cosmology - What has been proved about the big bang, and what has not?

Ok so the universe is in constant expansion, that has been proven, right? And that means that it was smaller in the past.. But what's the smallest size we can be sure the universe has ever had?

I just want to know what's the oldest thing we are sure about.

Answer

Spencer's comment is right: we never "prove" anything in science. This may sound like a minor point, but it's worth being careful about.

I might rephrase the question like this: What's the smallest size of the Universe for which we have substantial observational evidence in support of the standard big-bang picture?

People can disagree about what constitutes substantial evidence, but I'll nominate the epoch of nucleosynthesis as an answer to this question. This is the time when deuterium, helium, and lithium nuclei were formed via fusion in the early Universe. The observed abundances of those elements match the predictions of the theory, which is evidence that the theory works all the way back to that time.

The epoch of nucleosynthesis corresponds to a redshift of about $z=10^9$. The redshift (actually $1+z$) is just the factor by which the Universe has expanded in linear scale since the time in question, so nucleosynthesis occurred when the Universe was about a billion times smaller than it is today. The age of the Universe (according to the standard model) at that time was about one minute.

Other people may nominate different epochs for the title of "earliest epoch we are reasonably sure about." Even a hardened skeptic shouldn't go any later than the time of formation of the microwave background ($z=1100$, $t=400,000$ years). In the other direction, even the most credulous person shouldn't go any earlier than the time of electroweak symmetry breaking ($z=10^{15}$, $t=10^{-12}$ s.)

I vote for the nucleosynthesis epoch because I think it's the earliest period for which we have reliable astrophysical evidence.

The nucleosynthesis evidence was controversial as recently as about 10 or 15 years ago, but I don't think it is anymore. One way to think about it is that the theory of big-bang nucleosynthesis depends on essentially one parameter, namely the baryon density. If you use the nucleosynthesis observations to "measure" that parameter, you get the same answer as is obtained by quite a variety of other techniques.

The argument for an earlier epoch such as electroweak symmetry breaking is that we think we have a good understanding of the fundamental physical laws up to that energy scale. That's true, but we don't have direct observational tests of the cosmological application of those laws. I'd be very surprised if our standard theory turns out to be wrong on those scales, but we haven't tested it back to those times as directly as we've tested things back to nucleosynthesis.