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We examine the Hogan-Rees photoionization instability (Hogan 1992, Nature 359, 40) in the context of an $\Omega=1$ universe dominated by massive ($m_\nu\approx 30$ eV) decaying neutrinos. In a medium with a smoothly distributed source of ionizing radiation, the photoionization and heating rates on scales larger than the photon mean free path are independent of the local gas density. Thus, underdense regions receive more energy per particle and heat up faster; this nonadiabatic temperature change produces a pressure term which drives the growth of fluctuations.
Hogan (1992) showed that in a static medium this instability produces exponential growth, with growth rates which can be much larger than the expansion rate in the expanding universe. We have found that on small scales (comoving wavenumber $k > k_m$, where $k_m$ corresponds to $\lambda\sim 10^{-2}$ Mpc present-day), the growth remains exponential in an expanding universe. The instability growth rate is independent of scale for $k > k_m$, and declines rapidly with increasing scale, so the characteristic mass produced by the instability will correspond to $k\sim k_m$. For a neutrino energy above the Lyman limit $\Delta E$ $(\approx m_\nu/2-13.6$ eV) of a few eV and a decay lifetime $T\sim 10^{24}$ seconds, fluctuations at the Poisson level on the scale $k_m$ can grow to non-linearity between $z\sim 70$ (when Compton cooling inhibits the instability) and $z\sim 20$ (when the intergalactic medium becomes ionized).
