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Why was it necessary to think of the existence of a singularity during the big bang?

As per the current theory, during the big bang, the universe had an infinitesimally small radius, and just after the big bang, the rate of expansion was infinitely large.

My question: If the initial size had been finite, and had the rate of expansion just after been also some finite number,
then what physical observations that we make today would have been invalid? Is the assumption of a singularity related to any physical evidence we observe today at all?or is it just for mathematical convenience? Hopefully my question is clear.

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For the most part, or possibly totally, it seems that we can explain all of cosmology and the Big Bang if we start with the moment when inflation starts, and we assume the right kind of slow roll Inflaton field. Of course you can't explain that starting point.

You can also (again mostly or maybe totally) explain post-inflationary cosmology from assuming homogeneity and isotropy, with certain size perturbations, GR and the standard model.

So, yes, you don't need to start before inflation, or even right after inflation, if you are willing to ignore times before. If you remain satisfied that the universe started that way, and you don't need a cause.

But scientists tend to follow things to some logical conclusion, and then see if they can figure out a theory for that. Thus, if you follow it back, it is clear that until you get to the start of inflation (going back in time) thoe will mostly work without you needing any unknown theories, except for the Inflaton. Before you that also need either the singularity or quantum gravity. Without quantum gravity (I.e., some appropriate theory of gravity and the unified forces at the Planck scale), GR says you go back to a singularity.

Since singularities in a physical theory means the theory has broken down at that scale, it is understood we need to find some explanation and theory of what happens at Planck and pre-Planck scales.

If you start with the assumptions I listed, most or all our observations would still be right. However, there are still some uncertainties: 1) the Inflaton field has not been found nor well defined, there are possible different ways it could have been, if the inflation theory holds up. And we are getting more accurate and new observations from the cosmological microwave background (CMB), distributions and structure of galaxies, that still cause us some concern that the cosmological model we now have the numbers for may not be correctly predicting some features - like the density inhomogeneities and anisotropies we are analyzing. 2) we have still not seen incontrovertible proof of inflation. It explains a lot, but till we find some further evidence, such as remnantS of those times like gravitational radiation from then or some remnant particles, there is going to be doubts. 3) there is some hope of finding the dark matter particles, and of finding more about what dark energy is. Since those comprise about 95% of the universe's matter-energy, cosmology still needs more evidence

Thing is, even while we try to tackle the issue of the singularity and quantum gravity, we have other observations and issues in cosmology that also need more findings.

By the way, a spacetime that is homogeneous and isotropic, per General Relativity, has to be represented by the cosmological model, the FLRW solution, and the parameters then come form the observation of the cosmological expansion, etc, including the CMB. The solution predicts that going back in time the universe was more and more dense, and it has no way of stopping, going back, till we get to the singularity. Sort of the time reversed case of a Black Hole. It has to be so, if General Relativity and our observations are right. So, observe more, get more data, resolve the uncertainties and issues still of concern I noted, and then come up with a good inflation model, and a good Quantum Gravity Theory.

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