Inflationary tensor fossils in large-scale structure

TL;DR

This work analyzes the squeezed-limit tensor-scalar-scalar bispectrum from inflation and its observational imprints, focusing on single-clock consistency conditions and their possible violations. It demonstrates that non-attractor inflation can imprint model-dependent, $A_{ij}$-driven TSS signals that appear as a local quadrupole on the largest scales, yet decay rapidly toward smaller scales, while solid inflation naturally violates CCS due to non-adiabatic perturbations and anisotropic stresses, enabling potentially detectable clustering fossils in large-scale structure. The paper derives the perturbation theory for both scenarios, computes the local quadrupole and fossil signatures, and discusses how CMB quadrupole and galaxy/21-cm surveys can constrain or reveal these effects, highlighting that upcoming surveys could observe solid-inflation fossils or constrain the duration of non-attractor phases. Overall, it shows that tensor modes can leave accessible fingerprints in the late-time mass distribution even without direct detection, offering a pathway to probe inflationary dynamics through large-scale structure. The results emphasize the complementary role of CCS tests, large-scale quadrupole measurements, and clustering fossils in differentiating inflationary models with non-standard clock behavior.

Abstract

Inflation models make specific predictions for a tensor-scalar-scalar three-point correlation, or bispectrum, between one gravitational-wave (tensor) mode and two density-perturbation (scalar) modes. This tensor-scalar-scalar correlation leads to a local power quadrupole, an apparent departure from statistical isotropy in our Universe, as well as characteristic four-point correlations in the current mass distribution in the Universe. So far, the predictions for these observables have been worked out only for single-clock models in which certain consistency conditions between the tensor-scalar-scalar correlation and tensor and scalar power spectra are satisfied. Here we review the requirements on inflation models for these consistency conditions to be satisfied. We then consider several examples of inflation models, such as non-attractor and solid inflation models, in which these conditions are put to the test. In solid inflation the simplest consistency conditions are already violated whilst in the non-attractor model we find that, contrary to the standard scenario, the tensor-scalar-scalar correlator probes directly relevant model-dependent information. We work out the predictions for observables in these models. For non-attractor inflation we find an apparent local quadrupolar departure from statistical isotropy in large-scale structure but that this power quadrupole decreases very rapidly at smaller scales. The consistency of the CMB quadrupole with statistical isotropy then constrains the distance scale that corresponds to the transition from the non-attractor to attractor phase of inflation to be larger than the currently observable horizon. Solid inflation predicts clustering fossils signatures in the current galaxy distribution that may be large enough to be detectable with forthcoming, and possibly even current, galaxy surveys.

Figures (3)

• Figure 1: Squeezed configuration for the momenta: $k_{1}\ll k_{2}\simeq k_{3}$.
• Figure 2: A qualitative representation in the potential-scalar field plane of the non-attractor phase (a), and of the attractor phase, in the form of the (b) undershoot and overshoot (c) cases. The scalar field begins with a positive value. In the undershoot case, the field reaches a value $\phi_{*}>0$ at the point where its velocity becomes null, so it rolls down on the same side of the potential (b). In the overshoot case, the system has enough kinetic energy to go over the top of the potential and roll down the other side (c).
• Figure 3: Diagrammatic representation of the contribution in Eq. (\ref{['bisint']}) to the tensor-scalar-scalar correlator. The dashed line is the graviton propagator, continuous lines are the scalars.