_{1}

^{*}

In Part II of this study of spiral galaxy rotation curves we apply corrections and estimate all identified systematic uncertainties. We arrive at a detailed, precise, and self-consistent picture of dark matter.

Dark matter in the core of spiral galaxies can exceed 10^{7} times the mean dark matter density of the Universe. For this reason we have studied spiral galaxy rotation curves measured by the THINGS collaboration [

〈 v r h 2 〉 ′ ≡ 〈 v r h 2 〉 1 − κ h , (1)

and similarly for baryons. 〈 v r h 2 〉 1 / 2 is the root-mean-square of the radial component of the dark matter particle velocities, and 0 ≤ κ h ≤ 1 describes dark matter rotation, see [

In the present analysis we apply corrections and study all identified systematic uncertainties. We use the standard notation in cosmology as defined in [

The first measured point r min does not lie in the center of the spiral galaxy core, so we make a correction from ρ h ( r min ) to ρ h ( r → 0 ) by numerical integration with the same equations and parameters described above. These corrections are presented in

For each spiral galaxy we obtain the parameter

v h rms ( 1 ) 2 ≡ 3 〈 v r h 2 〉 ( Ω c ρ crit ρ h ( 0 ) ) 2 / 3 ≡ 3 k T h ( 1 ) m h . (2)

v h rms ( 1 ) is the dark matter particles root-mean-square velocity extrapolated to the present time with expansion parameter a = 1 in three dimensions, hence the factor 3. T h ( 1 ) is the temperature of dark matter of a homogeneous Universe at the present time. The parameter v h rms ( 1 ) is invariant with respect to adiabatic expansion of the dark matter. Note that for an ideal “noble” gas

Galaxy | r min [kpc] | ρ h ( r min ) [ 10 − 2 M ⊙ pc − 3 ] | ρ h ( r → 0 ) [ 10 − 2 M ⊙ pc − 3 ] |
---|---|---|---|

NGC 2403 | 0.5 | 7.5 ± 1.4 ( stat ) | 10.3 ± 1.4 ( stat ) ± 0.8 ( syst ) |

NGC 2841 | 4.0 | 9.3 ± 0.7 ( stat ) | 20.8 ± 0.7 ( stat ) ± 6.0 ( syst ) |

NGC 2903 | 1.0 | 14.6 ± 2.1 ( stat ) | 14.7 ± 2.1 ( stat ) ± 0.7 ( syst ) |

NGC 2976 | 0.1 | 4.0 ± 2.7 ( stat ) | 4.06 ± 2.70 ( stat ) ± 0.03 ( syst ) |

NGC 3198 | 1.0 | 4.5 ± 0.8 ( stat ) | 5.3 ± 0.8 ( stat ) ± 1.0 ( syst ) |

NGC 3521 | 1.0 | 22.9 ± 8.6 ( stat ) | 24.6 ± 8.6 ( stat ) ± 0.7 ( syst ) |

NGC 3621 | 0.5 | 2.6 ± 0.5 ( stat ) | 2.93 ± 0.50 ( stat ) ± 0.10 ( syst ) |

DDO 154 | 0.25 | 1.3 ± 0.3 ( stat ) | 1.36 ± 0.30 ( stat ) ± 0.10 ( syst ) |

NGC 5055 | 1.0 | 28.2 ± 6.8 ( stat ) | 37.3 ± 6.8 ( stat ) ± 9.0 ( syst ) |

NGC 7793 | 0.25 | 8.0 ± 1.6 ( stat ) | 8.98 ± 1.6 ( stat ) ± 0.5 ( syst ) |

T h V γ − 1 = constant with γ = 5 / 3 . By “noble” we mean that collisions (if any) between dark matter particles do not excite internal degrees of freedom (if any) of these particles. Alternatively, Equation (2) can be understood as v h ∝ 1 / a for non-relativistic particles in an expanding Universe. At expansion parameter a when perturbations are still linear, and after dark matter becomes non-relativistic, the root-mean-square velocity of dark matter particles is

v h rms ( a ) = v h rms ( 1 ) a ≡ ( 3 k T h ( a ) m h ) 1 / 2 . (3)

Results are presented in

v h rms ( 1 ) = 1.192 ± 0.109 ( tot ) km / s . (4)

This result is noteworthy since the 10 galaxies used for these measurements have masses spanning three orders of magnitude, and angular momenta spanning five orders of magnitude [

The expansion parameter a h NR at which dark matter becomes non-relativistic can be estimated from (3) as

Galaxy | 〈 v r h 2 〉 ′ 1 / 2 [km/s] | v h rms ( 1 ) [km/s] |
---|---|---|

NGC 2403 | 101 ± 3 | 1.103 ± 0.083 ( stat ) ± 0.088 ( syst ) |

NGC 2841 | 220 ± 3 | 1.900 ± 0.047 ( stat ) ± 0.232 ( syst ) |

NGC 2903 | 142 ± 3 | 1.377 ± 0.095 ( stat ) ± 0.106 ( syst ) |

NGC 2976 | 129 ± 177 | 1.921 ± 3.061 ( stat ) ± 0.144 ( syst ) |

NGC 3198 | 104 ± 3 | 1.417 ± 0.112 ( stat ) ± 0.139 ( syst ) |

NGC 3521 | 153 ± 10 | 1.250 ± 0.227 ( stat ) ± 0.095 ( syst ) |

NGC 3621 | 126 ± 5 | 2.092 ± 0.202 ( stat ) ± 0.159 ( syst ) |

DDO 154 | 36.5 ± 3.7 | 0.783 ± 0.137 ( stat ) ± 0.062 ( syst ) |

NGC 5055 | 144 ± 4 | 1.024 ± 0.091 ( stat ) ± 0.113 ( syst ) |

NGC 7793 | 85.5 ± 5.0 | 0.977 ± 0.115 ( stat ) ± 0.076 ( syst ) |

Average | 1.192 ± 0.109 ( tot ) |

a h NR ≈ v h rms ( 1 ) c . (5)

There are threshold factors of O(1) presented in Section 5.

We consider the scenario with dark matter dominated by a single type of particle (plus anti-particle) of mass m h . The mass density of a non-relativistic gas of fermions or bosons with chemical potential μ can be written as [

ρ h = 〈 v r h 2 〉 3 / 2 N f , b m h 4 ( 2 π ) 3 / 2 ℏ 3 Σ f , b , (6)

where the sums are

Σ f , b = e μ ′ 1 3 / 2 ∓ e 2 μ ′ 2 3 / 2 + e 3 μ ′ 3 3 / 2 ∓ e 4 μ ′ 4 3 / 2 + ⋯ , (7)

where μ ′ ≡ μ / ( k T h ) , with upper signs for fermions, and lower signs for bosons. The sums for fermions and bosons are Σ f = 0.76515 and Σ b = 2.612 for chemical potential μ = 0 . N f ( N b ) is the number of fermion (boson) degrees of freedom. From (2) and (6) we obtain

m h = [ ( 6 π ) 3 / 2 Ω c ρ crit ℏ 3 v h rms ( 1 ) 3 N f , b Σ f , b ] 1 / 4 . (8)

Note that the measured m h is independent of Ω c ρ crit , see (2). From (4) and (8) we obtain

m h = ( 53.5 ± 3.6 ( tot ) eV ) ⋅ ( 2 N f 0.76515 Σ f ) 1 / 4 , (9)

for fermions, and

m h = ( 46.8 ± 3.2 ( tot ) eV ) ⋅ ( 1 N b 2.612 Σ b ) 1 / 4 , (10)

for bosons. Note that we have obtained these results directly from the fits to the spiral galaxy rotation curves, with no input from cosmology. The uncertainties in (9) and (10) include all statistical and systematic uncertainties listed in

A non-relativistic non-degenerate ideal gas has

μ k T h = − ln ( ν ν Q ) , (11)

where ν ≡ V / N is the volume per particle, and ν Q ≡ [ 2 π ℏ 2 / ( m h k T h ) ] 3 / 2 is the “quantum volume”. For a non-degenerate ideal gas, ν / ν Q ≫ 1 so the chemical potential μ is negative, and increases logarithmically with particle concentration. Fermi-Dirac or Bose Einstein degeneracy sets in as μ → 0 . Note that in an adiabatic expansion μ / ( k T h ) is constant.

Fitting spiral galaxy rotation curves, we obtain limits m h > 16 eV for fermions, and m h > 45 eV for bosons, at 99% confidence [

Consider dark matter in statistical equilibrium with chemical potential μ and temperature T h . This assumption is justified by the observed Boltzmann distribution of the dark matter [

n h a 3 = N f , b ( 2 π ℏ ) 3 ∫ 0 ∞ 4 π p 2 d p 1 exp [ ( m h 2 c 4 + p 2 c 2 / a 2 − m h c 2 − μ ) / ( k T h ) ] ± 1 . (12)

The last factor is the average number of fermions (upper sign) or bosons (lower sign) in an orbital of momentum p / a .

Now let dark matter decouple while ultra-relativistic, and assume no self-annihilation. Then n h a 3 is conserved. In an adiabatic expansion, e.g. collisionless dark matter, the number of dark matter particles in an orbital is constant so μ and T h adjust accordingly. The problem has one degree of freedom, so we choose, without loss of generality, μ ′ ≡ μ / ( k T h ) constant. T h ∝ 1 / a in the ultra-relativistic limit ( k T h ≫ m c 2 ), and T h ∝ 1 / a 2 in the non-relativistic limit ( k T h ≪ m c 2 ). (In the transition between these two limits T h is momentum dependent.) Let us define x ≡ p c / ( a k T h ) , and y 2 ≡ p 2 / ( 2 m h a 2 k T h ) . In the ultra-relativistic limit

n h a 3 = A f , b N f , b ( k a T h ℏ c ) 3 , A f , b = 1 2 π 2 ∫ 0 ∞ x 2 d x exp [ x − μ ′ ] ± 1 . (13)

In the non-relativistic limit

n h a 3 = Σ f , b N f , b ( m h k a 2 T h 2 π ℏ 2 ) 3 / 2 , Σ f , b = 4 π 1 / 2 ∫ 0 ∞ y 2 d y exp [ y 2 − μ ′ ] ± 1 , (14)

as in (6). The intercept of these two asymptotes defines a h NR and T h NR ≡ T h ( a h NR ) = T h ( 1 ) / a h NR 2 :

m h c 2 = 2 π ( A f , b Σ f , b ) 2 / 3 k T h NR , (15)

a h NR = ( 2 π 3 ) 1 / 2 ( A f , b Σ f , b ) 1 / 3 v h rms ( 1 ) c . (16)

For μ = 0 , we obtain for fermions A f = 0.09135 , Σ f = 0.76515 , m h c 2 = 1.523 k T h NR , and a h NR = 0.7126 v h rms ( 1 ) / c ; and for bosons A b = 0.1218 , Σ b = 2.612 , m h c 2 = 0.8139 k T h NR , and a h NR = 0.5209 v h rms ( 1 ) / c . Einstein condensation sets in at μ = 0 .

For μ / ( k T h ) = − 1.5 we obtain for fermions A f = 0.0220 , Σ f = 0.2074 , m h c 2 = 1.409 k T h NR , and a h NR = 0.6852 v h rms ( 1 ) / c ; and for bosons A b = 0.02328 , Σ b = 0.2432 , m h c 2 = 1.315 k T h NR , and a h NR = 0.6620 v h rms ( 1 ) / c .

For μ / ( k T h ) = − 10.0 we obtain for both fermions and bosons A f , b = 4.6 × 10 − 6 , Σ f , b = 4.5 × 10 − 5 , m h c 2 = 1.366 k T h NR , and a h NR = 0.6747 v h rms ( 1 ) / c .

In summary, from the measured adiabatic invariant v h rms ( 1 ) we obtain m h and a h NR with (8) and (16) respectively. The ratio T h / T of dark matter-to-photon temperatures, after e + e − annihilation while dark matter is still ultra-relativistic, is

T h T = 1 2π ( Σ f , b A f , b ) 2 / 3 a h NR m h c 2 k T 0 , (17)

where the photon temperature is T = T 0 / a . Note that T h / T is proportional to v h rms ( 1 ) 1 / 4 , and is proportional to 1 / T 0 . The intercept of the two asymptotes that we implemented allows direct comparison of (17) with T h / T in

We now specialize to the case of zero chemical potential μ = 0 corresponding, in particular, to equal numbers of dark matter particles and anti-particles, or to Majorana sterile neutrinos [

m h = [ 53.5 ± 3.6 ( tot ) ] ⋅ ( 2 N f ) 1 / 4 eV , (18)

a h NR = [ 2.83 ± 0.26 ( tot ) ] × 10 − 6 , (19)

T h T = [ 0.423 ± 0.010 ( tot ) ] ⋅ ( 2 N f ) 1 / 4 (20)

for fermions, or

m h = [ 46.8 ± 3.2 ( tot ) ] ⋅ ( 1 N b ) 1 / 4 eV , (21)

a h NR = [ 2.07 ± 0.19 ( tot ) ] × 10 − 6 , (22)

T h T = [ 0.507 ± 0.012 ( tot ) ] ⋅ ( 1 N b ) 1 / 4 (23)

for bosons. These uncertainties are valid for the considered scenario and include statistical uncertainties and all identified systematic uncertainties listed in

These results can be compared with expectations in

Measurements with individual spiral galaxies for the case of fermions with N f = 2 , e.g. sterile Majorana neutrinos, are presented in

Non-spherical spiral galaxies: Equations (3) to (6) of [

Mixing of dark matter: So long as dark matter is assumed collisionless, the adiabatic invariant v h rms ( 1 ) should be exactly conserved, so we assign no systematic uncertainty to Equation (3).

New studies may require additional systematic uncertainties. However, at present we do not identify any.

Galaxy | 10 6 × a h NR | m h [eV] | T h / T |
---|---|---|---|

NGC 2403 | 2.62 ± 0.29 | 56.7 ± 4.6 | 0.415 ± 0.011 |

NGC 2841 | 4.52 ± 0.56 | 37.7 ± 3.5 | 0.476 ± 0.015 |

NGC 2903 | 3.27 ± 0.34 | 48.0 ± 3.7 | 0.439 ± 0.011 |

NGC 2976 | 4.57 ± 7.28 | 37.4 ± 44.7 | 0.477 ± 0.190 |

NGC 3198 | 3.37 ± 0.42 | 47.0 ± 4.4 | 0.442 ± 0.014 |

NGC 3521 | 2.97 ± 0.59 | 51.6 ± 7.6 | 0.428 ± 0.021 |

NGC 3621 | 4.97 ± 0.61 | 35.1 ± 3.2 | 0.487 ± 0.015 |

DDO 154 | 1.86 ± 0.36 | 73.3 ± 10.5 | 0.381 ± 0.018 |

NGC 5055 | 2.43 ± 0.34 | 59.9 ± 6.3 | 0.408 ± 0.014 |

NGC 7793 | 2.32 ± 0.33 | 62.1 ± 6.6 | 0.403 ± 0.014 |

Average | 2.83 ± 0.26 | 46.1 ± 3.3 | 0.432 ± 0.010 |

A numerical integration obtains rotation curves for spiral galaxies [

interest to the present analysis, are ρ h ( r min ) and 〈 v r h 2 〉 ′ 1 / 2 , and are presented in

v h rms ( 1 ) = 1.192 ± 0.109 ( tot ) km / s . (24)

This result is remarkable considering that the ten galaxies span three orders of magnitude in mass, and five orders of magnitude in angular momenta [

We consider dark matter that is dominated by a single type of particle of mass m h . We assume that dark matter decoupled from the Standard Model sector and from self-annihilation while still ultra-relativistic. Then from v h rms ( 1 ) we obtain directly the expansion parameter at which dark matter becomes non-relativistic:

a h NR ≈ v h rms ( 1 ) c , (25)

up to a threshold factor of O(1) presented in Section 5. From the adiabatic invariant v h rms ( 1 ) we also obtain the mass m h of dark matter particles, as a function of the chemical potential μ , with no input from cosmology, see (8).

The fits to spiral galaxy rotation curves allow us to set lower bounds to the dark matter particle mass m h [

To proceed, we need to know the chemical potential μ of dark matter. We consider the scenario with μ = 0 which is appropriate for equal numbers of dark matter particles and anti-particles, or Majorana sterile neutrinos [

The ratio T h / T is proportional to v h rms ( 1 ) 1 / 4 , and proportional to 1 / T 0 , so the result T h / T ≈ 0.4 is highly significant. A different measured adiabatic invariant v h rms ( 1 ) , or a different T 0 , could have obtained T h / T orders of magnitude different from unity, so the measurement T h / T ≈ 0.4 is strong evidence that dark matter was once in thermal equilibrium with the Standard Model sector, and gives added support to the scenario μ ≈ 0 .

We compare the measured T h / T and m h with expectations, see

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

Hoeneisen, B. (2019) A Study of Dark Matter with Spiral Galaxy Rotation Curves. Part II. International Journal of Astronomy and Astrophysics, 9, 133-141. https://doi.org/10.4236/ijaa.2019.92010