For generality we consider the following actionf (R) gravity but also other modified gravity theories such as Brans–Dicke theory, scalar-tensor theories, and dilaton gravity. We define the quantity . Varying the action (6.2) with respect to and , we obtain the following field equations
We decompose and into homogeneous and perturbed parts, and , respectively. In the following we omit the bar for simplicity. The energy-momentum tensor of an ideal fluid with perturbations is.
For the action (6.2) the background equations (without metric perturbations) are given by
For later convenience, we define the following perturbed quantities[316, 317] (see also [436, 566, 355, 438, 312, 313, 314, 492, 138, 33, 441, 328])
We shall solve the above equations in two different contexts: (i) inflation (Section 7), and (ii) the matter dominated epoch followed by the late-time cosmic acceleration (Section 8).
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