Dynamics of cosmological perturbations in the Horndeski theory

Despite the significant achievements of the inflationary model of the early Universe, alternative scenarios that can replace or complement the classical inflationary stage have been actively developed in the last decades. The aim of these studies is to obtain solutions that exclude singularities, which is not possible within the framework of general relativity theory. In such cases, the alternative is scalar-tensor theories of gravitation, among which the most general theory without senior derivatives in the equations of motion is the Horndeski theory.
However, the construction of fully stable solutions in the framework of Horndeski theory is restricted by the No-Go theorem. This paper presents a new approach to construct stable solutions in the general context of Horndeski theory. This approach is based on considering a subclass of the theory where the previously studied unitary gauge has singularity. As an example, a spatially flat stable solution described by the general theory of relativity in combination with non-canonical scalar fields is constructed. Then the stability of this solution to a possible anisotropy of the background is analyzed.
The second part of the talk is devoted to the problem of applicability of Horndeski theory in the context of late cosmology. An essential restriction is the strict coincidence of the propagation velocities of gravitational and electromagnetic waves. This requirement implies that a photon with minimal coupling is not subject to modification even at scales where the general relativity (GR) may require modification. It is shown that the four-dimensional Galileon, arising from the Kaluza-Klein compactification of its five-dimensional version, can be regarded as a natural modification of GR and electromagnetism. The class of theories in which gravitational waves propagate at the speed of light is considerably extended, which eliminates the necessity of a fine tuning of scalar potentials and extends the spectrum of admissible theories in comparison with the previous assumptions.