How Gaussian can our Universe be?

TL;DR

This work shows that in canonical single-field slow-roll inflation, the minimal level of primordial non-Gaussianity cannot be eliminated by a coordinate choice. Using Conformal Fermi Coordinates, the authors demonstrate a physical, tilt‑dependent contribution to local observables with squeezed-limit scaling $k_\ell^2/k_s^2$ and an amplitude of order $0.1\times(n_s-1)$. The result is established through two complementary approaches: transforming Maldacena's bispectrum to CFC and a direct action-based calculation, both converging on a nonzero $\eta$-dependent floor, with an additional $c_s$-dependent enhancement when the inflaton speed of sound differs from unity. These findings quantify a concrete, observable floor for primordial non-Gaussianity and clarify its dependence on slow-roll parameters and the sound speed, while highlighting observational channels and projection effects that can mimic local-type signals.

Abstract

Gravity is a non-linear theory, and hence, barring cancellations, the initial super-horizon perturbations produced by inflation must contain some minimum amount of mode coupling, or primordial non-Gaussianity. In single-field slow-roll models, where this lower bound is saturated, non-Gaussianity is controlled by two observables: the tensor-to-scalar ratio, which is uncertain by more than fifty orders of magnitude; and the scalar spectral index, or tilt, which is relatively well measured. It is well known that to leading and next-to-leading order in derivatives, the contributions proportional to the tilt disappear from any local observable, and suspicion has been raised that this might happen to all orders, allowing for an arbitrarily low amount of primordial non-Gaussianity. Employing Conformal Fermi Coordinates, we show explicitly that this is not the case. Instead, a contribution of order the tilt appears in local observables. In summary, the floor of physical primordial non-Gaussianity in our Universe has a squeezed-limit scaling of $k_\ell^2/k_s^2$, similar to equilateral and orthogonal shapes, and a dimensionless amplitude of order $0.1\times(n_\mathrm{s}-1)$.

Figures (2)

• Figure 1: Squeezed limit of $\zeta = \zeta_\ell + \zeta_s$ in real space: we compute how the correlation function of $\zeta_s$ (which we call $\braket{\zeta_s\zeta_s}(r)$, where $r\equiv\lvert\bm{x}_1-\bm{x}_2\rvert$) depends on the long-wavelength fluctuation $\zeta_\ell$. We can expand $\zeta_\ell$ in a Taylor series, since it is slowly varying inside $R$: any point inside of $R$ is as good as the other for the expansion Creminelli:2011rhPajer:2013ana, so we will choose the middle point $\bm{x}_c\equiv(\bm{x}_1+\bm{x}_2)/2$ for simplicity. We also stress that the choice of $R$ is immaterial in the squeezed limit, the only real requirement being that $k_\ell\ll k_s$Wagner:2015gva.
• Figure 2: For a local observer $U^\mu$, the effect of a long mode $k_\ell$ is that of making the short modes evolve in a separate universe of size $\sim 1/k_\ell$, described by a curved FLRW metric with time-dependent tidal corrections Dai:2015rdaDai:2015jaa. Long modes become classical on super-Hubble scales, and we can describe their effect on small-scale perturbations by going to CFC. We carry on our construction to the end of inflation, when all relevant modes are super-Hubble and time-independent. This gives the coupling between long- and short-scale perturbations measured by the observer $U^\mu$, that can be used as initial condition for the evolution of short modes as they re-enter the Hubble radius during the Hot Big Bang.