Context :

Aims : We explore potential 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 .

Methods : We compute models to simulate , in the low density regime , photoionization of interstellar gas by the radiation field of a luminous AGN .
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 compare our model results with recent observations of quasar Ly \alpha halos at z \sim 3.5 .

Results : Low ionization parameter , a relatively soft or filtered ionizing spectrum , low gas metallicity , \kappa -distributed electron energies , or reflection of Ly \alpha photons by neutral hydrogen can all result in significantly enhanced Ly \alpha relative to other lines ( \geq 10 % ) , with log Ly \alpha /HeII reaching values of 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 as high as 5.3 in an ‘ extreme case ’ model which combines several of these effects .
We find that at low gas metallicity ( e.g . Z / Z _ { \odot } =0.1 ) the production of Ly \alpha photons is predominantly via collisional excitation rather than by recombination .
In addition , we find that the collisional excitation of Ly \alpha becomes much more efficient if the ionizing continuum spectrum has been pre-filtered through an optically thin screen of gas closer to the AGN ( e.g. , by a wide-angle , feedback-driven outflow . )
We also show that the Ly \alpha and HeII emission line ratios of the z \sim 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 .
In addition , we present a set of UV-optical diagnostic diagrams to distinguish between photoionization by Pop III stars and photoionization by an AGN .

Conclusions :