Quasi Stellar Objects (QSOs) or quasars were defined originally as star-like objects of large redshift. They are powered by the accretion of matter onto supermassive black holes (SMBHs). The QSOs are considered to be the most luminous subclass of Active Galactic Nuclei (AGNs) [1] . Soon after the discovery of QSO [2] , their population was observed to evolve strongly with redshift. 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 QSO luminosity function and its evolution with redshift provide important clues about the demographics of the AGN population and strong constraints on physical models and evolutionary theories of AGN [4] - [6] . The faint-end slope of the luminosity function is a measure of how much time QSOs spend at relatively low accretion rates. On the other hand, the bright-end slope tells us about the intrinsic properties of the QSO population during the time when black holes were increasing in mass most rapidly (e.g. triggering rate, active black mass function, etc.) [7] .

The differential QSO luminosity function is defined as the number density of QSOs per unit comoving volume, and per unit luminosity as a function of luminosity and redshift [8] [9] . The luminosity function is usually derived by using the method (e.g. [10] - [14] ). The most common analytical representation for the shape of the QSO luminosity function in the literature is a double power-law (e.g. [6] [15] - [22] ). In this paper, we use the Schechter function model [23] to describe the shape of the QSO luminosity function. In earlier papers such as Goldschmidt et al. (1998) [24] , Warren et al. (1994) [25] and Singh et al. (2016) [26] the Schechter function is found to represent the shape of the QSO luminosity function. Using the Edinburgh UVX quasar survey, Goldschmidt et al. (1998) [24] used the Schechter function model with the evolution of the characteristic magnitude, to fit the quasar luminosity function at the redshift range 1.7 ≤ z ≤ 2.2. The fit is observed to be acceptable with a significance level for rejection of 10%. Warren et al. (1994) [25] used the Schechter function model with evolution of the characteristic magnitude of the form usinga wide-field multicolor survey for high redshift quasars (z ≥ 2.2), where τ is the look-back time. Singh et al. (2016) [26] has shown that the shape of the QSO luminosity functions is adequately represented by the Schechter function with second order polynomial evolution model from the 2QZ and 6QZ samples in the redshift range 0.3 ≤ z ≤ 2.4.

Historically, there are two fundamental models for the evolution of QSOs namely, pure luminosity evolution (PLE) and pure density evolution (PDE). If the redshift and luminosity dependence are separable, the evolution of the luminosity function can be modelled in terms of the PLE where QSO luminosities change with time, but the total number of QSO remains constant and the PDE where the number density of QSOs changes but their luminosities remain constant. Various hybrid models such as luminosity and density evolution model (LEDE) and luminosity-dependent density evolution model (LDDE) are also used to describe the evolution of QSOs with redshift. Ross et al. (2013) [21] and Croom et al. (2009) [7] presented the luminosity function evolved with LEDE where the bright-end and faint-end slopes have fixed values and normalization and characteristic luminosity evolve independently. Croom et al. (2009) [7] and Bongiorno et al. (2007) [27] used the LDDE to study the luminosity evolution of QSOs. The luminosity evolution of QSOs with redshift in this paper is described by the PLE.

In section 2 we give a brief description of the SDSS sample. The determination of the binned optical luminosity function and its evolutionary behaviors are presented in section 3. Finally in section 4 we give our conclusion. Throughout this paper we use a Ʌ cosmology with, , and H_o = 70.0 km×s⁻¹×Mpc⁻¹.

The Sloan Digital Sky Survey Data Release Seven (SDSS DR7) [28] uses a CCD camera [29] on a dedicated 2.5 m wide field telescope [30] located at Apache Point Observatory (APO) near the Sacramento peak in Southern New Mexico, to obtain images in five photometric bands: u, g, r, i and z [29] [31] over approximately 10,000 square degrees of high Galactic latitude sky in the Northern Hemisphere [32] . The survey data-processing 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 [33] - [35] . The photometricis calibrated to an AB system [36] and the photometric measurements are reported as asinh magnitudes [37] . The spectroscopy is performed by using a 640-fibre-fed pair of multiobject double spectrographs with coverage from 3800 Å to 9200 Å and a resolution of of roughly 2000 [28] .

The final SDSS DR7 quasars catalog from SDSS I/II was presented in Schneider et al. (2010) [38] which contains 105,783 spectroscopically confirmed quasars and the redshift distribution of these QSOs is also shown in Singh et al. (2014) [39] . The catalog consists of quasars that have a luminosity larger than M_i = −22.5 (calculated assuming Ʌ cosmology with, , and H_o = 70.0 km×s⁻¹×Mpc⁻¹) and have at least one emission line with full width at half-maximum (FWHM) larger than 1000 km×s⁻¹ or have interesting/complex absorption features, and also that are fainter than i = 15.0, and have highly reliable redshift [38] . About half of these objects are selected uniformly using the final quasar target selection algorithm described in Richards et al. (2002) [40] , and form a homogeneous, statistical quasar sample. In this homogeneous 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 deg² [41] .

The optical luminosity function of QSOs is determined by using the method [42] . It is given by

with a Poisson statistical uncertainty

where is the volume corresponding to the maximum distance that object j could be observed, and still be included in the sample. The summation is over all quasars within a redshift-magnitude bin.

Figure 1 shows the binned optical luminosity function of QSOs (indicated by filled circle points) for nine redshift intervals i.e. 0.3 ≤ 0.5; 0.5 ≤ z ≤ 0.7; 0.7 ≤ z ≤ 0.9; 0.9 ≤ z ≤ 1.1; 1.1 ≤ z ≤ 1.3; 1.3 ≤ z ≤ 1.5; 1.5 ≤ z ≤ 1.7; 1.7 ≤ z ≤ 1.9 and 1.9 ≤ z ≤ 2.4. The bin size of absolute magnitude, is 0.3 mag.

The shape of the QSO luminosity function is fitted by the Schechter function model [23] . The generalized form of the Schechter function model is given by

which, in terms of absolute magnitude, becomes

where is a normalization parameter whose dimension is the number density of objects, α is the faint-end slope of the luminosity function and is the characteristic luminosity (with an equivalent characteristic absolute magnitude,). The evolution of the luminosity function is described by the redshift dependence of the characteristic luminosity or magnitude. We have modelled this evolution as a second-order polynomial in redshift of the form

or in terms of absolute magnitude,

By using equations (3) and (4), we fit the PLE model to the binned optical luminosity function determined by Shen & Kelly (2012) [41] in various redshift ranges. The best-fitting parameters are determined from the PLE model by using the Levenberg-Marquardt method of nonlinear least square fit [43] . The resulting best-fitting parameters in various redshift ranges are listed in Table 1. In Figure 1, the solid lines represent the Schechter function model with PLE fit to the observed luminosity functions of QSOs. In assessing the goodness-of-fit, we measure the value by comparing the observed luminosity function and theoretical luminosity function predicted by best fit model. A comparison of the model luminosity function to the binned luminosity function gives for 0.3 ≤ z ≤ 2.4. But, if we restrict the redshift range being fit, we obtain significant improvement with for the redshift range 1.5 ≤ z ≤ 2.4. Thus, the Schechter function model can be regarded as simply one way of describing the basic shape of the QSO luminosity function which displays a steepening above the characteristic luminosity (or below a characteristic absolute magnitude). From Figure 1, it is clear that there is in general good agreement between the model and data. However, there are more bright QSOs than predicted by the model at the bright end of the luminosity function which is due to the exponential decrease in the Schechter function model at.

The shape of the luminosity function of QSOs and its evolution with redshift are studied by using the Schechter function model with PLE. The best fitting parameter values for the model are determined by using the Levenberg-Marquardt method of nonlinear least square fit. For the Schechter function model the dimensionless parameter α gives the slope of the luminosity function for QSOs fainter than the characteristic luminosity (i.e.); for QSOs more luminous than the characteristic luminosity (i.e.), the luminosity function drops exponentially with luminosity. A comparison of the values and the number of degrees of freedom shows that the Schechter function model with polynomial evolution of the characteristic magnitude provides acceptable fit to the QSO luminosity function.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web site is http://www.sdss.org/.

One of the authors (Salam Ajitkumar Singh) is grateful to the Indian Space Research Organization, Department of Space, Government of India for providing JRF under a RESPOND Project (ISRO/RES/2/385/2013-14).

Salam Ajitkumar Singh,I. Ablu Meitei,K. Yugindro Singh, (2016) Schechter Function Model for the QSO Luminosity Function from the SDSS DR7. International Journal of Astronomy and Astrophysics,06,247-253. doi: 10.4236/ijaa.2016.63020