Saddle point inflation from higher order corrections to Higgs/Starobinsky inflation

We explore two saddle point inflationary scenarios in the context of higher-order corrections related to different generalizations of general relativity. Firstly, we deal with the Jordan frame Starobinsky potential, for which we identify a portion of a parameter space of inflection point inflation, which can accommodate all the experimental results. Secondly, we analyze Higgs inflation and more specifically the influence of nonrenormalizible terms on the standard quartic potential. All results were verified with the Planck 2015 data.

Figure 1

$V(\varphi )$ for Starobinsky inflation and for saddle point inflation (dashed purple line and green lines respectively). Black points represent $\varphi ={\varphi }_s$ for ${\lambda }_1=10^k$, where $k\in \{-5,-4,\dots ,2\}$ (smallest ${\varphi }_s$ for biggest ${\lambda }_1$).

Figure 2

Left Panel: $M({\lambda }_1)$ for several values of $V^{\prime }({\varphi }_s)$. As expected, lower $V_{\varphi }({\varphi }_s)$ gives lower scale of inflation. Right panel: ${\varphi }_{\star }$ and ${\varphi }_s$ (dashed and solid lines respectively).

Figure 3

Left panel: $r({\varphi }_{\star })$ as a function of ${\lambda }_1$ for several values of $V_{\varphi }({\varphi }_s)$. All obtained values of $r$ are consistent with the Planck data. For ${\lambda }_1\gtrsim {10}^5$ one obtains $r\lesssim 0.002$, which gives $\mathrm{\Delta }\varphi <m_p$. Right panel: $n_s({\varphi }_{\star })$ as a function of ${\lambda }_1$ for several values of $V_{\varphi }({\varphi }_s)$. One, two and three $\sigma$ regimes of the Planck 2015 best fit of $n_s$ lay between green lines (solid, dashed and dotted respectively). For ${\lambda }_1\lesssim {10}^3$ one obtains correct value of $n_s$ even for $V_{\varphi }({\varphi }_s)=0$. In such a case, the last 60 $e$-folds of inflation are strongly influenced by the Starobinsky plateau.

Figure 4

The blue region represents the part of the $[{\lambda }_1,V_{\varphi }({\varphi }_s)]$ parameter space which fits $2\sigma$ constrains on $n_s$ from the Planck 2015 data. Smaller $V_{\varphi }({\varphi }_s)$ corresponds to the smaller scale of inflation. One could extrapolate the blue region for ${\lambda }_1>{10}^8$, which would give $M^2{\lambda }_1\ll 1$ for ${\lambda }_1<{10}^{20}$. For ${\lambda }_1\lesssim {10}^3$ any $V_{\varphi }({\varphi }_s)$ can fit the data, including the perfect saddle point case, however this corresponds to inflation on the plateau far from inflection point.

Figure 5

Left panel: $V(\mathrm{\Psi })$ with saddle point for ${\lambda }_8=1$, ${\lambda }_6\propto -{\lambda }^{3/2}$, $\lambda {\xi }^{-2}\sim 2\times {10}^{-9}$ and $\lambda \simeq \{0.15,0.4,1\}$ (dotted blue, dashed orange and solid red lines respectively). Dotted black line represents $V(\mathrm{\Psi })$ for ${\lambda }_6={\lambda }_8=0$. The saddle point inflation in this scenario will be analyzed in Sec. 4a. Right panel: $V$ as a function of Jordan frame field $\psi$ for a Higgs inflation with higher-order corrections to $W$ and $U$. ${\xi }_3={\mathrm{\Xi }}_3$ is an exact saddle. This scenario is briefly analyzed in Sec. 4b.

Figure 6

Left upper panel: spectral index $n_s=n_s(\lambda ,{\lambda }_6)$. One, two and three $\sigma$ regimes of the Planck 2015 best fit of $n_s$ are below green lines (solid, dashed and dotted, respectively), which correspond to ${\lambda }_6=-3.5{\lambda }^{3/2}$, ${\lambda }_6=-8.8{\lambda }^{3/2}$ and ${\lambda }_6=18{\lambda }^{3/2}$, respectively. Right upper panel: tensor-to-scalar ratio $r$. All values of $r$ presented in the plot are consistent with Planck results. Left lower panel: ${\lambda }_8$ as a function of $\lambda$ and ${\lambda }_6$. In order to obtain perturbative theory one requires ${\lambda }_8<10$, which gives ${\lambda }_6<0.1{\lambda }^{1/4}$. The allowed region lies under the red line. Right lower panel: ${\mathrm{\Psi }}_s/{\mathrm{\Psi }}_{\star }$, which determines how close to the saddle point one obtains freeze-out of given scale $k_{\star }$.

Figure 7

The nonminimal coupling parameter $\xi$ with constrains on ${\lambda }_8$ and $n_s$. For $\lambda \gtrsim 0.3$ the ${\lambda }_6$ dependence is very weak and therefore one recovers $\lambda /{\xi }^2=\text{const}$. Under the dashed green line (the Planck $2\sigma$ regime of $n_s$) the ${\lambda }_6$ dependence is weak for all $\lambda$ considered in the plot.