Statistical Anisotropies in Gravitational Waves in Solid Inflation

TL;DR

This work shows that solid inflation can sustain a long-lived anisotropic background, leading to statistical anisotropies in the scalar and tensor power spectra and a nonzero scalar–tensor cross-correlation. Using in‑in formalism on a nearly Bianchi I background, the authors compute leading anisotropies in $P_\zeta$, $P_h$, and $P_{\zeta h}$ in two limits, revealing quadrupolar scalar patterns and more intricate tensor patterns including a $\sin^4\theta$ component, with the cross-correlation often dominating observable TT anisotropies. They translate these primordial signals into CMB signatures via transfer functions, predicting off-diagonal TB/EB correlations and scale-dependent TT/TE/EE/BB features that differ from gauge-field–driven anisotropic inflation. The results highlight the scalar–tensor cross-correlation as a key observable and provide a framework to test anisotropic solid inflation with current and future CMB data, including considerations of IR anisotropies and the FRW attractor behavior of the background.

Abstract

Solid inflation can support a long period of anisotropic inflation. We calculate the statistical anisotropies in the scalar and tensor power spectra and their cross-correlation in anisotropic solid inflation. The tensor-scalar cross-correlation can either be positive or negative, which impacts the statistical anisotropies of the TT and TB spectra in CMB map more significantly compared with the tensor self-correlation. The tensor power spectrum contains potentially comparable contributions from quadrupole and octopole angular patterns, which is different from the power spectra of scalar, the cross-correlation or the scalar bispectrum, where the quadrupole type statistical anisotropy dominates over octopole.

Figures (11)

• Figure 1: Here we present the Fyenman diagrams. The top diagram corresponds to $\delta {\cal L}_{\zeta \zeta}$ which can be interpreted as the change in $\zeta$ effective mass. The second diagram corresponds to $\delta {\cal L}_{h h}$ which also is in the form of a mass insertion. The third diagram represents the corrections in $h_+$ propagator from the insertion of two exchange vertices. The last diagram represents the scalar-tensor cross-correlation. Note that the three couplings are indicated by $*, \bullet$ and $\times$.
• Figure 2: The TT correlation at $\ell_2=\ell_1$. The left panel is for $m=0$ and the right panel is for $m=\ell_1$. Here and hence after, the black curve represents the reference model with $g_*^\zeta = 0$. The blue line denotes the original solid inflation model with $F_Y, F_Z \sim F$, and the green line denotes the $F(X)$ model of solid inflation.
• Figure 3: The $m=0$ (left) and $m=\ell_1$ (right) plots for BB correlation with $\ell_2=\ell_1$.
• Figure 4: The $m=0$ (left) and $m=\ell_1$ (right) plots for TT correlation with $\ell_2=\ell_1+2$. Here and hence after, the dashed lines denote the plotted quantity (here $C_{l, l+2}^{TT}$) is negative along this line segment, and thus we plot $-C_{l, l+2}^{TT}$ on the logarithm scales.
• Figure 5: The $m=0$ (left) and $m=\ell_1$ (right) plots for BB correlation with $\ell_2=\ell_1+2$.