4. DEVIATIONS FROM HOMOGENEITY: THE PECULIAR VELOCITY FIELD 4.1 INTRODUCTION

Transcription

1 Section 4 4. DEVIATIONS FROM HOMOGENEITY: THE PECULIAR VELOCITY FIELD 4.1 INTRODUCTION In addition to the effects on the propagation of light rays and the graitational lensing effects (Sect. ), the cosological Large Scale Structure is supposed to leae iprints on the elocity fields of galaxies and other tracers. Indeed, the general hoogeneity and isotropy of the Unierse, and the Hubble law itself hae only alidity if considered on the largest spatial scales. On saller scales, the atter distribution shows fluctuations that we hae quantified in Sect. 1. And the Hubble flow itself reeals perturbations that go under the nae of uliar elocities. Systeatic easureents of the latter offer a fundaental cosological obserable. If we consider the spatial scales oer which we ext the Ω Λ paraeter operates, they are only of the order or larger than any hundreds or thousands Mps, the extreely large scales. As we know fro Sect.1, these are not the scales where large atter fluctuations are present. In conclusion it is unlikely that the Ω Λ paraeter strongly affects the cosic uliar elocity field on the scales of interest in a direct way (it does it indirectly by odifying the structure foration history, as we discussed in Sect. 3.4). On the other hand, it is ery likely that this is influenced by the Ω paraeter, and studies of uliar elocities can constrain it. Notice that the inforation that we will get fro uliar otions will be weakly affected by the cosological bias proble that instead influences other analyses based on the obsered galaxy distributions (Sect. 1). The knowledge and easureents of the elocity fields are also iportant for any other reasons, like for exaple correcting the distance deterinations based on redshift and the Hubble law with a large-scale odel for the uliar elocity flow. 4. PECULIAR MOTIONS FROM THE HUBBLE LAW AND A 3D MAPPING OF THE UNIVERSE As it is well known, the stral lines obsered in the stru of the Sun show changes in waelength due to the cobined otion of the obserer relatie to solar photosphere, and due to the otion of rotation of the Earth around its axis and the The Peculiar Velocity Field 4. 1

2 rotation of the Sun itself. Obsering other stars or galaxies, the stral lines will reflect their relatie otions to us. These otions are e.g. well easured coponents in the Local Group of galaxies, where they appear with a few exceptions shifts toward the blue (approaching otions). All these line shifts are due to the Doppler sial relatiistic effect, due to the relatie otion of source and obserer, ruled by the Lorentz transfor to be: [4.1] where λ and λ o are the eitted and obsered waelength. For sufficiently sall copared to the elocity of light, eq. [4.1] becoes [4.] Figure 4.1 Left: the graph published by Hubble with the expansion s discoery (by Hubble E. 199). Right: the odern ersion of the Hubble law fro obserations with the Hubble Space Telescope, with the different used distance indicators listed in the insert (by Freean K. et al. 001). This relation defines the redshift z in the local Unierse. For negatie (sources approaching the obserer, like in the Local Group) this is a blue-shift, while becoing a redshift for positie. The great discoery by Hubble in the 199 was that of the existence of a relation between the distance of a galaxy and the stroscopically easured redshift The Peculiar Velocity Field 4.

3 cz H d. [4.3] 0 The costant H 0 is the Hubble costant, expressed in K/sec/Mpc. Current easureents (ainly perfored easuring distances of galaxies in the Virgo cluster with the cefeids ethod, with an error <10%, thanks to the systeatic use of HST) indicate H 0 70 ± 6 Κ/sec/Μpc [4.3b] H 0 is known today with uch higher precision ( 1%) thanks to obserations of the Cosic Microwae Background, see Sect. 6). The source distances in [4.3] are easured fro distance indicators, whose sequence as a function of the distance scale is reported in the cosic distance ladder in Fig. 4.. Figure 4. Representation of the so-called cosic distance ladder. At each step of this, corresponding to a certain range of distances, different cosic sources are used as indicators. Each of these different categories are characterized by different luinosity classes. [Chart taken fro the book The cosological distance ladder (Rowan- Robinson, 1985]. The Hubble law applies on large scales (it is not alid, for exaple, in the Local Group, where we easure only blue-shifts). On saller scales, where the LSS is proinent, we hae to consider the effects of uliar otions (or proper otions) of galaxies, according to the relation: z costant d + / c [4.4] The Peculiar Velocity Field 4. 3

4 such that the uliar elocity field can be directly estiated fro the two independent easureents of the recession elocity redshift and distance d: redshift Hd 0 = [4.5] Vice-ersa, a correct 3D apping of the Unierse that we ight wish to obtain fro application of the Hubble law thanks to the relatie siplicity to obtain redshift easureents today, requires soe good odelling and correction for the systeatics inherent the uliar elocity flow d = ( ) / H [4.5b] redshift 0 Peculiar otions and the cosological Large Scale Structure When cosologists hae started to study the cosological large-scale structure, they hae also considered the possibility that the large oids, walls and sheets (see e.g. Figs. 1.1 and 1. of Sect. 1) could be siply due to the accuulation effects of the uliar otions on large scales. We hae seen that large oids, for exaple, ay extend up to scales of order of a hundred Mpc. The question then is: what elocity fields ight generate this structure during a Hubble tie? The siple answer is Mpc Mpc 7000 K/s t 1/ 70( K / s / Mpc). H Enorous elocities would then be required. The obserations of the uliar elocities, discussed later in this Sect., appear to be inconsistent with such large alues and indicate instead uch lower elocities, of order of a few to seeral hundred K/sec. The consequence is that the large scale structure should be rather interpreted as regions in which galaxies hae fored either ore (in walls, sheets, and filaents) or less (in oids) efficiently. Peculiar otions, on their side, are interpreted as the integrated effect of graitation by large scale in-hoogeneities in the atter distribution, operating during a tie coparable to the Hubble tie t H. The Cosic Virial Theore We ay start haing a sei-quantitatie assessent of the role of the large-scale structure in inducing perturbations in the galaxy elocity field by application of the irial theore to the atter distribution. The theore's alidity requires a kind of dynaical situation with soe leel of relaxation in the energy coponents at play (which strictly speaking we cannot really assue, but let us haing it as a first The Peculiar Velocity Field 4. 4

5 approxiation). These coponents are the self-graitation in the atter distribution and the kinetic energy of test particles (as galaxies are): GM. r As for the atter distribution, the effectie ter about uliar elocities is that corresponding to atter enhanceents aboe the aerage atter density of the Unierse, enhanceents that we can represent in ters of the -point spatial correlation function (see Sect. 1.3). The effectie excess ass producing the uliar elocity field can then be estiated as (the following is a particularly good assuption on sall enough scales that ξ () r > 1, but holds approxiately true also for larger r, ξ () r easures the excess density with rest to the underlying hoogeneous density field): 1.8 4p 3 4p 3 r M( r). r0r ξ( r). r0r such that h pG r 8pG r r0r ; 1 rcr. Ω 1 [4.7] 3 8.3h 3 8.3h 0.1 1/ r K Ω 1 (1000 / sec) h Mpc where we get a ery weak dependence on the spatial scale to which the uliar elocity is referred. This relation can be effectiely inerted to get a constraint on the density paraeter fro the easured elocities: 0. r Ω ( 1000 K / sec) Mpc [4.8] If we put the typical obsered alues, that are elocities of about 500 K/sec (approxiately the alue of the Local Group easured against the dipole coponent of the CMB, see Sect. 6) on scales of about 10 Mpc, we get Ω 0.16 In spite of the crudeness of the approach (the irialisation assuption is essentially incorrect except to first order), this is an interesting reference alue, not so far fro the correct accepted figure for the density paraeter. This result, although qualitatie at the oent, illustrates the fact that uliar elocities and the density paraeter are indeed tightly related. The adantage of this crude approach is its independence fro the bias paraeter (what enters the M ass is the total dynaical ass not the luinous one, galaxies are only treated as test particles). The Peculiar Velocity Field 4. 5

6 4.3 MODELLING THE PECULIAR VELOCITY FIELD We hae found in the preious Sect. 3.1 an iportant result about the uliar elocity field: in the absence of forces inducing perturbations in the elocity field, 1 any such perturbations quickly decay in tie (proportionally to at () for the proper elocities), as a result of the expanding ediu (eq. [3.]). The existence of uliar otions is direct consequence and eidence for a perturbation field in graitating atter continuously generating and aintaining at the sae tie these otions. In particular, eqs. [3.0], [3.38] and [3.39] are the ost general expressions about the eolution of structures in its arious coponents, both the relatiistic and nonrelatiistic ones (in particular these will be needed for treating perturbations before recobination in Sect. 6). Rotational otions. If we are interested, as we are now, in a physical odelling of the relationship between the structure in the atter distribution and the elocity field, things can be siplified significantly. Let us first of all consider eq. [3.0], where we can neglect the pressure ter at all, considering that galaxies and dark atter are pressure-less coponents (except in case the uliar otions ight becoe relatiistic, which is obiously not the case as we hae seen). That eq. then becoes d du a d 1 + u = cdφ [4.10] dt a a u being the uliar elocity coponent in cooing units. Now let us coe to the iportant point that these elocity perturbations, the uliar elocities, correspond essentially to potential otions, and are irrotational. Indeed, consider splitting the elocity into a coponent u parallel to the gradient of the perturbed graitational potential cδφ, and one perpendicular to it, u. This latter is ostly independent on the graitational potential, and corresponds to a field of rotational otions, like ortices. By definition, these otions obey our eqs. [3.1] and [3.] and feel the Hubble drag, because (4.10) becoes, in the absence of torque forces, d du a d d u ; u at = (). dt a The Peculiar Velocity Field 4. 6

7 These turbulent otions are not supported by any force, and then quickly decay 1 according to the rule of the Hubble drag [in proper units: at () ] 1. We ention just in passing that turbulence of priordial origin has been soeties considered as a echanis driing galaxy foration. Howeer, because these otions decay quickly, one would need to find soe echaniss aintaining the if they hae to surie. Since there are no candidates for such forces, cosological turbulence is not supposed to drie uliar otions in the Unierse. Indeed, forces potentially inducing shear fields and rotation ight be the tidal ones; howeer in our situation of sall assued perturbations these ters are negligible. These instead becoe significant during the non-linear phase of the collapse and are exted to be the source of rotational oentu in cosic objects (like galaxies) on sall spatial scales. Fro an obserational perstie, iportant coponents of rotational otions and orticity on large scales can be excluded fro the following consideration. Let us suppose that the elocity field consisted of rotational otion coponents with rando orientation of the rotational axes. Then one would ext strong discontinuities in the elocity field here and there to happen where two eddies with opposite rotation coe in touch. No such discontinuities are obsered in fact. Potential otions. Motions along the direction of the graitational field gradient are aintained and deeloped by the field itself. To start with, we can gie up using the whole eqs. [3.0], [3.38] and [3.39], and we can gie up the Euler equation, and just consider the Poisson and ass conseration conditions. Howeer, in addition, we ake use of our preious results about the eolution of the density contrast (the growth factor). This is an iportant eleent of our analysis, because the elocity field is not erely due to perturbations to the density field at the present tie, but it also integrates the operation of the perturbed graitational potential during the whole Hubble tie, hence it is sensitie on how the LSS has eoled with tie. Since we now hae fro Sect. 3.3 solutions for the eoling field of fluctuations in the atter distribution, we can easily calculate fro this the perturbed graitational potential that they generate. Let us start fro the perturbed Poisson eq. [3.16] either in proper or in cooing units: δφ = 4 π G( δr) = 4 π Gr ; or δφ = 4π Ga δr c Because the aerage atter density reads 3Ω H ρ =Ω ρc =, we get 8π G 1 As rearked by Longair (Galaxy Foration), this can be seen as a siple rule of conseration of the rotational angular oentu r = const. The Peculiar Velocity Field 4. 7

8 3Ω H 3 c δφ = 4π G a = ΩH a 8π G It ay also be useful to refer to the uliar graitational acceleration as δ δ 1 δ g = δφ = cδφ at () [4.13] [4.14] Now, since these otions are irrotational ( 0), we can express the graitational pull in ters of a new potential Φ inducing a uliar elocity field, like defined as: Φ c = = Φ [4.15] at () whose diergence gies: Φ = Φ = a () t c [4.16] Next, let us include the continuity equation for the perturbed ass distribution, eq. (3.13): d dρ d d d 1 d d = = = c [4.17] dt ρ0 dt a and putting together [4.16] and [4.17], d d d d Φ c = at () c p e c = a = a [4.18] dt Let us anipulate this by ultiplying and diiding by and H = a / a a Φ c = H a = Ht () f( Ω ) a [4.19] a where we hae introduced the function f ( Ω ) expressing the eolution of the atter density contrast (the growth factor discussed in Sect. 3.3), defined as a ad dln f ( Ω ) = =. [4.0] a da d ln a The Peculiar Velocity Field 4. 8

9 We can borrow fro the analysis ade in Sect. 3.3 all details about the function f ( Ω ). In particular, we hae seen in [3.5] that for an Einstein-de Sitter Unierse this eolution is siply δρ = ( 1 + z) 1 at ( ), such that f( Ω = 1) = 1. [3.5] ρ More in general, we hae also seen that the situation is ore coplex for Ω 1, in which case howeer a fair representation of the eolution is gien by (Peebles 1980) f ( Ω ). Ω [4.1] Note that this result is alost independent on Ω Λ, so it is surprisingly ery general. Fro [4.19] c Φ = a = Hf ( Ω) a and putting together this and the Poisson [4.13], by substituting the product a we get 1 δ 3 Φ c = Hf ( Ω) a = Hf ( Ω) Ω H c δf [4.3] 1 δ 3ΩH f( Ω) c Φ = c dφ, for which one solution is Φ = dφ, f( Ω) 3Ω H 0.6 and taking the gradient of this we get the iportant relation: δ δ f( Ω ) δ f( Ω) δ = Φ = δf = g. [4.4] 3ΩH0 3ΩH0 Note that a first integral of [4.3] ay be forally obtained as Φ c = p e c = δ = ( f 3 ΩH ) δf + const = ( f 3 Ω H ) g + const, but the constant elocity ter can be considered to quickly decay due to Hubble drag, hence [4.4] is obtained. We see in [4.4] that the action on the elocity fields is done by a (factorized) cobination of the uliar acceleration g and the growth factor f : because of the long ter action of graity, this latter factor is releant. We can also obtain a second useful expression by considering the diergence of the elocity field, through [4.18] and using [4.19]: = a Φ = a Ha f ( Ω = ) H ( t) f ( Ω ) [4.5] p e c c where the diergence on the elocity is here expressed in proper coordinates. Equations [4.4] and [4.5] are the iportant ones relating the uliar elocity field to the present-tie graitational pull and its tie eolution. These will be our workhorses in our later applications. The Peculiar Velocity Field 4. 9

10 4.4 OBSERVATIONS OF PECULIAR MOTIONS As we will see below, uliar otions reflect in-hoogeneities in the Unierse on large scales (<300 Mpc). The corresponding collapse phase is linear, so can be treated with good precision with the linear theory discussed in Sect. 3. The analysis of uliar elocity fields not only offers a solid confiration of the graitational instability scenario for the foration of the cosological large scale structure, but also a robust ethod to easure the aerage atter content of the Unierse through eqs. [4.4] and [4.5]. We will consider in the following two applications of our preious analysis to two scheatic situations, one in which the uliar elocity field feels a large ass concentration that ight be assued as point-like. The second one is associated to a large-scale extended ass distribution and the associated cosological dipoles Infall towards large point-like ass concentrations. Virgo infall. The situation we consider here is that of a circularly syetric atter concentration, like depicted in the schee below, whose graitational effects on the elocity fields ight be described as those of a point ass concentration. So with reference to eq. [4.4], we get for the graitational acceleration g : g 4π G r = 3 3 r0 r and for the uliar induced elocity This Section refers partly to the Chapter of the book "Physical Cosology" of J. Peacock, and takes aterial fro there. The Peculiar Velocity Field 4. 10

11 f( Ω) f( Ω) 4pG r0 8pG 3 H0 f( Ω) f( Ω) = g= r= r = Hr 0 3H Ω 3H 3 Ω 9 8p G H haing considered that ρ0 Ω = ρc = 3H0 8πG is the critical density, and noting that Hr= 0 Hubble is the Hubble recession elocity at distance r, we hae Hubble f ( Ω) Ω = =. [4.7] 3 3 Hubble Hubble This is a rearkable result: the percentage deiation of uliar elocities with rest to the Hubble flow does not depend on the spatial scale but only (linearly) on the percentage density fluctuation and on the density paraeter Ω through the growth function f. The ost obious application concerning obserations fro our reference frae is that about the perturbation of the Hubble flow induced by the Local Super-cluster centred on the Virgo cluster, as shown in Fig This ass concentration reduces the expansion elocity of our reference syste with rest to it, and consequently the recession elocity of Virgo is saller than that exted for its distance, gien the unperturbed Hubble flow. Fro the 3D distribution of galaxies in the Virgo enironent we can easure the aerage density contrast in the galaxy distribution copared to the aerage density, and this is approxiately.5. Figure 4.4 also shows the distribution of easured recession elocities of galaxies in the Virgo direction. Fro this, the obsered redshift is 1000 K/sec, while the distance (precisely easured with HST by Cepheid s distance indicators) is 18.5 Mpc, that translates onto an unperturbed recession elocity for H 0 = 70 of 1300 K/sec, corresponding to an infall uliar elocity of 300 K/sec. All this put together into eq. [4.7] brings to an estiate of the atter density of Ω if galaxies trace the underline dark atter field. This brings to an estiate of Ω (this includes a sall correction factor to account for soe non-linearity effect due to the relatiely large alue of ), which in any case confirs pointing towards an open Unierse and a atter density quite insufficient to closure. This results is still liited by arginal non-linearity on such sall scales and the fact that our frae is still too uch inside the perturbation to offer a reliable easure of the global Ω. The Peculiar Velocity Field 4. 11

12 Figure 4.4 Measured elocities of galaxies within 6 degrees of the Virgo cluster center. The Peculiar Velocity Field 4. 1

13 4.4. Cosological dipoles towards large scale structures. What is clearly needed fro Sect for a reliable easure of the global Ω is to expand the scale of our inestigation towards larger scales, which siultaneously becoes better representatie of the global alue and ore securely consistent with our linearity requireent (on larger scales is saller). The situation here is assued to represent the effects of ass concentration in a large cosic wall, as we see in 3D aps of the galaxy distribution and scheatically illustrated in the graph below. Again fro [4.4] we hae: 3 d θ r dr d r g = G r 0 = G r 0 r r 3 because θ r dr = d r and integrating oer the sheet d dr 3H 3 0 g = G Ω r 8 πg 3 3 d d f( Ω ) d fg dr 3H0 f3h0 dr ( ) ˆ x = g = Ω = r = 3H0Ω 3H0Ω r 8pG 3 8p r d [4.9] H0 f( Ω ) ( r ) 3 = rd ˆ r 4p r in a generic position x assued to be the origin of the reference syste for the integration. Note that the ass fluctuation field ( r ) appearing here, and responsible for inducing uliar otions, includes all non-relatiistic atter (baryons and dark The Peculiar Velocity Field 4. 13

14 atter), ass. Of course, to ealuate this ass reconstruction we need to use luinous atter, like galaxies, whose fluctuations are. The relation between the two is gien by the bias paraeter b that we introduced in Sect : such that d light light ass = [4.30] b 0.6 d d H0 Ω light ( r) ( ) 3 H0b light r 3 = rd ˆ r rd ˆ r 4p b = r 4p [4.31] r where the paraeter 0.6 Ω b. [4.3] b is often used, and akes the direct obserable fro this analysis. The local dipole. The flux-weighted dipole. So, gien soe kind of apping of the galaxy distribution () r, we can predict the induced elocity field. An interesting application concerns the one case in which we know in good detail and precision the uliar elocity in aplitude and direction: the well-deterined absolute otion of our reference frae with rest to the local fundaental obserer defined by the Cosic Microwae Background (CMB) photons. As we will see in detail in Sect. 6, obserations of the angular distribution of the CMB intensity allow us to define with high precision both the elocity and direction of the absolute otion of the Earth in the sky. Once we subtract fro this the elocity coponents of the Sun in the Galaxy and of the Galaxy in the Local Group, we get the absolute otion of the local fundaental obserer (the local frae). We defer to Sect. 6 for ore details. So, assuing that the ector at our position is known in [4.31], a odel for the 0.6 ass distribution () r can be used to infer estiates of the paraeter b = Ω b, and eentually for the density paraeter itself. In fact, a reasonable answer for the predicted dipole can be obtained without distance or een redshift data, but just using the angular projected distribution of galaxies (like for exaple discussed in Sect. 5.1 of the 3rd year Course). Any flux-liited surey will hae a radial distribution peaking at soe typical distance depending on the surey flux liit. So the anisotropies of the galaxy distributions calculated at different agnitude liits allow us in principle to infer an integrated estiate of the 3D ap. We can write that, in ters of the galaxy luinosity function Φ ( L), the The Peculiar Velocity Field 4. 14

15 nuber counts can be expressed as a function of the sky coordinates with integrals of the density fluctuation field () r weighted by Φ ( L) : dn[ RA, DEC] 3 1 d 3 = ( 1 +D [, r RA, DEC] ) Φ( Sr ) d r = r() r Φ( Sr ) d r ds r [4.33] This flux-weighted dipole is therefore roughly proportional to the graitating ass integral. An iportant question when perforing this analysis is at which radial distance fro us the integral [4.33] conerges, to get the total graitational pull. This obiously depends on how fast the cosic large-scale structure conerges to hoogeneity. This then sets a requireent on the depth of the surey used to deterine the counts. Fro what we found in Sect. 1, it is clear that surey depths approaching 300 Mpc are required for the uliar acceleration to conerge (Peacock 199). This approach has been followed by any authors, not always with consistent results. For exaple, the use of the IRAS extragalactic source catalogue (about galaxies, flux-liited saple, hoogeneous selection, aerage distance of 400 Mpc; see Sect. 5.1 of the 3rd year Course) has brought to rather high alues of the 0.6 paraeter b =Ω b ± 0.15, where b IRAS IRAS is the bias of the IRAS galaxies: that would ean an high alue of the density paraeter ( Ω 0.7), unless this bias factor is proportionally sall. Indeed this ight be the case if IRAS galaxies tend to aoid large atter concentrations, like the galaxy clusters and superclusters, because of lack of gas, dust and star-foration in the latter. Indeed this sees to be the case, as also reealed by clustering analyses of the IRAS galaxies, showing sall alues of the correlation length copared to optical galaxies. With reference to Fig. 6 Sect.1, the IRAS galaxies hae a -point correlation function like blue optical galaxies. If we assue b IRAS. 0.6, copletely consistent with the IRAS -point correlation function, we get Ω = ( 0.85b ) 1/ In conclusion, the high obsered alues for the β paraeter fro the IRAS galaxies is not a proble anyore (while it has been so in the past). IRAS Weighting the Unierse. The POTENT prograe. As we hae anticipated, studies of uliar elocity fields allow us to robustly constrain the aerage global atter density. We hae discussed in the preious paragraph that considerations of the atter distribution in the local Unierse can be used to predict the uliar otion of the local group, to be copared with easureents of the otion of our reference syste within the CMB photon field. We extend now this analysis to generalize to the elocity fields of galaxies on ery The Peculiar Velocity Field 4. 15

16 large scales. The present analysis has at least three otiations. a) One is to test the graitational instability odel for the origin of the otions. b) The second is to achiee ore precise ealuations of the aerage atter density. c) The third is to get a 3D apping of uliar elocities in the local Unierse to parallel the 3D atter distribution: with such elocity fields we can estiate precise corrections to redshift distance easureents to get the real distances fro stroscopic obserations, with good accuracy. Our reference relation in this analysis is still eq. [4.31], but we attept to use it here to relate the general elocity field with the atter distribution in a large olue of the Unierse. So the ector in [4.31] is the ector field as a function of local coordinates. Of course, the field is not known directly, except for a sall subsaple of galaxies haing distance easureents fro distance indicators. For the bulk of galaxies, distances are still elocity distances (redshift distances and Hubble law). Figure 4.5 A coparison of uliar elocities inferred fro the Tully-Fisher distance easureents (left), with predictions of the uliar elocity field based on the IRAS galaxy density ap, for galaxies within 30 degrees of the super-galactic plane and within 6000 k s -1. The coordinate syste is in Super-galactic coordinates. The odel assues [Dais et al. 1996]. In practice, eq. [4.31] is used in a classical iteratie way. A first guess of the density field is obtained ia Hubble law fro a large stroscopic surey of galaxies, with further constraints on a subset of those based on distance indicators. With this first-guess density odel we get a first-guess uliar elocity odel fro eq. The Peculiar Velocity Field 4. 16

17 [4.9], which is then used to get corrections to the recession elocities hence to the Hubble distances, and so an iproed odel for the density field. And the procedure continues so on until the ariations fro one step to the following in the iteration for both the elocity and density fields becoe sall enough. An exaple of such an analysis is reported in Figure 4.5. On the left panel the uliar elocity field after conergence appears, with distance easureents including those based on the Tully-Fisher ethod, in which the galaxy luinosity is related to its rotational elocity (and, fro luinosity, the luinosity distance is found). On the right panel the elocity field fro the iteration ethod is reported, with a best-fit alue for the beta paraeter as:. Since ost galaxies are concentrated towards the super-galactic plane defined by the local super-cluster, it akes sense to plot the elocity ectors projected onto this plane. Since thousands of galaxies are inoled, the elocity field is also shown aeraged onto a grid. Figure 4.6 The POTENT density field in the Super-galactic plane out to 80 h 1 Mpc, shown as landscape aps obsered fro two different directions. The height of the surface is proportional to. Note the attractors GA (Great Attractor) and PP (Perseus-Pisces super-cluster), and the extended oid in between. The figure shows that, within 5000 k s -1, the elocity field appears doinated by a few distinct regions in which the flow is coherent. The feature that has receied soe particular attention is at coordinates SGX: k s -1, SGY : k s -1. This suggests the existence of a single ass concentration, which has been naed the "Great Attractor" 3. This akes a large deiation fro the Hubble flow. The structure is explained with a large concentration of clusters and super-clusters at a distance range of about 47 to 80 Mpc (the latter being the ost recent estiate) fro 3 Popular reports in the journals years ago of this subject hae soeties gien the ipression of soe ysterious concentration of ass that is detected only through its graitational pull on surrounding galaxies. This is no ore a proble, the Great Attractor has been identified and resoled. The Peculiar Velocity Field 4. 17

18 the Milky Way, in the direction of the constellations Hydra and Centaurus. Objects in that direction lie in the Zone of Aoidance (the part of the night sky obscured by the Milky Way galaxy) and are thus difficult to study with isible waelengths, but X-ray obserations hae reealed that the region of space includes any assie clusters. One of the ost systeatic applications of the aboe concepts hae been achieed with the POTENT prograe (Dekel, Yahil, Burnstein, Faber, and others). Soe results about the density field reconstruction using the elocity fields and all other aailable inforation are reported in Figure The botto line of these analyses is about the alue of Ω. 0.5b± 0.07 based on an optically-selected saple of galaxies (Peacock et al. 001), corresponding to a density paraeter Ω. 0.3, unless the bias of optical galaxies is either ery larger or ery saller than unity, which is unlikely to be the case. The Peculiar Velocity Field 4. 18