_{1}

^{*}

We measure properties of dark matter in four well motivated scenarios: fermions with ultra-relativistic thermal equilibrium (URTE), bosons with URTE, fermions with non-relativistic thermal equilibrium (NRTE), and bosons with NRTE. We attempt to discriminate between these four scenarios with studies of spiral galaxy rotation curves, and galaxy stellar mass distributions. The measurements show evidence for boson dark matter with a significance of 3.5σ, and obtain no significant discrimination between URTE and NRTE.

Non-relativistic dark matter in the early universe has a density ρ h ( a ) that scales as a − 3 , and a particle root-mean-square (rms) velocity v h rms ( a ) that scales as a − 1 , where a is the expansion parameter. (Throughout, the sub-index “h” stands for the halo of dark matter.) Note that v h rms ( a ) / ρ h ( a ) 1 / 3 is an adiabatic invariant independent of a . Now consider a free observer in a density peak. This observer feels no gravity, observes dark matter expanding adiabatically, reaching maximum expansion, and then collapsing adiabatically into the core of a galaxy. Note that adiabatic expansion implies

v h rms ( a ) ρ h ( a ) 1 / 3 = v h rms ( 1 ) ( Ω c ρ crit ) 1 / 3 = 3 〈 v r h 2 〉 1 / 2 ρ h ( r → 0 ) 1 / 3 , (1)

where 3 〈 v r h 2 〉 1 / 2 is the root-mean-square velocity of dark matter particles in the core of the galaxy, and ρ h ( r → 0 ) is the density of dark matter in the core of the galaxy. (We use the standard notation in cosmology as defined in [

The adiabatic invariant v h rms ( 1 ) remains constant so long as the mean number of dark matter particles per orbital remains constant, as expected for non-interacting dark matter. The issue of possible phase-space dilution due to galaxy structure formation appears to be secondary, since measurements of v h rms ( 1 ) in 10 galaxies of the THINGS sample [

To obtain T ( a ) and m h separately, we need one more constraint, e.g. the chemical potential μ of dark matter. It turns out that the measured value of v h rms ( 1 ) corresponds to thermal equilibrium between dark matter and the standard model sector in the early universe if μ = 0 . This result is either a coincidence, or strong evidence that the chemical potential of dark matter has the very special value μ = 0 .

Thus we arrive at the following scenario: in the early universe dark matter is in diffusive and thermal equilibrium with the standard model sector, and decouples (from the standard model sector, and from self annihilation) while still ultra-relativistic. In particular, we assume that dark matter has zero chemical potential μ . This no freeze-in and no freeze-out scenario is the result of measurements presented in [

In the no freeze-in and no freeze-out scenario, the ultra-relativistic dark matter is in ultra-relativistic thermal equilibrium (URTE), either Fermi-Dirac, or Bose-Einstein. As the universe expands and cools, dark matter becomes non-relativistic. The momentum distribution of the non-relativistic dark matter particles approaches non-relativistic thermal equilibrium (NRTE) due to dark matter-dark matter elastic interactions [

The purpose of the present study is to try to discriminate between these four alternatives for non-relativistic dark matter with zero chemical potential: fermions with URTE, bosons with URTE, fermions with NRTE, or bosons with NRTE. We investigate the following observables: spiral galaxy rotation curves, and galaxy stellar mass distributions at large redshift z, and compare the results with the expectations of the no freeze-in and no freeze-out assumption.

In the following sections we study the dark matter equation of state, spiral galaxy rotation curves, dark matter free-streaming, the no freeze-in and no freeze-out scenario, and galaxy stellar mass distributions, and, finally, present the conclusions.

We analyze rotation curves of galaxies in the Spitzer Photometry and Accurate Rotation Curves (SPARC) catalog [

v ( r ) 2 = v b ( r ) 2 + v h ( r ) 2 , (2)

v b = | v gas | v gas + ϒ disk | V disk | V disk + ϒ bulge | V bulge | V bulge . (3)

V disk and V bulge are stellar contributions to the rotation velocity inferred from the 3.6 μm SPARC photometry [

Two differential equations of interest to dark matter are Newton’s equation for gravity, and the equation of conservation of the radial component of the momentum of the dark matter particles [

1 r 2 d ( r 2 g h ) d r = 4 π G ρ h ， (4)

d P h d r = − ρ h g ( 1 − κ h ) . (5)

g = v 2 / r = g h + g b is the gravitation field, and κ h is a correction due to dark matter rotation. The definition of pressure P h , for collisional or collision-less dark matter, is presented at the end of Appendix B. We take κ h = 0.15 − 0.15 + 0.35 [

From g h i at every measured radii r i , and the difference equation corresponding to (4), we obtain ρ h i at some point in each interval r i < r < r i + 1 . From the difference equation corresponding to (5), starting at r max , we obtain the accumulated pressure P h i down to radii r i . The root-mean-square of the radial component of the velocities of dark matter particles, v r h rms ( r i ) = [ 2 P h i / ( ρ h i + ρ h ( i − 1 ) ) ] 1 / 2 , is plotted in

In the case of thermal equilibrium, either URTE or NRTE, the equation of state of dark matter has the form P h ( r ) / ρ h ( r ) = f ( T h , μ ( r ) ) , see Appendix A and Appendix B. T h is the dark matter temperature, and μ ( r ) is the dark matter chemical potential. f ( T h , μ ( r ) ) = v r h rms 2 ( r ) = v h rms 2 ( r ) / 3 , where v h rms 2 ( r ) is the mean velocity squared of the dark matter particles (see Appendix B). Excellent fits to galaxy rotation curves are obtained assuming thermal equilibrium [

We define μ ′ ≡ μ / ( k T h ) . For μ ′ ≪ 0 , f ( T h , μ ( r ) ) becomes independent of r . For μ ′ ( 0 ) = 0 in the core of the galaxy, f ( T h , μ ( r ) ) increases (decreases) at the first two or three measured r i for fermions (bosons), see

To gain sensitivity, we integrate numerically (4) and (5), and two similar equations for baryons [

In the present analysis we use the following equivalent set of four boundary conditions: a ′ h NR ( r min ) , V ≡ 3 k T h / ( m h c 2 ) , ρ b ( r min ) , and v r b rms ( r min ) . We are therefore able to keep μ ′ ( r min ) = 0 fixed in the fits (for bosons we need to avoid the singularity at r min , so we start the integration with μ ′ ( r min ) = − 0.1 , −0.01, −0.001, or −0.0001). Furthermore, fitting a ′ h NR ( r min ) , and calculating a ′ h NR ( r ) , we are able to extrapolate to r → 0 and obtain a ′ h NR in the core of the galaxy. In the present analysis, we free the first measured rotation velocity v ( r min ) . The parameters a ′ h NR ( r min ) , V , ρ b ( r min ) , v r b rms ( r min ) , and v ( r min ) are varied to minimize the χ 2 .

Fits for galaxy F574-1 are presented in

Scenario | 10 6 a ′ h NR | 10 4 V | 〈 v r b 2 ( r min ) 〉 1 / 2 | 10 3 ρ b ( r min ) | v ( r min ) |
---|---|---|---|---|---|

[km/s] | [km/s] | [ M ⊙ / pc 3 ] | [km/s] | ||

Fermion URTE | 2.77 ± 0.15 | 164 ± 2 | 11.4 ± 0.9 | 2.3 ± 0.4 | 17.4 ± 4.0 |

Boson URTE | 2.40 ± 0.15 | 169 ± 2 | 11.3 ± 0.8 | 2.4 ± 0.4 | 17.0 ± 4.0 |

Fermion NRTE | 2.84 ± 0.15 | 3.07 ± 0.06 | 11.5 ± 0.9 | 2.3 ± 0.4 | 17.6 ± 4.1 |

Boson NRTE | 1.56 ± 0.12 | 3.45 ± 0.08 | 11.1 ± 0.8 | 2.5 ± 0.5 | 16.7 ± 3.9 |

A summary of fits to the rotation curves of several galaxies, selected for their very well measured v flat and core, and reaching deep into the core, are presented in

Let us consider the systematic uncertainties in

Let us examine the χ 2 ’s in

Galaxy | fermion URTE | boson URTE | fermion NRTE | boson NRTE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

10 6 a ′ h NR | χ 2 | 10 6 a ′ h NR | χ 2 | 10 6 a ′ h NR | χ 2 | 10 6 a ′ h NR | χ 2 | |||||

DDO161 | 4.5 ± 0.2 | 28.9 | 4.0 ± 0.2 | 29.8 | 4.6 ± 0.2 | 28.7 | 2.6 ± 0.2 | 30.6 | ||||

F568-1 * | 3.3 ± 0.3 | 14.2 | 2.9 ± 0.3 | 14.1 | 3.3 ± 0.3 | 14.3 | 2.0 ± 0.3 | 14.4 | ||||

F574-1 * | 2.8 ± 0.2 | 17.2 | 2.4 ± 0.2 | 15.4 | 2.8 ± 0.2 | 17.8 | 1.6 ± 0.1 | 15.4 | ||||

NGC0024 * | 1.8 ± 0.1 | 23.2 | 1.5 ± 0.1 | 19.7 | 1.8 ± 0.1 | 24.7 | 0.9 ± 0.1 | 20.7 | ||||

NGC3109 * | 2.9 ± 0.1 | 9.2 | 2.8 ± 0.2 | 8.5 | 2.9 ± 0.1 | 9.5 | 2.8 ± 0.2 | 8.3 | ||||

NGC3972 * | 3.4 ± 0.3 | 13.1 | 3.0 ± 0.3 | 11.6 | 3.5 ± 0.3 | 13.5 | 2.3 ± 0.3 | 10.3 | ||||

NGC4183 * | 3.3 ± 0.2 | 42.5 | 2.6 ± 0.2 | 43.5 | 3.4 ± 0.2 | 42.3 | 1.8 ± 0.1 | 44.9 | ||||

NGC4559 | 3.4 ± 0.2 | 33.7 | 2.7 ± 0.2 | 30.7 | 3.6 ± 0.2 | 34.6 | 2.0 ± 0.2 | 28.5 | ||||

NGC6503 | 2.1 ± 0.1 | 49.9 | 1.4 ± 0.1 | 51.2 | 2.2 ± 0.1 | 50.2 | 0.7 ± 0.1 | 55.6 | ||||

UGC00731 * | 2.1 ± 0.1 | 11.5 | 1.7 ± 0.1 | 9.9 | 2.2 ± 0.1 | 12.0 | 1.2 ± 0.1 | 8.9 | ||||

UGC06667 * | 2.5 ± 0.2 | 4.5 | 2.2 ± 0.2 | 4.8 | 2.6 ± 0.2 | 4.5 | 1.7 ± 0.2 | 5.6 | ||||

UGC07125 | 2.9 ± 0.3 | 25.1 | 2.3 ± 0.2 | 23.5 | 3.1 ± 0.3 | 25.7 | 1.5 ± 0.2 | 22.1 | ||||

UGC08490 | 1.3 ± 0.1 | 6.6 | 1.0 ± 0.1 | 4.3 | 1.4 ± 0.1 | 7.7 | 0.7 ± 0.1 | 4.5 | ||||

UGC11914 | 1.3 ± 0.1 | 130.5 | 1.1 ± 0.1 | 118.8 | 1.4 ± 0.1 | 134.6 | 0.95 ± 0.1 | 113.5 | ||||

UGC12632 * | 2.2 ± 0.1 | 6.9 | 1.8 ± 0.1 | 6.1 | 2.3 ± 0.1 | 7.3 | 1.4 ± 0.1 | 5.8 | ||||

Average | 2.65 | ∑ χ 2 | 2.23 | ∑ χ 2 | 2.75 | ∑ χ 2 | 1.61 | ∑ χ 2 | ||||

Standard dev. | 0.85 | 417.0 | 0.80 | 391.9 | 0.86 | 427.3 | 0.64 | 389.1 | ||||

9 galaxies * | ||||||||||||

Average | 2.69 | ∑ χ 2 | 2.34 | ∑ χ 2 | 2.77 | ∑ χ 2 | 1.74 | ∑ χ 2 | ||||

Standard dev. | 0.54 | 142.3 | 0.53 | 133.5 | 0.55 | 145.9 | 0.55 | 134.4 | ||||

Galaxy | 10 6 a ′ h NR | stat | ϒ * | k h | v ( r min ) | Total |
---|---|---|---|---|---|---|

0.2 to 0.5 | 0 to 0.5 | ±1σ | ||||

DDO161 | 4.52 | ±0.19 | ±0.20 | ±0.74 | ±0.033 | ±0.79 |

F568-1 | 3.25 | ±0.30 | ±0.08 | ±0.53 | ±0.001 | ±0.62 |

F574-1 | 2.77 | ±0.15 | ±0.03 | ±0.46 | ±0.001 | ±0.48 |

NGC0024 | 1.78 | ±0.06 | ±0.10 | ±0.29 | ±0.001 | ±0.31 |

UGC11914 | 1.25 | ±0.05 | ±0.63 | ±0.29 | ±0.035 | ±0.70 |

UGC12632 | 2.21 | ±0.13 | ±0.04 | ±0.39 | ±0.000 | ±0.41 |

Observable | v h rms ( 1 ) | 10 6 a ′ h NR | k fs | l o g 10 ( M fs / M ⊙ ) | m h |
---|---|---|---|---|---|

[km/s] | [Mpc^{−1}] | [eV] | |||

Fermions URTE | |||||

Spiral galaxies | 0.79 ± 0.26 | 2.65 ± 0.85 | 0.25 − 0.05 + 0.10 | 13.5 ± 0.4 | 107 − 20 + 36 |

No freeze-in/-out | 2.00 to 0.75 | 6.66 to 2.50 | 0.12 to 0.26 | 14.5 to 13.5 | 54 to 112 |

M s distribution | 0.90 − 0.34 + 0.44 | 11.9 ± 0.6 | |||

Bosons URTE | |||||

Spiral galaxies | 0.67 ± 0.24 | 2.23 ± 0.80 | 0.37 − 0.08 + 0.17 | 13.0 ± 0.4 | 124 − 25 + 50 |

No freeze-in/-out | 1.19 to 0.45 | 3.97 to 1.49 | 0.23 to 0.52 | 13.6 to 12.6 | 81 to 168 |

M s distribution | 0.90 − 0.40 + 0.44 | 11.9 ± 0.7 | |||

Fermions NRTE | |||||

Spiral galaxies | 0.82 ± 0.26 | 2.75 ± 0.86 | 0.21 − 0.04 + 0.07 | 13.8 ± 0.4 | 74 − 14 + 24 |

No freeze-in/-out | 1.04 to 0.39 | 3.46 to 1.30 | 0.17 to 0.38 | 14.0 to 13.0 | 62 to 130 |

M s distribution | 0.90 − 0.34 + 0.44 | 11.9 ± 0.6 | |||

Bosons NRTE | |||||

Spiral galaxies | 0.48 ± 0.19 | 1.61 ± 0.64 | 0.92 − 0.24 + 0.54 | 11.8 ± 0.5 | 73 − 17 + 33 |

No freeze-in/-out | 0.36 to 0.14 | 1.21 to 0.45 | 1.19 to 3.00 | 11.5 to 10.3 | 90 to 188 |

M s distribution | 0.90 − 0.40 + 0.44 | 11.9 ± 0.7 |

Note, in

We have made several measurements of a ′ h NR : a ′ h NR = ( 4.17 ± 2.52 ) × 10 − 6 with ten galaxies in the THINGS sample [

Free-streaming is important at expansion parameters of order a ′ h NR , long after dark matter has decoupled, see Section 5. We therefore consider collision-less dark matter with zero chemical potential. A density perturbation corresponds to a temperature fluctuation, i.e. to a change in the momentum distribution of the particles, see Appendix A and Appendix B. The comoving free-streaming distance of a dark matter particle of momentum p = p 1 / a is

d fs ( p 1 ) = ∫ 0 v h ( a ) ⋅ d t a = ∫ 0 c 1 + ( a / [ p 1 / ( m h c ) ] ) 2 d t a , = ∫ 0 dec c ⋅ d a 1 + ( a / [ p 1 / ( m h c ) ] ) 2 H 0 a 2 Ω r a − 4 + Ω m a − 3 . (6)

(We arbitrarily stop the integral at decoupling as further contributions are of order 5%.) Let P ( k ) be the power spectrum of linear density perturbations in the cold dark matter ΛCDM model. k is the comoving wavenumber. The power spectrum for warm dark matter is P ( k ) τ 2 ( k / k fs ) , where τ 2 ( k / k fs ) is a cut-off factor.

Let δ h ( x ) ≡ [ ρ ( x ) − ρ ¯ ] / ρ ¯ be the normalized dark matter density perturbation, and a h ( k ) its Fourier transform. We partition δ h ( x ) into parts that free-stream into different elements of solid angle d Ω . Due to free-streaming of dark matter, the corresponding part of a h ( k ) becomes multiplied by e x p [ i k c o s θ d fs ( p 1 ) ] , where θ is the angle between k and d fs ( p 1 ) . This factor needs to be averaged over Ω , and over the comoving momentum p 1 from 0 to ∞ . The average of the imaginary part is zero, so we need only average c o s ( k c o s θ d fs ( p 1 ) ) . The average of this term over Ω obtains s i n [ k d fs ( p 1 ) ] / [ k d fs ( p 1 ) ] . We take the p 1 average only over the perturbation of the momentum distribution of the dark matter particles (as other free-streaming cancels by detailed balance). The results are presented in

τ 2 ( k / k fs ) = e x p ( − k 2 / k fs 2 ) . (7)

We use τ 2 ( 1 ) ≡ e − 1 as our definition of the free-streaming cut-off wavenumber k fs . This precise definition supersedes the qualitative definition of the cut-off wavenumber in previous publications [

The approximation τ 2 ( k / k fs ) ≈ e x p ( − k 2 / k fs 2 ) is convenient: it allows the definition of the “free-streaming transition mass”

M fs = 4 3 π ( 1.555 k fs ) 3 Ω m ρ crit . (8)

The factor 1.555 comes from the Fourier transform of a 3-dimensional Gaussian. Free-streaming affects the distribution of halo masses with M < M fs . The cut-off wavenumbers k fs , and the free-streaming masses M fs , corresponding to the measured values of a ′ h NR , are summarized in

We assume that dark matter is in thermal and diffusive equilibrium with the standard model sector in the early universe, and decouples (from the standard model sector, and from self-annihilation) while still ultra-relativistic. As the universe expands and cools, standard model particles and anti-particles become non-relativistic and annihilate, heating the standard model sector, without heating dark matter if it has already decoupled. Let T h / T be the ratio of the dark matter-to-photon temperatures after e^{+}e^{−} annihilation (and, in the case NRTE, before dark matter becomes non-relativistic). If dark matter decouples at temperatures T > m t , then T h / T = [ 8 × 43 / ( 427 × 22 ) ] 1 / 3 = 0.332 . If dark matter decouples in the temperature range T C < T < m s , then T h / T = [ 8 × 43 / ( 205 × 22 ) ] 1 / 3 = 0.424 . These numbers can be found in Section 22.3.2 of [

From

Figures 10-13 compare galaxy stellar mass distribution predictions with observations. These figures were taken from Reference [

We repeat

τ 2 ( k / k fs ) = { e x p ( − k 2 / k fs 2 ) for k < k fs , e x p ( − k / k fs ) for k > k fs , (9)

and obtain

These results for k fs are in disagreement with studies of the Lyman-α forest. The Lyman-α forest allows measurements of the neutral hydrogen density profile along the line of sight to far away quasars (at redshifts z ≈ 5.5 ). From the analysis of these density profiles, with model dependent simulations of the inter-galactic medium (including the highly ionized hydrogen), the cut-off wavenumber k fs is excluded in the range from ≈0.4 Mpc^{−1} to ≈27 Mpc^{−1} [

From this and previous [

1) Each spiral galaxy allows a measurement of the adiabatic invariant a ′ h NR ≡ v h rms ( 1 ) / c . We find that a ′ h NR has the same value in the core of all measured relaxed steady state spiral galaxies (within statistical and systematic uncertainties). Therefore, we interpret a ′ h NR to be of cosmological origin: it is the expansion parameter at which dark matter particles become non-relativistic. a ′ h NR determines the ratio of dark matter temperature to mass T h ( a ) / m h in the early universe. To obtain T h ( a ) and m h separately, we need one more constraint, i.e. the value of μ ′ ≡ μ / ( k T h ) , where μ is the chemical potential.

2) The present dark matter density of the universe Ω c ρ crit determines the dark matter particle mass m h as a function of a ′ h NR and μ ′ , see Appendix A and Appendix B. Therefore, if a ′ h NR has the same value in the core of all relaxed steady state spiral galaxies, we can expect the same for μ ′ , so μ ′ may be of cosmological origin.

3) The measured value of v h rms ( 1 ) corresponds to thermal equilibrium of dark matter with the standard model sector in the early universe, with no freeze-in and no freeze-out, if μ ′ = 0 (see Section 5, and (16), (18), (24), and (26)). Thus, we have obtained either a coincidence, or strong evidence that μ ′ = 0 . Therefore, we assume μ ′ = 0 , and arrive at the four dark matter scenarios studied in this article.

4) With μ ′ = 0 , each spiral galaxy allows an independent measurement of the dark particle mass m h . The results are consistent within uncertainties.

5) The dark particle masses listed in

6) From the measured values of a ′ h NR , and μ ′ = 0 , we calculate the warm dark matter cut-off wavenumbers k fs due to free-streaming, see

7) Galaxy stellar mass distributions, presented in Figures 10-15, show convincing evidence that dark matter is warm with a cut-off wavenumber k fs = 0.90 − 0.34 + 0.44 for fermions, or k fs = 0.90 − 0.40 + 0.44 for bosons (the difference is due to the excess of low momentum bosons expected for μ ′ → − 0 , which produces a “tail” in the cut-off factor τ 2 ( k / k fs ) , see

8) Fits to spiral galaxy rotation curves generally favor boson dark matter, typically as shown in

9) From

10) In ^{−1} as required by fermions, see

11) To summarize, among the four well motivated dark matter scenarios studied in this article, measurements show evidence for boson dark matter with a significance of 3.5σ, see

The author declares no conflicts of interest regarding the publication of this paper.

Hoeneisen, B. (2020) Fermion or Boson Dark Matter? International Journal of Astronomy and Astrophysics, 10, 203-223. https://doi.org/10.4236/ijaa.2020.103011

In this Section we consider non-relativistic dark matter in thermal equilibrium with the non-relativistic Fermi-Dirac or Bose-Einstein distributions:

〈 n ( p , T h ) 〉 = 1 e x p [ p 2 / ( 2 m h k T h ) − μ ′ ] ± 1 . (10)

We define

V ≡ 3 k T h m h c 2 , μ ′ ≡ μ k T h . (11)

Note that V is proportional to a − 1 , and is independent of the galaxy dark matter halo radial coordinate r . Note that μ ′ is independent of a , but depends on r (it becomes more negative with increasing r ). We define

Σ f , b ≡ 4 π 1 / 2 ∫ 0 ∞ y 2 d y e x p [ y 2 − μ ′ ] ± 1 , B f , b ≡ ∫ 0 ∞ y 4 d y e x p [ y 2 − μ ′ ] ± 1 . (12)

Then, the density, mean-square velocity, and pressure are

ρ h = N f , b m h 5 / 2 ( k T h ) 3 / 2 Σ f , b 2 3 / 2 π 3 / 2 ℏ 3 , v h rms 2 = 8 k T h B f , b π 1 / 2 m h Σ f , b , P h = ρ h v h rms 2 3 . (13)

From these equations, applied to a homogeneous universe at the present time, we obtain

m h = [ 64 π 3 / 4 Ω c ρ crit ℏ 3 B f , b 3 / 2 v h rms ( 1 ) 3 N f , b Σ f , b 5 / 2 ] 1 / 4 . (14)

For fermions with μ ′ = 0 ,

m h = 78.8 ( 0.76 km / s v h rms ( 1 ) ) 3 / 4 ( 2 N f ) 1 / 4 eV , (15)

T h T = 0.392 ( v h rms ( 1 ) 0.76 km / s ) 1 / 4 ( 2 N f ) 1 / 4 , (16)

where T h / T is the dark matter-to-photon temperature ratio after e^{+}e^{−} annihilation, and before dark matter becomes non-relativistic. For bosons with μ ′ = − 0.001 ,

m h = 51.2 ( 0.76 km / s v h rms ( 1 ) ) 3 / 4 ( 1 N b ) 1 / 4 eV , (17)

T h T = 0.511 ( v h rms ( 1 ) 0.76 km / s ) 1 / 4 ( 1 N b ) 1 / 4 . (18)

In this Section we consider non-relativistic dark matter in thermal equilibrium with the ultra-relativistic Fermi-Dirac or Bose-Einstein distributions:

〈 n ( p , T h ) 〉 = 1 e x p [ p c / ( k T h ) − μ ′ ] ± 1 . (19)

Consider dark matter that is in thermal equilibrium with the standard model sector in the early universe, and decouples (from the standard model sector and from self interactions) while still ultra-relativistic. In this case, the number of dark matter particles per orbital remains unchanged during the transition to a non-relativistic gas. In this Appendix we assume that the dark matter-dark matter elastic interaction cross-section is sufficiently small that dark matter does not reach NRTE in the age of the universe. We define

A f , b ≡ 1 2 π 2 ∫ 0 ∞ x 2 d x e x p [ x − μ ′ ] ± 1 , C f , b ≡ ∫ 0 ∞ x 4 d x e x p [ x − μ ′ ] ± 1 . (20)

Then, the density, mean-square velocity, and pressure of the non-relativistic gas with the momentum distribution corresponding to ultra-relativistic thermal equilibrium (URTE), are

ρ h = m h N f , b A f , b ( k T h ℏ c ) 3 , v h rms 2 = C f , b 2 π 2 A f , b ( k T h m h c ) 2 , P h = ρ v h rms 2 3 . (21)

Note that T h ∝ 1 / a , ρ h ∝ 1 / a 3 , v h rms ∝ 1 / a , and P h ∝ 1 / a 5 . From these equations, applied to a homogeneous universe at the present time, we obtain

m h = [ Ω c ρ crit ℏ 3 C f , b 3 / 2 2 3 / 2 π 3 v h rms ( 1 ) 3 N f , b A f , b 5 / 2 ] 1 / 4 . (22)

For fermions with μ ′ = 0 ,

m h = 111 ( 0.76 km / s v h rms ( 1 ) ) 3 / 4 ( 2 N f ) 1 / 4 eV , (23)

T h T = 0.333 ( v h rms ( 1 ) 0.76 km / s ) 1 / 4 ( 2 N f ) 1 / 4 . (24)

For bosons with μ ′ = − 0.001 ,

m h = 113 ( 0.76 km / s v h rms ( 1 ) ) 3 / 4 ( 1 N b ) 1 / 4 eV , (25)

T h T = 0.379 ( v h rms ( 1 ) 0.76 km / s ) 1 / 4 ( 1 N b ) 1 / 4 . (26)

Let us recall [

v h rms 2 3 = v r h rms 2 ≡ 〈 v r h 2 〉 ≡ ∫ 0 ∞ v r h 2 f ( v r h 2 ) d ( v r h 2 ) ∫ 0 ∞ f ( v r h 2 ) d ( v r h 2 ) , (27)

where f ( v r h 2 ) d ( v r h 2 ) is proportional to the number of dark matter particles with v r h 2 between v r h 2 and v r h 2 + d ( v r h 2 ) with v r h > 0 .