= Solution
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.
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