Neutrinos of non-zero rest mass and the equivalence principle

Neutrinos of non-zero rest mass and the equivalence principle

by

D. G. Banhatti
Radio Astronomy Centre (TIFR), P O Box 8, Ootacamund 643001, India
and
TIFR Centre for Radio Astronomy, P O Box 1234, Bangalore 560012, India

(June 1983)
[…]

Abstract. Assuming that neutrinos of non-zero rest mass dominate the mass density in
the universe, and also the mass density on the scale of clusters of galaxies, one obtains
the upper limit m <≈ 20 eV/c2 on their mass, independent of the values of H0 and q0, and
the lower limit m >≈ 5 eV/c2 independent of q0 and almost independent of H0. Going one
step further, we allow neutrinos to have different gravitational and inertial masses so that
r = gravitational / inertial mass. Then using m for the inertial mass, we have m.r1/4 >≈ 5
eV/c2, m.r1/4 <≈ 14 eV/c2 and m.r <≈ 20 eV/c2, which together imply r <≈ 6.3. For a
specific value, say, 12 eV/c2, for m, we have 0.03 <≈ r <≈ 1.7.

Keywords: Neutrinos – rest mass – equivalence principle

Introduction
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.

The three limits
Upper limit from age of universe
Writing the total mass density ρ0 as the sum of the non-ν mass density ρm and the ν mass
density ρGν,

ρ0 = ρm + ρGν = ρm + r.ρν.

Subscript G refers to gravitational mass and ρGν = r.ρν (ρν 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,

ρ0 ≈ r.ρν = r.m.nν

where nν 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,

t0max = √(A / G. ρ0) ≈ √(A / G. r.m.nν);

A = 3π / 32 for Friedmann models, and 3π / 16 for general globally hyperbolic world-
models. Note that this limit is independent of the values of H0 and q0. (Since the limit for
the Friedmann models is tighter, we use that for numerical calculations below.) Let tU 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. tU is thus a lower limit on the age of the universe, independent of
H0- and q0-values (Symbalisty et al 1980). Therefore, t0max > tU, which implies

r.m < A / G.tU2.nν,
= 3π / 32. G.tU2.nν 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.ρν instead of ρν in the calculation of
Tremaine & Gunn to obtain

r.m4 > 9.h3 / (2π)5/2.N.gν.G.σcl.Rcl2 (2)

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

Fig.1. r ≡ gravitational / inertial mass of the neutrino, and m ≡ inertial mass of the
neutrino. The allowed region in the r-m plane is shown.
---------------------------------------------------------------

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 σB,SG and RB,SG the typical relative velocity and separation
between members in these astrophysical systems:

r.m4 < 9.h3 / (2π)5/2.N.gν.G.σB,SG.RB,SG2 (3)

Numerical results
We now put numerical values in (1)-(3). With tU = 20 Gyr and nν = 300 per cc in (1),

r.m < 20 eV/c2; (1’)

and taking N = 6 species of neutrinos (the e, µ and τ neutrinos and antineutrinos), each
with gν = 2 helicity states, (2) gives, for σcl = 103 km/sec, Rcl = 250.(50/H0) kpc, for the
typical cluster of galaxies,

r1/4.m > 5.√(H0/50) eV/c2. (2’)

Substituting σB,SG = 100 km/sec and RB,SG = 100 kpc in (3) as the typical orbital velocity
and separation between members of binary galaxies and small groups of galaxies, we get

r1/4.m > 14.√(H0/50) eV/c2. (3’)

The region delimited by (1’), (2’) and (3’) in the r-m plane is shown in Fig.1. From the
diagram, r <≈ 6.3 independent of the value of m. For a specific value, say, 12 eV/c2, for
m, we have 0.03 <≈ r <≈ 1.7.

Thus, we see that the dynamics of the universe and of clusters of galaxies, if dominated
by the neutrino, together with an estimate of the age of the universe from non-
cosmological considerations and the assumption that neutrinos of non-zero rest mass are
not necessary for collapse on a small enough scale (viz, binaries and small groups of
galaxies), are consistent with the neutrino satisfying the equivalence principle. This
conclusion is independent of the value of the cosmological deceleration parameter q0 and
only weakly dependent on that of the Hubble parameter H0. Should direct estimates of the
core radius of a nearby cluster of galaxies and of the separation between members of
small groups of galaxies become available, (2’) and (3’) will no longer have a
dependence on H0.

Acknowledgments
This work was spurred by the idea of making the calculations of Prof Arthur Halprin (on
similar lines (private communication) H0- and q0-independent. I also thank Pankaj Joshi
for encouragement and communicating his latest unpublished results on the general
globally-hyperbolic world-models. It is a pleasure to thank Chris Salter for moral support
and for reading the manuscript.

References
Joshi, P S & Chitre, S M 1981a Nature 293 679.
Joshi, P S & Chitre, S M 1981b Phys Lett 85A 131-5.
Schramm, D N & Steigman, G 1981a Astrophys J 243 1-7.
Schramm, D N & Steigman, G 1981b Gen Rel & Grav 13 101-7.
Symbalisty, E M D, Yang, J & Schramm, D N 1980 Nature 288 143-5.
Tremaine, S & Gunn, J E 1979 Phys Rev Lett 42 407-10.