At least some contemporary big bang theories begin with the entire universe packed into a very small, atom sized volume. Since black hole densities can be achieved by compressing the earth to the size of a marble; it seems that the early universe would have been dense enough to be a black hole and would have never expanded. Are the theories of the big bang and black holes at odds?
That is a very good question. The big bang theory and the existence of black holes are not at odds with each other. It's true that the early universe, shortly after the big bang, was much denser than it is now, but the formation of a black hole requires that there be a substantial excess of mass in one region relative to the average mass in that region, where the average is taken over the universe at that time.
If the distribution of mass in the early universe was very lumpy over large scales, with some regions overdense and others underdense relative to the mean, then black holes could have formed in the regions with excess mass, i.e., in the regions where the density was substantially higher than the average. But the formation of those primordial black holes would not have prevented the universe as a whole from expanding.
We have observational evidence that, at least after a certain epoch, the mass in the universe was very uniformly distributed, but it is nevertheless possible that small primordial black holes did form.
Your question does hint at another important point. We have evidence that, very shortly after the big bang, the expansion rate of the universe and the mean density of the universe were very finely tuned to each other, in the following sense: if, at a given epoch, the density of the universe had been slightly higher than it was, then the expansion of the universe indeed would have halted and it would have recollapsed shortly thereafter. If the density had been slightly lower, then the expansion would have continued but the universe would have become much emptier than it is now.
There are models that suggest why the universe reached this point of apparently delicate balance soon after the big bang, and observations in the coming years will refine the tests of those models.
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