Lecture 6

Question 1

The CMBR is described by a

Answer 1

Planck Spectrum - a black body - at a temperature T = 2.725 ± 0.001K

Question 2

the small error on the CMBR temperature indicates

Answer 2

a high degree of thermal equilibrium

Question 3

the energy density of a black body is given by

Answer 3

εrad = αT^4

where α is the radiation constant

Question 4

the mean energy per unit volume

Answer 4

u = αT^4

Question 5

the mean number of photons per unit volume

Answer 5

n ∝ T^3

Question 6

Each photon has energy

Answer 6

kT/2 (degree of freedom)

Question 7

the entropy per unit volume

Answer 7

s = 4/3 αT^3

Question 8

the equation of state for radiation

Answer 8

p = 1/3 ρc^2 = 1/3 ur

Question 9

for radiation ρr a^4 =

Answer 9

ρr0 a0^4

Question 10

the first law of thermodynamics shows that

Answer 10

aT = constant

Question 11

aT = constant shows

Answer 11

an inverse relationship between scale factor a and temperature T

at the big bang a -> 0 and the temperature becomes infinite T -> ∞

Question 12

If the universe was much hotter in the past, how did the thermal distribution evolve as the universe cooled

Answer 12

the black body form is preserved

Question 13

for a boson gas in thermal equilibrium at temperature T a Bose-Einstein distribution holds:

Answer 13

n(photons) = 1/(exp(hf/kT) -1)

Question 14

the number of available states in a box of volume V is,

Answer 14

n(states) = V g(eff) 4πf^2/c^3 df

where g(eff) represents the number of spin states which, for photons is two

Question 15

assuming an isentropic expansion we find

Answer 15

the reduction of frequency is proportional to 1/a is the consequence of the red-shifting of the universe as it expands.

Question 16

A black body spectrum at some point on the past will

Answer 16

persist at later times