Re: Old Stars

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

Concerning the "Old Stars" thread I previously wrote:
> There is also another further "horizon", beyond which light from those regions
> beyond it can never reach us in the future (if the Hubble expansion rate
> settled down to a finite non-zero value in the asymptotically distant future)
> since those sources are so far away that the stretching of space is causing
> them to move away from us faster than c.  We are literally outrunning the
> light from them.  This would always be the case in an open hyperbolic space
> with not enough "dark matter" to close the universe.  In the borderline flat
> space case which just barely has enough matter to halt the Hubble expansion, I
> believe I recall, the expansion rate will slow down more and more allowing the
> light from regions which would have been forever beyond the second horizon to
> eventually catch up with us.  This is because a place in a region which is
> receding from us faster than c now will eventually be receding from us slower
> than c in the distant future, allowing the light from that place to eventually
> reach us.  Whether or not all places, no matter how distant, will eventually
> become causally connected in the flat space borderline universe is something
> that I am not sure about right now.  I would have to consult a GR book, and
> none of mine are handy right now as I am posting from home and am not in my
> office at the moment.

Well, since writing this I have had time to look up the point addressed above
about the existence of a "forever" horizon beyond which there can never be any
causal influence on our part of the universe (since light rays from "there"
could *never* be able to reach us--even in the distant future).  It ends up
that I was partly right and partly wrong. I therefore thought I should not let
a possibly misleading error go uncorrected and post the correction below.

What follows below is only for a uniform, pressureless, matter-dominated
universe of "dust" which obeys Einstein's field equations without a
"cosmological constant".  Also words like "here", "there", "places", etc.
refer to fixed points in space in *comoving coordinates*.

I was *right* when I supposed that the borderline flat space case would allow
any two regions separated in space at a given time to eventually come into
causal contact.  I was *wrong*, however, in saying that the infinite open
hyperbolic space case would have such a "forever" horizon.  Even in this case
arbitrarily distant places "there" can eventually come into causal contact
with "here" after a sufficiently long waiting time.  This is because, even in
this case, the expansion rate of the universe slows down enough in the future
for the propagating light rays to "catch up" to any place--no matter how
distant.  In the uniformly (in time) expanding universe of the steady-state
model there *is* such a "forever" horizon however.  It also ends up that in
the finite closed 3-sphere surface case of the BB model there is sort of a
1-sided "forever" horizon.  It is just that in this case the universe doesn't
have forever to exist to wait for light signals from a distant "there" to
reach "here" before the Big Crunch occurs--and even in this case--the light
signal needs to go around the universe "the long way around".  In this model
if at some point in time two different places are sufficiently separated in
space the universe will recollapse in the Big Crunch before enough time
elapses for them to come into causal contact *if* that contact is via a light
ray taking the longest geodesic around the universe.  If the causal influences
always take the shortest geodesic path then *any* two places can be in causal
contact in the future as long as the influence (light ray) starts out from one
"place" to the other sometime during the *expansion* phase of the universe.
Also, related to this point, in this case, there is not enough time for a
light ray to "orbit around the circumference of the universe" before the Big
Crunch hits if it starts out *after* the BB.  If a light ray starts out in
some direction *at* the initial Big Bang singularity in time, it will just be
reaching the antipodal point of its starting point on the "other side" of the
universe just when the universe has reached its maximum "size".  As the
universe begins contracting the ray continues on around its Great Circle
geodesic and returns to its starting point just as the Big Crunch is occuring.
Thus the lifetime of the universe is just barely long enough for one complete
orbit *if* that orbit was started immediately at the BB.

This above orbital timing discussion assumes that the universe remains matter-
dominated throughout its history.  This is a bad approximation for times
sufficiently close to the initial BB and final BC singularities that the
temperature is so high that the universe is radiation-dominated.  The matter-
dominated approximation only breaks down for a very tiny fraction of the
universe's history however.  It should be noted that effects that happen 
during the very early history of the universe can have profound cosmological
consequences for the subsequent development of the universe.  For instance,
according to current modern theory there was an "inflationary era" from about
10^-35 to 10^-32 sec after the BB which effectively changed the subsequent
geometry of the universe to the borderline flat case--even if it was quite
different than this geometry before this time.

I hope this (sort of) clears things up.

David Bowman
dbowman@gtc.georgetown.ky.us