaming lengths [Equation (2)], the corresponding BEC phase-space density turns to be ridiculously small.
  Notice that the value Equation (35) is a maximal value for Q as evaluated from the minimum saturated quantum value of the momentum using the Heisenberg principle. That is to say, Equation (36) is a robust result. In conclusion, a gravitationnally bounded BEC cannot be the DM.

8. Thermal and Non-Thermal Axions

The main DM candidate for a scalar particle forming a BEC condensate is the axion [31,73,74,75,76]. For a report on axionic DM and axion-like particles (‘ALPs’), see for example [77].

In the usual scenario of DM axions, axions decouple soon after the QCD phase transition ($g_d\sim 25$) and then they are assumed (i) to become nonrelativistic, (ii) to thermalize and (iii) to form a BEC [2]. (Ref. [70] recently criticized this scenario). Hence, the bound Equation (22) clearly shows that no DM axion-like BEC can be formed.

For non-thermal axions, the canonical scenario is the axion vacuum misalignment scenario [30,31,34,76,77,78]; see also Refs. [79,80,81,82,83] for more recent discussions. In this scenario, the axion field, denote it $\overline{\theta }$, is not initially at the minimum of its potential and is thus “misaligned” with it. When the axion mass is around a temperature of $T\sim {\Lambda }_{QCD}$, the axion field will roll toward the minimum, overpass it and thereafter oscillate; these are like coherent oscillations.

The axion potential is negligibly small at Temperatures $T>>T_{QCD}$, the effective potential vanishing at a high temperature. As T decreases, the field starts to roll down from its initial value towards its zero value at the minimum of the potential ($\overline{\theta }=0$) and the axion mass starts to be generated. The temperature dependence of the axion mass follows the relation

$$m\left(T\right)\simeq 0.1m(T=0){\left(\frac{{\Lambda }_{QCD}}{T}\right)}^{3.7}$$

(37)

The characteristic axion momentum at birth is $m\left(T_1\right)$. $T_1$ is the temperature when oscillations begin, defined as $m\left(T_1\right)=3H\left(T_1\right)$. For an axion of mass $m\sim {10}^{-5}$ eV, $T_1$ is about 1 GeV, and in general $T_1$ scales as $m^{+0.18}$. The initial axion number density is given by

$$n\left(T_1\right)\simeq \rho \left(T_1\right)/m\left(T_1\right)\simeq m\left(T_1\right){\overline{\theta }}_1^2{(f/N)}^2/2$$

(38)

Without significant entropy production since the starting of axion oscillations, the axion’s contribution to the energy density today can be derived from the ratio $(n/s)$ of the initial axion number density to entropy density s multiplied by the entropy density today $s_0$ times the axion mass m. (If significant entropy is produced, the increasing entropy factor in a comoving volume will reduce the present axion mean density by the same factor). Taking into account anharmonic oscillation effects in the axion motion, the average axion’s contribution to the energy density today is given by

$${\rho }_{DM}={\rho }_{crit}0.13\times {10}^{\pm 0.4}{\Lambda }_{200}^{-0.7}F\left({\overline{\theta }}_1\right){\overline{\theta }}_1^2{\left(\frac{m}{{10}^{-5}\mathrm{eV}}\right)}^{-1.18}$$

(39)

where ${\Lambda }_{200}\equiv {\Lambda }_{QCD}/200$ MeV, ${\overline{\theta }}_1$ is the value of the axion phase field when axion oscillations begin. Its canonical value is ${\overline{\theta }}_1=\pi /\sqrt{3}.F\left({\overline{\theta }}_1\right)$ accounts for anharmonic effects, being $F(\pi /\sqrt{3})\simeq 1.3$.

${\Omega }_{DM}=0.22$ in Equation (39) yields for the axion mass:

$$m=3.77\times {10}^{-5}{J}^{-0.85}\mathrm{eV},J\equiv \frac{3{\overline{\theta }}_1^2{10}^{\pm 0.4}F\left({\overline{\theta }}_1\right)}{{\pi }^2{\Lambda }_{200}^{-0.7}1.3}\sim 1.$$

(40)

Axions are produced as semi-relativistic particles at a temperature much larger than their mass. Axion production from the misalignement scenario is highly non-thermal and forms a zero-momentum population; the phase occupancy is given by:

$$f(p=0)\simeq \frac{m\left(T_1\right){(f/N)}^2}{p_1^3}\sim {10}^{50}{\left(\frac{m}{eV}\right)}^{-2.7}$$

(41)

These axion field oscillations truly correspond to a Bose condensate and are like coherent oscillations.

Likewise, we can estimate the typical axion velocity today

$$v/c\sim {\left(\frac{p}{m}\right)}_{today}\sim \frac{p_1(3K/T_1)}{m}\sim {10}^{-22}{\left(\frac{m}{eV}\right)}^{-0.82}$$

(42)

Moreover, we can also estimate the phase-space density Q using the DM density Equation (39) and the axion velocity Equation (42) with the result

$$Q={\rho }_{crit}2.86\times {10}^{\pm 0.4}{10}^{66}{\Lambda }_{200}^{-0.7}{\left(\frac{m}{\mathrm{eV}}\right)}^{1.28},$$

(43)

which yields:

$$\left(\frac{Q}{{\mathrm{keV}}^4}\right)=3.21\times {10}^{49.4}{10}^{\pm 0.8}{\left(\frac{m}{\mathrm{eV}}\right)}^{1.28}$$

(44)

$$m=4.8\times {10}^{-39}\mathrm{eV}{10}^{\mp 0.6}{\left(\frac{Q}{{\mathrm{keV}}^4}\right)}^{0.78}$$

(45)

From the well-established different sets of galaxy observations [53,54,55,56,57,58,59,60,61,62] and their compilation [15] Equation (19), $Q_{\mathrm{today}}\le Q$ is in the range:

$$1.12\times {10}^{-8}<\left(\frac{Q_{\mathrm{today}}}{{\mathrm{keV}}^4}\right)<1.65,$$

(46)

which implies the axion mass values:

$$m=5\times {10}^{-45}\mathrm{eV}-7\times {10}^{-39}\mathrm{eV}$$

(47)

We see from Equations (40) and (47) that the axion mass in the misalignement scenario is hugely larger than the required values to satisfy the galaxy phase density data.

In addition, in order to reproduce the observed galaxy structures, from the non-linear regime computations, e.g., [1,2], Equation (2), the axion mass needs to have typically the order of magnitude $m\sim {10}^{-22}$ eV. We see from Equations (2) and (40) that the axion mass in the vacuum misalignment scenario is 17 orders of magnitude too large to reproduce the observed galactic structures.

In summary, the axion in the misalignment scenario cannot be the total dark matter, two robust constraints point to the same conclusion: On the one hand, the axion mass is too large by huge orders of magnitude to satisfy the observed phase space density galaxy data, and on the other hand the axion mass is too large too (by 17 orders of magnitude) to describe the observed galaxy structures.

We considered in this section the axion in the canonical misalignment scenario. More recent models can be considered by taking into account various effects, such as: (i) symmetry breaking with parametric resonance [79,80], (ii) anharmonicity effects for an initial value ${\overline{\theta }}_i$ approaching $\pi$ because of fine tuning or special inflationary dynamics [81,82], (iii) a “kinetic misalignment” mechanism [83], in which the axion initial kinetic energy can be larger than the potential energy, thus delaying the onset of axion field oscillations. However, the fundamental picture of axions in the basic misalignment scenario as a coherent zero-momentum condensate of coherent oscillations does not change in all these models (i), (ii), (iii) nor the main range of their velocity, and all these models produce axion masses in the range $(0.1-100)\times {10}^{-3}$ eV. These variations do not change at all the robust axion results found here, as these results do not depend on the details of the particle models, and the exclusion constraints on the axion masses are on huge orders of magnitude.

9. Conclusions

Present experimental limits leave as available a window for the axion mass [2]

$$6\times {10}^{-6}\mathrm{eV}<m_a<2\times {10}^{-3}\mathrm{eV}$$

(48)

• The window Equation (48) disagrees by many orders of magnitude both with the galaxy phase-space density constraint Equation (22) and with the required value $m\sim {10}^{-22}$ galaxy structure constraint in order for the axion to be DM.
• The existence of the axion particle is well motivated from QCD [31,73]. However, as we have observed, the axion cannot be the DM particle. The two observables: the average DM density ${\rho }_{DM}$ in real space and the phase space density Q robustly constrain in an inescapable way both the possibility to form a BEC, e.g., $(T_d/T_c)$, and the DM particle mass m ruling out BEC DM in general, and the BEC axion DM in particular.
• Moreover, the value $m\sim {10}^{-22}$ eV can only be obtained with a number of ultrarelativistic degrees of freedom at decoupling in the trillions, which is impossible for decoupling in the radiation-dominated era.
• In addition, we have also considered inhomogenous gravitationally bounded BECs supported by the bosonic quantum pressure independently of any particular particle physics scenario. For a typical size $R\sim$ kpc and compact object masses $M\sim {10}^7{M}_{\odot }$; they remarkably lead to the same particle mass $m\sim {10}^{-22}$ eV as the BEC free-streaming length. However, the phase-space density for the gravitationally bounded BECs turns out to be more than sixty orders of magnitude smaller than the galaxy observed values.
• We have provided here a generic treatment, independent of the particle physics model and which applies to all DM BEC, in both: in or out of equilibrium situations. We conclude that the BEC cannot be the total DM. The axion can be a candidate to only part of the DM of the universe.
• In all the DM BEC discussion here, it is assumed that axions represent the whole DM in the universe, as is usually the case to investigate the feasibility of a DM candidate. In mixed scenarios where particles other than axions could form a large part of the DM, one could have an axion DM BEC constituing a part of the universe DM.
• In supersymmetric models, the supersymmetric partner of the axion is a fermion called axino, degenerate in mass with the axion. An axino with mass in the keV scale would be a good warm dark matter (WDM) candidate. Actually, an axion (and hence an axino) with particle mass in the keV scale naturally appeared for a decoupling temperature $T_d\sim {10}^{11}$ GeV, [see Equations (22)–(24)].
• We would like to stress that although not being the DM, the axion may play a crucial role in cosmology. The observed dark energy density ${\rho }_{\Lambda }={\left(2.35\mathrm{meV}\right)}^4$ indicates an energy scale in the meV $={10}^{-3}$ eV. This energy value is in the allowed window of the axion masses. Therefore, the axion may be the source of the dark energy through the zero point cosmological quantum fluctuations as we derived in Ref. [35]. In addition, white dwarf stars observations would suggest axions in the range of 2–8 meV [36,37,38,39].
• Overall, a robust conclusion of this paper is that the BEC in general, and the BEC axion in particular, cannot be the total dark matter of the Universe. However, they can play an important role in astrophysics and cosmology. We see indications for an axion mass in the meV range from dwarf stars observations, e.g., [36,37,38,39], and mainly from the dark energy scale as we studied in Ref. [35]. In addition, the misalignment scenario [30,31,32,33,34] may be able to produce axions with mass in the meV range.