= Local cosmological tidal acceleration
{title2=$\ddot{\mathbf r}=(\ddot a/a)\mathbf r$}
Neighboring freely falling comoving particles in a homogeneous <Friedmann-Lemaître-Robertson-Walker metric> have separation $\mathbf r=a(t)\boldsymbol\chi$ and relative <acceleration> $\ddot{\mathbf r}=(\ddot a/a)\mathbf r$, where $t$ is <cosmic time> and $\boldsymbol\chi$ is constant. This is a local <tidal force> effect, of order $H^2r$ 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 $\Lambda c^2r/3$.
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