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I think in loop quantum gravity the rate of expansion of the universe is found from:

$H^2=\frac{8\pi G}{3}\rho\left(1-\frac{\rho}{\rho_c}\right)$

And I was wondering whether the critical density $p_c$ in the equation would actually mean that all of the matter was condensed to within its Schwarzschild radius, but quantum effects make it bounce anyway? Or is the Schwarzschild radius never reached? (I think it must be considering observation suggest that there are black holes in our and other galaxies, and if the critical density was not great enough to reach the Schwarzschild radius then loop quantum gravity would predict that ordinary black holes cannot exist either, right?)

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    $\begingroup$ The universe is not described by the Schwarzschild metric so it has no Schwarzschild radius. $\endgroup$ – John Rennie Oct 19 '14 at 9:20
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    $\begingroup$ No, because the matter distribution in the universe is (approximately) homogeneous so the gravitational potential is the same everywhere. There is simply no concept of a Schwarzschild radius in these circumstances. $\endgroup$ – John Rennie Oct 19 '14 at 9:34
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    $\begingroup$ The universe is described by the FLRW metric. This has a singularity at $t = 0$ (and for closed universes also at the recollapse time) however it has no horizons. A black hole is described by the Schwarzschild metric. This has a singularity at $r = 0$ and a horizon at $r = 2GM/c^2$. $\endgroup$ – John Rennie Oct 19 '14 at 9:48
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    $\begingroup$ A finite, i.e. closed, Friedmann universe is homogeneous and isotropic just like flat and open Friedmann universes. In the real universe at the current time the matter distribution is obviously not isotropic and locally black holes and their associated horizons can form. However if you look at the CMB (that dates from the recombination era) the universe was homogeneous to about one part in $10^5$. Because inhomogeneities grow with time we expect that near the Big Bang any conceivable universes would be almost exactly homogeneous. $\endgroup$ – John Rennie Oct 19 '14 at 9:52
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    $\begingroup$ Yes. If you want to understand this I'm afraid you'll have to get stuck in and learn about the FLRW metric. Without knowing your background I'm not sure what to recommend as a starting point. The Wikipedia article is not suitable for beginners. Incidentally your other question also stems from misunderstanding the FLRW metric. $\endgroup$ – John Rennie Oct 19 '14 at 10:03

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