Signs of Analyticity in Single-Field Inflation

TL;DR

This work develops IR/UV consistency relations for single-field inflation by exploiting analyticity of forward scattering to derive a sum rule that links low-energy EFT parameters, notably the sound speed $c_s$ and cubic and quartic couplings $c_3$ and $c_4$, to the high-energy spectrum of the theory. Under the assumption of a Lorentz-invariant UV completion, positivity of the sum-rule integral yields concrete bounds among $c_s$, $c_3$, and $c_4$, such as $c_4 > (2 c_3)^2$ in the large-$|c_4|$ limit and a general bound $c_4+1 > ((2 c_3 + 1) - a(c_s))^2 + b(c_s)$ with $a,b$ functions of $c_s$. The paper also analyzes perturbative unitarity to determine a critical sound speed $(c_s)_\star \approx 0.31$, below which the EFT becomes strongly coupled before the cutoff, and presents explicit UV-complete examples that satisfy the positivity constraints. A key conjecture is that $c_s=1$ (slow-roll) theories may be uniquely UV-completed by slow-roll inflation, implying strong theoretical ties between the absence of non-Gaussianity and the UV structure of inflation. Overall, the results place nontrivial, testable constraints on primordial non-Gaussianity and guide future explorations of non-forward sum rules to further pin down the microphysics of the inflationary epoch.

Abstract

The analyticity of response functions and scattering amplitudes implies powerful relations between low-energy observables and the underlying short-distance dynamics. These 'IR/UV' relations are rooted in basic physical principles, such as causality and unitarity. In this paper, we seek similar connections in inflation, relating cosmological observations to the physics responsible for the accelerated expansion. We assume that the inflationary theory is Lorentz invariant at short distances, but allow for non-relativistic interactions and a non-trivial speed of propagation at low energies. Focusing on forward scattering, we derive a 'sum rule' which equates a combination of low-energy parameters to an integral which is sensitive to the high-energy behavior of the theory. While for relativistic amplitudes unitarity is sufficient to prove positivity of the sum rule, this is not guaranteed in the non-relativistic case. We discuss the conditions under which positivity still applies, and show that they are satisfied by all known UV completions of single-field inflation. In that case, we obtain a consistency condition for primordial non-Gaussianity, which constrains the size and the sign of the equilateral four-point function in terms of the amplitude of the three-point function. The resulting bound rules out about half of the parameter space that is still allowed by current observations. Finding a violation of our consistency condition would point towards less conventional theories of inflation, or violations of basic physical principles.

Figures (4)

• Figure 1: Illustration of the relevant energy scales of the EFT. The flat space approximation applies for scattering energies above the Hubble scale, $\omega^2 > H^2$. The decoupling limit captures the regime $\omega^2 > |\dot H|$. The hierarchy $H^2 \ll \Lambda^2$ is guaranteed by the high degree of Gaussianity of the primordial perturbations, while $|\dot H| \ll H^2$ holds as a condition for inflation and is supported by observations of the spectral index.
• Figure 2: Illustration of the choice of contour in (\ref{['equ:A5']}).
• Figure 3: Illustration of the pole structure of the amplitude in (\ref{['equ:egamp1']}).
• Figure 4: Observational constraints on the EFT parameters $c_s$ and $c_3$PlanckNG.