Reconstructing the inflaton potential from the spectral index

Understand

Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\sim -2/N$, where $N$ is the number of e-foldings.

• Quadratic chaotic model, Starobinsky model and Higgs inflation or $\alpha$-attractors connecting them are typical examples predicting such a relation.
• We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(\phi)$.
• We find that for $n_s-1=-2/N$, $V(\phi)$ is either $\tanh^2(\gamma\phi/2)$ ("T-model") or $\phi^2$ (chaotic inflation) to the leading order in the slow-roll approximation.