If you want to understand physics (astronomy) you have to use terminilogy
independent of any human point of view.
IMO Max Tegmark does not follow this idea when he defines Our Universe
in his book "The Mathematical Universe" as: (page 120)
The part of the physical reality we can in principle observe.
Phillip Helbig in his document:
http://www.astro.multivax.de:8000/he..._universe.html
writes: Tegmark uses "our universe" to denote that which is more commonly
described as the "observable universe"
Tegmark also writes at page 121:
"Our Universe contains about 10^11 galaxies etc. This is certainly a
lot of stuff, but could there exist even more, farther away in space?
As we saw, inflation predicts that there is"

Alan Guth in his book "The Inflationary Universe" writes at page 186:
"We find that the entire universe is expected to be at least 10^23
times larger than the observed universe"
Alan Guth also uses the term "presently observed universe" which is
even more "difficult".

IMO when you study science from human perspectif you make a mistake.
The problem is that the "observable universe" from some one
living at the Andromeda Galaxy or 1 billion ly away at present
is different from ours. Part is overlapping, while we all have
a common origin: the Big Bang.
IMO it only makes sense to talk about "Our Universe" meaning all
that is "created" by the Big Bang and that is described
by Friedmann's equation (and not use the term observable universe)
The R(t) in that equation should express the size of "Our Universe"
during its evolution and not what is observed.

The simulations of the friedmann equation as described in
http://users.telenet.be/nicvroom/fri...20equation.htm
show that the distance of the events at the observed lightray is
small relative to the radius of the Universe at these events.
(except for the earliest events)

When we call "Our Universe" homogeneous it should be related
based on the above definition.
The same when we discuss inflation.

What the simulation also shows that the initial distance (v0 * dt)
of the Universe with the first iteration has almost no influence
on the final distance at present.

To define parallel Universes within "Our Universe" as defined
above, IMO is not realistic.

That does not mean that there could not be more Big Bangs.
I do not want to exclude that. A such you can define the
concept of Our Big Bang implying that there are more,
which happened at different moments.
The implications are speculations.

Nicolaas Vroom