The 2Df Galaxy Redshift Survey: Wiener Reconstruction of the Cosmic Web
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Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 20 January 2004 (MN LATEX style file v2.2) The 2dF Galaxy Redshift Survey: Wiener Reconstruction of the Cosmic Web Pirin Erdogdu˜ 1;2, Ofer Lahav1, Saleem Zaroubi3, George Efstathiou1, Steve Moody, John A. Peacock12, Matthew Colless17, Ivan K. Baldry9, Carlton M. Baugh16, Joss Bland- Hawthorn7, Terry Bridges7, Russell Cannon7, Shaun Cole16, Chris Collins4, Warrick Couch5, Gavin Dalton6;15, Roberto De Propris17, Simon P. Driver17, Richard S. Ellis8, Carlos S. Frenk16, Karl Glazebrook9, Carole Jackson17, Ian Lewis6, Stuart Lumsden10, Steve Maddox11, Darren Madgwick13, Peder Norberg14, Bruce A. Peterson17, Will Sutherland12, Keith Taylor8 (The 2dFGRS Team) 1Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 2Department of Physics, Middle East Technical University, 06531, Ankara, Turkey 3Max Planck Institut fur¨ Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany 4Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Birkenhead, L14 1LD, UK 5Department of Astrophysics, University of New South Wales, Sydney, NSW 2052, Australia 6Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 7Anglo-Australian Observatory, P.O. Box 296, Epping, NSW 2111, Australia 8Department of Astronomy, California Institute of Technology, Pasadena, CA 91025, USA 9Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21118-2686, USA 10Department of Physics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK 11School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD, UK 12Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 13Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 14ETHZ Institut fur¨ Astronomie, HPF G3.1, ETH Hong¨ gerberg, CH-8093 Zurich, Switzerland 15Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK 16Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK 17Research School of Astronomy & Astrophysics, The Australian National University, Weston Creek, ACT 2611, Australia 20 January 2004 ABSTRACT We reconstruct the underlying density field of the 2 degree Field Galaxy Redshift Survey (2dFGRS) for the redshift range 0:035 < z < 0:200 using the Wiener Filtering method. The 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 and the combination of matter density and bias parameters however the results are only slightly affected by changes to these parameters. We present maps of the density field in two different resolutions: 5h−1 Mpc and 10h−1 Mpc. We identify all major superclusters and voids in the survey. In particular, we find two large superclusters and two large local voids. The full set of colour maps can be viewed on the WWW at http://www.ast.cam.ac.uk/∼pirin. Key words: galaxies:distances and redshifts - cosmology: large-scale structure of Universe - methods: statistical 1 INTRODUCTION of fibre-fed spectrographs that can simultaneously measure spectra of hundreds of galaxies, cosmologists have at their fingertips large Historically, redshift surveys have provided the data and the test redshift surveys such as 2 degree Field (2dF) and Sloan Digital Sky Survey (SDSS). The analysis of these redshift surveys yield invalu- ground for much of the research on the nature of clustering and the able cosmological information. On the quantitative side, with the distribution of galaxies. In the past few years, observations of large scale structure have improved greatly. Today, with the development assumption that the galaxy distribution arises from the gravitational c 0000 RAS 2 Erdogdu˜ et al. instability of small fluctuations generated in the early universe, a view of the formalism of the Wiener Filter method. A summary wide range of statistical measurements can be obtained, such as of 2dFGRS data set, the survey mask and the selection function the power spectrum and bispectrum. Furthermore, a qualitative un- are given in Section 3. In section 4, we outline the scheme used derstanding of galaxy distribution provides insight into the mech- to pixelise the survey. In section 5, we give the formalism for the anisms of structure formation that generate the complex pattern of covariance matrix used in the analysis. After that, we describe the sheets and filaments comprising the ‘cosmic web’ (Bond, Kofman application of the Wiener Filter method to 2dFGRS and present de- & Pogosyan 1996) we observe and allows us to map a wide variety tailed maps of the reconstructed field. In section 7, we identify the of structure, including clusters, superclusters and voids. superclusters and voids in the survey. Today, many more redshifts are available for galaxies than di- Throughout this paper, we assume a ΛCDM cosmology with −1 −1 −1 rect distance measurements. This discrepancy inspired a great deal Ωm = 0:3 and ΩΛ = 0:7 and H0 = 100h kms Mpc . of work on methods for reconstruction of the real-space density field from that observed in redshift-space. These methods use a va- riety of functional representations (e.g. Cartesian, Fourier, spherical 2 WIENER FILTER harmonics or wavelets) and smoothing techniques (e.g. a Gaussian sphere or a Wiener Filter). There are physical as well as practical In this section, we give a brief description of the Wiener Fil- reasons why one would be interested in smoothing the observed ter method. For more details, we refer the reader to Zaroubi et density field. It is often assumed that the galaxy distribution sam- al. (1995). Let’s assume that we have a set of measurements, ples the underlying smooth density field and the two are related by a dα (α = 1; 2; : : : N) which are a linear convolution of the true f g proportionality constant, the so-called linear bias parameter, b. The underlying signal, sα, plus a contribution from statistical noise, α, finite sampling of the smooth underlying field introduces Poisson such that ‘shot noise’ 1. Any robust reconstruction technique must reliably (1) mitigate the statistical uncertainties due to shot noise. Moreover, in dα = sα + α: redshift surveys, the actual number of galaxies in a given volume is The Wiener Filter is defined as the linear combination of the ob- larger than the number observed, in particular in magnitude limited served data which is closest to the true signal in a minimum vari- W F samples where at large distances only very luminous galaxies can ance sense. More explicitly, the Wiener Filter estimate, sα , is W F be seen. given by sα = Fαβ dβ where the filter is chosen to minimise the In this paper, we analyse large scale structure in the 2 degree variance of the residual field, rα: Field Galaxy Redshift Survey (2dFGRS, Colless et al. 2001), which 2 W F 2 r = s sα : (2) has now obtained the redshifts for approximately 230,000 galax- hj αji hj α − j i ies. We recover the underlying density field, characterised by an It is straightforward to show that the Wiener Filter is assumed power spectrum of fluctuations, from the observed field y y −1 which suffers from incomplete sky coverage (described by the an- Fαβ = sαdγ dγ d ; (3) h ih β i gular mask) and incomplete galaxy sampling due to its magnitude where the first term on the right hand side is the signal-data corre- limit (described by the selection function). The filtering is achieved lation matrix; by a Wiener Filter (Wiener 1949, Press et al. 1992) within the y y framework of both linear and non-linear theory of density fluctu- sαdγ = sαsγ ; (4) ations. The Wiener Filter is optimal in the sense that the variance h i h i and the second term is the data-data correlation matrix; between the derived reconstruction and the underlying true density field is minimised. As opposed to ad hoc smoothing schemes, the y y y dαdβ = sγ sδ + αβ : (5) smoothing due to the Wiener Filter is determined by the data. In h i h i h i the limit of high signal-to-noise, the Wiener Filter modifies the ob- In the above equations, we have assumed that the signal and served data only weakly, whereas it suppresses the contribution of noise are uncorrelated. From equations 3 and 5, it is clear that, in the data contaminated by shot noise. order to implement the Wiener Filter, one must construct a prior The Wiener Filtering is a well known technique and has been which depends on the mean of the signal (which is 0 by construc- applied to many fields in astronomy (see Rybicki & Press 1992). tion) and the variance of the signal and noise. The assumption of a For example, the method was used to reconstruct the angular dis- prior may be alarming at first glance. However, slightly inaccurate tribution (Lahav et al. 1994), the real-space density, velocity and values of Wiener Filter will only introduce second order errors to gravitational potential fields of the 1.2 Jy-IRAS (Fisher et al. the full reconstruction (see Rybicki & Press 1992). The dependence of the Wiener Filter on the prior can be made clear by defining sig- 1995) and IRAS PSCz surveys (Schmoldt et al. 1999). The Wiener y y nal and noise matrices as Cαβ = sαs and Nαβ = α . With Filter was also applied to the reconstruction of the angular maps h β i h β i of the Cosmic Microwave Background temperature fluctuations this notation, we can rewrite the equations above so that sW F is (Bunn et al. 1994, Tegmark & Efstathiou 1995, Bouchet & Gis- given as pert 1999). A detailed formalism of the Wiener Filtering method as − sW F = C [C + N] 1 d: (6) it pertains to the large scale structure reconstruction can be found in Zaroubi et al. (1995). The mean square residual given in equation 2 can then be calculated This paper is structured as follows: we begin with a brief re- as rry = C [C + N]−1 N: (7) h i 1 Another popular model for galaxy clustering is the halo model where the Formulated in this way, we see that the purpose of the Wiener linear bias parameter depends on the mass of the dark matter halos where Filter is to attenuate the contribution of low signal-to-noise ra- the galaxies reside.