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From Wikipedia,

The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time,

implying that gravitationally bound regions are not affected. I have heard this statement here on StackExchange, too.

However, from Friedmann's second equation, $\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$, gravity is presumably factored in through the energy density term $ρ$.

Question: why are the influences of gravity and expansion of space asymmetric? Or they are and my logic is simply flawed? If so, how is it flawed?

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    $\begingroup$ You cannot use Friedman’s equation on local scales where the metric is not described by an FRLW universe. $\endgroup$ – gmarocco Nov 14 at 13:30
  • $\begingroup$ @gmarocco Could you expand on that just a bit and post that as a separate answer so that I can tick it? For example, is it true that local changes in gravity have no effect on the local rate of expansion of the universe? If the answer is 'yes', that would solve the problem of symmetry and answer my question. $\endgroup$ – Max Nov 14 at 13:34
  • $\begingroup$ They are affected, just not by much. Imagine the expansion of space trying to stretch a spring. $\endgroup$ – m4r35n357 Nov 14 at 13:37
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    $\begingroup$ Sure I’ll do so shortly. $\endgroup$ – gmarocco Nov 14 at 13:38
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    $\begingroup$ duplicate or near duplicate of physics.stackexchange.com/questions/70047/… $\endgroup$ – Ben Crowell Nov 14 at 15:00
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Friedmann equations treated a case where there is uniformly distributed matter and cosmological constant. This is a good approximation on a large enough scale. However, in any given galaxy or solar system you don't have uniformly distributed matter: you have clumps of matter (stars, planets) and near-vacuum in between. So some other solution of the field equation applies.

A study of the Schwarzschild and de Sitter-Schwarzschild solutions gives some good general guidance on what to expect within any given solar system. By employing Birkhoff's theorem one learns that for the case of a single spherical star at the middle of a spherical void inside a spherically symmetric matter distribution, the solution in the void (the region outside the star, and inside the rest) is Schwarzschild (or de Sitter-Schwarzschild) even if the further matter is expanding outwards. So this helps you to see why gravitationally bound systems do not expand, to first approximation, with the cosmic expansion. A planet orbiting such a star will have an orbit whose radius and period does not change with time.

Since in fact there is never perfect spherical symmetry nor perfect vacuum, the precise situation will not be quite like that, but it gives the correct starting-point for further calculation.

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