How can we predict a Big Crunch when all galaxies are moving further apart?

Question

I've read that the further away a Galaxy is from us the faster it moves away.

By this logic how can scientists predict that there will eventually be a big crunch when every piece of matter is seemingly getting further and further apart?

Answer 1

In a homogeneous and isotropic Universe (even if recent observations challenge this hypothesis), you can derive the Friedmann equations, which describe the evolution of the Hubble constant with time: $\frac{\dot{a}}{a} = H(t) = \frac{8 \pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}$ (with $c=1$) (Equation $1$)

where $a=a(t)$ is the scale factor, $\dot{a}$ its derivative, $G$ the gravitational constant, $\rho$ the matter density, $\frac{k}{a^2}$ the spatial curvature (a parameter that describes the metric of the Universe), and $\Lambda$ the cosmological constant (an integration constant added by Einstein). It could be useful to rewrite the equation as:

$H^2 = \frac{8 \pi G}{3}(\rho + \rho_{\Lambda}) - \frac{k}{a^2}$

where $\rho_{\Lambda} = \frac{\Lambda}{8 \pi G}$ is the "density of cosmological constant".

We can also expand the matter density as $\rho = \rho_{matter} + \rho_{radiation}$.

So we have a "total" density $\rho_{tot} = \rho_{matter} + \rho_{radiation} + \rho_{\Lambda}$. The destiny of the Universe depends on this amount.

In case of $\rho_{tot} > \rho_{crit}$, or equivalently a closed Universe ($k=+1$), the equation $(1)$ becomes:

$\dot a^2 = \frac{8 \pi G}{3}\rho a^2 -1$

Which points out that the scale factor must have an upper limit $a_{max}$ ($\dot a^2$ must be positive). This in turn means that the second derivative $\ddot a$ of the scale factor must be negative, when approaching $a_{max}$, that is the scale factor function inverts its behavior:

Look at here and here if you want to go deeper.

@Bardathehobo This figure shows what I mean when I say that a currently accelerating Universe can still crunch. This is because we are basically ignorant upon the dark energy issue.

Answer 2

Well, the original idea of the Big Crunch came about when it was thought that gravity was slowing down the expansion of the universe and that one day it will stop expanding (a finite universe). At which point the gravitational pull of all the objects in the universe would make it contract, growing smaller into itself until eventually it is just a singularity.

However, today we see that the universe is expanding and accelerating due to dark energy's effect on gravity making it an infinite universe. This makes the original idea of the Big Crunch not possible (since the universe would be infinite), but a big crunch can still occur in an alternate way if dark energy becomes too weak to counteract gravity which would set in motion the Big Crunch. However there is still so much unknown about dark energy that it is also possible that it will never weaken and the universe will just keep expanding, which opens different end of the universe theories.