# Farthest distance two objects are "gravitationally bounded", considering expansion of the Universe

The narrative is: "on greater scales the expansion of the Universe dominates, but on smaller scales gravitationally bounded objects still stay bounded". But how small is meant by "smaller scales"?

Galaxy superclusters are the biggest gravitationally connected objects in the recent (low z) Universe. How big (for given mass $$M$$) can they be before the expansion tears them apart?

For spherical mass $$M$$ and current epoch (and current value of expansion parameter $$\Lambda$$) - how far the closest unbounded object be?

Can a graph be built for "unbounding" distance as a function of $$M$$?

• You mean the Hubbles law
– user47732
Dec 10, 2022 at 16:56
• @ScienceAJ - Hublle's law describes expansion of the Universe (at least for galaxies no so close and not so far from us). But the law says nothing about gravity. My question is - at what distance the expansion overcomes gravity of galaxy superclusters? Or, by another words - how big (for given mass M) galaxy superclusters can be before the expansion tear them apart? Dec 12, 2022 at 9:45
• I'm not sure about that, Hubbles law states that the farther you are the faster they move, i think that's because of gravity. Classical Hubble expansion is characterized by a proportional increase in the rate of expansion groups based on the distance from the main center of gravity (I read this from a ScienceDirect article which could not be found now). Note: I'm not sure so kindly take this comment with a pinch of salt
– user47732
Dec 12, 2022 at 11:18
• @SienceAJ - Hubble's "constant" value depends on total mass and total energy of Universe, but not on local "gravity centers". Here are some basics explained by astrophysicist for public, without math: forbes.com/sites/startswithabang/2019/08/02/… Dec 12, 2022 at 12:19

If there were no dark energy, then a spherical region that is denser than the critical density $$3H^2/(8\pi G)$$, where $$H$$ is the Hubble rate (equal to about $$68~\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$$ today), can be said to be bound, since it will eventually collapse. From this you can estimate the radius $$r$$ of the "bound sphere" that surrounds a mass $$M$$ under the approximation that this mass dominates over its surroundings: it satisfies $$M/(4\pi r^3/3)=3H^2/(8\pi G)$$.
However, there is dark energy in our universe, which makes the calculation trickier. In general, to answer the question of whether an expanding region is bound in the presence of dark energy, we need to do numerical integrations. However, we can straightforwardly estimate the maximum radius of a bound static (not expanding) region by noting that its matter density must be at least twice the dark energy density. The second Friedmann equation tells us that "matter density is twice the dark energy density" is precisely the criterion under which matter's attraction counteracts dark energy's repulsion (and this was the tuning required for Einstein's static universe model). This means the largest possible bound system of mass $$M$$ has radius $$r$$ satisfying $$M/(4\pi r^3/3)=2\rho_\Lambda$$, where $$\rho_\Lambda\simeq 88~\mathrm{M}_\odot\mathrm{kpc}^{-3}$$ is the (constant) dark energy density. That is, $$r=[3M/(8\pi\rho_\Lambda)]^{1/3}\simeq 0.11~\mathrm{kpc}\left(\frac{M}{1~\mathrm{M}_\odot}\right)^{1/3}$$ represents the maximum radius of a bound spherical region of mass $$M$$.
For example, if we plug in the mass of the Milky Way and its dark matter halo, $$10^{12}~\mathrm{M}_\odot$$, then $$r=1.1~\mathrm{Mpc}$$. Note that this neglects all of the other mass that lies within $$r$$ (including the larger Andromeda galaxy), so the actual "maximum bound radius" around the Milky Way is somewhat larger than this. Nevertheless, the interpretation here is that in the distant future (assuming dark energy sticks around), we will retain a bound system a little over $$1~\mathrm{Mpc}$$ in radius, while everything else in the universe will recede indefinitely.