Returning Back to Mukhanov Parametrization of Inflationary Equation of State

We have re-examined Mukhanov parametrization for inflationary equation of state, $1+\omega=\frac{\beta}{({N}+1)^\alpha}$, in the light of Planck 2018 results and latest bound of tensor-to-scalar ratio employing Hamilton-Jacobi formalism. We have found that the current observational values of scalar spectral index and tensor-to-scalar ratio can be used efficiently to constrain the model parameters. The recent bound of $r<0.032$ has been used to put an upper bound on one of the model parameter. Whereas 1-$\sigma$ bound of the scalar spectral index $0.9607\leq n_{_S}\leq 0.9691$ along with the upper bound of tensor-to-scalar ratio provided restriction on the other model parameter $1.50<\alpha\leq2.20$. These bounds however depend on the number of e-foldings still left before the end of inflation and whenever $1.50<\alpha\leq2.20$ we can find appropriate values of the other model parameter $\beta$ so that the observational predictions are in tune with the latest available inflationary observables. We have further utilized the predictions from forthcoming CMB missions in the likes of CMB-S4 and LiteBIRD in order to obtain bounds on the model parameters. We find that detection of gravity waves would help us constrain the model parameters further. But in the absence of detection of primordial gravity wave signal by these CMB missions may rule out Mukhanov parametrization.