Download EXERCISES: Set 2 of 4 Q1: The absolute magnitude of the Sun in

M. Pettini: Introduction to Cosmology
EXERCISES: Set 2 of 4
Q1: The absolute magnitude of the Sun in the optical V band is MV = 4.8.
The brightness of the night sky at a dark astronomical site is 22 magnitudes
per square arcsec in the V band.
(a) At what distances would: (i) a star like the Sun, (ii) a globular cluster,
(iii) a galaxy like the Milky Way be as bright as the background sky?
You can assume that in all three cases the distances involved are sufficiently
large that the astronomical sources in question can be safely considered to
be point sources. You can also assume that atmospheric turbulence (which
astronomers refer to as ‘seeing’) spreads their light evenly over a circular
aperture of diameter 1.8 second of arc.
(b) The so-called K-correction accounts for the fact that, at sufficiently
high redshifts, the wavelengths recorded within the bandpass of a given
filter, in this case the V filter, are not the same as the wavelengths emitted
within that bandpass. In which of the three cases listed in part (a) do you
think that a K-correction should be applied? How would application of a
K-correction alter your answer?
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Q2(a): The space density of bright galaxies (as bright or brighter than
the Milky Way) is ngal ' 0.01 Mpc−3 . The average mass of a bright galaxy
(including the dark matter halo) is Mstars+gas+DM ' 2 × 1012 M . If all of
the matter in the universe is attached to bright galaxies, what is the value
of Ωm,0 ? Comment on the validity of your answer.
Q2(b): Using the same assumption as at 2(a), what is the ratio of the
matter energy density ρm,0 to the energy density ρr,0 of the 2.73 K background radiation? Is this ratio a function of redshift? If so, is there a
redshift at which ρm = ρr ?
Q2(c): The “Planck density”, ρPlanck = c5 /(h̄G2 ) is the only combination
of the fundamental constants c, G, and h̄ with the units of density. What
is the Planck density in cgs units (g cm−3 )? If ρΛ,0 were equal to ρPlanck ,
what would be the value of ΩΛ,0 ?
Q3(a): From the two Friedmann equations, verify that the equation linking the mass density ρ and pressure p may be written in the form:
d
p d
(ρR3 ) + 2 (R3 ) = 0
dt
c dt
where R is the scale factor (we have called it a in the lectures).
Q3(b): Assume that p = wρc2 where w is a constant. Show that:
ρR3(1+w) = constant
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Q4: The galaxy luminosity function can be approximated as follows:
α
n∗
L
φ(L) =
for L < L∗
L∗ L∗
n∗
exp(−L/L∗ )
for L > L∗
=
L∗
where for spiral galaxies L∗,s = 1011 L , αs = −1.5, while for elliptical
galaxies L∗,e = 1012 L , αe = −0.5, and n∗,e = 4n∗,s .
(a) A survey observes all galaxies that have a luminosity greater than L1 =
5 × 1011 L within some volume. Calculate the fraction of spiral galaxies
in the survey. Assume that all galaxies are either spirals or ellipticals.
(b) Repeat this calculation for a deeper survey that finds all galaxies down
to a luminosity L2 = 109 L , and comment on your result.
Q5(a): Derive an expression for the comoving particle horizon in terms of
the density parameters Ωm,0 and ΩΛ,0 in a matter dominated, flat universe
(Ωk,0 = 0).
Q5(b): A photon leaves today (t = t0 ) from the present particle horizon.
Assuming an Einstein-de Sitter universe (Ωm,0 = 1, ΩΛ,0 = Ωk,0 = 0), at
what time will the photon arrive at the Earth’s location? Express your
answer in terms of the present age of the universe.
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Q6(a): Explain what is meant in astronomy by the terms ‘standard candle’
and ‘standard ruler’ and explain their application to cosmological distance
determinations.
Q6(b): The surface brightness Σ of an astronomical object is defined as
its observed flux divided by its observed angular area; thus Σ ∝ f /(δθ)2 .
For a class of objects that are both standard candles and standard rulers,
deduce the functional dependence of Σ on redshift z. Would observing the
surface brightness of this class of objects be a useful way of determining
cosmological parameters such as Ωm,0 , Ωk,0 , and ΩΛ,0 (where Ωm,0 , Ωk,0 ,
and ΩΛ,0 are the present-day contributions to the critical density by, respectively, matter, curvature and the cosmological constant)? Give reasons
for your answer.
Q7: In today’s ‘concordance cosmology’:
(a): At what redshift did dark energy begin to dominate the cosmic expansion (as measured by the Hubble parameter)?
(b): At what redshift did the Universe begin to accelerate?
(c): Are your answers to parts (a) and (b) the same? Explain.
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Q8(a): Show that for a pressureless matter-dominated Universe with Λ =
0,
3H02
ρ0 =
q0 ,
4πG
and
kc2
= (2q0 − 1)H02 ,
2
a0
where q ≡ − aä
ȧ2 is the deceleration parameter, a is the scale factor, k is the
curvature, H is the Hubble parameter, ρ is the density, the subscript 0 indicates the present time, and the other symbols have their usual meanings.
Q8(b): Show that the age of the Universe, t0 , in such a model is given by:
−1/2
Z 1
2q0
1
1 − 2q0 +
dx
t0 =
H0 0
x
where x = a/a0 .
2q0 −1
q0
Q8(c): By making the substitution 1 − cos θ =
x, show that the
above integral can be written as:
√
Z θ∗
q0
sin θ 1 − cos θ
√
H0 t0 =
dθ
(2q0 − 1)3/2 0
1 + cos θ
2q0 −1
where (1 − cos θ∗ ) =
.
q0
Q8(d): From the evolution of the Hubble parameter with redshift, derive
an expression for H(z) in an empty Universe with Λ = 0.
Q8(e): What is the value of q0 in such a Universe?
Q8(f ): Is there an inconsistency between your answers to part (d) and
part (e) of this question?
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