Stars as Laboratories for Fundamental Physics - MPP Theory Group

472 Chapter 12
time structure of Eq. (12.22) reduces to Θ(t), i.e. its spectral form is
time independent except that it begins at t = 0. This is somewhat
surprising because the energy of a photon in the laboratory frame is
related to the angle of emission in the neutrino rest frame which in
turn determines the “detour” taken from the source to us (Fig. 12.10).
However, the first photons come from decays immediately at the source
and so any angle of emission leads to the same initial arrival time.
It must be stressed that the expression Eq. (12.22) depends on the
assumption of decays not too far from the source (d D ≪ d LMC ) and so
only the head of the photon pulse is correctly described while its tail
would require including decays even close to the Earth. Strictly speaking,
the photon burst never ends because even if the parent neutrinos
have passed the Earth, some photons will be received from backward
emission. However, because one is interested in neutrino masses so
large (m > ν ∼ 200 eV) that the photon burst is much longer than the
GRS measurement window, it is enough to account for the head of the
photon pulse.
12.4.5 Radiative Decay Limits: High-Mass Neutrinos
As a first explicit case for the distribution of photon energies and emission
angles I take the two-body decay ν → ν ′ γ with a massless daughter
neutrino and with the dipole angular distribution of Eq. (12.2) with
x = cos θ = − cos ϑ for a left-handed parent. This amounts to
f(ω, x) = 1 2 (1 + αx) δ(ω − 1 2 m ν) (12.23)
in Eq. (12.22). Because of the δ function it is trivial to integrate,
d 2 N γ
dE γ dt = 1 [
2E γ
1 + α 2E ]
γ − E ν
e −t/τ ∗
Θ(t − t env ), (12.24)
m ν τ γ p ν p ν
where
τ ∗ ≡ m ν
2E γ
τ tot , t env ≡ m2 ν
2E γ p ν
R env . (12.25)
Moreover, the flux vanishes if the chosen value for E γ does not fall
between 1(E 2 ν ± p ν ), or equivalently, unless
E ν > E γ + m 2 ν/4E γ , (12.26)
a condition on the minimum required neutrino energy.
For the simplest case when the neutrinos are relativistic (p ν = E ν ),
long-lived (e −t/τ∗ = 1), and absorption effects by the progenitor can be