Inflation in Motion: Unitarity Constraints in Effective Field Theories with Broken Lorentz Symmetry

TL;DR

This work develops radiative stability and perturbative unitarity bounds for effective field theories with spontaneously broken boosts in the inflationary context. By introducing a spherical-wave unitarity analysis, it reveals separate energy and momentum cutoffs and shows that cubic Lorentz-violating interactions force a mismatch between these cutoffs unless suppressed. Applying these constraints to the EFT of Inflation yields relations among leading coefficients (e.g., $c_s$, $\alpha_1$, $\beta_1$) and frames a theoretically informed prior that, when combined with Planck bispectrum data, tightens the allowed non-Gaussianity parameter space for subhorizon, single-field inflation. Overall, the paper demonstrates how Lorentz-violance–related EFT techniques can be ported to cosmology to sharpen theoretical priors and interpret current observational bounds.

Abstract

During inflation, there is a preferred reference frame in which the expansion of the background spacetime is spatially isotropic. In contrast to Minkowski spacetime, observables can depend on the velocity of the system with respect to this cosmic rest frame. We derive new constraints from radiative stability and unitarity on effective field theories with such spontaneously broken Lorentz symmetry. In addition to a maximum energy scale, there is now also a critical velocity at which the theory breaks down. The theory therefore has different resolving power in time and in space, and we show that these can only coincide if cubic Lorentz-violating interactions are absent. Applying these bounds to the Effective Field Theory of Inflation, we identify the region of parameter space in which inflation can be both single-field and weakly coupled on subhorizon scales. This can be implemented as a theoretical prior, and we illustrate this explicitly using Planck observational constraints on the primordial bispectrum.

Figures (7)

• Figure 1: Cartoon of the symmetry breaking. The scattering process can be described covariantly using two time-like vectors $n^\mu$ (the rest frame of the background) and $p_s^\mu$ (the center of mass motion). We then proceed along the top row, by first fixing our coordinates such that the background is at rest $n^\mu = (1, \mathbf{0})$ (removing $n^\mu$ in this way leaves a theory which is only manifestly invariant under spatial rotations), and then specifying the kinematics of the particles (the spacelike part of the CoM motion then breaks the 3 spatial rotations down to just 1). Since the underling physics is Lorentz invariant, a completely equivalent description is shown in bottom row, in which first one fixes coordinates such that the CoM is at rest, but at the price of now having a background spacetime which appears to expand anisotropically. In either case, once both the background and the CoM motion are fixed, there is only 1 rotational symmetry remaining.
• Figure 2: Left panel shows the highest EFT cutoff compatible with the radiative stability of \ref{['eqn:newEFT2']}, $\Lambda_{\rm max}^4 = 16 \pi^2 c_s^4 f_\pi^2 / g_n^2$ (where the minimum $g_n$ is given in \ref{['eqn:gnmin2']}), with dashed lines indicating where $\Lambda_{\rm max} = 3H$, $10H$ and $30H$ respectively (though note that these are order-of-magnitude estimates of $\Lambda_{\rm max}$ and may be subject to order unity corrections). Right panel shows this minimum $g_n$ (again subject to order unity corrections). The maximum energy identified in section \ref{['sec:3']} from unitary $2\to 2$ scattering, $\omega_{\rm max}^4 / f_\pi^4 = 480 \pi c_s^4 / (1- c_s^2 )$, is never larger than this $\Lambda_{\rm max}$ estimate by more than an $\mathcal{O}(1)$ factor, so it possible to neglect higher order corrections in a radiatively stable way.
• Figure 3: While the non-linear unitarity relation \ref{['eqn:Aunitary']} holds non-perturbatively, it can be applied in perturbation theory to relate the one-loop discontinuity (shown left) to the square of the tree-level amplitude (shown right), where the inequality holds whenever the kinematic states of particles 1 and 2 coincide with those of particles 3 and 4. Perturbative unitarity breaks down if the $| \mathcal{A}_{2\to 2}^{\rm loop}|$ required by this (perturbative) form of the unitarity condition is larger than the $| \mathcal{A}_{2 \to 2}^{\rm tree}|$ contribution.
• Figure 4: Properties of the $\dot \pi^4$ spherical-wave amplitudes. Left: The ratio $a^{00}_{\ell \ell} / a^{00}_{00}$ is plotted against $\gamma_s = 1/\sqrt{1-\rho_s^2}$ for $L=3$, $10$ and $30$, and gray grid lines show $\gamma_s = 3, 10$ and $30$. At small $\gamma_s$ it is $a^{00}_{00}$ which dominates, while at large $\gamma_s$ the $a^{00}_{\ell \ell}$ amplitude is larger by a factor of $2\ell +1$. The $\gamma_s$ at which $a^{00}_{\ell \ell}$ first exceeds $a^{00}_{00}$ scales $\propto \ell$, so terms with $\ell \gg \gamma_s$ can always be neglected from the sum in \ref{['eqn:Ub1']}. Right: The angular integral $\hat{I}_{\ell}$ is plotted against $\ell$ at fixed $\gamma_s = 3, 10, 30, 100$ and $300$. In general they display a large positive maximum at a finite $\ell$ near to $\gamma_s$, which is followed by a smaller negative minimum before they approach zero from below as $\ell \to \infty$ (sufficiently fast for the spherical wave expansion \ref{['eqn:pw_n']} to converge). Black dashed lines show the approximation \ref{['eqn:Ismallrsb1']} for $\gamma_s = 100$ and $\gamma_s = 300$, agreeing well with $\hat{I}_{\ell}$ at sufficiently small $\ell < \gamma_s/\sqrt{2}$.
• Figure 5: The values of $s$ and $|\mathbf{p}_s|$ consistent with perturbative unitarity for a $\dot \pi^4/f_\pi^4$ interaction (left) and a $\dot \pi^3/f_\pi^2$ interaction (right). Gray grid lines show $\gamma_s = 3, 10$ and $30$---the spherical waves with $\ell\approx \gamma_s$ are most responsible for the constraints. The cubic vertex has the special feature that at fixed $|\mathbf{p}_s|$ there is a minimum $s$, and the black dashed line shows the $s^{1/6}$ scaling expected from \ref{['eqn:smina1']}. The maximum $s_{\rm max}$ and $|\mathbf{p}_s|_{\rm max}$ are given in \ref{['eqn:smaxpmaxb1']} with $\alpha_1 =1$ and in \ref{['eqn:smaxpmaxa1']} with $\beta_1 =1$ respectively---note that the actual maximum $|\mathbf{p}_s|$ for the cubic vertex is set by $\ell \approx 1$ spherical-waves and is lower than the upper bound in \ref{['eqn:smaxpmaxa1']} from the $\ell = 0$ wave.