Abstract
In this paper we will illustrate how to constrain unavoidable K\"ahler corrections for ${\cal N}=1$ supergravity (SUGRA) inflation from the recent Planck data. We will show that the non-renormalizable K\"ahler operators will induce in general non-minimal kinetic term for the inflaton field, and two types of SUGRA corrections in the potential - the Hubble-induced mass ($c_{H}$), and the Hubble-induced A-term ($a_{H}$) correction. The entire SUGRA inflationary framework can now be constrained from (i) the speed of sound, $c_s$, and (ii) from the upper bound on the tensor to scalar ratio, $r_{\star}$. We will illustrate this by considering a heavy scalar degree of freedom at a scale, $M_s$, and a light inflationary field which is responsible for a slow-roll inflation. We will compute the corrections to the kinetic term and the potential for the light field explicitly. As an example, we will consider a visible sector inflationary model of inflation where inflation occurs at the point of inflection, which can match the density perturbations for the cosmic microwave background radiation, and also explain why the universe is filled with the Standard Model degrees of freedom. We will scan the parameter space of the non-renormalizable K\"ahler operators, which we find them to be order ${\cal O}(1)$, consistent with physical arguments. While the scale of heavy physics is found to be bounded by the tensor-to scalar ratio, and the speed of sound, $ {\cal O}(10^{11}\leq M_s\leq 10^{16}) $GeV, for $0.02\leq c_s\leq 1$ and $10^{-22}\leq r_\star \leq 0.12$.
Highlights
The aim of this paper is to place a generic bound on the Planck suppressed corrections to the Kahler potential on top of the minimal Kahler potential
The heavy field is coherently oscillating during the initial phases of inflation: in this case we will consider a very simple scenario, where we imagine that the heavy field is coherently oscillating at a VEV given by Ms with an amplitude Ms at the onset of inflation driven by the light field
We have shown that in any N = 1 SUGRA inflation model when ever there are more degrees of freedom, non-minimal Kahler corrections would induce three distinct types of corrections: (i) non-minimal kinetic term for the inflaton, (ii) Hubble-induced mass correction to the inflaton, and (iii) Hubble-induced A-term in the potential
Summary
For a sub-Plancikan slow-roll models of inflation, one can express the tensor-to-scalar ratio, r⋆, at the pivot scale, k⋆ ∼ 0.002 Mpc−1, in terms of the field displacement, ∆φ, during the observed ∆N ≈ 17 e-foldings of inflation, for cs = 1 [52, 53]: 25√3 cs r⋆ 0.12. Where a, b, c, d are dimensionless parameters.3 These corrections will inevitably lead to a departure from the minimal kinetic energy for both the fields. 2. Cross coupling terms between the flat direction induced Kahler derivative and the inflaton superpotential: KφK φφKφ. 3. Interaction between the Kahler derivative and superpotential of the inflaton, supergravity Kahler metric and Kahler potential of the flat direction: KφKφsDsW ∗
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