= Uniform-sphere collapse with a cosmological constant
For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3GM^2/(5R)$ and $W_\Lambda=-\Lambda MR^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The <virial theorem> reads $2T+W_G=2W_\Lambda$. Defining $x=R_f/R_{\rm ta}$ and $\eta=\Lambda/(4\pi G\rho_{\rm ta})$, <conservation of energy> gives $2\eta x^3-(2+\eta)x+1=0$. The collapsing branch has $x=1/2-\eta/8+O(\eta^2)$. For small positive <cosmological constant>, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.
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