# Calculating the "Turnaround Point," where expansion dominates over gravity

I'm trying to understand the point where the expansion of space dominates over gravity, but I'm coming up empty handed when searching the internet. Specifically, I'm trying to find the math for it. I'm wondering if there might be some sort of Hill sphere-like thing surrounding galaxies or clusters, inside which gravity keeps things together and there is no effect, or very little effect, from the expansion of space. Does anyone know where I can find the math for this or whether there is even such a point in space?

• This post from our sister site doesn't exactly answer your question, but it's got lots of great info (& references) about expansion: physics.stackexchange.com/a/63780/123208 Jul 26, 2022 at 16:55
• Are you asking for details of the turnaround radius? This is the surface around a mass where there is neither expansion away or infall towards the gravitating object. Jul 26, 2022 at 18:11
• Thanks @eshaya. This should be the answer. It's is exactly what I meant. I'm glad it has a name! Here's a reference paper arxiv.org/pdf/1912.08216.pdf. Jul 27, 2022 at 3:26
• Feel free to write your own answer, based on that paper. We encourage self learners! Jul 27, 2022 at 12:23
• This pair of papers on Perturbations of the Hubble flow provide a lot of math on growth of overdensities for professionals. In particular, the appendix of the second paper has turnaround equations. ui.adsabs.harvard.edu/abs/1980A&A....86..212O/abstract ui.adsabs.harvard.edu/abs/1982A&A...107..172O/abstract Jul 27, 2022 at 15:26