Not quite, Big Bang theory holds that in the beginning, all matter, space, and time was compressed into a singularity, which then expanded outward at great speed

(how time has a "beginning" is beyond me though

).
IMO, I think the whole notion of theorizing about multiple universes continually forming and expanding like bubbles in water is rather silly. It is fantasizing to an extreme degree I think. YES, it could maybe be true, but it's just so far out there. When Hawking says:
As recent advances in cosmology suggest, the laws of gravity and quantum theory allow universes to appear spontaneously from nothing.
Okay, so who wrote these "laws" of cosmology and gravity that allow universes to appear spontaneously from nothing? Are those spontaneous too? I think this is a prime example of how our brains are simply too limited to understand this subject beyond a small amount.
On God's existence, I think the arguments FOR a Creator are flawed (because who created the creator? The proponents say the Creator exists outside of these "rules," and thus doesn't need a creator, well why can't the universe then? The universe itself isn't subject to rules, things
within the universe are subject to rules). That is also inadverdently what Hawking is saying. HOWEVER, I also think the arguments against a Creator are flawed as well. To say that all the matter-space-time in the beginning was just "there" and no one or nothing created it, that nothing wrote the laws of physics, etc...that sounds silly as well.
Again I think it just shows the limits of our brains and the logic we use to understand the universe.
I believe if there is such a Creator, it is be something far more grandious and unimaginable then any of us can possibly conjure up. Right now I think humans trying to figure this stuff out is like chimpanzees trying to learn nuclear physics---it's beyond the capability of their brains.
