Exercises for
Introduction to Cosmology (WS2012/13)
Lecturers: Hans-Walter Rix & Simon Glover; Exercises: László Szűcs
Exercise sheet 10
Due (via e-mail to szucsl@uni-heidelberg.de): Jan. 15, 2013 18:00
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1. Characteristic scale on the surface of last scattering
In this problem, we will derive a simple estimate of the angular scale on which we
expect to find the first acoustic peak in the CMB fluctuation spectrum, using a highly
simplified model of recombination. For the purposes of the problem, we will assume
that recombination happens instantaneously at z = 1100, and hence that this redshift
also corresponds to the surface of last scattering. Furthermore, we will assume that
prior to recombination, matter and radiation are strongly
√ coupled and act as a single
fluid with sound-speed that we approximate as cs = c/ 3. After recombination, we
assume that there is no further coupling between matter and radiation.
(a) Consider a flat (Ωtot = 1) Universe containing only matter and radiation. Assume a sharp transition from radiation-dominated expansion to matter-dominated
expansion at z = zeq , where zeq is the redshift of matter-radiation equality, at
which Ωr (zeq ) = Ωm (zeq ). Calculate analytically the age of the Universe at the
time the CMB was formed.
(b) The conformal time η(t) is defined as
Z
η(t) =
0
t
dt0
.
a(t0 )
(1)
Using the same assumptions as in part (a), analytically calculate the conformal
time at which the CMB was formed.
(c) As discussed in the lecture, the first acoustic peak in the CMB occurs on the
scale of the “sound horizon”, the minimum distance that sound waves in the
photon-baryon fluid could have propagated before the CMB was formed. Estimate the size of the sound horizon in comoving units, giving your answer in
Mpc.
(d) Analytically calculate the angular size (in degrees) corresponding to this peak,
as observed at the present day.
(e) Now suppose that we live in an open Universe, with Ωm,0 = 0.3 and with
no cosmological constant. Numerically calculate the angular size on which we
expect to find the first acoustic peak in this case.
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2. Optical depth due to reionization
After the Universe becomes reionized, CMB photons can once again scatter off electrons. The Thomson scattering optical depth from an observer at z = 0 out to a
redshift of z can be written as
Z z
c
τ (z) = −
σT ne (z 0 )
dz 0 ,
(2)
0
0
(1 + z )H(z )
0
where σT is the Thomson scattering cross-section and ne (z 0 ) is the electron number
density at a redshift z = z 0 . One effect of this scattering is to suppress the size of
any CMB fluctuations by mixing together photons from cold and warm regions. To
a first approximation, the amplitudes of the perturbations are multiplied by a factor
e−τ , with τ given by the formula above.
Assume that the Universe reionizes instantly at a redshift zreion . Compute e−τ as
a function of zreion . What does the fact that we see any CMB fluctuations imply
regarding the range of possible values for zreion ?
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