Cosmic Ray Heating in the Early Universe: Joule Heating by Return Currents and its Impact on the Thermal Evolution of the Intergalactic Medium at Redshift around 10

We investigate heating mechanisms of the neutral intergalactic medium (IGM) around galaxies at $z\sim 10$ with a focus on cosmic ray (CR) effects. Besides conventional X-ray heating and direct CR heating via ionization and Coulomb collisions, we examine CR-driven resistive heating caused by return currents induced by streaming CRs. We include both Coulomb and charge-neutral collisions in modeling the resistivity and account for their dependence on temperature and ionization degree. Numerical results show that CR-driven resistive heating can dominate over the other mechanisms, rapidly increasing the IGM temperature to $10^4 \, \mathrm{K}$ near the galaxy and creating a hot region with $T\gtrsim 10^2 \, \mathrm{K}$ extending to $\sim 30 \, \mathrm{kpc}$. The size and ionization state of this region depend on the star formation rate, suggesting that future $21$-$\mathrm{cm}$ line observations could probe the impact of CRs on the early IGM.