The f (R) theories have one extra scalar degree of freedom compared to the CDM model. This feature results in more freedom for the background. As we have seen previously, a viable cosmological sequence of radiation, matter, and accelerated epochs is possible provided some conditions hold for f (R). In principle, however, one can specify any given and solve Eqs. (2.15) and (2.16) for those compatible with the given .
Therefore the background cosmological evolution is not in general enough to distinguish f (R) theories from other theories. Even worse, for the same , there may be some different forms of f (R) which fulfill the Friedmann equations. Hence other observables are needed in order to distinguish between different theories. In order to achieve this goal, perturbation theory turns out to be of fundamental importance. More than this, perturbations theory in cosmology has become as important as in particle physics, since it gives deep insight into these theories by providing information regarding the number of independent degrees of freedom, their speed of propagation, their time-evolution: all observables to be confronted with different data sets.
The main result of the perturbation analysis in f (R) gravity can be understood in the following way. Since it is possible to express this theory into a form of scalar-tensor theory, this should correspond to having a scalar-field degree of freedom which propagates with the speed of light. Therefore no extra vector or tensor modes come from the f (R) gravitational sector. Introducing matter fields will in general increase the number of degrees of freedom, e.g., a perfect fluid will only add another propagating scalar mode and a vector mode as well. In this section we shall provide perturbation equations for the general Lagrangian density including metric f (R) gravity as a special case.
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