Primordial trispectrum from inflation

TL;DR

This paper develops a comprehensive δN formalism framework to compute the leading-order power spectrum, bispectrum and trispectrum for multi-field inflation in slow-roll, linking non-Gaussianity to derivatives of the local expansion $N$ with respect to the field values. The authors derive explicit expressions for $P_\zeta$, $B_\zeta$, and $T_\zeta$ in terms of $N_A$, $N_{AB}$, and $N_{ABC}$ and the field correlators $C^{AB}$, $B^{ABC}$, and $T^{ABCD}$, introducing two key trispectrum parameters, $\tau_{NL}$ and $g_{NL}$, with $\tau_{NL}=N_{AB}N^{AC}N^B N_C/(N_D N^D)^3$ and $g_{NL}=(25/54) N_{ABC}N^A N^B N^C/(N_D N^D)^3$. In Gaussian-field scenarios, $f_{NL}$, $\tau_{NL}$ and $g_{NL}$ can be computed from second and third derivatives of $N$, yielding slow-roll limits for inflaton and curvaton models; notably, $\tau_{NL}$ generally scales as $f_{NL}^2$ in single-field cases, while $g_{NL}$ can be large even when $f_{NL}$ is small, allowing trispectrum observations to test beyond-slow-roll or non-traditional models. The results provide a practical framework for constraining inflationary models via CMB/LSS/21-cm probes by characterizing the distinct momentum-dependent shapes of the trispectrum through $\tau_{NL}$ and $g_{NL}$.

Abstract

We use the delta N-formalism to describe the leading order contributions to the primordial power spectrum, bispectrum and trispectrum in multiple-field models of inflation at leading order in a perturbative expansion. In slow-roll models where the initial field fluctuations at Hubble-exit are nearly Gaussian, any detectable non-Gaussianity is expected to come from super-Hubble evolution. We show that the contribution to the primordial trispectrum can be described by two non-linearity parameters, tau_{NL} and g_{NL}, which are dependent upon the second and third derivatives of the local expansion with respect to the field values during inflation.