 Turnaround bound with positive cosmological constant

A fixed-mass spherical shell has acceleration $\ddot{R}=-GM/R^2+\Lambda R/3$, in units $c=1$. At a finite maximum radius $R_m$, inward acceleration requires $\eta =\Lambda R_m^3/(3GM)<1$. Equality is the unstable static radius, approached only in the limiting separatrix case rather than a finite expansion-to-collapse turnaround. The effective potential reaches its maximum there, so an initially expanding bound inner-branch shell also needs energy below that maximum.