 Quartic hilltop inflation

Near a flat quartic maximum, $V\simeq \alpha {\sigma }^4-3\alpha {\varphi }^4$. With reduced Planck mass $M$, the slow-roll approximation gives $N\simeq {\sigma }^4/(24M^2{\varphi }_N^2)$, ${\eta }_V\simeq -3/(2N)$ and ${ϵ}_V\simeq {\sigma }^4/(192M^4{N}^3)$. Hence $n_s\simeq 1-3/N$, $n_T\simeq -2{ϵ}_V$, and ${\mathcal{P}}_{\mathcal{R}}\simeq 8\alpha N^3/{\pi }^2$. The result is dominated by the near-maximum part of the trajectory; extending slow roll to a distant minimum is unnecessary.

// Target: cosmic-inflation.bigb