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"Gravity holds galaxies and cluster of galaxies together, and they get farther away from each other--without themselves changing in size--as the universe expands." Quoted from the OpenStax astronomy textbook. This assertion is commonly made, but without a convincing explanation. After all, doesn't a large cluster occupy a significant chunk of (expanding) space? Would it be because gravity is continually shrinking the size of both galaxies and clusters, and the expansion of space just slows this shrinking a bit? (This question was asked 7 1/2 years ago, but did not get much expert response).

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  • $\begingroup$ Also see this answer on our sister site: physics.stackexchange.com/a/601323/123208 $\endgroup$
    – PM 2Ring
    Commented Oct 8, 2022 at 16:17
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    $\begingroup$ Most forces, including gravity, are significantly large enough to overcome the expansion, but only to a certain distance. It is also the same reason that e.g. you don't expand even though the space you are in expands; the expansion of space is not significant enough to, say, overcome the strong force and drive quarks apart, even though it is perfectly happy to increase the distance between extremely distant groups of quarks. Sort of like having a bunch of sand in a well mixed bucket of water; then you add water, and the sand particles just move farther apart; but stay the same individually. $\endgroup$
    – Jason C
    Commented Oct 9, 2022 at 3:17

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The idea of expanding space comes about as a solution to Einstein's field equations in General Relativity (the Friedmann-Lemaitre-Robertson-Walker metric) under the assumptions that all the matter and energy in the universe is uniformly and isotropically distributed throughout the volume of the universe.

Clearly if you have things like galaxies then this assumption cannot be strictly true. Whether it is approximately true depends on the scale at which you want to study the dynamics of the universe. For example, if you go to scales of 100 Mpc in the present-day universe, that is about 1000 times bigger than galaxies and about 100 times bigger than galaxy groups and clusters, and they might approximate to a uniformly distributed set of "sand grains".

On the other hand if you zoom in much closer then the non-uniform distribution of matter cannot be ignored, and on small scales you cannot just assume that the general expansion metric of the universe applies. Another way of looking at this is that if you have a structure that is in fact firmly gravitationally bound, then clearly it isn't participating in the universal expansion of space! In particular it is wrong to think of a galaxy as somehow sitting in a space that is expanding in the background. The space will not be expanding (much) in a structure that is strongly gravitationally bound.

On the other hand, in detail, it is also incorrect to think of a galaxy in total isolation and being somehow unconnected with the universal space expansion. The spacetime metric within a galaxy or galaxy cluster must ultimately "join onto" the FLRW metric at large scales.

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  • $\begingroup$ How does this answer square with the usual explanation of redshifts of galaxies; namely, that space itself is stretching, so that the greater the distance, the greater the recession speed? The "expanding breadloaf" analogy doesn't suggest that there are pockets within the loaf that don't expand. $\endgroup$ Commented Oct 8, 2022 at 16:40
  • $\begingroup$ @GulbenkianD The expanding loaf analogy works on large scales and "the greater the distance the greater the recession speed" is only true at large scales (larger than the local group). The galaxies would be currants within the loaf. $\endgroup$
    – ProfRob
    Commented Oct 8, 2022 at 16:50
  • $\begingroup$ So, relativity predicts that gravity not only 1) curves space, but also 2) counteracts or prevents the kind of expansion that is occurring in empty regions? #1 is repeated all the time. #2 I've not heard till now. $\endgroup$ Commented Oct 8, 2022 at 17:26
  • $\begingroup$ It is not so much that relativity explicitly predicts #2 as much as it is that #2 is just a consequence of mechanical behaviors: Any force (of which gravity is one) that attracts things locally at a rate that is greater than the expansion between those things, will lead to those things continuing to be attracted to eachother locally. If you jump out of an airplane, the expansion of the space between you and the ground isn't fast enough to stop you from hitting the ground, but we do not need relativity to predict that. If anything, it's almost just a consequence of Newtonian mechanics. $\endgroup$
    – Jason C
    Commented Oct 9, 2022 at 3:25
  • $\begingroup$ Btw, the "expanding breadloaf" analogy does indeed suggest that there are pockets within the loaf that don't expand. Bread expands because air volume inside it increases in isolated pockets of air, spreading out still-solid bread. The bread doesn't become uniformly less dense. When you look at a slice of bread you will see a porous material, not a completely uniform mass. The fact that the loaf stays cohesive at all is also evidence of this; the individual molecules don't become an even dispersion in air. As is the fact that e.g. expanding a loaf of bread certainly isn't splitting any atoms. $\endgroup$
    – Jason C
    Commented Oct 9, 2022 at 3:47
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The expansion of the universe is just the motion of the matter and radiation in it. See the first part of this answer.

As ProfRob said, the FLRW geometry is just the general-relativistic description of a universe that is uniformly filled with matter and radiation. It is the matter and radiation that expands, not space. There are vacuum FLRW solutions, but the scale factor in those is completely arbitrary: you can always cover any part of the spacetime with FLRW coordinates that are expanding or contracting at any rate you please. Only in the nonempty solutions is the rate of expansion well defined. Expansion of empty space is just not a thing in general relativity.

The FLRW geometry is widely misconstrued as a cosmological version of Minkowski spacetime, which is why you so often hear about "expanding space". People sometimes try to replace the Minkowski background with FLRW at small scales, not understanding the difference, and derive a space-expansion force that doesn't actually exist. See this answer.

That said, there is an outward force due to the cosmological constant. It is not the same as the space-expansion force you normally hear about, because it doesn't depend on the scale factor or Hubble parameter (which vary over time) but only on $Λ$ (which doesn't).

The force from the cosmological constant doesn't cause orbits to expand over time for the same reason the regular attractive force of gravity doesn't make orbits shrink over time. An inward/outward acceleration (Newtonian force) is quite different from an inward/outward velocity (Aristotelian force).

The force from the cosmological constant does slightly alter the orbital parameters. For a circular orbit at a radius $r$, the ratio of the orbital frequency with and without $Λ$ is $ω/ω_0 \approx \sqrt{1-Λr^3/3GM}$. Taking $M=M_\odot$ and $r=1\text{ AU}$ (mass and radius of the earth-sun system), the difference is about 1 part in $10^{22}$. Taking $M=10^{12} M_\odot$ and $r=27\text{ kpc}$ (approximate mass and radius of the Milky Way), the difference is about 1 part in $10^5$.

Note, though, that if $Λr^3/3GM>1$ then the equation has no solution, meaning that there is no such orbit with the nonzero $Λ$. For the Virgo supercluster, $Λr^3/3GM \sim 1$, so you shouldn't expect to find bound structures much larger than that.

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    $\begingroup$ I think this answer is partly confused (as are the answers you link to), e.g. "It is the matter that expands, not space" is wrong. The FLRW metric describes the spacetime manifold; the time evolution of the metric (= expansion or contraction) is determined by the energy content (not just "matter"; in the early universe, there is a phase when photon energies are dominant). $\endgroup$ Commented Oct 9, 2022 at 15:19
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    $\begingroup$ Your discussion of dark energy also seems confused, particularly the strange "Newtonian force versus Aristotelian force" comment. The accelerating cosmic expansion is consistent with an unvarying cosmological constant, but it is also consistent with a time-varying "dark energy"; a fair amount of research in cosmology is devoted to trying to identify -- or constrain -- possible time evolution of the dark energy. $\endgroup$ Commented Oct 9, 2022 at 15:22
  • $\begingroup$ I still wonder, for a star in stable orbit around a galactic center, or for a galaxy in orbit around a cluster's center, why any increase in separation wouldn't cause the object to start spiraling farther out. How is the star or galaxy "bound", when the farther it spirals out, the weaker the gravitational attraction between it and the center of mass of the galaxy or cluster? $\endgroup$ Commented Oct 9, 2022 at 17:56
  • $\begingroup$ @GulbenkianD You can ask the same question when the cosmological constant is zero. The answer is that it costs a lot of energy to escape to infinity, as you're fighting gravity the whole way. A positive cosmological constant reduces the energy needed, but not by very much (except at huge scales). $\endgroup$
    – benrg
    Commented Oct 9, 2022 at 20:22
  • $\begingroup$ @PeterErwin My answer is about the standard cosmological model, which has a cosmological constant and no quintessence. Of course it may turn out to be wrong in various ways. You're right that my use of "matter" wasn't standard for cosmology. I replaced it with "matter and radiation", but what I really meant by it is Lorentz-breaking field configurations (which includes all particles). $\endgroup$
    – benrg
    Commented Oct 9, 2022 at 20:43

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