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Old January 2nd 16, 09:01 AM posted to sci.astro.research,sci.physics.research
Phillip Helbig (undress to reply)[_2_]
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Default mathematical cosmology: general interpretation of alpha

In arXiv:astro-ph/0404319, Lake introduces a quantity called alpha and
points out that it essentially measures the ratio of the value of the
cosmological constant to that in Einstein's static universe (or those
asymptotic to it). This works only for k=+1 and lambda0. For k=+1,
there is another interpretation as well: it is proportional to the
product of the mass of the universe and the value of the cosmological
constant. (This conclusion is present, but not very explicit, at the
URL mentioned in reference [12] in Lake's paper.)

The interesting thing about alpha is that it is a combination of
parameters each of which, in general, is not constant during the
evolution of the universe, but the combination is constant. If one
thinks of the cosmological parameters Omega and lambda as constituting a
dynamical system, this is a constant of motion.

What about other cases, i.e. k=-1? Is there a physical interpretation
for this quantity? (See reference [14] in Lake's paper.)

Lake's paper is rather terse, but contains all the necessary
information. Some background, more than enough, is provided by the
references. In particular, [8] and [12] should be read by everyone even
remotely interested in such topics. (The notation is not uniform.
Sometimes it is just a simple change of variables or different
coefficients depending on the system of units used (so the same thing
denoted by different (combinations of) symbols). It becomes more
confusing when two people use the same symbol to denote different
things.)

If the universe collapses in the future, one can calculate the maximum
size of the scale factor from the values of the cosmological parameters
at any time. Obviously, along a trajectory in the Omega-lambda plane
representing the evolution of a cosmological model, this maximum value
is constant. However, there seems to be no simple relation between this
"constant of motion" and alpha. Or is there?

Note that there are models with k=+1 which collapse in the future, so in
these cases both constants of motion are present, though again without
any obvious connection.

It is no problem to calculate all the interesting quantities. The
question is whether there is any relation between these two constants of
motion and whether either or both can be generalized to the case k=-1.
Even better would be a generalization including the special cases
lambda=0 and Omega=0 (for which alpha=0) and k=0 (for which alpha is
infinite).