The recently discovered population of ultra-faint extended line emitters , with fluxes of a few times 10 ^ { -18 } ~ { } \textrm { erg s } ^ { -1 } ~ { } \textrm { cm } ^ { -2 } at z \sim 3 , can account for the majority of the incidence rate of Damped \textrm { Ly } \alpha systems ( DLAs ) at this redshift if the line emission is interpreted as \textrm { Ly } \alpha .
We show here that a model similar to that proposed by Haehnelt , Steinmetz , & Rauch ( 16 ) , which explains the incidence rate and kinematics of DLAs in the context of \Lambda CDM models for structure formation , also reproduces the size distribution of the new population of faint \textrm { Ly } \alpha emitters for plausible parameters .
This lends further support to the interpretation of the emission as \textrm { Ly } \alpha , as well as the identification of the emitters with the hitherto elusive population of DLA host galaxies .
The observed incidence rate of DLAs together with the observed space density and size distribution of the emitters suggest a duty cycle of \sim 0.2 - 0.4 for the \textrm { Ly } \alpha emission from DLA host galaxies .
We further show that \textrm { Ly } \alpha cooling is expected to contribute little to the \textrm { Ly } \alpha emission for the majority of emitters .
This leaves centrally concentrated star formation at a rate of a few tenths M _ { \odot } yr ^ { -1 } , surrounded by extended \textrm { Ly } \alpha halos with radii up to 30-50 kpc , as the most plausible explanation for the origin of the emission .
Both the luminosity function of \textrm { Ly } \alpha emission and the velocity width distribution of low ionization absorption require that galaxies inside Dark Matter ( DM ) halos with virial velocities \la 50 - 70 ~ { } \textrm { km } ~ { } \textrm { s } ^ { -1 } contribute little to the incidence rate of DLAs at z \sim 3 , suggesting that energy and momentum input due to star formation efficiently removes gas from these halos .
Galaxies with DM halos with virial velocities of 100 - 150 ~ { } \textrm { km } ~ { } \textrm { s } ^ { -1 } appear to account for the majority of DLA host galaxies .
DLA host galaxies at z \sim 3 should thus become the building blocks of typical present-day galaxies like our Milky Way .