Bispectrum at NLO in single field inflation: conservation and squeezed limit

In a recent paper, we computed the bispectrum of primordial density perturbations in CMB to second order in the slow-roll parameters of single field inflation, and found logarithmic infrared contributions that diverge in both large physical distances and squeezed limit where one momentum vanishes. In this work, we provide an independent test of the result by checking its conservation and the validity of the consistency relation between the squeezed limit of the bispectrum and the square of the power spectrum. Despite the violation of the main assumption for its general proofs which is the finiteness of the relevant observables in these limits, we find that the identity continues to hold in the vicinity of the squeezed limit and large time.

💡 Research Summary

This paper presents a detailed analysis of the bispectrum (three-point correlation function) of primordial density perturbations in single-field inflation models, calculated up to next-to-leading order (NLO) in the slow-roll expansion. The work serves as an independent verification and exploration of the properties of a previously derived NLO bispectrum result, focusing on its conservation over time and its behavior in the so-called squeezed limit.

The primary motivation stems from the discovery, in prior work by the authors, of logarithmic infrared (IR) divergences in the NLO bispectrum. These divergences appear in two forms: one proportional to ln(-k_t τ), which diverges at large physical distances (late times, τ → 0), and another proportional to ln(k_i/k_t), which diverges in the squeezed limit where one of the three momenta (k_i) becomes vanishingly small. Such divergences are physical, arising from the propagation of massless fields in a quasi-de Sitter space, and can give sizable corrections to the observable non-Gaussianity parameter f_NL.

The existence of these divergences potentially threatens a fundamental pillar of single-field inflation: the consistency relation. This relation connects the squeezed limit of the bispectrum to the square of the power spectrum, specifically B(k1,k2,k3) → -P(k1)