Finite-temperature stability from doublet inflation field with right-handed neutrinos

We study the augmentation of the Standard Model (SM) with another $SU(2)$ Higgs doublet and right-handed neutrinos. The second Higgs doublet ($\Phi_2$) is defined to be odd under the $Z_2$ symmetry, and hence, the lightest stable neutral particle from the additional doublet becomes the cold dark matter candidate. The right-handed neutrino field coupled to the Higgs field provides non-zero mass for the neutrinos. The inert doublet field coupled non-minimally to gravity as $\zeta_2 \Phi_2^\dagger \Phi_2 R$ also acts as an inflaton field. The inflationary bounds restrict the interaction couplings as $\lambda_2/\zeta_2^2 \approx 4\times 10^{-10}$. After inflation ends, the scalar bosonic degrees of freedom from the inert doublet can contribute to the electroweak phase transition. The strongly first-order phase transition bound, i.e., $\frac{\phi_{+}(T_c)}{T_c} \geq 1.0$ restricts the bare mass parameter of the additional doublet to $m_{22}=400.0$ GeV, demanding GUT scale perturbative unitarity for $Y_N=0.01$. The increase in $Y_N$ reduces the strength of phase transition, and it is no longer satisfied even for vanishing bare mass parameter. The Planck scale perturbative unitarity allows for the first-order phase transition, $\frac{\phi_{+}(T_c)}{T_c} \geq 0.6$, until $m_{22}=70.0$ GeV for $Y_N=0.01$, and none of the mass values satisfies the first-order phase transition for $Y_N=0.4$. The thermal corrections also affect the probability of tunneling from the false vacuum to the true vacuum, and hence, the finite temperature stability of the electroweak vacuum has been studied, including the finite-temperature effects.