f(R) Cosmology in the First Order Formalism

In the present work we consider those theories that are obtained from a Lagrangian density ℒT(R) = f(R)√{-g} + ℒM, that depends on the curvature scalar and a matter Lagrangian that does not depend on the connection, and apply Palatini's method to obtain the field equations. We start with a brief discussion of the field equations of the theory and apply them to a cosmological model described by the FRW metric. Then, we introduce an auxiliary metric to put the resultant equations into the form of GR with cosmological constant and coupling constant that are curvature depending. We show that we reproduce known results for the quadratic case. We find relations among the present values of the cosmological parameters q0, H0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathop {(G/G)}\limits^ \circ _0$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathop {(G/G)}\limits^{ \circ \circ } _0 $$ \end{document}. Next we use a simple perturbation scheme to find the departure in angular diameter distance with respect to General Relativity. Finally, we use the observational data to estimate the order of magnitude of what is essentially the departure of f(R) from linearity. The bound that we find for f″ (0) is so huge that permit almost any f(R). This is in the nature of things: the effect of higher order terms in f(R) are strongly suppressed by power of Planck's time 8πG0. In order to improve these bounds more research on mathematical aspects of these theories and experimental consequences is necessary.