A left and right truncated Schechter luminosity function for quasars

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.

luminous QSOs have absolute magnitude M b j ≈ −28 or the luminosity is not ∞ and the less luminous 23 QSOs have have absolute magnitude M b j ≈ −20 or the luminosity is not zero, see Figure 19 in [8] . 24 A physical source of truncation at the low luminosity boundary ( high absolute magnitude ) is the fact 25 that with increasing redshift the less luminous QSOs progressively disappear. In other words the upper 26 boundary in absolute magnitude for QSOs is function of the redshift. 27 The suggestion to introduce two boundaries in a probability density function (PDF) is not new and, as 28 an example, [9] considered a doubly-truncated gamma PDF restricted by both a lower (l) and upper (u) 29 truncation. A way to deduce a new truncated LF for galaxies or QSOs is to start from a truncated PDf 30 and then to derive the magnitude version. This approach has been used to deduce a left truncated beta 31 LF, see [10,11], and a truncated gamma LF, see [12]. 32 The main difference between LFs for galaxies and for QSOs is that in the first case, we have an LF The first definition of the luminosity distance, d L , in flat cosmology is where H 0 is the Hubble constant expressed in km s −1 Mpc −1 , c is the speed of light expressed in km s −1 , z is the redshift, a is the scale-factor, and Ω M is where G is the Newtonian gravitational constant and ρ 0 is the mass density at the present time, see eqn (2.1) in [15]. A second definition of the luminosity distance is The angular diameter distance, D A , after [17], is 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: Another useful distance is the transverse comoving distance, D M , with the connected total comoving volume V c which can be minimax-approximated as V c,3,2 = 3.01484 10 10 z 3 + 6.39699 10 10 z 2 − 1.26793 10 10 z + 4.10104 10 8 0.45999 − 0.01011 z + 0.093371 z 2 M pc 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 χ 2 is computed as 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 where y(i) theo is the theoretical value and y(i) astr is the astronomical value.

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A reduced merit function χ 2 red is evaluated by where N F = 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 [18], which involves the degrees of freedom and the χ 2 . According to [18], the fit "may be acceptable" if Q > 0.001. The Akaike information criterion (AIC), see [19], is defined by 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 χ 2 statistic L ∝ exp(− χ 2 2 ) where χ 2 has been computed by Equation (13) Let L be a random variable taking values in the closed interval [0, ∞]. The Schechter LF of galaxies, after [1], is where α sets the slope for low values of L, L * is the characteristic luminosity, and Φ * represents the number of galaxies per M pc 3 . The normalization is where is the gamma function. The average luminosity, L , is An equivalent form in absolute magnitude of the Schechter LF is where M * is the characteristic magnitude. The scaling with h is M * − 5 log 10 h and where Γ(a, z) is the incomplete Gamma function defined as see [22]. The normalization is the same as for the Schechter LF, see eqn (19), The average value is  (5) The averaged absolute magnitude is 3.3. The double power law 62 The double power law LF for QSOs is where L * is the characteristic luminosity, α models the low boundary, and β models the high boundary, see [8,13,[23][24][25][26]. The magnitude version is where the characteristic absolute magnitude, M * , and φ * are functions of the redshift. The exponential L 1/4 LF, or Pei LF, after [14], is  The K-correction for QSOs as f unction of the redshift can be parametrized as with −0.7 < α ν < −0.3, see [27]. Following [28], we have adopted α ν = −0.3. The corrected absolute magnitude, M K , is In the following, both the observed and the theoretical absolute magnitude will always be K-corrected.
see Figure 1.   We implemented the binned approximation of [32], φ est , as where N q is the number of quasars observed in the M i − z bin. The error is evaluated as The comoving volume in the flat cosmology is evaluated according to equation (11), 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 where A e is the effective area of the catalog in deg 2 .

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A typical example of the observed LF for QSOs when 0.3 < z < 0.5 is reported in Figure 2  The five parameters of the the best fit to the observed LF by the truncated Schechter LF can be found 87 with the Levenberg-Marquardt method and are reported in Table 1. The resulting fitted curve is displayed 88 in Figure 4.

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

90
As a first reference the fit with the double power LF, see equation (32), is displayed in Figure 5 with 91 parameters as in Table 3.

92
As a second reference the fit with the Pei LF, see equation (34), is displayed in Figure 6 with 93 parameters as in Table 3.

Evolutionary effects 95
In order to model the evolutionary effects, an empirical variable lower bound in absolute magnitude, M l , has been introduced, M l (z) = −24.5 − 10 × log 10 (1 + z) + K(z) . (42) The above empirical formula is classified as top line in Figure 5 of [28] 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 where M u (z) has been defined in eqn (37). Figure 7 reports a comparison between the theoretical and the 96 observed average absolute magnitudes; the value of M * reported in eqn (43) minimizes the difference 97 between the two curves.

98
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 and other parameters as in Table 3.

99
As a second reference Figure 9 reports a comparison between the theoretical and the observed average 100 absolute magnitude in the case of the Pei LF with parameters as in Table 4.

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In the above fit, the evolutionary correction for M * is absent.  The definition of the flux,f , is where r is the luminosity distance. The redshift is approximated as where z 2,1 has been introduced into eqn (9). The relation between dr and dz is dr = (2626.1 z + 821.99 z 2 + 804.33) 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 where δ is the Dirac delta function and S T (L; Ψ * , α, L * , L l , L u ) has been defined in eqn (23). The above formula has the following explicit version where 105 DL = (z + 1.59739) 6 L * Γ α + 1, The magnitude version is The number of galaxies in z and m as given by formula (52) has a maximum at z = z pos−max but there 109 is no analytical solution for such a position and a numerical analysis should be performed. Figure 10 Figure 6 in [37] where the 150 theoretical model is obtained by the generation of random catalogs.