Since posting my last post calculating Jack's "first fundamentalist
constant" last night I have since thought of an addendum to add to
it. I had worked out the value of the Cosmological Constant Lambda
and 'Omega_lambda' (i.e. O_L in my ASCII notation) for the case of a
spatially flat universe whose Big Bang happened at the origin of the
Jewish calendar. I also had estimated the order of magnitude of
this in the case of a universe that wasn't spatially flat but,
instead, had a realistic concentration of gravitating matter. In
both cases the Cosmological Constant came out with about the same
order of magnitude (i.e. negative in the trillions). In the first
(flat) case the concentration of gravitating matter ended up being
over 13 orders of magnitude greater than the observed value. Whereas
in the second case the universe instead had the property that space
was so strongly negatively curved (i.e. had an infinite hyperbolic
geometry) that its curvature length scale was on the order of only
about 10^4 ly.
What I noticed after posting all this is what also happens to the
rate of acceleration/deceleration of the universe's cosmic expansion
in these crazy scenarios. Cosmologists use a parameter called q_0 to
dimensionlessly parameterize the rate at which the expansion of the
universe is speeding up or slowing down. It is called the
deceleration parameter and has a positive value if the universe's
expansion is slowing down, and has a negative value if the expansion
is speeding up. Up until early 1998 most cosmologists has assumed
that the value of q_0 was positive. It was a shock to them to
discover that its value seems to be (convincingly) observed to be
negative (with a value of about - 1/2) signifying a universe whose
expansion is accelerating. We can calculate the value of q_0 in
terms of the values of the Omega parameters for the gravitating
matter density (i.e. O_M) and the Cosmological Constant (aka dark
energy) (i.e. O_L). The result is: q_0 = O_M/2 - O_L. This means
that in both of my assumed scenarios the value of q_0 comes out to be
of the order of about +10^13. This means that in a universe obeying
*either* of these scenarios the expansion of the universe is
currently decelerating over 13 orders of magnitude faster than it is
observed to be actually accelerating its expansion.
Although I haven't worked out the complete dynamics of these
universes in detail to be certain, I still strongly suspect that in
both cases these universes have a Big Crunch singularity in the
future at around 10^4 yrs from now--even though their spatial
topologies are unbounded and infinite in extent (remember the
dynamics of a universe with a hugely negative Cosmological Constant
is not the same as for a universe with a Cosmological Constant that
is tiny in absolute value). Such a huge current deceleration in the
present means that rate of initial expansion near the Big Bang
singularity would be unimaginably rapid all the while throughout
history that expansion rate would be slowing down *extremely* fast so
that by today it is *still* decelerating at a ridiculously fast rate.
We can think of a negative Cosmological Constant as giving space
an innate attractive tendency to contract--even in the absense of
any actual matter that would tend to further increase this attractive
tendency, whereas a positive Cosmological Constant gives space an
innate tendency repel itself and cause it to expand and which tends
to counteract the tendency of any gravitating matter in it to cause
it to tend to mutually attact itself.