Cosmic ray heating of intergalactic medium: patchy or uniform?

We study the heating of the intergalactic medium (IGM) surrounding high redshift star forming galaxies due to cosmic rays (CR). We take into account the diffusion of low energy cosmic rays and study the patchiness of the resulting heating. We discuss the case of IGM heating around a high redshift minihalo ($z\sim 10\hbox{--}20$, M$\sim 10^5\hbox{--}10^7$ M$_\odot$),and put an upper limit on the diffusion coefficient $D\le 1\times 10^{26}$ cm$^2$ s$^{-1}$ for the heating to be inhomogeneous at $z\sim 10$ and $D\le 5\hbox{--}6 \times 10^{26}$ cm$^2$ s$^{-1}$ at $z\sim 20$. For typical values of $D$, our results suggest uniform heating by CR at high redshift, although there are uncertainties in magnetic field and other CR parameters. We also discuss two cases with continuous star formation, one in which the star formation rate (SFR) of a galaxy is high enough to make the IGM in the vicinity photoionized, and another in which the SFR is low enough to keep it neutral but high enough to cause significant heating by cosmic ray protons. In the neutral case (low SFR), we find that the resulting heating can make the gas hotter than the cosmic microwave background (CMB) radiation for $D < 10^{30}$ cm$^2$ s$^{-1}$, within a few kpc of the galaxy, and unlikely to be probed by near future radio observations. In the case of photoionized IGM (high SFR), the resulting heating of the gas in the vicinity of high redshift ($z\sim 4)$ galaxies of mass $\ge 10^{12}$ M$_\odot$ can suppress gas infall into the galaxy. At lower redshifts ($z\sim 0$), an SFR of $\sim 1$ M$_\odot$ yr$^{-1}$ can suppress the infall into galaxies of mass $\le 10^{10}$ M$_\odot$.