Anisotropy in solid inflation

TL;DR

Solid inflation uses a set of scalars with $\langle \phi^i \rangle = x^i$ and a Lagrangian $F[X,Y,Z]$ that makes the background isotropic yet highly insensitive to volume and spatial deformations; this yields an isotropic FRW solution but with a second, slow-responding channel that supports a prolonged anisotropic phase. This anisotropy persists on a timescale $\Delta t = O(1/(\varepsilon H))$, circumventing Wald's cosmological no-hair theorem, and leads to a quadrupolar anisotropy in the curvature power spectrum $P_\zeta(\vec{k})=P_\zeta(k)[1+g_* P_2(\cos\theta)]$ with $g_* = O(\sigma)$. They perform a detailed in-in calculation of the anisotropic correction, showing it is dominated by a single $L L$–type interaction and yields a squeezed-limit bispectrum with nontrivial angular dependence, analogous to the $f(\phi)F^2$ model. The results imply observationally relevant bounds on residual anisotropy, motivate an effective description of symmetry breaking during inflation, and raise questions about reheating and infrared anisotropies.

Abstract

In the model of solid / elastic inflation, inflation is driven by a source that has the field theoretical description of a solid. To allow for prolonged slow roll inflation, the solid needs to be extremely insensitive to the spatial expansion. We point out that, because of this property, the solid is also rather inefficient in erasing anisotropic deformations of the geometry. This allows for a prolonged inflationary anisotropic solution, providing the first example with standard gravity and scalar fields only which evades the conditions of the so called cosmic no-hair conjecture. We compute the curvature perturbations on the anisotropic solution, and the corresponding phenomenological bound on the anisotropy. Finally, we discuss the analogy between this model and the f (phi) F^2 model, which also allows for anisotropic inflation thanks to a suitable coupling between the inflaton phi and a vector field. We remark that the bispectrum of the curvature perturbations in solid inflation is enhanced in the squeezed limit and presents a nontrivial angular dependence, as had previously been found for the f (phi) F^2 model.

Figures (2)

• Figure 1: Evolution of the anisotropy $\sigma$ (defined in eq. (\ref{['bianchi']})) as a function of the number of e-folds (of the average scale factor $\alpha$) for the model (\ref{['example-F']}), and for an initial approximately equal admixture of the two modes in (\ref{['bck-sol']}). The analytical solution (\ref{['bck-sol']}) shows a perfect agreement with the exact one. The line $\propto {\rm e}^{-3 N}$ shows the decrease of the fast decreasing mode. We note that this is also the rate at which the anisotropy $\dot{\sigma}$ decreases in standard inflationary models.
• Figure 2: Leading diagrams for $\langle {\hat{\zeta}}^2 \rangle$ on an anisotropic background. The first diagram is the FRW result, while the second diagram is the linear correction in the anisotropy. Only these two diagrams are computed in the main text. We disregard quadratic (the last two diagrams shown) and higher order corrections in the anisotropy.