We have modelled the evolution of the number of Ly \alpha absorbers with redshift , resulting from the evolution of the ionising background and the Hubble expansion .
The contribution of quasars ( QSOs ) and galaxies to the H i -ionising UV background has been estimated .
The QSOs emissivity is derived from recent fits of their luminosity function .
The galaxy emissivity is computed using a stellar population synthesis model , with a star-formation history scaled on observations of faint galaxies at \lambda \geq 1500 Å .
We allow for three values of the fraction of ionising photons that can escape the interstellar medium , f _ { \mathrm { esc } } = 0.05 , 0.1 and 0.4 .
The Intergalactic Medium is modelled as made of purely-absorbing clouds with the distribution in redshift and column density obtained from QSOs absorption lines .
For the adopted values of f _ { \mathrm { esc } } , the contribution of galaxies to the ionising UV background is comparable or greater than that of QSOs .
Accounting for the contribution of clouds to the UV emission , all models with f _ { \mathrm { esc } } \la 0.1 provide an ionising flux compatible with local and high- z determination , including those with a pure QSOs background .
The observed z \sim 1 break in the evolution can be better explained by a dominant contribution from galaxies .
We find that models in \Lambda -cosmology with \Omega _ { \mathrm { m } } =0.3 , \Omega _ { \Lambda } =0.7 describe the flat absorbers evolution for z \la 1.0 better than models for \Omega _ { \mathrm { m } } =1.0 .