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The Peculiar Velocity Field Predicted from the Distribution of fR IS Garages MICHAEL A. STRAUSS AND MARC DAVIS Umversity of California, Berkeley ABSTRACT We present recent results from our full-sly redshift survey of {RAS galaxies. After briefly describing our sample selection and observations, we use the linear theory relation between acceleration and peculiar velocity to make predictions of the peculiar velocity field in the Local Universe. We compare our predictions with directly measured peculiar velocities from the Local Supercluster spiral galaxy sample of Aaronson et al. (1982) and the elliptical galaxy sample of Lynden-Bell e! al. (1988~. In a reference frame at rest with respect to the Cosmic Microwave Background, there Is a systematic bunk flow in the residuals between observations and predictions. The Local Group itself takes part in this modon, so the bunk flow disappears in a frame at rest with respect to the Local Group. We show with the use of N-body techniques that this type of systematic effect is at least pardy due to small-number statistics in the tracing of the density field. We conclude with a discussion of the challenges facing us in the future. The N-body results show us that we have not yet achieved an optimum algorithm for translating from redshift space to real space; we briefly describe several alternative procedures. We compare the coherence of the observed and predicted peculiar velocity fields with predictions from several cosmogonical models. To IRAS SAMPLE Peebles (1976) describes the velours field expected at late times in 356

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HIGH-ENERGY ASTROPHYSICS 357 linear theory; the dipole anisotropy of the galaxy distn~ution around a given galaxy is directly proportional to the peculiar velocity of that galaxy: ., HoQ06 J [(z)x do Or x~ (1) where V is the peculiar velocity of the galaxy in question, Q is the cosmo- logical density parameter, and [(x) is the fractional density perturbation at position ~ The following year, a dipole anisotropy was discovered in the Cosmic Microwave Background (CMB) (e.g., Lubin and Held 1986), which was interpreted as due to a peculiar motion of the Local Group (LG) with respect to the rest frame of the CMB of 600 km sol. It was imme- diately realized that a measurement of the distribution of nearby galaxies could be used in concert with equation (1), and the measured LO peculiar velocity, to make an estimate of Q. This has been tried by many authors, usually with the simplifying assumption that the Virgo cluster is the only gravitating mass point (see, e.g., Davis and Peebles 1983; Huchra 1988~. ~ do better requires galaxy samples that cover a large fraction of the sky. Pioneering work using optically selected samples of galaxies has been done by Lahav and collaborators (Lynden-Bell et al. 1989, and references therein). In this paper we describe a survey of galaxies observed by the Infrared Astronomical Satellite (IRAS). We have discussed our work before In two other conference proceedings (Strauss and Davis 1988a, 1988b; Yahil 1988), and in this paper will concentrate on new results obtained since the writing of those articles. A much more detailed account can be found in Strauss (1989), and in several papers in preparation. The TRAS satellite surveyed the entire sky in four broad bands in the far-infrared, with a resolution of approximately 1' at 60pm. Because of the full-sly coverage and the lack of Galactic extinction in the far-infrared, the {RAS database offers a unique opportunity to study the large-scale distribution of galaxies. We have selected galaxy candidates hom the IRAS Point Source Catalog (1985) using the following criteria: 1. f60/f:2 ~ 3, where fx is the flux density at wavelength A. This discriminates effectively between stars and galaxies. 2. Ebb > 5, in a region not affected by confusion at 60pm. With these restrictions, our sample covers 87.6% of the sky. 3. fee > 1.936 Jy. We have obtained optical identifications and redshifts of the approx- imately 4,500 objects meeting these criteria, thus obtaining a sample of roughly 2,550 galaxies. The vast majority of the non-gal~es in the sample are associated win stars and Infrared cirrus at Ebb < 10 (see Yahil 19~ for a description of how optical identifications were made at these low latitudes). Below, we discuss some biases of the sample selection, and our

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358 AMERICAN AND SOVIET PERSPECTlYES [RAS Galaxies f60 > 1.936 dy ,,,.. .! ~ 4 ~ ~ \ `\~$ \ = * ~ ~ . .~* ! ~"'~'--~*';'')*'-*~'p~'-~* ~/''~"-'* ~ ~ A 2539 OBJECTS PLO ~ i Ed `~~ ~:'~'~ 360 FIGURE 1 Me distribution of galaxies in the surrey, plotted in Galactic coordinates. The sample does not include galaxies at |b| < 5, and there are other regions at high Galactic latitudes which are not covered due to confusion or incompleteness of the IRAS survey. attempts to correct for them. The distribution of all galaxies in the sample Is shown in Galactic coordinates in Figure 1. Note the strong continuity of structures across the Galactic plane, In particular, the overdensities asso- aated with Centaurus and the Pavo-Indus-~lescopium region (l ~ 330 appear to be physically associated. The Pisces-Perseus filament (l ~ 135 ~ and the Hydra region (l ~ 280 ~ also show continuity across the plane, as do the underdense regions centered at 1 = 90 and 1 = 220. Compare this figure with the corresponding Figure la in Yahil (1988~; note how much stronger this sense of continuity is now that we have identified galaxies in the two strips 5 < fib; < 10 . THE PREDICTED PECUI1AR VELOCITY f HELD As described In detail In Strauss and Davis (1988b) and Yahi! (19~), we have used equation (1) to predict the velocity flow field within 8,000 km -is of the Local Group. The procedure we follow is slightly modified from that of Yahil (1988~. We first fill the missing Galactic plane strip with random "galaxies" with number density given by an interpolation of the density in strips with 5 < Ebb < 20, under the assumption that the continuity across the plane descried in the last section is rear A selection function is derived using a mammum-l~elihood estimator (Yahil 1988), which is used to weigh the galaxies to compensate for the magnitude limit of the sample. A value of Q is derived by requiring that equation (1) reproduce the 600 km s~: peculiar velocity of the Local Group inferred from the CMB dipole.

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HIGH-ENERGY ASTROPHYSICS 359 For an adopted value of Q = 0.7, the predicted peculiar velocity of the Local Group is 650 lan s~i towards 1 = 250, b = +45, some 22 from the CMB dipole direction of 1 = 270, b = +30 (Lubin and Villela 1986~. Strauss and Davis (1988b) show that the majority of this acceleration is due to material within 4,000 km s~t of the Local Group.1 We can go further by using equation (1) to make estimates of the peculiar velocity of every galaxy in the sample. We restrict ourselves to objects within 8,000 km s-i of the Local Group, because the sample becomes prohibitively sparse at greater distances. In practice, our estimate of the peculiar velocity of a galaxy at position x Is given by: V( ~ HoQ0 6 41rn~ Iga axles ~ S(l z-z: D(z-z:) _ H Q 6 /3 +W(S) (2) Here n1 is the measured galaxy density, (x~) is the value of the selection function at distance x:, and S is a smoothing on small scales, to be described further below. The second term corrects for the fact that the sum is not spherically symmetric for a point displaced from the origin, and W(S) is an additional correction applied when the source is within one smoothing length of the edge of the sample. The measured redshift cz. of a oalaxv may be expressed as: , , 4~ ~ ~ cz = distance ~ (V- VEG) X, (3) where V and VLG are the peculiar velocity vectors of the galaxy and the Local Group, respectively, and x is the UDut vector pointing from the Local Group to the galaxy. We separate peculiar velocities and distances using an iterative procedure: galaxies are initially assumed to be at their redshift distances (V-0), and equation (2) is used to make a first estimate of the peculiar velocity field. Distances are then updated using equation (3) and the process is repeated until convergence. Q is kept fixed throughout this procedure, and the value of V[,G used is that derived Mom the IRAS sample itself on each iteration A "buffer zone" between 8,000 km s~: and 10,000 An s~i prevents galaxies from being lost from the sample if their peculiar velocities take them beyond 8,000 An sol. The resulting peculiar velocity field converges with a standard deviation of residuals less than 10 km s~i after eight iterations. There are several subtleties associated with this process. In regions that are Finalized, or even overdense by a factor of several, equation (1) no longer holds, and this iterative procedure may cause chaos. Thus we iWe will express distances in terms of radial velocities, as we do here, in order to bypass the necessity of spewing the Hubble Constant.

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360 AMERICAN AND SOVIET PERSPECTIVES will never be able to properly model the "Finger of God" effect seen in the cores of clusters. We have thus collapsed the galaxies in seven cluster cores (Virgo, Ursa Major, Fornax, Endanus, Centaurus, Hydra, and Perseus) to a common redshift, and have suppressed the mutual gravitational attraction of galaxies within each cluster. Another related problem is that equation (2) diverges at small separations. Note that because the selection function drops sharply with distance, a pair of galaxies separated by a small amount at a large distance from us can give each other a tremendous kick In order to suppress this, we adopt a smoothing law, S(r) = (r/r5~3 for r ~ r5, where r5, the smoothing length, is taken to be the mean interparticle spacing at that distance, or 500 km s-i, whichever is larger. The resulting velocity field is similar to that shown in Figure 10 of Yahil (1988), and will not be reproduced here. As those figures show, the predicted velocity field is dominated by two large mass concentrations in Hydra-Centaurus and Perseus-Pisces. We will-compare these predictions with the observed velocity field in the next section. COMPARISON WllI[ THE OBSERVED PECULIAR VELOCITY DATA Burstein (1989) reviews the current status of direct observations of the peculiar velocity now field. In this paper, we will make comparisons of our predictions with two data sets: the subset of the Aaronson et al. (1982) Local Supercluster spiral galaxies with high~uality observations defined by Faber and Burstein (19~), and the independent peculiar velocities of individual galaxies, groups, and clusters of the Seven Samurai elliptical galaxy sample, listed In Table 4 of Faber et al. (1989~. Figure 2 shows the distribution of peculiar velocities in a slice through the plane of the Local Supercluster, using the plotting technique introduced by Lynden-Bell et al. (1988~. All galaxies in the two samples within 22.5 of the plane are plotted, and the length of the line attached to each point is equal to the radial peculiar velocity. Solid symbols are used for positive peculiar velocities, and open symbols are used for negative. In the first panel of this figure, we show the positions of the well-lmown clusters that fall in this projection; the symbols are indicated in the figure caption. In panel (b), the observed peculiar velocities are plotted, panel (c) shows the IRAS predictions, and panel (d) shows the difference of (b) and (c). It is indeed a remarkable fact that is able to reproduce the qualitative nature of the velocity Dow field, perhaps the first direct indication that peculiar velocities are generated by gravity, and that galaxies at least approximated trace the mass distribution of the underlying dark matter. However, there are some systematic differences between observations and predictions. Note, for instance, that whereas the observed peculiar velocities (panel by to the upper right of the Ursa Major region are large and negative, the predicted

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4000 2000 -2000 -4000 4000 2000 -2000 -4000 HIGH-ENERGY ASTROPHYSICS _1 1, 1 1 1 1 1 ~ 1 1 1 1 1 ' ' ' 1 1 - (a) - ~ C VUm 0 0 361 ' 1 ' 1 1 1 ' ~ 1 1 1 1 1 1, 1 1 ~ 1 _ ~\\ o (b) ._ O ,/ Pa _ Pr - 1 1 1 1 1 1 1 1 1 ' 1 1 1 1 1 _ - ~: o 'C? ~O ;~ cs o q,. D~o ~ oo ~ ~ ~ ~, 8 ~- _ _ _ _ ' _ _ , _ I I I I I _ ~, I,,, I,,, I,, -4000-2000 0 2000 4000 -4000-2000 0 2000 4000 ~ '.0 -w _ 0 0 o - 1 1 1,, 1 1,,, 1 1 - ~1 .,,,,,. ,o ~ O O ~,~!,'.0' :0 7~0.\ . 2 , 1 , , , 1 , FIGURE 2 The velocity field of the Aaronson et al. and Faber et a1. gala~y samples in a slice through the Local Supercluster plane, in the CMB hame. (a) Ihe positions of the major clusters [ailing in the slice: V = V~rgo, Um = Ursa Major, C = Centaurus, Pa = Pavo, Pe = Perseus, and Pi = Pisces. (b) The observed veloci~ field. (c) Ihe IRAS predicted velocin~r field. (d) The dillerence between (b) and (c). motions (panel c3 are very quiet; predicts that the pulls of the Pisces-Perseus and Hydra-Centaurus superclusters appro~mately balance in this region. Note also the ve~y large observed peculiar velocities towards Centaurus; IRAS predicts motion of the same sigr~, but of smaller amplitude. In panel (d), there is a systematic motion seen in the difference between predicted and obsened peculiar velocities, as if there is a buLk Dow component to the motion that IRAS does not see. We find consistent solutions for the buLk flow using the Aaronson et al. and Faber e' al. galaxies separately as well as together: 250 km s~i pointing towards 1 = 320, b = -35 . In Figure 3, the same compar~son of observed and predicted peculiar velocities is made, this time in a frame at rest with respect to the Local

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362 4000 2000 o -2000 -4000 4000 2000 o -2000 -4000 AMETtICAN AND SOVIET PERSPECTIVES _1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ 1 1 1 - (a) - C VUm Pa Pi - 1' ~ ~ 1 ~ O.,, c, ,.,,,,,,0 ,,0 - Am,, 0 i": ' 0 0 _ jets ~ ~ WeI~.t 1 1 1 1 ~ 1 IS I ~ 1 1 1 1 1, 1 1 1 1 . (b) - . _ . . oo _ o i,, _ . o o. . _ o, - ; ,,.~"~2o'O' he ',,.r,l ~` : _ t ' T ~ I ' I l, ' I 1 1 ~ I I I _ ~-~ Dt2O _ o. O - 0 ' -. ..o ~ ~ ,, oso O O O O ~, ~ ..' .` _ (d) - . _ I , I I, l l l,, l'l l, l t l ,, l l l ' l ~ -4000-2000 0 2000 4000 -4000-2000 0 2000 4000 - FIGIJRE 3 As in F~gure 2, but plotted in the rest hame of the Local Group. Group. This is not a tribal change, because in order to transform the IRAS predicted motions, we use the Local Group motion as predicted by IRAN, while the observed LO frame peculiar velocities are merely the difference between their redshift and observed distance. Note now in panel (d) that the sense of an overall bunk flow to the residuals between observed and predicted peculiar velocities has vanished, and we are left with mostly small-scale motions that we have not modelled correctly. Thus Me Local Group itself takes part in the bulk Dow seen in the residuals in Figure 2d. The only significant systematic effect remaining in the LO frame is a strong infall in the direction of Hydra-Centaurus that IRAS is unable to reproduce. This plot should be compared with Figure 7b, panel 2, of Faber and Burste~n (1988), which shows residuals of observed peculiar velocities after subtracting away the predictions of their Great Attractor model. This comparison shows that there is a bunk motion within a sphere of 4,000 km s-i radius, which IRAS is not properly modelling. Moreover,

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HIGH-ENER~ ~TROP~SICS 363 the Local Group also takes part in this motion; most of the deviation of the IRAS calculated peculiar velocity for the Local Group from the value measured from the CMB dipole can be attributed to this motion. We show in the next section this may be explained as a purely statistical effect caused by dilute sampling of the density field. COMPARISON WITH N-BODY MODELS Our iteration scheme can be tested with N-body models for which we know the true distances of every point. We used N-body models consisting of 262,144 particles in a box 360 Mpc on a side (Ho = 50 km s~i Mpc-~) in an Q = 1 universe, with an initial power spectrum of perturbations appropriate for universe dominated by Cold Dark Matter, and evolved until the mars density contrast in a sphere of radius 16 Mpc is b-i = 0.5. The velocity field was smoothed with a Gaussian of width 100 lan sol. Particles with peculiar velocities of 600 km s~t, lying in regions of local overdensity ~ with -0.2 ~ ~ < 1.0, and with smooth local velocity field, were chosen as Local Group candidates. "IRAS" Dw`-limited catalogues were constructed around these Local Group candidates by selecting objects according to the number density and selection function of the real data; no biasing was applied in the galaxy selection, so these objects do trace the mass of the simulation. For more details, see Gorski e! al. (1989~. We then ran our iteration program on the N-body "IRAS" catalogue. That is, we gave the program only the Formation on the redshift of each particle as observed in the rest frame of the EG particle, and let it run to predict the true distances and the peculiar velocity field of the particles in the simulation. Q was set equal to unity for these simulations. We emphasize that the motivation is not so much to test the Cold Dark Matter cosmogony, but rather to test our iteration scheme of peculiar velocities in the presence of non-linear erects and shot noise. A total of 18 realizations were run: nine with different "Local Groups," and nine more with a single "Local Group," but different random numbers selecting the galaxies in the suney. In Figure 4, we show the direction for the predicted peculiar velocity of the Local Group in each of these models. They all have been rotated so that the true direction of the Local Group particle in each case is 1 = 270, b = +30 . Notice that the scatter is of the order of 15-20, comparable to the difference between the true and predicted motions of the Local Group. In one of these models, we selected a sample of points to match closely the distn~ution of galaxies with observed peculiar velocities in the samples used in the previous section. With this sample, we can make a direct comparison of true and predicted radial peculiar velocities, and see how these results compare with the real universe described in the

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364 AMERICAN AND SOVIET PERSPECTIVES 1 ' 1 9 LG's ~ 9 Realizations of 1 LO 50 = 2.0 /\ Eli, * _ ~_ ~_ * *_ *_ _ ~O _ C' ._ _ _ c) - * ~ -50 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~1 0 1 00 200 300 Galactic Longitude FIGURE 4 The distribution of predicted directions of the "Local Groups" in eighteen N-body realizations of the sample. The large star is placed at 1 = 270, b = 30 . previous section. The true peculiar velocities were perturbed by a gaussian "error" with a standard deviation of 15% of the true distance to mimic the observational situation. Unlike the real observations, we assume here that the distances are measured exactly. In Figure 5, we show the results. Panel (a) shows the distn~ution of Sue peculiar velocities in the CMB frame, in exact analogy with Figure 2b. Panel ~) is then We predicted peculiar velocity as given by the iteration procedure, and (c) is the difference between the two. FinaLl~r, (d) is the difference of "observed" and predicted, now in the L`G frame. Note the very strong bunk Dow seen in the residuals in panel (c), with an amplitude of ~ 200 km s-l. Unlike the real universe, however, there is still a significant bunk Dow in the LG frame residuals, with an amplitude of ~ 100 km s~i (panel d), which simply says that the deviation of the predicted LG motion from its true direction is at least partly due to small-scale effects in this model. We have done the same experiment in which we have assigned 15% errors to the true distances of the gal~es to more closely simulate the

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HIGH-ENERGY ASTROPHYSICS 4000 2000 o -2000 -4000 4000 2000 -2000 -4000 365 I I I I I \\ 1 i1 ~ I I I I I I I I I _ I I I I I I \\ I t1 ~ I I l I I I i I 1 ~(a) I I (b) O...o~ ~ O.,. '0 09~&o O & 0 o o t `, - 1 , 1 , 1 ~ ~ ~ ~ , ~ , 1 1 1 1 1 1 ool.l 1 1 1 1 1 ' 1 1 1 1, (C) ~i1, D O_ O - . ~ ._ _ _ 0'-'. Oo ~ - _ o & o ~ o oo e _ o" , ~ o _ - 1 1 1 ~ 1 ~ 1 1 1 1 1 1 !.1 `1 ~ ~ 1 1 1 1 1 1 1 1_ - , (d) - o O _ .0 _Oo 0"~ .o o ~ F ~, t o. ~ - ~o W. ,.o _ ~ o _ o ; O - o . o ~- _ , ~ ~ ~ oO J ~ _ ~' _ 1_ 1 1 1 1 1,,, 1, 1 1 1 - o ~_ . t ~_ _ t `. I I I `1' 1- -4000-2000 0 2000 4000 -4000-2000 0 2000 4000 FIGURE 5 The results of an N-body experiment simulating the comparison between obsenred and predicted peculiar velocities. (a) The true peculiar veloaty field in the CMB frame. (b) The predicted peculiar velocities for the same galaxie~ (c3 The difference between (a) and (b3. (d) l~ne difference between obsened and predicted in the LG ~me. Obsenations; we filld that this seriously degrades the good agreement between "measured" and predicted pec~'liar veloci~r. Why is it that there seems to be a missing component of granty in the CDM model, where the pO=tS are explicitly chosen to trace the mass? In these models, there is very little power on scales larger than the sample (8,000 km s~~) to cause the whole sample to translate (Juszkiewicz e! al. 1989~. Our interpretation is that this effect is caused by shot noise. Imagine a toy model in which the buLk of the modon of the region within 3,000 km s~: is caused by a large structure at 6,000 km s~~. Because of the extremely dilute sampling of the catalogue, the overdensity of this structure may be traced by only a small number of points, and this number will be subject to Poisson statisticse Thus, the inferred gravity from this structure can be

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366 AMERICAN AND SOVIET PERSPECTIVES IRAS Galaxies 1.2 1.936 Jy ~ ~ art. ~*~*~*~ . * ~ 2~6 OBJECTS PLOWED '=:*,, - At"' 360 FIGURE 6 Abe distribution of galaxies in Galactic coordinates of objects in a deeper IRAS redshitt surveyor, in preparation. Note the appreciable overdensities associated with the Perseus-Pisces ~ ~ 120 - 150, b ~ -30 - +30 ~ and Hydra-Centaurus (1 ~ 280 - O a 330, b ~ 0- +45 ~ regions. in error, affecting the predicted peculiar velocities III a systematic way. Detailed calculations of the noise in the acceleration of galaxies relative to the Local Group is consistent with the observed amplitude of the effect, which must be at least partly responsible for the missing bunk motions seen in the predicted velocity field of the real IRAS galaxies. UNRESOLVED ISSUES, AND THE WORK AEIEAD In the last section, we have identified shot noise as a stumbling block in the interpretation of the comparison of our predictions with observations. Thus we have started a deeper survey of IRAS galaxies, Dux-limited to 1.2 Jy, to rectify this situation. When the survey is completed, hopefully in early 1991, we shall have redshifts for ~ 5,500 IRAS galaxies selected uniformly over the sly. Figure 6 shows the distribution on the sly of the additional objects. Because this sample goes deeper, it emphasizes structures at distances of 3,000 - 5,000 lain sol; note in particular the strong enhancements in the Perseus-Pisces region and the Hydra-Centaurus region. We are in the process of using the IRAS predicted distances for the Aaronson e! al. galaxies to construct lblly-Fisher diagrams. We hope to make a more robust estimate of Q by finding that value that minimizes the scatter in the TF diagram. The recent work of Dekel and Bertschinger (1989) provides another powerful way of making comparison between TRAS predictions and observed peculiar velocities; we will soon be able to place

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HIGH-ENERGY ASTROPHYSICS 367 constraints on the relative overdensities of galaxies and dark matter different regions of space. ACKNOWLEDGMENTS We thank our collaborators in this project, Amos Yahil and John Huchra, for allowing us to use unpublished data, and for many helpful discussions. We are grateful to Dave Burstein for sending us machine- readable versions of me observed peculiar velocity data. MAS acknowledges the support of an NSF Graduate Fellowship. REFERENCES Aaronson, M., J. Huchra, J.R. Mould, P.L" Schechter, and RIB. Iblly. 198Z Ap. J. 258: 64. Aaronson, M. et al. 1982. Ap. J. Suppl. 50: 241. Bertschinger, E., and A. Dekel. (in press). In: Latham, D., and L. da Costa (eds.~. Urge Scale Structures and Peculiar Motions ~ the Universe. (ASP Conference Series). Burstein, D. 1989. Rep. Prog. Phys. in press. Davis, M., and PJ.E. Peebles. 1983. Ann. Rev. Astr. Ap. 21: 109. Faber, S.M., and D. Burstein. 1988. In: Rubin, V.C., and G.V. Coyne, S.J. (eds.~. Large Scale Motions in the Universe: A Vatican Study Weed Pnuceton University Press, Princeton. p. 116. Faber, S.M., G. Wegner, D. Burstein, AL Davies, A Dressler, D. Lynden-Bell, and RJ. Terlevich. 1989. Ap. J. Suppl. 69: 763. G.orski, K, M. Davis, M.A Strauss, S.D.M. White, and A Yahil. 1989. Ap. J. in press. Huchra, J.P. 1988. In: van den Bergh, S., and CJ. Pntchet (eds.~. The Extragalactic Distance Scale, A S.P. Conference Series Vol. 4. Astronomical Society of the Pacific, San Francisoo. p. 257. IRAS Catalogs and Atlases, Explanatory Supplement 1985. Beichman, CA, G. Neugebauer, H.J. Habing, P.E. Glegg, and IJ. Chester (eds.~. U.S. Government Printing Office, Washington, D.G IRAS Point Source Catalog 1985. Joint IRAS Science Working Group U.S. Government Printing Office, Washington, D.C (PSC3. Juszkiewicz, R. N. ~ttorio, and RJ. Wyse. 1989. Ap. J. in press. Lubin, P.M., and 1: Viola. 1986. In: Madore, B.F., and R.B. Billy (eds.~. Galaxy Distances and Deviations from Universal Expansion. Reidel, Dordrecht. p. 169. Lynden-Bell, D., S.M. Faber, D. 13u~stein, R.L. Davies, ~ Dressler, RJ. ltrlevich, and G. Wegner. 1988. Ap. J. 326: 19. Lynden-Bell, D., O. Lahav, and D. Burstein. 1989. prepnnt. Peebles, PJ.E. 1976. Astron. Astrophys. 53: 131. Peebles, PJ.E. 1980. The Large Scale Structure of the Universe. Princeton University Press, Princeton. Strauss, M.A 1989. Ph.D. thesis. University of California, Berkeley. Strauss, M^, and M. Davis. 1988a. In: Audouze, J., M.~. Pelletan, and ~ Szalay (eds.~. Prow IAU Symp. No. 130, Urge Scale Structures of the Universe. Reidel, Dordrecht. p. 191. Strauss, M4, and M. Davis. 198Sb. In: Rubin, V.C, and G. V. Coyne, SJ. teddy. Large Scale Motions in the Universe: A Vatican Study Weelc Princeton Umvemity Press, Pnaceton. p. 256. Yahil, A. 1988. In: Rubin, V.C, and G.V. Coyne, SJ. (eds.~. Large Scale Motions in the Universe: A Vatican Study Weed Princeton University Press, Princeton. p. 219.