  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.
