Fermion or Boson Dark Matter?

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.


Introduction
Non-relativistic dark matter in the early universe has a density ( ) h a ρ that scales as 3 a − , and a particle root-mean-square (rms) velocity ( ) rms h v a that scales as 1 a − , where a is the expansion parameter. (Throughout, the sub-index "h" stands for the halo of dark matter.) Note that ( ) ( ) 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 is the density of dark matter in the core of the galaxy. (We use the standard notation in cosmology as defined in [1].) The interest in Equation (1) lies in the ability to measure h v 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 ( ) rms 1 h v in 10 galaxies of the THINGS sample [2], and 46 different galaxies in the SPARC sample [3], obtain results consistent within statistical and systematic uncertainties [4] [5]. We therefore interpret 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 [4] [5] [6] [7] [8]. A convenient overview of these studies, and a discussion of the (apparent?) disagreements with current limits, are presented in [9].
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 [10].  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.

Dark Matter Equation of State
We analyze rotation curves of galaxies in the Spitzer Photometry and Accurate Rotation Curves (SPARC) catalog [3]. An example is presented in Figure 1 [3]. Estimates of * ϒ range from 0.5 to 0.2, see the discussion in Reference [3]. We take the stellar mass-to-light ratio equal to its fitted average * 0.32 ϒ = [5], except for galaxies F574-1 and UGC11914 for which we take * 0.2 ϒ = as in [5].  is consistent with being a constant, with large uncertainties that prevent us from distinguishing fermions from bosons by this direct method.

Fits to Spiral Galaxy Rotation Curves
To gain sensitivity, we integrate numerically (4) and (5), and two similar equations for baryons [4], starting at min r . To start these integrations we need four boundary conditions. We also require the equation of state of dark matter to obtain h ρ given h P (see Appendix A and Appendix B). In References [4] [5] we use these boundary conditions: . We vary these four parameters to minimize the 2 χ between the measured and calculated rotation curves. The mass h m of the dark matter particles is kept fixed in these fits.
In the present analysis we use the following equivalent set of four boundary conditions:   Fits for galaxy F574-1 are presented in Table 1, and Figure 1, Figure 2, and Figure 4. The 2 χ of the fits, as a function of µ′ , are presented in Figure 5.
Note, in Figure 5, that the fits for mions, but the difference in 2 χ is not statistically significant. Note also that the 2 χ increases for fermions as ( ) min r µ′ is raised above zero, so we obtain the following lower bound to the mass of dark matter particles if fermions: 48 eV at 3σ (or 99.7%) confidence, similarly to what we obtained in [4]. For bosons the lower bounds are the actual measurements summarized in Table 4. Finally, note in Figure 5 that the four dark matter scenarios studied in this article are extreme and well motivated cases of interest.     A summary of fits to the rotation curves of several galaxies, selected for their very well measured flat v and core, and reaching deep into the core, are presented in Table 2. The quality of these fits justifies the assumption of thermal equilibrium. For fermions with URTE we plot the distribution of the measured  Therefore, if NR h a′ is equal in the core of relaxed steady state galaxies, we might expect that µ′ is also equal in these galaxies, and hence is also of cosmological origin. If we set 0 µ′ = , each galaxy allows an independent measurement of h m .
The distribution of h m for fermions with URTE is shown in Figure 7. The consistency of these measurements is evidence that µ′ is indeed equal (within uncertainties) for all studied galaxies.
Let us consider the systematic uncertainties in Table 3. Galaxies DDO161 and in the core, so the systematic uncertainties due to the uncertainty of the mass/luminosity ratio * ϒ , is large. Galaxies F568-1, , so the dominant systematic uncertainty is due to the unknown dark matter rotation parameter h k . In addition to the known systematic uncertainties listed in Table 3, there are unknown systematic uncertainties including non-steady state galaxies, extraneous features of the rotation curves, phase space dilution, and systematic uncertainties of the observations. A summary of results for all galaxies listed in Table   2 is presented in rows "Spiral galaxies" of Table 4. In view of our incomplete understanding of systematic uncertainties, we assign the standard deviation of the distributions in Table 2 as the total uncertainties in rows "Spiral galaxies" in Table 4.
Let us examine the 2 χ 's in Table 2. There is generally a preference for boson dark matter over fermion dark matter, but the difference 2 χ ∆ is not statistical-ly significant for individual galaxies, except for UGC11914, see Figure 8. However, the core of UGC11914 is not dominated by dark matter, so the results from this galaxy need to be taken with caution. In Table 2 Table 2. Note, in Table 4, that we have measured dark matter particle masses of order 100 eV (with (15), (17), (23), or (25)). For fermion dark matter, these measurements are in disagreement with limits obtained from dwarf spheroidal (dSph) "satellites" of the Milky Way, assuming that they are dominated by dark matter, i.e. 410 eV h m > from the Pauli exclusion principle, and even more stringent limits with additional assumptions, e.g. the Tremaine-Gunn limit [11]. However, recent studies suggest that dwarf spheroidals are not satellites of the Milky Way, they are on their first entry to the Galaxy, and contain negligible amounts of dark matter [12]

Free-Streaming
Free-streaming is important at expansion parameters of order NR h a′ , 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  k . This precise definition supersedes the qualitative definition of the cut-off wavenumber in previous publications [7]. Equation (7) is consistent with the definition of fs k used in Figures 10-15 [8]. For bosons with URTE or NRTE, there is a tail at large k due to the excess of low momentum dark matter particles in the limit 0 µ′ → − . This tail depends on µ′ , and may have cosmological  Table 4. We verify that perturbations with fs k k < grow due to gravitational instability, i.e. fs where J k is the Jeans wavenumber for collision-less dark matter [9] [16]. Note that for fs k k ≈ it is the slower particles that survive free-streaming.        Figure 15. Same as Figure 13, except that the cut-off factor has a "tail" that applies to boson dark matter, see (9). 4.5 z = .

No Freeze-In and No Freeze-Out
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 h T 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  Table 4.
From Table 4 we conclude that all four extreme scenarios studied in this article, namely, fermions with URTE, bosons with URTE, fermions with NRTE, and bosons with NRTE, with 0 µ′ = , and with with each of 56 spiral galaxies [4] [5] [6], and the measurement of the cosmic microwave background temperature 0 T .  Figure 9 of Reference [20]. 3) We apply the cut-off factor (7) without the "tail", and 4) Update the cosmological parameters to [1]. From Figures 10-13

Galaxy Stellar Mass Distributions
as in Reference [8]. For completeness, we include Figure 14 for We repeat Figure 13 with the cut-off factor with a "tail" that corresponds to bosons with 0.01 µ′ ≈ − The Lyman-α forest allows measurements of the neutral hydrogen density profile along the line of sight to far away quasars (at redshifts 5.5 z ≈ ). 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 fs k is excluded in the range from ≈0.4 Mpc −1 to ≈27 Mpc −1 [24]. So, these two analysis, based on very different data sets, are in tension. For boson dark matter, the long free-streaming "tail" mitigates the tension. This discrepancy needs to be resolved.

Conclusions
From this and previous [4]- [9] studies we arrive at the following conclusions: , where µ is the chemical potential.
2) The present dark matter density of the universe 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 ( ) rms 1 h v 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 0 µ′ = (see Section 5,and (16), (18), (24), and (26)). Thus, we have obtained either a coincidence, or strong evidence that 0 µ′ = . Therefore, we assume 0 µ′ = , and arrive at the four dark matter scenarios studied in this article.

4) With
0 µ′ = , each spiral galaxy allows an independent measurement of the dark particle mass h m . The results are consistent within uncertainties. Table 4 were obtained from data, without reference to any particular extension of the standard model. These measurements are in tension with some limits. A comment on the Tremaine-Gunn limit is made in Section 3, and a comment on the Lyman-α forest limit is included in Section 6. Comments on limits from strong gravitational lensing, and from the UV luminosity function are addressed in [9]. These tensions need to be resolved.

5) The dark particle masses listed in
Nature will have the last word. 6) From the measured values of NR h a′ , and 0 µ′ = , we calculate the warm dark matter cut-off wavenumbers fs k due to free-streaming, see Table 4. 7) Galaxy stellar mass distributions, presented in Figures 10-15 for bosons (the difference is due to the excess of low momentum bosons expected for 0 µ′ → − , which produces a "tail" in the cut-off factor ( ) 2 fs k k τ , see Figure 9). 8) Fits to spiral galaxy rotation curves generally favor boson dark matter, typically as shown in Figure 1 and Figure 5, but the difference in 2 χ for individual galaxies is not statistically significant, see Table 2. An exception is galaxy UGC11914 (see Figure 8), but this case needs to be taken with caution because the core of UGC11914 is not dominated by dark matter. Among the galaxies with the core dominated by dark matter, the one with the largest 2 χ difference between fermions and bosons is NGC0024 with 2 4 χ ∆ = . From the sums of 2 χ 's of galaxies with the core dominated by dark matter, we obtain 2 8.8 χ ∆ = ∑ for URTE, and 11.5 for NRTE, corresponding to a discrimination of 3.0σ or 3.4σ, see Table 2. 9) From Table 4 we observe that fits to spiral galaxy rotation curves obtain agreement with the assumption of no freeze-in and no freeze-out, for each of the four scenarios with 0 µ′ = studied in this article. Table 4 we see that the cut-off wavenumber fs k , measured with the galaxy stellar mass distributions, is in some tension with fermion dark matter. In fact, from Figures 10-15 it is difficult to see how fs k can reach 0.38 or 0.26 Mpc −1 as required by fermions, see Table 4. 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 Table 2 and Table 4, and obtain no significant discrimination between URTE and NRTE.

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