in $R_{\Gamma}$ leads to a larger contribution of $\rho_{Y}$ and thus a
deviation from radiation domination. Towards smaller $\Gamma_{Y}$ the
dependence on $R_{\Gamma}$ stems from a deviation from matter domination; an
increase in $R_{\Gamma}$ leads to more DR production and a further deviation
from matter domination.

The dashed black line in Fig. 11 shows a numerical fit described by

$\frac{g-1}{R_{\Gamma}\xi}=\frac{x+y\left(x^{-1}\Gamma_{Y}\right)^{z}}{1+\left(x^{-1}\Gamma_{Y}\right)^{z}},$
(73)

for $x=1.18$, $y=1.25$, and $z=0.78$. From this fit, we can determine
$g(R_{\Gamma},\Gamma_{Y})$.

Figure 12: Post-decay $N_{\text{eff}}$ for varying values of $R_{\Gamma}$ and
$\Gamma_{Y}$. For short-lived cases in which the $Y$ particle decays during
radiation domination, $\Delta N_{\text{eff}}$ is simply a function of
$R_{\Gamma}$. Once the energy density of non-relativistic matter begins making
a non-negligible contribution to the total energy density, $\Delta
N_{\text{eff}}$ also becomes sensitive to $\Gamma_{Y}$.

If $N_{ur}$ is the effective number of relativistic species before injection,
excluding the massive neutrino, and $N^{\prime}_{ur}$ is the corresponding
number after the injection of DR, then it follows that

$g=\frac{(\rho_{sr,f}+\rho_{\mathrm{dr},f})a_{f}^{4}}{\rho_{sr,i}a_{i}^{4}}=\frac{\rho_{\gamma,f}a_{f}^{4}\left[1+\frac{7}{8}N^{\prime}_{ur}\left(\frac{4}{11}\right)^{4/3}\right]}{\rho_{\gamma,i}a_{i}^{4}\left[1+\frac{7}{8}N_{ur}\left(\frac{4}{11}\right)^{4/3}\right]}.$
(74)

The comoving photon energy density does not change and so this reduces to

$N^{\prime}_{ur}=N_{ur}+\left(g-1\right)\left[\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}+N_{ur}\right].$
(75)

The contribution that the single massive neutrino species makes to
$N_{\text{eff}}$ is $N_{ncdm}=(11/4)^{4/3}(0.71611)^{4}=1.0132$, and we
enforce that $N_{\text{eff}}=N_{ur}+N_{ncdm}$ is equal to 3.044 before the
injection of any DR. The post-decay $N_{\text{eff}}$ is then determined via
Eq. (75). Figure 12 shows the post-decay $N_{\text{eff}}$ values for a range
of $R_{\Gamma}$ and $\Gamma_{Y}$. For a fixed $\Gamma_{Y}$, a larger
$R_{\Gamma}$ corresponds to larger $\Delta N_{\text{eff}}$, as expected. If a
decay scenario occurs deep in radiation domination, then $\Delta
N_{\text{eff}}$ is solely a function of $R_{\Gamma}$. However, as we approach
$Y$ particle lifetimes that extend to matter domination, $\Delta
N_{\text{eff}}$ picks up an additional dependence on $\Gamma_{Y}$. This
dependence on $\Gamma_{Y}$ stems from how $R_{\Gamma}$ is defined.
$R_{\Gamma}$ is a measure of the energy density of the $Y$ particle compared
to the total energy density of the universe at the scale factor of
$a_{\Gamma}$. In other words,
$R_{\Gamma}\sim\rho_{Y}(a_{\Gamma})/\rho_{d}(a_{\Gamma})\sim\rho_{\mathrm{dr}}(a_{\Gamma})/\rho_{d}(a_{\Gamma})$,
where $\rho_{d}$ is the dominant species. Therefore, if $a_{\Gamma}\ll
a_{eq}$, $R_{\Gamma}\sim\rho_{\mathrm{dr}}(a_{\Gamma})/\rho_{sr}(a_{\Gamma})$
and thus $\Delta N_{\text{eff}}$ is solely dependent on $R_{\Gamma}$. However,
if $a_{\Gamma}>a_{eq}$, then
$R_{\Gamma}\sim\rho_{\mathrm{dr}}(a_{\Gamma})/\rho_{m}(a_{\Gamma})\sim[\rho_{\mathrm{dr}}(a_{\Gamma})/\rho_{sr}(a_{\Gamma})]\times[\rho_{sr}(a_{\Gamma})/\rho_{m}(a_{\Gamma})]$
and so $\Delta N_{\text{eff}}\sim
R_{\Gamma}\times[\rho_{m}(a_{\Gamma})/\rho_{sr}(a_{\Gamma})]$ for $Y$ particle
lifetimes extending into matter domination.

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