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

Abstract

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 $\eta \cdot (\varphi /2)\cdot G_{\mu \nu }{\nabla }^{\mu }{\nabla }^{\nu }\varphi$ in the Lagrangian. We assume that the space-times are filled a global unidirectional magnetic field that minimally interacts with the scalar field. We limit ourselves to the period before and during primary inflation. 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 $l=1+\epsilon \eta \Lambda /\mu$ determines the properties of cosmological models, where $\Lambda$ is the cosmological constant, $\mu =M_{PL}^2$ is the Planck mass squared. The sign $\epsilon =\pm 1$ defines the canonical scalar field and the phantom field, respectively. An inequality $\epsilon /\eta >0$ is a necessary condition for isotropization of models, but not sufficient. The model with $l>0$ has the necessary properties: isotropization during expansion, rapid transition to inflationary expansion ($a(t)\propto e^{\sqrt{\frac{\epsilon }{3\eta }}\cdot t}$), absence of ghost and Laplace instabilities. In other cases $l\ngtr 0$, the model has various disadvantages. Constraints on the tensor-to-scalar ratio, the conditions for avoidance of ghost and Laplacian instabilities lead to the inequalities: $\Lambda >0$, $\eta >0$, $\epsilon =1$, $1<\Lambda \eta /\mu <1.049$.

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