We explore mechanisms to produce extremely high Ly-alpha/HeII flux ratios, or to enhance the observed number of Ly-alpha photons per incident ionizing photon, in extended AGN-photoionized nebulae at high-redshift. Using photoionization models, we explore the impact of ionization parameter, gas metallicity, ionizing spectrum, electron energy distribution, and cloud viewing angle on the relative fluxes of Ly-alpha, HeII and other lines, and on the observed number of Ly-alpha photons per incident ionizing photon. We find that low ionization parameter, a relatively soft or filtered ionizing spectrum, low gas metallicity, kappa-distributed electron energies, or reflection of Ly-alpha photons by HI can all result in significantly enhanced Ly-alpha relative to other lines (>10%), with log Ly-alpha/HeII reaching values up to 4.6. In the cases of low gas metallicity, reflection by HI, or a hard or filtered ionizing spectrum, the observed number of Ly-alpha photons per incident ionizing photon is itself significantly enhanced above the nominal Case B value of 0.66 due to collisional excitation, reaching values up to 5.3 in our extreme case model. At low gas metallicity (e.g. 0.1 x Solar), the production of Ly-alpha is predominantly via collisional excitation rather than recombination. In addition, we find that collisional excitation of Ly-alpha becomes more efficient if the ionizing continuum is pre-filtered through an optically thin screen of gas closer to the AGN. We also show that Ly-alpha / HeII ratios of the z~3.5 quasars studied by Borisova et al. (2016) are consistent with AGN-photoionization of gas with moderate to low metallicity and/or low ionization parameter, without requiring exotic ionization/excitation mechanisms such as strong line-transfer effects. We also present UV-optical diagnostic diagrams to distinguish between photoionization by Pop III stars and AGN photoionization.
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