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## GR0877 Problem 20

PROBLEM STATEMENT: This problem is still being typed.

SOLUTION: (A) The temperature dependence of the radiation is $T_r \propto a^{-1}$, where $a$ is the cosmic scale factor. Thus, when the temperature was higher by a factor of 4 (12/3 = 4), the distances were four times smaller than today. See Misner, Thorne and Wheeler, “Gravitation”, Chapter 28.

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1. 11.03.2013 at 02:29

From an astronomy perspective this is a cosmological redshift: d_obs/d_emit = 1+z and z = (w_obs – w_emit)/w_obs, where d_obs and d_emit are the distance between objects at two different times and w_obs and w_emit are the wavelengths observed at those two times. From Wein’s law: wavelength = 10^(-3)/T. So for 3K, the peak of the blackbody is at 1 mm and for 12K the peak of the blackbody is at 0.25mm. So, if the observed values are for now (w_obs = 1mm) and the emitted values are for objects in the past (w_emit = 0.25 mm), then z = 3 and d_obs/d_emit = 4, so objects today are 4 times the distance they were in the past, or, objects in the past were 1/4 the distance that they are today.