Subluminal cosmological bounce beyond Horndeski

We address the issue of potential superluminal propagation of gravitational waves in backgrounds neighboring the previously suggested bounce [arXiv:1807.08361] in beyond Horndeski theory. We find that the bouncing solution lies right at the boundary of the region where the gravitational waves propagate at speed exceeding that of light, i.e. that solution suffers superluminality problem. We suggest a novel version of a completely stable bouncing model where both scalar and tensor perturbations remain safely subluminal not only on the solution itself but also in its neighbourhood. The model remains free of superluminality when extra matter in the form of radiation or, more generally, ideal fluid with equation of state parameter $w\leq 1/3$ (and also somewhat higher) is added. Superluminality reappears when extra matter is added whose sound velocity is equal or close to 1 in flat space; an example is scalar field minimally coupled to metric. The latter property is characteristic of all beyond Horndeski cosmologies; we briefly discuss its significance.