Cosmological perturbations on local systems

We study the effect of cosmological expansion on orbits—galactic, planetary, or atomic—subject to an inverse-square force law. We obtain the laws of motion for gravitational or electrical interactions from general relativity—in particular, we find the gravitational field of a mass distribution in an expanding universe by applying perturbation theory to the Robertson-Walker metric. Cosmological expansion induces an ($\ddot{a}/a)\overrightarrow{r}$ force where $a(t)$ is the cosmological scale factor. In a locally Newtonian framework, we show that the $(\ddot{a}/a)\overrightarrow{r}$ term represents the effect of a continuous distribution of cosmological material in Hubble flow, and that the total force on an object, due to the cosmological material plus the matter perturbation, can be represented as the negative gradient of a gravitational potential whose source is the material actually present. We also consider the effect on local dynamics of the cosmological constant. We calculate the perihelion precession of elliptical orbits due to the cosmological constant induced force, and work out generalizations to the rotation curve and virial relation applicable to clusters of galaxies.