Cosmology – HW 1 Solution Key
1. Ryden 2.2
bf Sol.
The density of stars is:
n∗
=
109 Mpc−3
=
3.4 × 10−59 m−3
The cross section of a star in this model is:
σ∗
= πR2
1.54 × 1018 m2
=
The mean free path before encountering a star is:
l∗ =
1
= 1.9 × 1040 m = 6.2 × 1017 Mpc
nσ
For galaxies:
ng
=
1 Mpc−3
and
σg = 1.26 × 10−5 Mpc2
so
lg =
1
= 80000 Mpc
nσ
2. Ryden 4.1
Sol.
The volume interior to Earth’s orbit is:
V =
4π
(AU )3 = 1.4 × 1034 m3
3
and so the dark energy (if it were at critical) is:
E = V = 7.3 × 1037 MeV
By contrast, the rest energy of the sun is:
E = M c2 = 1.8 × 1047 J = 1.1 × 1060 MeV
and thus:
EDE
' 6.6 × 10−23
E
No. Even if it were at critical, the dark energy density does not matter on solar system scales.
3. Ryden 5.3
Sol.
This is a fairly simply application of the result from problem 5.2. In a matter only universe:
h
i
dz
= H0 (1 + z) − (1 + z)3/2 = −0.83H0
dt0
for z = 1, and w = 0. So if |∆z| = 10−6 ,
∆t0 =
10−6
= 16, 800 yr
0.83H0
4. Consider a universe in which ΩM = 0.32, but which is otherwise empty.
(a) In units of the Hubble time (1/H0 ), how old is the universe now, t0 ? You will need to do a
numerical integration, so please include your code.
Sol. (in python)
from scipy.integrate import quad
import numpy as np
def AgeOfUniverse(Omega_M_0,Omega_L_0,Omega_G_0,Omega_k_0):
return quad(lambda a: 1/(a*np.sqrt(Omega_M_0/a**3 + Omega_L_0 +
Omega_G_0/a**4 + Omega_k_0/a**2)),0,1)[0]
print(AgeOfUniverse(0.32,0,0,0.68))
>>> 0.802280176649
t0 =
where we will hereafter note:
tH =
0.8
H0
1
= 9.78h−1 Gy
H0
(b) Now adjust your model so that it contains a matter component ΩM = 0.32, and a “cosmological
constant” component, ΩΛ = 0.68 (with w=-1). How old is this universe?
Sol.
Wasn’t I clever? My code above is ready to do all of the work.
print(AgeOfUniverse(0.32,0.68,0,0)
0.946769749747
so
t0 =
0.95
H0
(c) Grad Students only: For each of the above models, plot a (y-axis) versus t (x-axis). Be sure
that a = 1 and t =now represent the same point on the plot for both models, and that the two
models are superimposed. Time should be expressed in units of H0−1 .
Sol.
I have adjusted the time axis so that t0 is set at the present.
5. The currently accepted value of the Hubble constant is approximately 67km/s/M pc. Consider a galaxy
observed with a redshift, z = 0.05. Note that this is sufficiently small that you may assume H0 to be
a constant during the entire duration of the light travel time.
(a) How far away is the galaxy?
Sol.
d=
cz
3 × 105 km/s · 0.05
= 224 Mpc
=
H0
67 km/s/M pc
d = 220 Mpc
(b) How long ago was the light emitted from the galaxy?
Sol.
∆t =
d
2.2 × 108 pc 3.086 × 1013 km
=
·
= 0.72 Gy
c
3 × 105 km/s
pc
(c) Imagine a Balmer-α photon was emitted from this galaxy (If you don’t remember what this is, I
strongly recommend you look it up). At what wavelength will you observe it here on earth?
Sol.
The wavelength of an H-alpha photon is λ = 656.28 nm. Starting with the definition of redshift:
z=
λobs − λe
λe
so
λobs = λe (1 + z) = 656.28 nm · (1.05) = 689.09 nm
6. Imagine a matter only universe in which ΩM = 0.99999 at t = 1s. While we normally set t0 to “now,”
the “now” for this problem should be 1 second after the big bang.
(a) What was the Hubble constant 1s after the big bang in this universe? Given the value of ΩM ,
you may assume the universe behaved as an Einstein-deSitter universe up until that point.
Sol.
For an EdS universe, the scale factor as a function of time is:
a ∝ t2/3 = At2/3
No need to worry about the normalization constant; it will drop out soon.
H(a) = ȧa =
2
−1/3
3 At
At2/3
=
2
3t
so
H0 =
2 −1
s
3
(b) What was the critical density 1s after the big bang in this universe?
Sol.
ρcr =
3H02
= 7.9 × 108 kg/m3
8πG
(c) By what factor will the universe expand from t = 1s to t = 10Gy? You will need to do a numerical
integration.
Sol.
We essentially would like to know the upper limit of the following integral:
1
H0
Z
0
af
da
q
ΩM,0
a
+ (1 − ΩM,0 )
Z
10 Gy
=
dt
0
where the right side of the integral is equal to 3.156 × 1017 s, ΩM,0 = 0.99999, and H0 = (2/3) s−1 .
Solving numerically:
af = 6.65 × 1014
(d) What will ΩM be 10 billion years after the big bang?
Sol.
ΩM (a) =
ΩM,0
= 1.5 × 10−10
ΩM,0 + aΩk,0
7. Suppose you live in an expanding universe with the same physics as our own but (potentially) different
cosmological parameters. Through astronomical means, it is established that the universe is between
10 and 11 Gy old.
(a) Suppose the universe contains only matter, and thus there are only two relevant cosmological
parameters: ΩM and h. On a single plot relating the two parameters, illustrate all possible
combinations of these two parameters.
This should be done on a computer, not by hand, and ranges should include 0 < ΩM < 2 and
0 < h < 2.
(b) Grad Students only: Suppose instead, we consider universes that are geometrically flat and
have a cosmological constant:
ΩM + Ω Λ = 1
The universe can still be described by two parameters, ΩM and h. Do the same analysis in part
a, and overplot the ranges of parameters for these universes over the results from part a.
Sol.
We solve for the time in the standard way. The shaded curves represent the range in time from 10-11
Gy.