All of the macroscopic manifestations of the arrow of time - our ability to turn eggs into omelets but not vice versa, the tendency of milk to mix into coffee but never spontaneously unmix, the fact that we can remember the past but not the future - can be traced to the tendency of entropy to increase, in accordance with the Second Law of Thermodynamics.
In the 1870's, Boltzmann explained the microscopic underpinnings of the Second Law: Entropy counts the number of microstates corresponding to each macrostate, so if we start (for whatever reason) in a relatively low-entropy state, it's overwhelmingly likely that the entropy will increase toward the future.
However, due to the fundamental reversibility of the laws of physics, if the only thing we have to go on is the fact that the current state is low entropy, we would with equal legitimacy expect the entropy to have been larger in the past. But the real world doesn't seem to work that way, so we need something else to go on. That something else is the Past Hypothesis: the assumption that the very early universe found itself in an extremely low-entropy state, and we are currently witnessing its relaxation to a state of high entropy.
The question of why the Past Hypothesis is true belongs to the realm of cosmology. The anthropic principle is woefully inadequate for the task, since we could easily find ourselves constituted as random fluctuations (Boltzmann brains) in an otherwise empty de Sitter space. Likewise, inflation by itself doesn't address the question, as it requires an even lower-entropy starting point than the conventional Big Bang cosmology.
The way out is, that we can accept the Big Bang had a low entropy, but deny that the Big Bang was the beginning of the universe. The idea that the Big Bang is truly the beginning of the universe is simply a plausible hypothesis, not a result established by reasonable doubt. General relativity doesn't predict that space and time didn't exist before the Big Bang; it predicts that the curvature of spacetime in the very early universe became so large that general relativity itself ceases to be reliable. Therefore quantum gravity absolutely must be taken into account.
In quantum field theory it is natural to assume that de Sitter space itself should be fluctuating. The entropy associated with de Sitter space is (a) low when the energy density is high and (b) high when the energy density is low. (Therefore the decay of a high-energy de Sitter space into a state with lower vacuum energy is just the natural evolution of a low-entropy state into a high-entropy one.) When the de Sitter space starts at the bottom, where vacuum energy is very small, quantum fluctuations will occasionally push the field up the potential, from the true vacuum to the false vacuum - not everywhere at once, but in some small region of space.
What happens when a bubble of false vacuum fluctuates into existence in de Sitter space? Inside the bubble, where we've fluctuated into the false vacuum, space wants to expand; but the wall separating the inside from the outside of the bubble wants to shrink, and usually it shrinks away quickly before anything dramatic happens. Therefore, most of the time, the bubble will disappear again as the field will simply dissipate away back into its thermal surroundings.
However, every once in a while we could get lucky: We could find a fluctuation in a low-vacuum energy de Sitter space (creating a bubble of false vacuum, with the space inside wanting to expand), and simultaneously a fluctuation of space ifself (causing the wall separating the inside from the outside of the bubble to have a weaker tendency to shrink), creating a region that pinches off from the rest of the universe.
The tiny throat that initially still connects the two is a wormhole, which is unstable and will quickly collapse into nothing, leaving two disconnect spacetimes, the original parent universe and the tiny baby.
So, in quantum field theory, spacetime can not only bend or stretch (as in ordinary classical general relativity), but also split into multiple pieces. In particular, a tiny bit of space could branch off from a larger universe and go its own way. The separate bit of space is, naturally, known as a baby universe.
(Inspiration and extracts from Sean Carroll: From Eternity to Here)
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