Modelling cosmic ray electron physics in cosmological smoothed particle hydrodynamics simulation

Cosmic ray electron (CRE) acceleration and cooling are important physical processes in astrophysics. We develop an approximative framework to treat CRE physics in the parallel smoothed particle hydrodynamics code Gadget-3. In our methodology, the CRE spectrum of each fluid element is approximated by a single power-law distribution with spatially varying amplitude, upper cut-off, lower cut-off, and spectral index. We consider diffusive shock acceleration to be the source of injection, and oppositely the sinking processes is attributed to synchrotron radiation, inverse Compton scatters, and Coulomb scatters. The adiabatic gains and losses are also included. We show that our formalism produces the energy and pressure with an accuracy of $ > 90\%$ for a free cooling CRE spectrum. Both slope and intensity of the radio emission computed from the CRE population given by our method in cosmological hydro-simulation coincide well with observations, and our results also show that relaxed clusters have lower fluxes. Finally, we investigate several impacts of the CRE processes on the cosmological hydro-simulation, we find that: (1) the pressure of the CRE spectrum is very small and can be ignored in hydro-simulation, (2) the impacts of the CRE processes on the gas phase-space state of hydro-simulation is up to $3\%$, (3) the CRE processes induce a $5\%$ influence on the mass function in the mass range $10^{12} -10^{13} h^{-1} M_{\odot}$, (4) The gas temperature of massive galaxy cluster is influenced by the CRE processes up to $\sim 10\%$.