Chiral Cosmological Model of f(R) Gravity with a Kinetic Curvature Scalar

We consider modified f(R) gravity with a kinetic curvature scalar, which can be reduced to a chiral cosmological model of special kind. A detailed derivation is presented for the action of a chiral cosmological model as an equivalent to a gravitational model with higher derivatives with respect to the Ricci scalar using Lagrange multipliers and a transition from the Jordan frame to the Einstein one. The equations of the model are written in the spatially flat Friedmann-Robertson-Walker metric on the basis of the constructed chiral cosmological model. Examples of solutions are found, corresponding to a special choice of the field χ = χ* = const, and its fixed value χ0=−3/2ln2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi_0=-\sqrt{3/2}\;\text{ln}2$$\end{document}. For this value of χ0, in the case of the canonical inclusion of the kinetic component of the Ricci scalar, we have obtained a nonlinear second-order differential equation with respect to H, which is not amenable to analytic solution. Therefore we implement a transition to a noncanonical form of the kinetic term. Using a fixed value of χ0, an exact solution is obtained for power-law inflation. We have considered a transition from the de Sitter and power-law solutions specified in the Jordan frame to the Einstein frame for comparison with the results obtained in f(R) gravity with higher derivatives. It is proved that there is a Weyl conformal transformation which transforms the de Sitter and power-law solutions in one frame to similar solutions in the other.