7. THE QUASAR LUMINOSITY FUNCTION: CURRENT STATUS AND FUTURE PROSPECTS

The rationale for much survey work has been the determination of how the quasar population evolves as a function of look-back time. Following the pioneering work of Schmidt in the late 1960s, and the analysis of larger samples in the 1980s by Marshall, Boyle, Crampton, Usher and others, the qualitative picture is well established. For z 2, samples defined at radio, optical and X-ray wavelengths show a rapid increase in space density as a function of look-back time (Boyle 1993, Maccacaro et al. 1991, Dunlop and Peacock 1990). Over the redshift range 0 z 2 the rate of evolution inferred from optical samples corresponds to an increase in integral space density of a factor 200 (for quasars with M_B = -26), or, equivalently, to the characteristic luminosity of the quasar population increasing by a factor of ~ 40. A feature of all the analyses is the simplicity of the model consistent with the data at z 2: a two power-law luminosity function, of invariant shape, whose characteristic luminosity evolves as a function of redshift according to L* (1 + z)^k. The shape of the luminosity function shows quantitative differences between the radio, optical and X-ray, and the rate of evolution is somewhat different, with values of k = 3.0, 3.45, 2.75 (q₀ = 0.5) for radio, optical and X-ray samples respectively (Boyle 1993). However, the simplicity of the model and the similarity between samples defined at different wavelengths is striking. At redshifts z >2 few data are available in the radio and X-ray regimes, and only recently have suitable samples become available at optical wavelengths, however, it is clear that the rate of evolution at redshifts z > 2 slows dramatically. At z > 3, there is no consensus as to whether the space density continues to increase or whether a decline has begun. In the remainder of this section, we discuss areas where recent advances have produced new results, or where current work is expected to provide significantly improved constraints.

7.2 Luminous Quasars at Low Redshifts

For a decade the analysis of the Palomar-Green sample (Schmidt and Green 1983) has stood as the reference for studies of high-luminosity quasars at the brightest apparent magnitudes, m_B 16. The existence of redshift-dependent selection effects and substantial uncertainties in the Palomar-Green magnitudes have been known for some time (Wampler and Ponz 1985), but it is only recently that additional data at bright magnitudes have become available. The Edinburgh multicolor quasar survey, (Goldschmidt et al. 1992) shows that the surface density of for m_B 16 is a factor 2-3 higher than in the Palomar-Green sample. This conclusion is supported by the analysis of the LBQS (Hewett, Foltz and Chaffee 1993), which covers the magnitude range 16.5 m_BJ 18.85, and also indicates an excess of luminous quasars at low redshifts. In a second paper Miller et al. (1993) take the more extreme view that the space density of the most luminous quasars, M_B ~ -29, has not changed over the entire redshift range 0.2 < z < 4. While this interpretation is still controversial, a quantitative departure from the standard model evident from the Edinburgh survey and the LBQS is the systematic steepening of the bright end of the luminosity function with increasing redshift. These analyses do not include the selection function calculations so they should be considered preliminary, but the results appear relatively secure based on the agreement in the form of the source counts with previous work and the smooth redshift distributions of the samples. The publication of these two surveys has increased the number of quasars with 16.5 < m_B < 18.5 in well-defined samples by more than an order of magnitude.

7.3 High-redshifts

The motivation behind finding objects at high redshifts, derives from the large look-back times, , at redshift z = 4; for example, = 0.93 for q₀ = 0.5, = 0. Notwithstanding the recent advances made in the detection of radio galaxies at high-redshifts, quasars remain the only population detected in significant numbers at z > 4. The extreme nature of quasars makes their relevance to our understanding of galaxy formation a matter of debate, but Efstathiou and Rees (1988), and more recently Haehnelt and Rees (1993), have used the evolution of the quasar luminosity function to place constraints on the formation of massive, ~ 10¹¹-10¹² M, bound systems at early epochs. An important by-product of locating apparently bright quasars at z > 4 is the ability to study the behavior of intervening absorption systems at very high redshifts. Investigations of the Lyman- forest, Lyman-limit, and damped Lyman- systems are in progress, using quasars located by Irwin, McMahon and Hazard (1991).

At the highest redshifts, locating quasars at optical wavelengths becomes even more difficult. The surface densities relative to contaminating objects (Section 3.1) decrease by an additional one or two orders of magnitude. The shift of the ultraviolet continuum through the optical passbands means that shortward of 5500 Å the observed flux is reduced dramatically by the presence of the increasingly prevalent Lyman- forest lines and the higher column density metal-line absorption systems. Working in the R band ( ~ 6500 Å) or at longer wavelengths is a prerequisite for success, but even at red wavelengths, luminous quasars are apparently faint - an M_B = -26 quasar has an apparent magnitude m_R ~ 20.5 at z = 4.

At the time of the first searches for quasars with redshifts z ~ 4, it was not known that the strong evolution at redshifts z ~ 2 did not extend to higher redshifts. Based on the assumption that the evolution continued, large numbers of faint, high-redshift quasars were predicted, and searches concentrated on small areas of sky, probing to relatively faint magnitudes. Following Osmer's (1982) paper, the failure to identify high-redshift quasars from several surveys confirmed that the rapid rate of evolution observed at low-redshift must decrease above redshift z ~ 2. With this knowledge investigations probing brighter magnitudes over much larger areas of sky were initiated: Schmidt, Schneider and Gunn (1991) employing their CCD-based grism technique to a larger area, Warren et al. (1991a) using broadband multicolor selection, and Irwin, McMahon and Hazard (1991) utilizing a highly-specific version of the multicolor technique - essentially a high-redshift version of the ultraviolet excess technique. The parameters defining the latter of these, an area of 2500 deg² to an apparent magnitude m_R ~ 19, cf. <1 deg² to m_R ~ 20.5 (Schmidt, Schneider and Gunn 1986a), illustrate how the experimental design has evolved. The Irwin, McMahon and Hazard survey has proved outstandingly successful in identifying z > 4 quasars, with some 27 objects now confirmed. Coupled with the derivation of the first quantitative space density estimates by Schmidt et al. (1994) and Warren et al. (1994) the investigation of the high-redshift regime may be regarded as something of a success.

Warren et al. (1994) conclude there is a substantial decline in the space density between z ~ 3.3 and z 4. If the extent of the decline in space density at high redshift is as large as they claim, the prospects for identifying many quasars with z > 5 are poor. However, the constraints on the amplitude of the decline are still weak and taking the least extreme model consistent with the data suggests that searches of ~ 100 deg², to a limiting magnitude of m_I ~ 20, may detect a number of z > 5 quasars. Even if no objects were found, the improved constraints on the rate of decline would be very valuable. At redshifts much greater than z = 5 it is tempting to consider searches using near-infrared arrays; at z = 6, Lyman- appears at = 8500 Å and Mg II 2798 is nearly into the K-band. If evidence were found that observed quasar SEDs become significantly redder due (say) to extinction then the large I - K colors would make such an approach worth considering. However, such evidence remains elusive and, given the limited size of infrared arrays, the bright sky-background and the modest I - K colors of quasars with typical SEDs, the near-infrared is unlikely to provide the means to undertake major surveys over large areas in the near future.

7.4 The X-ray Luminosity Function

The understanding of the evolution of the quasar X-ray luminosity function has been revolutionized by the completion of two surveys: the Extended Medium Sensitivity Survey (EMSS), and very deep ROSAT observations of the Boyle et al. (1990) ultraviolet excess survey fields. Maccacaro and collaborators have defined a sample of 427 X-ray-selected quasars and AGN observed by the Einstein satellite. These objects were selected following the spectroscopic identification of the flux-limited catalog of the EMSS (Gioia et al. 1990). The Einstein satellite was sensitive to the energy range 0.3-3.5 keV and extends to a flux level of 10^-13 erg s^-1 cm^-2. This corresponds to the flux expected from a rather luminous X-ray quasar, L_x ~ 10⁴⁵ erg s^-1 at redshifts z > 1, but sensitivity at this flux level also enables the luminosity function to be probed as faint as L_x ~ 10⁴² erg s^-1 locally. The sample is the X-ray equivalent of the optical Palomar-Green survey, in that it has provided the reference for defining the properties of the X-ray luminosity function at low-redshifts and high luminosities. The EMSS is well-suited to the analysis of the quasar population, since the flux-limits are well-defined, and the spectroscopic identification is (almost) complete for objects within these limits. The analysis is described in Maccacaro et al. (1991) and Della Ceca and Maccacaro (1991).

For z 1, constraints on the evolution of the luminosity function require a sample extending to faint flux levels, such as that constructed by Shanks, Boyle and collaborators (Boyle et al. 1993), who exploited the excellent match between the ROSAT satellite's field of view and the constituent fields making up the Boyle et al. (1990) ultraviolet excess survey. Deep ROSAT exposures, with flux limits of ~ 6 x 10^-15 erg s^-1 cm^-2 in the 0.5-2.0 keV energy range, in several of their optical fields have produced an X-ray selected catalog of 42 confirmed quasars, the majority of which have z > 1. Twelve of the sources in the ROSAT exposures remain unidentified. Combining the EMSS and ROSAT samples Boyle et al. (1993) find that a two-power-law form for the luminosity function, that is invariant in shape and whose characteristic luminosity evolves as L_x* (1 + z)^{2.8 ± 0.1} (q₀ = 0.0) fits the data well. A significant slow-down in the rate of evolution at z ~ 2 is also required. The similarity with the form of the evolution in the optical is striking.

There are differences between the Boyle et al. (1993) analysis of the combined EMSS and ROSAT data and the Maccacaro et al. and Della Ceca and Maccacaro analyses of the EMSS sample alone. Boyle et al. address this problem explicitly, ascribing part of the difference to the methods of analysis and to the improved constraints arising from combining the samples. However, a key factor is how the two samples are related. The different energy ranges to which the Einstein and ROSAT detectors are sensitive, combined with the higher mean redshift of the ROSAT sample, means both the shape of the mean quasar SED at X-ray frequencies and the distribution of SEDs about the mean must be known. Approximating the X-ray SED of the quasars using a power-law in frequency, Boyle et al. test for the effect of a dispersion in the power-law slopes, and also whether inclusion of the unidentified objects could affect the results, and conclude that neither is important. The tests assume no correlation between redshift, quasar SED and the probabilities that a quasar remains unidentified. Thus, while the tests as described are valid, they are equivalent to assuming the array of probabilities P (M, z, SED) may be written P (M) · P (z) · P (SED), and further, that P (z), for example, is constant. Given the lack of information on the quasar SEDs it is sensible to assume that the selection function is separable, but there is no evidence to support such a contention. Once again, the critical factor limiting our knowledge is the lack of information about the form and distribution of quasar SEDs and the possible variations of the distribution as a function of M and z.

The difference in approach to the analysis of surveys propounded in this review and that of Boyle et al. is encapsulated in the statement at the end of their Section 4: ``Certainly, samples of QSOs at X-ray wavelengths are much more useful for providing an unequivocal determination of the existence of a cutoff at z ~ 2, since they are significantly less prone to the selection effects which bedevil optical samples of QSOs at z > 2.'' In contrast, we believe that the lack of information concerning quasar SEDs at both optical and X-ray wavelengths is the limiting factor, and that once a determination of the selection function is made for surveys at both wavelengths the conclusions regarding the evolution of the luminosity function and the intrinsic distribution of SEDs should be equally robust and entirely consistent.

7.5 Quasar Spectral Energy Distributions

Throughout this review we have stressed that a factor limiting our knowledge of the quasar population and its evolution is the lack of information concerning quasar SEDs. Approximating quasar SEDs by a power-law in frequency, , F () , the relation between the apparent magnitude, m_B, and the absolute magnitude, M_B, is:

M_B = m_B - 5 log [A (z) c / H₀] + 2.5 (1 + ) log(1 + z) - 25,

where A (z) c / H₀ is the luminosity distance (Schmidt and Green 1983). The discussion in this section assumes apparent and absolute magnitudes are referenced to the B band, but the principles apply equally to other wavebands, or indeed, to apparent magnitudes and absolute magnitudes referenced to different wavelengths, e.g., absolute magnitudes M_B derived from apparent magnitudes in the R band, m_R. An error, (), in the mean spectral index of the population translates into an error in the derived absolute magnitude. The size of the latter increases with redshift, resulting in a spurious ``evolution'' of the quasar population. Over the redshift interval 0 < z < 1, an error of () = 0.5 in the mean spectral index, will result in the absolute magnitude calculated for a quasar at z = 1 being in error by M_B = 0.37. At redshift z = 2 this increases to M_B = 0.60. If the characteristic luminosity of the luminosity function evolves as L* (1 + z)^k, the error in the spectral index, (), affects the evolution parameter, k, such that, k' = k - (), where k' is the inferred value of the evolution parameter. Alternatively, note that reaching 0.6 magnitudes fainter into the bright end of the quasar luminosity function, where N(L) L^-3.5, at z = 2 produces an increase of a factor ~ 7 in number, and hence the same factor in space density. Errors of several tenths in the effective mean power-law index of quasar SEDs are quite possible, and Boyle et al. (1993) note that such errors may in part be responsible for the differing rates of evolution observed in the optical and X-ray regimes, and also between samples defined at different X-ray frequencies.

This simple example assumes that all quasars have identical SEDs. A more realistic calculation would take into account a spread of SEDs, which corresponds to a range in the power-law index . The dispersion in the power-law index in the rest-frame ultraviolet is substantial (Sargent, Steidel and Boksenberg 1989, Francis et al. 1991, Schneider, Schmidt and Gunn 1991), with estimates of = 0.3-0.6. As a consequence, our view of the luminosity function is affected by an undesirable smoothing over absolute magnitude. Two quasars at z = 2, with equal apparent magnitudes, m_B, and power-law SEDs that differ in slope by = 1, have absolute magnitudes that differ by M_B = 1.2. Equivalently, a flux-limited sample would probe 1.2 magnitudes deeper into the luminosity function of the bluest objects relative to those with the reddest SEDs. In a flux-limited sample, intrinsically fainter blue quasars are preferentially brought into the sample relative to the mean SED and associated limiting absolute magnitude. Conversely, redder objects that are more luminous than the absolute magnitude limit are preferentially removed. Since the number of quasars increases as the absolute magnitude decreases, more objects are brought into the sample than are removed. The result is to increase the number observed (relative to a population with no dispersion in SED) and the rate of evolution as a function of redshift is overestimated as a consequence. Giallongo and Vagnetti (1992) made the first quantitative investigation of the effects of a spread in SEDs on the evolution by assuming an intrinsic spread in the power-law slope parameter of = 0.25, and considering two models for the evolution. Their quantitative results confirm the qualitative expectation that the rate of evolution decreases when account is taken of the dispersion in SEDs. Account has also been taken of the variation in quasar SEDs by Stocke et al. (1992). In this case a k-correction was derived specifically for BAL quasars to place their calculated absolute magnitudes on the same continuum-magnitude system as the control sample of non-BALs. Figure 1 of Stocke et al. illustrates that the corrections to the absolute magnitudes are significant, reaching M ~ 0.35 at z = 2.5, in their blue passband. Most recently Francis (1993) has considered the implications of a dispersion in SEDs on correlations between observed properties of quasars from flux limited samples.

If quasar SEDs can be individually characterized, then the distribution of SEDs may be incorporated into the calculation of the evolution explicitly as described by Warren et al. (1994) (although in that specific investigation the quality of the spectroscopic data is poor and their SED assignments are therefore often uncertain). Lacking information on individual quasar SEDs, the spread in SEDs must be included in the parametric models (e.g., Giallongo and Vagnetti 1992).