Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 337
Fig. 9.3. Contour plot for log(τθ0 2 ) with τ in seconds according to Eqs. (9.64)
and (9.68), taking F e = 0, F p = 1, CV 2 +3C2 A = 4, and µ ν e
/µ e = 1. (Adapted
from Raffelt and Sigl 1993.)
From Fig. 9.2 it is clear that effective NC scattering on electrons
slightly accelerates the initial rate of flavor conversion, but it does not
dramatically affect the overall time scale for achieving equilibrium. This
time scale is crudely estimated by ignoring F e entirely in Eqs. (9.64)
and (9.68) and by setting F p = 1 and CV 2 + 3CA 2 = 4. With µ νe ≈ µ e
one then finds results for τθ0 2 shown as contours in Fig. 9.3. The diagonal
band refers to the resonance condition of Eq. (9.57); there flavor
equilibrium would be established on a time scale nearly independent of
the vacuum mixing angle. However, the detailed behavior in this range
of parameters has not been determined.
Armed with these results it is straightforward to determine the range
of masses and mixing angles where ν x would achieve flavor equilibrium
and thus would effectively participate in β equilibrium ep ↔ nν. The
initially trapped lepton number escapes within a few seconds. Therefore,
τ < ∼ 1 s is adopted as a criterion for ν x to have any novel impact
on SN cooling or deleptonization. The relevant density is about three
times nuclear while Y p ≈ 0.35 so that Y p ρ = 3×10 14 g cm −3 is adopted.
Otherwise the same parameters are used as in Fig. 9.3. Then one finds
sin 2 2θ 0 ∼ > { 0.02 (keV 2 /∆m 2 ) 2 for ∆m 2 ∼ < (100 keV) 2
2×10 −10 for ∆m 2 ∼ > (100 keV) 2 (9.71)
as a requirement for ν x to reach chemical equilibrium. This range of
parameters is shown as a hatched region in Fig. 9.4.