Annu. Rev. Astron. Astrophys. 1994. 32: 371-418
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3.1 Distance indicators

Measuring redshift-independent distances to many galaxies at large distances is the key to large-scale dynamics (review: Jacoby et al. 1992). The simplest method assumes that a certain class of objects is a ``standard candle'', in the sense that a distance-dependent observable is distributed intrinsically at random with small variance about a universal mean. The luminosity of an object (propto r-2) or its diameter (propto r-1), can serve as this quantity. In a pioneering study, Rubin et al. (1976a, b) used the brightness of giant Sc spirals to discover a net motion for the shell at 35-60 h-1Mpc that agrees within the errors with more modern results, but the large uncertainties in this simple distance indicator made this result controversial at the time.

So far, the most useful distance indicators for LSS have been of the TF-kind, based on intrinsic relations between two quantities: a distance-dependent quantity such as the flux propto L / r2, and a distance-independent quantity sigma - the maximum rotation velocity of spirals (TF) or the velocity dispersion in ellipticals (FJ). The intrinsic relations are power laws, L propto sigma beta, i.e., M(eta) = a - beta, where M ident - 2.5log L + const is the absolute magnitude and eta ident logsigma. The slope b can be determined empirically in clusters, where all the galaxies are assumed to be at the same distance, typically yielding beta approx 3 - 4, depending on the luminosity band (e.g. betaI approx 3, betaH approx 4). Then, for any other galaxy with observed eta and apparent magnitude m ident -2.5 log (L / r2) + const, one can determine a relative distance via 5 log r = m - M(eta). There exists a fundamental freedom in determining the zero point, a, which fixes the distances at absolute values (in km s-1, not to be confused with H which translates to Mpc). Changing a, i.e., multiplying the distances by a factor (1 + epsilon) while the redshifts are fixed, is equivalent to adding a monopole Hubble-like component -epsilon r to v, and an offset 3epsilon to delta (Equation 4). It has been arbitrarily determined in several data sets, e.g. by assuming u = 0 for the Coma cluster, but a is better determined by minimizing the variance of the recovered peculiar velocity field in a large ``fair'' volume. The original TF technique has been improved by moving from blue to near-infrared photometry (H band, Aaronson et al. 1979) and recently to CCD R and I bands, where spiral galaxies are more transparent and therefore the intrinsic scatter is reduced to sigmam ~ 0.33 mag, corresponding to a relative distance error of Delta = (ln 10/5) sigmam approx 0.15.

A distance indicator of similar quality for ellipticals has proved harder to achieve. Minimum variance, corresponding to Delta = 0.21, was found for a revised FJ relation involving three physical quantities: D Ialpha propto sigma beta with D the diameter and I propto L / D2 the surface brightness (Dressler et al. 1987; Djorgovski & Davis 1987). The parameters were found to be alphaapprox 5/6 and betaapprox 4/3. By defining from the photometry a ``diameter'' at a fixed value of enclosed I, termed Dn, the relation returns to a simple form, Dn propto sigma beta, similar to FJ but with reduced variance.

The physical origin of the scaling relations is not fully understood, reflecting our limited understanding of galaxy formation. What matters for the purpose of distance measurements is the mean empirical relation and its variance. However, one can point at an important physical difference between the two relations (Gunn 1989), which is relevant to the testing for environmental effects (Section 6.3). The Dn - sigma relation is naturally explained by virial equilibrium, sigma2 propto M / D, and a smoothly varying M / L propto Mgamma, which together yield D I1 / (1+gamma) propto sigma2 (1-gamma) / (1+gamma), but the TF relation, involving only two of the three quantities entering the virial theorem, is more demanding - it requires an additional constraint which is probably imposed at galaxy formation.

There is some hope for reducing the error in the TF method to the ~ 10% range by certain modifications, e.g. by restricting attention to galaxies of normal morphology (Raychaudhury 1994). The most accurate to date uses the estimator based on surface-brightness fluctuations (SBF) in ellipticals (Tonry 1991), where the standard candle is the luminosity function of bright stars in the old population. These stars show up as distance-dependent fluctuations in sensitive surface-brightness measurements. The technique is being applied successfully out to ~ 30 h-1Mpc (e.g. Dressler 1994), with the improved accuracy of ~ 8% enabling high-resolution non-linear analysis, and it can be of great value for LSS if applied at larger distances. The need to remove sources of unwanted fluctuations such as globular clusters requires high resolution observations which could be achieved by HST or adaptive optics.

The prospects for the future can be evaluated by estimating the length scale over which LSS dynamics can be studied using a distance indicator of relative error Delta. The error in a velocity derived from N galaxies at a distance ~ r is sigmaV ~ r Delta / sqrtN. Let the mean sampling density be nbar. Let the desired quantity be the mean velocity V in spheres of radius R, and assume that its true rms value is V20 at R = 20 h-1Mpc and propto R-(n+1) on larger scales, with n the effective power index of the fluctuation spectrum near R. Then the relative error in V is

sigmaV / V approx 0.033 (nbar / 0.01)-1/2 (Delta / 0.15) (V20 / 500)-1 (R / 20)n+1/2 r / R, (9)

where distances are measured in h-1Mpc. The observations indicate that V20 ~ 500km s-1 and n ~ -0.5 for R = 20 - 60h-1Mpc (Section 7.1). Thus, with ideal sampling of nbar ~ 0.01, (h-1Mpc)-3, the relative error is always only a few percent of r / R. This means that LSS motions can in principle be meaningfully studied at all distances r with smoothing R ~ 0.1r, as long as n ~ -0.5 at the desired R. Since n seems to be negative out to ~ 100 h-1Mpc (Section 7.1), dense deep TF samples are potentially useful out to several hundred megaparsecs. However, several technical difficulties pose a serious challenge at such distances. For example, the calibration requires faint cluster galaxies which are harder to identify, aperture effects become severe, the spectroscopy capability is limited.

3.2 Malmquist Biases

The random scatter in the distance estimator is a source of severe systematic biases in the inferred distances and peculiar velocities, which are generally termed ``Malmquist'' biases but should carefully be distinguished from each other (e.g. Lynden-Bell et al. 1988; Willick 1994a, b).

The calibration of the TF relation is affected by the selection bias (or calibration bias). A magnitude limit in the selection of the sample used for calibration at a fixed true distance (e.g. in a cluster) tilts the ``forward'' TF regression line of M on eta towards bright M at small eta values. The bias extends to all values of eta when objects at a large range of distances are used for the calibration. This bias is inevitable when the dependent quantity is explicitly involved in the selection process, and it occurs to a certain extent even in the ``inverse'' relation eta (M) due to existing dependences of the selection on eta. Fortunately, the selection bias can be corrected once the selection function is known (e.g. Willick 1991; 1994a).

The TF inferred distance, d, and the mean peculiar velocity at a given d, suffer from an inferred-distance bias, which we term hereafter ``M'' bias. I comment later (Section 4.4) on a possible way to avoid the M bias by performing an inverse analysis in z-space, at the expense of a more complicated procedure and other biases. Here I focus on a statistical way for correcting the M bias within the simpler forward TF procedure in d-space. This bias can also be corrected in an inverse TF analysis in d-space, using the selection function S (d) which is in principle derivable from the sample itself (Landy & Szalay 1992).

The current POTENT procedure uses the forward TF relation in d-space. If M is distributed normally for a given eta, with standard deviation sigmam, then the TF-inferred distance d of a galaxy at a true distance r is distributed log-normally about r, with relative error Delta approx 0.46sigmam. Given d, the expectation value of r is (e.g. Willick 1991):

E (r | d) = integ0infty r P(r | d) dr / integ0infty P(r | d) dr =

{ integ0infty r3 n (r) exp{-[ln(r | d)]2 / 2Delta2}dr } /

{ integ0infty r2 n (r) exp{-[ln(r | d)]2 / 2Delta2}dr }, (10)

where n (r) is the number density in the underlying distribution from which galaxies were selected (by quantities that do not explicitly depend on r). The deviation of E(r | d) from d reflects the bias. The homogeneous part (HM) arises from the geometry of space - the inferred distance d underestimates r because it is more likely to have been scattered by errors from r > d than from r < d, the volume being propto r2. If n = const, Equation (10) reduces to E (r | d) = d exp(3.5 Delta2), in which the inferred distances are simply multiplied by a factor, 8% for Delta = 0.15, equivalent to changing the zero-point of the TF relation. The HM bias has been regularly corrected this way since Burstein et al. (1986).

Fluctuations in n (r) are responsible for the inhomogeneous bias (IM), which is worse because it systematically enhances the inferred density perturbations and the value of Omega inferred from them. If n (r) is varying slowly with r, and if Delta << 1, then Equation (10) reduces to E (r | d) = d [1 + 3.5Delta2 + Delta2 (d lnn / d lnr)r=d], showing the dependence on Delta and the gradients of n (r). To illustrate, consider a lump of galaxies at one point r with u = 0. Their inferred distances are randomly scattered to the foreground and background of r. With all galaxies having the same z, the inferred u on either side of r mimic a spurious infall towards r, which is interpreted dynamically as a spurious overdensity at r.

In the current data for POTENT analysis (Section 4.2) the IM bias is corrected in two steps. First, the galaxies are heavily grouped in z-space (Willick et al. 1994), reducing the distance error of each group of N members to Delta / sqrtN and thus significantly weakening the bias. Then, the noisy inferred distance of each object, d, is replaced by E (r | d) (Equation 10), with an assumed n (r) properly corrected for grouping. This procedure has been tested using realistic mock data from N-body simulations (Kolatt et al., in prep.), showing that IM bias can be reduced to a few percent. The practical uncertainty is in n (r), which can be approximated by the high-resolution density field of IRAS or optical galaxies (Section 5), or by the recovered mass-density itself in an iterative procedure under some assumption about how galaxies trace mass. The second-step correction to delta recovered by POTENT is < 20% even at the highest peaks (Dekel et al. 1994).

3.3 Homogenized Catalogs

Several samples of galaxies with TF or Dn - sigma measurements have accumulated in the last decade. Assuming that all galaxies trace the same underlying velocity field (Section 6.3), the analysis of large-scale motions greatly benefits from merging the different samples into one self-consistent catalog. The observers differ in their selection procedure, the quantities they measure, the method of measurement and the TF calibration techniques, which cause systematic errors and make the merger non-trivial. The original merged set, compiled by D. Burstein (Mark II) and used in the first application of POTENT (Bertschinger et al. 1990), consisted of 544 ellipticals and S0's (Lynden-Bell et al. 1988; Faber et al. 1989; Lucey & Carter 1988; Dressler & Faber 1991) and 429 spirals (Aaronson et al. 1982a; Aaronson et al. 1986; 1989; Bothun et al. 1984). The current merged set (Willick et al. 1994, Mark III of the Burstein series; Faber et al. 1994) consists of ~ 2850 spirals (Mark II plus Han & Mould 1990, 1992; Mould et al. 1991; Willick 1991; Courteau 1992; Mathewson et al. 1992) and the ellipticals of Mark II. This sample enables a reasonable recovery of the dynamical fields with ~ 12 h-1Mpc smoothing in a sphere of radius ~ 60 h-1Mpc about the LG, extending to ~ 80h-1Mpc in certain regions (Section 4). Part of the data are shown in Figure 2.

As carried out by this group, merger of catalogs involves the following major steps: (a) Standardizing the selection criteria, e.g. rejecting galaxies of high inclination or low eta which are suspected of large errors and sharpening any z cutoff. (b) Rederiving a provisional TF calibration for each data set using Willick's algorithm (1994a) which simultaneously groups, fits and corrects for selection bias, and then verifying that inverse-TF distances to clusters are similar to the forward-TF distances. (c) Starting with one data set, adding each new set in succession using the galaxies in common to adjust the TF parameters of the new set if necessary. (d) Using only one measurement per galaxy even if it was observed by more than one observer to ensure well defined errors, and using multiple observations for a ``cluster'' only if the overlap is small (e.g. < 50%). (e) Adding the ellipticals from Mark II, allowing for a slight zero-point shift (Section 6.3). Such a careful calibration and merger procedure is crucial for reliable results - in several cases it produced TF distances substantially different from those quoted by the original authors.

Figure 2. Inferred radial peculiar velocities of grouped galaxies in a ± 20° slice about the Supergalactic plane from the homogenized Mark III catalog (Willick et al. 1994). Distances and velocities are in 1000km s-1. The area of each circle marking the object position is proportional to the object richness. This slice contains 453 objects made of 1124 galaxies out of 1214 objects in the whole volume. Solid and dashed lines distinguish between outgoing and incoming objects. The positions and velocities are corrected for IM bias. Note the GA convergence (left) and the PP convergence (right-bottom).

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