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 $V(\phi)$. 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 $n_s(N)$.
Highlights
The latest Planck data [1] are in good agreement with the scalar spectral index ns with ns − 1 ∼ −2/N, where N is the number of e-foldings
Motivated by the relation ns − 1 −2/N indicated by recent cosmological observations, we derive the formulae to derive the inflaton potential V (φ) from ns(N )
Applied to ns −1 = −2/N, to the first order in the slow-roll approximation, we find that the potential is classified into two categories depending on the value of 1/V (N → ∞): T-model type (tanh2(γφ/2))
Summary
The latest Planck data [1] are in good agreement with the scalar spectral index ns with ns − 1 ∼ −2/N , where N is the number of e-foldings. In [8] the slow-roll parameter is given as a function of N to construct V (φ). In [9] the slow-roll parameters and η are given as functions of N to construct V (N ) and compute r. Ns − 1 = −p/N case is studied in [11] by solving for the slow-roll parameter. We first need to rewrite the slow-roll parameters as functions of N. From which V,N > 0 is required: V is larger in the past in the slow-roll approximation We note that this inequality follows from H < 0 which holds as long as the weak energy condition is satisfied:
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