Testing scalar-tensor theories and parametrized post-Newtonian parameters in Earth orbit

Abstract

We compute the parametrized post-Newtonian (PPN) parameters $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ for general scalar-tensor theories in the Einstein frame, which we compare to the existing PPN formulation in the Jordan frame for alternative theories of gravity. This computation is important for scalar-tensor theories that are expressed in the Einstein frame, such as chameleon and symmetron theories, which can incorporate hiding mechanisms that predict environment-dependent PPN parameters. We introduce a general formalism for scalar-tensor theories and constrain it using the limit on $\ensuremath{\gamma}$ given by the Cassini experiment. In particular, we discuss massive Brans-Dicke scalar fields for extended sources. Next, using a recently proposed Earth satellite experiment, in which atomic clocks are used for spacecraft tracking, we compute the observable perturbations in the redshift induced by PPN parameters deviating from their general relativistic values. Our estimates suggest that $|\ensuremath{\gamma}\ensuremath{-}1|\ensuremath{\sim}|\ensuremath{\beta}\ensuremath{-}1|\ensuremath{\sim}{10}^{\ensuremath{-}6}$ may be detectable by a satellite that carries a clock with fractional frequency uncertainty $\mathrm{\ensuremath{\Delta}}f/f\ensuremath{\sim}{10}^{\ensuremath{-}16}$ in an eccentric orbit around the Earth. Such space experiments are within reach of existing atomic clock technology. We discuss further the requirements necessary for such a mission to detect deviations from Einstein relativity.