🚀 go-pugleaf

In article <1992Aug4.183334.20946@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>In article <92216.155158DOCTORJ@SLACVM.SLAC.STANFORD.EDU> DOCTORJ@SLACVM.SLAC.STANFORD.EDU (Jon J Thaler) writes:
>>jbaez@riesz.mit.edu (John C. Baez) says:
>
>>If one is clever, one notices that the dimensionality of the Hubble
>>constant is inverse time.  In fact, in a big bang cosmology it tells you
>>to within a numerical factor the age of the universe.  
>
>Thanks.  I wasn't being clever!!!  So the Hubble constant times the age
>of the universe is a bit more "constant" (with time) than Hubble's
>constant, but deceleration means even it's not constant.  Let's call it
>the Hubble variable, how about.
>

In a couple of special cases, the quantity H_0 times t_0 (where H_0
is the Hubble parameter and t_0 is the age of the Universe) is constant.
There is a so-called "critical density" of the Universe, which is
defined to be the mass density the Universe would have to have in
order that the gravitational attraction be strong enough to stop the
expansion.  If the density of the Universe is much less than this 
critical density, then H_0 t_0 = 1.  (This is just the case of no
deceleration.)  If the density of the Universe exactly equals the
critical density, then H_0 t_0 = 2/3.  In all other cases you're
right that this quantity is not constant.

Note that the quantity H_0 t_0 measures something very different
from H_0.  H_0 is the expansion rate of the Universe, whereas
H_0 t_0 is largely independent of that rate.  (In the special cases
above, it is completely independent of it.)  

By the way, the notation for all of these quantities does have some
logic to it, which I haven't really been following.  
The subscript 0 generally refers to the present time, so that the
statement "H_0 t_0 is constant in time" really should just be "Ht is
constant in time."  The symbol H refers to the Hubble parameter as
a function of time, while H_0 refers to its present value.

One more thing:  The ratio of the actual density of the Universe
to the critical density is denoted by a capital Omega.  The symbol
Omega_0 refers to its value at the present time.  So the two
special cases I described are Omega << 1 and Omega = 1.  The quantity
Omega is an extremely useful number to know about the Universe, so
you'll see it mentioned a lot if you read about cosmology.
There are theoretical reasons for suspecting that Omega=1, but
direct measurements of the density (which is a tricky thing to do)
imply that Omega is about 0.1 or 0.2.  That's why people talk about 
the existence of large amounts of "dark matter," to bring up the
density to the critical density.

-Ted
(ted@physics.berkeley.edu)