Exercise 1
a) The so-called critical density is the value of ρ for which k = 0 in
Friedmann’s equations. Show that it is given by
3H 2
,
8πG
where H(t) = ȧ/a. Using the present value of the Hubble parameter,
H0 = 72 km s−1 Mpc−1 , calculate the present value of ρc . Give your
answer in units of g cm−3 and M Mpc−3 . Calculate the present value
of the energy density ρc c2 in units of GeV cm−3 .
ρc =
b) Starting with Friedmann’s equation with a cosmological constant:
8πG 2
1
ρa ,
ȧ2 + kc2 + Λc2 a2 =
3
3
and looking at the present epoch, t = t0 , show that it can be written
as
Ωm0 + ΩΛ0 + Ωk0 = 1,
where the so-called density parameters are given by Ωm0 = ρm0 /ρc0 ,
ΩΛ0 = −Λc2 /3H02 , and Ωk0 = −kc2 /a20 H02 .
c) The so-called deceleration parameter q0 is defined by
q0 = −
ä0 a0
.
ȧ20
Show that q0 > 0 for k = 0, Λ = 0. Assume that only dust (p = 0)
contributes to the density. What does this mean ?
d) Consider a model of the universe where the present value of the density
parameter for dust Ω0 < 1, Λ 6= 0, and k = 0. Find an expression for
the present deceleration parameter q0 in this case. What condition
must Ωm0 satisfy if the Universe is to expand at an accelerating rate ?
Exercise 2
‘Phantom energy’, a substance with equation of state parameter w < −1, has
been proposed as an alternative to the cosmological constant for explaining
the present accelerated phase of expansion. Assume that we live in a spatially
flat universe, dominated by phantom energy with w = −2.
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a) Determine how the energy density of this component varies with the
scale factor a.
b) Integrate the Friedmann equation for ȧ/a from our present epoch t0
(a(t0 ) = a0 ) and into the future to find a(t) for t > t0 .
c) What happens as t − t0 →
appropriate?
2
?
3H0
Does the expression ‘Big Rip’ seem
Exercise 3
Assume a spatially flat universe (k = 0) with scale factor given by
a(t) = a0
t
t0
2/3
.
Here t0 is the present cosmic time, and a0 is the present value of the scale
factor. We observe an object at cosmic redshift z = 3.
a) Calculate the comoving coordinate r of the object and its proper distance from us at t = t0 .
b) The radiation we receive from the object contains a message from an
advanced civilization. We wish to send a radio signal back to them.
If we send it at t0 , at what time (in units of t0 ) will our signal reach
them?
Exercise 4
Use the Friedmann equation for ä with a cosmological constant to find the
equation for the time evolution of a small, time
√ dependent perturbation η
around the Einstein static solution a = a0 = c/ Λ, and use this equation to
show that the Einstein model is unstable.
Exercise 5
The dutch astronomer Willem de Sitter originally published his universe
model as an alternative, static solution to Einstein’s model. In his original
solution, the line element is written as
r2
dr2
− r2 dθ2 − r2 sin2 θdφ2 ,
ds = 1 − 2 dt2 −
R
1 − r2 /R2
!
2
2
where R is a constant. Show that by transforming to a new set of coordinates,
r = q
r
1−
r2 /R2
e−t/R ,
1
r2
t = t + R ln 1 − 2 ,
2
R
!
the line element can be brought on the form
2
ds2 = dt − e2t/R (dr2 + r2 dθ2 + r2 sin2 θdφ2 ).
Comment on this result.
Exercise 6
a) Show that a(t) = ct is a solution of the Friedmann equations for a
completely empty universe (ρ = p = 0) if k = −1.
b) Find expressions for the proper distance dp and the angular diameter
distance dA as functions redshift z for this model.
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