1. (5 points) The cosmic microwave background is a “sea” of blackbody radiation filling all of space.

(1)

KTII - Exercise 3

Universit¨ at Z¨ urich

Due: 20 March 2020

For particle information, including hadronic quark content, particle masses, particle lifetimes, or other physical constants not given in the problem, please consult the Particle Data Group’s Review of Particle Physics. It is available for free on their website: http://pdg.lbl.gov/.

1. (5 points) The cosmic microwave background is a “sea” of blackbody radiation filling all of space.

Planck’s Law gives the energy per unit volume per unit frequency of such a blackbody spectrum as

S _ν = 8πh c ³

ν ³ exp

hν k

B

T

− 1 ,

where h is Planck’s constant, k _B is Boltzmann’s constant, ν is frequency, and T is the temperature of the radiation.

(a) Show that if the universe expands by a factor α, the result is to reduce the effective temperature of the radiation from T to T /α, while retaining the blackbody spectrum.

(b) An early alternative explanation for Hubble’s law was the idea of “tired light”, where the frequency of the light decreases by a factor proportional to distance travelled. Would this model also give a blackbody spectrum for the cosmic microwave background radiation? Explain your answer.

2. (5 points) Using this form of Friedmann equation R ˙

R

! 2

= 8π

3 G N ρ − k R ² ,

discuss how old the Universe is. Use the value of the Hubble constant H ₀ = 71 km/s/Mpc ×(1.00 ^+0.04 _−0.03 ) determined from the WMAP data on cosmic microwave background anisotropy.

(a) Assume the matter-dominated universe with flat geometry, and work out the age of the universe, and compare it with the estimated age of the old stars in the globular clusters in our Milky Way galaxy.

(b) Discuss if the combination of matter and curvature can resolve the paradox.

(c) Show that the paradox can be resolved if you allow for the vacuum energy without introducing the curvature term.

1

1. (5 points) The cosmic microwave background is a “sea” of blackbody radiation filling all of space.

KTII - Exercise 3

Universit¨ at Z¨ urich

Due: 20 March 2020

For particle information, including hadronic quark content, particle masses, particle lifetimes, or other physical constants not given in the problem, please consult the Particle Data Group’s Review of Particle Physics. It is available for free on their website: http://pdg.lbl.gov/.

1. (5 points) The cosmic microwave background is a “sea” of blackbody radiation filling all of space.

Planck’s Law gives the energy per unit volume per unit frequency of such a blackbody spectrum as

S ν = 8πh c 3

ν 3 exp

hν k

T

− 1 ,

where h is Planck’s constant, k B is Boltzmann’s constant, ν is frequency, and T is the temperature of the radiation.

(a) Show that if the universe expands by a factor α, the result is to reduce the effective temperature of the radiation from T to T /α, while retaining the blackbody spectrum.

(b) An early alternative explanation for Hubble’s law was the idea of “tired light”, where the frequency of the light decreases by a factor proportional to distance travelled. Would this model also give a blackbody spectrum for the cosmic microwave background radiation? Explain your answer.

2. (5 points) Using this form of Friedmann equation R ˙

R

! 2

= 8π

3 G N ρ − k R 2 ,

discuss how old the Universe is. Use the value of the Hubble constant H 0 = 71 km/s/Mpc ×(1.00 +0.04 −0.03 ) determined from the WMAP data on cosmic microwave background anisotropy.

(a) Assume the matter-dominated universe with flat geometry, and work out the age of the universe, and compare it with the estimated age of the old stars in the globular clusters in our Milky Way galaxy.

(b) Discuss if the combination of matter and curvature can resolve the paradox.

(c) Show that the paradox can be resolved if you allow for the vacuum energy without introducing the curvature term.

1

S _ν = 8πh c ³

ν ³ exp

where h is Planck’s constant, k _B is Boltzmann’s constant, ν is frequency, and T is the temperature of the radiation.

3 G N ρ − k R ² ,

discuss how old the Universe is. Use the value of the Hubble constant H ₀ = 71 km/s/Mpc ×(1.00 ^+0.04 _−0.03 ) determined from the WMAP data on cosmic microwave background anisotropy.