Galactic Haloes from Self-Interacting Neutrinos

The objective of this research is to provide an explanation of galactic haloes using established particles and forces using recent theoretical developments. Light fermions, with masses on the order of 1 eV/c 2 , are not a leading candi-date for dark matter because of their large free-streaming scale length and their violation of the Tremaine-Gunn bound. With a self-interaction of fermions, the free-streaming scaling length is reduced, and the tenets of the Tremaine-Gunn bound are not applicable. Binding of neutrinos via a feeble SU(3) force is considered as a model for such interactions. The assumed sum of masses of the three neutrino flavors is 0.07 eV/c 2 . The resulting form of matter for such bound neutrinos is found to be a degenerate Fermi fluid. Pressure-equilibrium approaches applied to this fluid provide cuspy solutions and match observationally-inferred profiles for galactic haloes. Such approaches also match the observed total enclosed mass for galaxies similar to the Milky Way. The computed structures are found to be stable. The hypothesis is considered in view of observationally-inferred halo-halo interactions and gives results that are consistent with the observed Bullet cluster halo interaction. The theory gives agreement with observationally-inferred properties of dark matter near earth. Questions related to interaction rates, consistency with SN1987a data, the cosmic microwave background, the issue of SU(3) interactions between neutrinos and quarks, free-streaming after neutrino decoupling, and dark-matter abundance are addressed in a companion paper.

The conventional picture of neutrinos as dark matter was ruled out early [10]. The best current model for CDM is the Λ-CDM model [11], which assumes a certain fraction of the matter in the universe is cold (non-relativistic), non-interacting, and stable [12]. However, the model has a number of open issues [13] [14]. In particular, there is no explanation for dark matter. Massive neutrinos as inserted in the Standard Model have been postulated for dark matter [1] but rejected because in the conventional view neutrinos are almost always relativistic particles so any structure would diffuse away quickly and could not lead to the structures observed in the universe today. There are currently many hypothetical explanations for dark matter. Extensions to the Standard Model have been proposed to explain dark matter, dark energy, and other aspects of cosmology, e.g. [15] [16] [17] [18] [19]. QCD-like and nuclear like forces have recently been suggested for self-interacting dark matter, e.g. [20] [21]. In this paper, an SU(3) force applied to neutrinos is hypothesized as an explanation for galactic haloes, and by extension, DM.
The standard view posits that DM was in thermal contact with visible matter in the early universe when the temperature was much greater than the DM mass. In those eras, the DM number density would be comparable to photon number density. If the DM number density was still comparable to the photon number density when it froze out, it would overproduce the observed amount of DM mass when the DM particle mass is more than about 1 eV/c 2 [22]. Thus, there is a need to deplete the abundance of any massive dark matter that is sufficiently cold at recombination. This is the path that leads to DM particles that are required to be largely annihilated in the early universe. An interaction energy of 100 GeV ~ kT leads to a weak interaction cross section that when multiplied by the density and velocity at that time leads to a decay rate comparable to the expansion rate. Here k is Boltzmann's constant and T is the temperature. This would result in freeze-out at that early time in the primordial universe for the corresponding particle mass. Hence DM masses of order 100 GeV would be candidates for DM under these standard assumptions. These would become extremely cold (non-relativistic) over the eons as the universe expanded.
Thus, the most anticipated masses of cold dark matter are associated with weakly interacting massive particles (WIMPs), with masses in the range of 10 to 100 GeV. However, in the absence of significant evidence of massive DM, the community is looking to lighter alternatives. To avoid the free-streaming issue with lighter DM, one approach is for DM to bind and/or cool early in the history of the universe. One might look for particles and/or forces for which such cooling and binding occurs. Such behavior might be analogous to the binding via SU(3) of quarks into hadrons in the early universe.
In the past 4 decades, computationally-intensive approaches have investigated the consistency of lighter DM with astronomical observations. Such investigations began with [10] regarding the possibility of neutrinos for dark matter. Reference [23] investigated consistency of dark matter with Lyman-α lines. Early  [24] with elastic collisions and [25] on gravothermal collapse. Reference [26] simulated subhaloes and [27] considered the impact of fermions. More recently, [28] [29] performed extensive modeling of galaxy formation within larger structures in the Universe, [30] investigated an effective theory for small scale structure, [20] [31] considered SIDM and halo interactions, [32] considered the impact of SIDM on structure and self-assembly history, and [33] modeled SIDM that includes inelastic scattering. There have been recent reviews of SIDM by [21] and of the larger topic of dark matter haloes and subhaloes by [34]. Most of the more recent papers consider particles with masses of the order of a keV/c 2 , as is consistent with observationally inferred values from the latest Lyman-α forest absorption measurements [35] and gravitational lensing measurements [36]. However, both these observationally-inferred mass bounds and the other aforementioned papers rely on assumptions that are not consistent with relativistic fermions with a mass of order 0.1 eV that bind into a number of species of heavier particles in the primordial era, which then further bind into macroscopic structures later.
Section 2 considers the basic application of the SU(3) hypothesis to neutrinos in the early universe. Section 3 compares the predictions of the hypothesis to the observed small-and large-scale structure of the universe. Section 4 discusses the hypothesis in the context of the Tremaine-Gunn bound. Section 5 computes the properties of the proposed bound neutrinos near earth. Section 6 discusses the self-interacting properties of the proposed DM in haloes and between haloes.
Section 7 summarizes the overall findings of this effort.

The Feeble SU(3) Hypothesis and Neutrinos
The hypothesis of a feeble form of SU(3) for the neutrino family is not immediately obvious from the standard model. From the standard model one might expect an interaction energy of the order of the QCD energy scale, ~200 MeV [37]. However, motivation can be found for a feeble SU(3) interaction between neutrinos in a modest extension of the standard model [38]. In this extension, SU(3) is not precluded for the neutrino family. In this theory, neutrino oscillations are direct evidence that neutrinos form bound states via SU (3). As shown in Appendix A, this extension also provides a means for estimating the neutrino interaction strength by scaling the quark interaction strength by ( )   (3) applied to the neutrino family will be denoted SU(3) νs . It will be seen that both forms of SU(3) are consistent with galactic halo data, but the former provides estimates of binding energies and particle masses whereas the standard SU(3) does not. What are the consequences of a feeble analogue of quark confinement with neutrinos? Neutrinos typically start out as ultra-relativistic isolated particles near infinite redshift. In the early universe, the neutrinos would have formed a neutrino/neutrino-gluon plasma, much as would have occurred with quarks and gluons. By analogy with quarks, neutrinos would be bound into "mesonic" or "baryonic" neutrinos, and they would then remain confined to the present day. When excited with sufficient energy, perhaps by hot stellar neutrinos, such bound states might "hadronize" to form additional bound neutrinos while remaining confined, analogous to the behavior of quarks we see today. Table 1 shows that the partition of energy using the standard approach, e.g. [40], applied to the hypothesis. It is seen that both forms of SU(3) can supply a fraction of the mass-energy that corresponds to the modern estimate of the fraction of mass-energy in the dark matter sector, about 84% [41]. The confined baryonic neutrinos would hold the vast reservoir of the mass-energy of SU(3) neutrino gluons shown in Table 1.
When would the above interactions terminate? Such neutrinos should "hadronize" when their total center-of-mass (CM) collision energy exceeds about 4m νμ c 2 , where m νμ is the mass of the muon neutrino, by analogy with the quark families, assuming the normal hierarchy. Hence this limits the maximum kinetic energy to about m νμ c 2 for each particle. Also, there is no interaction pathway to hadronize when the total CM energy of any neutrino state is less than 4m νe c 2 . Here m νe is the mass of the lowest-mass neutrino, the electron neutrino in the normal hierarchy. This gives a range of kinetic energies from m νe c 2 to m νμ c 2 in the CM frame, and about a factor of 2 more in the local rest frame in the early universe. From our current knowledge of neutrino masses in the normal hierarchy, m νe c 2 ~ 0.005 eV and m νμ c 2 ~ 0.01 eV [39]. The implied approximate range of steady-state mean thermal kinetic energy is (3/2)kT = 0.01 to 0.02 eV, i.e. kT = 0.007 to 0.014 eV. For the inverted hierarchy, the range of kT is 0.007 to 0.07 eV assuming appropriate adjustments of the above masses. These estimates for kinetic energies would apply at the end of the hypothesized period of neutrino binding into baryonic neutrinos; further evolution would be expected as the universe expands.

Equations and Inputs for Spatial Structure of DM
The spatial density profile derived from the hypothesis of bound neutrinos is compared to observationally-inferred galactic halo structures. The characteristic scales sizes of galaxies for ordinary radiant matter (OM) are 1 -200 kpc depending on the galaxy and somewhat larger for the associated haloes [42].
To analyze the spatial distribution of low-energy neutrinos or other weakly interacting particles, an N-body simulation [20] [31] or the Vlasov equation is typically preferred [43]. Herein, two simplified governing equations are considered for DM density profiles. The first is the standard equation for hydrostatic equilibrium in a spherically-symmetric geometry. This equation is known to be inadequate for dark matter haloes, as further shown below. This equation is given by Here ρ is the number density of DM, P is the pressure, r is the radius, bv m is the mass of a DM particle, ( ) enc M r is the enclosed mass, and G is the gravitational constant. This equation can be solved using the well-known Lane-Emden formulation [44] if pressure is a function of density only: where c γ is a constant for a given polytropic exponent γ. An inhomogeneous form of the Lane-Emden equation can be used when OM is present. As shown later in this section, initial calculations using Equation (1) where n s is the number of spin states and p is the fermion momentum. Note that the general form is used, applicable to both relativistic and non-relativistic states.
The chemical potential is denoted by μ ν and will be estimated later. Note that for trapped neutrinos the chemical potential may be non-zero, as with ordinary bound matter. Equation (5) yields the result ( ) , where T 0 is the temperature at the origin, T c is the temperature corresponding to the cutoff density c ρ , and the other variables are defined above. This notation will be used below.
To solve Equation (1) and Equation (2), an equation of state that relates pressure to density and temperature must be chosen. In many treatments in astronomy and astrophysics, the pressure is a function of density only as in Equation (2), yielding an implicit relationship between temperature and density. One natural choice that relates temperature and density is Equation (5). The relativistic and non-relativistic versions have been used to derive the equation for the density in dwarf stars [53] [54]. The relativistic and non-relativistic versions give polytropic exponents of 4/3 and 5/3, respectively. The latter choice might be expected to be a good one for a cool fermionic gas.
As another option for the polytropic exponent, one may look for physical models in the literature for chargeless baryonic particles in a gravitational field.
Such a model can be found in treatments of neutron stars. Treatments of neutron stars typically use a polytropic exponent ranging from 3/2 to 2, with near 2 as the most common and most likely choice [55] [56] [57]. Such an exponent is To complete the initial conditions, the density at r = 0 must be known or assumed. Given the discussions at the end of Section 2, one expects kT to be in the range of 0.007 to 0.07 eV near galactic centers at the time of halo formation. This range of temperatures corresponds to a range of densities given by Equation (5).
Such a range of mean kinetic energies is considered in combination with the possible range of masses of baryonic neutrinos. As given in Appendix A, the range of masses considered for baryonic neutrinos is 0.025 to 0.6 eV/c 2 . A possible range of key properties of a fermion gas is shown in Table 2. In particular, the table assumes a temperature of 0.0134 eV/k = 155 K, which is found to give good agreement with density profiles of galactic haloes inferred from data, as will be shown in the next section. From Equation (5) one then obtains a number density, shown in the second column (with the chemical potential initially set to zero). The third column shows the root-mean-square (RMS) velocity, v rms , which is obtained from the well-known relation between relativistic velocity and kinetic energy. The fourth column shows the energy density computed using ( ) From this table, one sees a range of mildly relativistic velocities. These velocities are not consistent with gaseous dark matter that is bound in galactic haloes solely by gravitational attraction, since the escape velocity for a galaxy is typically of the order of 500 km•sec −1 [41]. To address this issue, one may recall the analogous states of baryonic quarks, which form atomic nuclei or neutron stars, as mentioned above. In the absence of electrostatic repulsion, nuclei can be of unbounded size, according to the Weizsacker model [58]. Hence, one might surmise that baryonic neutrinos form a similar macroscopic state in which very weak binding occurs (but not binding with other forms of matter).
Such a binding energy between neutrinos in baryonic states would need to be at least the mean kinetic energy, i.e. the value of about 0.02 eV in order to avoid the free-streaming issue. To estimate such a binding energy, one might consider the nuclear binding energy of about 15 MeV per nucleon and use the ratio of the no-based baryons. This linear scaling of baryonic binding with particle mass is partially justified by Appendix A. One might also envision a binding between up to six neighboring baryonic neutrinos (divided by two since bonds are shared), leading to a total binding energy up to about 0.0192 eV, which is comparable to the mean kinetic energy of 0.02 eV given above. A more detailed ab initio calculation is beyond the scope of this paper. However, these estimates of binding energy and thermal kinetic energy are roughly consistent with what is required from the virial theorem for a medium in equilibrium ( . .
for massless gluons or gravity and ~1 n for a br inter-particle potential, where " . .
K E " denotes a time average of the total kinetic energy and " V " denotes the time average of the total potential energy).
With estimates for a binding energy and a range of densities, one may then estimate a chemical potential to use in Equation (5). The standard formula from the free electron model for the chemical potential for non-relativistic particles is given by and relevant values are shown in Table 3.
One may set the chemical potential equal to the Fermi energy and use Equation (5) with these chemical potentials and with the associated upper limit (2mε F ) 1/2 . The resulting densities at T = 155 K range from 72% to 60% of the results of Table 2 for the masses shown. The computed densities are not equal to the input densities because the integral does not converge with the aforementioned upper limit. For masses above 0.1 eV/c 2 , if the upper limit of the integral is extended until it converges, the resulting densities are consistent with the input densities from Table 2. This state of matter corresponds to Fermi temperatures varying from 162 to 172 K, and to thermodynamic temperatures (T in Equation (5)) of 2 to 15 K, with little variation in that range of T. The lower thermodynamic temperatures are expected from the theory of metals. For masses at or below 0.1 eV/c 2 , the particles are sufficiently relativistic that the Fermi temperatures need to be adjusted to maintain consistency with Table 2. For these  Table 2 for all masses shown.
Hence T will be set to 2 K in what follows, roughly consistent with the expected temperature of free neutrinos in the modern universe [12], p. 154. Given that the baryonic neutrinos were bound early in the history of universe when the mean kinetic energy was approximately 0.02 eV, the haloes comprising such matter should remain relatively stable after formation, much as bound ordinary matter. This expectation for halo evolution is justified in detail in Section 6. It is also possible that multiple species of baryonic neutrinos could result in lower average temperatures and lower RMS velocities that further reduce diffusion of the hypothesized matter away from galactic centers. There are 3 basic types of neutrinos, so there are expected to be at most 3 3 = 27 possible types of a colorless baryonic triplet, just as with the discrete SU(3) symmetry for (u, d, s) states in the quark sector. Accounting for antiparticles there may be as many as 54 species. To achieve the same total number density, one then requires a temperature of 12 K with 54 species rather than 155 K, assuming a Fermi-Dirac distribution and a mass of 0.4 eV/c 2 . The RMS velocity in such cases is about 0.09c, which is still sufficient to overcome a galactic escape velocity if there is no other form of binding. One might expect that all such species would be present in the hot early universe. Then, as the universe cools only a few species that are most stable would remain, matching the known behavior of the quark sector. With this in mind the analyses related to the modern era will assume one or two stable baryonic neutrino species. It is possible that a few mesonic neutrino states may be present. The net effect of two baryonic neutrino states would change the Fermi temperature from 155 K, for example, to about 99 K to achieve the same total density (at a mass of 0.4 eV/c 2 ).
The mid and upper range of energy densities in Table 2 is somewhat greater than the current nominal estimate of 0.3 GeV•cm −3 for the dark matter mass-energy near earth [41] [59] [60], which is a desirable property for dark matter near a galactic center.
Next, the distribution of ordinary matter must be specified. In recent years, the model of choice [47] for describing the projected density of elliptical galaxies is due to Sérsic [48]. An approximate de-projected form is given by [61]. A similar form for the density versus radius was developed by Einasto [49]. The center bulge of the Milky Way has a cluster with a Sersic exponent of n = 3 [50] for OM. As is well known, for spiral galaxies the overall Sersic exponent n is a measure of the balance between the disk and the bulge, two clearly distinct components. A typical Sersic exponent for a spiral galaxy might be 4 or more, but for the central bulge 2 is a common number [51]. For elliptical galaxies, 2 -4 is a common number [42] for OM. With these various results in mind, a Sersic exponent of 2 is initially chosen for ordinary matter. The projected form of the Sersic equation is where ρ rad0 , A, and α are constants. The de-projected Sersic (dpS) density distri- The parameter 0 rad ρ is obtained by setting the volume integral of Equation (8) equal to the measured or inferred ordinary mass of the galaxy. The variable R e denotes the radius which encloses 1/2 the total light of the galaxy. The other two parameters in Equation (8) are given conveniently and approximately from Equation (19) and Equation (27)  (9b) Figure 1 shows the n = 2 dpS radial profile and also a profile with n = 4. The assumed total OM mass of the galaxy is assumed to be M gal = 9 × 10 10 solar masses, approximating that of the Milky Way [60] [62]. It is assumed to have a black hole of 4 × 10 6 solar masses at its center.
The dpS profile is also used with some success for characterization of the DM density profile versus radius. Another density distribution used for characterization of DM is the Einasto distribution, which for the purposes of this paper is given by is the radius of the volume enclosing 1/2 the total of the galaxy, and d n is given approximately by Equation (24)   The radiant matter distribution is the source term for Equation (1) and Equation (2). To use these equations to compute a profile, a polytropic exponent must be chosen. Based on the discussions above, two polytropic exponents are given by [55] ( ) 2 2 Here R gh is the nominal radius of the galactic halo, chosen to be 92 kpc for the purposes of this paper. It will be checked for consistency in the following solutions. The pressure P 0 at the center of a galactic halo is defined by Equation (10) (11). Other approaches are available to estimate the value of c 2−ε , for example as given in [52]. Such an approach gives a value within 1.5 orders of magnitude of that given here. This section provided the equations and the input parameters needed to compute solutions for DM density. Variations about these input assumptions are also considered.

Galactic-Scale Solutions
Moving on to full numerical solutions of Equations (1)-(5), consider conditions corresponding to a galaxy similar to the Milky Way as mentioned above. To obtain a solution, a density at the origin must be specified. These are given in the second column of Table 2 for the respective masses. For example, for a baryonic neutrino mass of 0.4 eV/c 2 , the number density is ρ 0 = 5.34 × 10 15 m −3 . Table 4 summarizes the inputs used for the results of this section. Note that the quoted temperature is the thermodynamic temperature used in Equation (5), with chemical potentials μ ν given in Table 3.  (5) Fermi Temperature T 0 at origin 160 -210 K From Table 3 Masses of baryonic neutrinos, m bν 0.025 -0.6 eV/c 2 Upper limit of mass is from Appendix A The numerical integration of Equation (1) and Equation (4) for the density uses 4000 steps at 30.7 pc each. A simple finite-difference numerical approach proves adequate with the quoted step size. If the density is reduced below 10 9 m −3 , then the density is set to that value for display purposes.
The results of the calculations are shown in Figure 2 and Figure 3 for the standard and generalized equations of hydrostatic equilibrium, respectively. The horizontal axes in these figures are the dimensionless ratio radius/R gh , where R gh = 92 kpc as discussed above. A comprehensive (but not exhaustive) search was performed over fermionic mass, polytropic exponent, and mean kinetic energy within the ranges shown in Table 4. The search attempted to find the best match to the following properties of dark matter reported in the literature: 1) outer radius   Table 4, r 0 = 1 kpc, T out = 8 K. Right: T 0 varied as given in the legend to overlay curves for different masses. Also shown is a case with galactic halo radius set to 1.66 R gh = 153 kpc = r c , with m bν = 0.4 eV/c 2 , T 0 = 155 K, r 0 = 1 kpc, and T out = 5 K. It should also be noted that with a polytropic exponent near 2 and using Equation (11), the lower masses have less pressure and so result in smaller half-max radii. This differs from the case of a polytropic exponent of 5/3, which yields larger half-max radii for lower masses in view of the expression for fermions given after Equation (5). With smaller exponents such as 5/3, dramatically larger galactic haloes are computed using the standard hydrostatic equation, and so are not shown.
Several other metrics are worth discussion for Figure 2. First, the total enclosed mass is in the range of 4 to 60 times the ordinary matter for particle masses ranging from 0.3 to 0.5 eV/c 2 for the plots shown. For 0.4 eV/c 2 , the total enclosed mass is within 15% of 15 times ordinary matter. Second, the mass-energy density (as computed in Table 2) at the radius of Sol is 0.06 to 0.2 GeV•cm −3 for particle masses ranging from 0.4 to 0.6 eV/c 2 for Figure 2. Values of T 0 much larger or smaller than the range shown in Figure 2 do not show a radius consistent with the assumed halo radius for γ near 2. Overall, the results of Figure 2 are a match to the assumed halo radius and enclosed mass for some particle masses. However, such solutions have a relatively low energy density at earth compared to observational inferences and also do not have the cusp deduced from data.
Sample results for the generalized hydrostatic equation, Equation (4), are shown in Figure 3. In this case, as in Figure 2, the plots with the specified inputs also show a radius that is in the neighborhood of R gh = 92 kpc, consistent with the assumed radius. However, it is seen that the results using the generalized hydrostatic equation are a qualitative match to a dpS profile. A cusp is present out- The left plot of Figure 3 shows the sensitivity of the result to the mass of the hypothesized baryonic neutrino. It is seen that the range of masses that are consistent within 10% of R gh is from 0.2 to 0.6 eV/c 2 for the chosen inputs. Lower masses did not show such agreement. The right plot shows sensitivity to T 0 in the vicinity of the nominal 155 K, showing the masses required to obtain a similar density profile. The right plot also shows a sample case for the galactic radius set to 1.66R gh = 153 kpc with a baryonic neutrino mass of 0.4 eV/c 2 with appropriate choice of temperatures. The point of this plot is that other (self-consistent) galactic halo radii can be achieved with similar input parameters.
Further calculations, not shown, give consistent results for R gh with particle masses up to about 5.0 eV/c 2 for smaller values of r 0 , r 0 as low as 0.1 kpc. These typically require lower values of T 0 and T out . For larger particle masses, the temperature needed for T out is 0.1 K or less to match both galactic halo size (R gh ) and enclosed DM mass. For larger values of particle mass, above 5.0 eV/c 2 , consistency with both observationally-inferred galactic halo size and mass cannot be met simultaneously; either the computed radius matches R gh but the total enclosed DM mass is too large, or the total mass matches expectations but the radius is too small. Dwarf galaxies with masses of 10 7 to 10 10 solar masses can be obtained with this approach for particle masses of 0.2 eV/c 2 to 0.6 eV/c 2 . The required input values for dwarf galaxies are T 0 the same as in Table 4, with r 0 in the same range (0.1 to 1 kpc), R gh = 1 to 2.5 kpc, and T out = 30 to 95 K.  The plot on the right of Figure 4 shows the computed mass-energy density of DM at a radius of 7.9 kpc for plots on the left of Figure 3. 7.9 kpc corresponds to the distance of earth from the galactic center. As discussed above, the mass-energy density is given by ( ) As mentioned earlier, the estimated mass-energy density at earth is 0.3 GeV•cm −3 , "within a factor of 2 -3," as stated in the Astrophysical Constants section of the most recent PDG document [41]. The resulting lower bound is 0.1 GeV•cm −3 , and this lower bound is shown in the right plot as a dotted line. The particle masses consistent with these estimates from data are about 0.25 to 0.65 eV/c 2 .
The nominal computed density profile from Figure 3 with a baryonic neutrino mass of 0.40 eV/c 2 is fit to various model profiles as shown in Figure 5. In addition to this single-species computation, a two-species computation is also shown, and will be discussed below. The densities of Figure Table 5.  the table, which are from [42]. However, as expected, the central density parameters are quite different. This is expected because of the large derivatives of the model profiles at the origin, as discussed in [52], which do not match measured values [63]. Table 5 also shows two metrics for the quality of the fits. These metrics are The metrics are computed over the range of radii for which the densities are appreciable (U(r) = 1 for r < 0.9 R gh , 0 otherwise). The metrics for these model fits to the nominal curve vary from 20% to 91%, which is not particularly good, but are expected from visual inspection of Figure 5. It should be noted that both metrics 1 and 2 weight the agreement at the outer radii quite heavily, and this is a significant contributor, especially for the (α, β, γ) model. Table 5 additionally shows the total enclosed mass within a radius of 0.9 R gh .
For reference, the nominal density profile, the blue curve of Figure 5 has a total enclosed mass of 15.2M gal within this radius. The (α, β, γ) model has an enclosed mass close to expectations, about 19M gal . The profiles labelled dpS and Einasto have a total enclosed mass ratio (DM/OM) that is low compared to expected values of 15 or more. Also considered but not shown are dpS and Einasto profiles with an inner core that is about 2 kpc in radius, matching the mean radius of the bulge in the Milky Way. These had higher enclosed masses, of the order of 10 to 15M gal, but the fit metrics were no better than shown in the table. Journal of Modern Physics Overall, the differences between the best-fit dpS and Einasto models and the nominal computed solution are not negligible. This difference can be addressed via several physically reasonable approaches. Such approaches include (a) inclusion of angular momentum in the computation, (b) an allowance for non-equilibrium profiles at larger radii, or (c) inclusion of multiple particle species. These three approaches are discussed briefly in the following three paragraphs. As shown in Approach (b) involves appeal to density profiles that do not strictly adhere to the 1/r equilibrium profile at larger radii. Such non-equilibrium profiles are discussed in [34] and [52] and are attributed to the ongoing process of halo accretion and mergers. These profiles were considered in this effort but were found to differ from the model profiles even more than the nominal 1/r profile. This is because if the model profile and the 1/r profile agree at small radii and the computed non-equilibrium profiles are even flatter at intermediate radii then they are always above the 1/r profile. From Figure 5, one can see that such a profile would offer worse agreement at larger radii However, with a larger particle mass, e.g. about 0.6 eV/c 2 , a sharper decline near the center could support a flatter profile at larger radii, as shown in the 2-species profile of Figure 5.
Approach (c) was also considered and the result is shown in the black curve of Figure 5. It can be seen that this 2-species model is a particularly good match to the dpS and Einasto-1 models. Both model curves have a radius in the range of 700 to 800 kpc, corresponding to small galactic clusters. This 2-species approach can be described succinctly using the notation developed for Equation (4) and Equation (5). The mass density profile involving two species is denoted by ( )  eV/c 2 , lead to a better fit at smaller radii with a steeper mass-density slope, while still matching the total enclosed mass. Larger masses led to excessive total enclosed mass, and smaller masses led to insufficient total enclosed mass given the range of values for the chemical potentials and temperatures shown above. Note also that both species of masses lie within the range of values expected from Appendix A. Table 6 shows the inputs used to obtain the 2-species curve shown in Figure 5.
In summary, assuming the generalized hydrostatic Equation (4)  The above represents a summary of a search over multiple parameters, including baryonic-neutrino mass, particle temperature, polytropic exponents, and core radius. The standard and generalized hydrostatic equations are both considered. The generalized hydrostatic equation gives a better overall match for metrics derived from fits to representative models and data than does the standard equation for these ranges of values, when all 7 of the criteria mentioned in the previous paragraph are considered. However, the generalized hydrostatic equation of equilibrium does not fully trace to DM material properties. Only four of the six inputs to the solution are traceable to material properties: the particle mass and Fermi temperature of the constituent particle, both at the origin and outer radius. The other two inputs, the inner radius r 0 and the outer radius r c , are not traceable to fundamental physical properties and make it too easy to fit some of the measured parameters. Nonetheless, selection of these two parameters allows a simultaneous fit to multiple criteria, which seems more than fortuitous.

The Hypothesis and the Tremaine-Gunn Bound
Reference [72] identified a bound that assumes (a) that the subject particles are fermions, (b) that such particles are non-relativistic, and (c) that such particles Table 6. Parameters used for 2-species plot of Figure 5

The Hypothesis and Observation of DM near Earth
One might question whether such baryonic neutrinos might be observed at earth.
As seen in Figure  as seen at earth, and this is consistent with observations to date. Note that solar neutrinos are not initially bound to other neutrinos partly because of the way they are created, but also partly because they are ultra-relativistic, just as quarks were not initially bound in the hot early universe. It should be re-stated that the experimental fact of neutrino oscillations is direct evidence that neutrinos will form bound states via SU(3) νe within the context of the extended-color theory.
Possible observations for DM near earth also include an annual variation of the order of 1% of the time-average of scintillation in sodium iodide detectors at specific energies [74]. The observed energies of ~6 keV are not obviously related to the form of DM proposed herein. Measured values of the relative annual variation range from 0.0026 to 0.025 from Table 1 of that reference. The peak-to-valley velocity difference of earth relative to DM is about v Sol = 250 km•sec −1 , i.e. ~10 −3 c, assuming that DM is not rotating and the earth's orbital plane around the sun is oriented at 60˚ relative to the sun's velocity vector around the galactic center [41]. The RMS velocity v rms of the proposed DM is 0.03c to 0.05c with T = 2 K.
This gives an estimate of the relative annual variation of the flux of DM of v Sol /v rms = 10 −3 /0.03 to 10 −3 /0.05. Thus, the estimated range of relative flux variation is a factor of 0.8 to 1.33 times the maximum measured relative variation of 0.025 quoted above.
The standard cosmological theory of neutrinos [12] [75] indicates that cold relic neutrinos are also present. The presence of bound neutrino states is not inconsistent with neutrinos interspersed with ordinary and DM matter. It is thus expected that there is a substantial fraction of free neutrino states near earth. Because of their low relative velocity, the helicity of such neutrinos of mass 0.005 to 0.055 eV/c 2 would differ from those of high-energy free neutrinos observed in typical experiments, assuming that such neutrinos have mass and have Dirac wavefunctions. This would result in a relatively large fraction of right-handed neutrinos. Such neutrinos would be difficult to detect, however, because of the known inability to induce right-handed neutrinos to interact with normal matter [76]. Such relic neutrinos with a temperature of about 2 K would have electroweak cross sections comparable to those of bound neutrino states, and so would also be difficult to detect. This dense form of matter is justified with the core assumption that the constituent baryonic neutrinos interact and are bound via SU(3) ν and obey non-relativistic

The Hypothesis, Halo Stability, and Halo-Halo Interactions
Fermi-Dirac statistics. The SU(3) interaction with its approximate 1/E 2 dependence on CM energy E [77] gives a large cross-section and a vanishing mean free path as temperatures tend to zero. Further, the computed Fermi temperature of Table 3 is typically much greater than the thermodynamic temperature of 2 K, much as occurs with ordinary condensed matter. These considerations justify the simplest calculation for the mean free path,  : ( ) where ρ is the density appropriate to the specific location in the halo and d is the "size" of the particle. The values of ρ are given by the densities of Table 2 Table 7, along with the RMS velocity, v rms , which is 0.036c based on a temperature of 2 K and a particle mass of 0.4 eV/c 2 , in accord with the discussions following Table 3. The table also  Also shown is the medium pressure, assuming a polytropic exponent of 2 and assuming c 2 equals 2.08 × 10 −40 J•m 3 , as in the discussion surrounding Equation (11).
The which is quite high compared to conventional matter. However, this diffusivity leads to negligible mass or heat transfer over scale sizes of the order of 1 kpc over the age of the universe, as can be seen by the second-to-last column. This implies that the density and temperature distributions are expected to be relatively stable from the time of creation up to the present day, so that there is little evolution other than modest gravitational and SU(3)-based contraction and subhalo aggregation over most of the universe's lifetime. There are alternative formulations for the mean free path that differ from Equation (14), in which the cross section is calculated based on an interaction strength rather than the hard-sphere approximation. These estimates also lead to very long mass-transfer time constants, of the order of the age of the universe for haloes of 100 kpc in size. The above shows that the DM derived in this paper has some degree of self-consistency for a single halo. What does the above imply for halo-halo interactions? This is a very complex subject [34], involving baryonic feedback, dynamical friction, tidal stripping, and more. However, some calculations can be performed for the hypothesis under consideration. First, consider the kinetic energy of halo-halo interactions, which involve relative velocities v rel of 30 to 3000 m/sec, see e.g. [21]. The corresponding kinetic energy per particle m v υ ranges from 2 × 10 −9 to 2 × 10 −5 eV. Note that these values are less than both the estimated inter-baryonic-neutrino binding energy, 6.4 × 10 −3 eV, as well as the estimated intra-baryonic neutrino binding energy, 0.4 eV. Hence the medium is not expected to dissociate, but rather to maintain its form.
Given that the medium is a fluid, the interaction should be characterizable in terms of dimensionless parameters such as Reynolds number, Re, and Mach number, Ma. These are shown in Table 8 for three interaction velocities and for the same 4 densities shown in Table 7. The Reynolds number is computed assuming the value of κ in Table 7, the velocities shown, and a scale size of 1 kpc, which is assumed to be the typical diameter scale of a subhalo. The acoustic velocity is computed in the standard way assuming a polytropic exponent of 2 as in the discussion surrounding Equation (11).
From This, combined with the essentially incompressible properties of a quantum fluid satisfying Fermi-Dirac statistics, suggests that the fluid might approximate potential flow, which is known to have very low drag [78].
With such large Reynolds numbers, the accepted drag coefficient is about 0.2 for a sphere, as can be found in any textbook in fluid mechanics, neglecting possible quantum fluid effects. A drag coefficient as low as 0.1 occurs for an ellipsoid with a 2:1 aspect ratio, as may be found for subhaloes like the dwarf Sagittarius galaxy. With the above information, the drag-induced slowing of a subhalo in a larger medium can be computed using the simple differential rela- Here ρ sh is the DM density of the subhalo, set equal to 5 × 10 15 m −3 , ρ gal is the DM density of the galaxy in the vicinity of the subhalo, and v is the velocity of the subhalo relative to the galaxy. V is the volume of the subhalo, set equal to 3 3 sh r 4π , and r sh is the subhalo radius, set equal to 0.5 kpc. C d is the drag coefficient, and A is the cross-sectional area of the subhalo presented to the flow, set equal to 2 sh r π .
Equation (15) is readily solvable assuming the densities are constant over time (as a first approximation). Sample results are shown in Table 9 versus initial relative velocity v rel0 and local galaxy density, for times corresponding to 50% and 90% reductions in velocity.
The  (15), one sees that these numbers can be interpreted as the decay time per kpc diameter. So, for example, for a 10 kpc subhalo such as the Sagittarius dwarf spheroid, the predicted velocity decay time constant is about 1970 Myr for the 90% decay time with an initial velocity of 300 km•sec −1 , referring to the bottom of the middle column of Table 9. If the shape of the subhalo is elliptical with major axis in the flow direction, as is the case for Sagittarius, the drag coefficient might be a factor of 2 lower, resulting a time constant of about 4 Gyr. These numbers are within a factor of 2 of those inferred from observations [79] [80]. It is not clear that this result is consistent with such observations, due to the neglect of many other effects such as the type of orbit and dynamical friction.
Another well-known constraint on self-interacting dark matter is the observa- The net effect of the above calculations is decay of subhaloes into a larger halo, with associated erosion and assimilation. Scaling the results of Table 9, the slowing is most significant for the smaller haloes. From this, one sees a partial explanation for the unexplained dearth and diversity of smaller satellite haloes mentioned by other authors [21] [34]. Note that the decay rates for such haloes are significantly shorter than the time scales involving dynamical friction, so this offers observational means for assessing or informing this theory. Further, this picture of halo interaction is quantitatively consistent with the observed Bullet cluster halo interaction, provided the halo centers do not pass closer than about 10 kpc of each other, using the simplest relevant calculation. The basic picture for halo-halo interactions is that a smaller dense halo moves through the less-dense perimeter of the larger halo in a manner similar to that of a mercury ball as it moves through water under the influence of gravity, with additional mass stripping due to friction and diffusion. Much more could be said about halo-halo interactions as it relates to the self-interacting form of DM that derives from the hypothesis of this paper; hopefully the above is sufficient for an initial treatment.  [38], whereas the standard SU(3) provides no guidance on key parameters such as the bound state mass or the coupling parameter, as discussed in Section 2. As shown in Section 3, such baryonic neutrinos in haloes need a relatively high temperature to maintain the density and the associated total mass observed for haloes. Since the corresponding velocity exceeds the galactic gravitational escape velocity, some sort of additional binding is inferred. It is found that an SU(3) ν binding that is similar to the SU(3) binding of neutrons in neutron stars will produce sufficient binding. Because the resulting state of matter is similar to that for neutron stars, such haloes might be viewed as a form of "neutrino star". Such binding near galactic centers implies that the Tremaine-Gunn bound is not expected to apply as discussed in Section 4. The coupling constant g s of the feeble SU(3) ν is deduced to be as low as [(137) 2 × 10 −25 ] 1/4 = 6.58 × 10 −6 of that of the electric force (e) from theoretical estimates. This coupling strength applies for relativistic neutrinos and is greater at lower energies due to the running of the coupling parameter as well as the interaction probability as discussed in Appendix A.

Summary
As shown in Section 3, a generalized form of the equation for hydrostatic equilibrium provides a better match than the standard equation for the observationally-inferred cuspy behavior for DM near galactic centers. Such calculations provide a good match to the inferred total galactic-halo mass and to the DM mass-energy density near earth. Solutions to the generalized hydrostatic equation are found to have long spatial "tails" that are cut off based on a galactic-halo mass constraint. Other explanations for the details of a halo density profile are explored. These include (a) multiple species, (b) angular momentum, and (c) a transition to a condition in which the standard hydrostatic equation applies. An approach using 2 species provides a particularly good fit to sample Einasto and de-projected Sersic model profiles. Further, the generalized solution offers a resolution to the "core-cusp" problem in dwarf galaxies. The solutions must have a core due to the density-limiting Fermi-Dirac statistics of baryonic neutrinos, in the absence of a gravitational singularity. Dwarf galaxies are found to have the "core" portion of the solution in Section 3.2, but either lose the cusp portion of their DM or never accumulate it.
Section 5 discusses the prospects for observation of such DM near earth. Solar neutrinos are not expected to interact significantly with each other or with such dark matter because they are created mono-color (all solar neutrinos are "green" in the conventions of [38] Journal of Modern Physics diffusivity as discussed in Section 6. Also shown in Section 6 is that the fluid hypothesis is consistent with observed galactic halo interactions (particularly the Bullet Cluster interaction) via arguments put forth here and by other authors, e.g. [82]. The issue of SIDM cross-section is apparently not relevant for the same reasons. Also considered is the motion of smaller haloes, less than 1 kpc in diameter in a larger halo. It is found that the orbital decay is a factor of 1 to 10 times faster than expected from the standard DM model. Further observations of haloes and their interactions would provide helpful tests of the theory presented here. It remains to be seen if the hypothesis proposed herein has full consistency with the preponderance of observational evidence. Further work definitely remains.
A key issue for this form of dark matter is the free-streaming scale in the early universe. This issue is relegated to a companion paper. However, the calculations here show stability of haloes because of the relatively short mean free path of the hypothesized form of dark matter. This short mean free path applied to the early universe results in diffusive rather ballistic transport, vastly shortening the associated "free-streaming scale length" of such matter. Also covered in the companion paper is a discussion of the interaction strength of this form of DM with ordinary matter, dark-matter fraction of total matter, consistency with cosmic microwave background measurements, SN1987a data, neutrino accelerator anomalies, and the issue of SU(3) interactions between neutrinos and quarks