# A quantisation procedure in the presence of an initial Kasner singularity: primordial gravitational waves from triaxially anisotropic pre-inflation

## TL;DR

This paper proposes a new quantisation method for primordial gravitational waves in anisotropic Kasner-de Sitter space, overcoming initial singularity issues to predict a scale-invariant, isotropic spectrum similar to standard inflation.

## Contribution

It introduces a quantisation procedure that circumvents singularity-induced divergences, enabling meaningful predictions of gravitational wave spectra in anisotropic pre-inflationary models.

## Key findings

- The spectrum of primordial gravitational waves is scale-invariant on short wavelengths.
- The directional distribution of gravitational waves remains isotropic.
- The method aligns anisotropic models with standard inflation predictions.

## Abstract

In this paper, we discuss quantisation of cosmological tensor perturbations in the Kasner-de Sitter space-time as a model of (pre-)inflation. Quantisation in such an anisotropic background has been argued to be problematic based on the fact that the initial Kasner singularity, where the spatial anisotropy blows up, causes divergences to the effective frequencies squared for the perturbations, which render the standard quantisation procedure relying on the existence of an adiabatic vacuum state inexecutable. Here, an essential aspect of the problem is that the ability in determining the quantum spectra of the fields is restricted. Without its knowledge, one cannot even choose physically favourable states like the Bunch-Davies vacuum in de Sitter. We here argue that this difficulty may be circumvented if only there is a period, even if temporal, after the singularity where certain adiabatic conditions for the fields are met and the standard procedure of second quantisation can be carried out within the framework of the WKB approximation. We demonstrate that our prescription for determining the quantum energy spectrum is useful in making physically meaningful predictions for the primordial gravitational waves in triaxially anisotropic Kasner-de Sitter backgrounds. We confirm that, on short wave-length scales, the resulting spectrum and directional distribution of the primordial gravitational waves are the same as de Sitter inflation, namely, scale invariant and isotropic.