# Time Evolution of CMB

### Problem 1: temperature

Show that in the expanding Universe the quantity $aT$ is an approximate invariant.

### Problem 2: redshift

Show that the electromagnetic radiation frequency decreases with expansion of the Universe as $\omega(t)\propto a(t)^{-1}$.

### Problem 3: Planck spectrum

Show that if the radiation spectrum was equilibrium at some initial moment, then it will remain equilibrium during the following expansion.

### Problem 4: one second after Big Bang

Find the CMB temperature one second after the Big Bang.

### Problem 5: decoupling time

Show that creation of the relic radiation (the photon decoupling) took place in the matter-dominated epoch.

### Problem 6: color of the sky

What was the color of the sky at the recombination epoch?

### Problem 7: when the night sky appeared

When the night sky started to look black?

### Problem 8: CMB vs microwave

Estimate the moment of time when the CMB energy density was comparable to that in the microwave oven.

### Problem 9: wavelengths

Estimate the moment of time when the CMB wavelength will be comparable to that in the microwave oven, which is $\lambda=12.6\ cm$.

### Problem 10: micro or not

When the relic radiation obtained formal right to be called CMB? And for what period of time?

### Problem 11: photon number density

Calculate the presently observed density of photons for the CMB and express it in Planck units.

### Problem 12: $\gamma$ and $\nu$ backgrounds

Find the ratio of CMB photons' energy density to that of the neutrino background.

### Problem 13: energy of a photon

Determine the average energy of a CMB photon at present time.

### Problem 14: contribution to radiation's energy

Why, when calculating the energy density of electromagnetic radiation in the Universe, we can restrict ourselves to the CMB photons?

### Problem 15: conservation of photons' number

The relation $\rho_\gamma\propto a^{-4}$ assumes conservation of photon's number. Strictly speaking, this assumption is inaccurate. The Sun, for example, emits of the order of $10^{45}$ photons per second. Estimate the accuracy of this assumption regarding the photon's number conservation.

### Problem 16: thermonuclear sources

Can hydrogen burning in the thermonuclear reactions provide the observed energy density of the relic radiation?

### Problem 17: energy density evolution

Find the ratio of relic radiation energy density in the epoch of last scattering to the present one.

### Problem 18: photons and baryons

Find the ratio of average number densities of photons to baryons in the Universe.

### Problem 19: temperature at last scattering

Explain qualitatively why the temperature of photons at the surface of last scattering (0.3 eV) is considerably less than the ionization energy of the hydrogen atom (13.6 eV).

### Problem 20: recombination

Estimate the moment of the beginning of recombination: transition from ionized plasma to gas of neutral atoms.

### Problem 21: mean free path

Determine the moment of time when the mean free path of photons became of the same order as the current observable size of Universe).

### Problem 22: excited states

How will the moment of recombination and the time when mean free path of photons becomes comparable with the size of observable Universe change if one takes into account the possibility of creation of neutral hydrogen in excited states?

### Problem 23: what about neutrino?

Why is the cosmic neutrino background (CNB) temperature at present lower than the one for CMB?