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We know that binary systems slowly lose energy due to gravitational waves from the objects moving through spacetime and that if the objects are compact and massive enough, the mergers happen in time scales within the age of the universe, and LIGO and friends can pick up their gravitational wave signals.

We also know the universe is expanding, and that the geometry of this expansion is such that objects farther from us are expanding away at a higher relative velocity than objects close to us. We know that for gravitationally bound objects, this effect is not enough to keep objects from being in orbits or clusters.

So my question is this: If you have an isolated binary system of two small objects that wouldn't normally be treated with relativity, and take the system evolution over immense timescales, could the expansion of the universe (as minute and small effect as it has) have a counteracting effect as to negate the incredibly slow (but inevitably present) infalling of the objects due to orbital energy being radiated away by gravitational waves?

For starters, some ground rules, ideas and assumptions:

  1. Let's assume the binary system is composed of objects that aren't going to change a whole lot in time, maybe rouge exoplanets orbiting one another or a binary brown dwarf system, it doesn't really matter except that nothing about the bodies themselves are going to affect our solution here, and they are not intrinsically relativistic (e.g. no neutron stars or white dwarfs)

  2. Let's consider timescales much beyond the age of the universe; these processes to be considered only have measurable effects over immense time scales, but let's just say this system continues isolated and untouched by the rest of the universe for these time scales. There's no way this has ever happened, but I want to consider if it could in some distant, distant future

  3. I know gravitationally bound objects don't really expand away from each other practically, but it seems like the expansion would add just a little bit of energy to the system, and so I'm trying to figure out if that amount of energy negates the energy lost by gravitational waves. It definitely can't be more because we don't observe that, but we want to see if it's comparable.

  4. Let's ignore any ideas of a quantized gravitational field; let's assume a continuum of gravitational waves insomuch that these objects are emitting them from their orbit, as small as they may be.

  5. How these objects got here or anything leading up to this system is not relevant, although if I have made any physically incorrect assumptions, please point them out

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  • $\begingroup$ No, because the Hubble constant is ~70 km/s / Mpc, and typical stellar orbits are around 40 au, resulting in a 0.0000135747831 m/s change in velocity. Probably not very good logic, just a crude comment from someone who isn't very good at GR. $\endgroup$ Oct 19, 2021 at 18:27
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    $\begingroup$ #3 - You can't just assume the Hubble expansion because the little piece of the universe you are considering doesn't obey the approximations (being homogeneous and isotropic) that lead to the Hubble expansion. $\endgroup$
    – ProfRob
    Oct 19, 2021 at 21:39
  • $\begingroup$ It's probably not a great idea to ask GR questions on this site, because there are misunderstandings of GR circulating. An answer by benrg says: > There is no locally measurable force associated with the Hubble expansion. This is wrong. For a thorough discussion of this, see Cooperstock et al., arxiv.org/abs/astro-ph/9803097v1 . As an example, they calculate the rate of expansion for the earth-sun system at the end of section 4 (although of course other effects would completely overwhelm the effect due to cosmological expansion). It's not zero, and it's not what you would naively expect $\endgroup$
    – user44162
    Oct 20, 2021 at 14:37
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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – called2voyage
    Oct 20, 2021 at 16:20
  • $\begingroup$ @user44162 Indeed there seems to be some consensus that the effects are not zero, but they are much smaller than the naive expectation of applying an acceleration based on $H_0$. A more recent paper on the subject: ui.adsabs.harvard.edu/abs/2013MNRAS.429..915I/abstract $\endgroup$
    – ProfRob
    Oct 20, 2021 at 16:37

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There is no locally measurable force associated with the Hubble expansion. The short reason is in ProfRob's comment: "the little piece of the universe you are considering doesn't obey the approximations (being homogeneous and isotropic) that lead to the Hubble expansion." For more details, see this answer.

In ΛCDM cosmology, $Λ$ does cause an outward acceleration that is in principle measurable locally. Its magnitude is around $10^{-35}$ or $10^{-36}\text{ (m/s}^2\text{)/m}$ in any era (it only depends on $Λ$, which is constant). But you can define a conserved potential energy for this force, so it can't stabilize orbits against gravitational decay – otherwise you could build a machine that radiated power in the form of gravitational waves forever.

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  • $\begingroup$ Makes a lot of sense; Thanks!! $\endgroup$
    – Justin T
    Oct 20, 2021 at 0:24
  • $\begingroup$ So what exactly happens in the critical limit between a binary decaying due emission of gravitational waves, and it being torn apart cosmological constant driven expansion? $\endgroup$
    – mmeent
    Oct 20, 2021 at 7:59
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    $\begingroup$ @mmeent For circular orbits, the orbital speed goes to zero at a finite radius instead of asymptotically approaching zero as $r\to\infty$. At the critical distance you could have two massive bodies at relative rest (and not radiating), but it's an unstable equilibrium. $\endgroup$
    – benrg
    Oct 20, 2021 at 8:38
  • $\begingroup$ Playing with a gravitational acceleration calculator gives me a guess of about 10 000 km for 1 g masses. TBH I'm not sure if I did that right. $\endgroup$
    – JollyJoker
    Oct 20, 2021 at 11:36

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