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 CR and study the patchiness of the resulting heating. We discuss the case of IGM heating around a high redshift minihalo (z similar to 10-20, M similar to 10(5)-10(7) M-circle dot) and put an upper limit on the diffusion coefficient D <= 1 x 10(26) cm(2) s(-1) for the heating to be inhomogeneous at z similar to 10 and D <= 5-6 x 10(26) cm(2) s(-1) at z similar to 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 healing by CR protons. In the neutral case (low SFR), we find that the resulting heating can make the gas holler than the cosmic microwave background 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 similar to 4) galaxies of mass >= 10(12) M-circle dot can suppress gas infall into the galaxy. At lower redshifts (z similar to 0), an SFR of similar to 1 M-circle dot yr(-1) can suppress the infall into galaxies of mass <= 10(16) M-circle dot.