Constraining ${\cal N}=1$ supergravity inflationary framework with
  non-minimal K\"ahler operators
  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 $10^{-22}\leq r_\star \leq 0.12$.
