inflation $N_{\mathrm{new}}$ are written as $$\begin{split}
N_{\mathrm{H}}
&=
\int^{\sigma_{i}}_{\sigma_c}d\sigma
\frac{\mu^4}{\mu^4\left[
\frac{8}{27}\frac{\left(\mu M\right)^2}{\sigma^5}+\frac{\sigma^3}{2}
\right]}\\
&=
\int^{\sigma_{i}}_{
  0.89~\sigma_{i}
  \left[3~\sigma_{i}^2-\left(n_s-1\right)\right]^{1/6}
}d\sigma
\frac{1}{
  \frac{1}{10}\frac{
    \sigma_{i}^6\left(3\sigma_{i}^2-\left(n_s-1\right)\right)
  }{\sigma^5}
  + \frac{\sigma^3}{2}
},
\end{split}$$ $$\label{eq:Nn}
\begin{split}
N_{\mathrm{new}}
&=
N_{\mathrm{tot}}-N_{\mathrm{H}}\\
&=
54.3+\frac{1}{3}
\ln \left[
8\sigma_{i}^3-\left(n_s-1\right)\sigma_{i}
\right]\\[0.5\intextsep]
&\hspace{5ex}
-
\int^{\sigma_{i}}_{
  0.89~\sigma_{i}
  \left[3~\sigma_{i}^2-\left(n_s-1\right)\right]^{1/6}
}d\sigma
\frac{1}{
  \frac{1}{10}\frac{
    \sigma_{i}^6\left(3\sigma_{i}^2-\left(n_s-1\right)\right)
  }{\sigma^5}
  + \frac{\sigma^3}{2}
},
\end{split}$$ where we have used (\[eq:2:ntot\]) and (\[eq:mu\]) for $N_{\mathrm{tot}}$ and $\mu$. Thus, the parameters of the smooth hybrid new inflation are written in terms of $\sigma_{i}$ and $n_s$.

Since the effective potential in the supergravity is uncontrollable over the planck mass, $\sigma_{i}$ and $\mu$ must be smaller than the planck mass. In addition to this, $\sigma_i$ must be larger than $\sigma_c$. This condition guarantees that the hybrid inflation occurs. After the hybrid inflation, the fluctuations of the scalar fields are amplified through the parametric resonance, and then these fluctuations lead to a sharp peak at the scale corresponding to $k_{\rm peak}^{-1}$ in the spectrum of the curvature perturbation [@Kawasaki:2006zv]. The observations of the large scale structures have not seen such enhancement of the curvature perturbations so that the scale corresponding to $k_{\rm peak}^{-1}$ should be sufficiently small. Thus, $N_{\mathrm{H}}$ must be larger than about $10$. Furthermore, since we assume 