Re: The fundamendalist constant

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Jack U.'s suggestion:

>        I was at a cosmology lecture today on the "omega lambda" (dark
> energy) measurements.  It occurred to me that since tonight is the
> beginning of the High Holy days, this week would be a good time for
> someone to calculate the value of lambda for which the age of the universe
> would be 4000 years.
>        I would label this number: "the first fundamentalist constant."
>                               L'shanah tovah,
>                                                Jack

I don't know of any fundamentalists who would hold to a 4000 year old
universe (creation at approximately the traditional time of Abraham).
Maybe you meant 4000 BCE for the creation?  This would make about a
6 ky old universe.  It seems that the year 5763 is dawning according
to the Jewish calendar (since I'm not Jewish I had to look it up on
Google).  This would presumably make the youngest fundamentalist age
for the universe to be something like 5763 years old.

We can calculate the required value for 'Omega_lambda' from the
GR-predicted formula for the age of the universe (assuming it
remained a sufficiently spatially uniform and isotropic pressureless
dust of galaxies throughout nearly all its history) for a Friedmann-
Robertson-Walker universe with a cosmological constant.  If we let
t_0 be the age of the universe, H_0 be the Hubble parameter, O_M be
the 'Omega' parameter for the mean density of all gravitating
ordinary and dark matter, and O_L be the 'Omega' parameter for the
cosmological constant (i.e. dark energy) lambda, then the age of the
universe can be found to be predicted by GR to be given by:

t_0*H_0 =

 =  Integral{x=0 to 1 | dx/sqrt(1 + O_M*(1/x - 1) - O_L*(1 - x^2))}

Analytically evaluating this integral is a real pain for arbitrary
values for O_M and O_L.  In the special case of a spatially flat
universe (which observationally seems to be the case) we must have
1 = O_M + O_L.  In that case the integral boils down to
(2/3)*arctanh(sqrt(O_L))/sqrt(O_L) when O_L is positive, and
simplifies to (2/3)*arctan(sqrt(|O_L|))/sqrt(|O_L|) when O_L is
negative.

If we take the reasonable value for H_0 to be about 70 km/s/Mpc
and take t_0 to be 5763 years then the LHS of the equation
containing the above integral becomes numerically equal to
4.127 x 10^(-7).  If we make the above assumption that the universe
is spatially flat we find that the required value for O_L is hugely
*negative*.  It comes out to O_L = -6.44 x 10^12 .  Since
O_L + O_M = 1 this means that O_M is hugely positive (i.e.
+6.44 x 10^12).  This means that such a situation would require that
the mean matter density of the universe would be over 13 orders of
magnitude greater than the value it observationally seems to be
(i.e. ~0.35 or so).  This above value of O_L corresponds to a value
of the Cosmological Constant lambda = 3*O_L*(H_0/c)^2 =
= -9.91 x 10^(-8) ly^(-2) which happens to be the negative reciprocal
square of a distance of only 3177 ly.  Notice that a flat universe
that is only thousands of years old requires that the cosmological
constant be the reciprocal square of thousands of light years.  (The
thousand year length/time scale has to come from somewhere.)

OTOH, suppose we relax the assumption that the universe is spatially
flat and instead allow it to have any amount of spatial curvature on
its largest length scales, but we instead require that the value of
O_M be a realistic number somewhat less than 1 (cosmologists think
it is probably somewhere around 1/3).  In this case the above
integral becomes quite complicated to do in closed form but it is
still possible to do it sufficiently accurately enough to get an
order of magnitude estimate for what O_L needs to be to give a
5763 year old universe.  It ends up that in this second case of
a spatially curved universe whose matter density has a realistic
value whose Big Bang was 5763 years ago requires that the value of
O_L *still* be hugely negative on the order of trillions.  This makes
the value of the Cosmological Constant have about the same order of
magnitude as it did before in the first case of a flat universe.  But
in this second case since now O_M + O_L is hugely negative (instead
of being exactly 1) this requires that the universe have a negative
spatial curvature (open hyperbolic) with a ridiculously small value
for the curvature length scale R of about 10^4 ly.  This would make
the size of the Milky Way galaxy about an order of magnitude larger
than this curvature length scale.  Needless to say this would cause
our observations of the heavens to suffer gross geometric distortions
even for distances well inside our own galaxy.

In any event it seems that a universe of a few thousands of years old
requires a value of O_L to be in the realm of negative trillions
rather than the observed value of about +2/3.  Needless to say, a
universe of only a few thousands of years in age is quite
incompatible with observations, with current theory (i.e. GR), and/or
both.

David Bowman
dbowman@georgetowncollege.edu