CMB interaction with other components

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Problem 1: free electron is invisible

Show that an isolated free electron can neither emit nor absorb a photon.

Problem 2: Compton effect

A photon of frequency $\omega$ interacts with an electron at rest and is scattered at angle $\vartheta$. Find the change of the photon's frequency.

Problem 3: inverse Compton

When a relativistic charged particle is scattered on a photon, the process is called the inverse Compton scattering of the photon. Consider the inverse Compton scattering in the case when a charged particle with rest mass $m$ and total energy $E\gg m$ in the laboratory frame interacts with a photon of frequency $\nu$. What is the maximum energy that the particle can transfer to the photon?

Problem 4: secondary scatterings

Estimate the probability of the fact that a photon observed on Earth has already experienced Thomson scattering since the moment it left the surface of last scattering.

Problem 5: Sunyaev-Zel'dovich effect on cosmic protons

Cosmic rays contain protons with energies up to $10^{20}eV$. What energy can be transmitted from such a proton to a photon of temperature $3K$?

Problem 6: Sunyaev-Zel'dovich effect on intergalactic gas

Relic radiation passes through hot intergalactic gas and is scattered on the electrons. Estimate the CMB temperature variation due to the latter process (the Sunyaev-Zel'dovich effect).

Problem 7: photon in non-relativistic electron gas

A photon with energy $E\ll mc^2$ undergoes collisions and Compton scattering in the electron gas with temperature $kT\ll mc^2$. Show that to the leading order in $E$ and $T$ the average energy lost by photons in collisions takes the form \[\langle\Delta E\rangle=\frac{E}{mc^2}(E-4kT).\]

Problem 8: drag force

Find the force acting on an electron moving through the CMB with velocity $v\ll c$.

Problem 9: dissipation

Estimate the characteristic time of energy loss by high-energy electrons with energy of order $100GeV$ passing through the CMB.

Problem 10: ultra high-energy cosmic rays cut-off

Find the energy limit above which the $\gamma$-rays interacting with the CMB should not be observed.