Annu. Rev. Astron. Astrophys. 1992. 30: 613-52
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2.4 Redshift Distributions

Redshifts are the key measurement required to translate observable magnitudes, colors, and angles into physical luminosities, rest-frame spectral energy distributions, intrinsic sizes and projected separation distances. Depending on the spectral resolution and signal-to-noise ratio, the spectra used for redshifts may also yield a wealth of additional information on the physical conditions of line-emitting hot gas; limits to the age and metallicity of the component stellar populations; and a measure of mass from velocity widths of spectral features.

The pioneering effort in measuring redshifts of field galaxies faint enough to probe galaxy evolution (B ~ 20) was by E. Turner, J. Gunn, and W. Sargent in the late 1970's using the Palomar 5-m telescope and the SIT digital spectrograph. Their sample was selected visually by R. Kron to span a wide range in morphology and surface brightness. Their results were briefly described by Turner (1980) and Gunn (1982), and they generously allowed 58 redshifts to be published by Koo (1985). Since then, a number of fainter, better-defined redshift surveys of field galaxies have been undertaken as detailed in Section 2.4.2.

2.4.1 REDSHIFT SURVEYS 15 < B < 17 Bright redshift surveys provide a local measure of the luminosity function and other properties of galaxies for comparison to more distant samples. The best known field samples are the large (N ~ 9,000) optical and 21-cm redshift surveys limited at B ~ 16 or z ~ 0.05 (Giovanelli and Haynes 1991). Unfortunately, the general lack of good photometric colors; the use of small spectrograph entrance apertures that sample mainly the nuclear regions; the incompleteness of samples based on 21-cm redshifts; and the uncertainties due to effects of large- scale structure, have together restricted the usefulness of many of the brighter redshift surveys for faint galaxy studies. Good measures of magnitude, color, and radial profiles are needed, whereas in practice morphology (which has a poor relationship to color - see Figure 7 of Huchra 1977) and eye-estimated magnitudes from the Zwicky or UGC catalogs are often the main data available. Much more valuable has been the B ~ 16 survey of 164 galaxies by Kirshner, Oemler, and Schechter (1978) in eight patches of sky with colors and a well-defined magnitude limit; a similar but fainter study by Kirshner et al. (1983) of six fields with 280 redshifts; the B ~ 17 survey of 329 galaxies by Peterson et al. (1986) in five patches, with photographic blue magnitudes; and 264 galaxies, sampled 1 in 3, in nine patches by Metcalfe et al. (1989), also with photographic blue magnitudes. These surveys have been used to compute the general galaxy luminosity function; in addition, the Kirshner, Oemler, and Schechter (1978) and Kirshner et al. (1983) surveys have been important for establishing the color dependences, as discussed in Section 2.3.1.

2.4.2 REDSHIFT SURVEYS B > 17 Fainter surveys have been made practical with the development of multiple-object spectrographs. Fiber-fed spectrographs (Hill et al. 1980; Gray 1986) can be used to survey a wide field of view, and moreover, the formatting of the spectra on the detector can be arranged to take full advantage of the area of the CCD. Since more than 100 spectra can be measured simultaneously, and since CCD's are at least a factor of 5 times more efficient than the previous generation of photo-emissive detectors, thousand-fold gains in observing efficiency can be achieved. The multiple-fiber technique is complemented by placing a mask with multiple slits in the focal plane and reimaging the focal plane through a low-resolution spectrograph. The configuration must be arranged such that the spectra do not overlap. This technique is useful when the surface density of objects is high because the field of view is smaller, and the detector area would otherwise not be used effectively. Since the background sky is sampled locally along a slit passing through each object, better sky- subtraction can be achieved, and the multiple-slit technique is normally used at the faintest flux levels.

Several groups have attempted redshift surveys with hopes of pushing beyond B = 20 to z > 0.2 where evolution might be detectable. Most well known (and fully published) are the surveys of Broadhurst, Ellis, and Shanks (1988) of 187 galaxy redshifts to B = 21.7, and Colless et al. (1990) of 87 galaxies to B = 22.5. The former group used a multiple optical-fiber system, while the latter group used a mask with multiple slits and a wide-field spectrograph camera. Both surveys are roughly 80% complete. Lilly, Cowie, and Gardner (1991) reported six spectroscopic redshifts to B ~ 24, and Cowie, Songaila, and Hu (1991) have extended this survey to obtain an almost complete sample of 21 redshifts to the same limit. The largest survey is that undertaken at Kitt Peak National Observatory by Koo and Kron (1987) with over 400 redshifts complete for B < 20, but less than 50% complete at B ~ 22. This sample has yet to be published in detail, but some of the data will be presented for illustration. The Broadhurst, Ellis, and Shanks sample has only blue magnitudes; the Colless et al. sample has B - R colors as well; the Koo and Kron surveys have UbJrFI photographic photometry; the Lilly, Cowie, and Gardner survey has UBVI CCD photometry; and the Cowie, Songaila, and Hu sample has in addition very deep K-band photometry.

Figure 3. bJ - rF color of galaxies versus their spectroscopic redshifts. The dots are the authors' data; the triangles are from Colless et al. (1990), transformed in color according to Metcalfe et al. (1991). The curves show the colors of an average elliptical galaxy (gE) and NGC 4449 (Bruzual, private communication).

Figure 3 gives the bJ - rF colors versus redshift for a subsample of faint galaxies with redshifts and measured colors that can be reliably transformed to our system. The expected red and blue boundaries have been derived from redshifting appropriate empirical spectra through the bJ and rF passbands (these spectra are well matched by the model spectral energy distributions discussed in Section 3.3). In general the data do fall within the expected boundaries, but it is noteworthy that, if the colors are accurate, galaxies even bluer than NGC 4449 are present at all redshifts (see also Colless et al. 1990). Moreover, field galaxies at z > 0.5 can be found which appear similar to local elliptical galaxies, a result consistent with the findings of Oke (1983) and Hamilton (1985), who used the continuum break at 4000 Å instead of colors.

Figure 4. Log redshift versus blue magnitude of major redshift samples with magnitude limits fainter than B = 16. Completeness, reliability, and selection effects vary and have not been made homogeneous. The Dressler & Gunn (1992) sample consists of their interlopers, i.e. foreground and background galaxies in their rich cluster fields. The diagonal lines indicate simply the inverse-square law, B = 5 log z + 25 for the solid line, corresponding roughly to 0.1 L* (the dashed line is drawn for 0.01 L* to indicate the luminosities of true dwarfs). The lines correspond to q0 = 1 and zero effective K-correction. For a smaller value of q0 and a K-correction that applies to an average of non-evolving field galaxies, at high redshift the lines bend away from the naive inverse-square law towards fainter magnitudes, like the upper envelope of the distribution of data points.

Figure 4 shows the data in the form of redshift versus blue magnitude (the points from Dressler and Gunn 1992 are discussed in Section 2.5). Horizontal bands of points within a given survey reflect large-scale structure along the line of sight. Such structures can be seen in the authors' data at intermediate magnitudes, 18 < B < 20, even though five separate fields have been combined in this plot. It is evident from the distribution of points in Figure 4 that the average underlying redshift distribution cannot be accurately derived from samples of even hundreds of galaxies at large distances. The diagonal lines indicate approximate loci of constant luminosity (more accurately, the K-correction and cosmological factors make the corresponding apparent magnitudes fainter at a given redshift). It can also be seen that galaxies of quite low luminosity, i.e. true dwarf galaxies, appear at all magnitudes, but in relatively small numbers (cf. Cowie et al. 1992; Cowie, Songaila, and Hu 1991).

If field galaxies were overall more luminous at earlier epochs, one would expect the distribution of points in Figure 4 to be shifted progressively to brighter apparent magnitudes at higher redshifts. The early-type galaxies might have been brighter by approximately 0.5 mag at z = 0.5 (Tinsley 1972), roughly comparable (but of opposite sign) to the K-correction. The luminosity evolution of the later-type galaxies depends on the unknown change in the rate of star formation and may well have been less strong. One might conservatively expect that evolution in Figure 4 would be a subtle effect, especially considering the modulation of the data by the large-scale structure.

Figure 5. Cumulative distribution functions versus redshift for samples binned in steps of one magnitude. The top panel gives the homogeneous no-evolution model discussed in the text, and the bottom panel is derived from Figure 4 as follows. The number of galaxies contributing to each magnitude bin is indicated. The leftmost curve is for B = 15-16 (thicker solid line), the next B = 16-17 (thinner dotted line), and so on as indicated. The brightest two bins were derived from Peterson et al. (1986) and Metcalfe et al. (1989); the next three bins, to B = 20, are from the authors; the following bin (B = 20-21) includes both the Broadhurst et al. (1988) sample and that of the authors; B = 21-22 includes those two samples and that of Colless et al. (1990); B = 22-23 and B > 23 are from Colless et al. (1990), authors, Cowie, Songaila, and Hu (1991), and Dressler and Gunn (1992).

Figure 5 gives the cumulative redshift distributions for these same surveys, binned in apparent magnitudes as described in the caption. The normalized cumulative redshift distributions are useful in principle because even incomplete samples can be used, so long as the incompleteness is unbiased in redshift. Unfortunately, this qualification is expected to be difficult to achieve in practice for B > 21, and thus one should be cautious in accepting the B > 21 samples in Figure 5 as being representative. A priori we expect a smooth dependence on magnitude of the separate cumulative redshift distributions. Luminosity evolution may manifest itself as a relatively subtle change in the shape of the high-redshift tails, and the change should be smoothly progressive between intervals of apparent magnitude. On this basis, it seems likely that the faintest bin, B > 23, is incomplete at high redshifts.

Broadhurst, Ellis, and Shanks (1988) have argued that since the median redshift matches their no-evolution model while the observed counts are too steep, evolution in the shape of the luminosity function is required, a conclusion that was echoed by Colless et al. (1990). Galaxies that are currently of lower luminosity may have experienced more frequent bursts of star formation at the relevant redshifts, z > 0.1. This argument was supported by co-adding in the rest-frame the spectra of six galaxies with relatively strong W3727. The mean spectrum showed strong Balmer lines in absorption and a blue continuum, indicative of an episode of recent star formation. This is a promising technique, and with additional data it will be possible to look for correlations of Balmer absorption-line strength with absolute magnitude and with redshift.

The evolutionary picture of Broadhurst, Ellis, and Shanks (1988) and Colless et al. (1990) is however not unique, as Guiderdoni and Rocca-Volmerange (1990) have been able to account for the data without such luminosity- dependent luminosity evolution. We shall return to the interpretation of the redshift distributions later, but stress a number of cautionary points for accepting these faint redshift surveys as necessarily being representative. Of most importance is the completeness, especially for higher redshift galaxies, since these are expected from surface brightness considerations (Section 1) to be the most difficult to measure at a given magnitude. Unless surveys achieve close to 100% completeness, the missing galaxies are most likely to be exactly those of most interest for tests of evolution or cosmology. Thus far, not a single galaxy with z > 1 has been spectroscopically measured with high reliability in the above surveys (but see next section on samples selected in other ways). Unless this failure is due to a true lack of such galaxies in the samples, it may only reflect the unfortunate circumstance that at high redshifts, strong spectral features like the 4000 Å continuum break move into the very noisy red portion of the sky. For emission-line galaxies, only the [OII] 3727 doublet is visible at redshifts z ~ 0.5 when the redder set of confirming emission lines, Hbeta and [OIII] 5007, move into the red.

Independent samples can, however, be compared over similar magnitude ranges to check the completeness and to examine the distribution of the high redshift tail. It is possible that there has indeed been a systematic loss of high redshifts among the Broadhurst et al. (1988) and Colless et al. (1990) samples. The fainter portion of the Broadhurst, Ellis, and Shanks (1988) sample is relatively less complete but happens to overlap the bright and virtually complete portion of the Colless et al. (1990) sample between bJ = 21 and 21.7. Using the Kolmogorov-Smirnov test, we find, in this overlapping magnitude range, less than a 1% chance that the two distributions were drawn from the same parent population, with the more complete Colless et al. sample showing a more extended high-redshift tail (23% with z > 0.4, versus 4% for the less complete Broadhurst, Ellis, and Shanks sample).

As for the Colless et al. (1990) sample, one would expect a larger number of dimmer or weaker-emission-lined galaxies at z > 0.6 than are observed. Their incompleteness has recently been decreased from 19% to 5%, yet this paucity of high-redshift galaxies persists (Ellis, private communication). However, the galaxies selected for the redshift survey show an apparent deficit at the faint end, 22.2 < B < 22.5, of about a factor of two with respect to the expected count distribution. Another problem pointed out by Colless et al. (1990) is that one of their three fields has a higher redshift tail and appears to be inconsistent with the other two fields, which was attributed to clustering. Moreover, for a sample of only eight galaxies in the same magnitude range, B = 21 to 22.5, Cowie, Songaila, and Hu (1991) find one at z = 0.735 (with broad emission lines indicative of a Seyfert-like spectrum), another at z = 0.644, and one at z = 0.479. The published sample of Colless et al. (1990) is 11 times larger, but has only two galaxies with z > 0.6, and only 15 galaxies with z > 0.45. The various strong selection effects mentioned earlier must be better understood, and the surveys must reach a greater level of self-consistency, before more conventional expectations for the galaxy redshift distribution can be rejected with confidence.

Apart from the issues of incompleteness just discussed, the distribution of redshifts within an interval of apparent magnitude, and in particular the median redshift, is a more problematic test for evolution than it might first seem. The low end of the redshift distribution at a fixed apparent magnitude contains information only about the faint end of the luminosity function, and is particularly susceptible to the effects of large-scale structure because the volume sampled is small. On the other hand, the high-redshift end of the distribution contains information not only about the bright end of the luminosity function, but also about the volume element, the overall effective K-correction, and galaxy evolution. Tests for evolution should weight high redshifts more than low redshifts, because the unweighted median may be affected by irrelevant uncertainties at low redshift.

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