We have constrained the redshift-distortion parameter $beta equiv Omega^{0.6}/b$ and the real-space power spectrum of the IRAS PSCz survey using a spherical-harmonic redshift-distortion analysis combined with a data compression method which is designed to deal with correlated parameters. Our latest result, $beta=0.4 pm 0.1$, strongly rules out $beta=1$

Ballinger, WE

Heavens, AF

Tadros, H

Taylor, AN

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arXiv:astro-ph/0005094v1  4 May 2000 REDSHIFT DISTORTIONS AND CLUSTERING IN THE PSCZ SURVEYW.E. BALLINGERDepartment of Physics, Astrophysics, Keble Road,Oxford OX1 3RH, EnglandA.N. TAYLOR, A.F. HEAVENSInstitute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, ScotlandH. TADROSDepartment of Physics, Astrophysics, Keble Road,Oxford OX1 3RH, EnglandWe have constrained the redshift-distortion parameter β ≡ Ω0.6/b and the real-space powerspectrum of the IRAS PSCz survey using a spherical-harmonic redshift-distortion analysiscombined with a data compression method which is designed to deal with correlated param-eters. Our latest result, β = 0.4 ± 0.1, strongly rules out β = 1.1 IntroductionIt has been understood for a long time that redshift surveys are systematically distorted bypeculiar velocities 11 – so called redshift distortions, which can cause the observed distribution ofgalaxies to become anisotropic. On large scales where linear theory applies, redshift distortionsare characterised by the parameter a β ≡ Ω0.6m /b – see Hamilton8 for a detailed review.In a classic paper, Kaiser12 derived a simple formula for the effect of linear redshift distortionsfor a volume limited survey that subtends a small opening angle at the observer (the “distantobserver” approximation – all lines of sight are treated as parallel). He showed that power isboosted by a factor (1 + βµ2k)2, where µkis the cosine of the angle between wavevector and lineof sight. Hamilton 6 7 and Cole, Fisher & Weinberg 4 5 analysed all-sky IRAS surveys using amethod based on the Kaiser formalism, but were forced to break the survey up into sectionsbecause of the constraints of the distant observer approximation, losing information about thelargest (and most reliably linear) scales.Rather than fit a square peg into a round hole, Fisher et al 9 and Heavens & Taylor 10(HT) dropped the plane parallel approximation and used a spherical harmonic decompositionto match the spherical nature of the IRAS redshift surveys. HT used spherical Bessel functionsto decompose the density field radially; using eigenfunctions of the Laplacian retains all theadvantages of Fourier analysis. HT analysed the IRAS 1.2Jy survey, fitting β and the amplitudeof the power spectrum; Ballinger, Heavens & Taylor 2(BHT) extended this analysis to fit theshape of the power spectrum. Tadros et al 18 (T99) applied these techniques to the IRAS PSCzsurvey – section 4 includes a review of those results.aΩm is the contribution to the density parameter from matter. The bias parameter b depends on galaxy type,in this paper it refers to IRAS galaxies.2 Spherical Harmonic FormalismWe will briefly review the formalism for spherical harmonic analysis – see HT and T99 for details.The density field of the galaxy distribution ρ(s) is expanded in terms of spherical harmonics,Yℓm, and a discrete set of spherical Bessel functions, jℓ,ρ̂ℓmn = cℓn∫d3s ρ(s)w(s)jℓ (kℓns)Y∗ℓm (θ, φ) , (1)where the cℓn are normalization constants and kln are discrete wavenumbers.These observed coefficients can be related to those of the true underlying density field (δℓmn)by:ρ̂ℓmn = (ρ0)ℓmn +∑ℓ′m′n′∑n′′Snn′′W mm′ℓℓ′(Φn′′n′ℓℓ′ + βVn′′n′ℓℓ′)δℓ′m′n′ . (2)The transition matrices W, Φ, V and S describe the effects of the sky mask, the radial se-lection function, the linear redshift space distortion and the small scale distortion correctionrespectively; (ρ0)ℓmn is a mean term, non-zero because of partial sky coverage. The transitionmatrices are derived and defined in HT and T99.A likelihood approach is used to constrain β and the real-space power spectrum:− 2 lnL[D|β, P (k)] = ln(detC) + DC−1DT , (3)where C =< DDT > and elements of the data vector D are given by Dℓmn ≡ [ρ̂ℓmn− (ρ0)ℓmn]/ρ̄where ρ̄ is the mean number density – Gaussian statistics are assumed. Two different parame-terisations of P (k) are used: following HT a fixed shape is assumed and the amplitude is fitted,following BHT a stepwise maximum likelihood method is used, allowing the power spectrum toassume any shape (within bin discreteness).3 Data – The IRAS PSCz SurveyThe PSCz survey 14 is a redshift catalogue complete down to 0.6Jy over ∼ 83% of the sky witha total of ∼ 15, 000 redshifts – it is the largest all sky survey in existence. As our method isvery precise, we minimise systematic errors by using a more conservative flux cut (0.75Jy) andsky mask reducing the number of redshifts by a factor of roughly two – see T99.4 Previous ResultsBoth the fixed amplitude (HT) and stepwise P (k) (BHT) methods were applied to the PSCzby T99 for modes with k ≤ 0.13h Mpc−1. The first method produced β = 0.58 ± 0.26 and theamplitude of the real space power measured at wavenumber k = 0.1h Mpc−1 of ∆0.1 = 0.42±0.03– see Fig. 2 (dashed contours). Freeing the shape of the power spectrum we find the consistentresults β = 0.47 ± 0.16 (conditional error), and ∆0.1 = 0.47 ± 0.03 – Fig. 1. T99 also carriedout extensive tests on simulations, and the methods were found to be reliable. In addition, wecarried out a suite of tests for systematics effects in the data and found the cut catalogue gaveconsistent results.The method was restricted to a small range of wavenumber because the likelihood analysisinvolves the repeated inversion of a large n× n matrix; the time required for this process growsas n3. More importantly, the matrix rapidly became numerically unstable. The results above donot strongly rule out either high (∼ 1) or low (∼ 0.5) values of β. It would be nice to overcomethe matrix problem, extend the k-range and reduce the error bar.Figure 1: The real-space power spectrum of the PSCz redshift survey in dimensionless units (from T99). Thecurve is a CDM model with Γ = 0.2. The estimated value of β is 0.47 ± 0.16 (conditional error).5 Data CompressionIt is possible to transform the (length n) data vector D to create a new, smaller dataset,while retaining most of the information about the parameters of interest 20. A new dataset isconstructed which is a linear combination of the original:D′ = BD, (4)where D′ is a new data vector of length n′ < n, with a corresponding transformation for thecovariance matrix. TTH show that the optimal matrix B which minimises the conditional erroron a single parameter is made up from eigenvectors of bC,i b = λCb (5)where the comma refers to a derivative with respect to parameter i. The new covariance matrixis smaller and also close to diagonal – hence much more stable.TTH suggested extending this to multi-parameter problems by constructing a separate ma-trix B for each parameter and then using singular value decomposition to combine the matricesefficiently. However, Ballinger 1 (see also Taylor et al 19) tested this and found that error el-lipses/ellipsoids tended to grow along correlation axes – only conditional errors are constrained.Instead it was shown that equation (5) could be used to constrain the marginal errors by usingone or more linear combinations of the original parameters which lie directly along the correlationaxes.Unfortunately solving equation (5) involves manipulating uncompressed n×n matrices. Thisusually isn’t a timescale problem – it need only be done once – but still suffers from numericaldifficulties. To avoid this problem, we split the data into several sections and compressed themseparately, while still retaining full information about correlations between modes in differentsections. Strictly speaking this is slightly less optimal than compressing the whole dataset inone go, but it should make a negligible difference in practice and will not bias the result.6 New Results and conclusionsFig. 2 shows the old kmax = 0.13h Mpc−1 result together with the new kmax = 0.2h Mpc−1obtained using data compression. The original 4644 modes were compressed down to 2278. ThebThis, like the Wiener filter, is one of those marvellous data-handling methods which were invented by as-tronomers then sent back in time so that signal processing engineers could use them for the past fifty years.Figure 2: Likelihood contours for the two parameters β = Ω0.60 /b and the amplitude of the power spectrum for thePSCz survey. The new data-compression kmax = 0.2h Mpc−1 results (solid contours) are plotted along with theoriginal kmax = 0.13h Mpc−1 results from T99 (dashed contours). Contours are plotted at intervals of δ lnL = 0.5.new, lower, value of β = 0.4 ± 0.1 is consistent with the previous result but the error ellipseis considerably smaller; the best-fit amplitude of the power spectrum is essentially unchanged.β = 1, corresponding to Ωm = 1, b = 1, is now strongly disfavoured. More details will be inBallinger et al and Taylor et al (in preparation).The low value of β is consistent with currently popular cosmological models with a low valueof Ωmatter if the IRAS bias parameter is close to unity. The value is somewhat lower than thatfrom other analyses of the PSCz survey 3 13 15 16, but not inconsistent. It is consistent with therecent velocity-velocity comparison results from peculiar velocity catalogues 22, but somewhatlower than the corresponding density-density value 17. See Willick 21 for a discussion.References1. W.E. Ballinger, PhD thesis, University of Edinburgh, 1997.2. W.E. Ballinger, A.F. Heavens and A.N. Taylor, MNRAS 276, 59P (1995) (BHT).3. E. Branchini et al MNRAS 308, 1 (1999).4. S. Cole, K.B. Fisher and D.H. Weinberg, MNRAS 267, 785 (1994).5. S. Cole, K.B. Fisher and D.H. Weinberg, MNRAS 275, 515 (1995).6. A.J.S. Hamilton, ApJ 385, L5 (1992).7. A.J.S. Hamilton, ApJ 406, L47 (1993).8. A.J.S. Hamilton in The Evolving Universe, Ringberg Workshop on Large-Scale Structure 1996, ed.D. Hamilton (Kluwer Academic, Dordrecht, 1998), (astro-ph/9708102).9. K.B. Fisher, C. Scharf and O. Lahav, MNRAS 266, 219 (1994).10. A.F. Heavens and A.N. Taylor, MNRAS 275, 483 (1995) (HT).11. J. Jackson, MNRAS 156, 1P (1972).12. N. Kaiser, MNRAS 227, 1 (1987).13. M. Rowan-Robinson et al (astro-ph/9912223).14. W. Saunders et al, (astro-ph/0001117).15. I. Schmoldt et al, MNRAS 304, 893 (1999).16. I. Schmoldt et al, AJ 118, 1146 (1999).17. Y. Sigad, A. Eldar, A. Dekel, M. Strauss and A. Yahil, ApJ 495, 516 (1998).18. H. Tadros et al, MNRAS 305, 527 (1999) (T99).19. A.N. Taylor, A.F. Heavens, W.E. Ballinger and M. Tegmark, (astro-ph/9707265).20. M. Tegmark, A.N. Taylor and A.F. Heavens, ApJ 480, 22 (1997).21. J. Willick, these proceedings, (astro-ph/0003232)22. J. Willick, M. Strauss, A. Dekel and T. Kolatt, ApJ 486, 629 (1997).

Redshift Distortions and Clustering in the PSCz Survey

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