The anisotropic power spectrum and bispectrum in the f(phi) F^2 mechanism

TL;DR

The authors show that coupling the inflaton to a vector kinetic term $I^2(\,\varphi\,)F_{\mu\nu}F^{\mu\nu}$ generates a scale-invariant vector spectrum whose super-horizon modes create a classical, anisotropic background $\vec{E}_{\rm classical}$. Using in-in formalism and explicit expansion of the interaction Lagrangian, they derive anisotropic corrections to the curvature perturbation power spectrum and bispectrum, finding $P_\zeta(\vec{k}) = P^{(0)}[1 + \frac{24}{\epsilon}\frac{E_{\rm classical}^2}{V(\varphi)}N_k^2\sin^2\theta_{\hat{k},\hat{E}_{\rm classical}}]$ (to leading order) and a squeezed-limit bispectrum with an effective local $f_{\rm NL}$ of order a few to a few tens for plausible $|g_*|$. The results generalize to loop corrections and depend on the total inflationary duration, with backreaction and IR effects setting natural limits on the anisotropy; the framework also applies to anisotropic inflation, the waterfall mechanism, and magnetogenesis, predicting correlated TB/EB signals and direction-dependent non-Gaussianity. Overall, a percent-level or larger anisotropy in the primordial perturbations appears as a natural outcome of these high-spin field scenarios, with testable observational consequences.

Abstract

A suitable coupling of the inflaton phi to a vector kinetic term F^2 gives frozen and scale invariant vector perturbations. We compute the cosmological perturbations zeta that result from such coupling by taking into account the classical vector field that unavoidably gets generated at large scales during inflation. This generically results in a too anisotropic power spectrum of zeta. Specifically, the anisotropy exceeds the 1% level (10% level) if inflation lasted ~5 e-folds (~50 e-folds) more than the minimal amount required to produce the CMB modes. This conclusion applies, among others, to the application of this mechanism for magnetogenesis, for anisotropic inflation, and for the generation of anisotropic perturbations at the end of inflation through a waterfall field coupled to the vector (in this case, the unavoidable contribution that we obtain is effective all throughout inflation, and it is independent of the waterfall field). For a tuned duration of inflation, a 1% (10%) anisotropy in the power spectrum corresponds to an anisotropic bispectrum which is enhanced like the local one in the squeezed limit, and with an effective local f_{NL} ~3 (~30). More in general, a significant anisotropy of the perturbations may be a natural outcome of all models that sustain higher than 0 spin fields during inflation.

Figures (2)

• Figure 1: Leading diagrams for $\langle \zeta^2 \rangle$, with the vertices labelled as in (\ref{['Hint']}).
• Figure 2: Leading diagrams for $\langle \zeta^3 \rangle$, with the vertices labelled as in (\ref{['Hint']}).