New family of inhomogeneous $\gamma$-law cosmologies: example of gravitational waves in a homogeneous $p=\varrho \mathrm{/}3$ background

We present an explicit three-parameter class of $p=\gamma \varrho$, ($-1\mathrm{/}3\le \gamma <1$), cosmological models admitting a two-dimensional group $G_2$ of isometries acting on spacelike surfaces. The family is self-similar in the sense that it has a further homothetic vector field and it contains subfamilies of both (previously unknown) tilted and non-tilted Bianchi models with that equation of state. This is the first algebraically general class of solutions of this kind including dust inhomogeneous solutions. The whole class presents a universal spacelike big-bang singularity in the finite past. More interestingly, the case $p=\varrho \mathrm{/}3$ constitutes a new two-parameter inhomogeneous subfamily which can be viewed as a Bianchi V background with a gravitational wave travelling orthogonally to the surfaces of transitivity of the $G_2$ group. This wave generates the {\it inhomogeneity} of the spacetime and is related to the sound waves {\it tilting} the perfect fluid. It seems to be the first explicit exact example of a gravitational wave travelling along a homogeneous background that has a realistic equation of state $p=\varrho \mathrm{/}3$.