Intergalactic dust and its photoelectric heating
  We have examined the dust photoelectric heating in the intergalactic medium (IGM). The heating rate in a typical radiation field of the IGM is represented by $\Gamma_{\rm pe} = 1.2\times10^{-34}$ erg s$^{-1}$ cm$^{-3}$ $({\cal D}/10^{-4})(n_{\rm H}/10^{-5} {\rm cm^{-3}})^{4/3} (J_{\rm L}/10^{-21} {\rm erg s^{-1} cm^{-2} Hz^{-1} sr^{-1}})^{2/3} (T/10^4 {\rm K})^{-1/6}$, where ${\cal D}$ is the dust-to-gas mass ratio, $n_{\rm H}$ is the hydrogen number density, $J_{\rm L}$ is the mean intensity at the hydrogen Lyman limit of the background radiation, and $T$ is the gas temperature, if we assume the new X-ray photoelectric yield model by Weingartner et al. (2006) and the dust size distribution in the Milky Way by Mathis, Rumpl, & Nordsieck (1977). This heating rate dominates the HI and HeII photoionization heating rates when the hydrogen number density is less than $\sim10^{-6}$ cm$^{-3}$ if ${\cal D}=10^{-4}$ which is 1% of that in the Milky Way, although the heating rate is a factor of 2--4 smaller than that with the old yield model by Weingartner & Draine (2001). The grain size distribution is very important. If only large ($\ge0.1$ $\mu$m) grains exist in the IGM, the heating rate is reduced by a factor of $\simeq5$. Since the dust heating is more efficient in a lower density medium relative to the photoionization heating, it may cause an inverted temperature--density relation in the low density IGM suggested by Bolton et al. (2008). Finally, we have found that the dust heating is not very important in the mean IGM before the cosmic reionization. 
Akio K. Inoue (Osaka Sangyo Univ.) and Hideyuki Kamaya (National   Defense Academy of Japan)