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1) I don't know whether this question should belong to this site: please correct me if not.
In the earliest moments of the universe history probably all fundamental forces where indistinguishable: in particular there weren;t any massive particles. This time include the so called electroweak epoch which ended about $10^{-12}s$ after the Big Bang.

How do we understand this time, since when there is no massive particle the notion of time does not make sense?

2) Moreover we often speak about the size of the universe in that period, we say that universe was expanding (in some model like inflation, this expansion was very rapid).

Can we even discuss the size when there is no mass? No mass implies that everything travels with the speed of light and in the world where everything travels with the speed of light I would say that there is no notion of distance either.

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If there were a time when all energy were massless and moved at the speed of light, Relativity would provide no appearance of relative time, only the appearance of absolute time (time would seem to pass the same for all energy in the Universe). Only if space did not exist, if distance disappeared, would time also not exist. That would be the point of the singularity, where relativity does not work.

Two researchers, at Benha University in Egypt and at the University of Lethbridge in Canada, claim to have combined elements of quantum theory and general relativity to explain how the Universe could exist without a singularity. Their model, however, postulates a Universe of finite size and infinite age. In other words, they claim to have avoided the singularity.

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