Brans-Dicke Cosmologies in Arbitrary Units: Solutions in Flat Friedmann Universes

Assuming a Robertson-Walker metric and an equation of state $p=\epsilon \rho \left(0\mathrm{}\le \epsilon \le 1\right)$, the Brans-Dicke (BD) field equations are found to yield a first integral of the form $a^{3(1+\epsilon )}{\lambda }^{\frac{1}{2}(1-3\mathrm{}\epsilon )(1-\alpha )}=\mathrm{const}$, where $\alpha$ is the parameter denoting the particular units in which the field equations are expressed. This first integral is employed in the remaining field equations to yield polynomial solutions for $a$, $\rho$, and $\lambda$. For $\epsilon =0,\frac{1}{3}$, these solutions properly reduce to previously obtained pressure-free and radiation-filled universes, respectively.