_{1}

^{*}

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 candidate 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.

Dark matter (DM) has been postulated to take many forms, including hot dark matter [

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} [

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 [^{2}, as is consistent with observationally inferred values from the latest Lyman-α forest absorption measurements [

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 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 [_{ν} is the mass of the highest-mass neutrino and m_{q} is the mass of the highest-mass quark of the up or down families. This scaling applies when the interaction involves relativistic neutrinos. Using m_{ν} of about 0.055 eV/c^{2} for the tau neutrino for minimal neutrino masses and the normal hierarchy [^{−22} to 1.01 × 10^{−25} using the bottom quark or top quark, respectively, for m_{q}. This theory also has the property that in addition to the 8 massless gluons, there are 15 massive Goldstone bosons (massive gluons) for each family with gluon energies of the order of m_{ν}c^{2}. These massive gluons satisfy most of the criteria proposed by [_{ν}_{e} and the standard form of SU(3) applied to the neutrino family will be denoted SU(3)_{ν}_{s}. 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.

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_{ν}_{e}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_{ν}_{e}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,

Particle | Degrees of Freedom, SU(3)_{νs} | Degrees of Freedom, SU(3)_{νe} |
---|---|---|

Electron family | (1)(4) = 4 (×7/8) | (1)(4) = 4 (×7/8) |

Photons | 2 | 2 |

Neutrino family | (3)(2)(3) = 18 (×7/8) | (3)(2)(3) = 18 (×7/8) |

Neutrino family gluons | (8)(2) = 16 | (15)(3) + (8)(2) = 61 |

Total degrees of freedom | 37.25 | 82.25 |

Total degrees of freedom in neutrino sector | 31.75 | 76.75 |

% Degrees of freedom in neutrino sector | 83.9% | 93.3% |

m_{ν}_{e}c^{2} ~ 0.005 eV and m_{νμ}c^{2} ~ 0.01 eV [

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 [

To analyze the spatial distribution of low-energy neutrinos or other weakly interacting particles, an N-body simulation [

( 1 / ρ ) ( d P / d r ) = − m b v M e n c ( r ) G / r 2 . (1)

Here ρ is the number density of DM, P is the pressure, r is the radius, m b v is the mass of a DM particle, M e n c ( r ) is the enclosed mass, and G is the gravitational constant. This equation can be solved using the well-known Lane-Emden formulation [

P = c γ ρ γ , (2)

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) with a galaxy similar to the Milky Way for OM do not match key published results from simulations or inferences from observations for DM. In particular, the solutions have no cusp at the origin, and are a poor match to the standard de-projected Sersic or Einasto profiles based on observations [

To address this, a generalization of Equation (1) is used. The derivation and the properties of the resulting equation are given in [

( 1 / ρ ) d P / d r + ( 1 / ρ ) ( d ρ / d r ) [ m b v M e n c ( r ) G / r ] = − m b v M e n c ( r ) G / r 2 − ( d P / d ρ ) / r (3)

when pressure is assumed to be a function of density only. It can be seen that this equation reduces to Equation (1) when the second terms of both sides of the equation are negligible. When all terms are included, Equation (3) is seen to give a 1/r dependence for the density. Equation (3) with a total mass constraint and a density constraint is the most justifiable based on theoretical considerations, comparisons with data, and comparisons with others’ calculations and simulations. This best-justified result is given by

ρ ( r ) = { ρ 0 in a spherical region about the origin of radius r 0 , or ρ ( r ) satisfies d ln ρ d ln r = − 1 − 2 ( r r c ) ( ρ c ρ ) (4)

In this equation, r c is the cutoff radius and ρ c is the cutoff number density where the density drops due to the mass constraint. The last key equation used for calculations of haloes is the Fermi-Dirac equation for the number density given temperature T and particle mass m_{bν}:

ρ = n s / ( 2 π 2 ℏ 3 ) ∫ p 2 d p / [ exp ( { [ ( p c ) 2 + ( m v c 2 ) 2 ] 1 / 2 − m v c 2 − μ v } / k T ) + 1 ] , (5)

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 P = ( 1.914 ℏ 2 / m b v ) ρ 5 / 3 for non-relativistic fermions with two spin states. This equation is used to set the density at the origin, and then Equation (1) and Equation (2) or Equation (4) are used to generate a spatial profile. The resulting solution of Equation (4) for number density ρ ( r ) can be expressed as a function of its inputs, ρ ( r ) = ρ ( r , ρ 0 , r 0 , ρ c , r c ) . Using Equation (5), ρ ( r ) can also be expressed as a function of temperatures and mass, ρ ( r ) = ρ ( r , T 0 , r 0 , T c , r c , m b ν , μ ν ) , where T_{0} is the temperature at the origin, T_{c} is the temperature corresponding to the cutoff density ρ c , 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 [

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 [

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 _{rms}, which is obtained from the well-known relation between relativistic velocity and kinetic energy. The fourth column shows the energy density computed using ρ m b ν c 2 / [ 1 − ( v r m s / c ) 2 ] 1 / 2 .

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} [

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

Mass (eV/c^{2}) | Number Density, ρ_{0,mv} (×10^{15} m^{−3}) | RMS velocity/c | Energy Density (GeV∙cm^{−3}) |
---|---|---|---|

0.025 | 0.185 | 0.83 | 0.0083 |

0.05 | 0.36 | 0.70 | 0.0250 |

0.10 | 0.81 | 0.55 | 0.0966 |

0.2 | 2.01 | 0.42 | 0.441 |

0.3 | 3.55 | 0.35 | 1.13 |

0.4 | 5.34 | 0.30 | 2.23 |

0.6 | 9.58 | 0.25 | 5.91 |

^{a}Assumes a mean kinetic energy of 0.02 eV, corresponding to a temperature of 155 K as described in text.

mass of a baryonic neutrino to that of a neutron. Such a scaling gives a binding energy of about (15 MeV) × (0.4 eV)/(939 MeV) = 0.0064 eV between neutrino-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 ( 〈 K . E . 〉 = n 〈 V 〉 with n = − 1 for massless gluons or gravity and n ~ 1 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

μ ν 0 = ε F = [ ( ℏ c ) 2 / 2 m b v c 2 ] ( 3 π 2 ρ ) 2 / 3 , (6)

and relevant values are shown in

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 ^{2}, if the upper limit of the integral is extended until it converges, the resulting densities are consistent with the input densities from ^{2}, the particles are sufficiently relativistic that the Fermi temperatures need to be adjusted to maintain consistency with

Mass (eV/c^{2}) | |||||||
---|---|---|---|---|---|---|---|

0.025 | 0.05 | 0.1 | 0.2 | 0.3 | 0.4 | 0.6 | |

Fermi Energy ε_{F} (eV) | 0.0241 (0.018) | 0.0188 (0.016) | 0.0161 (0.015) | 0.0148 | 0.0144 | 0.0142 | 0.140 |

Fermi Temperature T_{F} (K) | 279 (206) | 218 (187) | 187 (174) | 172 | 167 | 164 | 162 |

lower masses, the Fermi temperature is set to 206, 187, and 174 K for masses of 0.025, 0.05, and 0.1 eV/c^{2}, respectively. The result is that a thermodynamic temperature of 2 to15 K still gives the densities of

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 ^{−3} for the dark matter mass-energy near earth [

Next, the distribution of ordinary matter must be specified. In recent years, the model of choice [

ρ r a d = ρ r a d 0 exp ( − A r a ) (7)

where ρ_{rad}_{0}, A, and α are constants. The de-projected Sersic (dpS) density distribution for radiant matter is approximated by [

ρ r a d = ρ r a d 0 ( r / R e ) − p n exp [ − b n ( r / R e ) 1 / n ] . (8)

The parameter ρ r a d 0 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) of [

p n = 1.0 − 0.6097 / n + 0.05463 / n 2 , (9a)

and

b n = 2 n − 1 / 3 + 0.009876 / n . (9b)

_{gal} = 9 × 10^{10} solar masses, approximating that of the Milky Way [^{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

ρ E i n = ρ 0 exp [ − d n ( r / R e , DM ) 1 / n ] , (10a)

where R e , DM is the radius of the volume enclosing 1/2 the total of the galaxy, and d_{n} is given approximately by Equation (24) of [

d n ≈ 3 n − 1 / 3 + 0.0079 / n , for n > 0.5 . (10b)

Both the dpS and Einasto distributions will be compared against the DM density profiles computed using Equation (1) through (5).

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 considered, γ = 5 / 3 and γ = 2 − ε , with ε a positive number much less than 1. The polytropic relation for γ = 5 / 3 is given above. For the case of γ = 2 − ε , one can use the polytropic relation P = c 2 − ε ρ 2 − ε with c 2 − ε given by [

c 2 − ε = ( 2 / π ) G m b υ 2 R g h 2 ρ 0 ε ≡ P 0 / ρ 0 2 − ε . (11)

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). So, for example, with a baryonic neutrino mass m_{bν} of 0.4 eV/c^{2} and Equation (11) one obtains c_{2} = 2.08 × 10^{−40} J∙m^{3}. In this case, P_{0} = 1.53 × 10^{−9} J∙m^{−3}. For ρ in units of m^{−3} one obtains the correct units in Equation (11). Other approaches are available to estimate the value of c_{2−}_{ε}, for example as given in [

This section provided the equations and the input parameters needed to compute solutions for DM density. Variations about these input assumptions are also considered.

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 ^{2}, the number density is ρ_{0} = 5.34 × 10^{15} m^{−3}.

_{ν} given in

Input Parameter | Values | Comment |
---|---|---|

Thermodynamic Temperature, T | 2 K | T of Equation (5) |

Fermi Temperature T_{0} at origin | 160 - 210 K | From |

Masses of baryonic neutrinos, m_{bν} | 0.025 - 0.6 eV/c^{2} | Upper limit of mass is from Appendix A |

Galactic halo radius, r_{c} | 92 kpc | r_{c} in Equation (4) |

Polytropic exponents | 5/3 to 2 | As discussed above |

Ordinary matter profile | dpS profile | Exponent = 2, radius = 15.3 kpc |

Total ordinary matter mass, M_{gal} | 9 × 10^{10} | Solar masses |

Fermi Temperature T_{out} at radius r_{c}, with chemical potential μ_{νc} º kT_{out} | 5 - 16 K | Sets density ρ_{c} at outer radius in Equation (4) using Equation (5) |

Inner scale r_{0} | 1 kpc | Sets radius of constant density region in Equation (4), approximate size of bulge of Milky Way |

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 _{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

consistent with the assumed radius of DM of 92 kpc; 2) cusp near the origin; 3) mass-energy density at radius of Sol; 4) ratio of DM in galactic halo to OM in galaxy; 5) quantitative shape consistent with published simulations and inferences from observations, and 6) consistency of temperature at outer radius with standard cosmology. The best or near-best fits considering all these criteria are shown in the figures.

Results for the standard hydrostatic equation, Equation (1), are shown in _{gh} = 92 kpc, for particle masses ranging from 0.3 to 0.6 eV/c^{2}. However, as expected, it is seen that the results using the standard hydrostatic equation are not a good match to a dpS profile, by comparison to

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 ^{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 ^{−3} for particle masses ranging from 0.4 to 0.6 eV/c^{2} for _{0} much larger or smaller than the range shown in

Sample results for the generalized hydrostatic equation, Equation (4), are shown in _{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 outside of a radius of 1 kpc with a logarithmic slope of about −1.

The left plot of _{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 _{0} in the same range (0.1 to 1 kpc), R_{gh} = 1 to 2.5 kpc, and T_{out} = 30 to 95 K.

^{2}.

The plot on the right of ^{−3}, “within a factor of 2 - 3,” as stated in the Astrophysical Constants section of the most recent PDG document [^{−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 ^{2} is fit to various model profiles as shown in _{bν} ρ_{0} = 3.76 × 10^{−21} kg∙m^{−3} within 1 kpc of the origin, consistent with the nominal computed profile for a baryonic neutrino mass of 0.40 eV/c^{2}. The vertical axis is normalized by this density. This limit at the origin is used and shown because it is in better accord with most measured data [_{gh}.

The chosen parameters for the model profiles are given in

Input Parameters and Metrics: | n | R_{e}_{,DM} (kpc) | ρ_{0}/ρ_{0,mv} | Metric 1 | Metric 2 | M/M_{gal} |
---|---|---|---|---|---|---|

(2, 3, 1), fit to nominal | - | r_{02} = 92 | 1 | 0.20 | 0.91 | 19.6 |

dpS, fit to nominal | 3.3 | 700 | 0.011 | 0.47 | 0.45 | 7.8 |

Einasto-2, fit to nominal | 5.5 | 400 | 220 | 0.61 | 0.68 | 4.9 |

dpS, fit to 2-species | 3.3 | 700 | 0.011 | 0.033 | 0.18 | 7.8 |

Einasto-1, fit to 2-species | 5.5 | 800 | 110 | 0.011 | 0.20 | 8.6 |

Einasto-2, fit to 2-species | 5.5 | 400 | 220 | 0.18 | 0.18 | 4.9 |

Galaxy-sized haloes, dpS | 3.1 - 4.6 | 110 - 230 | - | - | - | - |

Galaxy-sized haloes, Einasto | 5.3 - 7.8 | 190 - 400 | - | - | - | - |

Cluster-sized haloes, dpS | 2.2 - 3.5 | 700 - 4700 | - | - | - | - |

Cluster-sized haloes, Einasto | 3.9 - 7.4 | 1200 - 6000 | - | - | - | - |

^{a}Bottom four rows are from [

the table, which are from [

Metric 1 = { Σ r U ( r ) [ ρ M ( r ) − ρ C ( r ) ] 2 r 2 } 1 / 2 / { Σ r U ( r ) [ ρ C ( r ) ] 2 r 2 } 1 / 2 , (12a)

and

Metric 2 = { Σ r U ( r ) [ ρ M ( r ) / ρ C ( r ) − 1 ] 2 r 2 } 1 / 2 / { Σ r U ( r ) r 2 } 1 / 2 , (12b)

where ρ C ( r ) is the numerically-computed profile and ρ M ( r ) is the model profile. Three classes of model profiles are considered: (α, β, γ), dpS, and Einasto. 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

_{gh}. For reference, the nominal density profile, the blue curve of _{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.

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

Angular momentum profiles in CDM haloes have undergone considerable study using N-body simulations of particles that interact by gravity alone. Such studies include [^{−1} associated with angular momentum for the Milky Way [^{2}, such velocities correspond to kinetic energies of about 2 × 10^{−7} eV, which is much less than the estimated inter-particle binding energy of ~0.02 eV of the posited Fermi fluid. Hence, by analogy with other well-understood liquid or semi-solid celestial bodies, the primary impact of angular momentum would be distortion of the halo, with a limited impact on the density profile.

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 [^{2}, a sharper decline near the center could support a flatter profile at larger radii, as shown in the 2-species profile of

Approach (c) was also considered and the result is shown in the black curve of

ρ m , 2-species ( r ) = m 1 ρ ( r , T 01 , r 01 , T c 1 , r c 1 , m 1 , μ ν 1 ) + m 2 ρ ( r , T 02 , r 02 , T c 2 , r c 2 , m 2 , μ ν 2 ) (13)

Clearly, there are more parameters in Equation (13) that permit a better fit a model profile. Experimentation with the parameters indicated that values of m_{1} of 0.3 to 0.5 eV/c^{2} led to a good fit at larger radii. Larger masses, m_{2} = 0.6 to 0.8 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.

In summary, assuming the generalized hydrostatic Equation (4) and the posited baryonic neutrinos with masses of 0.4 eV/c^{2} ± 50%, the results are roughly consistent with the following observationally-inferred and simulated properties of galactic-scale or cluster-scale DM structure reported in the literature: 1) halo width consistent with the assumed radius of 92 kpc for a galaxy similar to the Milky Way; 2) relatively flat density profiles within a core radius of ~1 kpc; 3) cusp in the region outside of this core; 4) mass-energy density at radius of Sol; 5) ratio of DM in galactic halo to OM in a galaxy similar to the Milky Way; 6) qualitative shape; and with multiple species, 7) quantitative shape. Further, the temperature at the edge of the galactic halo is consistent with expectations from standard cosmology, with a temperature of 2 K (Fermi temperatures of 5 to 16 K).

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.

Reference [

Input Parameters | m (eV/c^{2}) | T_{0} = T_{c} (K) | r_{0} (kpc) | r_{c} (kpc) | μ_{ν}_{0}/k (K) | μ_{νc}/k (K) |
---|---|---|---|---|---|---|

Species 1 | 0.4 | 2 | 1 | 92 | 104 | 8 |

Species 2 | 0.6 | 2 | 1 | 9.2 | 51 | 16 |

are non-interacting. In this paper, (a) is true, (b) is approximately true, but (c) is not true near a galactic center. Regarding (a), neutrinos should form baryonic neutrinos with odd multiples of half-integer spin, as expected by analogy with quark-based baryons. Regarding (b), the estimated speed of baryonic neutrinos for a temperature of 2 K is about 0.03c to 0.05c depending on mass, and possibly less if there are multiple species. Regarding (c), the hypothesized baryonic neutrinos are weakly bound near a galactic center, based on the findings of Sections 3.1 and 3.2, and therefore interact. Hence (c) is not true. In the depths of space, far from any ordinary matter clusters, one might expect that neutrinos or the bound neutrino states considered herein are likely non-relativistic and not interacting and the Tremaine-Gunn bound is indeed expected to apply.

One might question whether such baryonic neutrinos might be observed at earth. As seen in ^{2}. This basic comparison supports the hypothesis. Somewhat larger masses are possible as well.

It is estimated that the flux of solar neutrinos from the sun at earth is about 7 × 10^{10} cm^{−2}∙sec^{−1}, see, e.g. [^{−3}. From the previous two sections, with the 2018 PDG mass-energy density of DM at the earth of ~0.3 GeV∙cm^{−3} one obtains a baryonic neutrino density of about 10^{9} cm^{−3}, assuming a baryonic neutrino mass of 0.3 eV/c^{2}. Based on a simple scaling of the electroweak force which goes as the square of the CM energy, DM baryonic neutrinos should interact much more weakly than solar neutrinos via the electroweak force by a factor of about 1/(3 × 10^{5})^{2} or less, because they have much lower energy than solar neutrinos in the earth reference frame (<1 eV for the former versus ~0.3 MeV for the latter). Hence, direct observation of such baryonic neutrinos seems challenging.

Further, solar neutrinos should interact in a very limited way with such DM via SU(3)_{νe}, because solar neutrinos are predominately electron neutrino states, which are mono-color based on the theory of [_{ν}. The latter would have similarities to high-energy electron scattering in bulk material. It will be seen in the companion paper that such elastic or quasielastic scattering should not significantly alter the composition, energy spectrum, or the flux of solar neutrinos 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 [_{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 [_{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 [^{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 [

The hypothesis of SU(3)_{ν} leads to an investigation of other self-interacting dark matter (SIDM) proposals. Recent papers [

With the self-interaction discussed here, DM is a dense form of matter that maintains its volume due to fermion degeneracy pressure of baryonic neutrinos. 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 Fermi-Dirac statistics. The SU(3) interaction with its approximate 1/E^{2} dependence on CM energy E [

l = ( 2 1 / 2 π ρ d 2 ) − 1 , (14)

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 _{F}, where p_{F} is the Fermi momentum of the particle. The value of mean free path is shown in _{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 _{F}, the diffusion constant, κ = l v r m s , the mean free time between collisions, τ = l / v r m s , and the time to diffuse 1 kpc, t 1 kpc = ( 1 kpc ) 2 / κ . 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 table shows a diffusion constant varying from about 200 to 2000 m^{2}∙sec^{−1}, 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.

Density ρ (m^{−3}) | T_{F} (K) | p_{F}/m_{bν}c | l (μm) | κ (m^{2}∙sec^{−1}) | τ (psec) | t_{1kpc} (yrs) | P (J∙m^{−3}) |
---|---|---|---|---|---|---|---|

5 × 10^{15} | 157 | 0.323 | 19.3 | 209 | 1.80 | 1.5 × 10^{29} | 4.4 × 10^{−9} |

5 × 10^{14} | 34 | 0.148 | 40.9 | 442 | 3.79 | 6.9 × 10^{28} | 4.4 × 10^{−11} |

5 × 10^{13} | 7.3 | 0.069 | 87.8 | 948 | 8.14 | 3.2 × 10^{28} | 4.4 × 10^{−13} |

5 × 10^{12} | 1.6 | 0.032 | 189 | 2040 | 17.5 | 1.5 × 10^{28} | 4.4 × 10^{−15} |

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 [_{rel} of 30 to 3000 m/sec, see e.g. [^{−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

From

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 relation

ρ s h d | v | / d t V = ( C d / 2 ) ρ g a l | v | 2 A . (15)

Here ρ_{sh} is the DM density of the subhalo, set equal to 5 × 10^{15} m^{−3}, ρ_{gal} is the DM

Density ρ (m^{−3}) | v_{rel} = 30 km∙sec^{−1} | v_{rel} = 300 km∙sec^{−1} | v_{rel} = 3000 km∙sec^{−1} |
---|---|---|---|

5 × 10^{15} | 4.4 × 10^{21}, 0.019 | 4.4 × 10^{22}, 0.19 | 4.4 × 10^{23}, 1.9 |

5 × 10^{14} | 2.1 × 10^{21}, 0.061 | 2.1 × 10^{22}, 0.61 | 2.1 × 10^{23}, 6.1 |

5 × 10^{13} | 9.8 × 10^{20}, 0.19 | 9.8 × 10^{21}, 1.9 | 9.8 × 10^{22}, 19.1 |

5 × 10^{12} | 4.6 × 10^{20}, 0.61 | 4.6 × 10^{21}, 6.1 | 4.6 × 10^{22}, 60.6 |

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 4 π r s h 3 / 3 , 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 π r s h 2 . Equation (15) is readily solvable assuming the densities are constant over time (as a first approximation). Sample results are shown in _{rel}_{0} and local galaxy density, for times corresponding to 50% and 90% reductions in velocity.

The table shows, for example, that the 50% and 90% times are about 22 and about 197 Myr, respectively, for a subhalo diameter of 1 kpc located at the outer edges of a galaxy where v_{rel}_{0} = 300 km∙sec^{−1}. For haloes of order 1 kpc in diameter and v_{rel}_{0} = 30 km∙sec^{−1}, the orbit decay times range from about 0.2 Gyr to 2 Gyr based on the lower left values of

Referring to Equation (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

Another well-known constraint on self-interacting dark matter is the observation of the Bullet Cluster halo collision. In this collision the r_{200} radius (the radius at which the galactic density is 200 times that of background) is about 2140 kpc for the main cluster and about 995 kpc for the smaller Bullet Cluster. The final observationally-estimated lag of DM behind stellar matter is 25 ± 29 kpc [

Density ρ_{gal} (m^{−3}) | v_{rel}_{0} = 30 km∙sec^{−1} | v_{rel}_{0} = 300 km∙sec^{−1} | v_{rel}_{0} = 3000 km∙sec^{−1} |
---|---|---|---|

5 × 10^{15} | 0.22, 1.97 | 0.022, 0.197 | 0.002, 0.020 |

5 × 10^{14} | 2.2, 19.7 | 0.22, 1.97 | 0.22, 0.197 |

5 × 10^{13} | 21.8, 197 | 2.2, 19.7 | 0.22, 1.97 |

5 × 10^{12} | 218, 1970 | 21.8, 197 | 2.2, 19.7 |

profiles are assumed as in [^{15} m^{−3} at a radius of 1 kpc is used for the main cluster, and number densities of 5 × 10^{15} to 5 × 10^{14} m^{−3} are used for central portion of the Bullet cluster, in accord with Section 3. The Bullet cluster radius is chosen to equal the Hernquist α-parameter, 279 kpc, as in [_{200} radius of 995 kpc results in smaller lags due to the larger mass and therefore lower acceleration. A drag coefficient of 0.2 is used. The resulting computed lags range from 0.35 to 7.7 kpc, when 700 kpc past closest approach, which is within error bars of that measured (25 kpc).

Using a strict 1/r density profile for the main cluster rather than the Hernquist profile gives a lag of as much as 15 kpc, because of the greater column density traversed by the Bullet Cluster. Also, a larger drag coefficient of 1.0 gives about 1.75 to 37 kpc of lag instead of 0.35 to 7.7 kpc. If these larger numbers are combined with a 1/r density profile, the result is about 73 kpc of lag as an extreme worst case, which is outside the error bars. Convergence of these calculations was checked; the computed lags are accurate to within 0.01% using up to 10^{4} steps in time. The lags are relatively small because of the large size of the Bullet cluster halo, as noted in the previous paragraph, so its large mass decelerates less. Evidently, this form of self-interacting dark matter is consistent with the measured lag of the Bullet cluster collision, based on the simplest relevant drag calculation for this state of matter.

The net effect of the above calculations is decay of subhaloes into a larger halo, with associated erosion and assimilation. Scaling the results of

Straightforward calculations of galactic haloes are performed for DM assuming an SU(3) interaction applies to neutrinos. Both SU(3)_{νe} and SU(3)_{νs} are largely consistent with observations if one allows SU(3)_{νs} to have a different strength than that of SU(3) for quarks. The key attributes of this force can be and are chosen consistent with SU(3)_{νe} given by [

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.

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 [

Halo solutions are stable and self-consistent, having low thermal and mass 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. [

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

Portions of this work were presented in Paper APR19-000356 at the 2019 April Meeting of the American Physical Society. The material of this paper represents significant improvements over that of the conference paper. The author also wishes to thank T. Slatyer, and A. Nelson (deceased) for helpful discussions and E. Vishniac for critical comments on earlier versions of this paper.

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

Holmes, R.B. (2020) Galactic Haloes from Self-Interacting Neutrinos. Journal of Modern Physics, 11, 854-885. https://doi.org/10.4236/jmp.2020.116053

This Appendix estimates the binding energy of baryonic and mesonic neutrinos as well as the SU(3)_{ν}_{e} interaction strength for relativistic neutrinos. Because the SU(3) binding energy is a large fraction of the mass-energy of bound quarks, one might expect that this would be the case for SU(3)-bound neutrinos as well (should they exist). This fact is utilized for estimation of the mass-energy of bound neutrino states.

The binding energy of baryonic neutrinos is estimated first. From equation (10.27a) of [_{b} of a baryonic neutrino can be approximated by

E b = β υ τ 2 ( ℏ c / 4 ) { 4 π α 3 ( m υ e 2 + m υ μ 2 + m υ τ 2 ) 1 / 2 c 2 / ( ℏ c ) } 2 | Δ x | , (A1)

where β υ τ 2 is the probability of an upper-mass tau neutrino state (assuming the normal hierarchy), and α_{3} is the dimensionless coupling parameter for the strong force, g s 2 / ( 4 π ћ c ) . The neutrino flavor masses are m_{νe}, m_{νμ}, and m_{ντ}. |Δx| is the characteristic size of an SU(3)-bound neutrino. The value of α_{3} is chosen to equal 1 in this calculation because for bound SU(3) states the coupling parameter is close to 1 for bound quark-quark interactions, and that should apply here as well. The probability of an upper-mass neutrino state from the same reference for a marginally relativistic bound state is given by

β υ τ 2 = m υ e / ( m υ e + m υ τ ) . (A2)

This probability is approximately 0.1 for m_{νe} ~ 0.005 eV/c^{2} and m_{ντ}~ 0.05 eV/c^{2}, assuming the normal hierarchy for neutrino masses, the known mass-squared differences, and the least possible mass for the tau neutrino. Under the same assumptions, the muon neutrino mass is about 0.01 eV/c^{2}. The last input to Equation (A1) is the characteristic size of SU(3)-bound neutrinos. For this, use an estimate based on the Heisenberg uncertainty principle:

| Δ x | ≥ ℏ c / ( p c ) ≈ ℏ c / ( m υ τ c 2 ) . (A3)

Using the nominal value of m_{ντ} given above, one obtains |Δx| ~ 3.3 microns. One might also use m_{νe} or m_{νμ} in Equation (A3), but the basis of Equation (A1) suggests that m_{ντ} should be used. Substituting the above into Equation (A1), one obtains an estimate of the binding energy of baryonic neutrinos.

E b ≥ 4 π 2 m υ e c 2 = 0.2 eV . (A4)

One can see that with these approximations and assumptions, the binding energy is roughly independent of the upper neutrino mass value. In Equation (A3), one might also use ( m υ τ m υ e ) 1 / 2 c 2 for the denominator based on Ch. 10 of [

E b ≥ 4 π 2 m υ e c 2 ( m υ τ / m υ e ) 1 / 2 = 4 π 2 ( m υ τ m υ e ) 1 / 2 c 2 = 0.62 eV . (A5)

To this range of binding energies, one must add the masses of the constituent neutrinos, which might range from 3m_{ν}_{e} to 3m_{ντ}. This then leads to a range of baryonic neutrino masses from about 0.22 eV/c^{2} to about 0.8 eV/c^{2}. Assuming the baryonic neutrinos comprise the lower-mass neutrino states as in quarks, a tighter range would be 0.22 to 0.64 eV/c^{2}. On the other hand, a baryonic neutrino mass as high as 0.8 eV/c^{2} should not immediately be ruled out. Equation (10.27b) of [^{2}.

The above mass-scaling analysis can also be applied to relativistic particles using Equation (10.13b) rather than (10.13a) of [_{b} is the bottom quark mass and m_{t} is the top quark mass. Note that the scaling factor between the down-quark family and up-quark family should be of order 1 because all hadrons bound by a strong quark interaction have sufficient energy for the presence of both u − u ¯ and d − d ¯ sea quarks. There is also the running of the coupling parameter that should be included; the standard calculation indicates that the correction is negligible when applied to neutrinos.