Complete density perturbations in the Jordan-Fierz-Brans-Dicke theory

In the context of scalar-tensor theories we study the evolution of the density contrast for Jordan-Fierz-Brans-Dicke theories in a Friedmann-Lemaitre-Robertson-Walker universe. Calculations are performed in the Einstein frame with the cosmological background described as Lambda-cold dark matter (Lambda CDM) and supplemented by a Jordan-Fierz-Brans-Dicke field. By using a completely general procedure valid for all scalar-tensor theories, we obtain the exact fourth-order differential equation for the density contrast evolution in modes of arbitrary size. In the case of sub-Hubble modes, the expression reduces to a simpler but still fourth-order equation that is then compared with the standard (quasistatic) approximation. Differences with respect to the evolution as predicted by the standard concordance Lambda CDM model are observed depending on the value of the coupling.