PX389- Cosmology
Problem Sheet
Note: The following equations might be useful:
Friedman
2
ȧ
κc2
Λ
8πG
− 2 2 + = H2
=
2
a
3c
R0 a
3
(1)
ȧ
˙ + 3 ( + P ) = 0
a
(2)
ä
4πG
Λ
= − 2 ( + 3P ) +
a
3c
3
(3)
P = ω
(4)
Fluid
Acceleration
Equation of state
Unless stated otherwise take H0 = 72 km s−1 Mpc−1 .
1a) A galaxy has a measured recession velocity of 5000 km s−1 , what is its;
i) redshift
ii) distance
b) Show that, in any uniformly expanding space the Hubble constant can be
attributed to the current rate of change of the scale factor ȧ/a.
c) Explain what is meant by the terms standard candle and standard ruler.
2) a) Show that in a universe consisting purely of matter, that the universe will
be flat if the matter density is
c =
3c2 H02
8πG
(5)
b) Using the fluid equation show that the energy density of matter varies as
1/a3 (you may assume P = ω, where ω = 0).
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c) Hence, solve the Friedman equation, assuming that the scale factor has a
power law dependence of a(t) ∝ tq . What is the value of q?
d) How does H evolve? What does this mean for the evolution of the critical density of the universe?
3) The Friedman equation, in terms of multiple component universes can be
re-written as
Ωr,0
(1 − Ω0 )
H2
Ωm,0
= 3 + 4 + ΩΛ +
H02
a
a
a2
(6)
a) Explain the origin of each term.
b) Given that the universe is spatially flat, with Ωm,0 = 0.27 and ΩΛ,0 = 0.73,
at what scale factor were the two equal?
c) What redshift does this correspond to?
d) Perform the same exercise for matter and radiation, assuming Ωr,0 = 8.4 ×
10−5 .
e) At very early times the universe was dominated by radiation only. Using
the equation above (or otherwise), show that in this phase its expansion was
governed by a(t) ∝ t1/2 .
4) a) Explain what is meant my the terms horizon problem and flatness problem.
b) Given that the Friedman equation can be written as
1 − Ω(t) = −
κc2
R02 a(t)2 H(t)2
(7)
Show how the density of the universe (1 − Ω(t)) evolves for universes dominated by matter, radiation and cosmological constant.
c) How does a brief period of acceleration in the early universe solve the horizon
and flatness problems?
d) If the energy density and pressure of the inflaton field are given by
1 2
φ̇ + V (φ)
2
1
Pφ = φ̇2 − V (φ)
2
φ =
(8)
2
Under what conditions will the inflaton field mimic a cosmological constant with
equation of state parameter ω ∼ −1.
e) Sketch a potential which may satisfy this criteria.
4) a) For a given component of the universe, with equation of state parameter
ω, show that the fluid equation can be re-written as
d
da
= −3(1 + ω) ,
a
(9)
= 0 a−3(1+ω) .
(10)
and hence that
b) By solving the Friedman equation for this generic equation of state, assuming
a power-law dependence (tq ), show that the scale factor evolves as
a(t) ∝ t2/(3+3ω) ,
(11)
assuming ω 6= −1.
5) a) Outline the evidence for dark matter in galaxies and clusters.
b) Show that the rotation curve of a galaxy should follow the form
r
GM
.
vorb =
R
(12)
c) Two possible origins of dark matter are WIMPS and MACHOS, what does
each term mean? Give an example of a possible WIMP and MACHO candidate.
d) Modified Newtonian Dynamics (MOND) states that Newton’s law should
be modified to the form,
a
F = ma.
(13)
a0
in very low accelerations. Show that in this case MOND predicts flat rotation
curves in galaxies.
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