F(R,..) theories from the point of view of the Hamiltonian approach: non-vacuum Anisotropic Bianchi type I cosmological model

TL;DR

This work develops a Hamiltonian formulation for generalized $F(R)$ gravity in an anisotropic $Bianchi I$ universe with a barotropic fluid, deriving a Lagrangian and Hamiltonian density and obtaining exact classical solutions in two gauges. It analyzes a broader $F(R,T,L_m)$ framework and its Bergmann-Wagoner specialization, then specializes to $F(R)$ to obtain exact solutions and inflationary scenarios, identifying a γ-dependent constraint that links the usual ansätze for $D$ and $H$ to the matter content. A key result is the explicit construction of scale factors and auxiliary functions in both the vacuum and standard-matter cases, including an inflationary regime in which the dynamics can be tracked through the Hamiltonian equations. The findings clarify how anisotropic cosmologies behave under $F(R)$ gravity, provide a concrete classical baseline for future quantum cosmology (Wheeler–DeWitt) studies, and lay groundwork for further extensions of the functional gravity framework.

Abstract

In this work, we will explore the effects of F(R) theories in the classical scheme using the anisotropic Bianchi Type I cosmological model with standard matter employing a barotropic fluid with equation of state $P=γρ$. In this work we present the classical solutions in two gauge, N=1 and $N=6ABCD=6η^3D$ obtaining some results that are usually used as ansatz to solve the Einstein field equation. For completeness, we present the solutions in vacuum as well.