Reconstructing a general inflationary action

TL;DR

This work addresses the problem of reconstructing the full single-field inflationary action $L(X,\phi)$ rather than restricting to a canonical kinetic term or a specific potential. It develops a Hamilton-Jacobi reconstruction framework and extends the inflationary flow formalism to general actions, connecting scalar/tensor power spectra and non-Gaussianity measurements to on-shell combinations of Lagrangian derivatives along the inflationary trajectory. The authors show that, under ideal slow-roll conditions, one can constrain $X\mathcal{L}_X$, $X^2\mathcal{L}_{XX}$ and the on-shell trajectory of $\mathcal{L}$ (up to gauge freely in $X$) across observable scales, using $\epsilon$, $\eta$, $\kappa$ and the sound speed $c_s$. They introduce three hierarchies of flow parameters to evolve general actions and illustrate how non-Gaussianity, particularly $f_{NL}$, furnishes crucial information about higher-derivative kinetic terms, with Planck-era data offering potent tests. Overall, the framework enables model-independent linkage between cosmological data and high-energy theory, clarifying how to distinguish kinetic-structure (e.g., DBI-like terms) from canonical inflation and guiding future data-driven action reconstruction.

Abstract

If inflation is to be considered in an unbiased way, as possibly originating from one of a wide range of underlying theories, then observations need not be simply applied to reconstructing the inflaton potential, V(φ), or a specific kinetic term, as in DBI inflation, but rather to reconstruct the inflationary action in its entirety. We discuss the constraints that can be placed on a general single field action from measurements of the primordial scalar and tensor fluctuation power spectra and non-Gaussianities. We also present the flow equation formalism for reconstructing a general inflationary Lagrangian, L(X,φ), with X={1/2}\partial_μφ\partial^μφ, in a general gauge, that reduces to canonical and DBI inflation in the specific gauge \partial L/\partial X = c_s^{-1}.

Figures (1)

• Figure 1: Dimension 8 kinetic operator interaction of $\delta\phi$ contribution to the 3-point function of $\langle\zeta(t,\vec{k}_{1})\zeta(t,\vec{k}_{2})\zeta(t,\vec{k}_{3})\rangle$.The small dots indicates the fact that the $\delta\phi$ propagator is a dS propagator (i.e. there is an interaction with the background classical homogeneous gravitational field leading to a time dependent mass). The blob on the right indicates that it is an interaction term (partly non-local) arising from the presence scalar metric fluctutations.