Re: Temp of Universe (was Definitions of Temperature)

[DSCHROEDER@cc.weber.edu]

James says:

> This reminds me of a problem given us in graduate stat. mech., on which I'd
> be interested to hear opinions.
>
> The problem was "What is the temperature of the universe?"  There were
> probably simplifying assumptions allowed, but I don't remember them, and
> they are important for the following.
>
> The "correct" (i.e., the one given by the grader) answer was that a
> temperature cannot be defined for the universe, because there is no object
> outside the universe with which you could check to see how willing the
> universe is to give up energy.
>
> This always bothered me.  Once you've derived a relation between energy and
> temperature, why isn't the temperature of the universe given by that
> relation (under simplifying assumptions and knowing the appropriate
> energies)?
>
> So I ask:  do you think that the universe has a temperature?

Well, it is kindof a vague question.  But I think we could refine it
enough to come up with an answer.  In cosmology it's customary to
assume that on large scales the universe is homogeneous.  So since
temperature is an intensive quantity, let's just ask about the
temperature of some large chunk of the universe and if we get an
answer that will be the "temperature of the universe".  I should
point out, though, that I don't necessarily buy into the idea that
the universe is the same everywhere; there could be other regions 
of the universe far beyond our "horizon" that are very different
from the parts we see.  So maybe we should only ask about the
"temperature of the observable universe".

The next problem we encounter is that the observable universe is
not in thermal equilibrium.  It's full of stars at all sorts of high
temperatures, and clouds of gas at lower temperatures, and a background
radiation of photons at 2.7 K, and a background radiation of neutrinos
at 2.0 K, and dark matter at unknown temperature(s).  However, the
ordinary matter contributes an utterly negligible fraction of the
entropy of the universe, while the neutrinos have such low energies 
that nobody even knows how to detect them, and the dark matter hasn't
been detected directly either, so if you force me to give you a number
for the temperature of the universe, I guess I'll pick the photon
temperature, 2.7 K.  I'll pick it despite the fact that there's no
longer any mechanism to keep the photons in equilibrium with each other
or with anything else (as they've been traveling freely through space,
redshifting all the way, since the temperature was 3000 K).  I'll pick
it because it can be measured, in a variety of ways, no matter where
you are in the universe (well, maybe not at the center of the sun).
For instance, there's the famous measurement from the 1940's based
on the rotational states of CN molecules in cold interstellar space,
from which astronomers concluded that something was keeping these
molecules in thermal equilibrium at a temperature of roughly 3 K.
In fact, if you take any chunk of ordinary matter and leave it in
interstellar space (i.e., at a randomly chosen location in the universe),
chances are it'll eventually come to an equilibrium temperature of 2.7 K.
So that's my choice for the temperature of the universe, based on an
ad hoc operational definition, subject to all the caveats mentioned.

The main thing that makes me queasy about all this is that while the
photons can exchange energy with other objects, they don't regain
equilibrium among themselves after doing so.  Thus we can look at
how readily the gas of photons will give up or accept energy, but
I can't think of a meaningful way to ask how the entropy of the
photon gas changes after such an exchange, since afterwards it will
no longer be in internal equilibrium.  I guess the reason that this
isn't a problem operationally is that the photon reservoir is so
huge, we can't really disturb it significantly no matter how much
energy we add or remove.

Well, those are my thoughts.  What do you think?

Dan