New smooth hybrid inflation

We consider the extension of the supersymmetric Pati-Salam model which solves the $b$-quark mass problem of supersymmetric grand unified models with exact Yukawa unification and universal boundary conditions and leads to the so-called new shifted hybrid inflationary scenario. We show that this model can also lead to a new version of smooth hybrid inflation based only on renormalizable interactions provided that a particular parameter of its superpotential is somewhat small. The potential possesses valleys of minima with classical inclination, which can be used as inflationary paths. The model is consistent with the fitting of the three-year Wilkinson microwave anisotropy probe data by the standard power-law cosmological model with cold dark matter and a cosmological constant. In particular, the spectral index turns out to be adequately small so that it is compatible with the data. Moreover, the Pati-Salam gauge group is broken to the standard model gauge group during inflation and, thus, no monopoles are formed at the end of inflation. Supergravity corrections based on a nonminimal Kähler potential with a convenient choice of a sign keep the spectral index comfortably within the allowed range without generating maxima and minima of the potential on the inflationary path. So, unnatural restrictions on the initial conditions for inflation can be avoided.

Figure 1

Relative error in $|\varphi |$ on the new smooth inflationary path in global SUSY for the values of the parameters in Eq. (46) and $\gamma =0$ when we use the expansion of Eq. (23) up to second order in $w$ with coefficient evaluated to leading order in $p$ (dashed line) or accurately (dot-dashed line) and up to fourth order in $w$ with coefficients evaluated to leading order in $p$ (dotted line) or accurately (solid line).

Figure 2

Number of e-foldings $N_{\mathrm{nsm}}$ along the new smooth inflationary path versus $\gamma$ in global SUSY when the system slowly rolls from $\sigma =0.95{\sigma }_c$ down to $\sigma ={\sigma }_f$. The other parameters of the model (except $\gamma$) take the values in Eq. (46).

Figure 3

Spectral index in new smooth hybrid inflation versus $\gamma$ in global SUSY for $p=\sqrt{2}\kappa M/m=1/\sqrt{2}$ and $\kappa =0.1$. The endpoint of the curve at $n_s\simeq 0.932$ corresponds to the case where our present horizon scale crosses outside the inflationary horizon when $\sigma =0.95{\sigma }_c$.

Figure 4

Spectral index in new smooth hybrid inflation versus $\gamma$ in minimal SUGRA for $p=\sqrt{2}\kappa M/m=1/\sqrt{2}$, $\kappa =0.1$. The endpoint of the curve at $\gamma \simeq 0.75\times {10}^{-6}$ ($n_s\simeq 1.0695$) corresponds to the case where our present horizon scale crosses outside the inflationary horizon when $\sigma =0.95{\sigma }_c$.

Figure 5

Spectral index in new smooth hybrid inflation in nonminimal SUGRA as a function of $\gamma$ for $p=\sqrt{2}\kappa M/m=1/\sqrt{2}$ and $\kappa =0.1$. The values of $k_S$, which are indicated on the curves, range from 0.008 to 0.0105 and $k_{SS}=0$.