Reconstructing the inflaton potential from the spectral index

Abstract

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 $V(\varphi )$. We find that for $n_s-1=-2/N$, $V(\varphi )$ is either ${\tanh }^2(\gamma \varphi /2)$ ("T-model") or ${\varphi }^2$ (chaotic inflation) to the leading order in the slow-roll approximation. $\gamma$ is the ratio of $1/V$ at $N\to \infty$ to the slope of $1/V$ at a finite $N$ and is related to "$\alpha$" in the $\alpha$-attractors by ${\gamma }^2=2/3\alpha$. The tensor-to-scalar ratio $r$ is $r=8/N({\gamma }^{2}N+1)$. The implications for the reheating temperature are also discussed. We also derive formulas for $n_s-1=-p/N$. We find that if the potential is bounded from above, only $p>1$ is allowed. Although $r$ depends on a parameter, the running of the spectral index is independent of it, which can be used as a consistency check of the assumed relation of $n_s(N)$.

15 pages, 3 figures, published version

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• Probing the era of reheating for reconstructed inflationary potential in the RS II braneworld
• Reconstructing Mimetic Cosmology
• Recent results and future prospects of kaonic nuclei at J-PARC
• Inflationary attractors from nonminimal coupling
• Constraints on the radiation temperature before inflation
• Search for Deeply Bound Kaonic Nuclear States with AMADEUS
• Primordial Perturbations Including Second-Order Derivatives of the Inflationary Potential