Kavli Affiliate: Misao Sasaki
| First 5 Authors: Shi Pi, Misao Sasaki, , ,
| Summary:
We study the comoving curvature perturbation $mathcal{R}$ in the
single-field inflation models whose potential can be approximated by a
piecewise quadratic potential $V(varphi)$ by using the $delta N$ formalism.
We find a general formula for $mathcal{R}(deltavarphi, deltapi)$,
consisting of a sum of logarithmic functions of the field perturbation
$deltavarphi$ and the velocity perturbation $deltapi$ at the point of
interest, as well as of $deltapi_*$ at the boundaries of each quadratic
piece, which are functions of ($deltavarphi, deltapi$) through the equation
of motion. Each logarithmic expression has an equivalent dual expression, due
to the second-order nature of the equation of motion for $varphi$. We also
clarify the condition under which $mathcal{R}(deltavarphi, deltapi)$
reduces to a single logarithm, which yields either the renowned “exponential
tail” of the probability distribution function of $mathcal{R}$ or a
Gumbel-distribution-like tail.
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