Revisiting Cosmic No-Hair Theorem for Inflationary Settings

TL;DR

This work extends Wald's cosmic no-hair theorem to inflationary settings by examining Bianchi cosmologies in Einstein gravity with a time-dependent cosmological term $\Lambda(t)$ and an energy-momentum decomposition $T_{\mu\nu}=-\Lambda(t) g_{\mu\nu}+\mathcal{T}_{\mu\nu}$, where $\mathcal{T}_{\mu\nu}$ satisfies SEC and WEC. The authors derive a general bound on the Hubble-normalized shear $|\sigma^i_{\; j}|/H$ during inflation and show that, while anisotropy can grow in slow-roll scenarios, it is limited by slow-roll parameters. In particular, a slow-roll refinement yields the end-of-inflation bound $|\sigma^i_{\; j}|/H|_{t_{sl}} \le \frac{8}{3}(\epsilon_0-\eta_0)$, with curvature damping and a quasi-de Sitter evolution shaping the dynamics. The paper applies these results to three classes: (i) scalar-driven models where anisotropy damps and no-hair holds, (ii) vector gauge-field models (Kanno–Soda) where anisotropic stress can drive brief anisotropy growth and saturation near the bound, and (iii) gauge-flation with non-Abelian fields where the isotropic FLRW attractor persists despite transient anisotropies. Altogether, the findings reveal that inflation does not erase anisotropy completely but enforces a small, controlled level of anisotropy tied to slow-roll parameters, with observable implications for CMB statistical anisotropy.

Abstract

In this work we revisit Wald's cosmic no-hair theorem in the context of accelerating Bianchi cosmologies for a generic cosmic fluid with non-vanishing anisotropic stress tensor and when the fluid energy momentum tensor is of the form of a cosmological constant term plus a piece which does not respect strong or dominant energy conditions. Such a fluid is the one appearing in inflationary models. We show that for such a system anisotropy may grow, in contrast to the cosmic no-hair conjecture. In particular, for a generic inflationary model we show that there is an upper bound on the growth of anisotropy. For slow-roll inflationary models our analysis can be refined further and the upper bound is found to be of the order of slow-roll parameters. We examine our general discussions and our extension of Wald's theorem for three classes of slow-roll inflationary models, generic multi-scalar field driven models, anisotropic models involving U(1) gauge fields and the gauge-flation scenario.

Figures (2)

• Figure 1: In the left panel we have the evolutions of Hubble-normalized shear $\frac{\Sigma}{H}:=\frac{\dot\sigma}{H}$ for various $c$ values when $\sqrt{c}\varphi_i=17$. As we see, growing during inflation and becoming of the order $\epsilon$, $\frac{\Sigma}{H}$ saturates the upper-bound value. This figure is taken from Kanno. Then, in the right panel and for a different initial values and parameters, we have the evolution of Hubble-normalized shear $\frac{h}{H}:=\frac{\dot\sigma}{H}$ during inflation when $c=2$ and $=\varphi_i=11M_{Pl}$. Again, anisotropies grow during inflation and eventually saturate their upper bound value. This figure is taken from Himmetoglu.
• Figure 2: Hubble-normalized anisotropic stress tensor, $\frac{\pi(t)}{H^2}$, and Hubble-normalized shear, $\frac{\Sigma(t)}{H^2}:=\frac{\dot\sigma(t)}{H^2}$ for a system with $\kappa=3.77\times10^{15}$, $g=10^{-1}$, $\psi_0=0.6\times10^{-3}$, $\dot\psi_0=10^{-10}$, $\lambda_0=10$, $\dot\lambda_0=-3.6$. As we see $\frac{\pi(t)}{H^2}$ increases (exponentially) for a very short time in the early stage of inflation (in small $H_0t$), saturating our upper bound \ref{['upperSigma']}. Then, it is damped exponentially fast to its isotropic fixed point.