Analytical Study on the Sunyaev-Zeldovich Effect for Clusters of Galaxies

Understand

Starting from a covariant formalism of the Sunyaev-Zeldovich effect for the thermal and non-thermal distributions, we derive the frequency redistribution function identical to Wright's method assuming the smallness of the photon energy (in the Thomson limit).

• We also derive the redistribution function in the covariant formalism in the Thomson limit.
• We show that two redistribution functions are mathematically equivalent in the Thomson limit which is fully valid for the cosmic microwave background photon energies.
• We will also extend the formalism to the kinematical Sunyaev-Zeldovich effect.