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Is it true that the Universe can not have an eternal past? It seems like Alexander Vilenkin tries to argue this? Does this imply that the Universe had a beginning?

Link to his publication regarding this question: https://arxiv.org/abs/1204.4658

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This is discussed in the paper's introduction.

It was addressed in the singularity theorems of Penrose and Hawking, with the conclusion that the initial singularity is not avoidable. These theorems rely on the strong energy condition and on certain assumptions about the global structure of spacetime.

In other words, if we accept the strong energy condition and the "certain assumptions" about the global structure of spacetime, then we must an initial singularity. In less technical terms, Hawking himself wrote in A Brief History of Time that the proof requires 1) that the universe contains as much matter as we observe it to contain and 2) that General Relativity is correct.

However, these theorems only hold if the assumptions are valid, and we just don't know. We haven't found any observation to suggest that GR is wrong yet, but given that massive unknowns of dark matter and dark energy, it certainly could be wrong. This allows models in which there is no initial singularity to survive, e.g. when I Googled for it just now I found this space.com article about bouncing cosmologies.

The other thing is that GR is almost certainly wrong at very high energy scales. In A Brief History of Time, Hawking said that what the theorems actually show is that there comes to a point where quantum effects must be taken into account. Since we don't have a quantum theory of gravity, we can't make any reliable statements once this point is reached.

tl; dr: we don't know. Maybe the universe did have a beginning, and maybe it didn't. We need better theories to be sure.

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