Friedmann Equation for Brans Dicke Cosmology

In the context of Brans-Dicke scalar tensor theory of gravitation, the cosmological Friedmann equation which relates the expansion rate $H$ of the universe to the various fractions of energy density is analyzed rigorously. It is shown that Brans-Dicke scalar tensor theory of gravitation brings a negligible correction to the matter density component of Friedmann equation. Besides, in addition to ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ and ${\mathrm{\Omega }}_M$ in standard Einstein cosmology, another density parameter, ${\mathrm{\Omega }}_{{}_{\mathrm{\Delta }}}$, is expected by the theory. This implies that if ${\mathrm{\Omega }}_{{}_{\mathrm{\Delta }}}$ is found to be nonzero, data will favor this model instead of the standard Einstein cosmological model with cosmological constant and will enable more accurate predictions for the rate of change of Newtonian gravitational constant in the future.