Assignment 3
AST121H
Due: March 16, 2022
You may work together with other students to solve these problem sets, but all solutions must
be written and submitted independently. Show your work clearly for full marks, including in-
termediate steps, and diagrams if necessary! Check the syllabus for reading recommendations.
Include units!
Problem 1:
[6 Points] For every baryon in the Universe, there are about 109 photons. The ratio of photons
to baryons has been constant since a few seconds after the big bang. This is a crucial number
that basically sets the stage for much of the future evolution of the Universe. If the number
was just a little different, the Universe would be a very different place, and perhaps life could
not exist. In this question we will use the photon-to-baryon ratio to work out the redshift at
which the Universe becomes dominated by matter, instead of by radiation.
Assume that the vast majority of the photons in the present Universe are cosmic microwave
radiation photons that are a relic of the big bang. (It turns out that this is not a bad
assumption). For simplicity, also assume that all the photons have the energy corresponding
to the wavelength of the peak of a 2.73 K black-body radiation curve. At approximately
what redshift will the energy density in radiation be equal to the energy density in matter?
(Hint: work out the energy density in photons at the present time. Then work it out for
baryons, assuming a proton for a typical baryon. We discussed this in class, but show your
work! Remember how the two quantities scale with redshift to work out when the energy
density is the same).
Problem 2:
[6 points] The size of the Universe is conveniently parameterized by a scale factor, a(t), which
simply describes how big the Universe is at other times relative to its present size (i.e. at
the present time we say that a is 1, and at some time in the past when the Universe was
half as big as it was today, then a was 0.5). We will derive in class that a matter-dominated
Universe grows with time as a ∝ t2/3. Assuming the Universe is 13.5 billion years old at
present, how old is the Universe at redshifts, z, of z = 0.5, z = 1.0, z = 3.0, z = 10, z = 100?
Assume (as is obviously true!) that we presently live in a matter-dominated Universe, and
that the Universe is matter-dominated out to redshifts of at least 100.
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Assignment 3
AST121H
Due: March 16, 2022
Problem 3:
[4 points] Suppose you were not held together by electromagnetic forces. How long would it
take you to grow one cm because of the expansion of the Universe?
(Hint: Apply Hubble’s law to your head as seen by your feet, or vice versa!)
Problem 4:
[4 points] At early times in the Universe, temperatures were so high that baryon-antibaryon
pairs could be created from the vacuum. For what value of T would kT equal the rest energy
of an electron-positron pair? A proton-antiproton pair? Here, k is the Boltzmann constant.
Recall the equivalence of energy and matter! Give references for the masses you use.
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