Gravity Waves Signatures from Anisotropic pre-Inflation

TL;DR

The paper shows that a pre-inflationary anisotropic Kasner stage can amplify long-wavelength gravitational waves in a polarization-dependent way, with the H_× mode growing during Kasner expansion while the coupled tensor-scalar sector remains largely non-amplified. By analyzing linear fluctuations on an axisymmetric Kasner background and quantifying the growth through the Weyl invariant C^2, the authors connect this instability to the BKL contraction picture and quantify observable consequences in the CMB. They derive how a residual classical GW can imprint a scale- and angle-dependent signature in the temperature anisotropy spectrum, constraining the extra inflation duration ΔN and the initial GW amplitude via the quadrupole C_2; longer inflation suppresses the signal. The work highlights a potential link between pre-inflationary anisotropy, GW growth, and late-time cosmological observables, and suggests avenues for studying backreaction and generic Kasner exponents in future research.

Abstract

We show that expanding or contracting Kasner universes are unstable due to the amplification of gravitational waves (GW). As an application of this general relativity effect, we consider a pre-inflationary anisotropic geometry characterized by a Kasner-like expansion, which is driven dynamically towards inflation by a scalar field. We investigate the evolution of linear metric fluctuations around this background, and calculate the amplification of the long-wavelength GW of a certain polarization during the anisotropic expansion (this effect is absent for another GW polarization, and for scalar fluctuations). These GW are superimposed to the usual tensor modes of quantum origin from inflation, and are potentially observable if the total number of inflationary e-folds exceeds the minimum required to homogenize the observable universe only by a small margin. Their contribution to the temperature anisotropy angular power spectrum decreases with the multipole l as l^(-p), where p depends on the slope of the initial GW power-spectrum. Constraints on the long-wavelength GW can be translated into limits on the total duration of inflation and the initial GW amplitude. The instability of classical GW (and zero-vacuum fluctuations of gravitons) during Kasner-like expansion (or contraction) may have other interesting applications. In particular, if GW become non-linear, they can significantly alter the geometry before the onset of inflation.

Figures (10)

• Figure 1: Power in different modes of the decoupled tensor polarization, normalized to its initial value ($k_L = k_T$ for all the cases shown). We show the time evolution starting from $h_0 = - 10^6\sqrt{V_0} / M$ (defined in eq. \ref{['hubbles']}). The universe isotropizes at $t_{\rm iso} = 1 / H_{\rm iso}$. The quantity $k_{\rm iso}$ is defined to be the momentum of the modes exiting the horizon at this time. Large scale modes do not exhibit oscillatory behavior, and grow during the anisotropic stage. Intermediate scale modes enter the horizons during the anisotropic era; this terminates their growth. Small scale modes are in the oscillatory regime all throughout the anisotropic phase, and do not experience any growth. All modes shown freeze during the inflationary stage. Notice the very different final values obtained in the three cases shown.
• Figure 2: Amplification of the power spectrum between the initial time (same as in the previous Figure) and some time during inflation for which the modes shown are outside the horizon, and the value of $P_{H_\times}$ is frozen. As explained in the main text, the growth is greater at small values of $k$ and $\theta$.
• Figure 3: Angular dependence of the power of different modes. We show the value reach by the mode once it is frozen in the inflationary stage, normalized to the initial value (the background evolution is identical to that of the previous two Figures). In the region $\theta \ll 1$ shown in the Figure ($k_T \ll k_L$) the final value exhibit a $1 / \sin \theta$ dependence.
• Figure 4: Contribution of different modes to the power spectrum of the tensor polarization $H_+$ (left panel) and of the comoving curvature perturbation $R$ (right panel), normalized to its initial value. The background evolution, and the momenta shown, are the same as in Figure \ref{['evoltcl']}. Contrary to the decoupled tensor mode, shown in Figure \ref{['evoltcl']}, the spectra do not grow while the modes are outside the horizons in the anisotropic regime.
• Figure 5: Decrease of the power spectra of the tensor polarization $H_+$ (left panel) and of the comoving curvature $R$ (right panel). The Figure shows the ratio between the power spectra at some time during inflation (when the modes are outside the horizon in the isotropic regime, and the spectra are frozen), and the initial time. The decrease should be compared with the growth experienced at large scales by $P_{H_\times} \,$, shown in Figure \ref{['powertcl']}.