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Different parametrizations of field equations in the $f(R)$ theories of gravity

In this paper we describe two different parametrizations of field equations in the $f(R)$ theories of gravity and two resulting parametrizations of the Friedmann Equations. In particular we show how these two parametrizations lead to two different results for the curvature pressure and the curvature density obtained respectively as the $i-i$ component and the $0-0$ component of the curvature stress-energy tensor. We introduce formulas in order to pass from one parametrization to another one for the curvature pressure and the curvature density. Furthermore, we analyse under which conditions the two parametrizations of the curvature stress-energy tensor lead to significantly different results.