In-in and $δN$ calculations of the bispectrum from non-attractor single-field inflation

TL;DR

This work analyzes non-attractor single-field inflation models that yield a scale-invariant power spectrum while generating a large local-type bispectrum, thereby violating Maldacena's consistency relation. It presents three independent calculations—in-in formalism in the comoving gauge, in-in formalism in the flat gauge, and the $\delta N$ formalism—and shows they all give the same result: $f_{NL}^{\rm local}=\frac{5(1+c_s^2)}{4c_s^2}$ for arbitrary sound speed $c_s$. The analysis demonstrates that non-Gaussianity arises from super-horizon interactions (e.g., $\dot{\cal R}^3$ and ${\cal R}\dot{\cal R}^2$) when ${\cal R}$ grows as $a^3$, and confirms the $\delta N$ approach remains valid in this non-attractor regime. Together, these results imply that a large local bispectrum does not rule out all single-field inflation models with a BD initial state; rather, it distinguishes attractor-based models from non-attractor histories.

Abstract

In non-attractor single-field inflation models producing a scale-invariant power spectrum, the curvature perturbation on super-horizon scales grows as ${\cal R}\propto a^3$. This is so far the only known class of self-consistent single-field models with a Bunch-Davies initial state that can produce a large squeezed-limit bispectrum violating Maldacena's consistency relation. Given the importance of this result, we calculate the bispectrum with three different methods: using quantum field theory calculations in two different gauges, and classical calculations (the $δN$ formalism). All the results agree, giving the local-form bispectrum parameter of $f^{local}_{NL}=5(1+c_s^2)/(4c_s^2)$. This result is valid for arbitrary values of the speed of sound parameter, $c_s$, for a particular non-attractor model we consider in this paper.

Figures (3)

• Figure 1: Evolution of $\phi (n)$ in the undershoot situation as a function of the number of $e$-folds, $n$, counted from the beginning of inflation. The inflaton field climbs up the potential, stops somewhere before the top of the potential ($\phi=0$), turns around and goes back to plus infinity. The dashed red curve is the analytic ansatz for the non-attractor phase, while the solid blue curve is the full numerical solution. The transition to a slow-roll inflation phase is sharp, and an extended slow-roll phase follows afterward. The parameters, consistent with Eqs. \ref{['sol2']} and \ref{['sol3']}, in units of $M_P$ are $V_0 \simeq 6.25 \times 10^{-4}$, $M=5 \times 10^{-5}$, $\alpha = 10$, and $\eta = -6$.
• Figure 2: Same as Fig. \ref{['stop']}, but for the overshoot situation. The inflaton field climbs up the potential, goes over the top of the potential, and rolls down on the other side of the potential.
• Figure 3: Phase-space diagram of the model given by Eq. \ref{['model']}. The black line separates two different trajectories: the undershoot (dashed blue lines) and overshoot (dot-dashed red lines) trajectories. The purple dashed line near the origin shows the asymptotic solution, $\phi \propto a^{-3}$. In the undershoot case, the inflaton field climbs up the potential, stops somewhere before reaching the top of the potential, and returns back. In the overshoot case, the field climbs up the potential, crosses the top of the potential, and rolls down on the other side of the potential. The symmetry in the plot reflects the fact that our Lagrangian given in Eq. \ref{['model']} is symmetric under the transformation $\phi \to - \phi$ and $\dot \phi \to - \dot \phi$.