# ©University of Maryland – Do not post or reproduce

```©University of Maryland – Do not post or reproduce
PROBLEM SET #6
ASTR340 SPRING 2016
Due Date: Thursday May 5th
Book Chapters 13 to 16
1. Some say the world will end in fire,
Some say in ice.
From what I’ve tasted of desire
I hold with those who favor fire.
But if I had to perish twice,
I think I know enough of hate
To say that for destruction ice
Is also great
And would suffice. [20 pts.]
Look closely at Figure 13.8 in the textbook. It describes the properties of the Universe for
a given combination of ΩM and ΩΛ: whether it is accelerating or decelerating, its
geometry, and whether it will expand forever or collapse again. The dashed lines also tell
you the relation between the age of the Universe and the Hubble time. Tell me what the
properties of the universe (accelerating/decelerating, geometry, and age in terms of the
Hubble time) in the following cases:
A. ΩM=0.6 and ΩΛ=0.1
B. ΩM=0.55 and ΩΛ=0.9
C. ΩM=0.25 and ΩΛ=0.75
D. ΩM=1 and ΩΛ=0
E. Who wrote the lines in the title? (3 pt toward the 20 pt)
2. The Cosmic Microwave Background [20 pts.]
A. What is the CMB, when was it discovered, and by whom?
B. What are its origins?
C. What temperature do we measure today, and why is it so cold?
D. What is the level of anisotropy, and what does that mean for the Cosmological
Principle?
3. The Cosmic Microwave Background strikes back [40 pts.]
A. The present day measured temperature of the CMB is about 2.73 K. We know that
protons and electrons recombine around a temperature of 3,000 K. What is the redshift of
the CMB surface? (Hint: remember that T(z)=T(now)(1+z)).
B. What is the ratio of the size of the universe now to the size of the universe when
hydrogen recombined? (Hint: if you know the redshift, you know all there is to know…)
C. The equation for the evolution of the Hubble parameter is, approximately,
H(t)=Ho(1+z)1.5. What was the value of the Hubble parameter at the epoch of
recombination? The current value is Ho=70 km/s/Mpc.
D. We discussed in class that the age of the universe at any given time is, approximately,
the Hubble time (the inverse of the Hubble parameter after working out the units
correctly). What is the current age of the universe, in years? What was the approximate
age of the universe at the time of recombination, in years? (Hint: the only issue is to cast
Ho in the correct units. You need to convert 70 km/s/Mpc into inverse yr. Remember that
1 Mpc=1,000,000 pc=3,260,000 light years, and that c=3×108 m/s. If this sounds familiar
it is because you have already done it in another problem set.)
E. Remember that the size of the particle horizon is the length that light can travel over
the age of the universe (in terms of space-time diagrams it is the size of the past light
cone). What was the size of the particle horizon at the time of the CMB, in light years?
5. The end of the dark ages [20 pts.]
Use the equation in 3.C above and the reasoning in 3.D to do the following calculations.
Assume the first stars formed at z~15, and the first galaxies at z~10. Recombination
happened at the redshift you computed in 3.A above. How many years did it take after
recombination to form the first stars and the first galaxies?
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