a) If (1+z)/T = constant regardless of z, and at z = 0 (the present day)
the temperature of the CMB is 2.7 K, then we can set up a ratio:

    (1 + 0)   (1 + 2.58)
    ------- = ----------  and thus T = (2.7 * 3.58) = 9.7 K
     2.7 K        T

Using Wien's law, (lambda) * T = 2.9 * 10^6 nm K, the peak wavelength is

  (lambda) = (2.9 * 10^6 nm K )/(9.7 K) = 3.0 * 10^5 nm.

This wavelength is still within the microwave radiation range, which we have
said is roughly 10^5 nm to 10^7 nm.


(b) We use Wien's law to compute the temperature where (lambda) = 700 nm.

  (700 nm) = (2.9 * 10^6 nm K)/T   so  T = 4100 K.

This occurred at a redshift where (1+z)/(4100 K)  = 1/(2.7 K)

  So (1+z) = 4100/2.7 = 1500

(c) At z = 1100, we can use the same ratio as in Parts (a) and (b) to say

    (1 + 0)   (1 + 1100)
    ------- = ----------  and thus T = 2.7 * 1101 = 2970 K (or about 3000 K)
     2.7 K        T


Curiously, cool main-sequence and red-giant stars, known as M-stars, have
surface temperatures of approximately 3000 K.  Is there a physical relationship
between the temperatures of the coolest stars and the temperature of the
universe at matter-energy decoupling?  

Usually,  the explanation given for decoupling (or "recombination") happening
when it does is that the ambient temperature of the universe has dropped to
the point where electrons and protons can come together without being knocked
apart again by hot photons flying all around.  This is often the cosmic epoch
where, it is said, the universe becomes "transparent," that photons can now
fly free of matter (and vice versa).  Coincidentally, the surface of a star is
where photons inside the star work their way free of the matter in the star,
and thus where the photons fly free.  So, perhaps, 3000 K is a "special"
temperature where photons can fly free?  It makes sense...

...except that in many stars, photons leaving the surface "fly free" at 
temperatures higher than 3000 K.  Our Sun, for example, has a surface 
temperature of 5800 K; O-stars are many times hotter at their surfaces.
So temperature alone must NOT be why decoupling occurs at z = 1100.  In fact,
graduate-level cosmology textbooks explain that the photon number density --
the number of photons per cubic meter -- that, combined with dropping 
temperature, tips the balance.  At z = 1100, the universe has expanded to
such a size that there aren't enough photons zipping around in any given
cubic meter of space to ionize the matter in that space.  So yes, the high
temperature does matter, but there was another equally important factor at
work in the early universe - expansion - which led to the decoupling of
matter and energy.  According to theory, decoupling begins to happen very
slowly at about z = 1600, is in full swing by z = 1500, is mostly complete
at z = 1100, and is all done by z = 1000.

Whether or not the answer links stellar matter with early-universe matter.
the "real" answer, as far as we know, is rather complicated.  This
particular question is an example of the kind of reasoning astronomers (and
all scientists, for that matter) must use and explore as we seek answers
to the mysteries that surround us.