# Problem Set 5 Solutions 1. FMC 12.3: = (1+z)

```Problem Set 5 Solutions
1. FMC 12.3:
The average energy density compared to now is ρ/ρ0 = (1+z)4 =(1+2)4 =81, where ρ is the
radiation density at the previous time and ρ0 energy density now. Radiation was 81 times
denser at z=2 compared to now.
The average matter density compared to now is ρ/ρ0=(1+z)3 =(1+2)3 = 27 , where ρ is the
matter density at the previous time and ρ0 matter density now. Radiation was 27 times denser
at z=2 compared to now.
The temperature of the background radiation T=T0(1+z) where T0 is the temperature of the
cosmic microwave background, T0=2.75 Kelvin; T=2.75(1+2)=8.25 Kelvin
2000(1+z)3=(1+z)4 divide both sides by (1+z)3 so:
2000=(1+z), now solve for z=1999
2. T=m0c2/3kB where m0 is the rest mass of the particle, kB is the Boltzmann constant and c is
the speed of light. Plug these values in to find
T=8.26x1011Kelvin
3. a) Deuterium is very sensitive to changes in the critical density of the universe. This can be
seen in figure 12.8 on page 367 in the text.
b) Pair production where a gamma ray decays into a particle and it’s antiparticle (ex. Electron
and positron). This can only happen if there is enough energy available to create these
particles; the energy must be equal to the rest masses of each particle. Nucleosynthesis
creates deuterium from existing protons electrons created from pair production.
c) Hydrogen and Helium; anything created in the very early universe would be immediately
destroyed because of the enormous energy and temperatures present during the early
universe.
4. Key points: missing mass in galaxies seen in the rotation curves of galaxies. We expect
galaxies rotations to slow where there are fewer stars, however, they do not slow in the expected way
leading to the conclusion that mass exists in the galaxies that we can’t see. Gravitational Lensing: clusters
of galaxies bend light from distant objects; the more massive they are the more they bend the light; the
mass of these clusters is much higher than the baryonic mass within them.
5. a) ρcrit=3H2/8πG where H is the Hubble constant and G is the gravitation constant (6.6726x10-11
) ; the Hubble constant is 72 km/s*Mpc; kilometers and mega parsecs are both units of length, so the first
step is to put them both into SI units of length: meters
H=72km/s*Mpc *1000m/1km *1Mpc/3.08x1018m =2.4x10-18 m/s*m
Plug this value into the ρcrit expression to find ρcrit=3.47x10-27 kg/m3
This is much less dense than water or air.
b) Derive the value of the critical density when the Universe was half its present age.
Using the hint in the problem and the fact that 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) 1/H(t) (and the relationship between R
The density of the universe is inversely proportional to the scale factor to the ‒3 power (R(t) -3).
In the text book there is a relationship between the time now t0 with a size now of R0 and the
time then t , R(t)=R0(t/ t0)2/3 for a flat universe, Ω0=1 , (eq 11.20 in the text) then when t/t0 =1/2
the size of the universe is 0.63 the size it is today and thus the density is 4 times larger.
Alternatively we can use the relationship between scale factor and z; 1+z=R(then)/R(now) and
the relation between z and the age of the universe (which I did not give in class and is not in the
book, but can be easily found) t(z)=2*((3H0Ω0(1+z)3/2)-1
6. a) Radiation density goes as (1+z)4 so as z increases radiation density increases very fast;
compared to matter density, radiation density increases much faster. Therefore at some point, radiation
density was greater than matter density.
b) The energy/photon E=hc/λ
the photon density is 4.11x106 photons/m3 so the energy per m3 is the energy/photon *photons/m3
energy/m3 = 4.11x106* 6.63x10-34*3.0x108/1.1x10-8 =9.4x10-16 Joules
7. The cosmological constant – the expanding universe. Einstein introduced this concept to
counteract gravity in his theory of general relativity. His addition of this constant to the equation made the
universe static. Today we know that the universe is actually expanding and the cosmological constant
represents the acceleration of the universe.