Exact cosmological solutions with magnetic field in the theory of gravity with non-minimal kinetic coupling

We investigate anisotropic and homogeneous cosmological models in the scalar-tensor theory of gravity with non-minimal kinetic coupling of a scalar field to the curvature given by the function η·(φ/2)· Gμν\,∇^μ ∇^ν φ in the Lagrangian. We assume that the space-times are filled a global unidirectional electromagnetic field that minimally interacts with the scalar field. The Horndeski theory allows anisotropy to grow over time. The question arises about isotropization. In the theory under consideration, a zero scalar charge imposes a condition on the anisotropy level, namely its dynamics develops in a limited region. This condition uniquely determines a viable branch of solutions of the field equations. The magnetic energy density that corresponds to this branch is a bounded function of time. The sign of parameter l=1+8π εη Λ determines the properties of cosmological models, where Λ is the cosmological constant. The sign ε=± 1 defines the canonical scalar field and the phantom field, respectively. An inequality ε/η>0 is a necessary condition for isotropization of models, but not sufficient. Three cases were considered: $l=0$, the isotropization does not occur; $l>0$, there is isotropization; $l<0$, there are two branches, one of which has the property of isotropization, and the other describes the Universe with collapse. The model with $l<0$ has an anisotropic bounce and the Universe evolution begins with a non-zero volume value. Models of an infinitely expanding Universe eventually go into inflation mode with the scale factor a(t)∝ eh_η· t, h_η=√ε/3η.