The 2M++ galaxy redshift catalogue
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 4 October 2018 (MN L A TEX style file v2.2)
The 2M++ galaxy redshift catalogue
Guilhem Lavaux , & Michael J. Hudson , , Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801-3080 Department of Physics and Astronomy, The Johns Hopkins University, 3701 San Martin Drive, Baltimore, MD 21218, USA Institut d’Astrophysique de Paris, UMR7095 CNRS, Univ. Pierre et Marie Curie, 98 bis Boulevard Arago, 75014 Paris, France Department of Physics & Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1 Canada Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada
ABSTRACT
Peculiar velocities arise from gravitational instability, and thus are linked to thesurrounding distribution of matter. In order to understand the motion of theLocal Group with respect to the Cosmic Microwave Background, a deep all-skymap of the galaxy distribution is required. Here we present a new redshift com-pilation of 69 160 galaxies, dubbed 2M++, to map large-scale structures of theLocal Universe over nearly the whole sky, and reaching depths of K ≤ .
5, or200 h − Mpc. The target catalogue is based on the Two-Micron-All-Sky ExtendedSource Catalog (2MASS-XSC). The primary sources of redshifts are the 2MASSRedshift Survey, the 6dF galaxy redshift survey and the Sloan Digital Sky Sur-vey (DR7). We assess redshift completeness in each region and compute theweights required to correct for redshift incompleteness and apparent magnitudelimits, and discuss corrections for incompleteness in the Zone of Avoidance. Wepresent the density field for this survey, and discuss the importance of large-scalestructures such as the Shapley Concentration.
Peculiar velocities remain the only method to mapthe distribution of dark matter on very largescales in the low redshift Universe. Recently, sev-eral intriguing measurements (Kashlinsky et al. 2008;Watkins et al. 2009; Lavaux et al. 2010; Kashlinsky et al.2010; Feldman et al. 2010) of the mean or “bulk” flowon scales larger than 100 h − Mpc suggest a high veloc-ity of our local ∼ h − Mpc volume with respectto the Cosmic Microwave Background (CMB) frame.In the standard cosmological framework, peculiar ve-locities are proportional to peculiar acceleration and soone expects the bulk flow to arise from fluctuations inthe distribution of matter, and hence presumably ofgalaxies, on very large scales. Another statistic for mea-suring such large-scale fluctuations is the convergenceof the gravity dipole as a function of distance. How-ever, the rate of convergence has been a subject of re-cent debate (Kocevski & Ebeling 2006a; Erdo˘gdu et al.2006a,b; Lavaux et al. 2010; Bilicki et al. 2011, and refer-ences therein). A closely-related topic is the gravitationalinfluence of the Shapley Concentration, the largest con-centration of galaxy clusters in the nearby Universe. Itis therefore important to have catalogues that are as fullsky and as deep as possible to understand whether thedistribution of matter in the nearby Universe may explainthe above-mentioned results.It is already possible with currently available datato build a redshift catalogue significantly deeper than previous full-sky galaxy redshift catalogues like PSCz(Saunders et al. 2000) or the Two-Micron-All-Sky Red-shift Survey (Huchra et al. 2005; Erdo˘gdu et al. 2006a;Huchra et al. 2011, 2MRS). We present here a new cata-logue called the 2M++ galaxy redshift compilation. Thephotometry for this compilation is based is on the Two-Micron-All-Sky-Survey (2MASS) Extended Source Cata-log (Skrutskie et al. 2006, 2MASS-XSC). We gather thehigh-quality redshifts from the 2MASS redshift survey(Huchra et al. 2005; Erdo˘gdu et al. 2006a; Huchra et al.2011) limited to K = 11 .
5, the 6dF galaxy redshift surveyData Release Three (6dFGRS Jones et al. 2009) and theSloan Digital Sky Survey Data Release Seven (SDSS-DR7Abazajian et al. 2009).A summary of this paper is as follows. In Section 2,we describe the steps in constructing the 2M++ red-shift galaxy catalogue: source selection, magnitude cor-rections, redshift incompleteness estimation and correc-tion, the luminosity function (LF) estimation and thefinal weight computation. In Section 3, we discuss theZone of Avoidance (ZoA) in our catalogue, and how itseffects can be mitigated. In Section 4, we define groupsof galaxies and check some of their overall properties. InSection 5, we compute and analyse the density field, pre-senting maps of the Supergalactic plane and three clusterdensity and velocity profiles. Section 6 summarizes ourkey results. c (cid:13) G. Lavaux & M. J. Hudson
In this Section, we describe the construction of the 2M++galaxy redshift catalogue from the different data sources.First, in Section 2.1, we describe the source data setsthat form the basis of our catalogue, as well as the pri-mary steps in the construction of the 2M++ catalogue.We then present the methodology used for merging thesedifferent sources. In Section 2.2, we describe the correc-tions applied to apparent magnitudes to homogenize thetarget selection. In Section 2.3, we test and apply the red-shift cloning procedure to our data to increase the overallredshift completeness. We then estimate redshift com-pleteness (Section 2.4.2) and present the number countsof galaxies as a function of redshift (Section 2.5). Finally,we compute the LF of our sample in Section 2.6 andcompute the total weights to apply to each galaxy inSection 2.7.
Our catalogue is based on the Two-Micron-All-Sky-Survey (Skrutskie et al. 2006, 2MASS) photometric cat-alogue for target selection, which has very high com-pleteness up to K S = 13 . K in place of K S . As notedabove, we will be using redshifts from the SDSS-DR7, the 6dfGRS and the 2MRS. In addition tothese main sources, we have gathered additional red-shifts from a number of other sources (Schneider et al.1990; de Vaucouleurs et al. 1991; Binggeli et al. 1993;Huchra et al. 1995; Falco et al. 1999; Conselice et al.2001; Rines et al. 2003; Koribalski et al. 2004) throughNED queries. Due to the inhomogeneity of the tar-get selection between the different redshift surveys, wethink that it is more appropriate to define a new tar-get selection rather than using existing target databasesfrom the above surveys. We used of the NYU-VAGC(Blanton et al. 2005) catalogue for matching the SDSSdata to the 2MASS Extended Source Catalog (2MASS-XSC). The NYU-VAGC provides the SDSS survey maskin MANGLE format (Hamilton & Tegmark 2004). Wesampled the mask on a
HEALPix grid at N side = 512( ∼
10 arcminutes resolution). This angular resolution cor-responds to ∼ h − Mpc at ∼ h − Mpc. Because ulti-mately we will be smoothing the density field on scalesof ∼ h − Mpc, the mask has sufficient resolution for ourpurposes. Additionally, we filter out from our target se-lection the extended sources that are known not to begalaxies. We aim to limit 2M++ at K ≃ . The NASA/IPAC Extragalactic Database (NED) is oper-ated by the Jet Propulsion Laboratory, California Institute ofTechnology, under contract with the National Aeronautics andSpace Administration. We use the file named lss combmask.dr72.ply , which givesthe geometry of the DR72 sample in terms of target selectionwith bright stars excised. We require that the visual code is not equal to two. to retain as much as possible information from the shal-lower 2MRS catalogue, we opt to follow closely the mag-nitude used by 2MRS for target selection. We define as K the magnitude of a galaxy measured in the K S band, within the circular isophote at 20 mag arcsec − , af-ter various corrections as described below (Section 2.2).Several of the steps taken to build the catalogue are de-scribed in greater detail in the following Sections. We nowoutline these steps:(i) We import the redshift information for 2MASS-XSC galaxies from the NYU-VAGC for SDSS-DR7, the6dF-DR3, and from the 2MASS Redshift Survey.(ii) We correct for small-scale redshift incompleteness(arising from fibre collisions) by ‘cloning’ the redshifts ofnearby galaxies (Section 2.3).(iii) We correct the apparent magnitudes for Galacticdust extinction (Section 2.2).(iv) We use the redshift to correct for galaxy evolution-ary effects and aperture corrections (Section 2.2). We callthis magnitude K . At those magnitudes, we assumethat the photometric completeness is one at Galactic lat-itudes higher than 10 ◦ .(v) We compute two sets of galaxy samples: a tar-get sample with K ≤ . K ≤ . We describe in this Section the corrections that must beapplied to apparent magnitudes to mitigate the effectsof cosmological surface brightness dimming, Galactic ex-tinction and stellar evolution. We choose to use a defini-tion of the magnitudes for target selection that is relatedto the one used for defining magnitudes in the 2MRS.This ensures that the final completeness is maximized inin the parts of the sky where only redshifts from 2MRSare available.The absolute magnitude M at redshift zero of agalaxy may be written as M = m − A K ( l, b ) − k ( z ) + e ( z ) − D L ( z ) (1)with m the apparent magnitude, A K ( l, b ) the absorptionby Milky Way’s dust in the direction ( l, b ), k ( z ) the k -correction due to the redshifting of the spectrum, e ( z ) isthe correction for evolution of the stellar population and c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue SDSS 6dF
Figure 1.
Error distribution due to the redshift cloning procedure – We give here the computed error of the redshifts of eitherSDSS or 6dF redshift catalogue. We removed the redshift information of half of the objects in these catalogues and tried to recoverthem using the cloning procedure. The difference is plotted as an histogram in the two plots above. The overlaid continuous curvecorrespond to a Cauchy-Lorentz distribution with a width equal to 2 . h − Mpc. D L is the luminosity distance. We convert the redshiftsinto luminosity distances assuming a ΛCDM cosmologywith a mean total matter density parameter Ω M = 0 . Λ = 0 .
70. All ab-solute magnitudes are computed assuming H = 100 kms − Mpc − .The absorption in K S band is related to the extinc-tion E B − V estimated using the maps of Schlegel et al.(1998) by the relation A K ( l, b ) = 0 . E ( B − V ) ( l, b ) (2)where the constant of proportionality is obtained fromthe relation between absorption in K band and in V band(Cardelli et al. 1989).The adopted k -correction is k ( z ) = − . z (3)from Bell et al. (2003a). Finally, the evolutionary correc-tion is e ( z ) = 0 . z (4)also from Bell et al. (2003a).The magnitudes adopted in this work are circularaperture magnitudes defined within a limiting surfacebrightness of 20 mag per square arcsec. However, vari-ous effects will cause not only the observed magnitudeto change, but also the observed surface brightness. Asthe surface brightness of the galaxy profile drops, theisophotal aperture will move inwards and so the aperturemagnitude will drop.The surface brightness will depend on redshift viathe usual (1 + z ) cosmological dimming effect as wellas the extinction, k − correction and evolutionary effectsdescribed above. Therefore the correction is∆SB = SB( z = 0) − SB( z )= −
10 log(1 + z ) − A K ( l, b ) − k ( z ) + e ( z ) . (5)Note that the k + e corrections have opposite sign to the cosmological surface brightness dimming, and so there issome cancellation of these effects. However, the cosmo-logical term still dominates, so the net effect is that asgalaxies are moved to higher redshift their surface bright-ness is dimmer.By simulating simple S´ersic profiles, we have esti-mated how much the aperture magnitude changes as aresult of surface brightness dimming. For a typical 2M++galaxy with S´ersic index n = 1 . h µ e i = 17 .
5, we findthat the correction to the magnitude due only to a changein aperture radius can be approximated by 0 . SB is the correction in the surface brightness.This term is only the shift in magnitude due to the shiftin isophotal radius, and does not include the “direct” ef-fect on the magnitude itself due to extinction and k + e corrections. Thus the total effect is: K = K ,c + 1 . − A K ( l, b ) − k ( z ) + e ( z )] − . z )] (6)Note that this is close, but not identical, to the 2MRScorrected magnitude.In some cases, only the magnitude K ,e , derivedfrom adjusting an ellipsoidal S´ersic profile, is available. Inthose cases, we have computed the corresponding K ,c using the following relation, obtained by fitting on thegalaxies for which the two magnitudes were available: K ,c = (0 . ± . K ,e + (0 . ± . . (7)The residual of the fit has a standard deviation equal to0.11. We also use this relation whenever the predicted K ,c and the actual K ,c from 2MASS-XSC differs by0.22 and calculate the K ,c from K ,e . We have usedthis relation for 7% of galaxies, both in the target andthe final redshift compilation. c (cid:13) , 000–000 G. Lavaux & M. J. Hudson
Within the 6dF and SDSS regions, there is small-scale in-completeness due primarily to fibre collisions. To improvethe redshift coverage of the catalogues we “clone” red-shifts of nearby galaxies within each survey region. Thisprocedure, which is related to another one described inBlanton et al. (2005), is as follows. Consider two targets T a and T b . If T a does not have a measured redshift and T b has one, and furthermore T b is the nearest target of T a with a angular distance less than ǫ , we copy the red-shift of T b to T a . ǫ is determined by the angular distancebetween two fibres of the measuring instrument, which is ǫ = 5 ′ . ǫ = 55 ′′ for SDSS(Blanton et al. 2003). We refer to redshifts cloned in thisway as “fibre-clones”.To assess the errors on redshifts for the fibre-clones,we randomly split the set of galaxies which have a mea-sured redshifts in two sets S keep and S test . We mark thegalaxies belonging to S test as having no redshift. We thenapply the fibre cloning procedure to these galaxies.The result of this test is shown in Figure 1for both SDSS galaxies and 6dF galaxies. We notethat the Cauchy-Lorentz distribution with width W =2 . h − Mpc is a good fit to the central part of the twodistributions. We used the formula P ( e ) = 1 πW
11 + ( e/W ) (8)for the modelled probability distribution function in bothpanels. We checked that a Gaussian distribution man-ages only to fit the central part of the distribution andis less adequate than a Cauchy-Lorentz distribution. Thefibre-clones are given a redshift error of 9 × − , whichcorresponds to ∼ . h − Mpc at redshift z = 0. Because of the different redshift catalogues used in2M++, we will separate the full sky into the follow-ing regions: K ≤ . K ≤ . The preliminary mask for the 6dF is as given inJones et al. (2004) : | b | > ◦ and in the southern hemi-sphere δ ≤ ◦ .For SDSS, from the full DR7, we select only the mosthomogeneous and contiguous portion of the redshift sur-vey. To obtain the geometry corresponding to this por-tion, we use the mask computed numerically from theMANGLE file, and impose an additional constraint onthe positions: we keep only galaxies within the region90 ◦ < α < ◦ , to which we add another region at250 ◦ ≤ α < ◦ and δ < ◦ . This selection retainsthe major contiguous piece of the SDSS in the North-ern Galactic cap, while removing the Southern Galactic Figure 2.
Molleweide projection of the SDSS spectroscopicmask in Equatorial coordinates. We removed unconnectedparts from the original mask. Note the presence of small holesin the mask due to both the presence of stars and not exactreconnection of the SDSS plates. This corresponds to an inter-section of the geometry described by lss geometry.ply andour selection criterion described in Section 2.4.
Figure 3.
A Molleweide projection in galactic coordinatesshowing the preliminary masks corresponding to the differ-ent redshift catalogues: the 2MASS Redshift Survey (red); the6dF galaxy redshift survey (green), and the Sloan Digital SkySurvey DR7 (blue). Regions with no redshift data are shownin grey. strips and the small disjoint piece with α ∼ | b | > ◦ , except in the region − ◦ < l < +30 ◦ , wherethe Galactic latitude is | b | > ◦ (Erdo˘gdu et al. 2006a),and excluding the regions covered by 6dF or SDSS. Werefer to this region as 2Mx6S.The combination of the three survey masks is shownin Fig. 3, in Galactic coordinates. The grey area near thegalactic plane is covered by none of the surveys. Thereis a small overlap between the SDSS and the 6dFGRS inthe north galactic cap (in cyan). c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue In order to determine LFs and weights needed for the den-sity field, it is first necessary to assess the completenessof the redshift catalogues on the sky. We will estimatethe redshift completeness in some direction of the skyfor two different magnitudes cuts: K ≤ . K ≤ . Based on the above analysis, we reject from 6dF andSDSS those regions where the completeness at K ≤ . K = 12 . K = 11 . K ≤ . K ≤ .
5, or in6dF or SDSS with K ≤ .
5, and contains 69 160galaxies with redshifts (including fibre-clones). Table 1summarizes the statistics and completeness for the differ-ent regions of the 2M++ compilation. We note that the2M++ compilation redshifts are nearly 90% complete,and so redshift completeness corrections are small.
Within the three regions outlined above, there are a totalof 69 160 galaxy redshifts (including fibre-clones). Fig. 5shows a histogram of all redshifts, as well as the cumula-tive counts starting from redshift z = 0. Conservatively,the catalogue appears totally complete up to z = 0 . ∼ h − Mpc). This is due to our use of the 2MRS forone part of the sky.In Fig. 6, we compare quantitatively the counts inthe 2MRS region with K ≤ . K ≤ .
5. Because our magnitude cor-rections are not precisely equivalent to those used for the2MRS catalogue, the increase of the magnitude cut to
Figure 5.
Redshift distribution of 2M++ galaxies. We show,in grey histogram, the number of galaxies within each redshiftbin δz = 0 . z is given by the red curve. The predictednumber of galaxies given by our fiducial LF given at the endof Section 2.6, is shown in solid green for the cumulative num-ber and in solid blue for the number of galaxies in each bin ofthe grey histogram. The LF has been fit using a subset of thecatalogue for which 5 ,
000 km s − ≤ cz ≤ − and − ≤ M ≤ − Figure 6. . Redshift galaxy distribution in 2M++ for K ≤ . K ≤ . K = 12 . . < ∼ z < ∼ . z ∼ .
05 corresponding to the Shapley Concentration isnot present for the subcatalogue K ≤ .
5, while itis clearly seen in the 6dF subcatalogue ( K ≤ . c (cid:13) , 000–000 G. Lavaux & M. J. Hudson
Figure 4.
The 2M++ compilation in Galactic coordinates. The top panels give the redshift completeness of 2M++ for the limitingmagnitude K = 11 . K = 12 . K ≤ . K ≤ . Table 1.
Summary statistics for the primary regions in the 2M++ compilationRegion m lim Area N m In order to correct for selection effects due to magni-tude limits, it is first necessary to measure the LF. Wetake into account the redshift completeness to measurethe LF of galaxies in the combined catalogue. We usea modified version of the likelihood formalism used tofind Schechter (1976) function parameters, as describedby Sandage et al. (1979). We assume that evolutionaryeffects on the luminosity of galaxies have been accountedfor by Eq. (6). The parametrization adopted is the usual Schechter function:Φ( L ) = n ∗ L ∗ (cid:18) LL ∗ (cid:19) α exp (cid:18) − LL ∗ (cid:19) , (9)with n ∗ the density normalization constant, L ∗ the char-acteristic luminosity break, or equivalently in terms ofabsolute magnitudesΦ( M ) =0 . n ∗ . α )( M ∗ − M ) exp (cid:16) − . M ∗ − M ) (cid:17) = n ∗ Φ ( M ) , (10) c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue Figure 7. Galaxy LF estimates. The left-hand panel shows the non-parametric LF estimated using the 1 /V max method is shownby the solid lines for the regions covered either by 2MRS, 6dF, SDSS or all together. For these plots we use data from 750 km s − to 20 000km s − . The dashed line shows the parametric LF using the likelihood method of Section 2.6 for galaxies with absolutemagnitudes in the range − ≤ M ≤ − 21, for redshift distances 5 000 km s − to 20 000 km s − . The error bars reflect only theuncertainties in galaxy counts and do not include cosmic variance effects. The right-hand panel shows the difference between the1 /V max LFs and the fitted parametric LF. with M ∗ the characteristic absolute magnitude breakin the Schechter function. Above, we have introducedΦ ( M ), which is the unnormalized Schechter function.We model the probability of observing a galaxy of abso-lute magnitude M i given its redshift z i as P ( M i | z i , α, M ∗ , n ∗ , c ) = c ( M i , ˆ u i , d i )Φ ( M i ) R M max M min c ( M, ˆ u i , d i )Φ ( M ) d M (11)with M min , M max the maximum absolute magnituderange from which the galaxies were selected in the cat-alogue, c ( M, ˆ u i , d i ) the completeness in the direction ˆ u i of the object i , at the absolute magnitude M , d i the lu-minosity distance of the galaxy i at redshift z i . This ex-pression is simplified using our assumption that redshiftincompleteness c ( M, ˆ u, r ) may be modelled by two mapsat two apparent magnitude cuts. c ( M, ˆ u, r ) is thus c ( M, ˆ u, r ) = c b (ˆ u ) if M + 5 log (cid:16) r 10 pc (cid:17) ≤ m b c f (ˆ u ) if m b < M + 5 log (cid:16) r 10 pc (cid:17) ≤ m f m b = 11 . m f = 12 . 5. The expression of theprobability (11) may thus be newly expressed as P ( M i | z i , α, M ∗ , n ∗ , c ) = c ( M i , ˆ u i , d i )Φ ( M i ) f ( d i , ˆ u i , M min , M max ) (13)with f ( r, ˆ r, M min , M max ) = c b (ˆ r )Γ M ∗ ,αM min ,M max ( m b , r )+ c f (ˆ r ) (cid:16) Γ M ∗ ,αM min ,M max ( m f , r ) − Γ M ∗ ,αM min ,M max ( m b , r ) (cid:17) , (14)the normalization coefficient for the direction ˆ r at dis-tance r , and r defined as the distance in units of 10 pc. In the above, we have also used the function Γ M ∗ ,αM min ,M max defined asΓ M ∗ ,αM min ,M max ( m, r ) =Γ inc (cid:16) α, . ( M ∗ − min(max( M ( m,r ) ,M min ) ,M max ) ) (cid:17) (15)with the absolute magnitude M ( m, r ) = m − ( r ) (16)and Γ inc ( a, y ) the incomplete Gamma functionΓ inc ( a, y ) = Z ∞ y x a − e − x d x . (17)We write the total probability of observing the galax-ies with intrinsic magnitude { M i } and redshift { z i } giventhe Schechter LF parameters: P ( { M i }|{ c i } , { z i } , α, M ∗ , n ∗ ) = N galaxies Y i =1 P ( M i | z i , α, M ∗ , n ∗ , c i ) . (18)Using Bayes theorem, we now estimate the most likelyvalue taken by α , M ∗ assuming a flat prior on these pa-rameters.The normalization constant n ∗ is determined us-ing the minimum variance estimator of Davis & Huchra(1982), but neglecting the effects of cosmic variance onthe weights by setting J = 0. While our estimate maybe biased relative to the galaxy mean density outside thecatalog, it is less noisy than the optimal case. The esti-mate also corresponds better to the density in the pieceof Universe that we consider than the density correspond-ing to the optimal weighing. Our choice leads also to asimplification of the mean density as the total number ofgalaxies divided by the effective volume of 2M++. Con-sequently, for a survey limited in the absolute magnitude c (cid:13) , 000–000 G. Lavaux & M. J. Hudson range [ M min , M max ] and with volume V we compute themean density of galaxy ¯ n by the following equation¯ n = N galaxies R V d r f ( r, ˆ r, M min , M max ) , (19)with a standard deviation only from Poisson noise σ ¯ n ¯ n = p N galaxies N galaxies . (20)because we have set J = 0. We then convert ¯ n into n ∗ using n ∗ = ¯ n R M max M min Φ ( M ) d M . (21)Similarly it is possible to define the luminosity density¯ L = n ∗ . ⊙ − M ∗ ) Γ(2 + α ) × (1L ⊙ ) , (22)with Γ( a ) = Γ inc ( a, α .To determine the LF parameters, we select a subsetof the galaxies in our catalogue. We have defined thesubset by the joint conditions:- Galaxies must have a redshift z such that 5,000 kms − ≤ cz ≤ − . The lower limit reduces theimpact of peculiar velocities on absolute magnitude esti-mation, which is derived using redshifts in the CMB restframe. By limiting to cz ≤ , 000 km s − , we avoid moredistant volumes with high incompleteness.- The absolute magnitude estimated from the red-shift in CMB rest frame is within the range [ M min = − , M max = − H = 100 h kms − Mpc − with h = 1 and we have assumed a flatΛCDMcosmology Ω m = 0 . 30 and Ω Λ = 0 . 70. We donot distinguish between early-type and late-type galax-ies, and so fit both populations with a single parameter. The derived LF parameters are summarized in Table 2 forour default choice of cuts discussed above as well as forother choices that we discuss below. The error-bars aregiven at 68% confidence limit, estimated using a Monte-Carlo-Markov-Chain method.Fig. 7 shows the LF for our default cuts in the CMBrest frame. We also show, for the entire data set and foreach subcatalogue, the non-parametric LFs estimated us-ing the unbiased 1 /V max method (Schmidt 1968; Felten1976). Note that for the 1 /V max LFs the volume and mag-nitude limits are different than for the parametric fit,which explains that the fitted parametric faint-end slopeis not a good fit to the 1 /V max . in the range [ − , − Table 2 also lists LF parameters from previous 2MASSstudies. Our fitted LF parameters are in agreementwith previous studies of the K-band LF (Bell et al.2003b; Eke et al. 2005) but are somewhat different thanthose found by Kochanek et al. (2001), Cole et al. (2001),Huchra et al. (2005) and Jones et al. (2006).The derived LF parameters are sensitive to a numberof systematic effects: the magnitude range used, the rest-frame used for the redshifts, and the fitting method itself.The Schechter function itself appears not to perfectfit over the whole range of magnitudes. Consequently,the fitted parameters depend in the magnitude (and dis-tance) range of the galaxies used in the fit. Our de-fault minimum distance r ≥ − correspondsto M K . − 21 for K = 12 . 5. However, the 1 /V max method seems to indicate an inflection in the LF at M K ∼− 21. This bend is also seen by Bell et al. (2003b) andEke et al. (2005). Indeed Bell et al. (2003b) attempted tofit the part at M K > − 21 with a power-law instead of aSchechter function.) Several studies (Biviano et al. 1995;Yagi et al. 2002) have noted a dip in the LF of clustergalaxies at a similar location (approximately 2 magni-tudes below M ∗ ), although other studies suggest that itis a flattening rather than a dip (Trentham 1998). In anycase, it seems clear that the choice of magnitude rangewill affect the Schechter LF parameters. In Jones et al.(2006) and Cole et al. (2001), the magnitude range usedin the fit is fainter than our default.A second issue, which arises when using galaxies withvery low redshifts, is the choice of flow model or rest-frame redshifts. Very nearby galaxies are likely to sharethe peculiar velocity of the Local Group (LG), so the red-shift in the LG frame is a better proxy distance than theCMB-frame redshift. For better understanding of the de-pendence of our results on both local flows and clustering,we have fit the parameters of the Schechter function intwo rest frames (CMB or LG). We find that, for samplesextending to M K ∼ − 17, the faint-end slope α steepens,but only by 0.06.Finally, the magnitudes, correct and the fittingmethod itself are probably the most important system-atics.(i) We note that the studies of Cole et al. (2001),Eke et al. (2005) and Bell et al. (2003b) are based onKron magnitudes, and that of Jones et al. (2006) is basedon total magnitudes, leading to a possible difference in M ∗ of 0 . ± . 04, as discussed by Kochanek et al. (2001).(ii) Another notable difference is that Jones et al.(2006) have tried to integrate the effect of uncertaintieson the determination of magnitudes, which we do not dohere.(iii) Bell et al. (2003b) matches SDSS redshifts to boththe 2MASS XSC and the PSC catalogues. Bell et al.(2003b) argues that selection effects bias the raw 2MASSLF compared to the true LF. However, whereas thoseauthors were interested in, for example, the total stellarmass density in the nearby Universe, our goal is rathera consistent magnitude system coupled with uniform se-lection across the sky. Since our primary method will beto weight by luminosity, the small missing contribution c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue Table 2. Summary of K-band Schechter LF parameters from this paper and selected results from the literature. Magnitude rangeswith a ∼ are estimated. n ∗ is in units of (10 − h Mpc − ) and the luminosity density ¯ L is in units of 10 h L ⊙ Mpc − , assumingM K, ⊙ = 3 . α M ∗ − h n ∗ ¯ L Kochanek et al. (2001) [ − − 20] [2000; 14000] − . ± . − . ± . 05 1 . ± . ∼ [ − − 20] ? − . ± . − . ± . 03 1 . ± . ∼ [ − − 18] ? − . ± . − . ± . 05 1 . ± . ∼ [ − − 20] ? − . ± . − . ± . 04 1 . ± . ∼ [ − . − 16] ? − . − . . − . − . 5] [750; + ∞ ] − . ± . − . ± . 03 0 . ± . This work CMB [ − − 21] [5000; 20000] − . ± . − . ± . 02 1 . ± . 02 3 . ± . − − 17] [750; 20000] − . ± . − . ± . 01 1 . ± . 02 4 . ± . − − 17] [750; 20000] − . ± . − . ± . 01 1 . ± . 02 4 . ± . − − 21] [5000; 20000] − . ± . − . ± . 01 1 . ± . 02 4 . ± . | b | > K < . − − 17] [300; 20000] − . ± . − . ± . /V max fit CMB [ − − 21] [750; 20000] − . ± . − . ± . 01 0 . ± . 06 4 . ± . from low surface brightness galaxies and the low surfacebrightness regions of catalogued galaxies is of little con-cern to us.(iv) The fitting method itself may also make a differ-ence. Our default parametric fit is pinned to the mag-nitude range where the formal Poisson errors are small-est, namely [ − , − /V max method, addedin quadrature the statistical error bars and the fluctua-tions from the different subcatalogues, and fitted thesedata with a Schechter LF. As indicated in Table 2, wehave obtained a steeper faint end slope and a brighter M ∗ , which are in better agreement with Kochanek et al.(2001), Cole et al. (2001) and Huchra et al. (2005), butstill discrepant with Jones et al. (2006).We conclude that, given all of these systematics, ourLFs are reasonably consistent with those that have beenfound previously. One aspect which can be improved ispeculiar velocity corrections, but we postpone a fully self-consistent treatment of peculiar velocities and the LFdetermination to a future paper.We confirm that the bright end part of the LF doesnot seem to follow a Schechter LF, as already seen bythe 6dfGRS (Jones et al. 2006). This effect is clearly seenin the SDSS, 2MRS and 6dfGRS subsamples separately.The deviation becomes significant at M K . − 25, ortwo magnitudes brighter than M ∗ , and is presumablydue to brightest cluster galaxies, which have typical K -band magnitudes of ∼ − 26 (Lin & Mohr 2004) and havelong been known to deviate from the extrapolation of aSchechter function Tremaine & Richstone (1977).We may check the consistency of this LF with thenumber of galaxies in the 2M++ catalogue. We predictthat the total number of galaxies of redshifts betweenthe distances r min ( z min ) and r max ( z max ), assuming the Schechter LF, is N = Z M max M min Φ( M )d M Z r max r min d r f ( r, ˆ r, M min , M max ) . (23)We plot this function as a solid green line in Fig. 5. Wealso show the predicted number of galaxies in each binof the grey histogram by a solid blue line. We see thatthe prediction in each redshift bin agrees well with theobserved number of galaxies, but the total is off by ∼ cz ≥ , 000 km s − and the high luminosity partfor which objects are not following a Schechter LF, as inFig. 7.In Table 2, we also give the mean luminosity density¯ L as derived from Eq. (22). ¯ L is a lot less sensitive than¯ n to the faint end of the luminosity function. As before,the errors are dominated by systematics due to the differ-ent corrections from peculiar velocities and the adequacyof the Schechter function to fit the observed luminosityfunction. Taking the average and computing the disper-sion in values for ¯ L for the four tests indicated in Table 2yields ¯ L = (4 . ± . h L ⊙ . Using the LF, we may now compute the appropriateweights to give to observed galaxies to account for incom-pleteness of the redshift catalogue. Our long-term goalis to reconstruct the dark matter density, under the as-sumption that galaxies trace the dark matter. There areseveral ways to link the galaxy density to the dark matterdensity: assuming that there is a linear relation betweenthe two fields, one might consider number-weighting,in which the DM density is assumed to be related tothe number-density of galaxies, or luminosity-weighting,which can serve as proxy for stellar mass, and so maybe a better tracer of DM density. We will consider bothof these schemes here. More complicated relationships, c (cid:13) , 000–000 G. Lavaux & M. J. Hudson for example based on a halo model (Marinoni & Hudson2002), will be considered in a future paper.We compute number-weighting based on the fractionof observed galaxies: f N observed ( r , M min , M max ) = N observed ( r ) N average = f ( r, ˆ r, M min , M max ) R M max M min Φ ( M ) d M . (24)The weight applied to each galaxy is then 1 /f N observed ( r ).This procedure is common and has been used previously(e.g. Davis & Huchra 1982; Pike & Hudson 2005)We follow a similar procedure for correcting the localluminosity density of galaxies by estimating how muchlight we are missing at the distance of each galaxy locatedat position r . The fraction f L observed ( r ) = L observed ( r ) L average =1 L average (cid:16) ( c b − c f )(ˆ r )Γ M ∗ , αM min ,M max ( m b , r ) + c f (ˆ r )Γ M ∗ , αM min ,M max ( m f , r ) (cid:17) (25)of luminosity with L observed is the mean luminosity ex-pected to be observed in a small volume at position r ,ˆ r = r / | r | and L average the mean luminosity emitted bygalaxies in the Universe. The value of L average is L average = Z M max M min L ( M )Φ ( M ) d M. (26)The weight to apply to each intrinsic luminosity of agalaxy is then 1 /f L observed ( r ). This procedure has alreadyalso been used with success with observation and mockcatalogues (e.g. Lavaux et al. 2008, 2010; Davis et al.2011).For our choice of absolute magnitudes, M min = − 25 and M max = − 20, the 2M++ is volume lim-ited up to r min ∼ h − Mpc, and extends up to r max =300 h − Mpc. We find that at a distance of ∼ h − Mpc, the galaxy number weights are typicallybetween 10 and 400, depending on whether the region islimited to K ≤ . K ≤ . 5, respectively.Similarly, the luminosity weights range between ∼ ∼ 40. So weighing by luminosity has the advantage thatit is less noisy at large distances. The “Zone of Avoidance” (ZoA) is the region of theGalactic plane where observations of galaxies are difficultdue to the extinction by Galactic dust and stellar confu-sion. We show in Fig. 8, the number of galaxies in 2M++with K ≤ . 5, in bins of sin( b ), and corrected forincompleteness effects. We see that the distribution isclose to flat as a function of Galactic latitude, except fora hole contained between the latitudes − ◦ ≤ b ≤ ◦ .We define the ZoA in 2M++ as this band for galacticlongitudes − ≤ l ≤ 30, but reduce it to 5 ◦ outside thisrange. In addition, we impose the constraint that the ab-sorption not to exceed A K = 0 . 25 in regions devoid ofgalaxies. Figure 8. The effect of the ZoA on 2M++. The weighednumber density of galaxies in each bin of sin( b ) is shown bythe thin solid histogram. The dashed line shows the numberdensity of galaxies once ZoA is filled with cloned galaxies. Thetwo thick vertical lines correspond to b = ± ◦ . Here we usedonly galaxies for which cz ≤ , 000 km s − . In order to reconstruct the density field over the fullsky, it is clearly necessary to fill the ZoA. One optionis to fill it with mock galaxies so that their density ofthese objects matches the mean density outside the ZoA.This option, would however, fail to interpolate large-scalestructure observed above and below the ZoA. The optionadopted here, following Lynden-Bell et al. (1989), is to“clone” galaxies immediately above and below the ZoA.The procedure of creating a galaxy clone at a latitude b c of a galaxy at latitude b is simply to shift the latitudesin( b c ) = sin( b zoa ) − sin( b ) (27)where b zoa = (cid:26) sign( b ) × ◦ if | b | > ◦ and | l | > ◦ sign( b ) × ◦ if | b | > ◦ and | l | < ◦ (28)We refer to these as “ZoA-clones”. Fig. 8 shows the dis-tribution of galaxies as a function of Galactic latitudebefore and after cloning. After cloning, the distributionshows no dependence on latitude. We use redshifts to estimate galaxy distances, but in thepresence of peculiar velocites this relationship is not per-fect. In addition to the so-called “Kaiser (1987) effect”which affects very large scales, there is also a contami-nation by the “Finger-of-god effect” due to the velocitydispersion of galaxies in clusters of galaxies. This causesa significant amount of noise on the redshift-estimateddistance. One way to deal with the problem is to groupgalaxies, which by simple averaging improves the dis-tances estimated from the redshifts. The grouping in-formation is also interesting to study the statistics andproperties of galaxy groups.In this Section, we describe the algorithm used toassign galaxies to groups and clusters. We use this infor-mation in the next sections for deriving a better density c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue Figure 9. Number of group members (richness) as a functionof redshift. The richness is is not corrected for incompleteness. Figure 10. Group luminosity as a function of redshift. Theluminosities are not corrected for incompleteness. field (Section 5.1) and, in a future work, peculiar veloci-ties. Grouping also allows a better determination of thecenter of mass of superclusters (Section 5.2) and their in-fall pattern (Section 5.3). As a byproduct of 2M++, weprovide the a catalog of groups and their properties inAppendix B. To assign galaxies to groups we use the standard per-colation, or “Friends-of-friends” algorithm developed byHuchra & Geller (1982). The algorithm is designed toidentify cone-like structures in redshift space. Two galax-ies are considered to be part of the same group if:- their estimated angular distance separation is lessthan D sep ,- their apparent total velocity separation separation isless than V . V is kept fixed for the whole volume of the catalogue. D sep is adapted such that the detected structures arealways significant compared to the apparent local number Group name cz min cz max θ sep l b Virgo −∞ − ◦ 279 74Fornax −∞ − ◦ 240 -50 Table 3. Parameters for manual grouping of galaxies. Allgalaxies which are in the direction ( l, b ) and within the red-shifts [ z min ; z max ] with a maximum angular separation to ( l, b )equal to θ sep , are considered part of the group indicated in thefirst column. density of galaxies, by explicitly accounting for selectioneffects. The constraint of a constant local overdensity ata redshift distance z leads to D sep = D R + ∞ L min ( z ) Φ( L ) d L R + ∞ L min ( z F ) Φ( L ) d L ! − / , (29)with L min ( z ) the minimum absolute luminosity observ-able at redshift z , Φ( L ) the galaxy LF, z F the fiducialredshift, D the selection angular distance at fiducial red-shift. The parameter D is linked to the sought overden-sity δ overdensity for group detection by the relation δ overdensity = π D Z + ∞ L min ( z F ) Φ( L ) d L ! − − . (30)This density is computed at the fiducial redshift distance z F . We have chosen the following parameters for definingour groups: V F = cz F = 1 , 000 km s − and δ overdensity =80. These parameters have been used in previous studies(Ramella et al. 1989). With the LF, for our choice of fidu-cial parameters, we compute that the transverse linkinglength is D = 0 . h − Mpc. We count 4 002 groups withthree or more members within 2M++, for redshift dis-tance less than 20,000 km s − . We do not group galaxiesfarther than 20,000 km s − where the catalogue becomessparse. For the very nearby Virgo and Fornax clusters,the FoF algorithm fails and so we manually assign galax-ies to nearby clusters according to the parameters givenin table 3. In Figure 9, we plot the richness of detected groups as afunction of redshift. In Figure 10, we have plotted thetotal luminosities of the same groups. Finally, in Fig-ure 11, we give the velocity dispersion of the galaxieswithin these groups. As expected the mean velocity dis-persion does not vary significantly with distance, and hasa mean value of ∼ 95 km s − . The richness is approxi-mately constant up to ∼ h − Mpc, which is a designfeature of the group finder. The minimal luminosity of thegroups increases with distance, as we are losing the fainterobjects at larger distances because the 2M++ catalogueis limited in apparent magnitude. The catalogue of groupproperties is given in Appendix B. We have checked thatthe parameters of the fitted Schechter LF do not changesignificantly after the grouping of the galaxies. c (cid:13) , 000–000 G. Lavaux & M. J. Hudson Figure 13. The 2M++ galaxy distribution and density field in three dimensions. The cube frame is in Galactic coordinates. TheGalactic plane cuts orthogonally through the middle of the back vertical red arrow. The length of a side of the cube is 200 h − Mpcand is centred on Milky Way. We highlight the iso-surface of number fluctuation, smoothed with a Gaussian kernel of radius1,000km s − , δ L = 2 with a shiny dark red surface. The position of some major structures in the Local Universe are indicatedby labelled arrows. We do not show isosurfaces beyond a distance of 150 h − Mpc, so Horologium-Reticulum is, for example, notpresent. Figure 11. Group velocity dispersions as a function of red-shift. The thick solid line indicates the trend of the evolutionof the average velocity dispersion with redshift. The scale ofthe variation is ∼ . h − Gpc, far larger than the depth ofthe 2M++ catalogue. In this Section, we consider some properties of the peaksin the three-dimensional density field obtained from thedistribution of galaxies in the 2M++ galaxy redshift cat-alogue. We assume that the number density and luminos-ity density of galaxies follow a Poisson distribution. Assuch, the mean smoothed density contrast ρ ( x ) given thegalaxy weights w i is ρ ( s ) = 1¯ ρ N galaxies X i =1 W ( s − s i ) w i (31)and the standard deviation σ ρ ( s ) = 1¯ ρ N galaxies X i =1 W ( s − s i ) w i , (32)with s the coordinate in redshift space, W ( x ) the smooth-ing kernel considered. To compute the position of thepeaks in this density field, we use an iterative sphericaloverdensity algorithm:(i) we initialize the algorithm with an approximation x of the expected position of the cluster; c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue Figure 12. The 2M++ number density field in Supergalacticplane. The density field is smoothed with a Gaussian kernel of1,000 km s − radius. Colour contours show the overdensity inunits of the mean density and are separated by 0 . (ii) we compute the barycenter x N +1c of the set ofgalaxies contained in a sphere centred on x N c and withradius R N ;(iii) we iterate (ii) until convergence, setting R N +1 = R N ;(iv) we reduce R N +1 = 0 . R N . If R N +1 > h − Mpc,then we go back to step (ii), in the other case we termi-nate the algorithm.We define the position of the structure as the one givenby the last step in the above algorithm. This position isused in the following sections to compute mean densitiesand infall velocities on clusters. Fig. 12 shows the galaxy number density field of our cata-logue in the Supergalactic Plane, smoothed to 10 h − Mpcwith a Gaussian kernel. The Shapley concentration(SC) in the upper-left corner, near (SGX , SGY) ≃ ( − , − , is particularly prominent and isthe largest density fluctuation in the 2M++ catalogue.The Shapley region is covered by the 6dF portion ofthe survey which extends to a depth K ≤ . K ≤ . 5. Whensmoothed with a Gaussian kernel of 10 h − Mpc radius,the Shapley concentration peaks at ( l, b ) = (312 , 30) and d = 152 h − Mpc with a density 1 + δ g = 8 . ± . 46, ingalaxy number density contrast, and 1 + δ L = 9 . ± . , SGY) ≃ (5000 , − − . Its highest redshift space density, smoothed with Gaussian kernel of 10 h − Mpc radius, isabout 1 + δ g = 4 . ± . 18 in terms of number densitycontrast, 1 + δ L = 4 . ± . 20 in terms of luminosity den-sity contrast. The position of the peak corresponds tothe Perseus cluster at ( l, b ) = (150 , − 13) , which is quitenear the ZoA, and a distance of 52 h − Mpc. It is quitepossible that the filling of the ZoA by galaxies clonedfrom the Perseus itself amplifies the overdensity of thissupercluster.The extended overdense structure in the central partof the Supergalactic plane, at about (SGX , SGY) ≃ ( − , 0) km s − , is the Hydra-Centaurus-Virgo (HC)supercluster. At 10 h − Mpc smoothing scale, the high-est peak, located at ( l, b ) = (302 , d = 38 h − Mpc,coincides with the Centaurus cluster and has a heightof 1 + δ g = 3 . ± . 08 in number density contrast and1 + δ L = 3 . ± . 14 is luminosity density contrast.Finally, Fig. 13 is a three-dimensional representationof the catalogue in Galactic coordinates, which meansthe Galactic plane goes through the middle of the ver-tical sides of the box, near the Norma cluster. We plotthe 2M++ galaxies as points. Strong overdensities arehighlighted by a transparent dark-red iso-surface of den-sity fluctuation of luminosity δ L = 2. This density hasbeen smoothed at 10 h − Mpc with a Gaussian kernelfrom the corrected number distribution. The Shapley su-percluster is located at the top-left corner of the cube.A number of overdensities in the right part of the cubearise from the high weights, as this region has a depth ofonly K ≤ . We show in Fig. 14 the mean overdensity and ex-cess mass within a sphere of 50 h − Mpc for fourimportant superclusters in the 2M++ catalogue: theShapley concentration, the Perseus-Pisces supercluster,the Horologium-Reticulum (HR) supercluster centred at( l, b ) = (265 , − 51) and a distance of 193 h − Mpc andthe Hydra-Centaurus supercluster. The profiles are cen-tred on the position where the density peaks for eachsupercluster.For the four superclusters, we note that the pro-files obtained through number weighing and luminosityweighing are nearly equivalent. The bumps in the meandensity, shown in the left panels, are reproduced in bothweighing schemes. This is particularly striking for the PPsupercluster, even for scales as small as 10 h − Mpc. In allcases, the luminosity weighted contrast is slightly lowerthan the number weighted density contrast. In the follow-ing discussion, we adopt the luminosity-weighted numbercontrast.The excess masses of all superclusters converge atradii of ∼ h − Mpc. While Shapley is the most mas-sive supercluster, we find that HR is very similar whenmeasured on scales of 50 h − Mpc. Both have massesclose to 10 h − M ⊙ . The PP and HC superclusters areless massive, but, being considerably closer, these havemore impact in the motion of the LG and nearby galax-ies, as we discuss below.Our estimate of Shapley’s mass and density contrast c (cid:13) , 000–000 G. Lavaux & M. J. Hudson (cid:0) Mpc)0246810 + (cid:1) ( r ) ShapleyHorologiumPerseusHydra-Centaurus (cid:2) Mpc)10 M ( < r ) ShapleyHorologiumPerseusHydra-Centaurus Figure 14. The cumulative average density profile and the excess mass as a function of radius from four major superclustersin the 2M++ redshift catalogue. In the two panels, we both show the profiles computed using the number weighed (solid lines)and the luminosity weighed (dashed lines) scheme. In the left panel, the horizontal black dashed line corresponds to the meandensity. In the right panel, the dotted lines indicates the mass of a sphere of the given radius at the mean density. Note thatsince we are plotting excess mass, to obtain the total mass one must add this value. The black, dark grey, light grey and red linescorrespond respectively to the Shapley concentration, the Horologium-Reticulum supercluster, the Perseus-Pisces supercluster andthe Hydra-Centaurus supercluster. The error bars are estimated assuming that galaxies follow Poisson distribution for samplingthe matter density field, as given by Eq. (32). is similar to that of Proust et al. (2006) who measureda density contrast δ n = 5 . ± . h − Mpc witha volume equivalent to a sphere of effective radius 30.3 h − Mpc. In a sphere of this radius centred on Shapley,we find a luminosity density contrast of δ K,L = 4 . ± . . ± . × h − M ⊙ within a sphere of35 h − Mpc. On the same scale, we obtain a mass of4 . ± . × h − M ⊙ , assuming Ω m = 0 . b K,L = 1 for K -band luminosity. Using similar argu-ments, Sheth & Diaferio (2011) quote a mass of 1 . × h − M ⊙ within a slightly smaller radius of 31 h − Mpc.On the same scale we find 4 . ± . × h − M ⊙ .These values could be brought into rough agreementif luminosity-weighted 2MASS galaxies are strongly bi-ased, with b = 2 – 3. Such a strong biasing would,however, conflict with the measurement b K,n = 1 . ± . m / . . by Pike & Hudson (2005) but may bemarginally consistent with the lower value b K,n = 1 . ± . m / . . found recently by Davis et al. (2011).A further caveat is that our density estimates arein redshift-space, and so are enhanced by a factor up to b s = 1 . compared to the estimates ofMu˜noz & Loeb (2008) and Sheth & Diaferio (2011). Ina future paper, we will reconstruct the density field inreal-space and calibrate the biasing factor directly usingpeculiar velocity data, so a detailed comparison of over-densities awaits future work.Hudson et al. (2004) studied the overdensity of theSC as traced by IRAS-selected galaxies. Within a 50 b s = p f/ f / 5, with f = 0 . (cid:3) Mpc)10 v i n f a ll ( k m . s (cid:4) ) ShapleyHorologiumPerseusHydra-Centaurus Figure 15. The infall velocities as a function of distancefor four major superclusters in the 2M++ redshift catalogue.Curves and error bars are as in Fig. 14. h − Mpc-radius sphere they found that the overdensityof IRAS-selected galaxies is only 0.2. Here we find thatthe overdensity of 2MASS-selected galaxies on the samescale is ∼ 1. Clearly, the relationship between IRAS and2MASS-selected galaxies is not well-described by a rela-tive linear bias, since a value of ∼ ∼ c (cid:13) , 000–000 he 2M++ galaxy redshift catalogue Supercluster Sphere centre 1 + δ L Massr l bh − Mpc ◦ ◦ h − M ⊙ Shapley 152 312 30 2 . ± . 05 8 . ± . − 51 2 . ± . 10 8 . ± . − 13 1 . ± . 03 6 . ± . . ± . 03 6 . ± . Table 4. Luminosity density contrast and estimated masses (assuming Ω m = 0 . b K,L = 1) of the four superclusters fromthe distribution of galaxy light within a sphere of radius 50 h − Mpc. We now discuss the impact these structures have on large-scale flows in the nearby Universe. We have estimated theinfall velocity onto each of these structures using lineartheory: v infall = 13 βH ¯ δ ( R ) R, (33)with H the Hubble constant, β ≡ f/b where f is thelinear density perturbation growth rate and b is a biasingparameter, and ¯ δ ( R ) the mean density inside a sphere ofradius R and centred on the supercluster. For a ΛCDMcosmology, f ≃ Ω / (Bouchet et al. 1995). We use β =0 . m ≃ . 30 and Ω Λ = 1 − Ω m with b = 1.Fig. 15 shows the infall velocity profiles of the foursuperclusters. Although we plot the linear theory infalldown to small radii ( R < ∼ h − Mpc), we note that lin-ear theory does not apply in these regions and focus thediscussion on distances R > ∼ h − Mpc. The infall veloc-ities at 10 h − Mpc are all at least 2,000 km s − , withShapley having the highest infall at nearly 4,000 km s − .At 50 h − Mpc, the Shapley and HR superclusters havean infall of ∼ 800 km s − . The average overdensity of theShapley concentration within a sphere of 50 h − Mpc is1 + δ L = 2 . ± . 05. Neglecting structures beyond 50 h − Mpc, linear theory implies that the supercluster isresponsible for attracting the LG with a peculiar veloc-ity of 90 ± − . This motion represents ∼ 15% ofthe total velocity of the LG with respect to the CMBrest frame. Although the excess mass of HR is similar toShapley, its effect on the LG’s motion is less than thatof Shapley due to its greater distance: we estimate 60km s − . Added vectorially, the net peculiar velocity fromthese two superclusters is approximately 110 km s − to-wards ( l, b ) = (297 , − − bulk flow foundby Watkins et al. (2009), but is lower in amplitude.Because they are closer to the LG, the HC and PPsuperclusters have a greater impact. Approximating HCas a sphere, the infall at the LG’s distance of 38 h − Mpcis 588 ± − . Whereas PP is denser, its greaterdistance of 52 h − Mpc puts it on the losing side of thegravitational tug-of-war with HC: the infall of the LGtowards PP is only 313 ± 24 km s − . Thus the net motionis towards HC.Note that it is likely that underdense regions also contribute a push. Kocevski & Ebeling (2006b) havenoted the deficit of rich clusters in the Northern sky, par-ticularly in the distance range 130 to 180 h − Mpc. Thusa full analysis of peculiar velocities requires integrationover the entire density field, a topic we defer to a laterpaper. We have compiled a new, nearly full-sky galaxy redshiftcatalogue, dubbed 2M++, based on the data from threeredshift surveys: the 2MASS Redshift Survey ( K ≤ . h − Mpc. Themost prominent structure within 200 h − Mpc is theShapley Concentration: its luminosity density within asphere of radius 50 h − Mpc is 2.05 times the mean, andis thus responsible for approximately 90 km s − of theLG’s motion with respect to the CMB rest frame. Wehave compared the density profile of four massive su-perclusters that are present in the 2M++ catalogue: theShapley Concentration, the Perseus-Pisces superclusterand the Horologium-Reticulum supercluster and Hydra-Centaurus. 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(km s − ) (km s − ) (km s − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)07345116-6917029 281.00 -21.54 7.90 1367 1486 69 4996 1.0 1.0 0 0 0 0 1 20096dF.....21100305-5448123 342.30 -41.63 12.31 18717 18571 0 1.0 0.9 0 0 0 0 1 20096dF.....20353522-4422308 356.19 -36.74 11.44 7066 6893 198 4388 1.0 1.0 0 0 0 0 1 20096dF.....13271270-2451409 313.15 37.30 12.04 12132 12429 0 1.0 1.0 0 0 0 0 1 20096dF.....21112498-0849375 41.49 -35.05 12.37 8296 7988 0 4177 1.0 0.9 0 0 0 0 1 20096dF.....02581778-0449064 182.17 -52.44 10.50 9235 9036 0 3733 1.0 0.9 0 0 0 0 1 20096dF.....14303940+0716300 357.34 59.22 9.97 1370 1601 10 1.0 0.8 0 0 0 1 0 1998AJ......00362801+1226414 117.21 -50.26 11.44 10339 9993 11 0.8 0.0 0 0 1 0 0 20112MRS....18490084+4739293 77.11 20.08 10.40 4671 4536 10 0.8 0.0 0 0 1 0 0 1991RC3.9...23054906-8545110 305.13 -30.91 12.45 19770 19788 270 1.0 0.7 0 1 0 0 1 none07511205-8540159 298.28 -25.94 12.44 25990 26053 270 1.0 0.6 0 1 0 0 1 none03355460-8537067 299.59 -30.39 11.90 12702 12736 270 3775 1.0 0.7 0 1 0 0 1 none08423963-8430223 297.59 -24.47 11.70 12284 12357 270 4039 1.0 1.0 0 1 0 0 1 none08431792-8429053 297.58 -24.44 12.35 12284 12357 270 4039 1.0 1.0 0 1 0 0 1 none08424060-8427453 297.55 -24.44 11.96 12284 12357 270 4039 1.0 1.0 0 1 0 0 1 noneZOA0000000 330.07 -0.41 10.21 5238 5387 0 0.9 0.7 1 0 0 0 1 zoaZOA0000001 330.49 -0.33 11.31 8841 8989 0 0.9 0.6 1 0 0 0 1 zoaZOA0000002 330.01 -1.18 11.96 11830 11983 0 1.0 0.9 1 0 0 0 1 zoaZOA0000003 331.29 -1.10 11.09 3158 3306 70 0.9 0.8 1 0 0 0 1 zoa07243410-8543223 298.20 -26.43 11.51 5301 5361 69 4638 1.0 0.7 0 0 0 0 1 20096dF.....03403012-8540119 299.56 -30.29 12.41 12714 12749 66 3775 1.0 0.6 0 0 0 0 1 20096dF.....07400785-8539307 298.20 -26.13 11.09 5184 5246 11 4638 1.0 0.7 0 0 0 0 1 2008ApJ.....03355460-8537067 299.59 -30.39 11.90 12702 12736 270 3775 1.0 0.7 0 1 0 0 1 none07420104-8525161 297.96 -26.04 10.31 5150 5213 14 4638 1.0 0.7 0 0 0 0 1 20096dF.....02090195-8520255 301.12 -31.51 11.52 12675 12699 11 3776 1.0 0.9 0 0 0 0 1 20112MRS.... Table A1. The 2M++ catalogue – Col. (1): the name of the galaxy as given in the 2MASS-XSC database. Col. (2): Galactic longitude in degrees. Col. (3): Galactic latitude in degrees.Col. (4): Apparent magnitude in band K S as defined in Section 2.2. Col. (5): Heliocentric total apparent velocity. Col. (6): Total apparent velocity in CMB rest frame, using relationfrom Kogut et al. (1993) and Tully et al. (2008). Col. (7): Total apparent velocity error (equal to zero if not measured). Col. (8): Unique group identifier obtained from the algorithm ofSection 4. Col. (9): Redshift incompleteness at magnitude K ≤ . 5. Col. (10): Redshift incompleteness at magnitude K ≤ . 5. It may be empty in that case the catalogueis limited to K ≤ . c (cid:13) R A S , M N R A S , h e M ++ g a l a x y r e d s h if t c a t a l o g u e Id l b K N galaxies V helio V CMB σ V (deg.) (deg.) (km s − ) (km s − ) (km s − )(1) (2) (3) (4) (5) (6) (7) (8)1 281.26 73.47 6.60 65 1223 1556 5991000 182.41 -13.06 10.65 4 3 -25 971001 91.75 51.01 6.15 4 691 752 511002 137.70 12.33 5.39 9 1139 1055 1181003 316.23 -10.88 10.42 3 -80 17 211004 123.60 74.51 3.80 9 -249 -33 1561005 184.67 83.05 4.96 24 675 956 3121006 144.05 66.22 3.59 69 845 1049 2271007 171.51 32.81 8.58 4 506 646 461008 108.51 58.06 5.83 6 186 302 1091009 319.07 -12.21 4.75 4 -146 -65 891010 144.70 36.20 3.55 5 92 157 821011 33.58 14.01 8.21 4 1865 1780 1011012 41.38 14.94 8.00 3 2291 2188 121013 134.90 32.70 6.94 4 1330 1348 511014 103.67 33.03 8.10 4 1174 1129 631015 41.06 12.68 9.01 3 2741 2626 931016 45.04 17.71 8.24 3 2261 2162 601017 150.98 5.93 8.69 4 5096 5028 1121018 147.84 7.90 8.10 6 4807 4736 2371019 129.32 8.92 7.37 5 3276 3146 2031020 103.21 12.39 7.94 5 2692 2523 1311021 50.91 6.89 8.34 4 4831 4659 1931022 69.13 8.13 8.96 4 4556 4359 811023 75.62 6.03 8.22 4 4711 4497 941024 269.12 5.57 9.91 3 5052 5324 751025 264.14 7.22 8.89 4 4685 4965 2341026 275.43 8.94 9.48 3 4008 4291 881027 264.36 8.40 8.29 7 4796 5081 1211028 295.46 8.81 8.36 4 4449 4701 113 Table B1. The 2M++ group catalogue – Col. (1): Group identifier in the catalogue. It corresponds to column 8 of Table A1. Col. (2): Galactic longitude Col. (3): Galactic latitudeCol. (4): Apparent magnitude in band K S as defined in Section 2.2. The magnitude is derived from the 2M++ galaxies. This is a magnitude uncorrected for incompleteness effect. Col.(5): Richness, uncorrected for incompleteness effect. Col. (6): Heliocentric total apparent velocity. Col. (7): Total apparent velocity in CMB rest frame, using relation from Kogut et al.(1993) and Tully et al. (2008). Col. (8): Velocity dispersion in the group. c (cid:13) R A S , M N R A S ,,