Dark matter in the core of spiral galaxies can exceed 107 times the mean dark matter density of the Universe. For this reason we have studied spiral galaxy rotation curves measured by the THINGS collaboration  with the hope of constraining the properties of dark matter  . In “Part I” of this study  we integrate numerically the equations that describe the mixture of two self-gravitating non-relativistic ideal gases, “baryons” and “dark matter”. These equations require four boundary conditions: the densities and of dark matter and baryons at the first measured point , and the “reduced” root-mean-square radial velocities and , defined as follows:
and similarly for baryons. is the root-mean-square of the radial component of the dark matter particle velocities, and describes dark matter rotation, see  for details. In the present analysis we take  . The four boundary parameters are fit to minimize the between the rotation curves and measured by the THINGS collaboration  , and the calculated rotation curves. The fits obtain rotation curves within the observational uncertainties. These fits are presented in Figures 1 to 10 of  , and the fitted parameters are presented in Table 1 of  .
In the present analysis we apply corrections and study all identified systematic uncertainties. We use the standard notation in cosmology as defined in  , and the values of the cosmological parameters therein. Occasionally we use units with and as is customary.
2. Corrections from to
The first measured point does not lie in the center of the spiral galaxy core, so we make a correction from to by numerical integration with the same equations and parameters described above. These corrections are presented in Table 1.
3. Measurement of the Adiabatic Invariant
For each spiral galaxy we obtain the parameter
is the dark matter particles root-mean-square velocity extrapolated to the present time with expansion parameter in three dimensions, hence the factor 3. is the temperature of dark matter of a homogeneous Universe at the present time. The parameter is invariant with respect to adiabatic expansion of the dark matter. Note that for an ideal “noble” gas
Table 1. Corrections from  to . The statistical uncertainty is from the fit  . The systematic uncertainty is from the extrapolation from to .
with . 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 for non-relativistic particles in an expanding Universe. At expansion parameter when perturbations are still linear, and after dark matter becomes non-relativistic, the root-mean-square velocity of dark matter particles is
Results are presented in Table 2. The average of of 10 complete and independent measurements is
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  . Note that the correction in Table 1 has allowed us to include galaxy NGC 2841 in the average (this galaxy was excluded in  because the first measured point at is at the edge of the galaxy core).
The expansion parameter at which dark matter becomes non-relativistic can be estimated from (3) as
Table 2. Presented are from Table 1 of  , and defined in (2). is taken from Table 1.  . The statistical uncertainties of and are correlated  . The systematic uncertainty includes contributions from Table 1 and from . The of these 10 measurements is , so the total uncertainty of the average has been multiplied by , as recommended in  .
There are threshold factors of O(1) presented in Section 5.
4. Dark Matter Mass
We consider the scenario with dark matter dominated by a single type of particle (plus anti-particle) of mass . The mass density of a non-relativistic gas of fermions or bosons with chemical potential can be written as 
where the sums are
where , with upper signs for fermions, and lower signs for bosons. The sums for fermions and bosons are and for chemical potential . ( ) is the number of fermion (boson) degrees of freedom. From (2) and (6) we obtain
Note that the measured is independent of , see (2). From (4) and (8) we obtain
for fermions, and
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 Table 1 and Table 2.
A non-relativistic non-degenerate ideal gas has
where is the volume per particle, and is the “quantum volume”. For a non-degenerate ideal gas, so the chemical potential is negative, and increases logarithmically with particle concentration. Fermi-Dirac or Bose Einstein degeneracy sets in as . Note that in an adiabatic expansion is constant.
Fitting spiral galaxy rotation curves, we obtain limits for fermions, and for bosons, at 99% confidence  . Equivalently, from (9) and (10), we obtain for , and for .
5. Transition from Ultra-Relativistic to Non-Relativistic Dark Matter
Consider dark matter in statistical equilibrium with chemical potential and temperature . This assumption is justified by the observed Boltzmann distribution of the dark matter  . We apply periodic boundary conditions in an expanding cube of volume . The comoving number density of dark matter particles is  :
The last factor is the average number of fermions (upper sign) or bosons (lower sign) in an orbital of momentum .
Now let dark matter decouple while ultra-relativistic, and assume no self-annihilation. Then is conserved. In an adiabatic expansion, e.g. collisionless dark matter, the number of dark matter particles in an orbital is constant so and adjust accordingly. The problem has one degree of freedom, so we choose, without loss of generality, constant. in the ultra-relativistic limit ( ), and in the non-relativistic limit ( ). (In the transition between these two limits is momentum dependent.) Let us define , and . In the ultra-relativistic limit
In the non-relativistic limit
as in (6). The intercept of these two asymptotes defines and :
For , we obtain for fermions , , , and ; and for bosons , , , and . Einstein condensation sets in at .
For we obtain for fermions , , , and ; and for bosons , , , and .
For we obtain for both fermions and bosons , , , and .
In summary, from the measured adiabatic invariant we obtain and with (8) and (16) respectively. The ratio of dark matter-to-photon temperatures, after annihilation while dark matter is still ultra-relativistic, is
where the photon temperature is . Note that is proportional to , and is proportional to . The intercept of the two asymptotes that we implemented allows direct comparison of (17) with in Table 7 of  .
6. Results for the Case
We now specialize to the case of zero chemical potential corresponding, in particular, to equal numbers of dark matter particles and anti-particles, or to Majorana sterile neutrinos  , or to dark matter that was once in diffusive equilibrium with the Standard Model sector. We obtain from the measured adiabatic invariant :
for fermions, or
for bosons. These uncertainties are valid for the considered scenario and include statistical uncertainties and all identified systematic uncertainties listed in Table 1 and Table 2. Systematic uncertainties unknown at present may be needed in the future.
These results can be compared with expectations in Table 7 of  (and its extensions for other and ). Note that is proportional to , and proportional to , so it is highly significant that the measured obtains for . A different measured , or a different , would have lead to the conclusion that and/or dark matter was never in thermal equilibrium with the Standard Model sector. In conclusion, the measured value of is strong evidence that and that dark matter was in thermal equilibrium with the Standard Model sector at some time in the early history of the Universe.
Measurements with individual spiral galaxies for the case of fermions with , e.g. sterile Majorana neutrinos, are presented in Table 3.
7. Additional Systematic Uncertainties?
Non-spherical spiral galaxies: Equations (3) to (6) of  are valid in general. So long as the numerical integration is along a radial direction in the plane of the galaxy, with and , and similarly for baryons, there is no approximation, and no systematic uncertainty is needed.
Mixing of dark matter: So long as dark matter is assumed collisionless, the adiabatic invariant 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.
Table 3. Measurements of the expansion parameter at which dark matter becomes non-relativistic, the dark matter particle mass , and the ratio of temperatures of dark matter-to-photons after annihilation and before dark matter becomes non-relativistic. In this table the particles of dark matter are assumed to be fermions with and . The total uncertainties include the statistical and systematic uncertainties of in Table 2. The 's are 36.4, 40.8, and 40.4 respectively, for 10 - 1 degrees of freedom, so the uncertainties of the averages have been multiplied by , as recommended in  .
A numerical integration obtains rotation curves for spiral galaxies  . This integration requires four parameters (boundary conditions). These parameters are obtained by a fit that minimizes the between the observed  and calculated rotation curves. The fits for ten spiral galaxies, as well as the fitted parameters, are presented in Reference  . The fits are in agreement with observations within observational uncertainties. Two of the measured parameters, that are of
interest to the present analysis, are and , and are presented in
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 . We assume that dark matter decoupled from the Standard Model sector and from self-annihilation while still ultra-relativistic. Then from we obtain directly the expansion parameter at which dark matter becomes non-relativistic:
up to a threshold factor of O(1) presented in Section 5. From the adiabatic invariant we also obtain the mass of dark matter particles, as a function of the chemical potential , 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  , and upper bounds to the dark matter chemical potential , that are not much greater than zero.
To proceed, we need to know the chemical potential of dark matter. We consider the scenario with which is appropriate for equal numbers of dark matter particles and anti-particles, or Majorana sterile neutrinos  , or dark matter that was once in diffusive equilibrium with the Standard Model sector. The upper bound to , obtained from the spiral galaxy rotation curves, is close to zero. A negative chemical potential would imply a dark matter temperature while ultra-relativistic higher than the temperature of the Standard Model sector, which seems implausible. In any case we proceed assuming , and obtain the results (18) to (23).
The ratio is proportional to , and proportional to , so the result is highly significant. A different measured adiabatic invariant , or a different , could have obtained orders of magnitude different from unity, so the measurement is strong evidence that dark matter was once in thermal equilibrium with the Standard Model sector, and gives added support to the scenario .
We compare the measured and with expectations, see Table 7 of  (and extensions with other and ), and find one very good match: fermion dark matter with that decoupled in the approximate temperature range from the confinement-deconfinement transition to , that suggests Majorana sterile neutrino dark matter  ; and one marginal match for a boson with that decoupled in the temperature range from to .
 Srivastava, Y.N., Widom, A. and Swain, J. (1997) Thermodynamic Equations of State for Dirac and Majorana Fermions. hep-ph/9709434.