Re: Old Stars/Olber's Paradox

[Bowman_David/tiger_mpc@tiger.gtc.georgetown.ky.us]

Jim G. wrote:

> David, we are talking about background photons here -- from the photon
> period -- when the Universe was something like the size of a basketball.
> How could it be that those photons haven't "filled the Universe"???  I don't
> understand you here.

These photons were emitted at the "decoupling time" when the temperature
cooled enough to form neutral atoms.  This time was on the order of about
300 kyr or 500 kyr since the BB.  The size of the universe only was some 10^3
times "smaller" then than it is now.  The universe today is *much* bigger than
10^3 basketballs large.

> Faster than c?!!!  Now I really need help.  Of course the stuff doesn't move
> at >c, but how is it that space expands >c?  Is this what you are saying?

Yes, faster than c.  This is what I am saying.  In a (spatially) uniformly
expanding universe the rate of separation of two different fixed (fixed in
comoving coordinates, that is) points in space is proportional to the
distance between them.  Thus, for sufficiently distant points their rate of
separation must be greater than c.  For illustration purposes consider the
steady-state model which has a (truly) constant Hubble Constant H_0.  Big Bang
cosmologies have the extra complication of a time-dependent expansion rate
which is slowing down, and possibly, (i.e. for a finite closed universe)
reversing itself.  In this steadily uniformly expanding universe consider two 
different points in space at the (cosmic standard) time t_0.  Let their proper
separation in space at this time be X_0.  Let H_0 be the Hubble Constant.
Then at time t the proper spatial separation between these points is given as:
X(t) = X_0 exp(H_0(t - t_0)).  These points are farther apart in the future
and closer together in the past.  The rate of change of their proper spatial
separation (effective velocity of recession) is dX(t)/dt = H_0 X(t).  Notice
that for large enough X(t) this rate is bigger than c.  This cosmology is
weird in that *any* two different points in space (beyond the local
gravitational influences of individual galaxies) will eventually be receding
from each other faster than c in the future and will therefore become
causually disconnected.

Big Bang cosmologies have this effect modified by a time-dependent Hubble
"Constant".  In the earliest stages of the Hubble expansion, when the universe
was *very* small the expansion rate was much faster than it is today.  This
made it easy for different (even relativly close) parts of the universe to
recede from each other faster than c.  As time went on the relative expansion
rate (the Hubble Constant) decreased so that regions which were previously
separating from each other faster than c before are now receding a a much
slower rate.  This slowing down of the expansion allows the old light from
those regions to eventually catch up with us.  This is why we just now can
see galaxies which were formed 10^9 yr after the BB and why now we can see
old cosmic background photons which were emitted from distant sources some
400 kyr after the BB.  We have slowed down our recession and allowed their
light to finally catch up with us.   Try studying the numbers I gave for the
4 different BB senarios.  (Remember that some of these numbers would need to
be modified somewhat in actual application to our universe because the
calculations that gave rise to them assume that the universe was always a
matter-dominated dust and ignored the effects in the *very* early universe
caused by it having a different equation of state (for a hot radiation-
dominated fluid) at that very early time.)

> What is the evidence for this?

Well, for one, Olber's paradox is not a problem (mechanism 2)).  Also the
cosmic background is quite uniform--so uniform that photons coming to us
from opposite directions in space have the same temperature to a high degree
of accuracy.  But since we are half way between the sources that emitted these
photons, they (the photons) have not had enough time to reach each other's
source.  Their uniform temperature seems to indicate that those regions of
space should have been in causal contact in order to equilibrate to a common
temperature.  Thus apparently those regions were in causal contact in the
past but since that time they have separated from each other so fast that they
have lost causal contact with each other and in the time since the decoupling
time between matter and radiation their photons have just barely had half the
time necessary to reach each other.  This phenomenon is naturally explained
by the inflationary model which has the universe inflating exponentially 
*very* rapidly in the time between 10-35 sec and 10-32 sec after the BB,
so that regions which were previously in intimate contact lost this contact
as they rapidly receded from each other faster than c.  It is just now that
the expansion has slowed enough, and enough time has elapsed for their photons
emitted at the decoupling time to reach half way to each other.  I'm sure
there must be other evidence for this effect but I can't think of it at the
moment.

David Bowman
dbowman@gtc.georgtown.ky.us