<p>Gravitational boundness is a tricky question. It's essentially asking, &quot;will these two objects come together at any point in the future?&quot; However, we can give approximate answers.</p>
<p>If there were no dark energy, then a spherical region that is denser than the critical density <span class="math-container">$3H^2/(8\pi G)$</span>, where <span class="math-container">$H$</span> is the Hubble rate (equal to about <span class="math-container">$68~\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$</span> today), can be said to be bound, since it will eventually collapse. From this you can estimate the radius <span class="math-container">$r$</span> of the &quot;bound sphere&quot; that surrounds a mass <span class="math-container">$M$</span> under the approximation that this mass dominates over its surroundings: it satisfies <span class="math-container">$M/(4\pi r^3/3)=3H^2/(8\pi G)$</span>.</p>
<p>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 <em>static</em> (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 &quot;matter density is twice the dark energy density&quot; 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 <span class="math-container">$M$</span> has radius <span class="math-container">$r$</span> satisfying <span class="math-container">$M/(4\pi r^3/3)=2\rho_\Lambda$</span>, where <span class="math-container">$\rho_\Lambda\simeq 88~\mathrm{M}_\odot\mathrm{kpc}^{-3}$</span> is the (constant) dark energy density. That is,
<span class="math-container">$$r=[3M/(8\pi\rho_\Lambda)]^{1/3}\simeq 0.11~\mathrm{kpc}\left(\frac{M}{1~\mathrm{M}_\odot}\right)^{1/3}$$</span>
represents the maximum radius of a bound spherical region of mass <span class="math-container">$M$</span>.</p>
<p>For example, if we plug in the mass of the Milky Way and its dark matter halo, <span class="math-container">$10^{12}~\mathrm{M}_\odot$</span>, then <span class="math-container">$r=1.1~\mathrm{Mpc}$</span>. Note that this neglects all of the other mass that lies within <span class="math-container">$r$</span> (including the larger Andromeda galaxy), so the actual &quot;maximum bound radius&quot; 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 <span class="math-container">$1~\mathrm{Mpc}$</span> in radius, while everything else in the universe will recede indefinitely.</p>

CC BY-SA 4.0
2022-12-12T17:52:43.890Z
2022-12-12T17:58:04.843Z
2022-12-12T17:58:04.843Z
Sten
<p>The narrative is: &quot;on greater scales the expansion of the Universe dominates, but on smaller scales gravitationally bounded objects still stay bounded&quot;. But how small is meant by &quot;smaller scales&quot;?</p>
<p>Galaxy superclusters are the biggest gravitationally connected objects in the recent (low z) Universe. How big (for given mass <span class="math-container">$M$</span>) can they be before the expansion tears them apart?</p>
<p>For spherical mass <span class="math-container">$M$</span> and current epoch (and current value of expansion parameter <span class="math-container">$\Lambda$</span>) - how far the closest unbounded object be?</p>
<p><strong>Can a graph be built for &quot;unbounding&quot; distance as a function of <span class="math-container">$M$</span>?</strong></p>

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2022-12-12T09:45:17.210Z
@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?
Heopps
CC BY-SA 4.0
2022-12-12T12:19:39.710Z
@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: https://www.forbes.com/sites/startswithabang/2019/08/02/sorry-astronomy-fans-the-hubble-constant-isnt-a-constant-at-all/
Heopps
<gravity><distances><expansion>
Farthest distance two objects are "gravitationally bounded", considering expansion of the Universe
Heopps
# Farthest distance two objects are "gravitationally bounded", considering expansion of the Universe

(Asked by: username_0 on 2022-11-25)

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 $M$?**

**Tags:** gravity, distances, expansion

> **Comments:**
> **username_0**: @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?  
> **username_0**: @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: https://www.forbes.com/sites/startswithabang/2019/08/02/sorry-astronomy-fans-the-hubble-constant-isnt-a-constant-at-all/  

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## 1 answers

### Answer by username_1 on 2022-12-12. Score: 5 | Accepted answer

Gravitational boundness is a tricky question. It's essentially asking, "will these two objects come together at any point in the future?" However, we can give approximate answers.

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.


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