Download EXERCISES: Set 1 of 4 Q1: Suppose the universe consisted of a

M. Pettini: Introduction to Cosmology
EXERCISES: Set 1 of 4
Q1: Suppose the universe consisted of a random, statistically uniform,
distribution of stars in space, all identical to the Sun, with radius R =
R = 7 × 1010 cm, and a space density of n = 1 pc−3 . Olber’s paradox
states that, if such a universe were infinitely old, the surface brightness of
the sky (energy received per unit time, per unit area, from unit solid angle
on the sky) as seen by an observer on Earth would be infinitely large.
(a) Would an observer on another planet of a different solar system also
see an infinitely bright sky?
(b) How far, on average, would you have to look in such a universe until
your line of sight struck the surface of a star? Express your answer in units
of the Hubble radius.
(c) Show that, if the universe has a finite age, the surface brightness on
Earth would be finite. Assume that the density of stars, n = 1 pc−3 , is
fixed in comoving coordinates. What would this surface brightness be in
an Einstein-deSitter universe with H0 = 70 km s−1 Mpc−1 ? Express your
answer in units of W m−2 sr−1 . The Sun’s luminosity is L = 3.8 × 1026 W.
Q2: Which of the following is homogeneous, isotropic, or both:
(a) The Earth’s atmosphere viewed from Cambridge on a cloudless day
(b) A field of wheat
(c) The crystal structure of (a) diamond, (b) graphite
1
Q3(a): Explain the difference between kinematic and cosmological redshifts.
Q3(b): Use the relativistic formula
ν0 =
v
u
u1
t
νe u
− vc
,
1 + vc
where νe and ν0 are the photon emission and reception frequencies respectively, to derive the kinematic redshift of a light source receding at velocity
v relative to the observer, in the limit v c.
Q3(c): Consider two galaxies, A and B. As viewed from Earth, galaxy A
is at redshift zA = 1 and galaxy B is at zB = 9. What is the redshift of
galaxy B as measured by a hypothetical observer on galaxy A?
Q3(d): When observing a distant galaxy, we measure a combination of
kinematic and cosmological redshifts, as galaxies respond to the local gravitational field which adds a ‘peculiar’ velocity (of either positive or negative
sign) on the uniform Hubble expansion.
(i) Show that the combined redshift due to these effects [kinematic and
cosmological] is
1 + ztot = (1 + zcosm )(1 + zkin )
(ii) Typical peculiar velocities of galaxies are vp = ±600 km s−1 . For a
constant Hubble parameter H0 = 100h km s−1 Mpc−1 , what is the minimum distance, rmin , at which a galaxy must be for its redshift to give an
estimate of its true distance accurate to better than 5%?
2
Q4(a): One of the most distant known bodies in the solar system is the minor planet Sedna (see http://web.gps.caltech.edu/∼mbrown/sedna/). Its
diameter has been estimated to be D ' 1500 km. If you drew a circle on
Sedna of radius 1 km, what difference would you find between the expected
and measured values of the circle’s circumference (assuming that Sedna is
perfectly spherical)?
Q4(b): The Robertson-Walker metric
dr2
2
2
2
2 
+
r
dθ
+
sin
θ
dφ
(ds)2 = (c dt)2 − a2 (t) 
1 − kr2


is the most general metric describing an expanding universe which obeys
the Cosmological Principle. Use this metric to show that the proper area
of a spherical surface, centred at the origin and passing through comoving
coordinate r, is 4π[a(t)r]2 .
Q5(a). Show that for a Newtonian expanding universe:
da
→ ∞ as t → 0
dt
Q5(b): In a universe with a zero cosmological constant, show that the
equation:
h
i
H 2 = H02 Ωm,0 (1 + z)3 + Ωk,0 (1 + z)2
can be rewritten as:


1
1
−1=
− 1 (1 + z)−1
Ωm
Ωm,0
What does this tell you about the difference between a closed, flat, and
open universe at early times?
3
Q6: Give the definition of the distance modulus. The galaxy Cam 300 has
a distance modulus of +30; what is its distance in parsecs? Knowing that
the Sun has an absolute magnitude in the B-band of MB ' +5, estimate
the approximate apparent magnitude mB of the galaxy Cam 300, if its stellar mass is similar to that of the Milky Way. State any assumptions made.
Q7: One of the most distant galaxies known whose redshift has been
spectroscopically confirmed is at z = 8.68 (see Zitrin et al. 2015:
http://arxiv.org/abs/1507.02679)
(a) Which spectral feature(s) were used to determine the redshift of the
galaxy?
(b) Why is the detection of such high-z galaxies challenging?
(c) In an Einstein-de Sitter cosmology, what is the interval of time from
the Big Bang available to form this galaxy?
(d) And in a Ωm,0 = 0.3, ΩΛ,0 = 0.7, Ωk,0 = 0 cosmology?
Q8: Elliptical galaxies are generally red. Irregular galaxies are generally
blue. Spiral galaxies are intermediate between ellipticals and irregulars.
What do these differences reflect? What other differences might you expect between these three classes of galaxies?
4
Q9(a): Show that the two Friedmann equations with Λ 6= 0, ρ = ρm ,
and p = pmat = 0 are equivalent to assuming Λ = 0, ρ = ρm + ρvac , and
p = pm + pvac = pvac = −ρvac c2 .
Q9(b): Show that a positive Λ term corresponds to a repulsive force whose
strength is proportional to distance. (In all the remaining parts of Q9, we
are going to assume Λ > 0).
Q9(c): Show that at late times all cosmological models with k = 0 and
k = −1, and some models with k = +1, behave as a ∝ exp[(Λc2 /3)1/2 t].
Q9(d): For the case k = +1, show that there is a static p = 0 solution for
a value of Λ = Λc such that: c2 Λc = 4πGρ0 = (c/a0 )2 .
Q9(e): Sketch the behaviour with time of the scale factor, a(t), for a
universe with Λ = Λc + , where Λc (keeping k = +1).
Q9(f ): Comment on the age of such a universe.
5