Local cosmological tidal acceleration ($\ddot{\mathbf{r}}=(\ddot{a}/a)\mathbf{r}$)

Neighboring freely falling comoving particles in a homogeneous Friedmann-Lemaître-Robertson-Walker metric have separation $\mathbf{r}=a(t)\mathbf{\chi }$ and relative acceleration $\ddot{\mathbf{r}}=(\ddot{a}/a)\mathbf{r}$, where $t$ is cosmic time and $\mathbf{\chi }$ is constant. This is a local tidal force effect, of order $H^{2}r$ when the dimensionless background parameters are of order one. In a weak-field central-mass model retaining the homogeneous background tide, one writes $\ddot{r}=-GM/r^2+{\ell }^2/r^3+(\ddot{a}/a)r$. The detailed residual tide depends on the cosmological embedding; a bound orbit is not required to satisfy $\dot{r}=Hr$. Comparing $H^2$ with $GM/r^3$ estimates whether that background tide can affect its dynamics. A locally vacuum region with a cosmological constant has the corresponding outward term $\Lambda c^{2}r/3$.