[harshavr]

I am no expert on this, but if you assume the cosmological principle which states that our universe is roughly isotropic, there is a natural set of observers called comoving observers who are at rest with respect to the expansion of the universe. One can define a universal time using the clocks of these observers. Presumably, this is what gives meaning to statements like the universe is 14 billion years old.

[Cushman]

Well, again, relativity doesn't violate causality. No one sees an effect before a cause. That translates into the cosmological principle in that all observers will see the big bang happen before anything else, all observers will see stars form before planets, et cetera, so it all comes out in the wash.

Nothing about that privileges a comoving observer with relation to unrelated events, though; they'll have their perspective just like everyone else, and that will be that. There's nothing "universal" about comoving time.