Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particles Interacting with Electrons and Baryons 115
mode p. The overall factor of 2 relative to Eq. (6.55) arises because
neutrinos and antineutrinos contribute equally.
The smallest k S (the largest radius over which leptonic fields remain
unscreened) arises if the background neutrinos have such small masses
that they are still relativistic today (v = 1). The neutrino number
density is given by n ν+ν = 2 ∫ f p d 3 p/(2π) 3 = ∫ ∞
0 f p p 2 dp/π 2 . Therefore,
in the relativistic limit one finds roughly
k −1
S
≈ e −1
L n −1/3
ν+ν ≈ e −1
L 0.2 cm, (3.53)
independently of details of the neutrino momentum distribution. In
this estimate the predicted number density of background neutrinos in
each flavor of n ν+ν ≈ 100 cm −3 was used.
The largest conceivable k S would obtain if neutrinos were nonrelativistic
today, and if some of them were bound to the galaxy. The
escape velocity from the galaxy is v esc ≈ 500 km s −1 so that the maximum
momentum of a gravitationally bound neutrino is p max = m ν v esc .
Because neutrinos obey Fermi statistics, the largest conceivable galactic
neutrino density is n max ≈ p 3 max ≈ m 3 ν vesc. 3 A typical neutrino velocity
is of order the galactic velocity dispersion, i.e. of order v esc . Therefore,
from Eq. (3.52) one estimates kS 2 ≈ e 2 Ln 2/3
maxvesc −1 ≈ e 2 Lm 2 νv esc . Because the
largest cosmologically allowed neutrino mass is about 30 eV one finds
that neutrino screening cannot operate on scales below e −1
L 10 −5 cm.
Therefore, the screening scale is reduced by no more than six orders
of magnitude by the fact that cosmic neutrinos could be nonrelativistic
today.
The stellar energy-loss result Eq. (3.50) informs us that for relativistic
neutrinos k > S ∼ 10 13 cm ≈ 1 AU where 1 AU = 1.5×10 13 cm
(astronomical unit) is the distance to the Sun. If neutrinos were nonrelativistic,
leptonic forces could be screened over distances six orders
of magnitude smaller, i.e. over about 100 km. However, it is probably
safe to assume that leptonic forces with a strength comparable to
gravity would have been noticed in terrestrial experiments searching
for a composition-dependent fifth force. Therefore, even with neutrino
screening it appears inconceivable that e L could exceed about 10 −19 .
In that case the screening scale would always exceed about 0.1 AU so
that terrestrial limits would easily apply. Thereby one could gain a
few orders of magnitude in the limit, and so one could even use solar
system constraints, taking one back to a result of order the baryonic
one Eq. (3.51). A similar conclusion was reached by Blinnikov
et al. (1995).