Topological inflation from the Starobinsky model in supergravity

We consider the ghost-free higher-order corrections to the Starobinsky model in the old-minimal supergravity, focusing on a sector among several scalar fields in the model that reproduces the scalaron potential in the original Starobinsky model. In general, higher-order corrections cannot be forbidden by symmetries, which likely violate the flatness of the scalaron potential and make inflation difficult in explaining the present Universe. We find a severe constraint on the dimensionless coupling of the $R^4$ correction as $-5.5\times {10}^{-8}<s<9.1\times {10}^{-8}$ from the recent results of the Planck observation. If we start from the chaotic initial condition, the constraint becomes much more severe. However, in the case in which the coupling of the $R^4$ correction is positive, the scalaron potential has a local maximum with two local minima at the origin and infinity, which admits topological inflation. In this case, inflation can take place naturally if the coupling satisfies the observational constraints.

Figure 1

The potential with higher-order correction for the Starobinsky model with various amplitudes of the corrections is shown. Here, we take ${\lambda }_1=8\times {10}^{10}(M_{\text{pl}}^{-2})$.

Figure 2

The cosmological perturbations generated from the supersymmetric Starobinsky model with higher-order corrections with respect to the model parameters are shown; (left) in the case with $s<0$ and (right) in the case with $s>0$. Curved (red) lines represent the contour lines of the amplitude of the scalar fluctuation $A_s$. The thick (light) shaded (blue) regions are the regions in which the values of the scalar spectral index are within the Planck $1\sigma (2\sigma )$ constraints. Horizontal (green) lines are the contour lines for the typical values of the tensor-to-scalar ratio $r$.

Figure 3

The potential (left) and its second derivative with respect to $\varphi$ (right) of both the Starobinsky model with higher order corrections $s=10^{-7}$ and the double well potential are shown. The double potential has the maximum at $\varphi =6$ and the amplitude is the same to the Starobinsky model. Around the potential maximum, the Starobinsky model is flatter than the double-well potential.

Figure 4

The ratio between the effective mass squared of the $C$ field and $V/3=H^2$ is shown as a function of inflaton $\varphi$ with $\zeta =0$, $-0.01$, and $-0.1$ and $s=10^{-7}$. The $C$ field is safely stabilized for $\zeta <-0.1$.

Figure 5

The ratio between the effective mass squared of the $b$ field and $V/3=H^2$ is shown as a function of inflaton $\varphi$ with $s=10^{-7}$ and $-10^{-7}$. The $b$ field is safely stabilized.