Optical Luminosity Function of Quasi Stellar Objects

We study the shape of the optical luminosity function of Quasi Stellar Objects (QSOs) from the Sloan Digital Sky Survey Data Release Seven (SDSS DR7) over the redshift range 0.3 ≤ ≤ 2.4. By using the Levenberg-Marquardt method of nonlinear least square fit, the observed QSO luminosity function is fitted by a double power-law model with luminosity evolution characterized by a second order polynomial in redshift. For a flat universe with Ωm=0.3 and ΩΛ=0.7, we determine the best-fitting optical luminosity function model parameters.


Introduction
Quasi Stellar Objects (QSOs) or quasars are intrinsically luminous subclass of Active Galactic Nuclei (AGNs) and these objects represent a fascinating and unique population of objects at the intersection of cosmology and astrophysics [1]. Soon after the discovery of QSOs their population was observed to be evolving [2]. As a result these objects provide a unique tool in the study of galaxies and large-scale structure formation throughout the history of the universe [3]. The luminosity function of QSOs describes the distribution of these objects in space as a function of their luminosity. Such a function also depends on other properties e.g. environment, evolutionary stage of the universe etc. [4]. If one specifies the dependence of mass on luminosity one can use the luminosity function to determine the local mass density [5]. Due to strong evolution of QSOs with cosmic time the luminosity function of the QSOs must be of particular importance in the understanding of formation and evolution of the QSOs. The QSO luminosity function and its evolution with redshift are the most important tools to constrain the accretion history of supermassive black holes (SMBHs) [6] and provide important clues about the demographics of the AGN population and constraints on physical models and evolutionary theories of AGN [7][8][9]. The faint-end slope of the luminosity function is a measure of how much time QSOs spend at relatively low accretion rates. The bright-end slope, on the other hand, tells us about the intrinsic properties of the QSO population during the time when black holes were increasing in mass most rapidly [10]. The QSOs are also strong X-ray sources, thus the QSO luminosity function can provide important constraints on the contribution of QSOs to the X-ray and UV background radiation [11][12][13][14].
The QSO luminosity function provides an essential constraint on how the population characteristics have changed with time [11]. The differential quasar luminosity function is defined as the number of quasars per unit comoving volume, per unit luminosity as a function of luminosity and redshift [15,16]. The luminosity function of QSOs is usually calculated by the classical 1 ⁄ method (e.g. [17][18][19][20]). The most common analytical representation for the shape of QSO luminosity function in the literature is a double power-law model (e.g. [1,[21][22][23][24]). The Schechter function model [25] can also be used to represent the shape of QSO luminosity function in the previous papers such as Goldschmidt & Miller (1998) [26], Warren et al. (1994) [27],   [28] and   [29].
The evolution of the luminosity function can be generally modeled as pure luminosity evolution (PLE) or pure density evolution (PDE) [2]. Under the PLE scenario the number density of QSOs remains constant with redshift, but their luminosities change with time and under the PDE scenario the number density of QSOs changes but their luminosities remain constant. More complex models are also used to describe the evolution of luminosity function namely the Luminosity Dependent Density Evolution (LDDE) and the Luminosity Evolution and Density Evolution (LEDE). Ross et al. (2013) [1] and Croom et al. (2009) [10] presented the LEDE where the bright-end and faint-end slopes have fixed values and normalization and characteristic luminosity evolve independently. Croom et al. (2009) [10] and Bongiorno (2007) [30] used the LDDE to study the evolution of QSO luminosity function. In this paper, the luminosity evolution with redshift is described by PLE. Throughout this paper we use a flat universe with Ω m =0.3, Ω Λ =0.7 and H 0 =70.0 km s -1 Mpc -1 .
In section 2, we give a brief description of the SDSS sample. The determination of the optical luminosity function of QSOs and its statistical analysis are described in section 3 and in section 4 we discuss the comparison between the SDSS QSO luminosity function and some other surveys of QSO luminosity function. Finally, in section 5 we give our conclusions.

The SDSS Survey
The Sloan Digital Sky Survey (SDSS) [31] is an optical survey that has mapped more than 10,000 square degrees of sky located in the northern galactic hemisphere and partially along the Celestial Equator. The SDSS uses a dedicated wide field 2.5 m altitude-azimuth telescope [32] located at Apache Point Observatory (APO) near the Sacramento peak in Southern New Mexico. The telescope has 3 0 diameter field of view, and it was equipped with a large-format moisac of 30 2048 2048 Tektronix CCD cameras [33] that took the images in five photometric bands: u, g, r, i and z [33,34]. The survey dataprocessing software measures the properties of each detected object in the imaging data in all five photometric bands and determines and applies both astrometric and photometric calibrations [35][36][37]. Photometric calibration is provided by simultaneous observations with a 20 inch (0.51 m) telescope at the same site [35,38]. The photometric system is based on the AB magnitude scale [39] and the photometric measurements are reported as asinh magnitudes [40].
The SDSS DR7 quasar catalog is described in [41]. It consists of 105,783 spectroscopically confirmed quasars that are brighter than =-22.0, have at least one broad emission line (FWHM 1000 km s -1 ), and have highly reliable redshifts [41]. The redshift distribution for SDSS DR7 quasars is also shown in Singh et al. (2014) [42]. About half of these objects were targeted by using the final quasar target selection algorithm described in [43], and form a homogeneous, statistical quasar sample. In this homogenous sample, quasars are flux-limited to i=19.1 for z 2.9 and to i=20.2 for z 2.9. The sky coverage of this uniform quasar sample is 6248 deg 2 [44].

The QSO Optical Luminosity Function and Model Fitting
The luminosity function of QSOs is determined by using the 1 ⁄ method [2]. It is given by and its Poisson statistical uncertainty is where , is the comoving volume within which the j th object would be included in the sample. The summation is over all quasars within a redshift-magnitude bin.
The QSO luminosity function is calculated at absolute magnitude intervals of 0.  The double power-law is used to fit the observed QSO luminosity function. The parametric form of the double powerlaw model, expressed in absolute magnitude, is given by 10 10 where α and β are respectively the faint-end and bright-end slopes of the luminosity function. * is the characteristic or break absolute magnitude. The evolution of the luminosity function is described by the redshift dependence of the characteristic magnitude ( * which is chosen as a secondorder polynomial in redshift such that Using equations (3) and (4), we fit the PLE model to the observed luminosity function of QSOs in various redshift ranges. The best fit parameter values are estimated by using the Levenberg-Marquardt method of non linear least square fit [45]. The resulting best-fitting parameters in various redshift ranges are given in Table 1. The double power-law model with PLE fitted to the observed luminosity function of QSOs is shown in Figure 1 (denoted by solid lines)

Comparison to other Luminosity Functions
In Figure  M z M = = − ± [10,18]. In Figure 2, the black lines in each panel are the best model fits to the SDSS DR7 QSO luminosity functions. The green, blue and red lines represent the best fitting double power-law model with a second order polynomial luminosity evolution from the 2QZ/6QZ, 2SLAQ and 2SLAQ+SDSS, respectively. In all redshift ranges, the SDSS DR7 predicts a higher density and steeper slope than the other surveys at the faint end of the luminosity function. At the bright end, the SDSS DR7 has slightly steeper slope of the luminosity function than the other surveys.

Conclusions
The luminosity function of the QSOs in the SDSS DR7 and its evolution with redshift are studied by using the double power-law model with PLE. Croom