Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Size of Universe



Regarding John Denker's comments:

At 08:06 AM 10/10/01 -0700, Wes Davis wrote:

I recently attended a lecture on GR.
I learned of recent data indicating that
the universe is flat. How does this affect
the size of the universe? Is the universe
finite, infinite, or is this not a sensible question?

There are several different questions in there.

First of all: Note that "flat" refers to intrinsic curvature. This is
the
only sort of curvature that makes sense when talking about the
universe...
but when visualizing models of the universe, you have to be careful not
to
be confused by any extrinsic curvature.

All true. But we probably ought to also mention that the curvature
or flatness being considered here is the intrinsic curvature of
*space* (i.e. the 3-d spacelike section of spacetime) at *each*
instant of cosmic time (i.e. the time parameter of comoving
coordinates). The curvature/flatness does *not* refer to all of
spacetime in this context. In particular, we can (& it seems we do)
have an expanding universe that is spatially flat. Yet the 4-d
spacetime of this universe is still (intrinsically) curved.
Spacetime as a whole would flat it the universe was not expanding or
contracting *and* its 3-d spatial sections were also flat.

In particular, take a flat piece
of paper. It really is flat, i.e. zero intrinsic curvature. Curl it
up
into a cylinder. It still has zero intrinsic curvature. Flatlanders
living on the cylinder can't measure anything non-flat about it, unless
they manage to circumnavigate it.

And even *then* what would be discovered is not so much a curvature
as a compactness of the spacelike geodesics in one or more directions
in space. Of course, we're assuming here (unphysically) that it
would be possible to instantaneously circumnavigate the spatial
universe so the relevant geodesics could be spacelike.

If we actually followed physical timelike or null geodesics in
spacetime in our spatial circumnavigation then we would have to
contend with the fact that *during* the circumnavigational voyage
the universe's circumference is increasing in time (in an expanding
universe) or decreasing in time (in a contracting universe). It is
even possible that the universe could be expanding so fast (and it
most probably *is*) that any timelike or null geodesic cannot
circumnavigate the compact dimensions because the circumference could
be expanding so much faster than c that the navigator could never
make any headway around the universe. It would sort of be like
trying to go up a down (looping Escheresque) escalator that was going
down faster than you could run up it. I guess maybe a better analogy
is that it would be like an ant trying to walk the long way around a
rubber band that is stretching much faster than the ant can walk. If
the stretching function in time is sufficiently rapid (such as the
asymptotic exponential growth of a truly constant Hubble constant
driven by a finite positive Cosmolgical Constant) then the ant can
*never* circumnavigate it as long as the rubber band's
circumference's initial expansion rate was faster than the ant's
limiting walking speed.

1) A flat universe could well be infinite.

2) A flat universe could be like a cylinder, or a torus, that has zero
intrinsic curvature everywhere but nevertheless closes back on itself.
We
would AFAICT be unable to distinguish this from case (1), unless we
managed
to circumnavigate the universe.

Actually, I suspect that we *could* distinguish such a situation.
This is because an intrinsically *flat* spatial universe that is
compact in one or more dimensions is *necessarily* *anisotropic*, and
this would violate the Copernican Cosmological Principle and that
presumably could have observable effects in our local neighborhood.
(If a spatial universe is both homogeneous *and* isotropic about
every point *and also* compact, then it must *not* be flat, as it
would then be a space of constant positive curvature, e.g. the 3-d
'surface' of a 4-d 'sphere'.) The universe could well be spatially
homogeneous, but still anisotropic about every spatial point. The
most symmetric situation we could have for a fully compact (in all
directions) intrinsically flat spatial universe is if it was a T^3
torus. This space has cubic symmetry--but *not* spherical symmetry
--about every point in space. In particular, the spatial
circumference about the universe is sqrt(3) times longer along a
'111' direction than along a '100' direction. It would presumably be
possible *in principle* to detect such anisotropies without actual
circumnavigation by observing the angular distribution of light from
distant galaxies in various directions. The different circumferences
in different directions would show up as an Olber's-type angular
modulation on the average amount of light coming in from various
directions since different directions have more radial depth of field
for the number distribution of the distant galaxies. We would expect
more light coming from more galaxies in the '111' directions than in
the '100' directions. Of course, this would only be possible to
observe this effect in this case if our observational horizon
distance was at least a significant fraction of a typical value for
the universe's circumference. Otherwise, we just couldn't see far
enough for the effect to begin to show up.

3) A lot of the time when people talk about infinite or non-infinite
universes, they are talking about the !time! duration not the spatial
size.

Consider a universe with enough mass to give itself a positive
curvature
(like a sphere) as opposed to having zero or negative curvature (like a
sombrero). According to standard theories, such a universe will have
enough gravity to arrest and reverse the Hubble expansion, leading to a
Big
Crunch at some finite time in the future.

So, if you buy the theory, a flat universe is special because it is
just
barely Crunch-free.

This is only if we can neglect the effect of any Cosmological
Constant. If the Cosmological Constant (also recently called 'dark
energy' for some reason) is turned on--as it *actually* seems to be
the case--then the universe can be spatially flat, and *still* *not*
be anywhere near a teetering condition for recollapse. Any finite
positive value for the Cosmological Constant will cause the universe
to expand *exponentially* fast in the asymptotic future as the Hubble
parameter will have a bounded finite positive future asymptotic value.
Recall that the Hubble parameter H is the logarithmic derivative in
time of the universe's spatial scale factor. If H settles down to a
finite constant then the universe settles on an exponential expansion
with a finite constant continuously compounded 'interest rate' for its
scale 'size'. If the Cosmological Constant was reduced (while
remaining positive) then this asympotic positive interest rate would,
likewise, be reduced, but it would *not* ever result in a negative
value needed for a recollapse.

If we an believe the recently measured numbers quoted in the
literature for the current Hubble parameter H_0, and the Cosmological
Constant OMEGA_lambda for our actual universe, then the universe will
have an asymptotic size "doubling time" in the distant future of
about 13 billion years. Such a universe is not "barely Crunch-free".

OTOH competing theories are a dime a dozen.

Maybe you considered a theory with a positive Cosmolgical Constant as
one of these cheap "competing theories"?

David Bowman
David_Bowman@georgetowncollege.edu