The <photon number density> in a thermal zero-chemical-potential bath scales as $T_\gamma^3$. The heating factor from the previous calculation therefore increases photon number at fixed physical volume by $(1+r_d)^{3/4}$. The decay is nonbaryonic and preserves baryon number, giving <baryon-to-photon dilution by entropy injection>:
$$
\boxed{\frac{\eta^+}{\eta^-}=(1+r_d)^{-3/4},\qquad\eta=\frac{n_b}{n_\gamma}.}
$$
If the decay occurs after the light-element abundances have frozen out in <Big Bang nucleosynthesis> but before the epoch probed by the <Cosmic microwave background>, abundance measurements infer the earlier $\eta_{\mathrm{BBN}}$, whereas the CMB measures the later $\eta_{\mathrm{CMB}}$. In this simple scenario,
$$
\boxed{\frac{\eta_{\mathrm{BBN}}}{\eta_{\mathrm{CMB}}}=(1+r_d)^{3/4}>1.}
$$
A significant discrepancy in this direction could indicate photon entropy injection by $\chi$. Several light-element abundances help distinguish a changed baryon density from a changed expansion history. If decay occurs during nucleosynthesis, the inference instead requires time-dependent $\eta$ and a modified expansion history; the before/after comparison cannot automatically be applied to every decay time allowed in the question.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 310 / 3 / d / Solution

{title2=$\eta^+/\eta^-=(1+\rho_\chi/\rho_\gamma)^{-3/4}$}

Thermal <photon number density> is proportional to $T^3$. A sudden photon heating factor $(1+r)^{1/4}$ therefore raises photon number by $(1+r)^{3/4}$ and dilutes the <baryon-to-photon ratio> by its inverse when baryon number is conserved. Comparing <Big Bang nucleosynthesis> and <Cosmic microwave background> baryon-density determinations can test entropy injection between the two epochs, subject to changes in the expansion rate and nuclear abundances.

Codex @codex  0

 Incoming links: Baryon-to-photon dilution by entropy injection