Cosmology – HW 3 Solution Key
1. Ryden 7.3
Sol.
Your textbook uses the phrase “current proper distance” in place of “comoving distance”, χ. Thus:
Z z
dz
c
p
χ =
H0 0
(1 + z)3(1+w)
Z z
3
c
=
dz(1 + z)− 2 (1+w)
H0 0
h
i
1
2c
=
(1 + z) 2 (−1+3w) − 1
H0 (−1 + 3w)
Likewise:
DA =
3
χ
2c
1
=
(1 + z) 2 (−1+w) −
1+z
H0 (−1 + 3w)
1+z
and
DL = (1 + z)χ =
h
i
1
2c
(1 + z) 2 (1+3w) − (1 + z)
H0 (−1 + 3w)
Finding the maximum angular diameter distance:
d(DA )
=0
dz
This is solved for:
2
(1 + z) =
3(−1 + w)
2/(−1+3w)
2. Ryden 8.2
Sol.
A dwarf galaxy is structurally similar to a cluster:
M=
hv 2 irh
αG
where in this case, α = 0.4, rh = 120 pc, and
hv 2 i = 3σr2 = 331
km2
s2
so:
M = 2.3 × 107 M
With a luminosity of L = 1.8 × 105 L , we have a mass to light ratio of:
M
= 128
L
Possible errors: Uncertainty in the profile, as well as measurement uncertainty. Of course, there’s also
the possibility that the galaxy isn’t fully relaxed, and isn’t actually spherical.
Cosmology – HW 3
HW 3
Due February 11, 2014
Please answer all questions clearly and concisely. You are strongly encouraged to discuss the homework with
your classmates, but you must complete the written homework by yourself, and of course, the material you
submit must be your own.
−1
3. A cluster has a typical radius of about 1.5h M pc. Assume that clusters have always existed, and
1
1. A physical
cluster has a size
typical(not
radius a
of good
about 1.5h
M pc. Assume thatand
clusters
have always
existed, and Ω
always had the same
assumption!),
further
assuming
M = 0.3, ΩΛ = 0.7:
always had the same physical size (not a good assumption!), and further assuming ⌦M = 0.3, ⌦⇤ = 0.7:
(a) will
At what
redshift
will the angular
minimum?
(a) At what redshift
the
angular
size size
bebeaa minimum?
Solution:
If
we
do
in
fact
assume
galaxy
clusters
have always been this size, the angle which
Sol.
it would subtend is only a function of the angular diameter distance. The angle is
minimum when
thisthe
quantity
is a maximum.
For a flat
universe: This is a flat universe, so Sk = χ
The angle is a minimum
when
angular
diameter
distance.
(z)
and thus:
dA =
1+z χ
DA =
Let’s plot this:
1+z
zpeak = 1.61
(b) What will the angular size of the cluster be if it were at that redshift?
Solution:
We have the intrinsic size of the cluster, l = 1.5h 1 Mpc, and the peak angular diameter
distance dA = 0.41·c
H0 .
This function has a maximum at:
z = 1.605
l
1.5h 1 Mpc
✓=
=
= 1.22 ⇥ 10
dA
1230h 1 Mpc
3
radat
= that
4.19 arcmin
(b) What will the angular size of the cluster be if it were
redshift?
Sol.
The angular diameter distance at the peak redshift:
DA = 0.408
c
H0
so
θ=
1.5h−1 Mpc
= 252” = 4.20
1224h−1 Mpc
(c) What is the luminosity distance to that redshift?
Sol.
The luminosity and angular diameter distances have a straightforward relationship:
DL = (1 + z)2 DA = 2.77
c
H0
4. Consider a cluster at a redshift of z = 0.05 which has a mass of 1014 h−1 M . For simplicity, you may
treat it as a point mass. Throughout this problem, please leave the h terms in.
(a) How far away is the cluster?
Sol.
d=
cz
= 150h−1 Mpc
H0
(b) What is the Einstein radius of the cluster?
Sol.
θE
=
r
4GM
c2 D
=
s
=
=
0.000358 rad
74”
4 · 6.67 × 10−11 Nm2 /kg2 · 1014 h−1 · 2 × 1030 kg
9 × 1016 m2 /s2 · 150h−1 (3.086 × 1022 Mpc)
5. Consider a possible series of universes, all of which have ΩM = 0.3, but which may be open or flat with
a cosmological constant:
ΩK = 1 − 0.3 − ΩΛ
You dealt with a similar problem in the last homework, but we’re generalizing this a bit. Suppose
you observe a series of SN 1a around z=0.2, and that the measurement uncertainty in the apparent
magnitude (and thus the distance modulus):
m − M = 5 log10 d(pc) − 5
is σm = 0.5.
Approximately how many supernovae would you have to measure at or near z = 0.2 to distinguish
between an open (ΩΛ = 0) and a flat√(ΩΛ = 0.7) universe?
Note: If you don’t know how to do N errors, please come and see me ASAP.
Sol.
The second plot shows us the difference in distance mouli of the extreme cases (the flat, ΩΛ = 0, and
the maximally
openplot
universe,
ΩΛ
0.7. At ainredshift
of z moduli
= 0.2, ofwethe
seeextreme
that the
difference
between
The second
shows us
the=di↵erence
the distance
cases
(the
open
universe
(⌦⇤∆.
= The
0.7) ).error
At ainredshift
of z = 0.2, is:
we see
⇤ = 0) and maximally
these twoflatare(⌦approximately
0.12; we
will
call this
N observations
that the di↵erence between these two are approximately 0.12; we will call this µ . As is
σ
often the case, the error in N observations
σµis:= √
N
µ
=p
Assuming we want to make a measurement at (say)N2 − σµ = ∆ significance:
But considering we have
N=
so
µ,
✓
and
µ
◆2
= 0.5:
σ
∆ = 2√
N
⇡ 17 Type Ia supernovae observations
2
σ
In case you were wondering, 42 Type Ia supernovae
were in fact used to discover that
N = 4 2 = 69
the universe was accelerating in its expansion∆(http://arxiv.org/abs/astroph/9812133)
a discovery worthy of a nobel prize in physics in 2011.
In case4. you
were wondering, 42 Type Ia supernovae were in fact used to discover that the universe was
I put a file called “hw3 cluster.txt” on the course website. In it, I’ve simulated a cluster of 100 galaxies.
accelerating
expansion
Each in
lineits
is of
the form: (http://arxiv.org/abs/astroph/9812133) - a discovery worthy of a nobel
prize in physics in 2011.
x(”), y(”), z
where the first two indicate the angular position of the galaxy with respect to the cluster center, and
the 3rd indicates the redshift of the galaxy (including both the Hubble flow and it’s motion within
6. I put a file called “hw3 cluster.txt” on the course website. In it, I’ve simulated a cluster of 100 galaxies.
Each line is of the form:
x(”), y(”), z
where the first two indicate the angular position of the galaxy with respect to the cluster center, and
the 3rd indicates the redshift of the galaxy (including both the Hubble flow and it’s motion within
the cluster). For reference, the galaxy positions are plotted below, so if you want to check that you’re
reading the file correctly, this is probably a good place to start. As always, using whatever numerical
package/language you like. Leave all calculations in terms of factors of h.
(a) Compute the mean redshift of the galaxies. Assume that this represents the cosmological redshift.
From that, compute the distance to the cluster.
Sol.
I find the mean redshift to be very close to z = 0.2 , thogh to defend against outliers (perhaps
galaxies unassociated with the cluster itself), one should generally use the median. From this, we
find the distance to the cluster as:
Z 1
c
da
q
χ =
H0 1/1.2 a2 0.32 + 0.68
a3
c
= 0.19
H0
=
590h−1 Mpc
(b) Using the distance above, calculate the outer radius of the cluster.
Sol.
The galaxy with the largest angle from the cluster center was found to be:
θ = 343” = 1.66 × 10−3 radians
The angular diameter distance is:
DA =
χ
= 492h−1 Mpc
1+z
So:
R = θ · DA = 0.82h−1 Mpc
(c) Compute the 1-d velocity scatter, σv , by computing the scatter (standard deviation) in redshift
and converting to velocity:
σv = cσz
Sol.
The standard deviation of the redshift column is:
σz = 0.00293
so
σv = 880 km/s
(d) The mass of a cluster may be estimated by:
M=
3σv2 R
αG
Undergraduates: You may assume α = 1.
Graduate Students: Determine a reasonable value of α based on the observed radial profile of
the galaxies themselves under the assumption that the galaxies are a fair sample. I want you to
base your estimate on the data (though clearly there is some room for error.
Sol.
Using the parameters, and α = 1
M=
3σv2 R
= 4.4 × 1014 M
G
Grad Student:
The first thing to do is to create a profile of the number of galaxies within a circular radius r.
Once this is done, fit this trend to a power law of the form:
Σ ∝ r−γ
I get the value of γ = 1.43. The mass will go as:
M = ΣA
so
so
M ∝ r−1.43 r2 ∝ r0.53
d log(M )
= 0.53
d log(r)
with this new value of α, we should get a mass of:
M = 8.32 × 1014 h−1 M