Quartic hilltop inflation ($n_s\simeq 1-3/N$)

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

Using the approximate large-angle CMB relation $T/S\simeq -7n_T$ gives $T/S\simeq 7{\sigma }^4/(96M^4{N}^3)$ in quartic hilltop inflation. It is suppressed by the fourth power of the small-field scale. This large-angle temperature ratio is not the primordial tensor-to-scalar ratio $r=16{ϵ}_V$; their transfer functions and normalization differ.

// Target: cosmology.bigb

For a negligible tensor contribution, the historical COBE horizon amplitude ${\delta }_H\sim 2\times 10^{-5}$ corresponds to curvature power ${\mathcal{P}}_{\mathcal{R}}\sim 25{\delta }_H^2/4$. In quartic hilltop inflation, $\alpha ={\pi }^2{\mathcal{P}}_{\mathcal{R}}/(8N^3)$ is of order $10^{-14}$ at $N=50$, independent of the small symmetry-breaking scale at this order. The exact inferred amplitude depends on the late cosmology and tilt used in the COBE transfer calculation.

// Target: cosmic-inflation.bigb