Third Year: Cosmology
2016 Problem Sheets (Version 1)
Prof. Jo Dunkley: [email protected]
(Problems by Prof. Pedro Ferreira)
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Problem Sheet 3b
1. Closed Universe
(a) Show that
η
)
η∗
η
t(η) = C(η − η∗ sin )
η∗
a(η) = C(1 − cos
satisfies the closed, matter dominated FRW equation and find an expression for C in terms
of H0 , Ω the curvature parameter, k and the scale factor today, a0 .
(b) If the parameter η that occurs there is used as a time coordinate, show that the metric
takes the form
ds2 = a2 (η)[−c2 dη 2 + dχ2 + sin2 χ(dθ2 + sin2 θdφ2 ]
2. Conformal time
We can define conformal time, η, in terms of physical time, t, through
dt = adη
where a is the scale factor, which is a function of t or η.
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(a) Show that η ∝ a 2 in a matter dominated universe and a in one dominated by radiation
(b) Consider a universe with only matter and radiation, with equality at aeq . Show that
q
2
√
η=q
( a + aeq − aeq )
2
ΩM H0
(c) What is the conformal time today? And at recombination?
3. Contributions to the dynamics of the Universe
(a) Suppose the Universe contains four different contributions to the Friedmann equation
namely dust, radiation, a cosmological constant and negative curvature. What is the
behaviour of each as a function of the scale-factor a(t)?
(b) Which will dominate at early times and which will dominate at late times?
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Problem Sheet 4
1. Hubble parameter
Assume that the Universe is dust-dominated. Take H0 = 100 km s−1 Mpc−1 .
(a) Give a rough estimate of the age of the Universe.
(b) How far can light have travelled in this time?
(c) The microwave background has been travelling towards us uninterrupted since decoupling,
when the Universe was 1/1000 of its current size. Compute the value of the Hubble
parameter H at the time of decoupling.
(d) How far could light have travelled in the time up to decoupling (assume that the Universe
was dominated by radiation until then)?
(e) Between decoupling and the present, the distance that light travelled up to the time of
decoupling has been stretched by the subsequent expansion. What would be its physical
size today?
(f) Assuming that the distance to the last-scattering surface is given by part b of this question,
what angle is subtended by the distance light could have travelled before decoupling?
(g) What is the physical significance of this value?
2. Redshift
Consider the Friedmann–Robertson–Walker metric for a homogeneous and isotropic Universe is
given by
#
"
dr2
2
2
2
2
2
2 2
2
+ r (dθ + sin θ dφ ) ,
ds = −c dt + a(t)
1 − kr2
where ds is the proper time interval between two events, t is the cosmic time, k measures the
spatial curvature, r, θ and φ are radial, polar and azimuthal co-ordinates respectively.
(a) Explain what is meant by a(t) and discuss its physical significance.
(b) Describe what is meant by redshift and how spectroscopic observations of extragalactic
objects may be used to deduce their redshifts.
(c) What does the above expression become in the case of a light-ray? Hence derive an
integral expression for a light-ray which leaves the origin at time tem and reaches a comoving distance r0 at time tobs . A second ray is emitted a time dt after the first. By
considering the two intervals as corresponding to successive wave crests, derive the relation
λobs
a(tobs )
=
≡ 1 + z,
λem
a(tem )
where z is the redshift and λem and λobs are the emitted and observed wavelengths respectively.
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(d) How does the separation of galaxies today compare with the separation of galaxies when
light left the galaxies we observe at redshift 1?
3. Horizons
(a) Describe the concept of our past and future light-cone. Explain the meaning of the terms
particle horizon distance, event horizon distance and world-line and discuss the difference
between time-like and space-like locations.
(b) Show that in an Einstein-de Sitter Universe in which the scale-factor a(t) at time t follows
a(t) ∝ t2/3 , the particle horizon is at 3ct and the event horizon is at infinity.
(c) Suppose that the scale-factor were given by a(t) ∝ exp(mt) where m is a positive constant.
Show that the event horizon is finite and that the particle horizon grows exponentially when
t 1/m.
(d) Explain how such behaviour of the particle horizon might be useful in explaining observations
of the cosmic microwave background.
4. The Big Bang and the acceleration of the Universe
(a) Give an account of the observational evidence for the hot Big Bang model of the Universe.
(b) The Friedmann and fluid equations respectively are given by
2
ȧ
a
=
and
8πG
kc2
ρ− 2
3
a
ȧ
p
ρ + 2 = 0,
a
c
where a is the scale factor, ρ is the density and p is the pressure. (ȧ and ρ̇ are the derivatives
of these quantities with respect to time.) Use these equations to derive the acceleration
equation for the Universe.
ρ̇ + 3
(c) Hence demonstrate that if the Universe is homogeneous and the strong energy condition
ρc2 + 3p > 0 holds, then the Universe must have undergone a Big Bang.
5. Recombination and the Surface of Last Scattering
(a) What is the ‘surface of last scattering’ ? Would the same surface be seen by any other
observer on a different galaxy?
(b) Estimate the radius of the last scattering surface, using the age of the Universe. Why
might this underestimate the true value?
(c) The present number density of electrons in the Universe is the same as that of protons,
namely about 0.2 m−3 . Consider a time long before decoupling when the Universe was a
year old and the scale factor was one millionth of its present value. Estimate the number
density of electrons at that time and comment on whether the electrons would be relativistic
or non-relativistic then.
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(d) Given that the mean free path of photons through an electron gas of number density ne is
d ≈ 1/[ne σe ], where the Thompson scattering cross-section σe = 6.7×10−29 m2 , determine
the mean free path for photons when the scale factor was one millionth its present value.
(e) From the mean free path, calculate the typical time between interactions between the
photons and electrons.
(f) Compare the interaction time with the age of the Universe at that time. What is the
significance of this comparison?
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