Re: Cosmology

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Bob Zannelli's comments:

> When we talk about R in our universe aren't we really talking about the scale
> factor or if you like radius of the observable Universe.

Not me. That is not what I meant by the symbol 'R' in the context I used
it.  Although it *is* traditional to use the symbol 'R' to denote the
curvature length of the universe's spatial sections when the topic of
discussion involves the universe's large scale spatial curvature a la the
Friedmann-Robertson-Walker-Lemaitre models, my use of the symbol was a
much more pedestrial one.

When I had used the symbol 'R' in my post objecting to John Denker's
claim that there was "nothing non-Newtonian" about the CC I was
discussing a universe that was spatially *flat* (i.e. its spatial
curvature length was infinite) and had only one or two masses of any
major significance in the *Newtonian limit* of slow speeds and weak
gravitational potentials. The meaning of "R" in that context was merely
the distance between the masses or the distance from an isolated mass to
a given spatial origin.  What I was discussing was the fact that a free
isolated mass will accelerate away (in the *absence* of any gravitational
effects from *anything else* in the universe *except* the CC) from that
origin according to a = B*R where B is proportional to the CC (i.e.
B = (1/3)*[LAMBDA]*c^2) and R is the proper distance of the mass from the
origin.  If the CC is zero then there is no such repulsive term appearing
in the Newtonian limit of GR, and only in this latter case does the
theory really boil down to Newton's theory in the nonrelativistic limit
of the relevant speeds being slow compared to c and relevant
gravitational potentials being small compared to c^2.

>   Would it not also be true, that assuming the cosmological constant is
> really some positive value, it would mean that each and every unbound
> cosmological structure would be accelerating away from each other at the same
> exact rate.

Not really.  The amount of the acceleration would depend on the distance
the unbound structures were from each other, the masses of those
structures, and the mean density of any background matter that might
fill the space between and around the structures.

> This acceleration would continue to increase because of the
> increase of the scale factor and the decrease of attractive component of the
> gravitational field as the average distance between the unbound structures
> increased.

The direction of the change in the scale factor is a separate issue.
Even if the universe was decelerating the scale factor would continue to
increase as the universe expanded (albeit at an ever slower rate).  If
you toss a rock up into the air it accelerates downward even when it is
ascending.

Nevertheless, it is true that in an expanding universe with a positive
CC the rate of expansion picks up as the spatial dilution of the
matter in it weakens any tendency to decelerate (in opposition to the
cosmic repulsion) and try to contract the universe because of the mutual
gravitation of that matter that is present.  In a universe with a
positive CC and which is spatially flat there will asymptotically (at
long times) result an expansion that eventually is governed by a constant
*in time* nonzero Hubble parameter, which has the spatial scale factor of
the universe increase exponentially with time.  This asymptotic behavior
is a version of an *inflating* universe, albeit one with a very slow
inflation rate (determined by the bare CC).  In this case the asymptotic
Hubble constant (being the logarithmic derivative of the spatial scale
factor) is the 'interest rate' for the 'size' of the universe which is
'compounding' continuously.  In the earlier times (such as now) before
the asymptotic behavior sets in, the Hubble constant varies in time
(because of a variable amount of mean matter density causing a variable
amount of mutual gravitational attraction) and that gives the universe a
'variable interest rate' for its growth.  Once the mean matter density
becomes so diluted that its gravitational effect is negligible compared
to the effect of the CC, then the asymptotic value of the Hubble
constant will have effectively been reached.

*If* the universe actually has the parameters I mentioned before (i.e.
spatially flat, H_0 = 65 km/s/Mpc, [OMEGA]_m = 0.35,
[OMEGA]_lambda = 0.65) and *if* GR is correct theory for gravitation,
then in the asymptotic future the value for the Hubble constant will
be 52.4 km/s/Mpc corresponding to a spatial doubling time of the 'size'
of the universe (i.e. scale factor) of 12.9 billion years.  Also, if we
take these parameters at face value this means that the universe is
currently 13.9 Gy old, and when it was 8.2 Gy old (some 5.7 billion years
ago) its scale factor crossed an inflection point.  Earlier than this
time (but well *after* any possible inflationary epoch) the universe had
a large expansion rate which decreased with time because of the mutual
gravitation of its constituent matter.  This rate of deceleration went to
zero at the inflection point, and after this time the universe has been
ever since *accelerating* its expansion rate.  The universe is now
accelerating its expansion rate because its matter is now diluted enough
that the effect of the cosmological constant is more significant than
that of the gravitating matter in the universe.

>  This line of thought would eliminate the need for any arbitrary point of
> origin which really has no meaning in the sense that it can be located at
> some particular place in space.

Whatever.

David Bowman
David_Bowman@georgetowncollege.edu