\rightarrow \psi \bar{\psi}}=\frac{Y^2_{\eta \psi}(m_{\eta}^2-4m_{\psi}^2)}{8\pi m_\eta}\sqrt{1-\frac{4m_{\psi}^2}{m_{\eta}^2}}
\label{decay:psipsi}$$ where $Y_{\eta \psi}$ is the effective Yukawa coupling, m$_\eta$ and m$_\psi$ are the masses of the mother particle $\eta$ and $\psi$ respectively.

However, this doublet scalar, apart from having electroweak gauge interactions, can have sizeable quartic interactions with other scalars like the standard model Higgs doublet and hence can be thermally produced in the early universe. Now, considering the mother particle $\eta$ to be in thermal equilibrium in the early universe which also decays into the dark matter particle $\psi$, we can write down the relevant Boltzmann equations for comoving number densities of $\eta, \psi$ as $$\begin{aligned}
\frac{dY_{\eta}}{dx} &=& -\frac{4 \pi^2}{45}\frac{M_{\rm Pl} M_{\rm sc}}{1.66}\frac{\sqrt{g_{\star}(x)}}{x^2}\Bigg[\sum_{p\,=\,\text{SM\, particles}}
\langle\sigma {\rm v} \rangle_{\eta \eta \rightarrow p\bar{p}}\left( Y_{\eta}^2-(Y_{\eta}^{\rm eq})^2\right) \Bigg]\nonumber \\
&& - \frac{M_{\rm Pl}}{1.66} \frac{x\sqrt{g_{\star}(x)}}{M_{\rm sc}^2\ g_s(x)} \Gamma_{\eta \rightarrow \bar{\psi} \psi}  \ Y_{\eta},
\label{BEeta}\end{aligned}$$ $$\begin{aligned}
\frac{dY_\psi}{dx} &=& \frac{2 M_{\rm Pl}}{1.66} \frac{x\sqrt{g_{\star}(x)}}{M_{\rm sc}^2 \ g_s(x)} \Gamma_{\eta \rightarrow \bar{\psi} \psi}
\ Y_{\eta}
\label{BEpsi}\end{aligned}$$ where $x=\dfrac{M_{\rm sc}}{T}$, is a dimensionless variable while $M_{\rm sc}$ is some arbitrary mass scale which we choose equal to the mass of $\eta$ and $M_{\rm Pl}$ is the Planck mass. Moreover, $g_s(x)$ is the number of effective degrees of freedom associated to the entropy density of 