Distance in cosmology

How can we calculate the change in distance over time between (say) two galaxies that they are at rest in respect to each other?

In other words, how do we modify the "classical" equation: $$\ddot{\mathbf{r}} = -G\frac{\mu}{|r|^2}\hat{\mathbf{r}}$$ to take into account expansion, G changes over time, etc?

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The answer depends on the exact details of the physical model you're considering.

The simplest model for the expanding Universe is a homogeneous one, in which the density is uniform in space (but varying in time, of course). If we imagine the two galaxies as "test particles" of negligible mass, living in such a Universe, then it's possible to work out the paths of both particles through space. If they are "at rest" with respect to each other at some initial time (meaning, to be precise, that their physical distance, measured at fixed cosmic time, is initially unchanging), then, roughly speaking, they start to fall towards each other if the Universe is dominated by "ordinary" matter (which pulls), and they start to fall away from each other if the Universe is dominated by "dark energy" (which pushes). This paper gives details.

But it looks to me like you want to consider a more complicated model, in which the galaxies themselves have non-negligible mass. That is, I think you're imagining that the density of the Universe is not uniform but has "lumps" at the locations of the galaxies. In that case, life gets complicated -- indeed, you can make it pretty much as complicated as you like! There's a large industry in astrophysics in simulating many-body gravitational systems like this.

There's one case I know of that can be treated analytically and that might give you a feel for how the answer goes. Suppose that the Universe has uniform density, except for one large galaxy, which can be treated as an extra pointlike "lump" of mass. Suppose that that galaxy is at rest in comoving coordinates (i.e., it's going with the flow of the expansion). Suppose that the other galaxy is a test particle of negligible mass. Finally, suppose that it's not too far away -- to be precise, its distance is small compared to $c/H$ where $H$ is the Hubble constant.

In that situation, the equation of motion of the test galaxy can be found from a roughly Newtonian analysis. The force on the test galaxy is determined by the gravitation of all of the stuff in a sphere centered on the big galaxy. To be precise, it's $$\ddot r = -{GM_{\rm eff}\over r^2},$$ where $r$ is the physical distance of the test galaxy (not the comoving distance), and the "effective mass" is $$M_{\rm eff}=M+(\rho+3p/c^2){4\over 3}\pi r^3.$$ Here $M$ is the mass of the big central galaxy, and $\rho,p$ are the average density and pressure. Without the $p$ term, this would be exactly the Newtonian result: $M_{\rm eff}$ would be the total mass in the sphere. The only general-relativistic correction is that pressure also gravitates.

In a Universe with lots of "dark energy," as ours appears to be, the pressure is negative and contributes more strongly than $\rho$, by the way.

Sorry for a bit of self-promotion, but here it is: John Baez and I wrote an article for the American Journal of Physics explaining Einstein's equation in relatively simple terms, and one of the main examples we worked out was the dynamics of the expanding Universe. If you want to know how to get from the exact general-relativity treatment to the quasi-Newtonian picture, you might want to check it out.

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A realistic consideration would be to take into account whether the two galaxies are close neighbors in the same cluster, because in that case they would be immerse in a "cloud" of dark matter whose density would be much higher than the average matter density of the universe. –  Alex Mar 30 '11 at 5:52

If the two objects were the only particles in the Universe, it would obviously not be our Universe which has something like $10^{80-90}$ particles (this number counting mostly CMB photons) in its visible portion. Because it would be a different Universe, you would have to explain its rules of the game, especially the cosmological constant, scales of inflation, etc. The answer to your question obviously depends on all these things. There's no universal concept of "the Universe" in which you may change absolutely everything but you still expect the events to be unique.

In particular, in any Universe with matter species, it is impossible for the Universe to only contain 2 particles after the inflation because the kinetic energy from the rolling inflaton field is always converted to gadzillions of particles when inflation is ending.

If you meant our Universe, with all the extra matter in it, and you're just asking how the distance between two objects has been changing, well, it's been changing proportionally to $a(t)$ that enters the Friedmann equations.

http://en.wikipedia.org/wiki/Friedmann_equations

If you want to compare the present and a particular moment in the past, you need to know what the moment is. Even if you say that it is the end of inflation, it depends on the energy scale where inflation occurred and we don't exactly know that.

One could calculate the answer for a GUT-scale inflation but the vagueness of your question isn't a good argument that such a calculation would be more than a waste of time. At any rate, such ratios of distances will be of the form "ten to a few dozens".

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I'll rephrase about the "only particles on the universe". –  Eelvex Mar 29 '11 at 6:16
I rephrased completely :) –  Eelvex Mar 29 '11 at 7:39

In line with what professor Bunn said, the galaxies being "at rest" with each other implies that either both of them (if their masses are similar) or one of them (if its mass is much smaller than the other's) is not at rest with respect to the local comoving background.

A metric commonly used to model a central gravitational mass in a flat FLRW spacetime with no accretion onto the central object is the McVittie metric, whose meaning and characteristics are still undergoing discussion. See Faraoni & Jacques (2007) "Cosmological expansion and local physics" arXiv:0707.1350 and Carrera & Giulini (2008) "Influence of global cosmological expansion on local dynamics and kinematics" arXiv:0810.2712. To note, if the FLRW spacetime is de Sitter (just cosmological constant with no matter), the McVittie solution is equivalent, through an appropriate coordinate transformation, to the Schwarzschild–de Sitter (Kottler) metric.

In words of the abstract of the second paper: "However, the general problem of gaining a qualitative and quantitative understanding of how the cosmological dynamics influences local systems remains challenging, with only partial clues being so far provided by exact solutions to the field equations of General Relativity."

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