Neutrinos of non-zero rest mass and the equivalence principle

The possibility that neutrinos may have non-zero rest mass has led to the investigation of their possible role in the dynamics of astrophysical systems in the universe, including their effect on the dynamics of the universe itself (Schramm & Steigman 1981a, b;Tremaine & Gunn 1979). If neutrinos dominate the mass density in the universe, estimates of the age of the universe from nucleocosmochronometry put an upper limit on the mass m of the neutrino (Joshi & Chitre 1981a, b). If they dominate the mass density on the scale of clusters of galaxies also, one gets a lower limit on the mass (Tremaine & Gunn 1979). The assumption that they do not dominate the mass in binaries and small groups of galaxies leads to an upper limit slightly less than that obtained from the age of the universe. Allowing the neutrino to have different gravitational and inertial masses, we assume that it does not necessarily obey the equivalence principle and let r denote the ratio of its gravitational to inertial mass. Incorporating the possibility that r ≠ 1 in the calculation for the (inertial) mass m spreads the three limits into a region in the r-m plane.

Upper limit from age of universe Writing the total mass density ρ 0 as the sum of the non-ν mass density ρ m and the ν mass density ρ Gν , ρ 0 = ρ m + ρ Gν = ρ m + r.ρ ν .

Subscript G refers to gravitational mass and ρ Gν = r.ρ ν (ρ ν being the inertial mass density) to allow for different coupling between gravitational and inertial masses for neutrinos as compared with non-ν matter. If neutrinos dominate, ρ 0 ≈ r.ρ ν = r.m.n ν where n ν is the number density of the three presently known types of neutrinos and m is the inertial mass of the neutrino (assumed same for all three types). Inserting this in the upper limit to the age of the universe as derived for both the Friedmann world models (Joshi & Chitre 1981a) and for the generally hyperbolic world models (Pankaj Joshi, private communication), of which the Friedmann models are a special case, t 0max = √(A / G. ρ 0 ) ≈ √(A / G. r.m.n ν ); A = 3π / 32 for Friedmann models, and 3π / 16 for general globally hyperbolic worldmodels. Note that this limit is independent of the values of H 0 and q 0 . (Since the limit for the Friedmann models is tighter, we use that for numerical calculations below.) Let t U be the maximum of the various estimates of the age of the universe obtained from the analysis of relative isotope abundances, helium abundance, dynamical considerations for globular clusters, etc. t U is thus a lower limit on the age of the universe, independent of H 0 -and q 0 -values (Symbalisty et al 1980). Therefore, t 0max > t U , which implies

.n ν for Friedmann models.

(1)

Lower limit from clusters of galaxies Examining the missing light problem on the scales of various astrophysical systems, Schramm & Steigman (1981a, b) found that its severity increases with size scale of the system (from galaxies to binaries to small groups to clusters). This, together with the ease with which heavier neutrinos can collapse on smaller scales compared to lighter neutrinos led them to conclude that non-ν matter (nucleons) can account for the mass density on all scales smaller than clusters of galaxies. Relic neutrinos could have collapsed on the scale of clusters of galaxies after they had cooled sufficiently, provided the gravitational potentials of the clusters were deep enough. A necessary condition for this collapse is that the maximum value of the phase-space density decreases in the transition from a free Fermi distribution (at a temperature of ≈ 1 MeV/k) to an isothermal distribution (at present) (Tremaine & Gunn 1979). We write r.ρ ν instead of ρ ν in the calculation of Tremaine & Gunn to obtain r.m 4 > 9.h 3 / (2π) 5/2 .N.g ν .G.σ cl .R cl 2

(2)

where N is the number of species of neutrinos (ν and ν-bar counted as two different species), g ν is the number of helicity states (assumed the same for all species) and σ cl and R cl are the one-dimensional velocity dispersion and core radius of the typical cluster of galaxies.

————————————————————————- —————————————————————Upper limit from binaries and small groups Coming down in size from clusters of galaxies, the next smaller astrophysical systems are binary galaxies and small groups of galaxies. Applying the same principles as above, if neutrinos are not to collapse on the scale of binary galaxies and small groups of galaxies, inequality (2) is reversed, with σ B,SG and R B,SG the typical relative velocity and separation between members in these astrophysical systems:

r.m 4 < 9.h 3 / (2π) 5/2 .N.g ν .G.σ B,SG .R B,SG 2

(3)

We now put numerical values in (1)-( 3). With t U = 20 Gyr and n ν = 300 per cc in (1), r.m < 20 eV/c 2 ;

(1')

and taking N = 6 species of neutrinos (the e, µ and τ neutrinos and antineutrinos), each with g ν = 2 helicity states, (2) gives, for σ cl = 10 3 km/sec, R cl

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