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

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## Abstract

**:**

**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 $M\sim {10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ 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

- 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 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.

## 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 $(p/{T}_{d}):\phantom{\rule{0.277778em}{0ex}}{f}_{d}\phantom{\rule{0.277778em}{0ex}}(p/{T}_{d})$, where ${T}_{d}$ is the covariant decoupling temperature.

## 3. The BEC Phase Space Density

- In the absence of self-gravity Q is the Liouville invariant because both $\rho $ and ${\u2329{\overrightarrow{P}}_{f}^{2}\u232a}^{3/2}$ 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

- 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.

**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.

## 5. Decoupling at Thermal Equilibrium

**no BEC forms for TE decoupling**.

- 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$$m>3.10\phantom{\rule{0.277778em}{0ex}}\mathrm{keV}\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{g}_{d}=1040\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{1.em}{0ex}}\mathrm{BEC},\phantom{\rule{3.33333pt}{0ex}}\mathrm{TE}\phantom{\rule{0.277778em}{0ex}}.$$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.

- •
**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]:$$6\times {10}^{-6}\phantom{\rule{0.277778em}{0ex}}\mathrm{eV}\phantom{\rule{0.277778em}{0ex}}<\phantom{\rule{0.277778em}{0ex}}{m}_{a}\phantom{\rule{0.277778em}{0ex}}<\phantom{\rule{0.277778em}{0ex}}2\times {10}^{-3}\phantom{\rule{0.277778em}{0ex}}\mathrm{eV},$$

**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].

## 6. Decoupling out of Thermal Equilibrium

**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 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 ${T}_{d}\le {T}_{c}$.

**decoupling out of TE**:

**decoupling out of TE**, we find:

**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

- 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:$$m\phantom{\rule{0.277778em}{0ex}}>\phantom{\rule{0.277778em}{0ex}}3\phantom{\rule{0.277778em}{0ex}}\mathrm{keV}\phantom{\rule{0.277778em}{0ex}}(\mathrm{in}\mathrm{TE})\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4pt}{0ex}}m\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.03\phantom{\rule{0.277778em}{0ex}}\mathrm{eV}\phantom{\rule{0.277778em}{0ex}}(\mathrm{out}\mathrm{of}\mathrm{TE}),$$
**exceedingly small**Jeans lengths, results which**strongly disfavour BEC DM**.

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

- 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

- 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

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

- BEC objects would correspond to compact halos, i.e., typically M about ${10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$, 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**.

## 8. Thermal and Non-Thermal Axions

**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.

**17 orders of magnitude too large**to reproduce the observed galactic structures.

## 9. Conclusions

- 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 $M\sim {10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$; 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 ${\rho}_{\Lambda}={\left(2.35\phantom{\rule{0.277778em}{0ex}}\mathrm{meV}\right)}^{4}$ 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.

## Author Contributions

## Funding

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

de Vega, H.J.; Sanchez, N.G.
Galaxy Phase-Space Density Data Preclude That Bose–Einstein Condensate Be the Total Dark Matter. *Universe* **2022**, *8*, 419.
https://doi.org/10.3390/universe8080419

**AMA Style**

de Vega HJ, Sanchez NG.
Galaxy Phase-Space Density Data Preclude That Bose–Einstein Condensate Be the Total Dark Matter. *Universe*. 2022; 8(8):419.
https://doi.org/10.3390/universe8080419

**Chicago/Turabian Style**

de Vega, Héctor J., and Norma G. Sanchez.
2022. "Galaxy Phase-Space Density Data Preclude That Bose–Einstein Condensate Be the Total Dark Matter" *Universe* 8, no. 8: 419.
https://doi.org/10.3390/universe8080419