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Re: The fundamendalist constant



Regarding Brian W.'s shameless request:

...
Still, since I have no shame and will not be taken aback by a null
response, let me ask what would be the effect
of a universal age of 1.2E13 years (i.e. about 1000X the age
currently estimated) and assuming other parameters which do
not place the present on an extremum of any kind.

It is possible to find the answer to Brian's request. If we take the
first scenario where we consider a (FRWL) universe that is taken a
priori to be spatially flat and for which the Hubble parameter (i.e.
current expansion rate) has a realistic value such as 70 km/s/MPC we
find that it is possible to have such a universe be arbitrarily old
by just requiring the matter density of the universe to be
infinitesimally small and having the Cosmological Constant be such
that the O_L value is infinitesimally less than 1. Such a universe
is very close to an eternal perpetually exponentially expanding
de Sitter universe. In the case of the 12 trillion year old
universe that Brian suggests above we find that we get a ridiculously
small value for the mean matter density. The 'Omega_M' value for the
matter in this case ends up being about O_M = 1.268 x 10^(-1119).
Yes, that's a -1119 in the exponent. This is one *really* rarified
universe. This universe has an O_L value of 1 - O_M which is
*really* close to exactly 1. The deceleration parameter q_0 for this
universe is only 1.902 x 10^(-1119) larger than exactly -1. If it
*was* exactly -1 it would be precisely exponentially expanding
forever with an e-folding time of about 14.0 Gyr and a doubling time
of about 9.68 Gyr. Needless to say, the amount of known accounted
for matter in the universe rules out such a tiny value for O_M.

If we consider our second scenario where we relax the requirement
that the universe be spatially flat and, instead, require that the
matter density have a realistic value (say by having O_M = 1/3) we
find that, again, it is possible to have the universe arbitrarily
old by sufficiently carefully fine tuning the value of O_L against
the presumed value of O_M. In this second case we find that we can
get a universe that is many times older (e.g. 1000 times older) than
the Hubble time (i.e. than 1/H_0) by taking O_L = 1.76547833939...
(for O_M = 1/3 exactly). In this case we get a deceleration
parameter of q_0 = -1.59791 corresponding to a universe whose
expansion is accelerating about 3.2 times faster than is currently
observed to be the case. If such an acceleration rate was to be
sustained over a very long time the universe would be expanding
*faster* than exponentially. This second case scenario has the
spatial universe positively curved into the 3-D 'surface' of a
hypersphere in embedded in 4 dimensions. The current radius for such
a 3-sphere 'surface' is 13.328 Gly (the radial direction for this
'sphere' is into the inaccessible 4th spatial dimension).

BTW, in case anyone is wondering what the meanings of O_M and O_L
are, they are convenient dimensionless measures of the mean density
of gravitating matter (including any dark matter) and of the
Cosmological Constant (aka dark energy) respectively. O_M is the
mean matter density measured in units of the critical density that
would be required have the universe be flat *and* to still expand at
the same rate it is observed to be expanding now (i.e. have the
proper value for H_0) in the *absense* of any nonzero value for the
Cosmological Constant. OTOH, O_L is the value of the Cosmological
Constant measured in units of the critical value it would have to
have to make the universe be flat *and* to expand at the currently
observed rate but without *any* gravitating matter in it (i.e. a
vacuum universe). Because of the definitions of O_M and of O_L and
because how they enter into the dynamical Friedman equation for the
universe it ends up that any spatially flat universe will always have
O_M + O_L = 1.

If some parameter becomes evidently untenable for these conditions,
I would like to think that this parameter is one that needs further study!

So are you hoping that only one such parameter becomes untenable?
Why? And why would you think it would require further study if
having a 12 trillion year old universe makes such a parameter
untenable? Do you have some other prior reason for wanting the
universe to be 12 trillion years old? If so, what is that reason?

Dave Bowman
dbowman@georgetowncollege.edu