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I understand that on galactic scales, the expansion of space time has no appreciable effect, gravity being dominant and thus distance between stars remains fixed despite universal expansion.

Can anyone help me see this mathematically? I have been unable to find equations to show me both gravity and expansion together. I am interested in whether individual mass objects far from the virial mass centre would start to be caught in the space time expansion and thus end up farther from the galactic centre, i.e. some boundary condition where expansion does overcome gravity due to gravitational effects being weaker at the furthest fringes of matter attracted by the central masses.

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  • $\begingroup$ while the purpose of the document is not mainstream, M. Villata analyses the local motions around the MW , Dark energy in the Local Void , chapter 3 Dynamics of the Local Sheet. This part is well referenced and credible enough. Before establishing an elegant mathematical relation, one must prove that he uses a model enough realistic, even toy. There are edges effects at this scale ... $\endgroup$
    – user46925
    Commented Jan 10, 2016 at 18:46

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One way to see this is simply to note the value of the Hubble constant, about $70\rm\,km/(s\cdot Mpc)$. The Milky Way galaxy is about 0.030 Mpc in diameter, so the Hubble "flow" from one end of the Milky Way to the other is only about 2 km/s. This is much smaller than the "peculiar" motions of objects within the galaxy. Heck, the eastward motion of Earth's surface near the equator is about 0.5 km/s.

Compare this to low-temperature phenomena in other areas of physics. If you're interested in milli-electronvolt physics, you can't explore it at room temperature ($kT = \rm25\,meV$); you have cool your material off to get rid of the random thermal motions.

Extending to the Local Group: the Andromeda galaxy and its satellites are approaching us at about 100 km/s, from sort-of a megaparsec away. So the "thermal motion" of galaxies in the Local Group is comparable to the Hubble flow, but still "hot" enough that the directions are more or less random.

Thriveth points out a subtle corollary in comment below: since there is no real distinction in GR between "expansion of matter within spacetime" and "expansion of spacetime carrying matter along," it is reasonable to conclude that within a galaxy, where motions are rapid and randomly oriented, there is no spacetime expansion at all. I admit to being somewhat out of my depth here.

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  • $\begingroup$ "However, it doesn't explain this presumed mechanism of gravity stronger than the expansion mechanism" ... I think it does. Within the galaxy, you have gravity requiring objects to move at hundreds of kilometers per second, while Hubble flow would induce speeds orders of magnitude smaller. Gravity wins. The expansion beyond the galaxy is a little bit iffier. $\endgroup$
    – rob ♦
    Commented Jan 13, 2016 at 18:02
  • $\begingroup$ @rob this answer is very helpful, but i suppose i should explain the reason i am looking at the threshold between galactic gravitation and expansion... I was considering a thought experiment that took as a premise the notion most often explain to me that galactic and other masses 'roll' on, or spacetime expands 'under' (if using a 2D rubber sheet analogy of spacetime), the surface of expanding space time remaining bound together by their own gravitation, hence, the universe expands whilst locally, the interstellar distances in galaxies appear to remain remain effectively fixed. $\endgroup$
    – Mentasm
    Commented Jan 14, 2016 at 3:51
  • $\begingroup$ ...It then occurred to me that if expansion was limited to flat(ish) spacetime, then this could lead to a model which could show a larger spacetime deformation than indicated by gravity alone. thus, lensed light would follow a path indicted by a hidden halo of mass, when in fact, the could be less mass than postulated due to expansion effects. $\endgroup$
    – Mentasm
    Commented Jan 14, 2016 at 3:52
  • $\begingroup$ Lacking the math to derive all of this (its been 30 years since i did nay serious calculus), but having very strong programming skills, i planned to model the experiment in a computer simulation and tweaks the parameters until i could get observed effects without the need for (so much) hidden mass. Hence, i am seeking the right equations or a pointer to where i can find/derive them, so i can start doing some computer modelling. $\endgroup$
    – Mentasm
    Commented Jan 14, 2016 at 3:52
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    $\begingroup$ I am not sure this answer is actually correct. It is not that gravtiational/thermal motion dominates over cosmological expansion in gravitationally bound systems, it simply turns it off. I suppose we are used to thinking of it like "expansion of Space drags galaxies along", but it is equivalent and equally valid to say "Motion of galaxies drags Space along". If you think of it like that, it is easy to see that within a system in which all peculiar motion averages to zero, there is no expansion of Space. Spacetime tells matter how to move, but matter tells Spacetime how to curve. $\endgroup$
    – Thriveth
    Commented Jan 14, 2016 at 16:37

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