I Mass in an expanding or static spherical distribution of matter

Ranku
Messages
433
Reaction score
18
In a spherical distribution of matter - such as with clusters of galaxies - how to calculate how much mass there should be for it to not expand with the expanding universe - in other word, for it to be a bound, static system?
 
Space news on Phys.org
I think that for a "ball" not to expand, the escape velocity on its surface needs to be greater than the velocity of the expansion there as per Hubble law.
 
I don't think there's a general answer to this. For example, in a Big Rip scenario no systems are bound at all.

I think the McVittie metric is the thing you need to look at.
 
Various models are described in Jones, Bernard J. T.. Precision Cosmology: The First Half Million Years. It starts with this:
1705662674334.png
 
Hill said:
I think that for a "ball" not to expand, the escape velocity on its surface needs to be greater than the velocity of the expansion there as per Hubble law.
I think this is basically correct, at least as a heuristic. We had a previous thread on this some time ago that referenced a paper which did the calculation in more detail. I'll see if I can find it.
 
  • Informative
  • Like
Likes Ranku and Hill
I claim this question is inherently complex, and that what approximations you make depends on what question you are really trying to answer. Cosmologically, the interesting question is what are the largest bound systems that might come to be in a universe broadly similar to ours, with various assumptions about initial inhomogeneities. But very different assumptions might be made as pure question of mathematical GR. To wit, I propose arguments that for one statement of the problem, the answer is completely determined by dark energy assumptions and can be answered without reference to an FLRW solution - using the same types of arguments used in the work referred to in post #4.

Consider an initially contracting ball of dust (pressureless perfect fluid) embedded in an empty, asymptotically flat spacetime. Basically, this is some initial state of an Oppenheimer-Snyder class of solution. Excise this just outside the ball, glue into an FLRW solution with a boundary shell where the FLRW perfect fluid density goes to zero (and there is no dark energy). This is needed for a smooth gluing. Now, by arguments based on Birkhoff, the evolution within the ball is unchanged, and it will contract to a BH no matter how large an instance of this you create.

Now consider dark energy. For simplicity, let's only discuss cosmological constant. Then the initial set up is an initially contracting dust ball in an otherwise empty universe with cosmological constant. I believe the result here is the for any choice of such constant and details of initial ball state, there is a minimum size such that the ball will eventually stop contracting and start expanding. Again, with the same gluing strategy as above, except that the at the inner glue shell boundary you have pure dark energy matching the ball solution (assumed to be the same as the universe at large), it is again true that the rest of the FLRW solution is irrelevant to the ball dynamics until well after reversal occurs (in the cases where it reverses). Thus, the question of whether the ball reverses and eventually joins the hubble flow is answerable with an isolated treatment of the ball.
 
  • Informative
  • Like
Likes Ranku and PeterDonis
https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
The formal paper is here. The Rutgers University news has published a story about an image being closely examined at their New Brunswick campus. Here is an excerpt: Computer modeling of the gravitational lens by Keeton and Eid showed that the four visible foreground galaxies causing the gravitational bending couldn’t explain the details of the five-image pattern. Only with the addition of a large, invisible mass, in this case, a dark matter halo, could the model match the observations...
Hi, I’m pretty new to cosmology and I’m trying to get my head around the Big Bang and the potential infinite extent of the universe as a whole. There’s lots of misleading info out there but this forum and a few others have helped me and I just wanted to check I have the right idea. The Big Bang was the creation of space and time. At this instant t=0 space was infinite in size but the scale factor was zero. I’m picturing it (hopefully correctly) like an excel spreadsheet with infinite...
Back
Top