Swinburne Research Bank httpresearchbankswinburneeduau

Erdogdu Pirin Lahav Ofer amp Zaroubi Saleem et al (2004)

The 2dF Galaxy Redshift Survey Wiener reconstruction of the cosmic web

Originally published in Monthly Notices of the Royal Astronomical Society 352 (3) 939ndash960 Available from httpdxdoiorg101111j1365-2966200407984x

Copyright copy 2003 The Authors

This is the authorrsquos version of the work It is posted here with the permission of the publisher for your personal use No further distribution is permitted If your library has a subscription to this

journal you may also be able to access the published version via the library catalogue

The definitive version is available at wwwintersciencewileycom

Accessed from Swinburne Research Bank httphdlhandlenet1959342754

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Mon Not R Astron Soc000 000ndash000 (0000) Printed 2 February 2008 (MN LATEX style file v22)

The 2dF Galaxy Redshift Survey Wiener Reconstruction of theCosmic Web

Pirin Erdogdu12 Ofer Lahav1 Saleem Zaroubi3 George Efstathiou1 Steve Moody JohnA Peacock12 Matthew Colless17 Ivan K Baldry9 Carlton M Baugh16 Joss Bland-Hawthorn7 Terry Bridges7 Russell Cannon7 Shaun Cole16 Chris Collins4 WarrickCouch5 Gavin Dalton615 Roberto De Propris17 Simon P Driver17 Richard S Ellis8Carlos S Frenk16 Karl Glazebrook9 Carole Jackson17 Ian Lewis6 Stuart Lumsden10Steve Maddox11 Darren Madgwick13 Peder Norberg14 Bruce A Peterson17 WillSutherland12 Keith Taylor8 (The 2dFGRS Team)1Institute of Astronomy Madingley Road Cambridge CB3 0HAUK2Department of Physics Middle East Technical University 06531 Ankara Turkey3Max Planck Institut fur Astrophysik Karl-Schwarzschild-Straszlige 1 85741 Garching Germany4Astrophysics Research Institute Liverpool John Moores University Twelve Quays House Birkenhead L14 1LD UK5Department of Astrophysics University of New South WalesSydney NSW 2052 Australia6Department of Physics University of Oxford Keble Road Oxford OX1 3RH UK7Anglo-Australian Observatory PO Box 296 Epping NSW 2111 Australia8Department of Astronomy California Institute of Technology Pasadena CA 91025 USA9Department of Physics amp Astronomy Johns Hopkins University Baltimore MD 21118-2686 USA10Department of Physics University of Leeds Woodhouse Lane Leeds LS2 9JT UK11School of Physics amp Astronomy University of Nottingham Nottingham NG7 2RD UK12Institute for Astronomy University of Edinburgh Royal Observatory Blackford Hill Edinburgh EH9 3HJ UK13Lawrence Berkeley National Laboratory 1 Cyclotron Road Berkeley CA 94720 USA14ETHZ Institut fur Astronomie HPF G31 ETH HonggerbergCH-8093 Zurich Switzerland15Rutherford Appleton Laboratory Chilton Didcot OX11 0QX UK16Department of Physics University of Durham South Road Durham DH1 3LE UK17Research School of Astronomy amp Astrophysics The Australian National University Weston Creek ACT 2611 Australia

2 February 2008

ABSTRACTWe reconstruct the underlying density field of the 2 degree Field Galaxy Redshift Survey(2dFGRS) for the redshift range0035 lt z lt 0200 using the Wiener Filtering methodThe Wiener Filter suppresses shot noise and accounts for selection and incompleteness ef-fects The method relies on prior knowledge of the 2dF power spectrum of fluctuations andthe combination of matter density and bias parameters however the results are only slightlyaffected by changes to these parameters We present maps of the density field in two differentresolutions5hminus1 Mpc and10hminus1 Mpc We identify all major superclusters and voids in thesurvey In particular we find two large superclusters and two large local voids A version ofthis paper with full set of colour maps can be found at httpwwwastcamacuksimpirin

Key words galaxiesdistances and redshifts - cosmology large-scale structure of Universe -methods statistical

1 INTRODUCTION

Historically redshift surveys have provided the data and the testground for much of the research on the nature of clustering and thedistribution of galaxies In the past few years observations of largescale structure have improved greatly Today with the development

of fibre-fed spectrographs that can simultaneously measurespectraof hundreds of galaxies cosmologists have at their fingertips largeredshift surveys such as 2 degree Field (2dF) and Sloan Digital SkySurvey (SDSS) The analysis of these redshift surveys yieldinvalu-able cosmological information On the quantitative side with theassumption that the galaxy distribution arises from the gravitational

ccopy 0000 RAS

arX

iva

stro

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0312

546v

1 1

9 D

ec 2

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Mon Not R Astron Soc000 000ndash000 (0000) Printed 2 February 2008 (MN LATEX style file v22)

The 2dF Galaxy Redshift Survey Wiener Reconstruction of theCosmic Web

Pirin Erdogdu12 Ofer Lahav1 Saleem Zaroubi3 George Efstathiou1 Steve Moody JohnA Peacock12 Matthew Colless17 Ivan K Baldry9 Carlton M Baugh16 Joss Bland-Hawthorn7 Terry Bridges7 Russell Cannon7 Shaun Cole16 Chris Collins4 WarrickCouch5 Gavin Dalton615 Roberto De Propris17 Simon P Driver17 Richard S Ellis8Carlos S Frenk16 Karl Glazebrook9 Carole Jackson17 Ian Lewis6 Stuart Lumsden10Steve Maddox11 Darren Madgwick13 Peder Norberg14 Bruce A Peterson17 WillSutherland12 Keith Taylor8 (The 2dFGRS Team)1Institute of Astronomy Madingley Road Cambridge CB3 0HAUK2Department of Physics Middle East Technical University 06531 Ankara Turkey3Max Planck Institut fur Astrophysik Karl-Schwarzschild-Straszlige 1 85741 Garching Germany4Astrophysics Research Institute Liverpool John Moores University Twelve Quays House Birkenhead L14 1LD UK5Department of Astrophysics University of New South WalesSydney NSW 2052 Australia6Department of Physics University of Oxford Keble Road Oxford OX1 3RH UK7Anglo-Australian Observatory PO Box 296 Epping NSW 2111 Australia8Department of Astronomy California Institute of Technology Pasadena CA 91025 USA9Department of Physics amp Astronomy Johns Hopkins University Baltimore MD 21118-2686 USA10Department of Physics University of Leeds Woodhouse Lane Leeds LS2 9JT UK11School of Physics amp Astronomy University of Nottingham Nottingham NG7 2RD UK12Institute for Astronomy University of Edinburgh Royal Observatory Blackford Hill Edinburgh EH9 3HJ UK13Lawrence Berkeley National Laboratory 1 Cyclotron Road Berkeley CA 94720 USA14ETHZ Institut fur Astronomie HPF G31 ETH HonggerbergCH-8093 Zurich Switzerland15Rutherford Appleton Laboratory Chilton Didcot OX11 0QX UK16Department of Physics University of Durham South Road Durham DH1 3LE UK17Research School of Astronomy amp Astrophysics The Australian National University Weston Creek ACT 2611 Australia

2 February 2008

ABSTRACTWe reconstruct the underlying density field of the 2 degree Field Galaxy Redshift Survey(2dFGRS) for the redshift range0035 lt z lt 0200 using the Wiener Filtering methodThe Wiener Filter suppresses shot noise and accounts for selection and incompleteness ef-fects The method relies on prior knowledge of the 2dF power spectrum of fluctuations andthe combination of matter density and bias parameters however the results are only slightlyaffected by changes to these parameters We present maps of the density field in two differentresolutions5hminus1 Mpc and10hminus1 Mpc We identify all major superclusters and voids in thesurvey In particular we find two large superclusters and two large local voids A version ofthis paper with full set of colour maps can be found at httpwwwastcamacuksimpirin

Key words galaxiesdistances and redshifts - cosmology large-scale structure of Universe -methods statistical

1 INTRODUCTION

Historically redshift surveys have provided the data and the testground for much of the research on the nature of clustering and thedistribution of galaxies In the past few years observations of largescale structure have improved greatly Today with the development

of fibre-fed spectrographs that can simultaneously measurespectraof hundreds of galaxies cosmologists have at their fingertips largeredshift surveys such as 2 degree Field (2dF) and Sloan Digital SkySurvey (SDSS) The analysis of these redshift surveys yieldinvalu-able cosmological information On the quantitative side with theassumption that the galaxy distribution arises from the gravitational

ccopy 0000 RAS

2 Erdogdu et al

instability of small fluctuations generated in the early universe awide range of statistical measurements can be obtained such asthe power spectrum and bispectrum Furthermore a qualitative un-derstanding of galaxy distribution provides insight into the mech-anisms of structure formation that generate the complex pattern ofsheets and filaments comprising the lsquocosmic webrsquo (Bond Kofmanamp Pogosyan 1996) we observe and allows us to map a wide varietyof structure including clusters superclusters and voids

Today many more redshifts are available for galaxies than di-rect distance measurements This discrepancy inspired a great dealof work on methods for reconstruction of the real-space densityfield from that observed in redshift-space These methods use a va-riety of functional representations (eg Cartesian Fourier sphericalharmonics or wavelets) and smoothing techniques (eg a Gaussiansphere or a Wiener Filter) There are physical as well as practicalreasons why one would be interested in smoothing the observeddensity field It is often assumed that the galaxy distribution sam-ples the underlying smooth density field and the two are related by aproportionality constant the so-called linear bias parameterb Thefinite sampling of the smooth underlying field introduces Poissonlsquoshot noisersquo1 Any robust reconstruction technique must reliablymitigate the statistical uncertainties due to shot noise Moreover inredshift surveys the actual number of galaxies in a given volume islarger than the number observed in particular in magnitudelimitedsamples where at large distances only very luminous galaxies canbe seen

In this paper we analyse large scale structure in the 2 degreeField Galaxy Redshift Survey (2dFGRS Collesset al2001) whichhas now obtained the redshifts for approximately 230000 galax-ies We recover the underlying density field characterisedby anassumed power spectrum of fluctuations from the observed fieldwhich suffers from incomplete sky coverage (described by the an-gular mask) and incomplete galaxy sampling due to its magnitudelimit (described by the selection function) The filtering is achievedby a Wiener Filter (Wiener 1949 Presset al 1992) within theframework of both linear and non-linear theory of density fluctu-ations The Wiener Filter is optimal in the sense that the variancebetween the derived reconstruction and the underlying truedensityfield is minimised As opposed toad hocsmoothing schemes thesmoothing due to the Wiener Filter is determined by the dataInthe limit of high signal-to-noise the Wiener Filter modifies the ob-served data only weakly whereas it suppresses the contribution ofthe data contaminated by shot noise

The Wiener Filtering is a well known technique and has beenapplied to many fields in astronomy (see Rybicki amp Press 1992)For example the method was used to reconstruct the angular dis-tribution (Lahavet al 1994) the real-space density velocity andgravitational potential fields of the 12 Jy-IRAS (Fisher et al1995) andIRAS PSCz surveys (Schmoldtet al1999) The WienerFilter was also applied to the reconstruction of the angularmapsof the Cosmic Microwave Background temperature fluctuations(Bunn et al 1994 Tegmark amp Efstathiou 1995 Bouchet amp Gis-pert 1999) A detailed formalism of the Wiener Filtering method asit pertains to the large scale structure reconstruction canbe foundin Zaroubiet al (1995)

This paper is structured as follows we begin with a brief re-

1 Another popular model for galaxy clustering is the halo model where thelinear bias parameter depends on the mass of the dark matter halos wherethe galaxies reside For this model the mean number of galaxy pairs in agiven halo is usually lower than the Poisson expectation

view of the formalism of the Wiener Filter method A summaryof 2dFGRS data set the survey mask and the selection functionare given in Section 3 In section 4 we outline the scheme usedto pixelise the survey In section 5 we give the formalism for thecovariance matrix used in the analysis After that we describe theapplication of the Wiener Filter method to 2dFGRS and present de-tailed maps of the reconstructed field In section 7 we identify thesuperclusters and voids in the survey

Throughout this paper we assume aΛCDM cosmology withΩm = 03 andΩΛ = 07 andH0 = 100hminus1kmsminus1Mpcminus1

2 WIENER FILTER

In this section we give a brief description of the Wiener Fil-ter method For more details we refer the reader to Zaroubietal (1995) Letrsquos assume that we have a set of measurementsdα (α = 1 2 N) which are a linear convolution of the trueunderlying signalsα plus a contribution from statistical noiseǫαsuch that

dα = sα + ǫα (1)

The Wiener Filter is defined as thelinear combination of the ob-served data which is closest to the true signal in a minimum vari-ance sense More explicitly the Wiener Filter estimatesWF

α isgiven bysWF

α = Fαβ dβ where the filter is chosen to minimise thevariance of the residual fieldrα

〈|r2α|〉 = 〈|sWF

α minus sα|2〉 (2)

It is straightforward to show that the Wiener Filter is givenby

Fαβ = 〈sαddaggerγ〉〈dγddagger

β〉minus1 (3)

where the first term on the right hand side is the signal-data corre-lation matrix

〈sαddaggerγ〉 = 〈sαsdaggerγ〉 (4)

and the second term is the data-data correlation matrix

〈dαddaggerβ〉 = 〈sγsdaggerδ〉 + 〈ǫαǫdaggerβ〉 (5)

In the above equations we have assumed that the signal andnoise are uncorrelated From equations 3 and 5 it is clear that inorder to implement the Wiener Filter one must construct apriorwhich depends on the mean of the signal (which is 0 by construc-tion) and the variance of the signal and noise The assumption of aprior may be alarming at first glance However slightly inaccuratevalues of Wiener Filter will only introduce second order errors tothe full reconstruction (see Rybicki amp Press 1992) The dependenceof the Wiener Filter on the prior can be made clear by defining sig-nal and noise matrices asCαβ = 〈sαsdaggerβ〉 andNαβ = 〈ǫαǫdaggerβ〉 With

this notation we can rewrite the equations above so thatsWF is

given as

sWF = C [C + N]minus1

d (6)

The mean square residual given in equation 2 can then be calculatedas

〈rrdagger〉 = C [C + N]minus1N (7)

Formulated in this way we see that the purpose of the WienerFilter is to attenuate the contribution of low signal-to-noise ra-tio data The derivation of the Wiener Filter given above followsfrom the sole requirement of minimum variance and requires only

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web3

a model for the variance of the signal and noise The Wiener Fil-ter can also be derived using the laws of conditional probability ifthe underlying distribution functions for the signal and noise areassumed to be Gaussian For the Gaussian prior the Wiener Filterestimate is both the maximumposteriorestimate and the mean field(see Zaroubiet al1995)

As several authors point out (eg Rybicki amp Press 1992Zaroubi 2002) the Wiener Filter is a biased estimator sinceit pre-dicts a null field in the absence of good data unless the field itselfhas zero mean Since we constructed the density field to have zeromean we are not worried about this bias However the observedfield deviates from zero due to selection effects and so one needs tobe aware of this bias in the reconstructions

It is well known that the peculiar velocities of galaxies distortclustering pattern in redshift space On small scales the randompeculiar velocity of each galaxy causes smearing along the line ofsight known asthe finger of God On larger scales there is com-pression of structures along the line of sight due to coherent infallvelocities of large-scale structure induced by gravity One of themajor difficulties in analysing redshift surveys is the transforma-tion of the position of galaxies from redshift space to real space Forall sky surveys this issue can be addressed using several methodsfor example the iterative method of Yahilet al(1991) and modifiedPoisson equation of Nusser amp Davis 1994 However these methodsare not applicable to surveys which are not all sky as they assumethat in linear theory the peculiar velocity of any galaxy isa result ofthe matter distribution around it and the gravitational field is dom-inated by the matter distribution inside the volume of the surveyFor a survey like 2dFGRS within the limitation of linear theorywhere the redshift space density is a linear transformationof thereal space density a Wiener Filter can be used to transform fromredshift space to real space (see Fisheret al (1995) and Zaroubietal (1995) for further details) This can be written as

sWF (rα) = 〈s(rα)d(sγ)〉〈d(sγ)d(sβ)〉minus1d(sβ) (8)

where the first term on the right hand side is the cross-correlationmatrix of real and redshift space densities ands(r) is the positionvector in redshift space It is worth emphasising that this methodis limited as it only recovers the peculiar velocity field generatedby the mass sources represented by the galaxies within the surveyboundaries It does not account for possible external forces Thislimitation can only be overcome by comparing the 2dF survey withall sky surveys

3 THE DATA

31 The 2dFGRS data

The 2dFGRS now completed is selected in the photometricbJ

band from the APM galaxy survey (Maddox Efstathiou amp Suther-land 1990) and its subsequent extensions (Maddoxet al 2003 inpreparation) The survey covers about2000 deg2 and is made upof two declination strips one in the South Galactic Pole region(SGP) covering approximatelyminus375 lt δ lt minus225 minus350 ltα lt 550 and the other in the direction of the North Galactic Pole(NGP) spanningminus75 lt δ lt 25 1475 lt α lt 2225 In ad-dition to these contiguous regions there are a number of randomlylocated circular 2-degree fields scattered over the full extent of thelow extinction regions of the southern APM galaxy survey

The magnitude limit at the start of the survey was set atbJ = 1945 but both the photometry of the input catalogue and the

dust extinction map have been revised since and so there are smallvariations in magnitude limit as a function of position overthe skywhich are taken into account using the magnitude limit maskTheeffective median magnitude limit over the area of the survey isbJ asymp 193 (Collesset al2001)

We use the data obtained prior to May 2002 when the sur-vey was nearly complete This includes 221 283 unique reliablegalaxy redshifts We analyse a magnitude-limited sample with red-shift limits zmin = 0035 andzmax = 020 The median redshift iszmed asymp 011 We use 167 305 galaxies in total 98 129 in the SGPand 69 176 in the NGP We do not include the random fields in ouranalysis

The 2dFGRS database and full documentation are availableon the WWW at httpwwwmsoanueduau2dFGRS

32 The Mask and The Radial Selection Function of 2dFGRS

The completeness of the survey varies according to the position inthe sky due to unobserved fields particularly at the survey edgesand unfibred objects in the observed fields because of collision con-straints or broken fibres

For our analysis we make use of two different masks (Collesset al 2001 Norberget al 2002) The first of these masks is theredshift completeness mask defined as the ratio of the numberofgalaxies for which redshifts have been measured to the totalnum-ber of objects in the parent catalogue This spatial incompletenessis illustrated in Figure 1 The second mask is the magnitude limitmask which gives the extinction corrected magnitude limit of thesurvey at each position

The radial selection function gives the probability of observ-ing a galaxy for a given redshift and can be readily calculated fromthe galaxy luminosity function

Φ(L)dL = Φlowast

(

L

Llowast

)α

exp

(

minus L

Llowast

)

dL

Llowast (9)

where for the concordance modelα = minus121plusmn 003 log10 Llowast =minus04(minus1966plusmn007+5 log10(h)) andΦlowast = 00161plusmn00008h3

(Norberget al2002)The selection function can then be expressed as

φ(r) =

int infin

L(r)Φ(L)dL

int infin

LminΦ(L)dL

(10)

where L(r) is the minimum luminosity detectable at luminos-ity distancer (assuming the concordance model) evaluated forthe concordance modelLmin = Min(L(r) Lcom) andLcom isthe minimum luminosity for which the catalogue is complete andvaries as a function of position over the sky For distances con-sidered in this paper where the deviations from the Hubble floware relatively small the selection function can be approximated asφ(r) asymp φ(zgal) Each galaxygal is then assigned the weight

w(gal) =1

φ(zgal)M(Ωi)(11)

whereφ(zgal) andM(Ωi) are the values of the selection functionfor each galaxy and angular survey mask for each celli (see Section4) respectively

4 SURVEY PIXELISATION

In order to form a data vector of overdensities the survey needsto be pixelised There are many ways to pixelise a survey equal

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web3

a model for the variance of the signal and noise The Wiener Fil-ter can also be derived using the laws of conditional probability ifthe underlying distribution functions for the signal and noise areassumed to be Gaussian For the Gaussian prior the Wiener Filterestimate is both the maximumposteriorestimate and the mean field(see Zaroubiet al1995)

As several authors point out (eg Rybicki amp Press 1992Zaroubi 2002) the Wiener Filter is a biased estimator sinceit pre-dicts a null field in the absence of good data unless the field itselfhas zero mean Since we constructed the density field to have zeromean we are not worried about this bias However the observedfield deviates from zero due to selection effects and so one needs tobe aware of this bias in the reconstructions

It is well known that the peculiar velocities of galaxies distortclustering pattern in redshift space On small scales the randompeculiar velocity of each galaxy causes smearing along the line ofsight known asthe finger of God On larger scales there is com-pression of structures along the line of sight due to coherent infallvelocities of large-scale structure induced by gravity One of themajor difficulties in analysing redshift surveys is the transforma-tion of the position of galaxies from redshift space to real space Forall sky surveys this issue can be addressed using several methodsfor example the iterative method of Yahilet al(1991) and modifiedPoisson equation of Nusser amp Davis 1994 However these methodsare not applicable to surveys which are not all sky as they assumethat in linear theory the peculiar velocity of any galaxy isa result ofthe matter distribution around it and the gravitational field is dom-inated by the matter distribution inside the volume of the surveyFor a survey like 2dFGRS within the limitation of linear theorywhere the redshift space density is a linear transformationof thereal space density a Wiener Filter can be used to transform fromredshift space to real space (see Fisheret al (1995) and Zaroubietal (1995) for further details) This can be written as

sWF (rα) = 〈s(rα)d(sγ)〉〈d(sγ)d(sβ)〉minus1d(sβ) (8)

where the first term on the right hand side is the cross-correlationmatrix of real and redshift space densities ands(r) is the positionvector in redshift space It is worth emphasising that this methodis limited as it only recovers the peculiar velocity field generatedby the mass sources represented by the galaxies within the surveyboundaries It does not account for possible external forces Thislimitation can only be overcome by comparing the 2dF survey withall sky surveys

3 THE DATA

31 The 2dFGRS data

The 2dFGRS now completed is selected in the photometricbJ

band from the APM galaxy survey (Maddox Efstathiou amp Suther-land 1990) and its subsequent extensions (Maddoxet al 2003 inpreparation) The survey covers about2000 deg2 and is made upof two declination strips one in the South Galactic Pole region(SGP) covering approximatelyminus375 lt δ lt minus225 minus350 ltα lt 550 and the other in the direction of the North Galactic Pole(NGP) spanningminus75 lt δ lt 25 1475 lt α lt 2225 In ad-dition to these contiguous regions there are a number of randomlylocated circular 2-degree fields scattered over the full extent of thelow extinction regions of the southern APM galaxy survey

The magnitude limit at the start of the survey was set atbJ = 1945 but both the photometry of the input catalogue and the

dust extinction map have been revised since and so there are smallvariations in magnitude limit as a function of position overthe skywhich are taken into account using the magnitude limit maskTheeffective median magnitude limit over the area of the survey isbJ asymp 193 (Collesset al2001)

We use the data obtained prior to May 2002 when the sur-vey was nearly complete This includes 221 283 unique reliablegalaxy redshifts We analyse a magnitude-limited sample with red-shift limits zmin = 0035 andzmax = 020 The median redshift iszmed asymp 011 We use 167 305 galaxies in total 98 129 in the SGPand 69 176 in the NGP We do not include the random fields in ouranalysis

The 2dFGRS database and full documentation are availableon the WWW at httpwwwmsoanueduau2dFGRS

32 The Mask and The Radial Selection Function of 2dFGRS

The completeness of the survey varies according to the position inthe sky due to unobserved fields particularly at the survey edgesand unfibred objects in the observed fields because of collision con-straints or broken fibres

For our analysis we make use of two different masks (Collesset al 2001 Norberget al 2002) The first of these masks is theredshift completeness mask defined as the ratio of the numberofgalaxies for which redshifts have been measured to the totalnum-ber of objects in the parent catalogue This spatial incompletenessis illustrated in Figure 1 The second mask is the magnitude limitmask which gives the extinction corrected magnitude limit of thesurvey at each position

The radial selection function gives the probability of observ-ing a galaxy for a given redshift and can be readily calculated fromthe galaxy luminosity function

Φ(L)dL = Φlowast

(

L

Llowast

)α

exp

(

minus L

Llowast

)

dL

Llowast (9)

where for the concordance modelα = minus121plusmn 003 log10 Llowast =minus04(minus1966plusmn007+5 log10(h)) andΦlowast = 00161plusmn00008h3

(Norberget al2002)The selection function can then be expressed as

φ(r) =

int infin

L(r)Φ(L)dL

int infin

LminΦ(L)dL

(10)

where L(r) is the minimum luminosity detectable at luminos-ity distancer (assuming the concordance model) evaluated forthe concordance modelLmin = Min(L(r) Lcom) andLcom isthe minimum luminosity for which the catalogue is complete andvaries as a function of position over the sky For distances con-sidered in this paper where the deviations from the Hubble floware relatively small the selection function can be approximated asφ(r) asymp φ(zgal) Each galaxygal is then assigned the weight

w(gal) =1

φ(zgal)M(Ωi)(11)

whereφ(zgal) andM(Ωi) are the values of the selection functionfor each galaxy and angular survey mask for each celli (see Section4) respectively

4 SURVEY PIXELISATION

In order to form a data vector of overdensities the survey needsto be pixelised There are many ways to pixelise a survey equal

ccopy 0000 RAS MNRAS000 000ndash000

4 Erdogdu et al

Figure 1 The redshift completeness masks for the NGP (top) and SGP (bottom) in Equatorial coordinates The gray scale shows the completeness fraction

Figure 2 An illustration of the survey pixelisation scheme used in the anal-ysis for10hminus1 Mpc (top) and5hminus1 Mpc (bottom) target cells widths Theredshift ranges given on top of each plot

sized cubes in redshift space igloo cells spherical harmonics De-launey tessellation methods wavelet decomposition etcEach ofthese methods have their own advantages and disadvantages andthey should be treated with care as they form functional bases inwhich all the statistical and physical properties of cosmicfields areretained

The pixelisation scheme used in this analysis is an lsquoigloorsquo gridwith wedge shaped pixels bounded in right ascension declinationand in redshift The pixelisation is constructed to keep theaveragenumber density per pixel approximately constant The advantageof using this pixelisation is that the number of pixels is minimisedsince the pixel volume is increased with redshift to counteract thedecrease in the selection function This is achieved by selecting alsquotarget cell widthrsquo for cells at the mean redshift of the survey andderiving the rest of the bin widths so as to match the shape of theselection function The target cell widths used in this analysis are10hminus1 Mpc and5hminus1 Mpc Once the redshift binning has been cal-culated each radial bin is split into declination bands and then eachband in declination is further divided into cells in right ascensionThe process is designed so as to make the cells roughly cubicalIn Figure 2 we show an illustration of this by plotting the cells inright ascension and declination for a given redshift strip

Although advantageous in many ways the pixelisationscheme used in this paper may complicate the interpretationof thereconstructed field By definition the Wiener Filter signalwill ap-proach to zero at the edges of the survey where the shot noise maydominate This means the true signal will be constructed in anon-uniform manner This effect will be amplified as the cell sizes getbigger at higher redshifts Hence both of these effects must be con-sidered when interpreting the results

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web5

5 ESTIMATING THE SIGNAL-SIGNAL CORRELATIONMATRIX OVER PIXELS

The signal covariance matrix can be accurately modelled by ananalytical approximation (Moody 2003) The calculation ofthecovariance matrix is similar to the analysis described by Efs-tathiou and Moody (2001) apart from the modification due to three-dimensionality of the survey The covariance matrix for thelsquonoisefreersquo density fluctuations is〈Cij〉 = 〈δiδj〉 whereδi = (ρi minus ρ)ρin the ith pixel It is estimated by first considering a pair of pixelswith volumesVi andVj separated by distancer so that

〈Cij〉 =

lang

1

ViVj

int

Celli

δ(x)dVi

int

Cellj

δ(x + r)dVj

rang

(12)

=1

ViVj

int

Celli

int

Cellj

〈δ(x)δ(x + r)〉dVidVj (13)

=1

ViVj

int

Celli

int

Cellj

ξ(r)dVidVj (14)

where the isotropic two point correlation functionξ(r) is given by

ξ(r) =1

(2π)3

int

P (k)eminusikmiddotrd3k (15)

and therefore

〈Cij〉 =1

(2π)3ViVj

int

P (k)d3k

timesint

Celli

int

Cellj

eminusik(riminusrj )dVidVj (16)

After performing the Fourier transform this equation can be writtenas

〈Cij〉 =1

(2π)3

int

P (k)S(kLi)S(kLj)C(k r)d3k (17)

where the functionsS andC are given by

S(kL) = sinc(kxLx2)sinc(kyLy2)sinc(kzLz2) (18)

C(k r) = cos(kxrx) cos(kyry) cos(kzrz) (19)

where the labelL describes the dimensions of the cell (Lx Ly Lz) the components ofr describes the separation between cell cen-tresk= (kx ky kz) is the wavevector andsinc(x) = sin(x)

x The

wavevectork is written in spherical co-ordinatesk θ φ to simplifythe evaluation ofC We define

kx = k sin(φ) cos(θ) (20)

ky = k sin(φ) sin(θ) (21)

kz = k cos(φ) (22)

Equation 17 can now be integrated overθ andφ to form the kernelGij(k) where

Gij(k) =1

π3

int π2

0

int π2

0

S(kLi)S(kLj)C(k r) sin(φ)dθdφ (23)

so that

〈Cij〉 =

int

P (k)Gij(k)k2dk (24)

In practice we evaluate

〈Cij〉 =sum

k

PkGijk (25)

wherePk is the binned bandpower spectrum andGijk is

Gijk =

int kmax

kmin

Gij(k)k2dk (26)

where the integral extends over the band corresponding to the bandpowerPk

For cells that are separated by a distance much larger than thecell dimensions the cell window functions can be ignored simpli-fying the calculation so that

Gijk =1

(2π)3

int kmax

kmin

sinc(kr)4πk2dk (27)

wherer is the separation between cell centres

6 THE APPLICATION

61 Reconstruction Using Linear Theory

In order to calculate the data vectord in equation 6 we first esti-mate the number of galaxiesNi in each pixeli

Ni =

Ngal(i)sum

gal

w(gal) (28)

where the sum is over all the observed galaxies in the pixel andw(gal) is the weight assigned to each galaxy (equation 11) Theboundaries of each pixel are defined by the scheme described inSection 4 using a target cell width of10hminus1 Mpc The mean num-ber of galaxies in pixeli is

Ni = nVi (29)

whereVi is the volume of the pixel and the mean galaxy densitynis estimated using the equation

n =

Ntotalsum

gal

w(gal)

int infin

0drr2φ(r)w(r)

(30)

where the sum is now over all the galaxies in the survey We notethat the value forn obtained using the equation above is consis-tent with the maximum estimator method proposed by Davis andHuchra (1982) Using these definitions we write theith compo-nent of the data vectord as

di =Ni minus Ni

Ni (31)

Note that the mean value ofd is zero by constructionReconstruction of the underlying signal given in equation 6

also requires the signal-signal and the inverse of the data-data cor-relation matrices The data-data correlation matrix (equation 5) isthe sum of noise-noise correlation matrixN and the signal-signalcorrelation matrixC formulated in the previous section The onlychange made is to the calculation ofC where the real space corre-lation functionξ(r) is now multiplied by Kaiserrsquos factor in order tocorrect for the redshift distortions on large scales So

ξs =1

(2π)3

int

P S(k) exp(ik middot (r2 minus r1))d3k (32)

ccopy 0000 RAS MNRAS000 000ndash000

6 Erdogdu et al

whereP S(k) is the galaxy power spectrum in redshift space

P S(k) = K[β]P R(k) (33)

derived in linear theory The subscriptsR andS in this equation(and hereafter) denote real and redshift space respectively

K[β] = 1 +2

3β +

1

5β2 (34)

is the direction averaged Kaiserrsquos (1987) factor derived using dis-tant observer approximation and with the assumption that the datasubtends a small solid angle with respect to the observer (the latterassumption is valid for the 2dFGRS but does not hold for a wideangle survey see Zaroubi and Hoffman 1996 for a full discussion)Equation 33 shows that in order to apply the Wiener Filter methodwe need a model for the galaxy power spectrum in redshift-spacewhich depends on the real-space power spectrum spectrum andonthe redshift distortion parameterβ equiv Ωm

06bThe real-space galaxy power spectrum is well described by a

scale invariant Cold Dark Matter power spectrum with shape pa-rameterΓ for the scales concerned in this analysis ForΓ we usethe value derived from the 2dF survey by Percivalet al(2001) whofitted the 2dFGRS power spectrum over the range of linear scalesusing the fitting formulae of Eisenstein and Hu (1998) Assuminga Gaussian prior on the Hubble constanth = 07 plusmn 007 (based onFreedmanet al2001) they findΓ = 02plusmn003 The normalisationof the power spectrum is conventionally expressed in terms of thevariance of the density field in spheres of8hminus1 Mpc σ8 Lahavetal (2002) use 2dFGRS data to deduceσS

8g(Ls zs) = 094 plusmn 002for the galaxies in redshift space assumingh = 07 plusmn 007 atzs asymp 017 andLs asymp 19Llowast We convert this result to real spaceusing the following equation

σR8g(Ls zs) = σS

8g(Ls zs)K12[β(Ls zs)] (35)

whereK[β] is Kaiserrsquos factor For our analysis we need to useσ8

evaluated at the mean redshift of the survey for galaxies with lumi-nosityLlowast However one needs to assume a model for the evolutionof galaxy clustering in order to findσ8 at different redshifts More-over the conversion fromLs to Llowast introduces uncertainties in thecalculation Therefore we choose an approximate valueσR

8g asymp 08to normalise the power spectrum Forβ we adopt the value foundby Hawkinset al (2002)β(Ls zs) = 049 plusmn 009 which is es-timated at the effective luminosityLs asymp 14Llowast and the effectiveredshiftzs asymp 015 of the survey sample Our results are not sen-sitive to minor changes inσ8 andβ

The other component of the data-data correlation matrix is thenoise correlation matrixN Assuming that the noise in differentcells is not correlated the only non-zero terms inN are the diag-onal terms given by the variance - the second central moment -ofthe density error in each cell

Nii =1

N2i

Ngal(i)sum

gal

w2(gal) (36)

The final aspect of the analysis is the reconstruction of thereal-space density field from the redshift-space observations Thisis achieved using equation 8 Following Kaiser (1987) using dis-tant observer and small-angle approximation the cross-correlationmatrix in equation 8 for the linear regime can be written as

〈s(r)d(s)〉 = 〈δrδs〉 = ξ(r)(1 +1

3β) (37)

wheres and r are position vectors in redshift and real space re-

spectively The term(1 + 13β) is easily obtained by integrating

the direction dependent density field in redshift space Using equa-tion 37 the transformation from redshift space to real space simpli-fies to

sWF (r) =1 + 1

3β

K[β]C [K[β]C + N]minus1

d (38)

As mentioned earlier the equation above is calculated for linearscales only and hence small scale distortions (iefingers of God)are not corrected for For this reason we collapse in redshift spacethe fingers seen in 2dF groups (Ekeet al2003) with more than 75members 25 groups in total (11 in NGP and 14 in SGP) All thegalaxies in these groups are assigned the same coordinatesAs ex-pected correcting these small scale distortions does not change theconstructed fields substantially as these distortions are practicallysmoothed out because of the cell size used in binning the data

The maps shown in this section were derived by the techniquedetailed above There are 80 sets of plots which show the densityfields as strips inRA andDec 40 maps for SGP and 40 maps forNGP Here we just show some examples the rest of the plots canbefound in url httpwwwastcamacuksimpirin For comparison thetop plots of Figures 10 11 14 and 15 show the redshift spaceden-sity field weighted by the selection function and the angularmaskThe contours are spaced at∆δ = 05 with solid (dashed) lines de-noting positive (negative) contours the heavy solid contours cor-respond toδ = 0 Also plotted for comparison are the galaxies(red dots) and the groups withNgr number of members (Ekeetal 2003) and9 6 Ngr 6 17 (green circles)18 6 Ngr 6 44(blue squares) and45 6 Ngr (magenta stars) We also show thenumber of Abell APM and EDCC clusters studied by De Propriset al (2002) (black upside down triangles) The middle plots showthe redshift space density shown in top plots after the Wiener Filterapplied As expected the Wiener Filter suppresses the noise Thesmoothing performed by the Wiener Filter is variable and increaseswith distance The bottom plots show the reconstructed realdensityfield s

WF (r) after correcting for the redshift distortions Here theamplitude of density contrast is reduced slightly We also plot thereconstructed fields in declination slices These plots areshown inFigures 7 and 8

We also plot the square root of the variance of the residualfield (equation 2) which defines the scatter around the mean recon-structed field We plot the residual fields corresponding to some ofthe redshift slices shown in this paper (Figures 3 and 4) For bettercomparison plots are made so that the cell number increaseswithincreasingRA If the volume of the cells used to pixelise the surveywas constant we would expect the square root of the variance∆δto increase due to the increase in shot noise (equation 7) Howeversince the pixelisation was constructed to keep the shot noise perpixel approximately constant∆δ also stays constant (∆δ asymp 023for both NGP and SGP) but the average density contrast in eachpixel decreases with increasing redshift This means that althoughthe variance of the residual in each cell is roughly equal the rel-ative variance (represented by∆δ

δ) increases with increasing red-

shift This increase is clearly evident in Figures 3 and 4 Anotherconclusion that can be drawn from the figures is that the bumpsin the density field are due to real features not due to error inthereconstruction even at higher redshifts

62 Reconstruction Using Non-linear Theory

In order to increase the resolution of the density field mapswe re-duce the target cell width to5hminus1 Mpc A volume of a cubic cell

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web5

5 ESTIMATING THE SIGNAL-SIGNAL CORRELATIONMATRIX OVER PIXELS

The signal covariance matrix can be accurately modelled by ananalytical approximation (Moody 2003) The calculation ofthecovariance matrix is similar to the analysis described by Efs-tathiou and Moody (2001) apart from the modification due to three-dimensionality of the survey The covariance matrix for thelsquonoisefreersquo density fluctuations is〈Cij〉 = 〈δiδj〉 whereδi = (ρi minus ρ)ρin the ith pixel It is estimated by first considering a pair of pixelswith volumesVi andVj separated by distancer so that

〈Cij〉 =

lang

1

ViVj

int

Celli

δ(x)dVi

int

Cellj

δ(x + r)dVj

rang

(12)

=1

ViVj

int

Celli

int

Cellj

〈δ(x)δ(x + r)〉dVidVj (13)

=1

ViVj

int

Celli

int

Cellj

ξ(r)dVidVj (14)

where the isotropic two point correlation functionξ(r) is given by

ξ(r) =1

(2π)3

int

P (k)eminusikmiddotrd3k (15)

and therefore

〈Cij〉 =1

(2π)3ViVj

int

P (k)d3k

timesint

Celli

int

Cellj

eminusik(riminusrj )dVidVj (16)

After performing the Fourier transform this equation can be writtenas

〈Cij〉 =1

(2π)3

int

P (k)S(kLi)S(kLj)C(k r)d3k (17)

where the functionsS andC are given by

S(kL) = sinc(kxLx2)sinc(kyLy2)sinc(kzLz2) (18)

C(k r) = cos(kxrx) cos(kyry) cos(kzrz) (19)

where the labelL describes the dimensions of the cell (Lx Ly Lz) the components ofr describes the separation between cell cen-tresk= (kx ky kz) is the wavevector andsinc(x) = sin(x)

x The

wavevectork is written in spherical co-ordinatesk θ φ to simplifythe evaluation ofC We define

kx = k sin(φ) cos(θ) (20)

ky = k sin(φ) sin(θ) (21)

kz = k cos(φ) (22)

Equation 17 can now be integrated overθ andφ to form the kernelGij(k) where

Gij(k) =1

π3

int π2

0

int π2

0

S(kLi)S(kLj)C(k r) sin(φ)dθdφ (23)

so that

〈Cij〉 =

int

P (k)Gij(k)k2dk (24)

In practice we evaluate

〈Cij〉 =sum

k

PkGijk (25)

wherePk is the binned bandpower spectrum andGijk is

Gijk =

int kmax

kmin

Gij(k)k2dk (26)

where the integral extends over the band corresponding to the bandpowerPk

For cells that are separated by a distance much larger than thecell dimensions the cell window functions can be ignored simpli-fying the calculation so that

Gijk =1

(2π)3

int kmax

kmin

sinc(kr)4πk2dk (27)

wherer is the separation between cell centres

6 THE APPLICATION

61 Reconstruction Using Linear Theory

In order to calculate the data vectord in equation 6 we first esti-mate the number of galaxiesNi in each pixeli

Ni =

Ngal(i)sum

gal

w(gal) (28)

where the sum is over all the observed galaxies in the pixel andw(gal) is the weight assigned to each galaxy (equation 11) Theboundaries of each pixel are defined by the scheme described inSection 4 using a target cell width of10hminus1 Mpc The mean num-ber of galaxies in pixeli is

Ni = nVi (29)

whereVi is the volume of the pixel and the mean galaxy densitynis estimated using the equation

n =

Ntotalsum

gal

w(gal)

int infin

0drr2φ(r)w(r)

(30)

where the sum is now over all the galaxies in the survey We notethat the value forn obtained using the equation above is consis-tent with the maximum estimator method proposed by Davis andHuchra (1982) Using these definitions we write theith compo-nent of the data vectord as

di =Ni minus Ni

Ni (31)

Note that the mean value ofd is zero by constructionReconstruction of the underlying signal given in equation 6

also requires the signal-signal and the inverse of the data-data cor-relation matrices The data-data correlation matrix (equation 5) isthe sum of noise-noise correlation matrixN and the signal-signalcorrelation matrixC formulated in the previous section The onlychange made is to the calculation ofC where the real space corre-lation functionξ(r) is now multiplied by Kaiserrsquos factor in order tocorrect for the redshift distortions on large scales So

ξs =1

(2π)3

int

P S(k) exp(ik middot (r2 minus r1))d3k (32)

ccopy 0000 RAS MNRAS000 000ndash000

6 Erdogdu et al

whereP S(k) is the galaxy power spectrum in redshift space

P S(k) = K[β]P R(k) (33)

derived in linear theory The subscriptsR andS in this equation(and hereafter) denote real and redshift space respectively

K[β] = 1 +2

3β +

1

5β2 (34)

is the direction averaged Kaiserrsquos (1987) factor derived using dis-tant observer approximation and with the assumption that the datasubtends a small solid angle with respect to the observer (the latterassumption is valid for the 2dFGRS but does not hold for a wideangle survey see Zaroubi and Hoffman 1996 for a full discussion)Equation 33 shows that in order to apply the Wiener Filter methodwe need a model for the galaxy power spectrum in redshift-spacewhich depends on the real-space power spectrum spectrum andonthe redshift distortion parameterβ equiv Ωm

06bThe real-space galaxy power spectrum is well described by a

scale invariant Cold Dark Matter power spectrum with shape pa-rameterΓ for the scales concerned in this analysis ForΓ we usethe value derived from the 2dF survey by Percivalet al(2001) whofitted the 2dFGRS power spectrum over the range of linear scalesusing the fitting formulae of Eisenstein and Hu (1998) Assuminga Gaussian prior on the Hubble constanth = 07 plusmn 007 (based onFreedmanet al2001) they findΓ = 02plusmn003 The normalisationof the power spectrum is conventionally expressed in terms of thevariance of the density field in spheres of8hminus1 Mpc σ8 Lahavetal (2002) use 2dFGRS data to deduceσS

8g(Ls zs) = 094 plusmn 002for the galaxies in redshift space assumingh = 07 plusmn 007 atzs asymp 017 andLs asymp 19Llowast We convert this result to real spaceusing the following equation

σR8g(Ls zs) = σS

8g(Ls zs)K12[β(Ls zs)] (35)

whereK[β] is Kaiserrsquos factor For our analysis we need to useσ8

evaluated at the mean redshift of the survey for galaxies with lumi-nosityLlowast However one needs to assume a model for the evolutionof galaxy clustering in order to findσ8 at different redshifts More-over the conversion fromLs to Llowast introduces uncertainties in thecalculation Therefore we choose an approximate valueσR

8g asymp 08to normalise the power spectrum Forβ we adopt the value foundby Hawkinset al (2002)β(Ls zs) = 049 plusmn 009 which is es-timated at the effective luminosityLs asymp 14Llowast and the effectiveredshiftzs asymp 015 of the survey sample Our results are not sen-sitive to minor changes inσ8 andβ

The other component of the data-data correlation matrix is thenoise correlation matrixN Assuming that the noise in differentcells is not correlated the only non-zero terms inN are the diag-onal terms given by the variance - the second central moment -ofthe density error in each cell

Nii =1

N2i

Ngal(i)sum

gal

w2(gal) (36)

The final aspect of the analysis is the reconstruction of thereal-space density field from the redshift-space observations Thisis achieved using equation 8 Following Kaiser (1987) using dis-tant observer and small-angle approximation the cross-correlationmatrix in equation 8 for the linear regime can be written as

〈s(r)d(s)〉 = 〈δrδs〉 = ξ(r)(1 +1

3β) (37)

wheres and r are position vectors in redshift and real space re-

spectively The term(1 + 13β) is easily obtained by integrating

the direction dependent density field in redshift space Using equa-tion 37 the transformation from redshift space to real space simpli-fies to

sWF (r) =1 + 1

3β

K[β]C [K[β]C + N]minus1

d (38)

As mentioned earlier the equation above is calculated for linearscales only and hence small scale distortions (iefingers of God)are not corrected for For this reason we collapse in redshift spacethe fingers seen in 2dF groups (Ekeet al2003) with more than 75members 25 groups in total (11 in NGP and 14 in SGP) All thegalaxies in these groups are assigned the same coordinatesAs ex-pected correcting these small scale distortions does not change theconstructed fields substantially as these distortions are practicallysmoothed out because of the cell size used in binning the data

The maps shown in this section were derived by the techniquedetailed above There are 80 sets of plots which show the densityfields as strips inRA andDec 40 maps for SGP and 40 maps forNGP Here we just show some examples the rest of the plots canbefound in url httpwwwastcamacuksimpirin For comparison thetop plots of Figures 10 11 14 and 15 show the redshift spaceden-sity field weighted by the selection function and the angularmaskThe contours are spaced at∆δ = 05 with solid (dashed) lines de-noting positive (negative) contours the heavy solid contours cor-respond toδ = 0 Also plotted for comparison are the galaxies(red dots) and the groups withNgr number of members (Ekeetal 2003) and9 6 Ngr 6 17 (green circles)18 6 Ngr 6 44(blue squares) and45 6 Ngr (magenta stars) We also show thenumber of Abell APM and EDCC clusters studied by De Propriset al (2002) (black upside down triangles) The middle plots showthe redshift space density shown in top plots after the Wiener Filterapplied As expected the Wiener Filter suppresses the noise Thesmoothing performed by the Wiener Filter is variable and increaseswith distance The bottom plots show the reconstructed realdensityfield s

WF (r) after correcting for the redshift distortions Here theamplitude of density contrast is reduced slightly We also plot thereconstructed fields in declination slices These plots areshown inFigures 7 and 8

We also plot the square root of the variance of the residualfield (equation 2) which defines the scatter around the mean recon-structed field We plot the residual fields corresponding to some ofthe redshift slices shown in this paper (Figures 3 and 4) For bettercomparison plots are made so that the cell number increaseswithincreasingRA If the volume of the cells used to pixelise the surveywas constant we would expect the square root of the variance∆δto increase due to the increase in shot noise (equation 7) Howeversince the pixelisation was constructed to keep the shot noise perpixel approximately constant∆δ also stays constant (∆δ asymp 023for both NGP and SGP) but the average density contrast in eachpixel decreases with increasing redshift This means that althoughthe variance of the residual in each cell is roughly equal the rel-ative variance (represented by∆δ

δ) increases with increasing red-

shift This increase is clearly evident in Figures 3 and 4 Anotherconclusion that can be drawn from the figures is that the bumpsin the density field are due to real features not due to error inthereconstruction even at higher redshifts

62 Reconstruction Using Non-linear Theory

In order to increase the resolution of the density field mapswe re-duce the target cell width to5hminus1 Mpc A volume of a cubic cell

ccopy 0000 RAS MNRAS000 000ndash000

6 Erdogdu et al

whereP S(k) is the galaxy power spectrum in redshift space

P S(k) = K[β]P R(k) (33)

derived in linear theory The subscriptsR andS in this equation(and hereafter) denote real and redshift space respectively

K[β] = 1 +2

3β +

1

5β2 (34)

is the direction averaged Kaiserrsquos (1987) factor derived using dis-tant observer approximation and with the assumption that the datasubtends a small solid angle with respect to the observer (the latterassumption is valid for the 2dFGRS but does not hold for a wideangle survey see Zaroubi and Hoffman 1996 for a full discussion)Equation 33 shows that in order to apply the Wiener Filter methodwe need a model for the galaxy power spectrum in redshift-spacewhich depends on the real-space power spectrum spectrum andonthe redshift distortion parameterβ equiv Ωm

06bThe real-space galaxy power spectrum is well described by a

scale invariant Cold Dark Matter power spectrum with shape pa-rameterΓ for the scales concerned in this analysis ForΓ we usethe value derived from the 2dF survey by Percivalet al(2001) whofitted the 2dFGRS power spectrum over the range of linear scalesusing the fitting formulae of Eisenstein and Hu (1998) Assuminga Gaussian prior on the Hubble constanth = 07 plusmn 007 (based onFreedmanet al2001) they findΓ = 02plusmn003 The normalisationof the power spectrum is conventionally expressed in terms of thevariance of the density field in spheres of8hminus1 Mpc σ8 Lahavetal (2002) use 2dFGRS data to deduceσS

8g(Ls zs) = 094 plusmn 002for the galaxies in redshift space assumingh = 07 plusmn 007 atzs asymp 017 andLs asymp 19Llowast We convert this result to real spaceusing the following equation

σR8g(Ls zs) = σS

8g(Ls zs)K12[β(Ls zs)] (35)

whereK[β] is Kaiserrsquos factor For our analysis we need to useσ8

evaluated at the mean redshift of the survey for galaxies with lumi-nosityLlowast However one needs to assume a model for the evolutionof galaxy clustering in order to findσ8 at different redshifts More-over the conversion fromLs to Llowast introduces uncertainties in thecalculation Therefore we choose an approximate valueσR

8g asymp 08to normalise the power spectrum Forβ we adopt the value foundby Hawkinset al (2002)β(Ls zs) = 049 plusmn 009 which is es-timated at the effective luminosityLs asymp 14Llowast and the effectiveredshiftzs asymp 015 of the survey sample Our results are not sen-sitive to minor changes inσ8 andβ

The other component of the data-data correlation matrix is thenoise correlation matrixN Assuming that the noise in differentcells is not correlated the only non-zero terms inN are the diag-onal terms given by the variance - the second central moment -ofthe density error in each cell

Nii =1

N2i

Ngal(i)sum

gal

w2(gal) (36)

The final aspect of the analysis is the reconstruction of thereal-space density field from the redshift-space observations Thisis achieved using equation 8 Following Kaiser (1987) using dis-tant observer and small-angle approximation the cross-correlationmatrix in equation 8 for the linear regime can be written as

〈s(r)d(s)〉 = 〈δrδs〉 = ξ(r)(1 +1

3β) (37)

wheres and r are position vectors in redshift and real space re-

spectively The term(1 + 13β) is easily obtained by integrating

the direction dependent density field in redshift space Using equa-tion 37 the transformation from redshift space to real space simpli-fies to

sWF (r) =1 + 1

3β

K[β]C [K[β]C + N]minus1

d (38)

As mentioned earlier the equation above is calculated for linearscales only and hence small scale distortions (iefingers of God)are not corrected for For this reason we collapse in redshift spacethe fingers seen in 2dF groups (Ekeet al2003) with more than 75members 25 groups in total (11 in NGP and 14 in SGP) All thegalaxies in these groups are assigned the same coordinatesAs ex-pected correcting these small scale distortions does not change theconstructed fields substantially as these distortions are practicallysmoothed out because of the cell size used in binning the data

The maps shown in this section were derived by the techniquedetailed above There are 80 sets of plots which show the densityfields as strips inRA andDec 40 maps for SGP and 40 maps forNGP Here we just show some examples the rest of the plots canbefound in url httpwwwastcamacuksimpirin For comparison thetop plots of Figures 10 11 14 and 15 show the redshift spaceden-sity field weighted by the selection function and the angularmaskThe contours are spaced at∆δ = 05 with solid (dashed) lines de-noting positive (negative) contours the heavy solid contours cor-respond toδ = 0 Also plotted for comparison are the galaxies(red dots) and the groups withNgr number of members (Ekeetal 2003) and9 6 Ngr 6 17 (green circles)18 6 Ngr 6 44(blue squares) and45 6 Ngr (magenta stars) We also show thenumber of Abell APM and EDCC clusters studied by De Propriset al (2002) (black upside down triangles) The middle plots showthe redshift space density shown in top plots after the Wiener Filterapplied As expected the Wiener Filter suppresses the noise Thesmoothing performed by the Wiener Filter is variable and increaseswith distance The bottom plots show the reconstructed realdensityfield s

WF (r) after correcting for the redshift distortions Here theamplitude of density contrast is reduced slightly We also plot thereconstructed fields in declination slices These plots areshown inFigures 7 and 8

We also plot the square root of the variance of the residualfield (equation 2) which defines the scatter around the mean recon-structed field We plot the residual fields corresponding to some ofthe redshift slices shown in this paper (Figures 3 and 4) For bettercomparison plots are made so that the cell number increaseswithincreasingRA If the volume of the cells used to pixelise the surveywas constant we would expect the square root of the variance∆δto increase due to the increase in shot noise (equation 7) Howeversince the pixelisation was constructed to keep the shot noise perpixel approximately constant∆δ also stays constant (∆δ asymp 023for both NGP and SGP) but the average density contrast in eachpixel decreases with increasing redshift This means that althoughthe variance of the residual in each cell is roughly equal the rel-ative variance (represented by∆δ

δ) increases with increasing red-

shift This increase is clearly evident in Figures 3 and 4 Anotherconclusion that can be drawn from the figures is that the bumpsin the density field are due to real features not due to error inthereconstruction even at higher redshifts

62 Reconstruction Using Non-linear Theory

In order to increase the resolution of the density field mapswe re-duce the target cell width to5hminus1 Mpc A volume of a cubic cell

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web7

Figure 3 The plot of overdensities in SGP for the each redshift slice for 10hminus1 Mpc target cell size shown above Also plotted are the variances of the residualassociated for each cellThe increase in cell number indicates the increaseRA in each redshift slice

Figure 4 Same as in figure 3 but for the redshift slices in NGP shown above

of side5hminus1 Mpc is roughly equal to a top-hat sphere of radiusof about3hminus1 Mpc The variance of the mass density field in thissphere isσ3 asymp 17 which corresponds to non-linear scales To re-construct the density field on these scales we require accurate de-scriptions of non-linear galaxy power spectrum and the non-linearredshift space distortions

For the non-linear matter power spectrumP Rnl (k) we adopt

the empirical fitting formula of Smithet al (2003) This formuladerived using the lsquohalo modelrsquo for galaxy clustering is more accu-rate than the widely used Peacock amp Dodds (1996) fitting formulawhich is based on the assumption of lsquostable clusteringrsquo of virial-ized halos We note that for the scales concerned in this paper (upto k asymp 10hMpcminus1) Smith et al (2003) and Peacock amp Dodds(1996) fitting formulae give very similar results For simplicity weassume linear scale independent biasing in order to determine thegalaxy power spectrum from the mass power spectrum wherebmeasures the ratio between galaxy and mass distribution

P Rnl = b2P m

nl (39)

whereP Rnl is the galaxy andP m

nl is the matter power spectrum Weassume thatb = 10 for our analysis While this value is in agree-ment with the result obtained from 2dFGRS (Lahavet al 2002Verdeet al 2002) for scales of tens of Mpc it does not hold truefor the scales of5hminus1 Mpc on which different galaxy populationsshow different clustering patterns (Norberget al2002 Madgwicket al2002 Zehaviet al2002) More realistic models exist wherebiasing is scale dependent (eg Seljak 2000 and Peacock amp Smith2000) but since the Wiener Filtering method is not sensitiveto smallerrors in the prior parameters and the reconstruction scales are nothighly non-linear the simple assumption of no bias will still giveaccurate reconstructions

The main effect of redshift distortions on non-linear scales isthe reduction of power as a result of radial smearing due to viri-alized motions The density profile in redshift space is thentheconvolution of its real space counterpart with the peculiarveloc-ity distribution along the line of sight leading to dampingof poweron small scales This effect is known to be reasonably well approxi-

mated by treating the pairwise peculiar velocity field as Gaussian orbetter still as an exponential in real space (superpositions of Gaus-sians) with dispersionσp (eg Peacock amp Dodds 1994 Ballingeret al1996 amp Kanget al2002) Therefore the galaxy power spec-trum in redshift space is written as

P Snl(k micro) = P R

nl (k micro)(1 + βmicro2)2D(kσpmicro) (40)

wheremicro is the cosine of the wave vector to the line of sightσp

has the unit ofhminus1 Mpc and the damping function ink-space is aLoretzian

D(kσpmicro) =1

1 + (k2σ2pmicro2)2

(41)

Integrating equation 40 overmicro we obtain the direction-averagedpower spectrum in redshift space

P Snl(k)

P Rnl (k)

=4(σ2

pk2 minus β)β

σ4pk4

+2β2

3σ2pk2

+

radic2(k2σ2

p minus 2β)2arctan(kσpradic

2)

k5σ5p

(42)

For the non-linear reconstruction we use equation 42 instead ofequation 33 when deriving the correlation function in redshiftspace Figure 5 shows how the non-linear power spectrum isdamped in redshift space (dashed line) and compared to the lin-ear power spectrum (dotted line) In this plot and throughout thispaper we adopt theσp value derived by Hawkinset al (2002)σp = 506plusmn52kmsminus1 Interestingly by coincidence the non-linearand linear power spectra look very similar in redshift space So ifwe had used the linear power spectrum instead of its non-linearcounterpart we still would have obtained physically accurate re-constructions of the density field in redshift space

The optimal density field in real space is calculated usingequation 8 The cross-correlation matrix in equation 38 cannowbe approximated as

〈s(r)d(s)〉 = ξ(r micro)(1 + βmicro2)radic

D(kσpmicro) (43)

ccopy 0000 RAS MNRAS000 000ndash000

8 Erdogdu et al

Figure 5 Non-linear power spectra for z=0 and the concordance modelwith σp = 506kmsminus1 in real space (solid line) in redshift space fromequation 42 (dashed line) both derived using the fitting formulae of Smithet al (2003) and linear power spectra in redshift space derived using lineartheory and Kaiserrsquos factor (dotted line)

Again integrating the equation above overmicro the direction aver-aged cross-correlation matrix of the density field in real space andthe density field in redshift space can be written as

〈s(r)d(s)〉〈s(r)s(r)〉 =

1

2k2σ2p

ln(k2σ2p(1 +

radic

1 + 1k2σ2p))

+β

k2σ2p

radic

1 + k2σ2p +

β

k3σ3p

arcsinh(k2σ2p) (44)

At the end of this paper we show some examples of the non-linear reconstructions (Figures 9 12 13 and 16) As can be seenfrom these plots the resolution of the reconstructions improves rad-ically down to the scale of large clusters Comparing Figure 15 andFigure 16 where the redshift ranges of the maps are similar we con-clude that 10hminus1 Mpc and 5hminus1 Mpc resolutions give consistentreconstructions

Due to the very large number of cells we reconstruct fourseparate density fields for redshift ranges0035simltzsimlt005 and009simltzsimlt011 in SGP0035simltzsimlt005 and009simltzsimlt012 in NGPThere are 46 redshift slices in total The number of cells forthelatter reconstructions are in the order of 10 000

To investigate the effects of using different non-linear redshiftdistortion approximations we also reconstruct one field without thedamping function but just collapsing the fingers of god Althoughincluding the damping function results in a more physicallyaccu-rate reconstruction of the density field in real space it still is notenough to account for the elongation of the richest clustersalongthe line of sight Collapsing the fingers of god as well as includ-ing the damping function underestimates power resulting innoisyreconstruction of the density field We choose to use only thedamp-ing function for the non-linear scales obtaining stable results for thedensity field reconstruction

The theory of gravitational instability states that as the dynam-ics evolve away from the linear regime the initial field deviatesfrom Gaussianity and skewness develops The Wiener Filterin theform presented here only minimises the variance and it ignores thehigher moments that describe the skewness of the underlyingdis-tribution However since the scales of reconstructions presented inthis section are not highly non-linear and the signal-to-noise ratio

is high the assumptions involved in this analysis are not severelyviolated for the non-linear reconstruction of the density field in red-shift space The real space reconstructions are more sensitive to thechoice of cell size and the power spectrum since the Wiener Filteris used not only for noise suppression but also for transformationfrom redshift space Therefore reconstructions in redshift space aremore reliable on these non-linear scales than those in real space

A different approach to the non-linear density field reconstruc-tion presented in this paper is to apply the Wiener Filter to the re-construction of the logarithm of the density field as there isgoodevidence that the statistical properties of the perturbation field in thequasi-linear regime is well approximated by a log-normal distribu-tion A detailed analysis of the application of the Wiener Filteringtechnique to log-normal fields is given in Sheth (1995)

7 MAPPING THE LARGE SCALE STRUCTURE OF THE2dFGRS

One of the main goals of this paper is to use the reconstructeddensity field to identify the major superclusters and voids in the2dFGRS We define superclusters (voids) as regions of large over-density (underdensity) which is above (below) a certain thresh-old This approach has been used successfully by several authors(eg Einastoet al 2002 amp 2003 Kolokotronis Basilakos amp Plio-nis 2002 Plionis amp Basilakos 2002 Saunderset al1991)

71 The Superclusters

In order to find the superclusters listed in Table 1 we define thedensity contrast thresholdδth as distance dependent We use avarying density threshold for two reasons that are related to the waythe density field is reconstructed Firstly the adaptive gridding weuse implies that the cells get bigger with increasing redshift Thismeans that the density contrast in each cell decreases The secondeffect that decreases the density contrast arises due to theWienerFilter signal tending to zero towards the edges of the survey There-fore for each redshift slice we find the mean and the standarddevi-ation of the density field (averaged over 113 cells for NGP and223for SGP) then we calculateδth as twice the standard deviation ofthe field added to its mean averaged over SGP and NGP for eachredshift bin (see Figure 6) In order to account for the clustering ef-fects we fit a smooth curve toδth usingχ2 minimisation The bestfit also shown Figure 6 is a quadratic equation withχ2(number ofdegrees of freedom) = 16 We use this fit when selecting the over-densities We note that choosing smoothed or unsmoothed densitythreshold does not change the selection of the superclusters

The list of superclusters in SGP and NGP region are givenin Table 1 This table is structured as follows column 1 is theidentification columns 2 and 3 are the minimum and the maxi-mum redshift columns 4 and 5 are the minimum and maximumRA columns 6 and 7 are the minimum and maximumDec overwhich the density contours aboveδth extend In column 8 weshow the number of Durham groups with more than 5 membersthat the supercluster contains In column 9 we show the numberof Abell APM and EDCC clusters studied by De Propriset al(2002) the supercluster has In the last column we show the totalnumber of groups and clusters Note that most of the Abell clus-ters are counted in the Durham group catalogue We observe thatthe rich groups (groups with more than 9 members) almost alwaysreside in superclusters whereas poorer groups are more dispersed

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web9

Figure 6 The density thresholdδth as a function of redshiftz and the bestfit model used to select the superclusters in Table 1

We note that the superclusters that contain Abell clusters are on av-erage richer than superclusters that do not contain Abell clustersin agreement with Einastoet al 2003 However we also note thatthe number of Abell clusters in a rich supercluster can be equal tothe number of Abell clusters in a poorer supercluster whereas thenumber of Durham groups increase as the overdensity increasesThus we conclude that the Durham groups are in general betterrepresentatives of the underlying density distribution of2dFGRSthan Abell clusters

The superclusters SCSGP03 SCSGP04 and SCSGP05 can beseen in Figure 10 SCSGP04 is part the rich Pisces-Cetus Super-cluster which was first described by Tully (1987) SCSGP01 SC-SGP02 SCSGP03 and SCSGP04 are all filamentary structures con-nected to each other forming a multi branching system SCSGP05seems to be a more isolated system possibly connected to SC-SGP06 and SCSGP07 SCSGP06 (figure 11) is the upper part of thegigantic Horoglium Reticulum Supercluster SCSGP07 (Figure 11)is the extended part of the Leo-Coma Supercluster The richest su-percluster in the SGP region is SCSGP16 which can be seen inthe middle of the plots in Figure 12 Also shown in the same fig-ure is SCSGP15 one of the richest superclusters in SGP and theedge of SCSGP17 In fact as evident from Figure 12 SCSGP14SCSGP15 SCSGP16 and SCSGP17 are branches of one big fila-mentary structure In Figure 13 we see SCNGP01 this structure ispart of the upper edge of the Shapley Supercluster The richest su-percluster in NGP is SCNGP06 shown in Figure 14 Around thisin the neighbouring redshift slices there are two rich filamentarysuperclusters SCNGP06 SCNGP08 (figure 15) SCNGP07 seemsto be the node point of these filamentary structures

Table 1 shows only the major overdensities in the surveyThese tend to be filamentary structures that are mostly connectedto each other

72 The Voids

For the catalogue of the largest voids in Table 2 and Table 3 weonly consider regions with 80 percent completeness or more andgo up to a redshift of 015 Following the previous studies (egEl-Ad amp Piran 2000 Bensonet al 2003 amp Shethet al 2003)our chosen underdensity threshold isδth = minus085 The lists ofgalaxies and their properties that are in these voids and in the

other smaller voids that are not in Tables 2 and 3 are given in urlhttpwwwastcamacuksimpirin The tables of voids is structuredin a similar way as the table of superclusters only we give sep-arate tables for NGP and SGP voids The most intriguing struc-tures seen are the voids VSGP01 VSGP2 VSGP03 VSGP4 andVSGP05 These voids clearly visible in the radial number den-sity function of 2dFGRS are actually part of a big superholebro-ken up by the low value of the void density threshold This super-hole has been observed and investigated by several authors (egCrosset al2002 De Propiset al2002 Norberget al2002 Frithet al 2003) The local NGP region also has excess underdensity(VNGP01VNGP02 VNGP03 VNGP04 VNGP05 VNGP06 andVNGP07 again part of a big superhole) but the voids in this areaare not as big or as empty as the voids in local SGP In fact bycombining the results from 2 Micron All Sky Survey Las Cam-panas Survey and 2dFGRS Frithet al (2003) conclude that theseunderdensities suggest that there is a contiguous void stretchingfrom north to south If such a void does exist then it is unexpectedlylarge for our present understanding of large scale structure whereon large enough scales the Universe is isotropic and homogeneous

8 CONCLUSIONS

In this paper we use the Wiener Filtering technique to reconstructthe density field of the 2dF galaxy redshift survey We pixelise thesurvey into igloo cells bounded byRA Dec and redshift The cellsize varies in order to keep the number of galaxies per cell roughlyconstant and is approximately 10hminus1 for Mpc high and 5hminus1

Mpc for the low resolution maps at the median redshift of the sur-vey Assuming a prior based on parametersΩm = 03 ΩΛ = 07β = 049 σ8 = 08 andΓ = 02 we find that the reconstructeddensity field clearly picks out the groups catalogue built byEkeet al (2003) and Abell APM and EDCC clusters investigated byDe Propriset al (2002) We also reconstruct four separate densityfields with different redshift ranges for a smaller cell sizeof 5 hminus1

Mpc at the median redshift For these reconstructions we assume anon-linear power spectrum fit developed by Smithet al(2003) andlinear biasing The resolution of the density field improvesdramat-ically down to the size of big clusters The derived high resolutiondensity fields is in agreement with the lower resolution versions

We use the reconstructed fields to identify the major super-clusters and voids in SGP and NGP We find that the richest su-perclusters are filamentary and multi branching in agreement withEinastoet al 2002 We also find that the rich clusters always re-side in superclusters whereas poor clusters are more dispersed Wepresent the major superclusters in 2dFGRS in Table 1 We pickouttwo very rich superclusters one in SGP and one in NGP We alsoidentify voids as underdensities that are belowδ asymp minus085 and thatlie in regions with more that 80 completeness These underdensi-ties are presented in Table 2 (SGP) and Table 3 (NGP) We pick outtwo big voids one in SGP and one in NGP Unfortunately we can-not measure the sizes and masses of the large structures we observein 2dF as most of these structures continue beyond the boundariesof the survey

The detailed maps and lists of all of the reconstructed den-sity fields plots of the residual fields and the lists of the galax-ies that are in underdense regions can be found on WWW athttpwwwastcamacuksimpirin

One of the main aims of this paper is to identify the largescale structure in 2dFGRS The Wiener Filtering technique pro-vides a rigorous methodology for variable smoothing and noise

ccopy 0000 RAS MNRAS000 000ndash000

8 Erdogdu et al

Figure 5 Non-linear power spectra for z=0 and the concordance modelwith σp = 506kmsminus1 in real space (solid line) in redshift space fromequation 42 (dashed line) both derived using the fitting formulae of Smithet al (2003) and linear power spectra in redshift space derived using lineartheory and Kaiserrsquos factor (dotted line)

Again integrating the equation above overmicro the direction aver-aged cross-correlation matrix of the density field in real space andthe density field in redshift space can be written as

〈s(r)d(s)〉〈s(r)s(r)〉 =

1

2k2σ2p

ln(k2σ2p(1 +

radic

1 + 1k2σ2p))

+β

k2σ2p

radic

1 + k2σ2p +

β

k3σ3p

arcsinh(k2σ2p) (44)

At the end of this paper we show some examples of the non-linear reconstructions (Figures 9 12 13 and 16) As can be seenfrom these plots the resolution of the reconstructions improves rad-ically down to the scale of large clusters Comparing Figure 15 andFigure 16 where the redshift ranges of the maps are similar we con-clude that 10hminus1 Mpc and 5hminus1 Mpc resolutions give consistentreconstructions

Due to the very large number of cells we reconstruct fourseparate density fields for redshift ranges0035simltzsimlt005 and009simltzsimlt011 in SGP0035simltzsimlt005 and009simltzsimlt012 in NGPThere are 46 redshift slices in total The number of cells forthelatter reconstructions are in the order of 10 000

To investigate the effects of using different non-linear redshiftdistortion approximations we also reconstruct one field without thedamping function but just collapsing the fingers of god Althoughincluding the damping function results in a more physicallyaccu-rate reconstruction of the density field in real space it still is notenough to account for the elongation of the richest clustersalongthe line of sight Collapsing the fingers of god as well as includ-ing the damping function underestimates power resulting innoisyreconstruction of the density field We choose to use only thedamp-ing function for the non-linear scales obtaining stable results for thedensity field reconstruction

The theory of gravitational instability states that as the dynam-ics evolve away from the linear regime the initial field deviatesfrom Gaussianity and skewness develops The Wiener Filterin theform presented here only minimises the variance and it ignores thehigher moments that describe the skewness of the underlyingdis-tribution However since the scales of reconstructions presented inthis section are not highly non-linear and the signal-to-noise ratio

is high the assumptions involved in this analysis are not severelyviolated for the non-linear reconstruction of the density field in red-shift space The real space reconstructions are more sensitive to thechoice of cell size and the power spectrum since the Wiener Filteris used not only for noise suppression but also for transformationfrom redshift space Therefore reconstructions in redshift space aremore reliable on these non-linear scales than those in real space

A different approach to the non-linear density field reconstruc-tion presented in this paper is to apply the Wiener Filter to the re-construction of the logarithm of the density field as there isgoodevidence that the statistical properties of the perturbation field in thequasi-linear regime is well approximated by a log-normal distribu-tion A detailed analysis of the application of the Wiener Filteringtechnique to log-normal fields is given in Sheth (1995)

7 MAPPING THE LARGE SCALE STRUCTURE OF THE2dFGRS

One of the main goals of this paper is to use the reconstructeddensity field to identify the major superclusters and voids in the2dFGRS We define superclusters (voids) as regions of large over-density (underdensity) which is above (below) a certain thresh-old This approach has been used successfully by several authors(eg Einastoet al 2002 amp 2003 Kolokotronis Basilakos amp Plio-nis 2002 Plionis amp Basilakos 2002 Saunderset al1991)

71 The Superclusters

In order to find the superclusters listed in Table 1 we define thedensity contrast thresholdδth as distance dependent We use avarying density threshold for two reasons that are related to the waythe density field is reconstructed Firstly the adaptive gridding weuse implies that the cells get bigger with increasing redshift Thismeans that the density contrast in each cell decreases The secondeffect that decreases the density contrast arises due to theWienerFilter signal tending to zero towards the edges of the survey There-fore for each redshift slice we find the mean and the standarddevi-ation of the density field (averaged over 113 cells for NGP and223for SGP) then we calculateδth as twice the standard deviation ofthe field added to its mean averaged over SGP and NGP for eachredshift bin (see Figure 6) In order to account for the clustering ef-fects we fit a smooth curve toδth usingχ2 minimisation The bestfit also shown Figure 6 is a quadratic equation withχ2(number ofdegrees of freedom) = 16 We use this fit when selecting the over-densities We note that choosing smoothed or unsmoothed densitythreshold does not change the selection of the superclusters

The list of superclusters in SGP and NGP region are givenin Table 1 This table is structured as follows column 1 is theidentification columns 2 and 3 are the minimum and the maxi-mum redshift columns 4 and 5 are the minimum and maximumRA columns 6 and 7 are the minimum and maximumDec overwhich the density contours aboveδth extend In column 8 weshow the number of Durham groups with more than 5 membersthat the supercluster contains In column 9 we show the numberof Abell APM and EDCC clusters studied by De Propriset al(2002) the supercluster has In the last column we show the totalnumber of groups and clusters Note that most of the Abell clus-ters are counted in the Durham group catalogue We observe thatthe rich groups (groups with more than 9 members) almost alwaysreside in superclusters whereas poorer groups are more dispersed

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web9

Figure 6 The density thresholdδth as a function of redshiftz and the bestfit model used to select the superclusters in Table 1

We note that the superclusters that contain Abell clusters are on av-erage richer than superclusters that do not contain Abell clustersin agreement with Einastoet al 2003 However we also note thatthe number of Abell clusters in a rich supercluster can be equal tothe number of Abell clusters in a poorer supercluster whereas thenumber of Durham groups increase as the overdensity increasesThus we conclude that the Durham groups are in general betterrepresentatives of the underlying density distribution of2dFGRSthan Abell clusters

The superclusters SCSGP03 SCSGP04 and SCSGP05 can beseen in Figure 10 SCSGP04 is part the rich Pisces-Cetus Super-cluster which was first described by Tully (1987) SCSGP01 SC-SGP02 SCSGP03 and SCSGP04 are all filamentary structures con-nected to each other forming a multi branching system SCSGP05seems to be a more isolated system possibly connected to SC-SGP06 and SCSGP07 SCSGP06 (figure 11) is the upper part of thegigantic Horoglium Reticulum Supercluster SCSGP07 (Figure 11)is the extended part of the Leo-Coma Supercluster The richest su-percluster in the SGP region is SCSGP16 which can be seen inthe middle of the plots in Figure 12 Also shown in the same fig-ure is SCSGP15 one of the richest superclusters in SGP and theedge of SCSGP17 In fact as evident from Figure 12 SCSGP14SCSGP15 SCSGP16 and SCSGP17 are branches of one big fila-mentary structure In Figure 13 we see SCNGP01 this structure ispart of the upper edge of the Shapley Supercluster The richest su-percluster in NGP is SCNGP06 shown in Figure 14 Around thisin the neighbouring redshift slices there are two rich filamentarysuperclusters SCNGP06 SCNGP08 (figure 15) SCNGP07 seemsto be the node point of these filamentary structures

Table 1 shows only the major overdensities in the surveyThese tend to be filamentary structures that are mostly connectedto each other

72 The Voids

For the catalogue of the largest voids in Table 2 and Table 3 weonly consider regions with 80 percent completeness or more andgo up to a redshift of 015 Following the previous studies (egEl-Ad amp Piran 2000 Bensonet al 2003 amp Shethet al 2003)our chosen underdensity threshold isδth = minus085 The lists ofgalaxies and their properties that are in these voids and in the

other smaller voids that are not in Tables 2 and 3 are given in urlhttpwwwastcamacuksimpirin The tables of voids is structuredin a similar way as the table of superclusters only we give sep-arate tables for NGP and SGP voids The most intriguing struc-tures seen are the voids VSGP01 VSGP2 VSGP03 VSGP4 andVSGP05 These voids clearly visible in the radial number den-sity function of 2dFGRS are actually part of a big superholebro-ken up by the low value of the void density threshold This super-hole has been observed and investigated by several authors (egCrosset al2002 De Propiset al2002 Norberget al2002 Frithet al 2003) The local NGP region also has excess underdensity(VNGP01VNGP02 VNGP03 VNGP04 VNGP05 VNGP06 andVNGP07 again part of a big superhole) but the voids in this areaare not as big or as empty as the voids in local SGP In fact bycombining the results from 2 Micron All Sky Survey Las Cam-panas Survey and 2dFGRS Frithet al (2003) conclude that theseunderdensities suggest that there is a contiguous void stretchingfrom north to south If such a void does exist then it is unexpectedlylarge for our present understanding of large scale structure whereon large enough scales the Universe is isotropic and homogeneous

8 CONCLUSIONS

In this paper we use the Wiener Filtering technique to reconstructthe density field of the 2dF galaxy redshift survey We pixelise thesurvey into igloo cells bounded byRA Dec and redshift The cellsize varies in order to keep the number of galaxies per cell roughlyconstant and is approximately 10hminus1 for Mpc high and 5hminus1

Mpc for the low resolution maps at the median redshift of the sur-vey Assuming a prior based on parametersΩm = 03 ΩΛ = 07β = 049 σ8 = 08 andΓ = 02 we find that the reconstructeddensity field clearly picks out the groups catalogue built byEkeet al (2003) and Abell APM and EDCC clusters investigated byDe Propriset al (2002) We also reconstruct four separate densityfields with different redshift ranges for a smaller cell sizeof 5 hminus1

Mpc at the median redshift For these reconstructions we assume anon-linear power spectrum fit developed by Smithet al(2003) andlinear biasing The resolution of the density field improvesdramat-ically down to the size of big clusters The derived high resolutiondensity fields is in agreement with the lower resolution versions

We use the reconstructed fields to identify the major super-clusters and voids in SGP and NGP We find that the richest su-perclusters are filamentary and multi branching in agreement withEinastoet al 2002 We also find that the rich clusters always re-side in superclusters whereas poor clusters are more dispersed Wepresent the major superclusters in 2dFGRS in Table 1 We pickouttwo very rich superclusters one in SGP and one in NGP We alsoidentify voids as underdensities that are belowδ asymp minus085 and thatlie in regions with more that 80 completeness These underdensi-ties are presented in Table 2 (SGP) and Table 3 (NGP) We pick outtwo big voids one in SGP and one in NGP Unfortunately we can-not measure the sizes and masses of the large structures we observein 2dF as most of these structures continue beyond the boundariesof the survey

The detailed maps and lists of all of the reconstructed den-sity fields plots of the residual fields and the lists of the galax-ies that are in underdense regions can be found on WWW athttpwwwastcamacuksimpirin

One of the main aims of this paper is to identify the largescale structure in 2dFGRS The Wiener Filtering technique pro-vides a rigorous methodology for variable smoothing and noise

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web9

Figure 6 The density thresholdδth as a function of redshiftz and the bestfit model used to select the superclusters in Table 1

We note that the superclusters that contain Abell clusters are on av-erage richer than superclusters that do not contain Abell clustersin agreement with Einastoet al 2003 However we also note thatthe number of Abell clusters in a rich supercluster can be equal tothe number of Abell clusters in a poorer supercluster whereas thenumber of Durham groups increase as the overdensity increasesThus we conclude that the Durham groups are in general betterrepresentatives of the underlying density distribution of2dFGRSthan Abell clusters

The superclusters SCSGP03 SCSGP04 and SCSGP05 can beseen in Figure 10 SCSGP04 is part the rich Pisces-Cetus Super-cluster which was first described by Tully (1987) SCSGP01 SC-SGP02 SCSGP03 and SCSGP04 are all filamentary structures con-nected to each other forming a multi branching system SCSGP05seems to be a more isolated system possibly connected to SC-SGP06 and SCSGP07 SCSGP06 (figure 11) is the upper part of thegigantic Horoglium Reticulum Supercluster SCSGP07 (Figure 11)is the extended part of the Leo-Coma Supercluster The richest su-percluster in the SGP region is SCSGP16 which can be seen inthe middle of the plots in Figure 12 Also shown in the same fig-ure is SCSGP15 one of the richest superclusters in SGP and theedge of SCSGP17 In fact as evident from Figure 12 SCSGP14SCSGP15 SCSGP16 and SCSGP17 are branches of one big fila-mentary structure In Figure 13 we see SCNGP01 this structure ispart of the upper edge of the Shapley Supercluster The richest su-percluster in NGP is SCNGP06 shown in Figure 14 Around thisin the neighbouring redshift slices there are two rich filamentarysuperclusters SCNGP06 SCNGP08 (figure 15) SCNGP07 seemsto be the node point of these filamentary structures

Table 1 shows only the major overdensities in the surveyThese tend to be filamentary structures that are mostly connectedto each other

72 The Voids

For the catalogue of the largest voids in Table 2 and Table 3 weonly consider regions with 80 percent completeness or more andgo up to a redshift of 015 Following the previous studies (egEl-Ad amp Piran 2000 Bensonet al 2003 amp Shethet al 2003)our chosen underdensity threshold isδth = minus085 The lists ofgalaxies and their properties that are in these voids and in the

other smaller voids that are not in Tables 2 and 3 are given in urlhttpwwwastcamacuksimpirin The tables of voids is structuredin a similar way as the table of superclusters only we give sep-arate tables for NGP and SGP voids The most intriguing struc-tures seen are the voids VSGP01 VSGP2 VSGP03 VSGP4 andVSGP05 These voids clearly visible in the radial number den-sity function of 2dFGRS are actually part of a big superholebro-ken up by the low value of the void density threshold This super-hole has been observed and investigated by several authors (egCrosset al2002 De Propiset al2002 Norberget al2002 Frithet al 2003) The local NGP region also has excess underdensity(VNGP01VNGP02 VNGP03 VNGP04 VNGP05 VNGP06 andVNGP07 again part of a big superhole) but the voids in this areaare not as big or as empty as the voids in local SGP In fact bycombining the results from 2 Micron All Sky Survey Las Cam-panas Survey and 2dFGRS Frithet al (2003) conclude that theseunderdensities suggest that there is a contiguous void stretchingfrom north to south If such a void does exist then it is unexpectedlylarge for our present understanding of large scale structure whereon large enough scales the Universe is isotropic and homogeneous

8 CONCLUSIONS

In this paper we use the Wiener Filtering technique to reconstructthe density field of the 2dF galaxy redshift survey We pixelise thesurvey into igloo cells bounded byRA Dec and redshift The cellsize varies in order to keep the number of galaxies per cell roughlyconstant and is approximately 10hminus1 for Mpc high and 5hminus1

Mpc for the low resolution maps at the median redshift of the sur-vey Assuming a prior based on parametersΩm = 03 ΩΛ = 07β = 049 σ8 = 08 andΓ = 02 we find that the reconstructeddensity field clearly picks out the groups catalogue built byEkeet al (2003) and Abell APM and EDCC clusters investigated byDe Propriset al (2002) We also reconstruct four separate densityfields with different redshift ranges for a smaller cell sizeof 5 hminus1

Mpc at the median redshift For these reconstructions we assume anon-linear power spectrum fit developed by Smithet al(2003) andlinear biasing The resolution of the density field improvesdramat-ically down to the size of big clusters The derived high resolutiondensity fields is in agreement with the lower resolution versions

We use the reconstructed fields to identify the major super-clusters and voids in SGP and NGP We find that the richest su-perclusters are filamentary and multi branching in agreement withEinastoet al 2002 We also find that the rich clusters always re-side in superclusters whereas poor clusters are more dispersed Wepresent the major superclusters in 2dFGRS in Table 1 We pickouttwo very rich superclusters one in SGP and one in NGP We alsoidentify voids as underdensities that are belowδ asymp minus085 and thatlie in regions with more that 80 completeness These underdensi-ties are presented in Table 2 (SGP) and Table 3 (NGP) We pick outtwo big voids one in SGP and one in NGP Unfortunately we can-not measure the sizes and masses of the large structures we observein 2dF as most of these structures continue beyond the boundariesof the survey

The detailed maps and lists of all of the reconstructed den-sity fields plots of the residual fields and the lists of the galax-ies that are in underdense regions can be found on WWW athttpwwwastcamacuksimpirin

One of the main aims of this paper is to identify the largescale structure in 2dFGRS The Wiener Filtering technique pro-vides a rigorous methodology for variable smoothing and noise

ccopy 0000 RAS MNRAS000 000ndash000

10 Erdogdu et al

suppression As such a natural continuation of this work isus-ing the Wiener filtering method in conjunction with other meth-ods to further investigate the geometry and the topology of thesupercluster-void network Shethet al (2003) developed a pow-erful surface modelling scheme SURFGEN in order to calculatethe Minkowski Functionals of the surface generated from theden-sity field The four Minkowski functionals ndash the area the volumethe extrinsic curvature and the genus ndash contain informationaboutthe geometry connectivity and topology of the surface (cfMeckeBuchert amp Wagner 1994 and Shethet al 2003) and thus they willprovide a detailed morphological analysis of the superclusters andvoids in 2dFGRS

Although not applicable to 2dFGRS the Wiener reconstruc-tion technique is well suited to recovering the velocity fields frompeculiar velocity catalogues Comparisons of galaxy density andvelocity fields allow direct estimations of the cosmological param-eters such as the bias parameter and the mean mass density Thesecomparisons will be possible with the upcoming 6dF Galaxy Sur-vey (httpwwwmsoanueduau6dFGS) which will measure theredshifts of 170000 galaxies and the peculiar velocities of 15000galaxies by June 2005 Compared to 2dFGRS the 6dF surveyhas a much higher sky-coverage (the entire southern sky downto|b| gt 10) This wide survey area would allow a full hemisphericWiener reconstruction of large scale structure so that the sizes andmasses of the superclusters and voids can be determined

ACKNOWLEDGEMENTS

The 2dF Galaxy Redshift Survey was made possible through thededicated efforts of the staff at Anglo-Australian Observatory bothin crediting the two-degree field instrument and supportingit on thetelescope PE acknowledges support from the Middle East Tech-nical University Ankara Turkey the Turkish Higher EducationCouncil and Overseas Research Trust This research was conductedutilising the super computer COSMOS at DAMTP Cambridge

REFERENCES

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432 L75Cross M and 2dFGRS team 2001 MNRAS 324 825Colles M and 2dFGRS team 2001 MNRAS 328 1039Davis M Huchra JP 1982 ApJ 254 437Davis M Geller MJ 1976 ApJ 208 13De Propris R amp the 2dFGRS team 2002 MNRAS 329 87Dressler A 1980 ApJ 236 351El-Ad H Piran T 2000 MNRAS 313 553Einasto J Hutsi G Einasto G Saar E Tucker DL Muller V Heinamaki

P Allam SS 2002 astro-ph02123112Einasto J Einasto G Hrdquoutsi G Saar E Tucker DL TagoE Mrdquouller V

Heinrdquoamrdquoaki P Allam SS 2003 astro-ph0304546Eisenstein DJ Hu W 1998 ApJ 518 2Efstathiou G Moody S 2001 MNRAS 325 1603Eke V amp 2dFGRS Team 2003 submitted to MNRASFrith WJ Busswell GS Fong R Metcalfe N Shanks T 2003 submitted

to MNRASFisher KB Lahav O Hoffman Y Lynden-Bell and ZaroubiS 1995

MNRAS 272 885Freedman WL 2001 ApJ 553 47

Hawkins E amp the 2dFGRS team 2002 MNRAS in pressHogg DWet al 2003 ApJ 585 L5Kaiser N 1987 MNRAS 227 1Kang X Jing YP Mo HJ Borner G 2002 MNRAS 336 892Kolokotronis V Basilakos S amp Plionis M 2002 MNRAS 331 1020Lahav O Fisher KB Hoffman Y Scharf CA and Zaroubi S Wiener

Reconstruction of Galaxy Surveys in Spherical Harmonics 1994 TheAstrophysical Journal Letters 423 L93

Lahav O amp the 2dFGRS team 2002 MNRAS 333 961Maddox SJ Efstathiou G Sutherland WJ 1990 MNRAS 246 433Maddox SJ 2003 in preperationMadgwick D and 2dFGRS team 2002 333133Madgwick D and 2dFGRS team 2003 MNRAS 344 847Mecke KR Buchert T amp Wagner 1994 AampA 288 697Moody S PhD Thesis 2003Nusser A amp Davis M 1994 ApJ 421 L1Norberg P and 2dFGRS team 2001 MNRAS 328 64Norberg P and 2dFGRS team 2002 MNRAS 332 827Peacock JA amp Dodds SJ 1994 MNRAS 267 1020Peacock JA amp Dodds SJ 1996 MNRAS 280 L19Peacock JA amp Smith RE 2000 MNRAS 318 1144Peacock JA amp the 2dFGRS team 2001 Nature 410 169Percival W amp the 2dFGRS team 2001 MNRAS 327 1297Plionis M Basilakos S 2002 MNRAS 330 339Press WH Vetterling WT Teukolsky SA Flannery B P 1992 Numeri-

cal Recipes in Fortran 77 2nd edn Cambridge University Press Cam-bridge

Rybicki GB amp Press WH1992 ApJ 398 169Saunders W Frenk C Rowan-Robinson M Lawrence A Efstathiou G

1991 Nature 349 32Scharf CA Hoffman Y Lahav O Lyden-Bell D 1992 MNRAS 264

439Schmoldt IMet al 1999 ApJ 118 1146Sheth JV Sahni V Shandarin SF Sathyaprakah BS 2003 MNRAS in

pressSeljak U 2000 MNRAS 318 119Sheth JV 2003 submitted to MNRASSheth RK 1995 MNRAS 264 439Smith RE et al 2003 MNRAS 341 1311Tegmark M Efstathiou G 1996 MNRAS 2811297Tully B 1987Atlas of Nearby Galaxies Cambridge University PressWiener N 1949 inExtrapolation and Smoothing of Stationary Time Series

(New York Wiley)Yahil A Strauss MA amp Huchra JP 1991 ApJ 372 380Zaroubi S Hoffman Y Fisher K and Lahav O Wiener Reconstruction

of the Large Scale Structure 1995 The Astrophysical Journal 449446

Zaroubi S Hoffman Y 1996 ApJ 462 25Zaroubi S 2002 MNRAS 331 901Zehavi I amp the SDSS collaboration 2002 ApJ 501 172

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The 2dFGRS Wiener Reconstruction of the Cosmic Web11

Figure 7 Reconstructions of the 2dFGRS SGP region in slices of declination for10hminus1 Mpc target cell size The declination range is given on top each plotThe contours are spaced at∆δ = 10 with solid (dashed) lines denoting positive (negative) contours the heavy solid contours correspond toδ = 0

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12 Erdogdu et al

Figure 8 Reconstructions of the 2dFGRS NGP region in slices of declination for10hminus1 Mpc target cell size The declination range is given on top each plotThe contours are spaced at∆δ = 10 with solid (dashed) lines denoting positive (negative) contours the heavy solid contours correspond toδ = 0

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The 2dFGRS Wiener Reconstruction of the Cosmic Web13

Figure 9 Reconstructions of the 2dFGRS SGP region for the redshift range0047 6 z 6 0049 for 5hminus1 Mpc target cell size The contours are spacedat ∆δ = 05 with solid (dashed) lines denoting positive (negative) contours the heavy solid contours correspond toδ = 0 The red dots denote the galaxieswith redshifts in the plotted range a) Redshift space density field weighted by the selection function and the angular mask b) Same as in a) but smoothed bya Wiener Filter c) Same as in b) but corrected for the redshiftdistortion The overdensity centred on 1)RA asymp minus235 Dec asymp minus300 is SCSGP03 (see Table1) 2) RA asymp 00 Dec asymp minus300 is SCSGP04 3)RA asymp 360 Dec asymp minus293 is SCSGP05 The underdensity centred onRA asymp minus205 Dec asymp minus300 isVSGP01 2)RA asymp minus85 Dec asymp minus293 is VSGP02 3)RA asymp 180 Dec asymp minus285 is VSGP04 4)RA asymp 322 Dec asymp minus295 is VSGP05 (see Table 2)

Figure 10 Reconstructions of the 2dFGRS SGP region for the redshift range0057 6 z 6 0061 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp minus235 Dec asymp minus300 is SCSGP03 (see Table 1) 2)RA asymp 00 Dec asymp minus300 is SCSGP04 this overdensity is part of thePisces-Cetus Supercluster 3)RA asymp 360 Dec asymp minus293 is SCSGP05 The underdensity centred onRA asymp minus100 Dec asymp minus300 is VSGP12 (see Table 2)

Figure 11 Reconstructions of the 2dFGRS SGP region for the redshift range0068 6 z 6 0071 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 390 Dec asymp minus345 is SCSGP07 (see Table 1) and is part of Leo-Coma Supercluster 2) RA asymp 00 Dec asymp minus300 isSCSGP06 and is part of Horoglium Reticulum Supercluster (see Table 1)

Figure 12 Reconstructions of the 2dFGRS SGP region for the redshift range0107 6 z 6 0108 for 5hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 17 Dec asymp minus310 is SCSGP16 (see Table 1) 2)RA asymp 363 Dec asymp minus300 is SCSGP15 3)RA asymp minus150Dec asymp minus300 is SCSGP17The underdensity centred on 1)RA asymp minus250 Dec asymp minus352 is VSGP25 (see Table 2) 2)RA asymp 113 Dec asymp minus245 isVSGP22 3)RA asymp 480 Dec asymp minus305 is VSGP20

Figure 13 Reconstructions of the 2dFGRS NGP region for the redshift range0039 6 z 6 0041 for 5hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred onRA asymp 1530 Dec asymp minus40 is SCNGP01 and is part of the Shapley Supercluster (see Table1) The underdensities are VNGP01VNGP02 VNGP03 VNGP04 VNGP05 VNGP06 and VNGP07 (see Table 3)

Figure 14 Reconstructions of the 2dFGRS NGP region for the redshift range0082 6 z 6 0086 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred onRA asymp 1940 Dec asymp minus25 is SCNGP06 (see Table 1)

Figure 15 Reconstructions of the 2dFGRS NGP region for the redshift range0100 6 z 6 0104 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 1700 Dec asymp minus10 is SCNGP08 (see Table 1) The underdensity centred on 1)RA asymp 1500 Dec asymp minus15 is VNGP18(see Table 3) 2)RA asymp 1925 Dec asymp 05 is VNGP19 2)RA asymp 2090 Dec asymp minus15 is VNGP17

Figure 16 Reconstructions of the 2dFGRS NGP region for the redshift range0103 6 z 6 0108 for 5hminus1 Mpc target cell size Same as in figure 9

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14 Erdogdu et al

Table 1 The list of superclusters

No zmin zmax RAmin RAmax Decmin Decmax Ngr gt 9 Nclus Ntotal

(1950) deg (1950) deg (1950) deg (1950) deg

SCSGP01 0048 0054 minus115 minus32 minus375 minus255 23 9 27SCSGP02 0054 0057 minus223 00 minus355 minus330 9 1 9SCSGP03 0057 0064 minus294 minus175 minus375 minus225 23 8 26SCSGP04 0057 0068 minus100 +100 minus355 minus240 34 14 37SCSGP05 0054 0064 325 390 minus330 minus255 14 7 18SCSGP06 0064 0082 455 550 minus350 minus240 26 7 31SCSGP07 0064 0071 185 400 minus350 minus310 10 4 11SCSGP08 0068 0075 minus385 minus330 minus350 minus225 9 3 11SCSGP09 0075 0082 minus270 minus175 minus375 minus225 20 9 24SCSGP10 0082 0093 minus175 minus95 minus360 minus290 18 9 21SCSGP11 0086 0097 minus330 minus270 minus345 minus240 14 4 15SCSGP12 0093 0097 minus220 minus174 minus360 minus348 4 1 5SCSGP13 0093 0097 00 38 minus360 minus348 3 0 3SCSGP14 0093 0104 160 246 minus330 minus282 8 1 8SCSGP15 0097 0115 280 446 minus352 minus246 34 18 41SCSGP16 0100 0119 15 219 minus352 minus265 93 20 98SCSGP17 0100 0108 minus271 minus30 minus352 minus265 38 7 43SCSGP18 0150 0166 15 318 minus330 minus265 18 6 22SCSGP19 0162 0177 228 345 minus352 minus246 13 5 15SCSGP20 0181 0202 minus175 minus315 minus355 minus246 13 4 14

SCNGP01 0035 0068 1475 1745 minus65 25 59 6 61SCNGP02 0048 0061 2100 2185 minus50 00 19 0 19SCNGP03 0068 0075 1500 1550 minus10 25 6 0 6SCNGP04 0068 0075 1580 1665 minus10 25 8 1 9SCNGP05 0071 0082 1710 1840 minus35 2 5 27 6 28SCNGP06 0079 0094 1850 2025 minus75 25 77 7 79SCNGP07 0086 0101 1470 1815 minus55 25 31 4 34SCNGP08 0090 0131 1650 1850 minus63 25 51 10 57SCNGP09 0103 0114 1585 1630 minus13 25 8 1 9SCNGP10 0103 0118 1970 2030 minus375 125 22 0 22SCNGP11 0123 0131 1470 1730 00 25 3 2 3SCNGP12 0119 0142 2100 2190 minus40 13 19 2 20SCNGP13 0131 0142 1730 1790 minus625 minus13 5 2 6SCNGP14 0131 0142 1850 1930 minus625 13 7 0 7SCNGP15 0131 0138 1550 1600 minus63 13 9 1 9SCNGP16 0142 0146 1660 1740 minus03 13 5 0 5SCNGP17 0142 0146 1940 1990 minus25 13 1 0 1SCNGP18 0146 0150 2040 2100 minus25 13 4 0 4SCNGP19 0173 0177 1810 1840 minus10 minus40 3 0 3SCNGP20 0181 0185 1855 1930 minus10 minus40 1 0 1

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The 2dFGRS Wiener Reconstruction of the Cosmic Web15

Table 2 The list of voids in SGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VSGP01 0035 0051 minus275 minus173 minus375 minus225VSGP02 0035 0051 minus173 00 minus330 minus255VSGP03 0035 0048 09 115 minus345 minus285VSGP04 0035 0051 124 236 minus305 minus260VSGP05 0035 0054 236 408 minus335 minus260VSGP06 0035 0042 328 373 minus335 minus285VSGP07 0035 0051 395 464 minus315 minus285VSGP08 0035 0042 464 510 minus315 minus285VSGP09 0039 0044 minus355 minus300 minus345 minus285VSGP10 0039 0041 100 175 minus335 minus295VSGP11 0041 0044 155 225 minus335 minus285VSGP12 0057 0061 minus128 minus82 minus315 minus285VSGP13 0082 0086 54 80 minus315 minus285VSGP15 0082 0089 418 493 minus315 minus285VSGP16 0086 0089 125 163 minus315 minus285VSGP17 0086 0089 310 335 minus325 minus285VSGP18 0089 0097 minus82 minus595 minus305 minus295VSGP19 0089 0093 427 455 minus305 minus295VSGP20 0093 0104 464 525 minus325 minus282VSGP21 0093 0100 282 328 minus315 minus285VSGP22 0097 0104 77 125 minus320 minus285VSGP23 0097 0100 minus128 minus115 minus320 minus285VSGP24 0100 0108 minus318 minus253 minus285 minus255VSGP25 0100 0112 minus253 minus219 minus345 minus330VSGP26 0112 0119 minus70 minus37 minus315 minus285VSGP27 0112 0115 15 54 minus300 minus270VSGP28 0112 0115 259 282 minus300 minus270VSGP29 0112 0115 410 464 minus300 minus270VSGP30 0115 0119 262 305 minus345 minus315VSGP31 0119 0123 minus350 minus308 minus285 minus255VSGP32 0119 0123 minus270 minus255 minus285 minus265VSGP33 0119 0123 minus128 minus100 minus285 minus255VSGP34 0119 0123 415 493 minus388 minus318VSGP35 0119 0127 112 165 minus282 minus275VSGP36 0123 0131 minus264 minus224 minus315 minus285VSGP37 0123 0127 minus128 minus82 minus300 minus270VSGP38 0131 0142 minus264 minus195 minus345 minus315VSGP39 0131 0138 323 375 minus352 minus248VSGP40 0138 0142 303 335 minus335 minus290

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16 Erdogdu et al

Table 3 The list of voids in NGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VNGP01 0035 0054 1470 1600 minus40 15VNGP02 0035 0041 1660 1710 minus10 25VNGP03 0035 0046 1710 1812 minus15 25VNGP04 0035 0051 1830 1890 minus15 25VNGP05 0035 0046 1890 1970 minus15 20VNGP06 0035 0046 2022 2060 minus15 20VNGP07 0037 0046 2042 2156 minus20 00VNGP08 0041 0051 1638 1698 minus45 25VNGP09 0049 0061 1770 1964 minus35 25VNGP10 0054 0057 1655 1730 minus15 25VNGP11 0057 0061 1635 1660 00 15VNGP12 0054 0061 2060 2080 minus15 05VNGP13 0071 0075 1760 1790 minus10 05VNGP14 0079 0094 1865 2042 minus75 05VNGP15 0092 0097 1508 1602 minus75 minus60VNGP16 0093 0100 1470 1495 minus6 15VNGP17 0094 0101 2040 2115 minus35 25VNGP18 0099 0115 1475 1536 minus40 10VNGP19 0099 0120 1888 1970 minus30 25VNGP20 0112 0120 1766 1780 05 25VNGP21 0110 0115 2116 2170 minus25 minus05VNGP22 0110 0123 2002 2050 minus325 25VNGP23 0123 0131 1690 1740 minus40 minus125VNGP24 0127 0131 1825 1880 00 25VNGP25 0127 0131 1975 2020 minus125 05VNGP26 0134 0142 1795 1875 minus20 25VNGP27 0134 0138 2040 2070 minus15 00VNGP28 0138 0150 1470 1546 minus35 25VNGP29 0142 0150 1620 1660 minus40 15VNGP30 0142 0150 1726 1780 minus10 25

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• INTRODUCTION
• Wiener Filter
• The Data
• • The 2dFGRS data
  • The Mask and The Radial Selection Function of 2dFGRS
  • • Survey Pixelisation
    • Estimating The Signal-Signal Correlation Matrix Over Pixels
    • The Application
    • • Reconstruction Using Linear Theory
      • Reconstruction Using Non-linear Theory
      • • Mapping the Large Scale Structure of the 2dFGRS
        • • The Superclusters
          • The Voids
          • • Conclusions
            • REFERENCES

12 Erdogdu et al

Figure 8 Reconstructions of the 2dFGRS NGP region in slices of declination for10hminus1 Mpc target cell size The declination range is given on top each plotThe contours are spaced at∆δ = 10 with solid (dashed) lines denoting positive (negative) contours the heavy solid contours correspond toδ = 0

ccopy 0000 RAS MNRAS000 000ndash000

The 2dFGRS Wiener Reconstruction of the Cosmic Web13

Figure 9 Reconstructions of the 2dFGRS SGP region for the redshift range0047 6 z 6 0049 for 5hminus1 Mpc target cell size The contours are spacedat ∆δ = 05 with solid (dashed) lines denoting positive (negative) contours the heavy solid contours correspond toδ = 0 The red dots denote the galaxieswith redshifts in the plotted range a) Redshift space density field weighted by the selection function and the angular mask b) Same as in a) but smoothed bya Wiener Filter c) Same as in b) but corrected for the redshiftdistortion The overdensity centred on 1)RA asymp minus235 Dec asymp minus300 is SCSGP03 (see Table1) 2) RA asymp 00 Dec asymp minus300 is SCSGP04 3)RA asymp 360 Dec asymp minus293 is SCSGP05 The underdensity centred onRA asymp minus205 Dec asymp minus300 isVSGP01 2)RA asymp minus85 Dec asymp minus293 is VSGP02 3)RA asymp 180 Dec asymp minus285 is VSGP04 4)RA asymp 322 Dec asymp minus295 is VSGP05 (see Table 2)

Figure 10 Reconstructions of the 2dFGRS SGP region for the redshift range0057 6 z 6 0061 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp minus235 Dec asymp minus300 is SCSGP03 (see Table 1) 2)RA asymp 00 Dec asymp minus300 is SCSGP04 this overdensity is part of thePisces-Cetus Supercluster 3)RA asymp 360 Dec asymp minus293 is SCSGP05 The underdensity centred onRA asymp minus100 Dec asymp minus300 is VSGP12 (see Table 2)

Figure 11 Reconstructions of the 2dFGRS SGP region for the redshift range0068 6 z 6 0071 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 390 Dec asymp minus345 is SCSGP07 (see Table 1) and is part of Leo-Coma Supercluster 2) RA asymp 00 Dec asymp minus300 isSCSGP06 and is part of Horoglium Reticulum Supercluster (see Table 1)

Figure 12 Reconstructions of the 2dFGRS SGP region for the redshift range0107 6 z 6 0108 for 5hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 17 Dec asymp minus310 is SCSGP16 (see Table 1) 2)RA asymp 363 Dec asymp minus300 is SCSGP15 3)RA asymp minus150Dec asymp minus300 is SCSGP17The underdensity centred on 1)RA asymp minus250 Dec asymp minus352 is VSGP25 (see Table 2) 2)RA asymp 113 Dec asymp minus245 isVSGP22 3)RA asymp 480 Dec asymp minus305 is VSGP20

Figure 13 Reconstructions of the 2dFGRS NGP region for the redshift range0039 6 z 6 0041 for 5hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred onRA asymp 1530 Dec asymp minus40 is SCNGP01 and is part of the Shapley Supercluster (see Table1) The underdensities are VNGP01VNGP02 VNGP03 VNGP04 VNGP05 VNGP06 and VNGP07 (see Table 3)

Figure 14 Reconstructions of the 2dFGRS NGP region for the redshift range0082 6 z 6 0086 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred onRA asymp 1940 Dec asymp minus25 is SCNGP06 (see Table 1)

Figure 15 Reconstructions of the 2dFGRS NGP region for the redshift range0100 6 z 6 0104 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 1700 Dec asymp minus10 is SCNGP08 (see Table 1) The underdensity centred on 1)RA asymp 1500 Dec asymp minus15 is VNGP18(see Table 3) 2)RA asymp 1925 Dec asymp 05 is VNGP19 2)RA asymp 2090 Dec asymp minus15 is VNGP17

Figure 16 Reconstructions of the 2dFGRS NGP region for the redshift range0103 6 z 6 0108 for 5hminus1 Mpc target cell size Same as in figure 9

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14 Erdogdu et al

Table 1 The list of superclusters

No zmin zmax RAmin RAmax Decmin Decmax Ngr gt 9 Nclus Ntotal

(1950) deg (1950) deg (1950) deg (1950) deg

SCSGP01 0048 0054 minus115 minus32 minus375 minus255 23 9 27SCSGP02 0054 0057 minus223 00 minus355 minus330 9 1 9SCSGP03 0057 0064 minus294 minus175 minus375 minus225 23 8 26SCSGP04 0057 0068 minus100 +100 minus355 minus240 34 14 37SCSGP05 0054 0064 325 390 minus330 minus255 14 7 18SCSGP06 0064 0082 455 550 minus350 minus240 26 7 31SCSGP07 0064 0071 185 400 minus350 minus310 10 4 11SCSGP08 0068 0075 minus385 minus330 minus350 minus225 9 3 11SCSGP09 0075 0082 minus270 minus175 minus375 minus225 20 9 24SCSGP10 0082 0093 minus175 minus95 minus360 minus290 18 9 21SCSGP11 0086 0097 minus330 minus270 minus345 minus240 14 4 15SCSGP12 0093 0097 minus220 minus174 minus360 minus348 4 1 5SCSGP13 0093 0097 00 38 minus360 minus348 3 0 3SCSGP14 0093 0104 160 246 minus330 minus282 8 1 8SCSGP15 0097 0115 280 446 minus352 minus246 34 18 41SCSGP16 0100 0119 15 219 minus352 minus265 93 20 98SCSGP17 0100 0108 minus271 minus30 minus352 minus265 38 7 43SCSGP18 0150 0166 15 318 minus330 minus265 18 6 22SCSGP19 0162 0177 228 345 minus352 minus246 13 5 15SCSGP20 0181 0202 minus175 minus315 minus355 minus246 13 4 14

SCNGP01 0035 0068 1475 1745 minus65 25 59 6 61SCNGP02 0048 0061 2100 2185 minus50 00 19 0 19SCNGP03 0068 0075 1500 1550 minus10 25 6 0 6SCNGP04 0068 0075 1580 1665 minus10 25 8 1 9SCNGP05 0071 0082 1710 1840 minus35 2 5 27 6 28SCNGP06 0079 0094 1850 2025 minus75 25 77 7 79SCNGP07 0086 0101 1470 1815 minus55 25 31 4 34SCNGP08 0090 0131 1650 1850 minus63 25 51 10 57SCNGP09 0103 0114 1585 1630 minus13 25 8 1 9SCNGP10 0103 0118 1970 2030 minus375 125 22 0 22SCNGP11 0123 0131 1470 1730 00 25 3 2 3SCNGP12 0119 0142 2100 2190 minus40 13 19 2 20SCNGP13 0131 0142 1730 1790 minus625 minus13 5 2 6SCNGP14 0131 0142 1850 1930 minus625 13 7 0 7SCNGP15 0131 0138 1550 1600 minus63 13 9 1 9SCNGP16 0142 0146 1660 1740 minus03 13 5 0 5SCNGP17 0142 0146 1940 1990 minus25 13 1 0 1SCNGP18 0146 0150 2040 2100 minus25 13 4 0 4SCNGP19 0173 0177 1810 1840 minus10 minus40 3 0 3SCNGP20 0181 0185 1855 1930 minus10 minus40 1 0 1

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The 2dFGRS Wiener Reconstruction of the Cosmic Web15

Table 2 The list of voids in SGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VSGP01 0035 0051 minus275 minus173 minus375 minus225VSGP02 0035 0051 minus173 00 minus330 minus255VSGP03 0035 0048 09 115 minus345 minus285VSGP04 0035 0051 124 236 minus305 minus260VSGP05 0035 0054 236 408 minus335 minus260VSGP06 0035 0042 328 373 minus335 minus285VSGP07 0035 0051 395 464 minus315 minus285VSGP08 0035 0042 464 510 minus315 minus285VSGP09 0039 0044 minus355 minus300 minus345 minus285VSGP10 0039 0041 100 175 minus335 minus295VSGP11 0041 0044 155 225 minus335 minus285VSGP12 0057 0061 minus128 minus82 minus315 minus285VSGP13 0082 0086 54 80 minus315 minus285VSGP15 0082 0089 418 493 minus315 minus285VSGP16 0086 0089 125 163 minus315 minus285VSGP17 0086 0089 310 335 minus325 minus285VSGP18 0089 0097 minus82 minus595 minus305 minus295VSGP19 0089 0093 427 455 minus305 minus295VSGP20 0093 0104 464 525 minus325 minus282VSGP21 0093 0100 282 328 minus315 minus285VSGP22 0097 0104 77 125 minus320 minus285VSGP23 0097 0100 minus128 minus115 minus320 minus285VSGP24 0100 0108 minus318 minus253 minus285 minus255VSGP25 0100 0112 minus253 minus219 minus345 minus330VSGP26 0112 0119 minus70 minus37 minus315 minus285VSGP27 0112 0115 15 54 minus300 minus270VSGP28 0112 0115 259 282 minus300 minus270VSGP29 0112 0115 410 464 minus300 minus270VSGP30 0115 0119 262 305 minus345 minus315VSGP31 0119 0123 minus350 minus308 minus285 minus255VSGP32 0119 0123 minus270 minus255 minus285 minus265VSGP33 0119 0123 minus128 minus100 minus285 minus255VSGP34 0119 0123 415 493 minus388 minus318VSGP35 0119 0127 112 165 minus282 minus275VSGP36 0123 0131 minus264 minus224 minus315 minus285VSGP37 0123 0127 minus128 minus82 minus300 minus270VSGP38 0131 0142 minus264 minus195 minus345 minus315VSGP39 0131 0138 323 375 minus352 minus248VSGP40 0138 0142 303 335 minus335 minus290

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16 Erdogdu et al

Table 3 The list of voids in NGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VNGP01 0035 0054 1470 1600 minus40 15VNGP02 0035 0041 1660 1710 minus10 25VNGP03 0035 0046 1710 1812 minus15 25VNGP04 0035 0051 1830 1890 minus15 25VNGP05 0035 0046 1890 1970 minus15 20VNGP06 0035 0046 2022 2060 minus15 20VNGP07 0037 0046 2042 2156 minus20 00VNGP08 0041 0051 1638 1698 minus45 25VNGP09 0049 0061 1770 1964 minus35 25VNGP10 0054 0057 1655 1730 minus15 25VNGP11 0057 0061 1635 1660 00 15VNGP12 0054 0061 2060 2080 minus15 05VNGP13 0071 0075 1760 1790 minus10 05VNGP14 0079 0094 1865 2042 minus75 05VNGP15 0092 0097 1508 1602 minus75 minus60VNGP16 0093 0100 1470 1495 minus6 15VNGP17 0094 0101 2040 2115 minus35 25VNGP18 0099 0115 1475 1536 minus40 10VNGP19 0099 0120 1888 1970 minus30 25VNGP20 0112 0120 1766 1780 05 25VNGP21 0110 0115 2116 2170 minus25 minus05VNGP22 0110 0123 2002 2050 minus325 25VNGP23 0123 0131 1690 1740 minus40 minus125VNGP24 0127 0131 1825 1880 00 25VNGP25 0127 0131 1975 2020 minus125 05VNGP26 0134 0142 1795 1875 minus20 25VNGP27 0134 0138 2040 2070 minus15 00VNGP28 0138 0150 1470 1546 minus35 25VNGP29 0142 0150 1620 1660 minus40 15VNGP30 0142 0150 1726 1780 minus10 25

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• INTRODUCTION
• Wiener Filter
• The Data
• • The 2dFGRS data
  • The Mask and The Radial Selection Function of 2dFGRS
  • • Survey Pixelisation
    • Estimating The Signal-Signal Correlation Matrix Over Pixels
    • The Application
    • • Reconstruction Using Linear Theory
      • Reconstruction Using Non-linear Theory
      • • Mapping the Large Scale Structure of the 2dFGRS
        • • The Superclusters
          • The Voids
          • • Conclusions
            • REFERENCES

The 2dFGRS Wiener Reconstruction of the Cosmic Web13

Figure 9 Reconstructions of the 2dFGRS SGP region for the redshift range0047 6 z 6 0049 for 5hminus1 Mpc target cell size The contours are spacedat ∆δ = 05 with solid (dashed) lines denoting positive (negative) contours the heavy solid contours correspond toδ = 0 The red dots denote the galaxieswith redshifts in the plotted range a) Redshift space density field weighted by the selection function and the angular mask b) Same as in a) but smoothed bya Wiener Filter c) Same as in b) but corrected for the redshiftdistortion The overdensity centred on 1)RA asymp minus235 Dec asymp minus300 is SCSGP03 (see Table1) 2) RA asymp 00 Dec asymp minus300 is SCSGP04 3)RA asymp 360 Dec asymp minus293 is SCSGP05 The underdensity centred onRA asymp minus205 Dec asymp minus300 isVSGP01 2)RA asymp minus85 Dec asymp minus293 is VSGP02 3)RA asymp 180 Dec asymp minus285 is VSGP04 4)RA asymp 322 Dec asymp minus295 is VSGP05 (see Table 2)

Figure 10 Reconstructions of the 2dFGRS SGP region for the redshift range0057 6 z 6 0061 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp minus235 Dec asymp minus300 is SCSGP03 (see Table 1) 2)RA asymp 00 Dec asymp minus300 is SCSGP04 this overdensity is part of thePisces-Cetus Supercluster 3)RA asymp 360 Dec asymp minus293 is SCSGP05 The underdensity centred onRA asymp minus100 Dec asymp minus300 is VSGP12 (see Table 2)

Figure 11 Reconstructions of the 2dFGRS SGP region for the redshift range0068 6 z 6 0071 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 390 Dec asymp minus345 is SCSGP07 (see Table 1) and is part of Leo-Coma Supercluster 2) RA asymp 00 Dec asymp minus300 isSCSGP06 and is part of Horoglium Reticulum Supercluster (see Table 1)

Figure 12 Reconstructions of the 2dFGRS SGP region for the redshift range0107 6 z 6 0108 for 5hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 17 Dec asymp minus310 is SCSGP16 (see Table 1) 2)RA asymp 363 Dec asymp minus300 is SCSGP15 3)RA asymp minus150Dec asymp minus300 is SCSGP17The underdensity centred on 1)RA asymp minus250 Dec asymp minus352 is VSGP25 (see Table 2) 2)RA asymp 113 Dec asymp minus245 isVSGP22 3)RA asymp 480 Dec asymp minus305 is VSGP20

Figure 13 Reconstructions of the 2dFGRS NGP region for the redshift range0039 6 z 6 0041 for 5hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred onRA asymp 1530 Dec asymp minus40 is SCNGP01 and is part of the Shapley Supercluster (see Table1) The underdensities are VNGP01VNGP02 VNGP03 VNGP04 VNGP05 VNGP06 and VNGP07 (see Table 3)

Figure 14 Reconstructions of the 2dFGRS NGP region for the redshift range0082 6 z 6 0086 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred onRA asymp 1940 Dec asymp minus25 is SCNGP06 (see Table 1)

Figure 15 Reconstructions of the 2dFGRS NGP region for the redshift range0100 6 z 6 0104 for 10hminus1 Mpc target cell size Same as in figure 9Theoverdensity centred on 1)RA asymp 1700 Dec asymp minus10 is SCNGP08 (see Table 1) The underdensity centred on 1)RA asymp 1500 Dec asymp minus15 is VNGP18(see Table 3) 2)RA asymp 1925 Dec asymp 05 is VNGP19 2)RA asymp 2090 Dec asymp minus15 is VNGP17

Figure 16 Reconstructions of the 2dFGRS NGP region for the redshift range0103 6 z 6 0108 for 5hminus1 Mpc target cell size Same as in figure 9

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14 Erdogdu et al

Table 1 The list of superclusters

No zmin zmax RAmin RAmax Decmin Decmax Ngr gt 9 Nclus Ntotal

(1950) deg (1950) deg (1950) deg (1950) deg

SCSGP01 0048 0054 minus115 minus32 minus375 minus255 23 9 27SCSGP02 0054 0057 minus223 00 minus355 minus330 9 1 9SCSGP03 0057 0064 minus294 minus175 minus375 minus225 23 8 26SCSGP04 0057 0068 minus100 +100 minus355 minus240 34 14 37SCSGP05 0054 0064 325 390 minus330 minus255 14 7 18SCSGP06 0064 0082 455 550 minus350 minus240 26 7 31SCSGP07 0064 0071 185 400 minus350 minus310 10 4 11SCSGP08 0068 0075 minus385 minus330 minus350 minus225 9 3 11SCSGP09 0075 0082 minus270 minus175 minus375 minus225 20 9 24SCSGP10 0082 0093 minus175 minus95 minus360 minus290 18 9 21SCSGP11 0086 0097 minus330 minus270 minus345 minus240 14 4 15SCSGP12 0093 0097 minus220 minus174 minus360 minus348 4 1 5SCSGP13 0093 0097 00 38 minus360 minus348 3 0 3SCSGP14 0093 0104 160 246 minus330 minus282 8 1 8SCSGP15 0097 0115 280 446 minus352 minus246 34 18 41SCSGP16 0100 0119 15 219 minus352 minus265 93 20 98SCSGP17 0100 0108 minus271 minus30 minus352 minus265 38 7 43SCSGP18 0150 0166 15 318 minus330 minus265 18 6 22SCSGP19 0162 0177 228 345 minus352 minus246 13 5 15SCSGP20 0181 0202 minus175 minus315 minus355 minus246 13 4 14

SCNGP01 0035 0068 1475 1745 minus65 25 59 6 61SCNGP02 0048 0061 2100 2185 minus50 00 19 0 19SCNGP03 0068 0075 1500 1550 minus10 25 6 0 6SCNGP04 0068 0075 1580 1665 minus10 25 8 1 9SCNGP05 0071 0082 1710 1840 minus35 2 5 27 6 28SCNGP06 0079 0094 1850 2025 minus75 25 77 7 79SCNGP07 0086 0101 1470 1815 minus55 25 31 4 34SCNGP08 0090 0131 1650 1850 minus63 25 51 10 57SCNGP09 0103 0114 1585 1630 minus13 25 8 1 9SCNGP10 0103 0118 1970 2030 minus375 125 22 0 22SCNGP11 0123 0131 1470 1730 00 25 3 2 3SCNGP12 0119 0142 2100 2190 minus40 13 19 2 20SCNGP13 0131 0142 1730 1790 minus625 minus13 5 2 6SCNGP14 0131 0142 1850 1930 minus625 13 7 0 7SCNGP15 0131 0138 1550 1600 minus63 13 9 1 9SCNGP16 0142 0146 1660 1740 minus03 13 5 0 5SCNGP17 0142 0146 1940 1990 minus25 13 1 0 1SCNGP18 0146 0150 2040 2100 minus25 13 4 0 4SCNGP19 0173 0177 1810 1840 minus10 minus40 3 0 3SCNGP20 0181 0185 1855 1930 minus10 minus40 1 0 1

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The 2dFGRS Wiener Reconstruction of the Cosmic Web15

Table 2 The list of voids in SGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VSGP01 0035 0051 minus275 minus173 minus375 minus225VSGP02 0035 0051 minus173 00 minus330 minus255VSGP03 0035 0048 09 115 minus345 minus285VSGP04 0035 0051 124 236 minus305 minus260VSGP05 0035 0054 236 408 minus335 minus260VSGP06 0035 0042 328 373 minus335 minus285VSGP07 0035 0051 395 464 minus315 minus285VSGP08 0035 0042 464 510 minus315 minus285VSGP09 0039 0044 minus355 minus300 minus345 minus285VSGP10 0039 0041 100 175 minus335 minus295VSGP11 0041 0044 155 225 minus335 minus285VSGP12 0057 0061 minus128 minus82 minus315 minus285VSGP13 0082 0086 54 80 minus315 minus285VSGP15 0082 0089 418 493 minus315 minus285VSGP16 0086 0089 125 163 minus315 minus285VSGP17 0086 0089 310 335 minus325 minus285VSGP18 0089 0097 minus82 minus595 minus305 minus295VSGP19 0089 0093 427 455 minus305 minus295VSGP20 0093 0104 464 525 minus325 minus282VSGP21 0093 0100 282 328 minus315 minus285VSGP22 0097 0104 77 125 minus320 minus285VSGP23 0097 0100 minus128 minus115 minus320 minus285VSGP24 0100 0108 minus318 minus253 minus285 minus255VSGP25 0100 0112 minus253 minus219 minus345 minus330VSGP26 0112 0119 minus70 minus37 minus315 minus285VSGP27 0112 0115 15 54 minus300 minus270VSGP28 0112 0115 259 282 minus300 minus270VSGP29 0112 0115 410 464 minus300 minus270VSGP30 0115 0119 262 305 minus345 minus315VSGP31 0119 0123 minus350 minus308 minus285 minus255VSGP32 0119 0123 minus270 minus255 minus285 minus265VSGP33 0119 0123 minus128 minus100 minus285 minus255VSGP34 0119 0123 415 493 minus388 minus318VSGP35 0119 0127 112 165 minus282 minus275VSGP36 0123 0131 minus264 minus224 minus315 minus285VSGP37 0123 0127 minus128 minus82 minus300 minus270VSGP38 0131 0142 minus264 minus195 minus345 minus315VSGP39 0131 0138 323 375 minus352 minus248VSGP40 0138 0142 303 335 minus335 minus290

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16 Erdogdu et al

Table 3 The list of voids in NGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VNGP01 0035 0054 1470 1600 minus40 15VNGP02 0035 0041 1660 1710 minus10 25VNGP03 0035 0046 1710 1812 minus15 25VNGP04 0035 0051 1830 1890 minus15 25VNGP05 0035 0046 1890 1970 minus15 20VNGP06 0035 0046 2022 2060 minus15 20VNGP07 0037 0046 2042 2156 minus20 00VNGP08 0041 0051 1638 1698 minus45 25VNGP09 0049 0061 1770 1964 minus35 25VNGP10 0054 0057 1655 1730 minus15 25VNGP11 0057 0061 1635 1660 00 15VNGP12 0054 0061 2060 2080 minus15 05VNGP13 0071 0075 1760 1790 minus10 05VNGP14 0079 0094 1865 2042 minus75 05VNGP15 0092 0097 1508 1602 minus75 minus60VNGP16 0093 0100 1470 1495 minus6 15VNGP17 0094 0101 2040 2115 minus35 25VNGP18 0099 0115 1475 1536 minus40 10VNGP19 0099 0120 1888 1970 minus30 25VNGP20 0112 0120 1766 1780 05 25VNGP21 0110 0115 2116 2170 minus25 minus05VNGP22 0110 0123 2002 2050 minus325 25VNGP23 0123 0131 1690 1740 minus40 minus125VNGP24 0127 0131 1825 1880 00 25VNGP25 0127 0131 1975 2020 minus125 05VNGP26 0134 0142 1795 1875 minus20 25VNGP27 0134 0138 2040 2070 minus15 00VNGP28 0138 0150 1470 1546 minus35 25VNGP29 0142 0150 1620 1660 minus40 15VNGP30 0142 0150 1726 1780 minus10 25

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• INTRODUCTION
• Wiener Filter
• The Data
• • The 2dFGRS data
  • The Mask and The Radial Selection Function of 2dFGRS
  • • Survey Pixelisation
    • Estimating The Signal-Signal Correlation Matrix Over Pixels
    • The Application
    • • Reconstruction Using Linear Theory
      • Reconstruction Using Non-linear Theory
      • • Mapping the Large Scale Structure of the 2dFGRS
        • • The Superclusters
          • The Voids
          • • Conclusions
            • REFERENCES

14 Erdogdu et al

Table 1 The list of superclusters

No zmin zmax RAmin RAmax Decmin Decmax Ngr gt 9 Nclus Ntotal

(1950) deg (1950) deg (1950) deg (1950) deg

SCSGP01 0048 0054 minus115 minus32 minus375 minus255 23 9 27SCSGP02 0054 0057 minus223 00 minus355 minus330 9 1 9SCSGP03 0057 0064 minus294 minus175 minus375 minus225 23 8 26SCSGP04 0057 0068 minus100 +100 minus355 minus240 34 14 37SCSGP05 0054 0064 325 390 minus330 minus255 14 7 18SCSGP06 0064 0082 455 550 minus350 minus240 26 7 31SCSGP07 0064 0071 185 400 minus350 minus310 10 4 11SCSGP08 0068 0075 minus385 minus330 minus350 minus225 9 3 11SCSGP09 0075 0082 minus270 minus175 minus375 minus225 20 9 24SCSGP10 0082 0093 minus175 minus95 minus360 minus290 18 9 21SCSGP11 0086 0097 minus330 minus270 minus345 minus240 14 4 15SCSGP12 0093 0097 minus220 minus174 minus360 minus348 4 1 5SCSGP13 0093 0097 00 38 minus360 minus348 3 0 3SCSGP14 0093 0104 160 246 minus330 minus282 8 1 8SCSGP15 0097 0115 280 446 minus352 minus246 34 18 41SCSGP16 0100 0119 15 219 minus352 minus265 93 20 98SCSGP17 0100 0108 minus271 minus30 minus352 minus265 38 7 43SCSGP18 0150 0166 15 318 minus330 minus265 18 6 22SCSGP19 0162 0177 228 345 minus352 minus246 13 5 15SCSGP20 0181 0202 minus175 minus315 minus355 minus246 13 4 14

SCNGP01 0035 0068 1475 1745 minus65 25 59 6 61SCNGP02 0048 0061 2100 2185 minus50 00 19 0 19SCNGP03 0068 0075 1500 1550 minus10 25 6 0 6SCNGP04 0068 0075 1580 1665 minus10 25 8 1 9SCNGP05 0071 0082 1710 1840 minus35 2 5 27 6 28SCNGP06 0079 0094 1850 2025 minus75 25 77 7 79SCNGP07 0086 0101 1470 1815 minus55 25 31 4 34SCNGP08 0090 0131 1650 1850 minus63 25 51 10 57SCNGP09 0103 0114 1585 1630 minus13 25 8 1 9SCNGP10 0103 0118 1970 2030 minus375 125 22 0 22SCNGP11 0123 0131 1470 1730 00 25 3 2 3SCNGP12 0119 0142 2100 2190 minus40 13 19 2 20SCNGP13 0131 0142 1730 1790 minus625 minus13 5 2 6SCNGP14 0131 0142 1850 1930 minus625 13 7 0 7SCNGP15 0131 0138 1550 1600 minus63 13 9 1 9SCNGP16 0142 0146 1660 1740 minus03 13 5 0 5SCNGP17 0142 0146 1940 1990 minus25 13 1 0 1SCNGP18 0146 0150 2040 2100 minus25 13 4 0 4SCNGP19 0173 0177 1810 1840 minus10 minus40 3 0 3SCNGP20 0181 0185 1855 1930 minus10 minus40 1 0 1

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The 2dFGRS Wiener Reconstruction of the Cosmic Web15

Table 2 The list of voids in SGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VSGP01 0035 0051 minus275 minus173 minus375 minus225VSGP02 0035 0051 minus173 00 minus330 minus255VSGP03 0035 0048 09 115 minus345 minus285VSGP04 0035 0051 124 236 minus305 minus260VSGP05 0035 0054 236 408 minus335 minus260VSGP06 0035 0042 328 373 minus335 minus285VSGP07 0035 0051 395 464 minus315 minus285VSGP08 0035 0042 464 510 minus315 minus285VSGP09 0039 0044 minus355 minus300 minus345 minus285VSGP10 0039 0041 100 175 minus335 minus295VSGP11 0041 0044 155 225 minus335 minus285VSGP12 0057 0061 minus128 minus82 minus315 minus285VSGP13 0082 0086 54 80 minus315 minus285VSGP15 0082 0089 418 493 minus315 minus285VSGP16 0086 0089 125 163 minus315 minus285VSGP17 0086 0089 310 335 minus325 minus285VSGP18 0089 0097 minus82 minus595 minus305 minus295VSGP19 0089 0093 427 455 minus305 minus295VSGP20 0093 0104 464 525 minus325 minus282VSGP21 0093 0100 282 328 minus315 minus285VSGP22 0097 0104 77 125 minus320 minus285VSGP23 0097 0100 minus128 minus115 minus320 minus285VSGP24 0100 0108 minus318 minus253 minus285 minus255VSGP25 0100 0112 minus253 minus219 minus345 minus330VSGP26 0112 0119 minus70 minus37 minus315 minus285VSGP27 0112 0115 15 54 minus300 minus270VSGP28 0112 0115 259 282 minus300 minus270VSGP29 0112 0115 410 464 minus300 minus270VSGP30 0115 0119 262 305 minus345 minus315VSGP31 0119 0123 minus350 minus308 minus285 minus255VSGP32 0119 0123 minus270 minus255 minus285 minus265VSGP33 0119 0123 minus128 minus100 minus285 minus255VSGP34 0119 0123 415 493 minus388 minus318VSGP35 0119 0127 112 165 minus282 minus275VSGP36 0123 0131 minus264 minus224 minus315 minus285VSGP37 0123 0127 minus128 minus82 minus300 minus270VSGP38 0131 0142 minus264 minus195 minus345 minus315VSGP39 0131 0138 323 375 minus352 minus248VSGP40 0138 0142 303 335 minus335 minus290

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16 Erdogdu et al

Table 3 The list of voids in NGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VNGP01 0035 0054 1470 1600 minus40 15VNGP02 0035 0041 1660 1710 minus10 25VNGP03 0035 0046 1710 1812 minus15 25VNGP04 0035 0051 1830 1890 minus15 25VNGP05 0035 0046 1890 1970 minus15 20VNGP06 0035 0046 2022 2060 minus15 20VNGP07 0037 0046 2042 2156 minus20 00VNGP08 0041 0051 1638 1698 minus45 25VNGP09 0049 0061 1770 1964 minus35 25VNGP10 0054 0057 1655 1730 minus15 25VNGP11 0057 0061 1635 1660 00 15VNGP12 0054 0061 2060 2080 minus15 05VNGP13 0071 0075 1760 1790 minus10 05VNGP14 0079 0094 1865 2042 minus75 05VNGP15 0092 0097 1508 1602 minus75 minus60VNGP16 0093 0100 1470 1495 minus6 15VNGP17 0094 0101 2040 2115 minus35 25VNGP18 0099 0115 1475 1536 minus40 10VNGP19 0099 0120 1888 1970 minus30 25VNGP20 0112 0120 1766 1780 05 25VNGP21 0110 0115 2116 2170 minus25 minus05VNGP22 0110 0123 2002 2050 minus325 25VNGP23 0123 0131 1690 1740 minus40 minus125VNGP24 0127 0131 1825 1880 00 25VNGP25 0127 0131 1975 2020 minus125 05VNGP26 0134 0142 1795 1875 minus20 25VNGP27 0134 0138 2040 2070 minus15 00VNGP28 0138 0150 1470 1546 minus35 25VNGP29 0142 0150 1620 1660 minus40 15VNGP30 0142 0150 1726 1780 minus10 25

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• INTRODUCTION
• Wiener Filter
• The Data
• • The 2dFGRS data
  • The Mask and The Radial Selection Function of 2dFGRS
  • • Survey Pixelisation
    • Estimating The Signal-Signal Correlation Matrix Over Pixels
    • The Application
    • • Reconstruction Using Linear Theory
      • Reconstruction Using Non-linear Theory
      • • Mapping the Large Scale Structure of the 2dFGRS
        • • The Superclusters
          • The Voids
          • • Conclusions
            • REFERENCES

The 2dFGRS Wiener Reconstruction of the Cosmic Web15

Table 2 The list of voids in SGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VSGP01 0035 0051 minus275 minus173 minus375 minus225VSGP02 0035 0051 minus173 00 minus330 minus255VSGP03 0035 0048 09 115 minus345 minus285VSGP04 0035 0051 124 236 minus305 minus260VSGP05 0035 0054 236 408 minus335 minus260VSGP06 0035 0042 328 373 minus335 minus285VSGP07 0035 0051 395 464 minus315 minus285VSGP08 0035 0042 464 510 minus315 minus285VSGP09 0039 0044 minus355 minus300 minus345 minus285VSGP10 0039 0041 100 175 minus335 minus295VSGP11 0041 0044 155 225 minus335 minus285VSGP12 0057 0061 minus128 minus82 minus315 minus285VSGP13 0082 0086 54 80 minus315 minus285VSGP15 0082 0089 418 493 minus315 minus285VSGP16 0086 0089 125 163 minus315 minus285VSGP17 0086 0089 310 335 minus325 minus285VSGP18 0089 0097 minus82 minus595 minus305 minus295VSGP19 0089 0093 427 455 minus305 minus295VSGP20 0093 0104 464 525 minus325 minus282VSGP21 0093 0100 282 328 minus315 minus285VSGP22 0097 0104 77 125 minus320 minus285VSGP23 0097 0100 minus128 minus115 minus320 minus285VSGP24 0100 0108 minus318 minus253 minus285 minus255VSGP25 0100 0112 minus253 minus219 minus345 minus330VSGP26 0112 0119 minus70 minus37 minus315 minus285VSGP27 0112 0115 15 54 minus300 minus270VSGP28 0112 0115 259 282 minus300 minus270VSGP29 0112 0115 410 464 minus300 minus270VSGP30 0115 0119 262 305 minus345 minus315VSGP31 0119 0123 minus350 minus308 minus285 minus255VSGP32 0119 0123 minus270 minus255 minus285 minus265VSGP33 0119 0123 minus128 minus100 minus285 minus255VSGP34 0119 0123 415 493 minus388 minus318VSGP35 0119 0127 112 165 minus282 minus275VSGP36 0123 0131 minus264 minus224 minus315 minus285VSGP37 0123 0127 minus128 minus82 minus300 minus270VSGP38 0131 0142 minus264 minus195 minus345 minus315VSGP39 0131 0138 323 375 minus352 minus248VSGP40 0138 0142 303 335 minus335 minus290

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16 Erdogdu et al

Table 3 The list of voids in NGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VNGP01 0035 0054 1470 1600 minus40 15VNGP02 0035 0041 1660 1710 minus10 25VNGP03 0035 0046 1710 1812 minus15 25VNGP04 0035 0051 1830 1890 minus15 25VNGP05 0035 0046 1890 1970 minus15 20VNGP06 0035 0046 2022 2060 minus15 20VNGP07 0037 0046 2042 2156 minus20 00VNGP08 0041 0051 1638 1698 minus45 25VNGP09 0049 0061 1770 1964 minus35 25VNGP10 0054 0057 1655 1730 minus15 25VNGP11 0057 0061 1635 1660 00 15VNGP12 0054 0061 2060 2080 minus15 05VNGP13 0071 0075 1760 1790 minus10 05VNGP14 0079 0094 1865 2042 minus75 05VNGP15 0092 0097 1508 1602 minus75 minus60VNGP16 0093 0100 1470 1495 minus6 15VNGP17 0094 0101 2040 2115 minus35 25VNGP18 0099 0115 1475 1536 minus40 10VNGP19 0099 0120 1888 1970 minus30 25VNGP20 0112 0120 1766 1780 05 25VNGP21 0110 0115 2116 2170 minus25 minus05VNGP22 0110 0123 2002 2050 minus325 25VNGP23 0123 0131 1690 1740 minus40 minus125VNGP24 0127 0131 1825 1880 00 25VNGP25 0127 0131 1975 2020 minus125 05VNGP26 0134 0142 1795 1875 minus20 25VNGP27 0134 0138 2040 2070 minus15 00VNGP28 0138 0150 1470 1546 minus35 25VNGP29 0142 0150 1620 1660 minus40 15VNGP30 0142 0150 1726 1780 minus10 25

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• INTRODUCTION
• Wiener Filter
• The Data
• • The 2dFGRS data
  • The Mask and The Radial Selection Function of 2dFGRS
  • • Survey Pixelisation
    • Estimating The Signal-Signal Correlation Matrix Over Pixels
    • The Application
    • • Reconstruction Using Linear Theory
      • Reconstruction Using Non-linear Theory
      • • Mapping the Large Scale Structure of the 2dFGRS
        • • The Superclusters
          • The Voids
          • • Conclusions
            • REFERENCES

16 Erdogdu et al

Table 3 The list of voids in NGP

No zmin zmax RAmin RAmax Decmin Decmax

(1950) deg (1950) deg (1950) deg (1950) deg

VNGP01 0035 0054 1470 1600 minus40 15VNGP02 0035 0041 1660 1710 minus10 25VNGP03 0035 0046 1710 1812 minus15 25VNGP04 0035 0051 1830 1890 minus15 25VNGP05 0035 0046 1890 1970 minus15 20VNGP06 0035 0046 2022 2060 minus15 20VNGP07 0037 0046 2042 2156 minus20 00VNGP08 0041 0051 1638 1698 minus45 25VNGP09 0049 0061 1770 1964 minus35 25VNGP10 0054 0057 1655 1730 minus15 25VNGP11 0057 0061 1635 1660 00 15VNGP12 0054 0061 2060 2080 minus15 05VNGP13 0071 0075 1760 1790 minus10 05VNGP14 0079 0094 1865 2042 minus75 05VNGP15 0092 0097 1508 1602 minus75 minus60VNGP16 0093 0100 1470 1495 minus6 15VNGP17 0094 0101 2040 2115 minus35 25VNGP18 0099 0115 1475 1536 minus40 10VNGP19 0099 0120 1888 1970 minus30 25VNGP20 0112 0120 1766 1780 05 25VNGP21 0110 0115 2116 2170 minus25 minus05VNGP22 0110 0123 2002 2050 minus325 25VNGP23 0123 0131 1690 1740 minus40 minus125VNGP24 0127 0131 1825 1880 00 25VNGP25 0127 0131 1975 2020 minus125 05VNGP26 0134 0142 1795 1875 minus20 25VNGP27 0134 0138 2040 2070 minus15 00VNGP28 0138 0150 1470 1546 minus35 25VNGP29 0142 0150 1620 1660 minus40 15VNGP30 0142 0150 1726 1780 minus10 25

ccopy 0000 RAS MNRAS000 000ndash000

• INTRODUCTION
• Wiener Filter
• The Data
• • The 2dFGRS data
  • The Mask and The Radial Selection Function of 2dFGRS
  • • Survey Pixelisation
    • Estimating The Signal-Signal Correlation Matrix Over Pixels
    • The Application
    • • Reconstruction Using Linear Theory
      • Reconstruction Using Non-linear Theory
      • • Mapping the Large Scale Structure of the 2dFGRS
        • • The Superclusters
          • The Voids
          • • Conclusions
            • REFERENCES