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Register a new userAnalytical studies on the Sunyaev-Zeldovich effect in the cluster of galaxies for three Lorentz frames II: single integral formula
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We study the Sunyaev-Zeldovich effect for clusters of galaxies. The Boltzmann equations for the cosmic microwave background photon distribution function are studied in three Lorentz frames. We extend the previous work and derive analytic expressions for the integrated photon redistribution functions over the photon frequency. We also derive analytic expressions in the power series expansion approximation. By combining two formulas, we offer a simple and accurate tool to analyse observation data. These formulas are applicable to the non-thermal electron distributions as well as the standard thermal distribution. The Boltzmann equation is reduced to a single integral form of the electron velocity.
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We study the Sunyaev-Zeldovich effect for clusters of galaxies. The Boltzmann equations for the CMB photon distribution function are studied in three Lorentz frames. We clarify the relations of the SZ effects among the different Lorentz frames. We derive analytic expressions for the photon redistribution functions. These formulas are applicable to the nonthermal electron distributions as well as the standard thermal distribution. We show that the Fokker-Planck expansion of the Boltzmann equation can be expanded by the power series of the diffusion operator of the original Kompaneets equation.
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 Wrights 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. With the present formalism we will clarify the situation for the discrepancy existed in the higher order terms of the kinematical Sunyaev-Zeldovich effect.
We study a covariant formalism for the Sunyaev-Zeldovich effects developed in the previous papers by the present authors, and derive analytic expressions for the redistribution functions in the Thomson approximation. We also explore another covariant formalism recently developed by Poutanen and Vurm. We show that the two formalisms are mathematically equivalent in the Thomson approximation which is fully valid for the cosmic microwave background photon energies. The present finding will establish a theoretical foundation for the analysis of the Sunyaev-Zeldovich effects for the clusters of galaxies.
Based upon the rate equations for the photon distribution function obtained in the previous paper, we study the formal solutions in three different representation forms for the Sunyaev-Zeldovich effect. By expanding the formal solution in the operator representation in powers of both the derivative operator and electron velocity, we derive a formal solution that is equivalent to the Fokker-Planck expansion approximation. We extend the present formalism to the kinematical Sunyaev-Zeldovich effect. The properties of the frequency redistribution functions are studied. We find that the kinematical Sunyaev-Zeldovich effect is described by the redistribution function related to the electron pressure. We also solve the rate equations numerically. We obtain the exact numerical solutions, which include the full-order terms in powers of the optical depth.
Studying galaxy clusters through their Sunyaev-Zeldovich (SZ) imprint on the Cosmic Microwave Background has many important advantages. The total SZ signal is an accurate and precise tracer of the total pressure in the intra-cluster medium and of cluster mass, the key observable for using clusters as cosmological probes. Band 5 observations with SKA-MID towards cluster surveys from the next generation of X-ray telescopes such as e-ROSITA and from Euclid will provide the robust mass estimates required to exploit these samples. This will be especially important for high redshift systems, arising from the SZs unique independence to redshift. In addition, galaxy clusters are very interesting astrophysical systems in their own right, and the SKAs excellent surface brightness sensitivity down to small angular scales will allow us to explore the detailed gas physics of the intra-cluster medium.
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