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Tucker Hiatt wrote:
...
How is our
universe -- with, as it seems, an infinite amount of space --
consistent with a big bang?
This is tricky business. I will answer in fairly
formal terms, because that is the only way I know
of being confident of the answer.
Formally, the words "infinite" or "infinity" are
shorthand, referring to a limit.
We shouldn't
speak of N=infinity, but rather the limit as
N->infinity. Specifically, the infinite universe
is the limit of a sequence of universes of size N,
in the limit as N grows without bound.
The problem with the origin of the infinite universe
arises only when we play the movie backwards and take
the limit as t->0. Another limit.
We know that it is not possible in general to
interchange the order of limits, not without changing
the meaning of the expression.
So suppose our universe consists of a set of
particles, with a neighbor-to-neighbor spacing
that grows in proportion to time, i.e. k*t.
For N particles, the total extent is N*k*t.
So let us consider the expression
lim(N->infinity) lim(t->0) N k t.
This is zero. In particular,
lim(t->0) N k t
is zero, and then when we take the other limit
it is just
lim(N->infinity) 0,
which is manifestly zero.
Next, let us contrast that with the taking the
limits in the other order, namely
lim(t->0) lim(N->infinity) N k t.
This is infinite. In particular,
lim(N->infinity) N k t
is infinite for any nonzero t, so when we take
the other limit it is just
lim(t->0) infinity,
which is manifestly infinite.
I believe the latter expression is the correct
way to think about the past history of the
infinite universe. At every past moment right
up to but not including t=0, the universe was
infinite. No matter how much you shrink it,
there's always more. Infinity divided by two
is still infinity. Infinity divided by a million
is still infinity.
If you start out by assuming that the universe
had size=0 at t=0 and ask yourself what sort
of expansion factor you need to get from 0
to infinity, it would appear that you need an
expansion factor of infinity squared or something
like that. But wait a minute! What is the
evidence that the universe had size=0 at t=0?
I don't think there is any such evidence. All
we have is evidence that the universe had pretty
high density and pretty high temperature back when
t was small, but more than that we cannot say.
AFAIK the evidence is perfectly compatible with
the notion that the high-temperature high-density
universe was already infinite in extent.
If anybody has evidence to the contrary, please
explain.