Real-space Power Spectrum Research Articles

Spherical harmonic analysis of the PSCz galaxy catalogue: redshift distortions and the real-space power spectrum

We apply the formalism of spherical harmonic decomposition to the galaxy density field of the IRAS PSCz redshift survey. The PSCz redshift survey has almost all-sky coverage and includes IRAS galaxies to a flux limit of 0.6 Jy. Using maximum likelihood methods to examine (to first order) the distortion of the galaxy pattern resulting from redshift coordinates, we have measured the parameter β≡Ω{0.6}/b. We also simultaneously measure either (a) the undistorted amplitude of perturbations in the galaxy distribution when a parametrized power spectrum is assumed, or (b) the shape and amplitude of the real-space power spectrum if the band-power in a set of passbands is measured in a step-wise fashion. These methods are extensively tested on a series of CDM, Λ CDM and MDM simulations and are found to be unbiased. We obtain consistent results for the subset of the PSCz catalogue with flux above 0.75 Jy, but inclusion of galaxies to the formal flux limit of the catalogue gives variations which are larger than our internal errors. For the 0.75-Jy catalogue we find, in the case of a parametrized power spectrum, β=0.58±0.26 and the amplitude of the real-space power measured at wavenumber k=0.1h Mpc−1 is Δ0.1=0.42±0.03. Freeing the shape of the power spectrum we find that β=0.47±0.16 (conditional error) and Δ0.1=0.47±0.03. The shape of the real-space power spectrum is consistent with a Γ=0.2 CDM-like model, but does not strongly rule out a number of other models. Finally by combining our estimate of the amplitude of galaxy clustering and the distortion parameter we find the amplitude of mass fluctuations on a scale k=0.1h Mpc−1 is Δρ=0.24Ω0−0.6, with an uncertainty of 50 per cent.

Design and analysis of redshift surveys

In this paper we consider methods of analysis and optimal design of redshift surveys. In the first part, we develop a formalism for analysing galaxy redshift surveys which are essentially two-dimensional, such as thin declination slices. The formalism is a power spectrum method, using spherical coordinates, allowing the distorting effects of galaxy peculiar velocities to be calculated to linear order on the assumption of statistical isotropy but without further approximation. In this paper, we calculate the measured two-dimensional power for a constant declination strip, widely used in redshift surveys. We present a likelihood method for estimating the three-dimensional real-space power spectrum and the redshift distortion simultaneously, and show that for thin surveys of reasonable depth, the large-scale 3D power cannot be measured with high accuracy. The redshift distortion may be estimated successfully, and with higher accuracy if the 3D power spectrum can be measured independently, for example from a large-scale sky-projected catalogue. In the second part, we show how a 3D survey design can be optimized to measure the power spectrum, considering whether areal coverage is more important than depth, and whether the survey should be sampled sparsely or not. We show quite generally that width is better than depth, and show how the optimal sparse-sampling fraction, f, depends on the power, P, to be measured. For a Schechter luminosity function, a simple optimization fP \simeq 500 h^{-3} Mpc^3 is found.

Power spectrum analysis of the Stromlo—APM redshift survey

We test estimators of the galaxy power spectrum $P(k)$ against simulated galaxy catalogues constructed from N-body simulations and we derive formulae to correct for biases. These estimators are then applied to compute the power spectrum of galaxies in the Stromlo-APM redshift survey. We test whether the amplitude of $P(k)$ depends on galaxy luminosity, but find no significant luminosity dependence except at absolute magnitudes brighter than $M_{\bj} = -20.3$, ($H_{0} = 100 \kms$) where there is some evidence for a rise in the amplitude of $P(k)$. By comparing the redshift space power spectrum of the Stromlo-APM survey with the real space power spectrum determined from the parent APM Galaxy Survey, we attempt to measure the distortion in the shape of $P(k)$ caused by galaxy peculiar motions. We find some evidence for an effect, but the errors are large and do not exclude a value of $\beta = \Omega^{0.6}/b = 1$, where $\Omega$ is the cosmological density parameter and $b$ is the linear biasing parameter relating galaxy fluctuations to those in the mass, $\left(\delta \rho/\rho\right)_{gal} = b \left(\delta \rho/\rho\right)_{m}$. The shape of the Stromlo-APM power spectrum is consistent with that determined from the CfA-2 survey, but has a slightly higher amplitude by a factor of about 1.4 than the power spectrum of IRAS galaxies.

The real-space power spectrum of IRAS galaxies on large scales and the redshift distortion

Galaxy redshift surveys contain valuable cosmological information on the power spectrum of galaxy clustering, but the information is not easily obtained because of distortion of the map arising from the use of redshift as a distance indicator. The distortion arises because of peculiar velocities, the magnitude of which depends on the mean density of the Universe. Using a spherical transform of the data which correctly accounts for the radial nature of the distortion, we calculate simultaneously the real-space power spectrum and the distortion, for the near-all-sky Infrared Astronomical Satellite (IRAS) 1.2-Jy survey. For the distortion, we find |$\beta=1.04\pm0.3$|⁠, where |$\beta\equiv\Omega^{0.6}_0/b$|⁠, and b is the IRAS linear bias parameter. We constrain the power spectrum on large scales with wavenumbers 0.01 ≲ k ≲ 0.1 h Mpc−1 where the ‘distant-observer’ Fourier approximation should not be used. Power is not convincingly detected at wavenumbers less than 0.03 h Mpc−1, but the power spectrum is consistent with a change in slope from the k−1.5 behaviour observed on smaller scales to Zeldovich spectrum on large scales. In the wavenumber range considered, we find somewhat less power than in the optically selected APM survey.