Predictions from an anisotropic inflationary era

Abstract

This article investigates the predictions of an inflationary phase starting from a homogeneous and anisotropic universe of the Bianchi I type. After discussing the evolution of the background spacetime, focusing on the number of e-folds and the isotropization, we solve the perturbation equations and predict the power spectra of the curvature perturbations and gravity waves at the end of inflation. The main features of the early anisotropic phase is (1) a dependence of the spectra on the direction of the modes, (2) a coupling between curvature perturbations and gravity waves, and (3) the fact that the two gravity waves polarisations do not share the same spectrum on large scales. All these effects are significant only on large scales and die out on small scales where isotropy is recovered. They depend on a characteristic scale that can, but a priori must not, be tuned to some observable scale. To fix the initial conditions, we propose a procedure that generalises the one standardly used in inflation but that takes into account the fact that the WKB regime is violated at early times when the shear dominates. We stress that there exist modes that do not satisfy the WKB condition during the shear-dominated regime and for which the amplitude at the end of inflation depends on unknown initial conditions. On such scales, inflation loses its predictability. This study paves the way to the determination of the cosmological signature of a primordial shear, whatever the Bianchi I spacetime. It thus stresses the importance of the WKB regime to draw inflationary predictions and demonstrates that when the number of e-folds is large enough, the predictions converge toward those of inflation in a Friedmann-Lemaitre spacetime but that they are less robust in the case of an inflationary era with a small number of e-folds.

Figures (24)

• Figure 1: Evolution of the scale factors according to the value of the parameter $\alpha$. We depict the three directional scale factors and the average scale factor $S$ (dashed line), all in units of $S_*$. The three directional scale factors are permuted when $\alpha$ is changed by $2\pi/3$. Note that there exist two particular cases in which the spacetime has an extra rotational symmetry when $\alpha=\pi/6$ or $\alpha=\pi/2$. The latter case is even more peculiar since this is the only Bianchi $I$ universe for which none of the direction is bouncing.
• Figure 2: (left) Time at which the direction $i$ bounces (blue: $i=3$), (red: $i=2$). At $\pi/2$, none of the direction is contracting and a change of the contracting direction occurs. (right) Value of the Kasner exponents as a function of the parameter $\alpha$ characterising the Bianchi $I$ model close to the singularity.
• Figure 3: Comparison of the phase portraits of a Friedmann-Lemaı̂tre (left) and a Bianchi $I$ inflationary phases. The curves with same colour corresponds to same initial conditions for the scalar field but starting with different initial shear (see Fig. \ref{['fig:phase3D']} for a 3-dimensional representation).
• Figure 4: Phase portrait of a Bianchi $I$ inflationary phases in the space $\{\varphi,\dot\varphi,\sigma\}$. The plane $\sigma=0$ corresponds to the FL-limit (see Fig. \ref{['fig:dynaFLvB']}). This illustrates the double attraction mechanism of the spacetime toward FL and of the solution toward the slow-roll attractor.
• Figure 5: Evolution of the average scale factor (left) and scalar field (right) as a function of time for both the Bianchi (red, solid line) and FL cases (blue, dashed line). The Bianchi solution is characterised by $\varphi_0=16 M_p$. We have normalized the solutions such that the scale factors have the same value at the end of the inflation when the shear is negligible. The inner left figure shows the logarithmic evolution of the velocity of the expansion in both cases, the minimum of which indicates the time at which $\ddot S=0$ for the Bianchi case.