Nucleosynthesis constraints on scalar-tensor theories of gravity

We study the cosmological evolution of massless single-field scalar-tensor theories of gravitation from the time before the onset of $e^{+}{e}^{-}$ annihilation and nucleosynthesis up to the present. The cosmological evolution together with the observational bounds on the abundances of the lightest elements (those mostly produced in the early universe) place constraints on the coefficients of the Taylor series expansion of $a\left(\phi \right)$, which specifies the coupling of the scalar field to matter and is the only free function in the theory. In the case when $a\left(\phi \right)$ has a minimum (i.e., when the theory evolves towards general relativity) these constraints translate into a stronger limit on the post-Newtonian parameters $\gamma$ and $\beta$ than any other observational test. Moreover, our bounds imply that, even at the epoch of annihilation and nucleosynthesis, the evolution of the universe must be very close to that predicted by general relativity if we do not want to over- or underproduce ${}^4$He. Thus the amount of scalar field contribution to gravity is very small even at such an early epoch.