A left and right truncated Schechter luminosity function for quasars

TL;DR

This paper introduces a truncated Schechter luminosity function for quasars that accounts for redshift-dependent luminosity boundaries, providing improved modeling of quasar distributions across different redshifts.

Contribution

The paper develops an analytical form for a truncated Schechter function for quasars, incorporating lower and upper luminosity boundaries and comparing it with existing models.

Findings

Derived an analytical average value for the truncated Schechter function.

Applied the model to low and high redshift quasar data.

Compared the truncated Schechter function with other luminosity functions.

Abstract

The luminosity function for quasars (QSOs) is usually fitted by a Schechter function. The dependence of the number of quasars on the redshift, both in the low and high luminosity regions, requires the inclusion of a lower and upper boundary in the Schechter function. The normalization of the truncated Schechter function is forced to be the same as that for the Schechter function, and an analytical form for the average value is derived. Three astrophysical applications for QSOs are provided: deduction of the parameters at low redshifts, behavior of the average absolute magnitude at high redshifts, and the location (in redshift) of the photometric maximum as a function of the selected apparent magnitude. The truncated Schechter function with the double power law and an improved Schechter function are compared as luminosity functions for QSOs. The chosen cosmological framework is that of the flat cosmology, for which we provided the luminosity distance, the inverse relation for the luminosity distance, and the distance modulus.

Tables

Table 1. Table 1: Parameters of the truncated Schechter LF in the range of redshifts [ 0.3 , 0.5 ] 0.3 0.5 [0.3,0.5] when n=10 and k=5.

Table 2. Table 2: Parameters of the Schechter LF in the range [ 0.3 , 0.5 ] 0.3 0.5 [0.3,0.5] when k=3 and n=10.

Table 3. Table 3: Parameters of the double power LF in the range of redshifts [ 0.3 , 0.5 ] 0.3 0.5 [0.3,0.5] when n=10 and k=4.

Table 4. Table 4: Parameters of the Pei LF in the range of redshifts [ 0.3 , 0.5 ] 0.3 0.5 [0.3,0.5] with k=3 and n=10.

Equations

$$d_{\mathrm{L}}(z;c,H_0,{\Omega }_{\mathrm{M}})=\frac{c}{H_0}(1+z){\int }_{\frac{1}{1+z}}^1\frac{da}{\sqrt{{\Omega }_{\mathrm{M}}a+(1-{\Omega }_{\mathrm{M}})a^4}},$$

$${\Omega }_{\mathrm{M}}=\frac{8\pi G{\rho }_0}{3H_0^2},$$

$$d_{\mathrm{L}}(z;c,H_0,{\Omega }_{\mathrm{M}})=\frac{c}{H_0}(1+z){\int }_0^z\frac{1}{\sqrt{{\left(1+z\right)}^3{\Omega }_{\mathrm{M}}+1-{\Omega }_{\mathrm{M}}}}\mathrm{d}z,$$

$$d_{\mathrm{L}}(z)=4282.74(1+z)(2.45214+0.01506(1+z{)}^{-8}-0.06633(1+z{)}^{-7}$$

$$-0.01619(1+z{)}^{-6}+0.60913(1+z{)}^{-5}-1.29912(1+z{)}^{-4}+0.406124(1+z{)}^{-3}$$

$$+2.47428(1+z{)}^{-2}-4.57509(1+z{)}^{-1})Mpc$$

$$(m-M)=F(z)=43.15861+2.17147\ln (7.77498z^2+2.45214z^8+15.0420z^7+39.1085z^6$$

$$+56.4947z^5+49.3673z^4+26.1512z^3+0.99999z+1.7310^{-7})-15.2003\ln (1+z).$$

$$M=m-F(z).$$

$$D_A=\frac{D_L}{(1+z{)}^2}.$$

$$d_{L,2,1}(z)=\frac{4.10871+1813.96z+2957.04z^2}{0.44404+0.27797z}$$

$$z_{2,1}(d_{\mathrm{L}})=0.000047d_{\mathrm{L}}-0.306718$$

$$+3.38175\times 10^{-14}\sqrt{1.931810^{18}{d_{\mathrm{L}}}^2+1.06093\times 10^{23}{d}_{\mathrm{L}}+8.10464\times 10^{25}}.$$

$$D_M=\frac{D_L}{1+z},$$

$$V_c=\frac{4}{3}\pi D_M^3,$$

$$V_{c,3,2}=\frac{3.0148410^{10}{z}^3+6.3969910^{10}{z}^2-1.2679310^{10}z+4.1010410^8}{0.45999-0.01011z+0.093371z^2}Mp{c}^3.$$

$${\chi }^2=\sum _{j=1}^n(\frac{L{F}_{theo}-L{F}_{astr}}{{\sigma }_{L{F}_{astr}}}{)}^2,$$

$$RSS=\sum _{j=1}^n(y(i{)}_{theo}-y(i{)}_{astr}{)}^2,$$

$${\chi }_{red}^2={\chi }^2/NF,$$

$$AIC=2k-2ln(L),$$

$$AIC=2k+{\chi }^2.$$

$$\Phi(L;\Phi^{*},\alpha,L^{*})dL=(\frac{\Phi^{*}}{L^{*}})(\frac{L}{L^{*}})^{\alpha}\exp\bigl{(}{-\frac{L}{L^{*}}}\bigr{)}dL\quad,$$

$${\int }_0^{\infty }\Phi (L;{\Phi }^{\ast },\alpha ,L^{\ast })dL={\mathrm{\Phi }}^{\ast }\mathrm{\Gamma }\left(\alpha +1\right),$$

$$\mathrm{\Gamma }(\mathrm{z})={\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{t}}{\mathrm{t}}^{\mathrm{z}-1}\mathrm{dt},$$

$$⟨L({\Phi }^{\ast },\alpha ,L^{\ast })⟩={\mathrm{L}}^{\ast }{\mathrm{\Phi }}^{\ast }\mathrm{\Gamma }\left(\alpha +2\right).$$

$$\rm\Phi(M;\Phi^{*},\alpha,M^{*})dM=0.921\Phi^{*}10^{0.4(\alpha+1)(M^{*}-M)}\exp\bigl{(}{-10^{0.4(M^{*}-M)}}\bigr{)}dM\,,$$

$$S_T(L;{\Psi }^{\ast },\alpha ,L^{\ast },L_l,L_u)=\frac{-{\left(\frac{L}{L^{\ast }}\right)}^{\alpha }{\mathrm{e}}^{-\frac{L}{L^{\ast }}}{\Psi }^{\ast }\Gamma \left(\alpha +1\right)}{L^{\ast }\left(\Gamma \left(\alpha +1,\frac{L_u}{L^{\ast }}\right)-\Gamma \left(\alpha +1,\frac{L_l}{L^{\ast }}\right)\right)},$$

$$\Gamma \text{/}\left(a,z\right)={\int }_z^{\infty }{t}^{a-1}{e}^{-t}dt,$$

$${\int }_0^{\infty }{S}_T(L;{\Psi }^{\ast },\alpha ,L^{\ast },L_l,L_u)dL={\mathrm{\Psi }}^{\ast }\mathrm{\Gamma }\left(\alpha +1\right).$$

$$⟨L({\Psi }^{\ast },\alpha ,L^{\ast },L_l,L_u)⟩=\frac{N}{L^{\ast }\left(\Gamma \left(\alpha +1,\frac{L_u}{L^{\ast }}\right)-\Gamma \left(\alpha +1,\frac{L_l}{L^{\ast }}\right)\right)}$$

$$\displaystyle N={\it\Psi^{*}}\,\Bigg{(}{{\it L^{*}}}^{2}\Gamma\big{(}\alpha+1,{\frac{L_{{u}}}{{\it L^{*}}}}\big{)}\alpha-{{\it L^{*}}}^{2}\Gamma\big{(}\alpha+1,{\frac{L_{{l}}}{{\it L^{*}}}}\big{)}\alpha+{{\it L^{*}}}^{2}\Gamma\big{(}\alpha+1,{\frac{L_{{u}}}{{\it L^{*}}}}\big{)}$$

$$\displaystyle-{{\it L^{*}}}^{2}\Gamma\big{(}\alpha+1,{\frac{L_{{l}}}{{\it L^{*}}}}\big{)}-{{\it L^{*}}}^{-\alpha+1}{{\rm e}^{-{\frac{L_{{l}}}{{\it L^{*}}}}}}{L_{{l}}}^{\alpha+1}+{{\it L^{*}}}^{-\alpha+1}{{\rm e}^{-{\frac{L_{{u}}}{{\it L^{*}}}}}}{L_{{u}}}^{\alpha+1}\Bigg{)}\Gamma\big{(}\alpha+1\big{)}\quad.$$

$$\frac{L}{L_{\odot }}=10^{0.4(M_{\odot }-M)},\frac{L_l}{L_{\odot }}=10^{0.4(M_{\odot }-M_u)},\frac{L^{\ast }}{L_{\odot }}=10^{0.4(M_{\odot }-M^{\ast })},\frac{L_u}{L_{\odot }}=10^{0.4(M_{\odot }-M_l)}$$

Full text

\doinum

10.3390/—— \pubvolumexx

\historyReceived: xx / Accepted: xx / Published: xx \Title A left and right truncated Schechter luminosity function for quasars

\AuthorLorenzo Zaninetti 1,

\corres zaninetti@ph.unito.it

\PACS 98.54.-h 98.80.-k

1 Introduction

The Schechter function was first introduced in order to model the luminosity function (LF) for galaxies, see Schechter (1976), and later was used to model the LF for quasars (QSOs), see Warren et al. (1994); Goldschmidt and Miller (1998). Over the years, other LFs for galaxies have been suggested, such as a two-component Schechter-like LF, see Driver and Phillipps (1996), the hybrid Schechter+power-law LF to fit the faint end of the K-band, see Bell et al. (2003), and the double Schechter LF, see Blanton et al. (2005). In order to improve the flexibility at the bright end, a new parameter $\eta$ was introduced in the Schechter LF, see Alcaniz and Lima (2004). The above discussion suggests the introduction of finite boundaries for the Schechter LF rather than the usual zero and infinity. As a practical example the most luminous QSOs have absolute magnitude $M_{b_{j}}\approx-28$ or the luminosity is not $\infty$ and the less luminous QSOs have have absolute magnitude $M_{b_{j}}\approx-20$ or the luminosity is not zero, see Figure 19 in Croom et al. (2004) . A physical source of truncation at the low luminosity boundary ( high absolute magnitude ) is the fact that with increasing redshift the less luminous QSOs progressively disappear. In other words the upper boundary in absolute magnitude for QSOs is function of the redshift.

The suggestion to introduce two boundaries in a probability density function (PDF) is not new and, as an example, Coffey and Muller (2000) considered a doubly-truncated gamma PDF restricted by both a lower (l) and upper (u) truncation. A way to deduce a new truncated LF for galaxies or QSOs is to start from a truncated PDf and then to derive the magnitude version. This approach has been used to deduce a left truncated beta LF, see Zaninetti (2014, 2015), and a truncated gamma LF, see Zaninetti (2016).

The main difference between LFs for galaxies and for QSOs is that in the first case, we have an LF for a unit volume of 1 $Mpc^{3}$ and in the second case we are speaking of an LF for unit volume but with a redshift dependence. The dependence on the redshift complicates an analytical approach, because the number of observed QSOs at low luminosity decreases with the redshift and the highest observed luminosity increases with the redshift. The first effect is connected with the Malmquist bias, i.e. the average luminosity increases with the redshift, and the second one can be modeled by an empirical law. The above redshift dependence in the case of QSOs can be modeled by the double power law LF, see Boyle et al. (1988), or by an improved Schechter function, see Pei (1995). The present paper derives, in Section 2, the luminosity distance and the distance modulus in a flat cosmology. Section 3 derives a truncated version of the Schechter LF. Section 4 applies the truncated Schechter LF to QSOs, deriving the parameters of the LF in the range of redshift $[0.3,0.5]$, modeling the average absolute magnitude as a function of the redshift, and deriving the photometric maximum for a given apparent magnitude as a function of the redshift.

2 The flat cosmology

The first definition of the luminosity distance, $d_{\rm{L}}$, in flat cosmology is

[TABLE]

where $H_{0}$ is the Hubble constant expressed in $\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $c$ is the speed of light expressed in $\mathrm{km\,s^{-1}}$, $z$ is the redshift, $a$ is the scale-factor, and $\Omega_{\rm M}$ is

[TABLE]

where $G$ is the Newtonian gravitational constant and $\rho_{0}$ is the mass density at the present time, see eqn (2.1) in Adachi and Kasai (2012). A second definition of the luminosity distance is

[TABLE]

see eqn (2) in Mészáros and Řípa (2013). The change of variable $z=-1+1/a$ in the second definition allows finding the first definition. An analytical expression for the integral (1) is here reported as a Taylor series of order 8 when $\Omega_{\rm M}=0.3$ and $H_{0}=70\mathrm{km\,s^{-1}\,Mpc^{-1}}$

[TABLE]

and the distance modulus as a function of $z$, $F(z)$,

[TABLE]

As a consequence, the absolute magnitude, $M$, is

[TABLE]

The angular diameter distance, $D_{A}$, after Etherington (1933), is

[TABLE]

We may approximate the luminosity distance as given by eqn (4) by the minimax rational approximation, $d_{L,2,1}$, with the degree of the numerator $p=2$ and the degree of the denominator $q=1$:

[TABLE]

which allows deriving the inverse formula, the redshift as a function of the luminosity distance:

[TABLE]

Another useful distance is the transverse comoving distance, $D_{M}$,

[TABLE]

with the connected total comoving volume $V_{c}$

[TABLE]

which can be minimax-approximated as

[TABLE]

3 The adopted LFs

This section reviews the Schechter LF, the double power law LF, and the Pei LF for QSOs. The truncated version of the Schechter LF is derived. The merit function $\chi^{2}$ is computed as

[TABLE]

where $n$ is the number of bins for LF of QSOs and the two indices $theo$ and $astr$ stand for ‘theoretical’ and ‘astronomical’, respectively. The residual sum of squares (RSS) is

[TABLE]

where $y(i)_{theo}$ is the theoretical value and $y(i)_{astr}$ is the astronomical value.

A reduced merit function $\chi_{red}^{2}$ is evaluated by

[TABLE]

where $NF=n-k$ is the number of degrees of freedom and $k$ is the number of parameters. The goodness of the fit can be expressed by the probability $Q$, see equation 15.2.12 in Press et al. (1992), which involves the degrees of freedom and the $\chi^{2}$. According to Press et al. (1992), the fit “may be acceptable” if $Q>0.001$. The Akaike information criterion (AIC), see Akaike (1974), is defined by

[TABLE]

where $L$ is the likelihood function and $k$ is the number of free parameters in the model. We assume a Gaussian distribution for the errors and the likelihood function can be derived from the $\chi^{2}$ statistic $L\propto\exp(-\frac{\chi^{2}}{2})$ where $\chi^{2}$ has been computed by Equation (13), see Liddle (2004), Godlowski and Szydowski (2005). Now the AIC becomes

[TABLE]

3.1 The Schechter LF

Let $L$ be a random variable taking values in the closed interval $[0,\infty]$. The Schechter LF of galaxies, after Schechter (1976), is

[TABLE]

where $\alpha$ sets the slope for low values of $L$, $L^{*}$ is the characteristic luminosity, and $\Phi^{*}$ represents the number of galaxies per $Mpc^{3}$. The normalization is

[TABLE]

where

[TABLE]

is the gamma function. The average luminosity, ${\langle L\rangle}$, is

[TABLE]

An equivalent form in absolute magnitude of the Schechter LF is

[TABLE]

where $M^{*}$ is the characteristic magnitude. The scaling with $h$ is $M^{*}-5\log_{10}h$ and

$\Phi^{*}h^{3}~{}[Mpc^{-3}]$.

3.2 The truncated Schechter LF

We assume that the luminosity $L$ takes values in the interval $[L_{l},L_{u}]$, where the indices $l$ and $u$ mean ‘lower’ and ‘upper’; the truncated Schechter LF, $S_{T}$, is

[TABLE]

where $\Gamma(a,z)$ is the incomplete Gamma function defined as

[TABLE]

see Olver et al. (2010). The normalization is the same as for the Schechter LF, see eqn (19),

[TABLE]

The average value is

[TABLE]

with

[TABLE]

The four luminosities $L,L_{l},L^{*}$ and $L_{u}$ are connected with the absolute magnitudes $M$, $M_{l}$, $M_{u}$ and $M^{*}$ through the following relationship

[TABLE]

where the indices $u$ and $l$ are inverted in the transformation from luminosity to absolute magnitude and $L_{\odot}$ and $M_{\odot}$ are the luminosity and absolute magnitude of the sun in the considered band. The equivalent form in absolute magnitude of the truncated Schechter LF is therefore

[TABLE]

The averaged absolute magnitude is

[TABLE]

3.3 The double power law

The double power law LF for QSOs is

[TABLE]

where $L^{*}$ is the characteristic luminosity, $\alpha$ models the low boundary, and $\beta$ models the high boundary, see Boyle et al. (1988, 2000); Croom et al. (2004); Richards et al. (2006); Ross et al. (2013); Singh (2016). The magnitude version is

[TABLE]

where the characteristic absolute magnitude, $M^{*}$, and $\phi^{*}$ are functions of the redshift.

3.4 The Pei function

The exponential $L^{1/4}$ LF, or Pei LF, after Pei (1995), is

[TABLE]

and the magnitude version is

[TABLE]

4 The astrophysical applications

This section explains the K-correction for QSOs, introduces the sample of QSOs on which the various tests are performed, finds the parameters of the new LF in the range of redshift $[0.3,0.5]$, and finds the number of QSOs as a function of the redshift.

4.1 K-correction

The K-correction for QSOs as f unction of the redshift can be parametrized as

[TABLE]

with $-0.7<\alpha_{\nu}<-0.3$, see Wisotzki (2000). Following Croom et al. (2009), we have adopted $\alpha_{\nu}=-0.3$. The corrected absolute magnitude, $M_{K}$, is

[TABLE]

In the following, both the observed and the theoretical absolute magnitude will always be K-corrected.

4.2 The sample of QSO

We selected the catalog of the 2dF QSO Redshift Survey (2QZ), which contains 22431 redshifts of QSOs with $18.25<b_{J}<20.85$, a total survey area of 721.6 $\deg^{2}$, and an effective area of 673.4 $\deg^{2}$, see Croom et al. (2004) 111 Data at http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=VII/241.

. Section 3 in Croom et al. (2004) discusses four separate types of completeness which characterize the 2QZ and 6QZ surveys: (i) morphological completeness, $f_{m}(bJ,z)$, (ii) photometric completeness, $f_{p}(bJ,z)$, (iii)coverage completeness $f_{c}(\theta)$ and (iv) spectroscopic completeness, $f_{s}(bJ,z,\theta θ)$. The first test can be done on the upper limit of the maximum absolute magnitude, $M_{u}(z)$, which can be observed in a catalog of QSOs characterized by a given limiting magnitude, in our case $b_{j}=20.85$, where $F(z)$ has been defined by eqn (5):

[TABLE]

see Figure 1.

A careful examination of Figure 1 allows concluding that all the QSOs are in the region over the border line, the number of observed QSOs decreases with increasing $z$, and the average absolute magnitude decreases with increasing $z$. The previous comments can be connected with the Malmquist bias, see Malmquist (1920, 1922), which was originally applied to the stars and later on to the galaxies by Behr (1951).

4.3 The luminosity function for QSOs

A binned luminosity function for quasars can be built in one of the two methods suggested by Page and Carrera (2000): the $\frac{1}{V_{a}}$ method , see Avni and Bahcall (1980); Eales (1993); Ellis et al. (1996), and a binned approximation. Notably, Yuan and Wang (2013), argued that both the $\frac{1}{V_{a}}$ and the binned approximation can produce bias at the faint end of the LF due to the arbitrary choosing of redshift and luminosity intervals.

We implemented the binned approximation of Page and Carrera (2000), $\phi_{est}$, as

[TABLE]

where $N_{q}$ is the number of quasars observed in the $M_{i}-z$ bin. The error is evaluated as

[TABLE]

The comoving volume in the flat cosmology is evaluated according to equation (11),

[TABLE]

where $D_{M,upp}$ and $D_{M,low}$ are, respectively, the upper and lower comoving distance. A correction for the effective volume of the catalog, $V_{q}$, gives

[TABLE]

where $A_{e}$ is the effective area of the catalog in $deg^{2}$.

A typical example of the observed LF for QSOs when $0.3<z<0.5$ is reported in Figure 2

and Figure 3 reports the LF for QSOs in four ranges of redshift.

The variable lower bound in absolute magnitude, $M_{l}$ can be connected with evolutionary effects, and the upper bound, $M_{u}$, is fixed by the physics, see the nonlinear Eq. (37), see Section 4.4.

The five parameters of the the best fit to the observed LF by the truncated Schechter LF can be found with the Levenberg–Marquardt method and are reported in Table 1. The resulting fitted curve is displayed in Figure 4.

For the sake of comparison, Table 2 reports the three parameters of the Schechter LF.

As a first reference the fit with the double power LF, see equation (32), is displayed in Figure 5 with parameters as in Table 3.

As a second reference the fit with the Pei LF, see equation (34), is displayed in Figure 6 with parameters as in Table 3.

4.4 Evolutionary effects

In order to model the evolutionary effects, an empirical variable lower bound in absolute magnitude, $M_{l}$, has been introduced,

[TABLE]

The above empirical formula is classified as top line in Figure 5 of Croom et al. (2009) and connected with the limits in magnitude. Conversely the upper bound, $M_{u}$ was already fixed by the nonlinear Eq. (37). A second evolutionary correction is

[TABLE]

where $M_{u}(z)$ has been defined in eqn (37). Figure 7 reports a comparison between the theoretical and the observed average absolute magnitudes; the value of $M^{*}$ reported in eqn (43) minimizes the difference between the two curves.

As a first reference Figure 8 reports a comparison between the theoretical and the observed average absolute magnitudes in the case of the double power LF; the value of $M^{*}$ which minimizes the difference between the two curves

[TABLE]

and other parameters as in Table 3.

As a *second * reference Figure 9 reports a comparison between the theoretical and the observed average absolute magnitude in the case of the Pei LF with parameters as in Table 4.

In the above fit, the evolutionary correction for $M^{*}$ is absent.

4.5 The photometric maximum

The definition of the flux,$f$, is

[TABLE]

where $r$ is the luminosity distance. The redshift is approximated as

[TABLE]

where $z_{2,1}$ has been introduced into eqn (9). The relation between $dr$ and $dz$ is

[TABLE]

where $r$ has been defined as $d_{L,2,1}$ by the minimax rational approximation, see eqn (8). The joint distribution in z and f for the number of galaxies is

[TABLE]

where $\delta$ is the Dirac delta function and $S_{T}(L;\Psi^{*},\alpha,L^{*},L_{l},L_{u})$ has been defined in eqn (23). The above formula has the following explicit version

[TABLE]

where

[TABLE]

where

[TABLE]

The magnitude version is

[TABLE]

with

[TABLE]

and

[TABLE]

where $m$ is the apparent magnitude of the catalog, the absolute magnitudes $M_{l}$, $M_{u}$,$M^{*}$ and $M_{\odot}$ have been defined in Section 3.2. The conversion from flux, $f$, to apparent magnitude, $m$, in the above formula is obtained from the usual formula

[TABLE]

and

[TABLE]

The number of galaxies in $z$ and $m$ as given by formula (52) has a maximum at $z=z_{pos-max}$ but there is no analytical solution for such a position and a numerical analysis should be performed. Figure 10 reports the observed and the theoretical number of QSOs as functions of the redshift at a given apparent magnitude when $M_{l}(z)$ is given by eqn (42) and $M_{u}(z)$ is given by eqn (37). Here we adopted the law of rare events, i.e. the Poisson distribution, in which the variance is equal to the mean, i.e. the error bar is given by the square root of the frequency.

In the above fit the observed position of the maximum , $z=1.478$, and the theoretical prediction , $z=1.491$, have approximately the same value. In the two regions surrounding the maximum, the degree of prediction is not as accurate, due to the fact that the three absolute magnitudes $M_{l}$, $M_{u}$ and $M^{*}$ are functions of $z$.

5 Conclusions

**Absolute Magnitude **

The evaluation of the absolute magnitude of a QSO is connected with the distance modulus, which, in the case of the flat cosmology, ($\Omega_{\rm M}=0.3$, $H_{0}=70\mathrm{km\,s^{-1}\,Mpc^{-1}}$) is reported in eqn (5) as a Taylor series of order 8 with range in $z$, $[0-4]$. As an application of the above series, we derived an inverse formula for the redshift as a function of the luminosity distance and an approximate formula for the total comoving volume.

Truncated Schechter LF

The Schechter LF is characterized by three parameters: $\Phi^{*}$, $\alpha$ and $M^{*}$. The truncated Schechter LF is characterized by five parameters: $\Psi^{*}$, $\alpha$, $M^{*}$, $M_{l}$ and $M_{u}$. The reference LF for QSOs, the double power law LF, is characterized by four parameters: $\phi^{*}$, $\alpha$, $\beta$ and $M^{*}$. An application of the above LFs in the range of z $[0.3,0.5]$ gives the following reduced chi-square $\chi_{red}^{2}=$ 2.57 for the truncated Schechter LF, $\chi_{red}^{2}=$ 1.49 for the Schechter LF, $\chi_{red}^{2}=$ 1.57 for the double power LF, and $\chi_{red}^{2}=$ 2.05 for the Pei LF. The other statistical such as the AIC are reported in Tables 1, 2, 3, and 4. We can therefore speak of minimum differences between the four LFs here analyzed in the nearby universe defined by redshifts $[0.3,0.5]$.

Evolutionary effects

The evolution of the LF for QSOs as a function of the redshift is here modeled by an upper and lower truncated Schechter function. This choice allows modeling the lower bound in luminosity (the higher bound in absolute magnitude) according to the evolution of the absolute magnitude, see Eq. (37). The evaluation of the upper bound in luminosity (the lower bound in absolute magnitude) is empirical and is reported in eqn (42). A variable value of $M^{*}$ with $z$ in the case of the truncated Schechter LF, see eqn (43), allows matching the evolution of the observed average value of absolute magnitude with the theoretical average value of absolute magnitude, see Figure 7. A comparison is done with the theoretical average value in absolute magnitude for the case of a double power law and the Pei function, see Figures 8 and 9.

Maximum in magnitude

The joint distribution in redshift and energy flux density is here modeled in the case of a flat universe, see formula 48. The position in redshift of the maximum in the number of galaxies for a given flux or apparent magnitude does not have an analytical expression and is therefore found numerically, see Figure 10. A comparison can be done with the number of galaxies as a function of the redshift in $[0-0.3]$ for the 2dF Galaxy Redshift Survey in the South and North galactic poles, see Figure 6 in Cole et al. (2005) where the theoretical model is obtained by the generation of random catalogs.