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M. Pettini: Introduction to Cosmology
EXERCISES: Set 4 of 4
Q1: (You will need a ruler and a calculator)
The attached figure shows sections of the Lyman alpha forest in three
quasars. Three lines are marked.
(a) Indicate on which portion of the Curve-of-Growth each line falls.
(b) Measure the equivalent widths of lines (1) and (2). Why is it not
possible to measure the equivalent width of line (3) accurately?
(c) Assuming that lines (1) and (2) are Lyα lines, deduce the corresponding
column densities of neutral hydrogen, N (H i), if it is possible to do so. If
it is not possible to do so, state what additional information you require.
How would you measure N (H i) in cloud (3)?
(d) Does cloud (1) correspond to an over- or under-density of matter?
What is its likely linear size? And its density (in units cm−3 ) of neutral
gas? What is the angle subtended by this Lyα cloud on the sky? (You
may assume an Einstein-de Sitter universe).
Figure 4.1: Portions of the Lyα forest. The x-axis is Observed Wavelength, while the
y-axis is Relative Intensity.
Q2(a): Using the Friedmann equation for a pressureless cosmological
ȧ2 kc2
show that for a population of absorbing gas clouds with constant number
density per comoving volume and with constant physical cross-section, the
probability that the line of sight to a distant source intersects such a cloud
per unit redshift at redshift z is proportional to:
(1 + z)2
[Ωm,0 (1 + z)3 + Ωk,0 (1 + z)2 + ΩΛ,0 ]1/2
where Ωm,0 + ΩΛ,0 + Ωk,0 = 1.
Q2(b): Surveys have shown that, on average, the spectrum of a quasar at
redshift zem = 2 shows five sets of absorption lines from elements heavier
than helium (i.e. five metal line systems) at redshifts zabs < zem . If such
systems are produced by intervening galaxies, randomly distributed along
the line of sight to the quasar, estimate the gaseous size of galaxies required
to satisfy the absorption line statistics in a ΩΛ,0 = 0, Ωm,0 = 0.3 cosmology.
(Hint: Use the average density of galaxies implied by the galaxy luminosity
function). Comment on the implications of your answer.
Q3(a): Derive the functional form of the rotation curve expected for the
outer regions of our Galaxy, where the density of stars is negligible, in the
absence of dark matter.
Q3(b): The rotation curve of the Milky Way is found to remain flat beyond
r ∼ 10 kpc, where v ' 220 km s−1 . Deduce M (r) in the outer regions of
the Galaxy, including the constant of proportionality in your answer.
Q3(c): Instead of appealing to dark matter as the explanation for the
flat rotation curves of spiral galaxies, it has been proposed that the law
of gravity should be modified on large scales. If the gravitational force is
given by F = GM m/rγ , what value of γ is required to remove the need
for dark matter?
Q3(d): Derive the density ρ(r) in the outer regions of the Milky Way
corresponding to the value of M (r) found in part (b). Include the constant
of proportionality in your answer.
Q3(e): Under the assumption that the Milky Way has a constant density
in its inner regions, derive an expression for the rotation curve close to the
Q4: The nature of dark matter has yet to be established. Suppose that
all the DM in the halo of our Galaxy consisted of black holes of mass
m = 10−8 M .
(a) How far away would you expect the nearest such black hole to be?
(b) How frequently would you expect such a black hole to pass within 1 AU
of the Sun?
(c) Repeat both (a) and (b) for the case where the dark matter in the
Galactic halo consists of Jupiter-mass planets, with m = 10−3 M .
(d) Can the above considerations help exclude either candidate for Galactic
Note: approximate estimates will suffice. You can assume that the average
density of matter near the Sun is ρm = 0.06 M pc−3 .
Q5(a): A light ray just grazes the surface of the Earth. Ignoring atmospheric refraction, through what angle α is the light bent by gravitational
lensing? Repeat your calculation for a solar-mass white dwarf and for a
neutron star, and comment on the results.
Q5(b): In lecture 16.4.1 we saw that the bending of the images of stars
whose light rays just graze the surface of the Sun observed during the total
solar eclipse of 29 May 1919 confirmed one of the predictions of General
Relativity. It was also stated that the first empirical confirmation of strong
lensing was the realisation, in 1979, that the two quasars, 0957+561A and
0957+561B are in fact two images of the same object.
Does gravitational lensing by the Sun cause two images? If so, why was
only one (displaced) image of the background stars seen during the total
solar eclipse of May 1919?
Q6: The Nobel Prize in Physics 2015 was awarded jointly to Takaaki
Kajita and Arthur B. McDonald “for the discovery of neutrino oscillations,
which shows that neutrinos have mass”.
An oscillation is the transmutation of one flavour of neutrino into another.
The rate at which two neutrino flavours oscillate is proportional to the
difference of the squares of their masses. Current data indicate that:
m(νµ )2 − m(νe )2 c4 = 5 × 10−5 eV2
m(ντ )2 − m(νµ )2 c4 = 3 × 10−3 eV2
What values of m(νe ), m(νµ ), and m(ντ ) minimize the sum m(νe )+m(νµ )+
m(ντ ), given the above constraints?
Q7: The pair of quasars 1146+111B,C are separated by 157 arcseconds on
the sky. Both quasars have magnitude mV = 18.5, and their spectra exhibit
broad Mg ii λ2798 emission lines at the same redshift z = 1.012±0.001 and
with the same width FWHM = 64 ± 4 Å.
Discuss the alternative possibilities that this pair is:
(a) the same quasar gravitationally lensed into two images, or
(b) two separate quasars.
Propose a set of observations that would distinguish between the two possibilities.
Note: Q8 is at an advanced level. However, try your best: even partial
attempts will be useful ahead of the supervision.
Q8(a): Give the definition and the physical interpretation of the power
spectrum P (k) and of the correlation function ξ(r) of the density fluctuation field δ.
Q8(b): Write an expression for the rms mass fluctuation σM within a
sphere of radius R using the power spectrum P (k). Assuming a top-hat
filtering of density fluctuations, recover the expression for the window function in Fourier space W (kR). [N.B. you should use the expansion in spherical harmonics of the quantity exp[ik · x] or alternatively you can Fourier
transform the top-hat filter].
Q8(c): Sketch the function W (kR) and find its behaviour for small values
of x = kR and large values of x.
Q8(d): Let us assume that P (k) = Ak n and that W (kR) has the two
asymptotic behaviours found in 7(c) for k < km and for k > km . Find out
a power law expression, dependent on km only, which relates σM to R and
Calculate the ratio of rms mass fluctuations between the particle horizon
now and the fluctuations at R = 1 Mpc [You can assume an EdS cosmology
here] adopting n = 1 (a scale invariant power spectrum).
Q8(e): The correlation function of star forming galaxies at redshift z = 3
can be well approximated by ξ(r) = (r/r0 )−1.6 with r0 = 4 Mpc. Write
down the probability of finding a pair of galaxies at a generic separation
r. Let us assume that there is another set of cosmological objects—Lyman
alpha clouds—with a uniform distribution, which sample the same volume and the same redshifts, and whose number density is twice of that of
the galaxies. At which r is the probability of finding two galaxies in the
infinitesimal volumes dV1 and dV2 the same as that of finding two Lymanalpha clouds in the same volumes?