In article , Aidan Karley
.group writes:

A discussion elsewhere lead me to wonder what the temperature
of the CMB was at about the time that Jupiter was forming. (I was
wondering how the different background temperature would affect the
rate at which the initial gravitational infall energy was radiated.)
I'd guess that the temperature is declining exponentially,

Why? Exponential decline of temperature is associated with flow of heat
from a warmer to a cooler body (IIRC this is known as "Newton's law of
cooling"), which is not the case here.

so I
should be able to work out an expression from the decomposition
temperature of hydrogen (ca. 5,000 K ? ), the present age of the
universe (13.7 Gyr) and the present CMB temperature (3.7 K). But it
would only be a guess.

What is the relation between the decomposition temperature of hydrogen
and the formation of Jupiter? Jupiter didn't form as soon as hydrogen
could exist.

Does anyone have an expression that would give the CMB's
temperature as a function of time - either time after the hydrogen
recombination/ big bang, or back from the present day? Doesn't have to
be too accurate - give or take a factor of two would be fine.

The temperature is inversely proportional to the scale factor. Thus, if
the universe doubles in size, the temperature is half of what it was.
If you know the epoch at which Jupiter was being formed, you can then
work out the increase in the scale factor since then, and from that can
easily calculate the temperature.

The difficult part is the increase of scale factor with time. This
depends on the cosmological parameters. However, they are relatively
well known.

Ned Wright's cosmology calculator might be interesting to look at.

For some theory on how the temperature of the CMB evolves, see the first
part of chapter 8 of Berry's COSMOLOGY AND GRAVITATION.

Interestingly, since spectra are influenced by the ambient temperature,
one can actually OBSERVE what the temperature was at high redshift,
rather than just calculating it.

The present temperature is approximately 2.7, not 3.7.

Very roughly, Jupiter formed when the universe was about 2/3 of its
present age. Since then, the rate of expansion has, on average, been
roughly linear (to a first approximation). So, roughly, 4 K.