Big Bang and Penrose-Hawking singularity theorems

Roger Penrose proved in 1965 that if in a collapse there forms a "trapped surface" then general relativity develops the system toward a "singularity". A weakness of the result is that no one has been able to prove mathematically that a realistic collapse of a star produces a trapped surface.

A singularity at the start of the Big Bang?

What about the hypothetical singularity in the Big Bang? Do we know that there has to be a singularity in out past?

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1970.0021

In the link we have the 1970 paper by Hawking and Penrose about singularity theorems.

Let us assume that the universe at a very large scale is close to Minkowski space. The universe has an infinite spatial volume. Let us reverse time and develop the universe backward in time. Does our local observable universe collapse into a singularity?

The density of visible matter and dark matter seems to be so low that we are not yet inside a trapped surface. Thus, the problem is qualitatively equivalent to a collapse of a star. It looks likely that the collapse leads to a trapped surface, but there is no mathematical proof.

Hawking and Penrose also prove that if the universe contains a spacelike compact hypersurface, then a singularity must exist in our past. Such a surface means that the "current" spatial volume of the universe is finite. This assumption is close to assuming the existence of a trapped surface, because for someone living inside a trapped surface, the volume of space that he can reach is finite.

The Milne model

In this blog we have suggested that the Milne model describes the observable universe. That is, the amount of dark matter whose gravity charge is negative, exactly cancels the positive gravity charge of other matter.

Then there is no need for a singularity to exist in the history of observable universe. We may live inside a "Milne explosion" cloud in a universe which is essentially Minkowski. Note that our Milne model breaks the weak energy condition which Hawking and Penrose assume, since we assume the existence of negative gravity charges.

Conclusions

The existence of a singularity in our past is a problem quite similar to the existence of a singularity in a star collapse. There is no mathematical proof in general relativity of the existence of a singularity for a realistic case where matter is not uniformly distributed.