6. PECULIAR VELOCITY FIELDS: TECHNIQUES OF MEASUREMENT AND ANALYSIS

We have learned a great deal about the large-scale structure of the Universe from the redshift surveys discussed in preceding chapters. In these surveys we have generally taken the measured redshift cz as a distance indicator, enabling us to construct three-dimensional maps of the distribution of galaxies. We have also gone one step beyond this, combining theory with observations to use the redshift maps as a means of studying the origin of large scale structure, and the value of the cosmological density parameter ₀. These analyses have been premised on the gravitational instability paradigm, according to which structure grows due to gravitational amplification of initial density perturbations (Section 2.2). But redshift measurements do not, by themselves, lead to a conclusive test of the gravitational instability paradigm. Such a test requires the additional information we gain from what we broadly refer to as peculiar velocity surveys.

The limitations of redshift surveys stem from their inability to separate the two contributions to a galaxy's observed redshift: the cosmological component associated with the expansion of the Universe, and the peculiar component associated with the galaxy's proper motion with respect to the local rest frame (see Eq. 2). Such a separation is possible only if, in addition to the redshift cz, one has a redshift-independent distance measurement d. Peculiar velocity surveys are based on data sets consisting of redshifts and redshift-independent distance measurements, or "redshift-distance samples." From these two measurements one readily obtains the radial component of each galaxy's peculiar velocity,

(147)

where we have used Eq. (2) and defined

(148)

Note it is only the radial component of the peculiar velocity that we can measure. We discuss an indirect means of reconstructing the transverse components of the peculiar velocity field in Section 7.5 below.

The gravitational instability paradigm requires that the linear peculiar velocity and density fluctuation fields be related to one another according to the local expression Eq. (30), or the global expression Eq. (33). Those formulations of the velocity-density relation contain the Hubble constant H₀. In actual peculiar velocity analyses, explicit dependence on the Hubble constant is avoided, by working in a system of units in which distances are expressed as velocities - i.e., in which the Hubble constant is defined to be unity. Eqs. (30) and (33) also involve actual mass density fluctuations, whereas redshift surveys provide galaxy density fluctuations only. We thus rewrite the two equations as they appear in an actual analysis:

(149)

and

(150)

where = f(₀, ) / b (Eq. 51) and b is the biasing parameter. It is understood that the derivatives in Eq. (149) and the integral in Eq. (150) are with respect to spatial variables expressed in km s^-1.

A proper analysis of redshift-distance samples will yield v(r), while we derive _g(r) from redshift survey data. We can thus test the validity of the gravitational instability paradigm by seeing whether these fields obey Eqs. (149) and (150) ⁽²⁰⁾ . Furthermore, to the extent that the data are consistent with the gravitational instability picture, the ratio of the galaxy density and velocity divergence fields gives a direct measure of . Indeed, with data of sufficient quality and quantity, we might hope eventually to go beyond the linear velocity-density relations, and thus break the degeneracy between ₀ and b.

While confirmation of the gravitational instability picture and determination of ₀ are their most important goals, peculiar velocity surveys serve other purposes as well. One is to characterize the mass distribution on very large scales. Because of shot noise, existing wide-angle redshift surveys cannot accurately measure number density fluctuations at distances much greater than ~ 15,000 km s^-1. However, large-amplitude ( 500 km s^-1) coherent peculiar velocities on very large scales can be detected at such distances with relatively modest samples (e.g., Lauer & Postman 1994 ; Dekel 1994). Velocity perturbations drop off more slowly with increasing scale than do density perturbations, as we saw in Fig. 1; mass density fluctuations of a few percent on a 20,000 km s^-1 scale - undetectable in present redshift surveys - can give rise to detectable bulk motions if ₀ is not much less than unity. For this reason, peculiar velocity measurements constitute one of the best methods of measuring mass density fluctuations on very large scales; this quality makes them a powerful discriminant of rival cosmogonic theories (Section 7.1).

A related virtue of peculiar velocity surveys is their capacity to reveal mass fluctuations independently of bias. Through redshift surveys we can map the number density fluctuations _g, and derive the mass density field through an assumed model of bias. But if the relationship between luminous and dark mass is, in reality, far different from the linear model we usually adopt - or worse, if that relationship has large variance (Cen & Ostriker 1992b) or is stochastic - redshift surveys may in fact tell us little about mass fluctuations. If the gravitational instability picture is valid, however, peculiar velocities uniquely reflect the mass density field. This point has been emphasized in recent years by Dekel, Bertschinger, and coworkers (e.g., Bertschinger & Dekel 1989 ; Dekel et al. 1990; Dekel 1994), who have developed a method to reconstruct the mass density field from peculiar velocity measurements (Section 7.5).

The use of redshift-distance samples as described above appears straightforward in principle. In reality, redshift-independent distances are far too inaccurate for Eq. (147) to be applied as written to real data. Much of the complexity (and controversy) surrounding peculiar velocity surveys arises from the steps needed to account for the consequences of distance measurement errors. In what follows we describe how workers in the field have attempted to deal with this complexity, and discuss the issues that remain unresolved. Still, it is worth emphasizing at the outset our optimism about the program of comparing the peculiar velocity and galaxy density fields in order to estimate . Much progress has been made to date using existing large, fairly complete redshift-distance samples, and still larger and improved redshift-distance samples will soon become available. Redshift survey data yielding _g(r) for the local region are similarly improving. We consider it likely that within a decade or so, peculiar velocity analyses will have provided not only convincing substantiation of the gravitational instability picture, but also a believable estimate of , and quite possibly, further insights into the subtle relationship between luminous and dark mass.

We begin this chapter with a discussion of distance indicator relations, with emphasis on the Tully-Fisher and D_n- relations for spiral and elliptical galaxies, respectively. Section 6.2 briefly discusses the evidence we have that these distance indicator relations are universal, and Section 6.3 covers other distance indicator relations that are likely to become important in coming years. The rest of the chapter is devoted to the types of analyses one does with peculiar velocity data, emphasizing the inherent pitfalls and biases. This is first done at a qualitative level in Section 6.4, followed by a mathematical treatment in Section 6.5. The scientific results of peculiar velocity work are discussed in Chapters 7 and 8.

6.1. Galaxian Distance Indicator Relations

We obtain redshift-independent galaxy distances by means of distance-indicator relations, or DIs. The essence of a DI is an empirical correlation between two intrinsic properties of a galaxy, one of which is distance-dependent (e.g., diameter) and one of which is distance-independent (e.g., surface brightness). ⁽²¹⁾ Thus, for example, surface brightness might be known to correlate with absolute diameter; a comparison of the measured angular diameter with the inferred absolute diameter, obtained from this correlation, yields a distance estimate. In some cases the distance-independent property is simply the identification of an object as belonging to a particular class, such as Type 1a Supernovae (Section 6.3.3); in that case a DI may be called a "standard candle" or a "standard ruler," according to whether the distance-dependent property is luminosity or diameter.

Long before their use in peculiar velocity surveys, DIs were employed in the quest to measure the Hubble constant H₀ and the acceleration parameter q₀ (e.g., Weinberg 1972). Determination of H₀ requires absolute calibration of a DI, which involves referencing DI measurements to a set of fiducial galaxies whose distances in Mpc are known. For peculiar velocity work, absolute calibration is unnecessary; the DI measurements are instead referenced to a fiducial set whose distances in km s^-1 are known. (We discuss how this is done in practice in Section 7.2.) In this review, we specifically do not discuss efforts to absolutely calibrate DIs and apply them toward determination of H₀; this subject has been recently reviewed by Jacoby et al. (1992) . Estimation of q₀ does not require absolute calibration, but does entail observations of galaxies at substantial (z 0.2) redshifts. The peculiar velocity surveys we will discuss are strictly confined to z 0.06 and thus have no bearing on the q₀ problem.

While many empirical correlations between distance-independent and distance-dependent galaxian properties exist, not all are equally useful as DIs. In what follows, we describe in detail those DIs which have been most fruitfully applied to the study of peculiar velocities, as well as some likely to be so employed in the future.

Both elliptical and spiral galaxies show correlations between galaxian luminosity and a relevant measure of internal velocity dispersion. In Section 6.1.1, we discuss this correlation for spirals, the Tully-Fisher (TF) relation; Section 6.1.2 discusses the corresponding correlation for ellipticals.

6.1.1. The Tully-Fisher Relation for Spiral Galaxies

Ordinary spiral galaxies are, for practical purposes, the most suitable objects to use as tracers of the peculiar velocity field in the local Universe. They are more numerous and more uniformly distributed than their luminous counterparts, ordinary elliptical galaxies (which tend to congregate in the cores of dense clusters). They are also bright enough to be carefully studied at fairly large distances, unlike the still more numerous but faint dwarf galaxies. As a result, the TF relation, which is the main DI for spirals, has been the workhorse in peculiar velocity surveys to date.

Since spiral galaxies are flattened systems supported by rotation, the relevant measure of internal velocity for Tully-Fisher is rotation velocity. Because spiral galaxies have "flat" rotation curves, their rotation velocities are well-defined, as we discuss below. In practice, the luminosity-rotation velocity correlation is well modeled as a power law,

(151)

Using the conventional notation of astronomers, which we will follow here, this equation may be rewritten

(152)

where M = const. - 2.5log L is the absolute magnitude, and

(153)

where v_rot is measured in km s^-1, is a convenient logarithmic measure of the rotation velocity. The normalization of is such that its value lies roughly between ± 0.3 for most spirals. The quantities A and b are known as the "zeropoint" and "slope" of the TF relation; the slope is found to lie in the range b ~ 5-10, depending on the details of how v_rot and M are measured. Thus, the power-law exponent of the luminosity-rotation velocity relation is ~ 2-4. The apparent magnitude m of a galaxy is the observed quantity; it is related to the absolute magnitude via the inverse-square law:

(154)

which defines the distance modulus µ(r).

The measurement of rotation velocities in TF studies has been carried out mainly through analysis of HI 21 cm profiles, especially prior to 1990. These profiles are derived from spatially unresolved data, and therefore do not give rotation velocity per se, but rather total 21 cm intensity as a function of velocity. This has led to a variety of efforts to define a 21 cm profile width which most faithfully reflects rotation velocity (e.g., Tully & Fouqué 1985 ; Aaronson et al. 1986 ; Bicay & Giovanelli 1986a , b). In recent years some workers have employed optical measures of rotation velocity, measured from the H emission line (Dressler & Faber 1990b ; Courteau 1992 ; Mathewson et al. 1992a, b; Mathewson & Ford 1994), using a long slit oriented along the major axis of the galaxy. Such measurements have become increasingly useful as TF studies have been extended to the Southern Hemisphere, where fewer radio telescopes exist, and to larger distances, where confusion and sensitivity problems become severe in the radio. An alternative is to employ imaging Fabry-Perot spectroscopy in the H line (Schommer et al. 1993), which allows one to map out the rotation velocity field in two-dimensional detail. The optical, unlike the HI, measurements are spatially resolved. Because 21 cm and optical measures of rotation can differ systematically (Tully & Fouqué 1985 ; Pierce & Tully 1988), it cannot be assumed that the slope or zeropoint of the HI and optical TF relations will be the same; recent work on combining the two types of data have addressed this issue (Mathewson et al. 1992b; Willick et al. 1995b).

Regardless of how velocity width is measured, the raw measurement differs from the true width (which enters into the TF relation) because of projection on the sky. The true width is estimated by dividing the raw value by the sine of the estimated inclination angle of the galaxy, obtained from the ellipticity of its optical image.

The existence of a correlation between optical luminosity and HI velocity width had been recognized since at least the early 1970s (see, for example, Bottinelli et al. 1971 ; Balkowski et al. 1973 ; Sandage & Tammann 1976), but Tully & Fisher (1977) were the first to characterize this correlation by the linear relation (Eq. 152), finding a slope b ~ 6. Their work brought its potential for use as an important cosmological tool to the attention of the astronomical community. Early work with the TF relation used blue-bandpass (B) measurements of luminosity, which were by far the most readily available in the late 1970s. It was recognized early on, however, that such measurements were subject to extinction of starlight by dust in both the source galaxy and in our own Galaxy. These effects represented systematic uncertainties in early applications of the TF relation.

An important advance was made in the late 1970's and early 1980's through the work of Aaronson and collaborators (e.g., Aaronson, Huchra, & Mould 1979 ; Aaronson et al. 1982a, b). They recognized that long-wavelength photometry would suffer less from internal and Galactic extinction than B band photographic or photoelectric data. The Aaronson group undertook a program of photoelectric H band ( = 1.6µm) photometry of spiral galaxies in the Local Supercluster. These measurements were made through apertures whose size was dictated by instrumental constraints, and were subsequently scaled to a fiducial aperture roughly one-third the optical size of the galaxy. This awkward procedure was a drawback of the method, but the Aaronson group nonetheless found that the new infrared TF relation (IRTF) was tighter than the older short-wavelength version. They also found that the IRTF required little or no correction for internal and Galactic extinction, and that its slope b was almost exactly 10, corresponding to a power-law exponent = 4.

The v⁴_rot dependence of the IRTF was seen as favorable for two reasons. First, as we discuss further below, it drops out naturally from a simple theoretical "derivation" of the TF relation based on considerations of centrifugal equilibrium. Second, a v⁴ power law also described the analogous Faber-Jackson relation for elliptical galaxies (Section 6.1.2). Since elliptical galaxies generally lack ongoing star formation, their radiant output in the blue is produced by an old stellar population. The blue-light luminosities of spiral galaxies, by contrast, may in some cases be dominated by the hot, massive stars which result from recent bursts of star formation. Aaronson et al. (1979) suggested that the H band photometry measured the light of the old stellar population, presumably the most faithful tracer of the mass, and therefore correlated better with the gravitational potential - and through it, the observed rotation velocity. The agreement in slope with that of the Faber-Jackson relation seemed to confirm this notion.

Attempts to explain the TF relation in terms of the composition and dynamics of spiral galaxies begin with the equation of centrifugal equilibrium,

(155)

If one then assumes a universal mass-to-light ratio (M/L), and a constant mean surface brightness for spirals, so that

(156)

one obtains

(157)

or

(158)

precisely the exponent characterizing the IRTF. Aaronson et al. (1979) cited this consistency as evidence that the near-infrared TF relation was more fundamental than shorter wavelength versions.

While this explanation of the TF slope is not unreasonable, plausible "derivations" of quite different slopes have been presented. For example, Sandage & Tammann (1976) began with Eq. (155), and also invoked the presumed constancy of (M/L). They then supposed that the characteristic radius R at which the equation is to be interpreted is independent of luminosity, on average. This yields L v_rot², corresponding to a TF slope b = 5, similar to the earliest estimates for the B bandpass (Tully & Fisher 1977 ; Bottinelli et al. 1983). Moreover, Burstein (1982) has argued on the basis of the observed slope of the TF relation and the relation between surface brightness and rotation velocity, that the assumption of a uniform mass-to-light ratio cannot hold true. The notion that one can straightforwardly derive the TF slope from first principles is now, in fact, considered dubious. It has become clear that the TF slope has more to do with technical issues of measurement than with the constancy of mass-to-light ratios or surface brightnesses. In particular, the size of the aperture and the photometric bandpass used materially affect the observed slope (Bottinelli et al. 1983 ; Pierce & Tully 1988 ; Willick 1991 ; Jacoby et al. 1992 ; cf. Fig. 12). Bernstein et al. (1994) , for example, found an H band TF slope of ~ 6, rather than the canonical 10, when they used large apertures which encompassed essentially all of the light from a galaxy. Moreover, the assumption that spirals possess constant (M/L) in H band light, has been challenged. Bothun (1986) , for example, has pointed out that small variations in heavy-element abundances can greatly alter the infrared energy output of a stellar population, in which case the age of a galaxy should affect (M/L). Pierce (1988, 1991) has shown that, as measured in I band light, (M/L) ratios show strong variations with luminosity. In short, it appears that poorly understood issues pertaining to the nature and spatial distribution of stellar populations will be central to an eventual understanding of the TF relation.

Beyond this, there is a fatal flaw in any simple model of the TF relation based on the equation of centrifugal support. Spiral galaxies possess roughly exponential light profiles (Freeman 1970) with a characteristic scale length. However, they have rotation curves which rise quickly from zero to a characteristic rotation speed that is observed to be remarkably constant out to many exponential scale lengths (e.g., Rubin et al. 1980, 1982, 1985). This constancy of the rotation speed is one of the main pieces of evidence for the existence of dark halos in spiral galaxies (Faber & Gallagher 1979). Indeed, it is only because of such flat rotation curves that a well-defined rotation velocity of spiral galaxies exists. Thus the characteristic scale of the mass distribution in spiral galaxies is appreciably larger than is that of the light, although the two were equated in Eqs. (155)-(157). Moreover, the TF relation applies equally well to all galaxies of a given rotation speed, although their exponential scale lengths, as well as their bulge-to-disk ratios and morphologies, may vary considerably (Han 1991 ; Willick 1991). Evidently, there is a complex interplay between luminous and dark mass which determines the final state of spiral galaxies. While some progress has been made toward understanding these effects (e.g., Blumenthal et al. 1986), our understanding of the physical basis of the TF relation remains schematic at best. The origin of the TF relation is shrouded in the mysteries of spiral galaxy formation.

With the advent of CCD detectors in the 1980s, highly accurate galaxy photometry has become routine. Two-dimensional digital detectors such as CCDs permit one to tailor the photometric aperture to the galaxy image in a natural way, in contrast to the somewhat arbitrary scalings required in fixed-aperture photoelectric photometry. CCDs are very sensitive in the R ( ~ 0.7µm) and I ( ~ 0.8µm) bandpasses, and most recent TF studies have been carried out in one of these bands (Bothun & Mould 1987 ; Pierce & Tully 1988 ; Willick 1991 ; Han 1991 ; Courteau 1992 ; Mathewson et al. 1992a, b; Roth 1993 ; Mould et al. 1993 ; Mathewson & Ford 1994 ; Haynes et al. 1995). Dust extinction is less of a problem in R or I than in B, though not negligible as it is at 1.6µm (a detailed discussion of models of internal extinction is given by Giovanelli et al. 1994). With the advent of imaging detectors sensitive at ~ 2µm, a few workers are returning to the near infrared for TF work (Peletier & Willner 1993 ; Bernstein et al. 1994).

In Fig. 12 we plot B, R, and I TF relations obtained from the large data set of Mathewson et al. (1992b), and an H band TF relation from the Aaronson et al. (1982b) data set, as updated by Tormen & Burstein (1994, 1995). In each case a Hubble flow model of distances has been used; thus, the diagrams contain extra scatter due to the inaccuracy of the distance assignments, as discussed below. Nonetheless, the relation is well defined in all of the bandpasses, although the scatter in the R, I and H bandpasses is noticeably smaller than in the B bandpass. The B and R band data are photoelectric measurements taken from the RC3 and Lauberts & Valentijn (1989) catalogs, respectively, as compiled by D. Burstein (private communication). The slopes of the fitted TF relations are indicated in the figure; these TF fits are corrected for sample selection bias as described in Section 6.5. As can be seen, the slope increases with increasing wavelength. The very large change from I to H is due only partially to wavelength increase; it stems mainly from the small apertures within which the H band magnitudes are measured (the I band magnitudes are "total" magnitudes) and the different definitions of velocity width used by Mathewson et al. vs. Aaronson et al.In all cases, the absolute magnitudes are expressed in the units suitable to peculiar velocity work, in which distances are measured in km s^-1.

Figure 12. TF relations in the B, R, I, and H bandpasses. The B, R, and I TF relations are obtained from the data set of Mathewson et al. (1992b). The B and R magnitudes for the Mathewson galaxies are photoelectric measurements obtained from the RC3 and Lauberts-Valentijn (1989) catalogs respectively; the I band magnitudes are from the CCD photometry, and the velocity widths from a combination of HI and optical measurements, carried out by the Mathewson group. The H band magnitudes are from the local sample of Aaronson et al. (1982b) , as updated by Tormen & Burstein (1994, 1995). In each case the plotted absolute magnitudes are computed as M = m - 5log(cz), where cz is the observed redshift, and are thus expressed in the units appropriate to peculiar velocity studies. Further details are given in the text.

An important and unresolved issue is the rms scatter of galaxies around the mean TF relation, usually quoted in magnitude units and denoted . The earliest estimates of the TF scatter were ~ 0.5-0.6 mag in B (Tully & Fisher 1977). The Aaronson group later found a somewhat tighter correlation, 0.45-0.50 mag, in the near infrared (e.g., Aaronson et al. 1986). However, Sandage, Tammann, and coworkers were critical of these relatively low estimates, arguing that sample selection effects (Section 6.4) were introducing gross biases into estimates of the scatter. By enlarging the data sets studied by the Aaronson group in the Virgo cluster, Sandage (1988) and Kraan-Korteweg, Cameron, & Tammann (1988) derived values ranging from ~ 0.7-1.0 mag, although Fukugita, Okamura, & Yasuda (1993) explain much of this increased scatter as due to appreciable depth effects in the cluster. More recently, Federspiel, Sandage, & Tammann (1994) have claimed that while the most luminous galaxies may have = 0.4 mag, the TF scatter for a typical galaxy is ~ 0.6-0.7 mag.

On the other hand, recent years have seen ever-decreasing estimates of the TF scatter by groups employing CCD photometry. Bothun & Mould (1987) suggested that an appropriate adjustment of the photometric aperture with rotation velocity width could reduce the scatter of the I band TF relation to ~ 0.20-0.25 mag. Pierce & Tully (1988) observed a scatter in the nearby Ursa Major cluster of ~ 0.30 mag, and calculated that the contribution of true cluster depth was such that the true scatter was 0.25 mag. Willick (1990, 1991) studied a sample of relatively distant clusters and found an R band scatter of ~ 0.30 mag. Courteau (1992) identified a region of "quiet Hubble flow" within a larger field sample in which the R band TF scatter was 0.32 mag. Schommer et al. (1993) found a scatter of 0.25-0.3 mag for their I band Fabry-Perot sample (see above). Very recently, Bernstein et al. (1994) have studied a carefully selected I band CCD sample in the Coma Supercluster region which exhibits an apparent TF scatter of ~ 0.10 mag.

The observed TF scatter comes from three sources: intrinsic or "cosmic" scatter, photometric and velocity width measurement errors, and incorrect distance assignments. Cosmic scatter is the quantity of greatest interest astrophysically; because of uncertainties in the other two sources of scatter, however, it is not well-constrained at present. Photometric errors per se are very small ( 0.05 mag) when CCDs are used. However, the extinction corrections made to the apparent magnitudes, while accurate in the mean, may be incorrect by ~ 0.1 mag in individual cases due to variations in the dust content of spiral galaxies. Raw velocity width measurements contain ~ 5% errors, and comparable errors are incurred in making the inclination correction; resulting errors in the width parameter must then be multiplied by the TF slope. Overall, measurement errors can account for ~ 0.15-0.30 mag, depending on bandpass, inclination, and quality of the HI or optical width determination. Cosmic scatter and measurement errors, added in quadrature, are what the quantity is meant to signify. However, when a TF sample is used to estimate scatter, galaxies must be assigned distances; errors in these assignments contribute to the observed scatter. Minimizing this last contribution, and subtracting it from the observed scatter, is a tricky but necessary part of estimating the true TF scatter.

The observed value of the TF scatter is important not only in determining the size of random errors, but also of systematic errors due to statistical bias effects (Section 6.4). The discrepancy among estimates of the TF scatter discussed above probably arises from a combination of two factors: small number statistics, and different approaches to the problem of distance assignment errors. Large TF residuals that one author attributes to an erroneous distance model, such as the misassignment of a galaxy to a cluster, another might attribute to true scatter. In the first case the galaxy in question would be eliminated from the analysis and a relatively small scatter obtained, while in the second it is included and a higher value results. In many cases such a decision is subjective, and may depend on the worker's own biases with regard to the true scatter. As larger and more complete samples are studied, the importance of such subjective factors will diminish, and it is likely that scatter estimates will converge. A recent study using a data set consisting of over 3000 galaxies culled from five samples (Willick et al. 1995a , b, c; cf. Section 7.2) finds the CCD R and I band scatters to be in the range 0.35-0.40 mag, and the H band scatter to be ~ 0.45 mag. A partly overlapping sample of ~ 2500 spirals analyzed by Mathewson & Ford (1994) estimated the I band scatter at 0.42 mag. Still, the issue is far from closed; claims of very small (~ 0.25 mag) scatter, large TF data sets continue to appear in the literature (e.g., Haynes et al. 1995).

6.1.2. The Faber-Jackson and D_n - for Elliptical Galaxies

The elliptical galaxy DI first used in peculiar velocity studies was the Faber-Jackson (FJ) relation (Faber & Jackson 1976). Like the TF relation, FJ expresses a power-law relationship between luminosity and internal velocity. Unlike spirals, luminous ellipticals do not rotate (Kormendy & Djorgovski 1989), and the relevant measure of internal velocity is the rms dispersion of radial stellar velocities in the central regions of the galaxy, _e. Thus the FJ relation is expressed as

(159)

Unlike the case with TF, early work with FJ (Faber & Jackson 1976 ; Schechter 1980 ; Tonry & Davis 1981) produced reasonable agreement that the exponent of the power law was 4, corresponding to the infrared TF slope found by Aaronson and collaborators (see above). However, while early estimates of the TF scatter were ~ 0.45 mag, the scatter of the FJ relation was found to be roughly twice that. For this reason, although FJ was used with some success in studies of the Virgocentric motion of nearby galaxies (Schechter 1980 ; Tonry & Davis 1981), it did not seem to hold great promise as a DI.

The situation changed dramatically when two independent groups discovered that the addition of a third parameter significantly improved the accuracy of the FJ relation (Djorgovski & Davis 1987 ; Dressler et al. 1987b ⁽²²⁾ ). Working with newly acquired photometric and kinematic data for large samples of ellipticals, each group found that elliptical galaxies populate a planar region in a three dimensional parameter space. One expression of this fundamental plane describes the connection between the effective radius R_e and the average stellar velocity dispersion and surface brightness interior to that radius:

(160)

the dual power law maps to a plane when logarithmic quantities are considered. The two groups both found that 1.4, 0.9 (Faber et al. 1987). When using the fundamental plane as a DI, the 7S group preferred to incorporate the surface brightness and radius terms into a single photometric diameter they called D_n, the diameter interior to which the average surface brightness attained a fiducial value. In a subsequent study (Lynden-Bell et al. 1988a), the group provided a final calibration of the new DI:

(161)

where the exponent 1.20± 0.1. In this form the modified FJ relation is known as the D_n- relation. Dressler (1987a) has shown that the D_n- relation also holds for the bulges of early-type spirals, although this has not been exploited beyond his initial paper.

Lynden-Bell et al. (1988a) found that the scatter in the D_n- relation was 0.10 dex in logD_n, corresponding to about 23% accuracy as a distance predictor. Thus, D_n- was found to be about twice as accurate as FJ, and compared favorably with TF. Indeed, the 7S group used the D_n- distances to make one of the earliest inferences of very large-scale deviation from Hubble flow (Dressler et al. 1987a), as we discuss in greater detail in Section 7.1. It is interesting to note, however, that while a fundamental plane led to a tighter DI, it also implied less insight into elliptical galaxy formation. For, while a one-dimensional relationship (such as TF) requires special conditions to be imposed at galaxy formation, a fundamental plane is simply a reflection of gravitational equilibrium, with the additional (albeit ill-understood; cf. Renzini & Ciotti 1993) assumption that (M/L) be a function only of the other parameters in the relation (Faber et al. 1987 ; Gunn 1988). Thus the fundamental plane relations tell us little more than that elliptical galaxies are gravitationally bound structures, which is hardly surprising.

6.2. Universality of the Distance Indicator Relations

The measurement of redshift-independent distances by means of DIs such as TF and D_n- assumes that the galaxian properties they embody are universal, i.e., independent of spatial location or local environment. If, say, the zeropoint of the TF relation varied systematically from one place to another, then the distances inferred from it would carry a position-dependent error, giving rise to spurious peculiar velocity measurements. Recognizing this, some astronomers have questioned the reality of detections of large-scale departures from Hubble flow (Section 7.1) - or as one theorist asked in the provocative title to a paper on the subject, "Is Cosmic Drift a Cosmic Myth?" (Silk 1989).

Silk (1989) suggested, on the basis of plausible (but speculative) scenarios of galaxy formation, that both the TF and D_n- relations would show a systematic cluster vs. field offset. Because the DIs are typically calibrated using cluster observations, the measured peculiar velocities of field galaxies (which dominate most analyses of large-scale motions) might then be invalid. de Carvalho & Djorgovski (1992) carried out a comparison of the properties of field and cluster ellipticals. They argued that as measured by the relationship between several distance-independent indices (surface brightness, color, metallicity, and velocity dispersion) the field and cluster galaxies differed systematically. The sense of these differences was that field galaxies were bluer, had higher surface brightness, and were more metal-poor than their field counterparts of similar velocity dispersion. They proposed that these differences would follow logically from star-formation histories if the field ellipticals were typically younger than cluster ellipticals. A related study was carried out by Guzmàn and coworkers (Guzmàn 1994 ; Guzmàn & Lucey 1993 ; Guzmàn, Lucey, & Bower 1993 ; cf., Gregg 1992), who also concluded that metallicity influences the fundamental plane relations (Section 6.1.2). Guzmàn (1994) explicitly argued that this "age effect" induces a significant difference between D_n- distances for cluster and field galaxies, and that with appropriate corrections, the large peculiar velocities obtained from D_n- data (Section 7.1) vanish.

There are, however, good reasons to view the results of the previous paragraph with considerable caution. The study by de Carvalho & Djorgovski (1992) mixed two data sets, those of Djorgovski & Davis (1987) and of the 7S group (Faber et al. 1989), which might well differ systematically from one another. Indeed, it is primarily the Djorgovski & Davis (1987) data set which manifests a clear cluster/field offset as measured by the distance-independent indices; no significant effect is found in the 7S data set. The work of Guzmàn and coworkers has been questioned on a variety of grounds (D. Burstein, private communication), in particular the use of an erroneous metallicity-velocity dispersion correlation and outdated stellar population models. Finally, it should be noted that the 7S group tested for metallicity and color dependences of the D_n- relations and found them to be statistically insignificant (Lynden-Bell et al. 1988b). The 7S group recognized the existence of a fraction (~ 10-20%) of the total elliptical population in which stellar population anomalies might be present, but showed that these objects did not strongly affect their conclusions about the systematics of the flows (Burstein et al. 1990).

Another means of assessing whether measured peculiar velocities could be due to DI non-universality is to compare the velocity field in a given region of space obtained from different DIs. Kolatt & Dekel (1994) compared the peculiar velocity fields reconstructed by the POTENT technique (Section 7.5) from spiral (TF) and elliptical (D_n-) data separately, and found no statistically meaningful differences between the two. It is unlikely that zeropoint gradients would affect TF and D_n- in precisely the same way. Another consistency test derives from a comparison of D_n- distances with distances obtained for the same galaxies using the Surface Brightness Fluctuation (SBF) technique (Dressler 1994 ; see Section 6.3.1). This comparison has validated D_n- peculiar velocity estimates for ~ 15 ellipticals in the Southern sky. Since the SBF technique is thought to be highly accurate, and involves measurements entirely unrelated to D_n-, these results represent another encouraging (albeit preliminary) indication of D_n- universality.

In summary, then, there is no compelling evidence at present for substantial spatial or environmental variations in the TF or D_n- relations. While there exist some indications that metallicity and/or age effects enter into the fundamental plane relations for a fraction of elliptical galaxies, these effects apparently do not influence the distance estimates obtained from D_n- in a systematic way. As more data are acquired in the coming years, using a multiplicity of DIs, intercomparisons will permit more sensitive tests of DI universality.

6.3. Beyond TF and D_n - : A Look to the Future?

While the TF and D_n- methods have been the workhorses in peculiar velocity surveys, there a number of other promising distance indicators likely to play a role in coming years.

6.3.1. Surface Brightness Fluctuations

Over the last several years Tonry and collaborators (Tonry & Schneider 1988 ; Tonry, Ahjar, & Luppino 1989, 1990; Tonry & Schechter 1990 ; Tonry 1991) have begun to exploit an old idea first discussed in detail by Baum (1955) : a sufficiently high resolution image of an elliptical galaxy will reveal the discrete nature of the ultimate sources of luminosity - billions of individual stars. A finite number of stars are included in a single pixel of an image; this number is subject to Poisson fluctuations, and therefore so is the pixel intensity. The farther away the galaxy is, the greater is the number of stars per pixel, and the smaller the Poisson fluctuations; thus the fluctuations are a measure of the distance of the galaxy. Tonry measures the surface brightness fluctuation (SBF) which characterizes an image. The mean SBF can in turn be related to the luminosity of the giant branch stars which dominate the light, which depends weakly on metallicity. The technique is applied to ellipticals rather than spirals because the latter are multicomponent, often dusty systems in which the purely Poisson fluctuations cannot easily be isolated.

The uncertainty in SBF distance estimates varies in inverse proportion with the resolution of the images obtained. For galaxies within ~ 3000 km s^-1, SBF distances are estimated to have ~ 5% accuracy under the best observing conditions. Ground-based resolution is limited by microturbulence in the atmosphere ("seeing"). In collaboration with Tonry, Dressler (1994) has undertaken a program to obtain SBF distances for ellipticals studied with the D_n- method using CCD images acquired at the du Pont Telescope in Chile, where the seeing is often 1". This program promises to yield accurate distances for nearby ellipticals, and may significantly improve our knowledge of the local peculiar velocity field. Extension of the SBF technique to distances of ~ 5000 km s^-1 and beyond will require either a dedicated program of space-based imaging, or the use of adaptive optics technology for enhancing ground-based resolution.

6.3.2. BCG L- relation

The DIs discussed thus far apply to ordinary spiral or elliptical galaxies. There is also a class of extraordinary galaxies, the most luminous objects in the centers of rich clusters (called "brightest cluster galaxies" or BCGs), which can be used as DIs. BCGs were originally of interest because they are easily identified out to fairly large (z 0.2) redshifts. As a result, they were thought to be ideal tools for studying problems such as the linearity of the Hubble expansion (Sandage 1972 ; Sandage & Hardy 1973 ; Gunn & Oke 1975 ) ⁽²³⁾ In addition, because they are highly luminous objects and are unique in their environment, their study is not subject to the statistical biases associated with sample incompleteness we discuss in Section 6.4.

The studies just mentioned considered BCGs as standard candles, i.e., they assumed their luminosities were independent of any other galaxian property. Later, Hoessel (1980) showed that BCG luminosity correlates with the shape of the luminosity profile. Specifically, Hoessel defined a metric luminosity L_m as the total light interior to the metric radius r_m = 10 h^-1 kpc, and a luminosity shape parameter

(162)

and found a linear correlation between L_m and . Recently, the L- relation has been exploited by Lauer & Postman (1994; LP), and Postman & Lauer (1995) , who studied the BCGs in the 119 Abell (1958) and Abell, Corwin, & Olowin (1989) clusters with redshifts less than 15,000 km s^-1. LP found a quadratic L- relation with an equivalent distance error of ~ 16%, comparable to the TF scatter. The BCG L- relation is perhaps the most promising technique for extending peculiar velocity studies to much larger scales (z 0.1) than is currently possible, although it requires further study to test its universality.

6.3.3. Possible Methods of Future Peculiar Velocity Work

Two other DIs have recently been the focus of intense observational work, but thus far in the context of the distance scale (i.e., Hubble constant determination) rather than the velocity field. The first of these is fitting the planetary nebula luminosity function (PNLF; Jacoby et al. 1989 ; Ciardullo et al. 1989a, b; Jacoby et al. 1990; Ciardullo, Jacoby, & Harris 1991). "Planetary nebulae" are the luminescent ejectae of dying stars soon to become white dwarfs. Planetary nebulae may be detected and photometered in nearby ( cz 1500 km s^-1) elliptical galaxies using narrow-band imaging in the light of twice-ionized oxygen at = 5007 Å. The PNLF in this emission line is observed to have a universal form, with a sharp cutoff at the highest luminosities (Jacoby 1989 ; Ciardullo et al. 1989a). Fitting the observed distribution of planetary nebula brightnesses to the universal PNLF yields the distance to a galaxy with ~ 5% accuracy. The PNLF and SBF methods yield consistent distances for Virgo Cluster galaxies (Ciardullo, Jacoby, & Tonry 1993). At distances much greater than that of Virgo it is quite difficult to reliably identify even luminous planetary nebulae using present techniques (Jacoby et al. 1992). As a result, the usefulness of the PNLF method for peculiar velocity work is uncertain.

A DI which has been around for some time, but which is currently undergoing a kind of renaissance, is the mean absolute luminosity of Type Ia Supernovae (Sne Ia). Of all the DIs discussed here, Sne Ia have the best-understood physical basis. It is generally agreed that Sne Ia result from thermonuclear disruption of a carbon-oxygen white dwarf that has accreted enough mass from a companion star to approach the Chandrasekhar mass of ~ 1.4 M (e.g., Wheeler & Harkness 1990 and references therein). The radioactive decay of ⁵⁶Ni and ⁵⁶Co following the explosion power the subsequent observable light curve. Theoretical models suggest that the mass of the nickel ejectae is in the range 0.6-0.8 M (Arnett et al. 1985); when combined with the observed rise time to maximum light of ~ 17 days, an absolute luminosity of M_B - 19.6 mag is predicted.

There has been some controversy over their intrinsic scatter as standard candles. The discovery that the peak luminosity of SN 1a correlates with their rate of decline (Pskovskii 1977 , 1984; Phillips 1993 ; Hamuy et al. 1995 , but see Tammann & Sandage 1995) may tighten up the scatter considerably, to as small as 0.1-0.2 mag (Hamuy et al. 1995 ; Reiss, Press, & Kirshner 1995).

Sne Ia have been used almost exclusively for Hubble constant determination (e.g., Saha et al. 1994 a, b). Indeed, their idiosyncratic nature as DIs - one must wait for a supernova to appear in a galaxy in order to estimate its distance, and their occurrence is both unpredictable and rare - limits their utility for peculiar velocity work. However, they can be used as DIs to extremely large distances with relatively little loss of accuracy (e.g., Perlmutter et al. 1993), and may thus prove useful in extensions of peculiar velocity analyses to large distances.

Type II supernovae, which mark the end-point of the life of a massive star, can also be used as a distance indicator (Schmidt, Kirshner, & Eastman 1992 ; 1994). Spectroscopic observations yield the velocity of the ejecta and the temperature of the outer layers, from which the absolute brightness can be inferred with the use of model atmospheres. Comparison with the apparent brightness yields a distance. This method requires extensive model calculations and detailed observations, and has thus far been applied only to a handful of objects.

6.4. Statistical Bias and Methods of Peculiar Velocity Analysis

With the possible exception of the SBF technique, the DIs used in peculiar velocity surveys are not very accurate. The TF and D_n- relations, for example, predict galaxy distances with only ~ 20% accuracy. In the volume within which we hope to study the peculiar velocity field in reasonable detail, typical galaxies may lie at distances of ~ 3000 km s^-1. The rms peculiar velocity error for such a galaxy is thus on the order of 600 km s^-1, which also happens to be the amplitude of typical peculiar velocities. At interesting distances, then, we cannot measure with any precision the peculiar velocity of a single galaxy. Meaningful analyses must use statistical techniques applied to large samples.

At first glance, this might not be considered a major problem. There are, after all, thousands of galaxies in the local volume for which we have obtained, or soon hope to obtain, distance indicator data. We might expect that by virtue of N statistics alone the signal-to-noise ratio of any statistical analysis could be made rather high. However, this expectation is not realized in practice. The analysis of DI data is instead subject to statistical bias effects; these effects result in random errors dropping more slowly than 1/N, and also in the possibility of large systematic errors if the biases are not properly corrected for. While there are a number of bias effects, all originate in a coupling of the DI scatter with external influences on the makeup and spatial distribution of the redshift-distance sample. The goal of this section is to explore the nature and consequences of this coupling in a variety of circumstances.

A peculiar velocity analysis has, logically if not sequentially, two steps. The first is calibration of the DI being used. If we assume for definiteness that it is the TF relation, calibration means determination of the zeropoint A, slope b, and scatter which quantify the M- relationship (Section 6.1.1). The second step in the analysis is the inference ⁽²⁴⁾ of distances and thus peculiar velocities for sample objects. Each step is potentially subject to biases. Those which enter into the calibration phase, if not corrected for, can lead to erroneous determinations of any or all of the parameters A, b, or . This incorrect TF relation will produce incorrect distances and peculiar velocities. Even with a properly calibrated TF relation, however, uncorrected biases in the distance-inference phase have the same result.

The separation of bias effects into "calibration" and "inferred-distance" problems (Willick 1994) is useful in emphasizing the different ways bias can affect peculiar velocity analyses. However, it does not distinguish the root causes of bias. These lie not in the goal but in the underlying assumptions of the analytic approach, as we will clarify in Section 6.4.1. With one set of assumptions, an analysis will be subject to selection bias, which is intimately related (as the name implies) to observational selection criteria such as a flux limit. With the second set of assumptions, Malmquist bias will result. As defined here (but see next paragraph), Malmquist bias is not fundamentally related to a flux limit or any other selection criteria. It occurs because the true distance of a galaxy cannot be estimated from DI information alone, but requires knowledge of the actual line of sight density distribution as well. Selection bias can affect either DI calibration or peculiar velocity measurement - or both, as they often are carried out simultaneously (Section 6.4.3). Malmquist bias typically enters only when a calibrated DI is used to infer peculiar velocities; approaches to DI calibration which are subject to Malmquist bias are rarely used in practice and will not concern us here.

We organize our discussion mainly around the selection/Malmquist dichotomy, but we note that this distinction has rarely been maintained in the existing literature, where much confusion reigns. One source of confusion is the ambiguity surrounding the term "Malmquist bias," which historically has meant several different things. The term originated with the work of Malmquist (1924) , who showed that flux limited samples were subject to a luminosity bias - one which in fact resembles our selection bias more closely than our Malmquist bias. Thus, early discussions of DI bias (Sandage, Tammann, & Yahil 1979 ; Teerikorpi 1984 , 1987; Aaronson et al. 1986) typically refer to bias effects associated with flux limits as "Malmquist biases." However, the distinct biases related to the line of sight galaxy density distribution were also referred to as "Malmquist" effects by Lynden-Bell et al. (1988a) , who were the first to discuss them quantitatively in the extragalactic context. The latter usage has gained widespread acceptance, and it is the one we adopt here. Even so the impression has persisted in some quarters that Malmquist bias thus defined relates to a flux limit. One goal of the discussion to follow is to dispel this and related misperceptions brought about by conflicting uses of this catch-all phrase.

6.4.1. Selection vs. Malmquist Bias - Method I vs. Method II

Because of DI errors, data from a large number of galaxies are needed if the estimated peculiar velocity at a given point r is to be of value. Thus, in devising a method of analysis we must ask the question, "for what particular set of objects should data be averaged ⁽²⁵⁾ to estimate peculiar velocity at position r?". There are two methodologically orthogonal answers to this question:

1. the set of objects which redshift space information suggests are "near" r;
2. the set of objects which the DI information suggests are "near" r.

A method based on the first answer will be subject to selection bias; one based on the second answer will experience Malmquist bias. This holds true whether our goals are DI calibration, peculiar velocity measurement, or both.

It will prove useful to have terms describing methods of peculiar velocity analysis, in addition to terms describing the types of bias to which they are subject. When redshift-space information is used as the a priori indicator of true proximity, we will describe the analysis as belonging to "Method II." When DI information is used as the a priori indicator of true proximity, we will describe the analysis as belonging to "Method I." This terminology was originally introduced by Faber & Burstein (1988) , although their usage was somewhat less general than ours. In Section 6.4.2 and Section 6.4.3, we will show, using prototypical examples, why Method II is subject to selection bias and Method I to Malmquist bias, and clarify how these biases affect the analysis. Later, in Section 6.5 we will show how to quantify and correct for these biases. We will assume throughout that the DI in question is the TF relation, but the concepts apply equally to DIs such as D_n-.

6.4.2. Bias in a Cluster Sample

Suppose that a flux-limited Tully-Fisher sample consists of spiral galaxies thought to belong to an isolated rich cluster. A reasonable assumption to adopt in analyzing such a sample is that its members all lie at the same distance. This is an example of a Method II analysis: redshift-space information - from which we infer cluster membership - is treated as the a priori indicator of distance; the TF information has not entered into the assumption in any way. Let us further suppose that our goal is to calibrate the TF relation. Using the common distance assumption we may write the expected apparent magnitude of a cluster galaxy with velocity width parameter as <m()> = A - b + µ (see Eq. 152), where µ = 5logr_c is the cluster distance modulus; for now suppose that the cluster distance r_c, in km s^-1, is independently known. We also expect that the observed apparent magnitudes will scatter about <m()> with rms dispersion . These considerations seem to suggest that linear regression of m on for the cluster sample will correctly give the TF slope and zeropoint (after subtracting the distance modulus), and that the scatter about that fit will yield .

However, because we have made a priori distance assignments based on redshift-space information, the above exercise will be subject to selection bias. How, specifically, does the bias enter in? Let m denote the magnitude equivalent of the flux limit. Fig. 13 illustrates a Monte-Carlo simulation of a Tully-Fisher sample at a common distance, with a scatter of 0.35 mag, and a magnitude selection at m = 16. The dashed line is the input TF relation. Consider a subset of sample galaxies with similar and relatively large velocity width parameters ₁ = 0.2 such that they are typically much brighter than the flux limit, i.e., <m(₁)> << m. Their observed apparent magnitudes m will be distributed uniformly about <m(₁)> with rms dispersion . Now consider a second subset with relatively small width parameters ₂ = - 0.2 such that <m(₂)> m. They will exhibit a range of apparent magnitudes m <m(₂)>. However, none will have m > <m(₂)>. Such galaxies exist in the cluster, but are "missing" from the sample because they are fainter than the magnitude limit. The mean apparent magnitude of the second subset will therefore be brighter than <m(₂)>, as the solid line in the figure shows. This is one definition of bias: the mean apparent magnitude does not conform to the TF prediction. A linear fit to the points which does not take into account the magnitude limit will be biased; as illustrated by the stars in Fig. 13: the slope of the fitted line is flatter (by 2 in this example), and the fitted zeropoint is brighter (by 0.3 mag in this example) than the true TF relation (the dashed line).

Figure 13. A Monte-Carlo simulation illustrating the effects of a magnitude limit on the derived TF relation. Points were assumed to be at a common distance, and assigned a TF relation (indicated by the dashed line) with a scatter of 0.35 mag. The sample was limited at apparent magnitude 16. The stars indicate the best-fit linear TF relation which does not take the magnitude limit into account; it is shallower, with a brighter zeropoint.

When galaxies scatter from r to d because of DI errors, they "bring with them" the correct redshifts of their true positions. That is, DI errors do not affect redshifts, but only the distances at which they are evaluated and converted to peculiar velocities. Thus, if _d (say) is the mean true distance of galaxies with inferred distance d, the mean redshift of these galaxies will to first order be <cz(d)> = _d + u_t(_d), where the subscript t signifies true as opposed to inferred radial peculiar velocity ⁽²⁶⁾ . From this redshift one would infer a peculiar velocity u(d) = u_t(_d) + (_d - d). Thus, the distance bias (d _d) translates directly into a peculiar velocity bias. In the case that the galaxies truly constitute a cluster (i.e., _d = r_c for all d), we see that u > u_t when d < r_c, and u < u_t when d > r_c. It follows that the inferred radial peculiar velocity field u(d ) would exhibit a compressional inflow into the cluster center, even if there were no real peculiar velocities. This last effect - spurious peculiar velocities associated with strong line of sight density gradients, of which a cluster is the most extreme case - is the most serious consequence of Malmquist bias. As density gradients generally are not known with great precision, Malmquist bias can be difficult to correct for reliably (Section 6.5.2).

6.4.3. Bias in a Field Sample

We used the example of a cluster to distinguish selection and Malmquist bias with maximum clarity. However, peculiar velocity analyses more often involve field galaxy samples. It is therefore important to realize that the two kinds of bias occur, for analogous reasons, in field sample analyses as well. In the cluster example, the Method II approach seemed "natural" in comparison with Method I. With a field sample, the choice of method is not clear cut. We illustrate these issues in what follows.

A Method II analysis of field galaxies usually entails the adoption of a model of the peculiar velocity field, v(r; a); the quantity a is a vector of free parameters which specifies the model. An important example is a velocity field obtained from Eq. (150) using redshift survey data, with as the only free parameter. Using the model, one obtains a distance r for a galaxy of redshift cz as the solution of the equation

(163)

Such an expression of course assumes a unique redshift-distance mapping, which may not obtain in the vicinity of large mass concentrations (see Section 5.9), an issue we will return to later. From Eq. (163) one can predict the galaxy's apparent magnitude, given the value of its velocity width parameter :

(164)

this predicted apparent magnitude depends, as indicated, not only on the velocity field parameter vector a, but also on the TF parameters A and b, whose determination might also be part of the analysis. One can then form a ² statistic of observed minus predicted apparent magnitudes:

(165)

(the individual object scatters are usually assumed equal in the simplest implementations of Method II). Minimization of this statistic might then be expected to yield the "best" values of the velocity field parameters a and (if necessary) the TF parameters A and b.

A bit of reflection shows, however, that such an exercise will be subject to the same bias that affected the first TF analysis of cluster galaxies discussed in Section 6.4.2. In particular, there is again the issue of "missing" galaxies, although in a more abstract sense than earlier. There is in fact only one galaxy at redshift cz_i, direction _i, and width parameter _i. However, the validity of the ²-minimization procedure requires that this object be representative of the ensemble of all possible such objects. Because of the flux limit, however, it cannot be: if the predicted apparent magnitude m(_i,cz_i; A, b, a) is close to or fainter than the flux limit, the fainter members of this hypothetical ensemble cannot belong to the sample. As a result, the observed apparent magnitude m_i is likely to be brighter than the predicted value; minimization of _II²(A, b, a) consequently does not yield the correct values of the parameters on which it depends. A Method II analysis of a field sample is thus affected by selection bias, just as was the earlier one based on the assumption of equidistant cluster members. We note that although we have assumed the existence of a model peculiar velocity field, this is not fundamental to a Method II field sample analysis. For example, if pure Hubble flow were assumed (v(r;a) vanishes identically), an exercise identical to that outlined above could be used just to calibrate the TF relation; its results also would be affected by selection bias.

A Method I analysis of the same sample would proceed differently. To make the comparison as direct as possible, we again suppose that the purpose is to fit a model velocity field v(r; a). Now, however, there is no need to solve Eq. (163) to obtain a distance. Instead we use the inferred distance d as the best indicator of true distance, and thus as the position at which we evaluate the velocity model. Inferred distances are given as described in the Method I cluster analysis (Section 6.4.2), and the TF relation is again assumed to have been independently calibrated ⁽²⁷⁾ . Using the model velocity field we write a predicted redshift for a galaxy with inferred distance d as

(166)

where d = d . This in turn motivates a ² statistic of the form

(167)

where _i is a suitable measure of redshift and distance errors. Again one might suppose that minimization of this statistic leads to the best values of the velocity model parameters a.

However, this naïve approach will lead to biased results. The validity of the ² minimization rests on the assumption that cz(d_i; a) is an unbiased estimator of cz_i when the velocity model is correct. As discussed in Section 6.4.2, the object with inferred distance d_i can come from a range of true distances with varying probability. Let the expected true distance of this object be _di. Its observed redshift then has a corresponding expected value <cz_i> = _di + u_t(_di), where as in Section 6.4.2 u_t denotes the true radial peculiar velocity field and we neglect second order effects. From this it follows (compare with Eq. 166) that, even if the velocity model is correct, <cz_i> cz(d_i, a) unless _di = d_i. But we know from our discussion of the Method I cluster analysis (Section 6.4.2) that this last condition does not hold in general; the relationship between _di and d_i depends on the details of the underlying density distribution n(r), which is the essence of Malmquist bias. Thus cz(d_i, a) is a biased measure of cz_i, and as a result the parameters which minimize _I² yield an incorrect velocity field. Again, while we have framed the problem in terms of a velocity model, Method I includes any analysis in which galaxies with similar inferred distances are assumed to be physically proximate and, therefore, to share a peculiar velocity vector. In particular, nonparametric methods, such as the POTENT reconstruction (Bertschinger & Dekel 1989 ; see Section 7.5), which smooth in inferred-distance space, belong to Method I and suffer from Malmquist bias.

6.4.4. "Inverse" Distance Indicator Relations

There is another basic division between methods of peculiar velocity analysis, one which has sometimes been subsumed under the Method I/II dichotomy but in fact is entirely distinct. Until now we have considered only the "forward" form of DIs, in which the distance-independent quantity ( in the case of TF) is thought of as predicting the distance-dependent quantity (M in the case of TF). But we can just as easily "turn the relation on its side" and consider the TF relation as a predictor of velocity width given absolute magnitude ⁽²⁸⁾ . We quantify this prediction as ₀(M) (strictly speaking, the expectation value of given M), and call it the "inverse" TF relation. Like its forward counterpart ₀(M) is well-described by a linear relation

(168)

we denote its rms dispersion . Because of this finite scatter the forward and inverse forms are not mathematically inverse to one another, i.e., e b^-1 and D A. The forward and inverse forms of DIs are alternative representations of a single physical phenomenon; nonetheless, their statistical properties differ markedly.

We illustrate with the following example both this difference in bias properties, and why the forward/inverse and Method I/II distinctions have been conflated. Suppose that we wish to carry out a Method II analysis of the field galaxy sample of Section 6.4.3 using the inverse TF relation. In that case we use Eq. (163) to predict the velocity width parameter of an object with redshift cz and apparent magnitude m as

(169)

this equation motivates a new ² statistic

(170)

apparently a perfect analogue to its forward counterpart, Eq. (165). The analogy breaks down, however, when one considers the bias properties of this inverse statistic. In the forward case bias arose because the sample galaxy with redshift cz and velocity width parameter was not representative of the ensemble of all possible such objects; the presence of a flux limit meant that the faint end was excluded. In the inverse case the relevant question is whether the sample galaxy with redshift cz and apparent magnitude m is representative of all such objects. If we assume that sample selection is independent of the value of for a given m, then the answer to this question is evidently "yes," since m must be brighter than the flux limit. There now are no "missing" galaxies, regardless of how close m is to the flux limit, and (m, cz; D, e, a) is an unbiased estimator of _i; minimization of _II²(D, e, a) may thus be expected to yield essentially unbiased values of the free parameters of the model.

This felicitous property of inverse DIs was first noticed by Schechter (1980) in a study of the Virgocentric flow field, and was subsequently exploited to great advantage by Aaronson et al. (1982b) and by Faber & Burstein (1988) . These studies enshrined the notion that the inverse relation was free of selection bias, and also cemented the association between Method II and the inverse form of DIs. They also, perhaps because of the catch-all nature of the term "Malmquist," led to the assertion (e.g., Tully 1988) that inverse DIs are free of Malmquist bias. None of these conventional notions are accurate. The above example shows only that an inverse Method II analysis is unaffected by selection bias when sample selection is independent of velocity width. The paradigmatic form of sample selection, a photometric flux limit, which we have used here for illustrative purposes, is -independent. However, as we discuss below (Section 6.5.4), in real TF samples selection criteria are more complex and may depend, usually weakly, on velocity width. In such cases, even an inverse Method II analysis will be subject to mild selection bias. Moreover, there is no inherent connection between inverse DIs and Method II; the essence of Method II is its a priori assignment of distances from redshift, after which either form of the DI can be employed.

Finally, because of the conventional association between Method II and inverse DIs, it has not generally been recognized that inverse DIs can be used in Method I analyses. We defer the details to Section 6.5.5, and note here only that an inverse inferred distance is defined (in the case of TF) by the implicit relation ₀(m - 5logd )= . A consequence of this definition, as will be come clear, is the fact that inverse and forward inferred distances differ, in general, for a given galaxy. Nonetheless, the fundamental source of Malmquist bias in forward Method I analyses - that the likelihood an object will scatter from true distance r to inferred distance d depends on the density n(r) - is equally present in an inverse Method I analysis. While the use of forward vs. inverse affects its details, Malmquist bias inevitably accompanies a Method I analysis.

6.4.5. The "Method Matrix" of Peculiar Velocity Analysis

We provide here a succinct summary of the multiple and to some degree overlapping methods of peculiar velocity analysis in the form of a "method matrix." The notion of a matrix derives from the two-dimensional character of the methodology discussed above: peculiar velocity analyses can belong to either Method I or II, and can employ either the forward or inverse form of the relevant DI. The statistical biases present are specific to each case. Table 2 summarizes these ideas; two terms in that table not yet defined, the "Gould effect" and the "Landy-Szalay correction," are discussed in Section 6.5.

Table 2. "Method Matrix" of Peculiar Velocity Analysis

DI type/Method Type	Method I	Method II
	DI-inferred distance best	Redshift-space data best
	indicator of true distance	indicator of true distance
Forward	Malmquist bias	Strong selection bias
dist-dep (e.g.mag) predicted	(selection-independent	(depends on observational
by dist-indep (e.g. ) quantity	unless "Gould effect")	selection criteria)
Inverse	Malmquist bias	Weak or no selection bias
dist-indep predicted	(selection-dependent,	(bias present if selection
by dist-dep quantity	Landy-Szalay possible)	related to dist-indep quantity)

We have defined Methods I and II and discussed the biases inherent in each; we have not yet indicated why we might choose one over the other. Method I has the advantage of being unburdened by preconceived notions about the relation between redshift and distance. Method II, which requires that we invoke at least some form of redshift-distance relation, is necessarily subject to our theoretical prejudice. Ideally, we would like to know where the galaxies really are, and only later compare these real-space positions with the observed redshifts. The most compelling motivation for this is the possibility of triple-valued zones in dense regions (Fig. 11), where infall patterns along the line-of-sight can result in widely separated galaxies having the same observed redshift. Since Method I considers redshift only after it has placed the galaxies at their DI-inferred distances, this presents no special problem. By contrast, Method II is inherently incapable of dealing with multivaluedness in the redshift-distance mapping. It assumes that redshift uniquely specifies distance and thus presents an overly smooth picture of a velocity field which in reality is rapidly changing. (In Section 8.1.3 we will present a recent variant of Method II which circumvents this particular flaw.)

On the other hand, Method II has the advantage that its statistical aspects are comparatively well-constrained. If we suppose that the velocity field model v(r; a) is not unreasonable, then the dominant errors in the assignment of distances (Eq. 163) come from "noise" in the velocity field, whose rms value in the field appears to be at most ~ 200 km s^-1 (Section 5.2.1). The dominant errors in Method I distance assignments come from DI scatter, and are therefore several times larger at interesting distances. Moreover, as we will see more clearly in the following section, we can correct for selection bias in terms of knowable information, namely, observational selection criteria. With inverse DIs we can realistically expect selection bias to be quite small. Correction for Malmquist bias, however, depends on density gradients along the line-of-sight, which are not knowable a priori. We can estimate such gradients from redshift data, but this involves additional assumptions concerning peculiar velocities (needed to map redshift- to real-space density) and smoothing methods. Working with inverse DIs gains us nothing in terms of Malmquist bias. In summary, both Method I and Method II have advantages and disadvantages. As we describe in Chapter 8, both are being used in contemporary peculiar velocity analyses. It should be pointed out, however, that many of the analyses we describe in Chapter 8 were done without proper attention to the biases we describe here, and of the differences between the different entries in the Method Matrix of Table 2. Indeed, there remains much work to be done in exploring all these methods; for example, there is no analysis beyond that of Landy & Szalay (1992) which exploits Method I with an inverse DI.

6.5. Quantifying Statistical Bias

In the last section we presented a qualitative outline of the main statistical biases affecting distance indicator and peculiar velocity analysis. In order to correct for these effects, the biases must be quantified and correction formulae and methods developed. This task has been the focus of a growing number of workers over the last decade or so. Sandage and coworkers (e.g., Sandage, Tammann, & Yahil 1979 ; Sandage 1988 ; Federspiel, Sandage, & Tammann 1994) have provided useful, though mainly qualitative, suggestions for treating selection bias. A quantitative approach to the subject was pioneered by Teerikorpi, Bottinelli, and collaborators (e.g., Teerikorpi 1984 , 1987; Bottinelli et al. 1986), and extended by Willick (1991, 1994). Malmquist bias (in the DI context) was first treated in detail by Lynden-Bell et al. (1988a) ; this work has been followed up and extended by Willick (1991, 1995), Landy & Szalay (1992) , and Gould (1993) . The bias properties of inverse DIs have been studied by Schechter (1980) , Aaronson et al. (1982b) , Faber & Burstein (1988) , Tully (1988) , Han (1991) , Roth (1993) , Willick (1994, 1995), and Triay, Lachièze-Rey, & Rauzy (1994) . We will use the formalism developed by Willick (1991, 1994, 1995) who has advocated a comprehensive approach to the problem. We assume in what follows that the distance indicator in question is the TF relation, but the discussion is readily modified to treat comparable DIs such as D_n-.

The material discussed in this section remains controversial, and the formal mathematical approach we have adopted has not been accepted by all of the workers in the field. For example, Mathewson & Ford (1994) have argued that approaches like ours require idealized assumptions about sample selection that are invalid for realistic data sets. They instead separate their sample into "control" and "target" regions, using the former to calibrate the biases in the latter. We believe, however, that such qualitative or semiquantitative approaches are subject to their own caveats (are the control and target regions the same in their bias properties?), and run the risk of obscuring rather than illuminating the principal issues. Unlike much of the rest of the review, we have not attempted here to summarize all the analyses extant in the literature, but rather aim to give a comprehensive coverage of our own approach. This section can be skipped on a first reading.

6.5.1. Forward DIs: Selection Bias

We consider first the formalism appropriate to the forward relation; the inverse relation is treated in Section 6.5.4. A starting point in the quantification of bias is the empirical fact that it is the TF observables, and not the inferred distances, which exhibit a roughly Gaussian distribution (e.g., Willick et al. 1995c). Thus, the apparent magnitude of a galaxy with width parameter and true distance r is normally distributed about a mean value M() + 5log r with rms dispersion , when selection effects may be neglected. We combine this Gaussian distribution with the a priori distributions of galaxy distances and width parameters, and with a function describing sample selection probability, to obtain the joint probability distribution of the observables (m, ) and the true distance r for a galaxy:

(171)

where µ(r) = 5log r, () is the -distribution function (similar to the luminosity function), and n(r) is the galaxy number density along the line of sight in question. The quantity S(m, ) is the sample selection function, defined as the probability an object with observables (m, ) will be included in the sample irrespective of other considerations. Note that we express the selection function here in terms of the observables, rather than in terms of distance as we did in redshift survey analyses (Section 3.4). An idealized form of S(m, ) describes a sample with a strict magnitude limit,

(172)

Selection and Malmquist bias effects are each implicit in Eq. (171). Which of the two comes into play depends, as already noted, on the assumptions underlying the method of data analysis; these assumptions determine which of several conditional probability distributions derivable from Eq. (171) are used. We first discuss selection bias; in Section 6.5.2 we discuss Malmquist bias.

We know from Section 6.4.1 that selection biases arise in Method II analyses, in which we take redshift-space information as the a priori distance indicator, either through the assumption of a common distance for cluster galaxies, or via a redshift-real space mapping such as Eq. (163). To the extent this mapping is a good one, we are in effect taking the true distance as "known" - dependent, perhaps, on model free parameters, but entirely independent of the TF observables. ⁽²⁹⁾ In mathematical terms, this is equivalent to using probability distributions which are conditioned on the value of the true distance r. In the forward TF case, the relevant distribution is P(m|, r), which is readily derived from the joint probability distribution Eq. 171:

(173)

Eq. (173) is the mathematical basis of a forward Method II analysis. The distance r would be replaced by a redshift-space prediction, such as Eq. (163), in an actual implementation of Method II.

If sample selection were independent of apparent magnitude, S(m, ) would drop out and Eq. (173) would reduce to the familiar normal distribution centered on the naïve expected apparent magnitude M() + µ(r). This is the condition that a simple ² minimization, such as that suggested by Eq. (165), be valid for determining values of any unknown model parameters. In a real study, however, sample selection almost always depends on apparent magnitude, although the dependence is generally indirect, as we explain below. For simplicity, we first suppose that in fact a strict magnitude limit, Eq. (172), applies. In that case Eq. (173) obviously does not reduce to a normal distribution centered on M() + µ(r). It is, rather, straightforward to show that the expected apparent magnitude is given by

(174)

where

(175)

The last term on the right hand side of Eq. (174) is the apparent magnitude bias, the difference between actual and naïve expectation values. Its significance is determined by the parameter (, r;m), a dimensionless measure of how "close" the object is expected to be to the magnitude limit. For 1, the bias is small; in that case few galaxies are "missing" due to the magnitude limit. But for 0, the hypothetical ensemble of galaxies of given and r is strongly truncated by the magnitude limit, and the bias is comparable to or greater than the TF scatter. Note that the sense of the bias is that objects are typically brighter than the naïve TF prediction, as we saw in Section 6.4.1. In any magnitude-limited sample, a large fraction of the objects are likely to be near the magnitude limit. Hence, the bias represented by Eq. (174) is a problematic issue in most Method II analyses ⁽³⁰⁾ .

In real samples, the situation is more complicated. TF and D_n- samples are drawn from catalogs culled from photographic plate material. Such catalogs (see Section 3.1) typically list galaxy apparent magnitudes and/or angular diameters on a measurement system different from that used by the DI. The catalog might give, for example, blue-bandpass photographic magnitudes (m_B), while the TF study uses I band CCD magnitudes (m_I). Candidates for the TF sample might be required to satisfy m_B m, but once thus selected, the allowed values of m_I are unrestricted (the CCD photometry is far deeper than the catalog). Nonetheless, it is clear that the distribution of I band magnitudes must affected by this selection procedure, as m_I is sure to be correlated with m_B. Analogous statements hold if the sample is limited by catalog diameter rather than magnitude.

Willick (1994) has shown how this problem of an "indirect" magnitude limit may be dealt with in several practical situations. Let us refer to the quantity determining sample inclusion as , which might be a photographic magnitude or logarithmic diameter, and assume it is required to satisfy . The selection probability for an object with TF observables (m, ) is then given by

(176)

The quantity on the right hand side may be determined by first assuming that P( | m,) is a Gaussian centered on a mean relation (m, ) and with rms dispersion . Both may be determined empirically from the data; Willick et al. (1995a, b) have shown that (m, ) is well modeled as linear in m and when is a photographic apparent magnitude or logarithmic diameter. The resulting Gaussian probability distribution is then integrated over the allowed values of :

(177)

This selection function may be substituted into Eq. (173), and the expected apparent magnitude, and thus the bias, may then be obtained in the usual way. The results, given by Willick (1994) , are complicated and need not be reproduced here. We note only that, as in the case of a strict magnitude limit, the bias again depends on a "limit closeness parameter," given in the present case by

(178)

and is also proportional to a "coupling" parameter / , which measures the tightness of the TF relation relative to that of the -(m, ) correlation. In particular, when the latter correlation is weak, the bias is small, since selection criteria only weakly constrain the apparent magnitude used in the TF analysis. In most real samples, however, the -(m, ) correlation is strong enough that even the bias due to "indirect" limits is significant. Another feature of an indirect limit is that, if is significantly correlated with as well as with m, sample selection probability is -dependent; the consequences of this are discussed in Section 6.5.4.

The formulae given by Willick (1994) for the magnitude and dispersion biases allow one to correct for selection bias. The basic idea is to iterate least-squares fits of the underlying model, using the parameter estimates from one iteration to correct the next for bias. "Corrected" apparent magnitudes m^(c), defined as the observed magnitudes plus the computed bias, are used in the iterated fits. This procedure typically produces convergent parameter values in ~ 10 iterations, and has been used by Willick et al. (1995a, b) to calibrate TF relations for recent peculiar velocity analyses. An alternative to iteration is direct likelihood maximization, using individual object probability functions P(m | , r) given by Eq. (173).

In Fig. 14 we illustrate the effects of selection bias on both DI calibration and distance determination using numerically simulated data. Each simulated data set consisted of two clusters with 25 galaxies each. The distance to the first cluster was fixed at r₁ = 1500 km s^-1, which was assumed known to the "observer." The second cluster was stepped out in distance, as shown on the horizontal axis; its distance r₂ was assumed unknown to the "observer." Determination of its distance modulus µ₂ was assumed to be one goal of the analysis; the other was determination of the TF parameters A and b. The simulated data obey a TF relation with slope b = 7.5 and scatter = 0.35 mag; sample selection was imposed by requiring galaxies to have photographic diameters 1'. These diameters were generated from the TF observables (m, ) using empirical relations from real data sets (Willick et al. 1995a b).

Figure 14. The main effects of selection bias on a Method II TF analysis are shown in the left-hand panels; the same results following application of the iterative correction procedure (see text) are shown in the right-hand panels. Data sets consisting of 25 galaxies in each of two clusters were numerically simulated. The simulated galaxies obey a true TF relation with slope b = 7.5 and scatter = 0.35 mag, and are subject to a diameter limit. Each simulated data set consisted of one cluster at fixed distance r1 = 1500 km s-1, assumed known, and another at variable distance r2 shown on the horizontal axis, which was taken as an unknown in the analysis. Each plotted point represents the mean of ~ 25 simulations within bins of 100 km s-1. A TF relation was fitted to each data set; the free parameters were the zeropoint A (not shown), slope b, and the distance modulus µ2 of the second cluster. The left hand panels exhibit the principal effects of selection bias: the fitted slope and scatter are too small, and the distance of the second cluster is underestimated. See text for details.

For each simulated data set the TF parameters A and b, and the distance modulus of the second cluster µ₂, were solved for using standard least-squares techniques. An rms scatter about the fit was computed separately in each cluster. The results of these least-squares fits are shown in the left-hand panels of Fig. 14. As the second cluster moves farther away, so that an ever-larger fraction of its member galaxies are near the diameter limit, the bias increases: the top and middle panels show that the fitted slope and rms scatter (in the second cluster) become progressively smaller as the second cluster moves farther away. (The fitted slope levels out at large distances because it is strongly constrained by data in the first cluster, which is close enough to be nearly unbiased.) The bottom panel shows the error in the least-squares determination of the distance modulus of the second cluster, µ₂ = µ₂ - 5logr₂. This error is in the sense that the distance is underestimated, and grows fractionally worse with increasing distance. The right hand panel of Fig. 14 show the same quantities following application of the iterative bias correction scheme discussed above. As can be seen, the iterative method yields unbiased final values of the slope, scatter, and distance of the second cluster.

6.5.2. Forward DIs: Malmquist Bias

Malmquist bias arises in Method I analyses, in which we first assign distances directly from the DI, and consider redshift information only afterward. As discussed above, the large scatter of DIs such as TF or D_n- guarantees that objects with inferred distance d come, with varying probability, from a wide range of true distances. The probability distribution of true distance r given inferred distance d is thus the mathematical basis for treating Malmquist bias.

The simplest way to arrive at P(r| d ) is first to consider P(r, µ(d)), which we obtain from Eq. (171) as follows:

(179)

In Eq. (179) is the Dirac delta function, and we have defined

(180)

Converting to natural logarithms we derive

(181)

where (ln 10 / 5) 0.46 is a measure of the fractional distance uncertainty of the DI. The desired conditional probability is then given by

(182)

We see that S(m, ) has dropped out entirely in Eq. (182); selection bias does not play a role in forward Method I analyses. This may appear odd at first glance; one would think that because selection depletes the sample more at r > d than at r < d, it must affect the ratio of near and far objects at inferred distance d. The reason it does not may be understood heuristically as follows. Objects at r < d must be intrinsically fainter than expected for their velocity width (M > M()) in order to scatter out to d; objects at r > d must be brighter than expected (M < M()) in order to scatter in to d. Thus, although the sample as a whole is progressively depleted with distance, the set of objects with inferred distance d is atypical: its closer members are unusually faint, and its more distant members are unusually bright. This discrepancy exactly balances the overall decrease in sampling density with distance. It is important to note, however, that this argument is specific to the forward form of the DI (we show how inverse DIs differ in this respect in Section 6.5.5) and to the absence of an explicit distance-dependence in the sample selection function (in Section 6.5.3 we demonstrate how such a dependence might arise).

Although independent of observational selection, forward Malmquist bias is nonetheless a complicated effect. It is best illustrated by considering the expectation value of r given d :

(183)

where x = r / d. Eq. (183) shows that, in general E(r | d) d; that is, the inferred distance is biased. It also shows that Malmquist bias depends on the number density of galaxies all along the line of sight, and thus is a nonlocal effect.

The simplest (though usually unrealistic) example of Malmquist bias occurs when density gradients are negligible on the scale of distance errors (i.e., n' / n << (d)^-1). In that case, the density term drops out of Eq. (183). The integrals are then easily evaluated with the result

(184)

Thus even in the case of uniform density E(r| d) d. This effect, uniform density Malmquist bias, is due to the increasing volume element at larger distances: there are more galaxies at r > d than at r < d, and thus E(r| d ) > d; for a typical distance indicator ( ~ 0.2) it is a ~ 15% effect. More generally, density gradients cannot be neglected, and can either reinforce or work against the volume effect. This is most easily seen in the approximation that n(r) is slowly varying on the scale of DI errors. Expanding n(r) to first order in Eq. (183) then yields

(185)

where

(186)

where we have further assumed ² << 1. Eqs. (184) and (185) were first derived in the extragalactic context by Lynden-Bell et al. (1988a; cf., Malmquist 1920).

The slowly-varying approximation Eq. (185) illustrates in a general way the effect of density gradients on the statistical properties of inferred distances. This effect is also known as inhomogeneous Malmquist bias or IHM. When (d ) > 0, IHM reinforces the uniform density bias; when (d ) < 0 it works against it. While the essence of IHM is shown by the slowly-varying density approximation, Eq. (185) is in fact a rather poor approximation to the true IHM for the density fields seen in the real universe (see below). A proper calculation of IHM requires the full numerical integration indicated by Eq. (183).

Figure 15. Malmquist bias induced by a density peak at a distance of 5000 km s-1. Simulated galaxies were generated along a line of sight with a Gaussian perturbation of full width ~ 1200 km s-1 and maximum overdensity max = 25, atop a uniform distribution. The upper panel shows the distribution of simulated galaxies as a function of their true distance. The lower panel plots, as open squares, average true distance vs. inferred distance for objects in 150 km s-1 wide bins. The meaning of the various lines drawn through the points is indicated in the panel.

These points are illustrated using simulated data in Fig. 15, which shows the Malmquist bias associated with a very strong density perturbation. Galaxies were simulated along a line of sight in which the density is given by n₀ + n_p(r), where n_p(r) is a Gaussian perturbation centered at 5000 km s^-1, with full width ~ 1200 km s^-1 and maximum overdensity _max = 25. Galaxies were assumed to be moving with Hubble flow; there are no true peculiar velocities in this simulation. The number of galaxies per unit true distance along the line of sight is shown in the upper panel; the effect of the density peak on the distribution is clear. The simulated galaxies were required to have diameters 1.2', although this diameter limit does not affect the character of Malmquist bias, for reasons described above. The simulated galaxies obey a TF relation with scatter = 0.35 mag; the "observer" of the simulation assigned inferred distances d with a properly calibrated TF relation. In the lower panel the galaxies have been binned according to the values of their inferred distances. The average true distances of the galaxies in each bin are plotted as open squares.

Nowhere do the simulated data follow the Hubble line. Very far from the density peak (d 2500 km s^-1 and d 7500 km s^-1) the simulated data follow the uniform density Malmquist bias prediction Eq. (184). However, within ~ 2000 km s^-1 of the peak, the data depart strongly from the uniform density prediction; this is the IHM effect. The sense is that in the foreground of the peak distances are underestimated (the squares lie above the long dashed line) and in the background they are overestimated. The amplitude of these effects is 500-1000 km s^-1. The short dashed line is the prediction of the slowly varying approximation Eq. (185), using the exact density gradient. It is apparent that this approximation is extremely poor; this is because the density varies rapidly on the scale of distance errors. The heavy solid line is obtained from the exact formula Eq. (183) by numerical integration of the true density field, and correctly predicts the bias at all distances.

Fig. 15 gives further insight into an issue touched upon in Section 6.4.2, namely, why IHM can lead to spurious peculiar velocities. Suppose that the galaxies depicted in the figure followed the Hubble flow. Their redshift vs. inferred distance diagram would then closely resemble the lower panel. The "S-wave" seen in the figure - measured with respect to the uniform density prediction - would be interpreted as peculiar velocities if IHM were neglected. In particular, motions of many hundreds of km s^-1 toward the density peak would be seen in both the foreground and the background. Such motions are what is expected from gravitational instability, but in this example they are merely an artifact of Malmquist bias. More generally, if gravitational instability is correct, there are real motions but IHM will cause the observed motions to be larger. This bias mimics the effect of having more mass in the density enhancement than actually exists. It follows that, if not properly accounted for, IHM will result in an overestimate of (e.g., Eq. 149).

6.5.3. Distance-Dependent Selection Functions: The "Gould Effect"

An important feature of the Malmquist bias equations in Section 6.5.2 was their independence of observational selection. This greatly simplifies their application in peculiar velocity analyses; in particular, it makes the uniform density correction trivial. However, as was first pointed out by Gould (1993) , under certain circumstances this desirable attribute of Malmquist bias might break down. The origin of this effect is best understood in terms of the selection function, which we have assumed to depend explicitly only on the observables (m, ), not on distance. As a result, in the steps leading from Eq. (179) to Eq. (182) S(m, ) naturally cancelled out. Had the selection function actually been of the form S(m,, r), it would not have dropped out, and the result would have been a selection-dependent equation for Malmquist bias.

To see how distance-dependent selection might arise, suppose that selection is based on a blue apparent magnitude which must satisfy m_B m, while the TF analysis involves a distinct magnitude m. Now, instead of empirically relating m_B to the TF observables and deriving S(m, ) according to Eq. (176), let us consider the TF relation M_B() satisfied by the absolute magnitude corresponding to m_B. We can then relate the probability that m_B m, for a galaxy of width parameter and distance r, to a probability involving the absolute magnitude M_B:

(187)

where we have assumed a normally distributed blue TF relation with scatter _B. Eq. (187) is apparently nothing but the sample selection function as we have defined it. However, it is independent of the TF magnitude m, and explicitly dependent on r: rather than writing it S(m, ) we must write it S(, r).

How can this derivation be reconciled with Eq. (176), whose premises appear equally valid but which yields a qualitatively different result? The answer is quite subtle, and we refer the reader to Willick (1995) for a detailed explanation. There it is shown that Eq. (176) is valid to the extent that the residuals of the TF relations M_B() and M() are strongly correlated, as is normally the case. Eq. (187) is valid in the opposite limit of uncorrelated residuals. A rigorous derivation of sample selection must take this effect into account; the general result is a selection function S(m,, r), with the strength of the r-dependence inversely related to the degree of correlation between the two TF relations.

There are two methods to determine whether the Gould effect is present is real samples. The first is to directly examine correlations between the M_B() and M() TF residuals. The second is to test whether m_B is statistically best described as a function of m and , of and r, or of all three. In the first of the three cases, there is no Gould effect; otherwise it is present at some level. Willick et al. (1995a, b) used the second test (with r modeled by redshift) to show that the effect is significant for samples selected on the ESO angular diameter (e.g., Mathewson et al. 1992b). Peculiar velocity analyses using such samples (e.g., Hudson et al. 1995) rely heavily on redshift-space grouping to minimize the effect.

6.5.4. Inverse DIs: Selection Bias

A formalism quite analogous to that developed above may be presented for inverse DIs; we assume, as usual, that the DI in question is TF. The central element of this formalism is the Gaussian distribution of observed width parameters about the expectation value ₀(m - µ(r)) when selection effects are neglected. Adding in the remaining elements we arrive at the joint distribution of m, , and r written in terms of the inverse TF relation:

(188)

We note that, in contrast to its forward analogue Eq. (171), this equation involves the luminosity function (M).

Both selection and Malmquist bias are implicit in Eq. (188). As before the two are distinguished by the assumptions of data analysis. In a Method II analysis we (implicitly) condition on the true distance (see Section 6.5.1). The probability distribution which underlies such an analysis is thus P( | m, r), obtained from Eq. (188) according to

(189)

This equation is the inverse counterpart of Eq. (173). In the forward case the condition for selection bias to vanish was that S(m, ) be independent of m, which is almost always unrealistic. Now the condition is that S(m, ) be independent of . We can expect that, to a good approximation, this condition will frequently hold. To the extent that it does, Eq. (189) reduces to a Gaussian distribution of about ₀(m - µ(r). It follows that a standard ² minimization procedure will yield unbiased values of model parameters, as discussed in Section 6.4.4.

However, the assumption that inverse Method II analyses are free of selection bias is subject to several caveats which are often overlooked:

1. As noted in Section 6.5.1, using different photometric quantities for sample selection than for DI analysis can result in an -dependence of the function S(m, ). Willick (1994) has quantified and described correction techniques for this effect.

2. Direct dependence of sample selection on velocity width can occur in studies in which is measured from the HI 21 cm line. Signal-to-noise ratio considerations can mean that large velocity widths more likely to be undetected at low 21 cm flux levels (Roth 1993). Willick et al. (1995a) have attempted to estimate this effect for the Mould et al. (1991) TF sample, and found it small but non-negligible.
3. The slope of the inverse relation - the quantity e in Eq. (168) - is affected by the accuracy of the redshift-distance model used in calibrating the relation (Willick 1991). This fitting bias does not affect the forward relation, in which a poor model increases scatter but does not bias the slope.

4. Finally and perhaps most importantly, while inverse DIs can avoid selection bias in the case of velocity width-independent selection, they have no special advantage over forward DIs in terms of susceptibility to Malmquist bias. Indeed, inverse Malmquist bias is inherently more complicated than forward, as we describe in the following section.

6.5.5. Inverse DIs: Malmquist Bias

An inverse Method I analysis is fully analogous to a forward Method I analysis, except that the inferred distances are obtained by solving the equation ₀(m - 5logd )= implicitly for d. For the linear inverse TF relation Eq. (168), this gives d = 10^{0.2[m - (D - /e)]}, which may be compared with the forward inferred distance d = 10^{0.2[m - (A - b]}. Although we have used the symbol d for both the forward and inverse inferred distance, the two are not equal in general; the non-equality of the forward and inverse TF parameters (Section 6.4.4) guarantees this. In fact, inverse DIs are steeper than their forward counterparts when plotted in the usual way (M vs. ). Inverse inferred distances are thus larger for luminous, and smaller for faint, galaxies. Since luminous objects dominate a sample at large redshifts, it is clear that a redshift vs. inferred distance diagram will look different depending on whether the forward or inverse relation has been used, with quite different biases in the two cases. Let us now quantify these ideas.

If we begin with Eq. (188) and go through a series of steps analogous to those which took us from Eq. (171) through Eq. (182), we obtain the following for the probability distribution of the true distance of a galaxy with inverse inferred distance d:

(190)

Here /e, the effective fractional inverse distance error, is roughly but not precisely equal to its forward counterpart. Derivation of Eq. (190) required the assumption of a linear inverse TF, and also of a selection function at most weakly dependent upon (details are given in Willick 1995). With these (realistic) assumptions Eq. (190) resembles its forward analogue Eq. (182), but with a crucial difference: the presence of the real space selection function

(191)

the probability that a sample object lies at true distance r irrespective of the observables (m, ).

Eq. (190) shows that the inverse Malmquist effect quantitatively resembles the forward, with the exception that now the product n(r)s(r) replaces n(r) alone. The differences between inverse and forward Malmquist bias are most clearly illustrated by considering the uniform density case, with the further assumption that s(r) varies slowly on scales d. By analogy with Eq. (185) it is then easy to derive the result

(192)

where

(193)

We see again that, like their forward counterparts, inverse inferred distances are biased: E(r| d) d. However, in contrast with the forward case, the effect is not independent of sample selection or the luminosity function. Even in the case of uniform density, it is not simple to obtain a Malmquist correction for an inverse distance. One must not only accurately model the selection function S(m, ), but integrate it with respect to a believable luminosity function (M). These difficulties are only compounded by the realistic requirement of estimating n(r) in order to account for IHM. The net effect of these considerations is that inverse DIs are less suitable than forward DIs for Method I analyses. This point must be kept in mind when one speaks of the "unbiased" character of inverse DIs. The quantity s(r) might be modeled by the observables s(z) or s(d ), and thus perhaps be dealt with easily. There is, however, a method which, at least in principle, permits effective use of inverse inferred distances, as we discuss in the next section.

6.5.6. The Method of Landy and Szalay

A severe obstacle to correcting for IHM is the requirement of knowing the number density n(r) along any given line of sight. One way around this obstacle is to estimate n(r) from redshift survey information. This has been done, for example, in comparisons of the IRAS density field with the POTENT mass reconstruction (Dekel et al. 1993 ; Hudson 1994a ; Hudson et al. 1995). Although reasonable, this procedure has unsatisfactory features, such as uncertainties introduced by smoothing and the need for an assumed peculiar velocity field to convert redshift- to real-space density. One would like, if possible, to derive the IHM correction self-consistently from the DI data themselves. Such a method was outlined by Landy & Szalay (1992; LS), and independently by Willick (1991) . The LS technique relies on the mathematical relationship between E(r| d) and a quantity which is in principle observable: the number of objects per unit inferred distance.

Let us define f (d) as the number of objects, per unit solid angle along a particular line of sight, per unit inferred distance. If the galaxy distribution is densely enough sampled, one can imagine measuring f (d) by binning objects in inferred distance. The resulting histogram representation of f (d) is proportional to the probability, P(d), that an object along the given line of sight has inferred distance d. This probability may in turn be calculated by integrating P(r, d ), given by Eq. (181) in the forward case and derivable from Eq. (188) in the inverse case, over the true distance r:

(194)

At this point, the expressions for the forward and cases diverge, and we present separately the result for each:

(195)

and

(196)

In the forward case, the quantity (d), which dropped out of the conditional expectation E(r| d), is present here in the expression for f (d). In the inverse case, however, f (d) contains a dependence on selection identical to that seen in E(r| d). Thus, it is only for the inverse case that the relationship between f (d) and E(r| d) will be straightforward (cf., Teerikorpi 1993), and we consider only the inverse case (now without a subscript) in what follows.

Taking as usual x = r/d we have

(197)

for inverse distances. To relate this to E(r| d) we rely on the mathematical identity (e.g., LS or Willick 1995) that

(198)

where g(r) is an arbitrary function. Combining Eqs. (183), (197), and (198) one readily obtains

(199)

Eq. (199) is the basis of the Landy-Szalay method of Malmquist bias correction. It says that we can obtain an unbiased estimate of the true distance, given an inverse inferred distance d, from the observable distribution f (d).

The LS method is attractive for several reasons. First, as already noted, it enables us to correct for IHM using the DI data alone. Second, because LS involves inverse DI distances, it is based on a DI relation whose calibration was very nearly free of selection bias. In this sense, inverse DIs can be viewed as (nearly) bias-free, but only when the LS method is used. Finally, while we have not made this explicit in the above derivation, the LS method is applicable even when the Gould effect is present. This is because even when the selection function is of the form S(m, , r), one arrives at the expression Eq. 190.

With this rosy view of LS must go several realistic caveats. Accurate determination of the quantity f (d) presents a number of technical problems (Willick 1995). Basically, the issue is one of resolution: one needs a sufficient number of objects per bin that the histogram representation of f (d) is not overwhelmed by shot noise; to obtain this number one must make bigger bins in both angle and distance. The larger the bin, of course, the less accurately one determines the actual IHM correction. For these reasons, LS has yet to be implemented for a real sample. An effort is presently underway (Dekel et al. 1995) to do so for the large TF samples currently available (Section 7.2). It remains unclear at this time whether implementation of the LS method will constitute a major advance in dealing with Malmquist bias.

²⁰ Babul et al. (1994) point out that in certain scenarios of non-gravitational growth of structure, Eqs. (149) and (150) might apparently be satisfied, yielding an incorrect value of (Section 8.2). Back.

²¹ Often the "distance-independent" quantity is in fact weakly distance-dependent. An example is surface brightness, which falls off with increasing redshift as (1 + z)^-4. However, as in that example, it is usually the case that the weak distance-dependence is either exactly or adequately modeled in terms of the redshift alone, and thus is easy to compensate for. Back.

²² The members of this second collaboration were D. Burstein, R. Davies, A. Dressler, S. Faber, D. Lynden-Bell, R. Terlevich, and G. Wegner. As a group they became widely known among astronomers as the "7 Samurai" (hereafter 7S). Back.

²³ However, it was discovered that evolutionary effects are equally as important as those due to cosmology. See Sandage (1988) for a review. Back.

²⁴ We use the term "inference," rather than the stronger "prediction" (as is used, for example, by Burstein [1990]), to underscore the fact that redshift-independent distances are merely statistical estimators, obtained from more basic measurements, of true distances--and not very accurate ones at that. Back.

²⁵ Here we use "averaged" in its widest sense, including smoothing or modeling techniques. Back.

²⁶ This last statement neglects second-order changes in the peculiar velocity field; cf. Willick 1991 for details. Back.

²⁷ It is not fundamental to our definition of Method I that the TF relation be calibrated a priori. However, unnecessary complications arise if the TF parameters are considered "free." Back.

²⁸ For pure standard candles, there is no distinction between forward and inverse methods; the material discussed in this section is only relevant for distance indicators which depend on a distance-independent observable such as line width. Back.

²⁹ It is not, of course, literally correct that the true distance is specified in a Method II analysis. Errors in the velocity model being fit to the data, as well as small-scale "noise" in the actual velocity field, mean that r is only approximately specified by redshift information. However, to first order this only increases the variance, but does not bias, a Method II solution. The formalism presented above is correct in this approximation, but the quantity should be regarded as the quadrature sum of DI scatter and model errors. In Section 8.1.3 we will describe a method which aims to go beyond this first order approximation and include the effect of model errors and velocity noise. Back.

³⁰ A related complication is that not only the expected apparent magnitude, but also its variance is biased, in the sense that the actual variance is strictly < ². The causes of this effect, known as dispersion bias, were discussed in Section 6.4.2. Quantitative expressions for dispersion bias are given by Willick (1994) . Back.