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I am trying to understand the effects of gravity in the Cosmos without complications of Dark Matter/Dark energy issues. So my question is, assuming that a galaxy (for example, the Milky Way) does not contain any dark matter, does anyone know the equation that would explain how long it would take for the galaxy to collapse into the massive black hole standing in the center?

If we were to extend the question to the Universe, would the equations be different and why? (again no dark energy or $\Lambda$ or a Hubble Constant would be involved so that I can easily get my head around the question of gravity at a very large scale).

I've been trying to get information on this issue for a while, so any help with equations will be welcome. I will also appreciate some good references especially a good book on the subject.

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You've got a number of misconceptions in there, and they're going to make it very hard to answer the question as asked...

1) Our galaxy will not naturally collapse into the central BH. Some material will fall in, but most of it just keeps orbiting. See this question, for instance: How will the super massive black hole affect our galaxy?

2) It is believed (from a lot of compelling evidence) that galaxies are primarily made up of dark matter, it's very difficult to just ignore it on galaxy scales, since forming a realistic stable galaxy without it is nigh-impossible. And dark matter is actually easier to understand than ordinary "baryonic" matter because the only thing it does is gravitate, so you can ignore all the complications that arise when you need to deal with electromagnetism. You're better off ignoring "normal" matter and considering a "dark matter only galaxy" (though this will only bring about a spherical blob component of the galaxy, with more little spherical blobs surrounding it, none of the nice spiral shapes you see in pictures - best to start simple though!).

3) At very large scales (much larger than galaxies), $\Lambda$ is the dominant "force" driving evolution in the standard model of cosmology. Again, can't really be ignored (though you can reasonably ignore $\Lambda$ for a galaxy-sized system).

So my suggestion would be to focus your efforts on learning about DM and DE, instead of wasting your efforts trying to learn about a Universe without them. I've got to run for now, but will try and post a couple of useful links to get you started a little later.

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  • $\begingroup$ Yes, I appreciate everything you are saying. I actually know a bit abut DE and I am trying to build a model (computer programme) that would show how the universe would collapse without DE. So DE is actually getting on the way. Using any Standard Equations that I find for the universe evolution I run into problems, because they already include DE, and I am not sure how to strip it out of the equation, without breaking the equation. So for me to understand gravity at a large scale and at a pure form, it would be good if DE could be hypothetically ignored. $\endgroup$
    – Luis
    Commented Jan 25, 2014 at 0:03
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    $\begingroup$ Can you post some of these equations you're working with, and where you're running into trouble? (maybe as a new question, or extension to this one) $\endgroup$
    – Kyle Oman
    Commented Jan 25, 2014 at 0:11
  • $\begingroup$ Okay, fair enough. I think the best way to look at it is by looking at galaxy clusters. Galaxies in a cluster experience mutual attraction. I suppose in a galaxy clusters we don't have to worry so much about DE or make distinctions between dark and baryonic matter. It just matter clumping with each other over time due to the force of gravity. So, what are the equations that describe the time that it would take for a galaxy cluster to eventually clump together into one single object? $\endgroup$
    – Luis
    Commented Jan 25, 2014 at 17:51
  • $\begingroup$ Ah now you're getting somewhere. A short answer/good place to start is with Press-Schechter theory en.wikipedia.org/wiki/Press%E2%80%93Schechter_formalism. There's a good section on the technical details in Binney & Tremaine's classic text "Galactic Dynamics". $\endgroup$
    – Kyle Oman
    Commented Jan 26, 2014 at 6:59
  • $\begingroup$ Yes, thank you. That might lead me to a good start. For now, looking at the equation I can see that the number of objects formed depend on $\rho_m$. If the formula is consistent with evolution of the universe, I would hope if $\rho_m$ increases (i.e. $\rho_{\Lambda}$ decreases) then the number of objects will decrease, leading to the collapse by gravity. Would you agree and do you think I can use the equation in such a fashion? $\endgroup$
    – Luis
    Commented Jan 26, 2014 at 19:45
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Just as there are other forces involved at the subatomic scale where gravity has little influence there are different influences at the macro scale. One side of the aisle says Einstein is wrong and the other creates a new set of unknowns expressed as DE and DM. either way there is a speed limit in the universe. I envision a relative slow compression up to the point of effectual gravity and a cascading event where upon gravity takes the wheel so to speak. there is no natural means by which such an event can occur without folding space

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