We present the result of our low-luminosity quasar survey in the redshift range of 4.5 \lesssim z \lesssim 5.5 in the COSMOS field .
Using the COSMOS photometric catalog , we selected 15 quasar candidates with 22 < i′ < 24 at z \sim 5 , that are \sim 3 mag fainter than the SDSS quasars in the same redshift range .
We obtained optical spectra for 14 of the 15 candidates using FOCAS on the Subaru Telescope and did not identify any low-luminosity type-1 quasars at z \sim 5 while a low-luminosity type-2 quasar at z \sim 5.07 was discovered .
In order to constrain the faint end of the quasar luminosity function at z \sim 5 , we calculated the 1 \sigma confidence upper limits of the space density of type-1 quasars .
As a result , the 1 \sigma confidence upper limits on the quasar space density are \Phi < 1.33 \times 10 ^ { -7 } Mpc ^ { -3 } mag ^ { -1 } for -24.52 < M _ { 1450 } < -23.52 and \Phi < 2.88 \times 10 ^ { -7 } Mpc ^ { -3 } mag ^ { -1 } for -23.52 < M _ { 1450 } < -22.52 .
The inferred 1 \sigma confidence upper limits of the space density are then used to provide constrains on the faint-end slope and the break absolute magnitude of the quasar luminosity function at z \sim 5 .
We find that the quasar space density decreases gradually as a function of redshift at low luminosity ( M _ { 1450 } \sim - 23 ) , being similar to the trend found for quasars with high luminosity ( M _ { 1450 } < -26 ) .
This result is consistent with the so-called downsizing evolution of quasars seen at lower redshifts .