The expected anisotropy in solid inflation

TL;DR

This work investigates how infrared, superhorizon perturbations in solid inflation generate a background anisotropy via a random-walk accumulation of the three scalar field perturbations. The authors compute the resulting anisotropy in the curvature power spectrum and relate it to both the duration of inflation beyond the CMB-scale e-folds, $N_{\rm tot}-N_{\rm CMB}$, and the amplitude of the squeezed bispectrum, encapsulated in the non-Gaussianity coefficient $c_2$. They derive an explicit RMS estimate $\sigma_{\rm expected}$ for the background anisotropy and translate it into the CMB power-spectrum anisotropy parameter $g_* = 3 c_2 \sigma$, showing that for $N_{\rm tot}-N_{\rm CMB}\sim 20$–$30$ and near-Planck limits on $c_2$, the predicted anisotropy can be at the few percent level and potentially observable. The analysis also discusses the dependence on the slow-roll parameter $\epsilon$, the scalar and vector sound speeds, and notes similarities to $f(\phi)F^2$ vector-field models, highlighting concrete observational signatures for solid inflation.

Abstract

Solid inflation is an effective field theory of inflation in which isotropy and homogeneity are accomplished via a specific combination of anisotropic sources (three scalar fields that individually break isotropy). This results in specific observational signatures that are not found in standard models of inflation: a non-trivial angular dependence for the squeezed bispectrum, and a possibly long period of anisotropic inflation (to drive inflation, the "solid" must be very insensitive to any deformation, and thus background anisotropies are very slowly erased). In this paper we compute the expected level of statistical anisotropy in the power spectrum of the curvature perturbations of this model. To do so, we account for the classical background values of the three scalar fields that are generated on large (superhorizon) scales during inflation via a random walk sum, as the perturbation modes leave the horizon. Such an anisotropy is unavoidably generated, even starting from perfectly isotropic classical initial conditions. The expected level of anisotropy is related to the duration of inflation and to the amplitude of the squeezed bispectrum. If this amplitude is close to its current observational limit (so that one of the most interesting predictions of the model can be observed in the near future), we find that a level of statistical anisotropy $\gtrsim 3\%$ in the power spectrum is to be expected, if inflation lasted $\gtrsim 20-30$ e-folds more than the final $50-60$ efolds required to generare the CMB modes. We also comment and point out various similarities between solid inflation and models of inflation where a suitable coupling of the inflaton to a vector kinetic term $F^{2}$ gives frozen and scale invariant vector perturbations on superhorizon scales.

Figures (2)

• Figure 1: Blue solid lines: ratio (\ref{['gs-exp']}). The expected anisotropy in solid inflation is given by this quantity times $\frac{c_2}{60}$ (where $c_2$ parametrizes the amount of non-gaussianity in the model, and $\frac{c_2}{60} \leq 1$ at $95\%$ C.L.). Black dashed lines: tensor-to-scalar ratio $r$. The $x-$ axis is the standard slow roll parameter $\epsilon$, while the $y$ axis is the duration of inflation in addition to the final $N_{\rm CMB} \simeq 50-60$ e-folds. The value $c_L = \frac{1}{\sqrt{3}}$ is assumed in this figure.
• Figure 2: Asymptotic value of the ratio (\ref{['gs-exp']}) at asymptotically large $N_{\rm tot} - N_{\rm CMB} \rightarrow \infty$, as a function of the slow roll parameter $\epsilon$, and for the two limiting values of $c_L$. This asymptotic value diverges at $\epsilon < \frac{1-n_s}{2 \left( 1 + c_L^2 \right)}$.