FRW Model Q: Why Does Positive (ρ+3p) Prove Big Bang?

[Auburnman]

So I've been reading a bunch about the FRW metric and doing FRW dynamics. I had a quick theory question involving the FRW Model in general. So in the Friedmann equations why is it that if the combination of (ρ + 3p) is always positive then it somehow proves the existence of a big bang like singularity in the past?

 

[PeterDonis]

I assume you're referring to the second Friedmann equation, which (in units where G = c = 1, and with zero cosmological constant) looks like this:

$$\frac{1}{a} \frac{d^2 a}{dt^2} = - \frac{4 \pi}{3} \left( \rho + 3 p \right)$$

Suppose that at some time $t_0$, which we could call "now" , we have that $a > 0$ and $da /dt > 0$. This says that the Universe has some nonzero "size" (the scale factor a is a measure of "how large" the Universe is, though there are some technicalities with that that we probably don't need to get into here), and that it is expanding--i.e., to the future of $t_0$, a(t) will increase.

Now follow the history of the Universe into the past, given those conditions at $t_0$. In the past direction, the Universe is contracting; and if $\rho + 3 p$ is positive, then $d^2 a / dt^2$ is negative, meaning that as we go into the past, the contraction of the universe "accelerates". That is enough to ensure that at some finite time in the past, we will reach a = 0, which is the Big Bang singularity.

 

[bcrowell]

Although this is fine as far as it goes, you have to be careful not to oversell it, because it's only an argument that applies to the perfectly symmetric FRW model you're working with. The real universe lacks that perfect symmetry, so you really need the Hawking singularity theorem plus some observational constraints to prove that there was a singularity in the past.