The Cosmological Phonon: Symmetries and Amplitudes on Sub-Horizon Scales

TL;DR

This paper develops an algebraic classification of scalar EFTs in cosmology where Lorentz boosts are spontaneously broken, using the coset construction and inverse Higgs constraints to organize possible non-linear symmetries realized by a single phonon π. It identifies a rich zoo of theories, including the new extended galileid, and shows that exceptional EFTs like DBI and the special galileon do not admit a broken-boost phase; notably, DBI is secretly Lorentz invariant in its flat-space limit, with consequences for cosmological correlators. The analysis extends to scattering amplitudes, proving that cubic vertices are generally inevitable when boosts are broken, and derives soft-theorem and weak-coupling constraints for sub-horizon physics. The results have significant implications for inflationary model-building and the interpretation of cosmological correlators, clarifying when flat-space amplitudes map to boundary observables and highlighting refined symmetry-based criteria for viable EFTs.

Abstract

In contrast to massless spinning particles, scalars are not heavily constrained by unitarity and locality. Off-shell, no gauge symmetries are required to write down manifestly local theories, while on-shell consistent factorisation is trivial. Instead a useful classification scheme for scalars is based on the symmetries they can non-linearly realise. Motivated by the breaking of Lorentz boosts in cosmology, in this paper we classify the possible symmetries of a shift-symmetric scalar that is assumed to non-linearly realise Lorentz boosts as, for example, in the EFT of inflation. Our classification method is algebraic; guided by the coset construction and inverse Higgs constraints. We rediscover some known phonon theories within the superfluid and galileid classes, and discover a new galileid theory which we call the $\textit{extended galileid}$. Generic galileids correspond to the broken phase of galileon scalar EFTs and our extended galileids correspond to special subsets where each galileon coupling is fixed by an additional symmetry. We discuss the broken phase of theories that also admit a perturbation theory around Poincaré invariant vacua and we show that the so-called exceptional EFTs, the DBI scalar and special galileon, do not admit such a broken phase. Concentrating on DBI we provide a detailed account of this showing that the scattering amplitudes are secretly Poincaré invariant when the theory is expanded around the superfluid background used in the EFT of inflation. We point out that DBI is an exception to the common lore that the residue of the total energy pole of cosmological correlators is proportional to the amplitude. We also discuss the inevitability of poles in $2 \rightarrow 2$ scattering amplitudes when boost are spontaneously broken meaning that such theories do not admit Adler zeros and generalisations even in the presence of a shift symmetry.

Figures (7)

• Figure 1: Inverse Higgs tree for a single essential scalar in a Lorentz invariant theory, from RSW1.
• Figure 2: Level-1 Inverse Higgs Trees
• Figure 3: Level-2 Inverse Higgs Trees
• Figure 4: The soft limit of the scattering of phonons is dominated by diagrams within which the soft momentum is attached to an external leg.
• Figure 5: Bounds on the perturbative unitary regions of the scaling superfluid for spherical wave coefficient up to $\ell=2$. Each colored region corresponds a different value of $c_s$ with the conformal superfluid corresponding to $c_s=1/\sqrt{3}$. All colored regions extend to arbitrarily low $|\mathbf{p}_s|$, $E_s$ and $s$, as spherical wave coefficients get smaller. The dashed enclosed region represents the subhorizon regime of scattering. Left: Weakly-coupled regions for various scattering energies $E_s$ and momenta $|\mathbf{p}_s|$. The patterned region is forbidden kinematically. Right: Same weakly-coupled regions, this time in terms of the Mandelstam variable $s$ and momentum $|\mathbf{p}_s|$. For a given $c_s$, the maximum $s=s_{\text{max}}$ is reached at low $|\mathbf{p}_s|$ while $|\mathbf{p}_s|_{\text{max}}$ is reached at an intermediate $s$.