In article , Nicolaas Vroom
writes:

At page 120 Max Tegmark writes:
Our Universe: The spherical region of space from which light has had time
to reach us during the 14 billion years since our Big Bang - basically
this:

Right; this is his definition (what some call the "observable
universe").

Beside that there is sphere from the image of the front page of his book:
See: http://space.mit.edu/home/tegmark/mathematical.html

The problem is that the radius of this sphere at the time of the creation
of the CMB 300000 years after the BB was a very small.

Right. However, what Tegmark is referring to is the proper distance NOW
of the most distant objects we can see, not the distance when the
radiation we detect now was emitted. (Of course, in co-moving
coordinates there is no distinction.)

7 b years after the BB the radius was roughly 5 b lightyears.
At present the radius is again very small.

It doesn't shrink in an expanding universe, neither in co-moving nor in
proper distance. What you are thinking of is that there is a maximum
distance AT THE TIME OF EMISSION. Yes, that's true. Don't confuse this
with the proper distance NOW.

We can only observe a tiny
bit of Our universe (All what is created after the BB) at present.

It depends on what you mean by "at present". We observe events on our
backward light-cone (and have other evidence for events within it). The
CURRENT radius of the observable universe is measured in dozens of
light-years (more than the 13.7 which is the age of the universe---that
is the light-travel time, but expansion increases the distance NOW,
though of course we cannot observe EVENTS, only OBJECTS, which are NOW
at that distance). If the radius of curvature of the universe is much
larger than the Hubble radius (as seems to be the case), then it is
indeed the case that the universe is much larger (perhaps infinitely
larger) than the observable universe. (Note that, in general, the
Hubble sphere does not correspond to any sort of horizon and so, in
general, is not equivalent to the observable universe. However, except
for rather special cosmological models, it is larger or smaller by at
most an order of magnitude, so is useful as a ball-park figure.)

Mak Tegmark also calls this our observable Universe.
IMO that name is correct and is mainly used to study Supernovae data
and to calculate the cosmological parameters.

I'm not sure why you are dwelling on the supernovae data here.
Obviously, all our empirical information about the universe comes from
the observable universe. This includes CMB, galaxy surveys, QSOs etc
and not just supernovae.

To call this Our Universe is wrong because IMO that name should refer
to all what is created after the Big Bang and is described by the
Friedmann equation which Radius roughly speaking increases lineair with time

Again, different authors have different terminologies. Tegmark is not
the first to use a different terminology. As long as it is consistent,
it is just a matter of taste. On a similar not, some authors have a
dimensionless scale factor, some don't. However, all agree when
discussing actually observable quantities.

As such our observable Universe is much more a subset of Our Universe.
In Tegmark his terminology:
Our Universe is much more a subset of Level 1 multiverse.

Right. I would say "observable universe" and "universe" for his
"universe" and "Level I multiverse". There is no confusion about
concepts, though---only different terminology.

The important part is that this is direct consequence of the Big Bang and
expanding space concept and has almost nothing to do with the inflation theory

Right.