Cosmic Velocity Fields and Their Interpretation

Proc. Natl. Acad. Sci. USA Vol. 95, pp. 78–81, January 1998 Colloquium Paper

This paper was presented at a colloquium entitled ‘‘The Age of the Universe, Dark Matter, and Structure Formation,’’ organized by David N. Schramm, held March 21–23, 1997, sponsored by the National Academy of Sciences at the Beckman Center in Irvine, CA.

Cosmic velocity fields and their interpretation

MARC DAVIS

Departments of Astronomy and Physics, University of California, Berkeley, CA 94720

ABSTRACT We review the current status of our knowl- survey (1), has been the standard method of estimating the edge of cosmic velocity fields, on both small and large scales. thermal state of the distribution of galaxies. It is well known A new statistic is described that characterizes the incoherent, that this statistic is dominated by the pairs contributed by rare, thermal component of the velocity field on scales less than rich clusters of galaxies and is thus an unstable measure. Its ؊1 ؊1 ؊1 2h Mpc (h is H0͞100 km⅐s ⅐Mpc , where H0 is the Hubble interpretation in terms of the cosmic virial theorem is com- ؋ 1022 m) and smaller. The derived plicated by the difficulties of evaluating the necessary integral 3.09 ؍ constant and 1 Mpc velocity is found to be quite stable across different catalogs over the three-point correlation function of the mass distri- and is of remarkably low amplitude, consistent with an bution (2, 3). effective ⍀ ϳ 0.15 on this scale. We advocate the use of this Recently Davis, Miller, and White (4) have suggested an statistic as a standard diagnostic of the small-scale kinetic alternative measure of the thermal state of the galaxy distri- ␴ energy of the galaxy distribution. The analysis of large-scale bution, which they label 1. This statistic is the rms motion of flows probes the velocity field on scales of 10–60 h؊1 Mpc and galaxies relative to their neighbors within projected cylinders Ϫ1 ␴ should be adequately described by linear perturbation theory. of radius 2h Mpc; it is similar to the traditional [12], with the Recent work has focused on the comparison of gravity or major difference being that each galaxy is given equal weight density fields derived from whole-sky redshift surveys of in the computed distribution function of the redshift separa- galaxies [e.g., the Infrared Astronomical Satellite (IRAS)] tion of neighbors, rather than each pair of galaxies. Either of with velocity fields derived from a variety of sources. All the these statistics is evaluated using only redshift space informa- algorithms that directly compare the gravity and velocity tion, and thus they can be applied to large distant redshift ␴ fields suggest low values of the density parameter, while the surveys of galaxies. The 1 measure can be interpreted in terms POTENT analysis, using the same data but comparing the of a filtered version of the cosmic-energy equation, which is derived IRAS galaxy density field with the Mark-III derived lower order than the cosmic virial theorem, because it depends ␰ matter density field, leads to much higher estimates of the on an integral of the two-point correlation function , rather ␨ inferred density. Since the IRAS and Mark-III fields are not than the three-point function . This is much easier to evaluate fully consistent with each other, the present discrepancies and is expected to be much more stable in different samples. might result from the very different weighting applied to the Indeed, tests of this statistic within mock catalogs and with data in the competing methods. different samples of real galaxies confirm remarkable stability. ␴ ϭ Ϯ ͞ Davis et al. (4) report 1 96 16 km s for the Infrared Astronomical Satellite (IRAS) 1.2-Jansky (Jy; 1 Jy ϭ 10Ϫ26 The deviations of the local galaxy distribution from smooth Ϫ Ϫ W⅐m 2⅐Hz 1) sample of galaxies, and ␴ ϭ 130 Ϯ 15 km͞s for Hubble flow, known as peculiar velocities, can be character- 1 Ϫ a redshift sample drawn from the Uppsala General Catalog ized in a number of ways. On scales of order 1h 1 megaparsec Ϫ Ϫ (UGC) sample of galaxies. Analysis of the Las Campanas (Mpc; 1 Mpc ϭ 3.09 ϫ 1022 m; h is H ͞100 km⅐s 1⅐Mpc 1, 0 Redshift Survey (M.D., H. Lin, and R. Kirshner, unpublished H where 0 is the Hubble constant), the galaxy clustering is work) leads to similar conclusions as for the UGC, with known to be highly nonlinear and the peculiar velocities near consistency observed for the six individual slices of the Las most galaxies can be expected to be incoherent and random. Campanas Redshift Survey. This must be compared with Details will of course depend on the local environment, but it results derived from mock catalogs extracted from N-body is of interest to estimate the rms amplitude of the peculiar simulations with nearly identical clustering amplitude. In velocity field averaged over all galaxies. On much larger scales, ⍀ϭ ␴ ␴ unbiased 1 simulations, the measured 1 dispersion is 1 linear theory should apply and there should exist a well defined ϭ 325 km͞s, nearly three times higher than observed in velocity field. We discuss the present state of our knowledge of optically selected galaxy catalogs! If galaxies are unbiased mass both of these components of peculiar velocity and their tracers, the inferred density parameter from this test is ⍀ϭ comparison to the field predicted on the basis of the observed 0.15 Ϯ 0.02. galaxy distribution. Because this comparison is the best ⍀ This test is important because it measures the thermal method of measuring the cosmological parameter ,itisof environment of a typical galaxy and is not biased by the rare considerable interest to fully understand the systematics of the rich clusters of galaxies. It confirms that the quiet thermal analysis. As we shall see below, the current status of the environment of a typical galaxy is well constrained and is very analysis is somewhat murky. different from the hot thermal environment characteristic on small scales in N-body simulations. Such a discrepancy is very Small-Scale Fields difficult to reconcile with high values of the density parameter, with or without bias in the galaxy distribution. For more than a decade, the pair weighted velocity dispersion ␴ Peebles (5) has long argued that the ‘‘coldness’’ of the local [12](r), first employed by Davis and Peebles on the CfA1 flow of galaxies is a serious problem for high-density models

© 1998 by The National Academy of Sciences 0027-8424͞98͞9578-4$2.00͞0 Abbreviations: Mpc, megaparsec; IRAS, Infrared Astronomical Sat- PNAS is available online at http:͞͞www.pnas.org. ellite; Jy, Jansky; ITF, inverse Tully–Fisher.

78 Downloaded by guest on September 25, 2021 Colloquium Paper: Davis Proc. Natl. Acad. Sci. USA 95 (1998) 79

of structure formation. Schlegel et al. (6) measure deviations background radiation (CMBR) dipole is a difficult, dangerous from Hubble flow of only 60 km͞s for galaxies within 5 Mpc game (22, 23). The amplitude and direction of the CMBR of the Local Group. Govertano et al. (7) show that candidate dipole are rather precisely known, but the coherence of the Local Groups within high-resolution N-body models never flow is completely uncertain. How large a region is flowing at exhibit such cold local flows, either in ⍀ϭ1or⍀ϭ0.3 models. 620 km͞s along with the Local Group of galaxies? Should a full Typical local group candidates in their simulations have much reflex of this dipole be detectable within 50, 100, or 3,000 Mpc? higher local velocity dispersions, including objects with blue- Without an answer to this question—i.e., an assumption of the shifts, which are not observed for the nearby galaxies outside nature of the large-scale power spectrum—it is not possible to our own Local Group. infer the cosmological density parameter by comparing the The small-scale ‘‘coldness’’ of the galaxy distribution is well CMBR dipole to the gravitational dipole inferred from galaxy known by alternative expressions: the high ‘‘mach number’’ of catalogs such as the 1.2-Jy IRAS survey. the cosmic velocity field (8) or the thinness of the observed However, the nonlocality does not apply to all aspects of the sheets of galaxies in redshift space (9). We want to emphasize velocity–gravity field comparison. Recall Einstein’s gedanken that the problems presented by such low amplitude peculiar experiment of an observer within an elevator. He cannot velocities are real and that they indicate a serious gap in our distinguish whether he is in an accelerating frame or is understanding of structure formation. stationary in an external gravitational field. If the elevator goes into freefall, he can detect the presence of an external gravi- Large-Scale Flows tational field only by its tidal influence on the matter within the elevator. Exactly the same considerations apply for the grav- Ϫ On scales greater than 5h 1 Mpc the deviations from Hubble itational field estimated from an imperfectly sampled galaxy flow are expected to be smaller than the Hubble flow itself, so distribution. A poorly sampled, distant cluster of galaxies will that in most regions galaxies are physically expanding from induce coherent errors in the nearby gravity field, but they will each other. In such a situation it is reasonable to expect the be tidal in nature. Working in the Local Group frame of velocity field to be largely irrotational. On slightly larger scales, reference rather than the cosmic microwave background frame one can expect linear perturbation theory to be a reasonably is conceptually cleaner for this analysis. Recall Newton’s iron accurate guide to the expected peculiar velocity field. This is sphere theorem, which states that for spherical symmetry, the a fortunate circumstance because it allows comparison of the acceleration of a point at radius r is sensitive only to the mass observed deviations from Hubble flow with the flow predicted interior to that point. In terms of a spherical harmonic on the basis of full-sky redshift catalogs of galaxies. The most expansion of the external gravity field, the l ϭ 0 component of recently compiled large datasets used in these analyses are the the field at radius r is insensitive to the mass distribution at R Ͼ peculiar velocity data of the Mark III consortium (10) and the r. The general solution of the Poisson equation for an exterior sample of Tully–Fisher galaxies presented by Giovanelli and mass at radius R leads to an acceleration proportional to colleagues (11). For comparison to the gravity field predicted (r͞R)lϪ1͞R2. Tidal effects are described by l ϭ 2, and grow by the observed galaxy distribution, most recent work has used linearly with r, as we know well. But we tend to forget that for the full-sky 1.2-Jy IRAS flux limited redshift survey (12), but the dipole term, l ϭ 1, the external field produces an accel- an optically selected sample of galaxies combined with the eration independent of r, which means that by moving to the IRAS survey has also been used in recent work (J. Baker, freely falling Local Group frame, the l ϭ 1 gravity field at M.D., M. Strauss, and O. Lahav, unpublished work). radius r is sensitive only to the mass distribution interior to that Recent reviews and methodological details are given by point (24). This is a critical point, demonstrating why a Dekel (13, 14) and by Strauss and Willick (15, 16). Although comparison of the peculiar velocity field with the gravity field, there are many uses for the peculiar velocity data, the most if limited to the monopole and dipole terms, is effectively a powerful, model-independent, tests are those that keep phase local test that is totally insensitive to the matter distribution at information. There are two broad categories of tests of this large distances. sort, the first of which compares the observed galaxy density The ITF method of Nusser and Davis (24) is ideal for this field with the mass density field inferred from the divergence type of comparison, because it allows expansion of the gravity of the observed velocity field [also known as POTENT (13, and velocity fields in terms of whatever functional expansion 14)], and the second of which compares the observed velocity is most convenient. This method expands the radial compo- field with the gravity field inferred from the galaxy distribution nent of the peculiar velocity field in terms of a set of orthogonal [e.g., the VELMOD analysis (15–17), the inverse Tully–Fisher functions, characterized by coefficients that are determined by (ITF) method (18), or the least-action method (19)]. The minimizing the scatter in the linewidth direction for a uni- search for reflex dipole flows [e.g., the local motion relative to formly calibrated set of Tully–Fisher data. Details are given by brightest cluster galaxies (20) or relative to nearby supernovae Davis et al. (18). The method is merely a filtering tool that can (21)] are similar in spirit to the second form of these tests. smooth both the gravity and velocity fields to the same The density–density comparison of POTENT is inherently resolution, which can be dependent on position; it smoothes a local comparison, which is a tremendous virtue, because the data without binning it, and the derived coefficients of both neither the galaxy density field nor the peculiar velocity field the gravity and velocity fields, for which one can compute a full are well known at large distance. The velocity–gravity com- covariance matrix, are a complete description of the field to a parison, on the other hand, suffers from coherent errors given resolution. In a typical application, 50–100 coefficients induced by nonlocality, since the estimated peculiar gravity are fit. Tests demonstrate that the method works and recovers field at a given point in space is computed by effectively the true velocity field with minimal bias. These tests with mock summing over neighbors at all distances, weighted inversely by data show that the residual field, the difference of the inferred the square of the distance. Poisson shot noise of the galaxy velocity and gravity fields, has negligible dipole errors, and is distribution in one locale therefore generates a coherent error dominated by l ϭ 2, 3 components due to the dilutely sampled in the gravity field over all space. This nonlocality makes the exterior mass field. statistics of the field comparison quite complicated. As previously discussed in detail (18), the comparison of the It has been argued that this sensitivity to the uncertain IRAS gravity field with the Mark III Tully–Fisher data is not far-field galaxy distribution precludes the second class of tests nearly as successful. For no value of the density parameter is as a useful diagnostic of gravitational instability or as a it possible to eliminate the dipole residual. The comparison of measure of the cosmic density parameter. For example, com- the modal coefficients demonstrates that for no value of the parison of the gravitational dipole for the cosmic microwave density parameter are the gravity and velocity fields statisti- Downloaded by guest on September 25, 2021 80 Colloquium Paper: Davis Proc. Natl. Acad. Sci. USA 95 (1998)

cally consistent with each other. Thus, although the qualitative to an inferred density ⍀ϭ0.3 if IRAS galaxies trace the mass ϭ ⍀ϭ comparison of the two fields is quite impressive, the fields are distribution (bI 1), or 0.2 if optical galaxies trace the ϭ inconsistent with each other at the 4-␴ level or worse. A value mass distribution (bo 1). A plot of the ITF inferred velocity of ␤ ϭ 0.6 fits best, but it is not an acceptable fit. Until we and gravity field (for ␤ ϭ 0.5) for this subset of the Mark III better understand the reason for the large dipole residual, data is shown in Fig. 1. Note that the two fields are remarkably which grows nearly linearly with redshift, it is premature to use well aligned, and that the sign of the predicted vs. observed these methods to measure the density parameter. A dipole peculiar velocity matches for virtually every galaxy in the residual that grows with redshift is not physically reasonable sample. The residual field is also shown, and at this redshift limit, it is not overly dominated by the dipole residual. Perhaps, and strongly suggests a calibration error that is nonuniform on ␤ the sky, either in the IRAS catalog or in the Mark III database. therefore, the value inferred by VELMOD is an acceptable fit. Willick et al. (17) report a more successful comparison of the Similar values for ␤ emerge from the use of the ITF IRAS gravity field to a subset of the Mark III data limited to ͞ ͞ algorithm and the SFI sample of Tully–Fisher data (11), which redshift of 3,000 km s, vs. the 6,000 km s limit used by Davis, has many galaxies and much data in common with Mark III but Nusser, and Willick. Their analysis procedure, VELMOD, is in which the data were processed somewhat differently. Da capable of treating the nonlinear, multivalued zones around Costa et al. (25) report no anomalous dipole residuals in their clusters of galaxies, whereas the ITF method is a redshift space fits, which seems quite encouraging. procedure that assumes a single-valued relationship between Yet additional, independent confirmation of this relatively distance and redshift. The gravity field computed for the ITF low value of ␤ is given by a direct comparison of the observed procedure is similarly a redshift space algorithm that works vs. predicted peculiar velocities for a sample of 24 supernovae only in the linear theory limit. VELMOD does not produce a 1A within a redshift limit of 10,000 km͞s (26). They find a good visual image of the flow field but it yields a maximum fit and a 95% confidence limit in the range 0.15 Ͻ ␤ Ͻ 0.7, a likelihood solution for ␤ ' ⍀0.6͞b based on a series of IRAS constraint which will tighten as the sample of supernovae with gravity fields. Willick et al. report ␤ ϭ 0.5 Ϯ 0.1, which leads good data increases over the coming years.

FIG. 1. A full-sky plot of the peculiar velocity field of the Mark III galaxies within 3,000 km͞s. Galaxies with positive (negative) peculiar velocity in the Local Group frame are shown as stars (circles) with symbol size proportional to velocity. (Top) ITF peculiar velocity inferred from the Tully–Fisher data. (Middle) IRAS predicted peculiar velocities for these galaxies, assuming ␤ ϭ 0.5, filtered by the same set of functions that were used for the ITF fit. (Bottom) The difference of the ITF and IRAS fields. Downloaded by guest on September 25, 2021 Colloquium Paper: Davis Proc. Natl. Acad. Sci. USA 95 (1998) 81

On the other hand, the same Mark III and IRAS data, when 3. Bartlett, J. & Blanchard, A. (1996) Astron. Astrophys. 307, 1–7. analyzed by the POTENT procedure, lead to estimates of ␤ 4. Davis, M., Miller, A. & White, S. D. M. (1997) preprint astro- consistent with unity (27), well above the limits from the ph͞9705224. velocity–gravity field comparisons. The POTENT algorithm 5. Peebles, P. J. E. (1993) Principles of Physical Cosmology (Prince- has had at least as much validation studies applied to it as any ton Univ. Press, Princeton, NJ). of the competing algorithms, and tests show that it is fully 6. Schlegel, D., Davis, M., Summers, F. & Holtzman, J. (1994) capable of recovering the underlying true value of ␤ in a noisy Astrophys. J. 427, 527–532. 7. Govertano, F., Moore, B., Cen, R., Stadel, J., Lake, G. & Quinn, field of data. How is it that the same data can lead to such T. (1997) New Astron. 2, 91–106. divergent conclusions when analyzed using different proce- 8. Suto, Y., Cen, R. Y. & Ostriker, J. (1992) Astrophys. J. 395, 1–20. dures? 9. Bothun, G., Geller, M. J., Kurtz, M., Huchra, J. & Schild, R. The answer is currently unknown but might result from the (1992) Astrophys. J. 395, 347–359. very different weighting applied to the density–density algo- 10. Willick, J., Courteau, S., Faber, S., Burstein, D., Dekel, A. & rithms vs. the velocity–gravity algorithms. In the former case, Strauss, M. (1996) Astrophys. J. 457, 460–489. the procedure is most sensitive to the divergence of the velocity 11. Haynes, M., Giovanelli, R., Heter, T., Vogt, N., Freundling, W., field and is insensitive to the largest scale components of the Maia, M., Salzer, J. & Wegner, G. (1997) Astron. J. 113, 1197– field. In the Fourier domain, the divergence of velocity is 1211. proportional to (kvk), so the highest Fourier modes remaining 12. Fisher, K., Huchra, J., Strauss, M., Davis, M., Yahil, A. & after the smoothing are given maximal weight. The velocity– Schlegel, D. (1995) Astrophys. J. Suppl. Ser. 100, 69–103. gravity algorithms, on the other hand, are seeking the reflex 13. Dekel, A. (1994) Annu. Rev. Astron. Astrophys. 32, 371–418. ͞ dipole of the motion of the local group and are dominated by 14. Dekel, A. (1997), preprint astro-ph 9705033. 15. Strauss, M. & Willick, J. (1995) Phys. Rep. 261, 271–431. the largest wavelength Fourier modes in the volume. Thus the ͞ weightings of the alternative procedures could not be more 16. Strauss, M. (1996) preprint astro-ph 9610033. different! 17. Willick, J., Strauss, M., Dekel, A. & Kolatt, T. (1997) preprint astro-ph͞9612240. Because, as argued above, the Mark III data do not really fit 18. Davis, M., Nusser, A. & Willick, J. (1996) Astrophys. J. 473, 22–41. the predictions of the IRAS inferred gravity field for any value ␤ 19. Shaya, E., Peebles, P. & Tully, B. (1995) Astrophys. J. 454, 15–31. of , it should therefore not be a surprise when very differently 20. Lauer, T. & Postman, M. (1995) Astrophys. J. 440, 28–47. weighted analyses of the data lead to differing conclusions. The 21. Riess, A., Press, W. & Kirshner, R. (1995) Astrophys. J. 445, prudent course of action is to work to understand why the L91–L94. fields are so inconsistent, to search for systematic errors in one 22. Strauss, M., Yahil, A., Davis, M., Huchra, J. & Fisher, K. (1992) or both of the fields, and to restrain from drawing overly strong Astrophys. J. 397, 395–419. cosmological conclusions until the competing algorithms lead 23. Juszkiewicz, R., Vittorio, N. & Wyse, R. (1990) Astrophys. J. 349, to consistent solutions. 408–414. 24. Nusser, A. & Davis, M. (1995) Mon. Not. R. Astron. Soc. 276, This work was supported by National Science Foundation Grant 1391–1401. AST95-28340 and National Aeronautics and Space Administration 25. da Costa, L., Nusser, A., Freundling, W., Giovanelli, R., Haynes, Grant NAG 5-1360. M., Salzer, J. & Wegner, G. (1997) preprint astro-ph͞9707299. 26. Riess, A., Davis, M., Baker, J. & Kirshner, R. (1997) Astrophys. 1. Davis, M. & Peebles, P. J. E. (1995) Astrophys. J. 267, 465–482. J. 488, L1–L5. 2. Peebles, P. J. E. (1980) The Large-Scale Structure of the Universe 27. Sigad, Y., Dekel, A., Strauss, M. & Yahil, A. (1997) preprint (Princeton Univ. Press, Princeton, NJ). astro-ph͞9708141. Downloaded by guest on September 25, 2021