How can gravity lead to the Big Crunch scenario?

Question

According to modern cosmology, space is expanding, causing proper distances (but not comoving distances) to increase between galaxies. In the Big Crunch hypothesis, gravity halts and reverses the expansion of the Universe, causing all matter to collide and eventually form a single black hole. This gives way to other oscillating universe hypotheses, which generally propose that the conditions in a compressed Universe would be the same as those during the Big Bang, leading a cycle of expanding and contracting universes.

Ignoring the problems with entropy in reverting the Universe to Big Bang conditions, how can gravity be the cause of a Big Crunch in the first place? Specifically, gravity (to my knowledge) only curves space; the idea that it can revert the Universe to the conditions of the Big Bang seems to imply that gravity actually can contract space. Is this actually the case?

If not, the gravitating objects should be moving through a comoving coordinate system, so space itself would not actually be contracting. As far as I can tell, we would have all the Universe's matter compressed within a single point in space, instead of space itself contracting. This should be completely unlike the Big Bang, when space was far less dilated than it is now. If this is in fact what the Big Crunch hypothesis describes, than I'm utterly confused as to how an oscillating universe could work in such a situation.

Am I mistaken, or do the Big Crunch and oscillating universe hypotheses imply that gravity actually contracts space (as in, the comoving distances of gravitationally attracting objects would not change)? If not, how could gravity possibly lead to these scenarios?

Answer 1

The amount of matter in the universe is directly related to the curvature of space itself. We can look at the Friedmann Equations to see how this works: $$ H(t)^2 = \frac{R'(t)^2}{R(t)^2} = \frac{8\pi G}{3}(\rho_{m} + \rho_{r}) + \frac{1}{3}\Lambda - \frac{c^2}{R^2\mathscr{R}^2} $$ This is the equation for determining the scale factor for finding distances in the universe at any given time. Here, $H$ is the Hubble Constant, $R$ is the scale factor, $G$ is the gravitational constant, $\rho_m$ is the density of matter (both dark matter and baryonic matter), $\rho_r$ is the density of radiation (photons), $\Lambda$ is the cosmological constant due to dark energy, and $\mathscr{R}$ is the radius of curvature of space.

Next, we define a quantity we call the "critical density" ($\rho_c$). This is the density of matter needed to make the universe go from an "open", hyperbolic geometry to a "closed", spherical geometry. $\rho_c=\frac{3H^2}{8\pi G}$. We create a value $\Omega=\frac{\rho}{\rho_c}=\frac{8\pi G \rho}{3H^2}$ to take into account this critical density, making our equation:

$$ H^2 = \Omega_m + \Omega_r + \Omega_\Lambda - \frac{c^2}{R^2\mathscr{R}^2} $$

From here, we input our present-day values for everything. We define $R$ to be 1 and $H$ to be $H_0$ at the present day. We also choose to neglect the density of radiation and dark energy. Simplifying the equation a little gives us:

$$ \frac{c^2}{\mathscr{R}^2} = H_0^2(\Omega_{m,0} - 1) $$

From here, we can see that the curvature of the universe is dependent on $\Omega_{m,0}$, which is directly related to the density of matter in the universe. Specifically, for $\Omega_{m,0}\gt1$, $\mathscr{R}$ will be positive, meaning this will be a spherical, closed universe. If $\Omega_{m,0}\lt1$, $\mathscr{R}$ will be negative, meaning the universe will expand forever, having hyperbolic curvature. If $\Omega_{m,0}=1$, then $\mathscr{R}=\infty$, which is a flat universe.

So, we can see that having a dense enough universe, gravity will curve space and make it spherical. You can also use the Friedmann Equation to calculate the deceleration parameter of space - how quickly the expansion is slowing or speeding up:

$$ q_0=\frac{\Omega_{m,0}}{2} $$

Here, we can see that in a positively curved universe, $q_0>0$, meaning that the universe's expansion will be decelerating. Eventually, the expansion rate will be negative, and then it will begin collapsing in on itself.

One thing to keep in mind when thinking about this is that gravity can, in a way, be considered a 'fictitious force'. Gravity is the force objects feel as they attempt to travel in straight lines through curved spacetime. Matter is what is responsible for the curving of spacetime, so by increasing the density of matter in the universe, you increase the curvature, and thus make objects want to move closer together, which will increase the density, which increases the curvature. Thus you have a feedback cycle, where you will ultimately collect all matter at a single point, and $\mathscr{R}=0$, so spacetime will have a 0 curvature radius, meaning spacetime has collapsed as well.

Answer 2

In less mathy words, the Big Crunch scenario occurs if the ratio of the total density of the Universe to its expansion rate is sufficiently large$^\dagger$.

As I understand your question, you're basically asking, "Why would a sufficiently dense universe not simply contract into a clump? Why does it have to pull space itself with it?"

And you basically answer that question yourself: Yes, space is indeed "tied" to matter. This is the essence of the Friedmann equation, and of general relativity in general. To my knowledge, there's no "proof" of this other than, well, it's one of the foundations of GR, which has so far proved an immensely successful theory. In moderately overdense regions (galaxy clusters) space expands slower than in underdense regions (voids). In very overdense regions (galaxies, stars, cats, etc.), it doesn't expand at all. And in extremely overdense regions (black holes), space contracts. In the case of a black hole, space only contracts locally, but in principle the whole of the Universe could do the same. Only expansion prevents this, and seems to be able to prevent it forever.

We believe that the Universe is homogeneous and isotropic; if this is indeed the case, matter wouldn't be able to contract to a point within the Universe and make one giant black hole in an otherwise expanding Universe, since every bit of matter is attracted the same in all directions. You could perhaps envision clumps of matter contract on very large, yet sub-Universal, scales, to create many supersupermassive black holes, but as it turns out, the expansion rate was simply too large in the early Universe for this to happen, and now it's too late.

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$^\dagger$_{This is analogous to a rock thrown upwards falling back again if the ratio of the gravitational attraction between Earth and rock to the speed with which it is thrown is suffiently large.}