Year: 2015
Type: article
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. The 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. |$\gamma$| is the ratio of |$1/V$| at |$N\rightarrow \infty$| 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 |$n_s(N)$|.
DOI: 10.1093/ptep/ptv090
Author: Takeshi Chiba
Institution: Nihon University
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $979
OpenAlex ID: W1926022807