From: ted@rosencrantz.stcloudstate.edu
Subject: Re: Expansion of the Universe and Ionization
Date: 16 Aug 2000 21:56:16 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <200008161948.PAA24550@bbn.com>,
Thomas Colthurst  <thomasc@bbn.com> wrote:

>The mass ( ~ 10^51 kg ) of this non-evaporating black hole is curiously
>close to the mass estimates ( ~ 10^52 kg, e.g. ) one occasionally sees
>for the observable universe.  Is this near identity just a consequence
>of the same sort of dimensional analysis that Ted Bunn used to get his
>rough estimate, or are more hoaky anthropic arguments necessary to
>explain it?

It's just the miracle of dimensional analysis, combined with the
assumption that the matter density is close (in order of magnitude) to
the critical density.

Let's use units where c = hbar = k = 1 and all dimensionful quantities
are expressed in mass units: [Energy] = [Mass] = 1/[Length] = 1/[Time]
= [Temperature].  Then the Unruh temperature is equal to the Hubble
constant H (in order of magnitude), and the temperature of a black
hole of mass M is Mp^2 / M, where Mp is the Planck mass, so a
non-evaporating black hole has mass Mp^2 / H.

The mass of the observable Universe, on the other hand, is of order
rho / H^3, where rho is the density.  The critical density is of order
H^2 Mp^2, so the mass of the observable Universe is of order Mp^2 / H.

Looking up some numbers in the ever-so-useful Appendix A of Kolb and
Turner's "The Early Universe," we get Mp = 1.22 x 10^19 GeV and H ~
10^(-42) GeV, so

Mp^2 / H ~ 10^80 GeV ~ 10^53 kg.

In reality, the matter density is down by an order of magnitude from
the critical density, so this number agrees quite well with your
estimate of 10^52 kg for the mass of the observable Universe.
I'm not sure why the mass of the non-evaporating black hole came out
a couple of orders of magnitude smaller, but we were being sufficiently
sloppy with the calculations at every stage that I'm not too worried
about it.  

(I'm sure it's still Mp^2/H in order of magnitude.  After all, the
only dimensionful things it can depend on are Mp and H, so it's got to
be Mp times some power of Mp/H.  If that power were anything other
than one, the numerical value would be much more than one or two
orders of magnitude off from 10^53 kg!)

-Ted