Redshift-space Power Spectrum Research Articles

Redshift-space distortions, pairwise velocities, and nonlinearities

We derive the exact relationship, including all nonlinearities, between real-space and redshift-space two-point statistics through the pairwise velocity distribution function. We show using numerical simulations that the pairwise velocity probability distribution function is strongly non-Gaussian at all scales and explain why this is so. We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive. We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able tomore » recover the large-scale real-space power spectrum rather well.« less

Redshift distortions in one-dimensional power spectra

We present a model for one-dimensional (1D) matter power spectra in redshift space as estimated from data provided along individual lines of sight. We derive analytic expressions for these power spectra in the linear and nonlinear regimes, focusing on redshift distortions arising from peculiar velocities. In the linear regime, redshift distortions enhance the 1D power spectra only on small scales, and do not affect the power on large scales. This is in contrast to the effect of distortions on three-dimensional (3D) power spectra estimated from data in 3D space, where the enhancement is independent of scale. For CDM cosmologies, the 1D power spectra in redshift and real space are similar for wavenumbers $q<0.1h/Mpc$ where both have a spectral index close to unity, independent of the details of the 3D power spectrum. Nonlinear corrections drive the 1D power spectrum in redshift space into a nearly universal shape over scale $q<10h/Mpc$, and suppress the power on small scales as a result of the strong velocity shear and random motions. The redshift space, 1D power spectrum is mostly sensitive to the amplitude of the initial density perturbations. Our results are useful in particular for power spectra computed from the SDSS quasars sample.

Redshifting rings of power

The cosmic microwave background (CMB) has provided a precise template for features in the linear power spectrum: the matter-radiation turnover, sound horizon drop, and acoustic oscillations. In a two-dimensional power spectrum in redshift and angular space, the features appear as distorted rings, and yield simultaneous, purely geometric, measures of the Hubble parameter $H(z)$ and angular diameter distance ${D}_{A}(z)$ via an absolute version of the Alcock-Paczynski test. Employing a simple Fisher matrix tool, we explore how future surveys can exploit these rings of power for dark energy studies. High-z CMB determinations of H and ${D}_{A}$ are best complemented at moderate to low redshift $(z\ensuremath{\lesssim}0.5)$ with a population of objects that are at least as abundant as clusters of galaxies. We find that a sample similar to that of the ongoing SDSS luminous red galaxy survey can achieve statistical errors at the $\ensuremath{\sim}5%$ level for ${D}_{A}(z)$ and $H(z)$ in several redshift bins. This, in turn, implies errors of $\ensuremath{\sigma}(w)=0.03--0.05$ for a constant dark energy equation of state in a flat universe. Deep galaxy cluster surveys such as the planned South Pole Telescope survey can extend this test out to $z\ensuremath{\sim}1$ or as far as redshift follow-up is available. We find that the expected constraints are at the $\ensuremath{\sigma}(w)=0.04--0.08$ level, comparable to those of galaxies and complementary in redshift coverage.

Large scale magnetic fields: Density power spectrum in redshift space

We compute the density redshift-space power spectrum in the presence of tangled magnetic fields and compare it with existing observations. Our analysis shows that if these magnetic fields originated in the early universe then it is possible to construct models for which the shape of the power spectrum agrees with the large scale slope of the observed power spectrum. However requiring compatibility with observed CMBR anisotropies, the normalization of the power spectrum is too low for magnetic fields to have significant impact on the large scale structure at present. Magnetic fields of a more recent origin generically give density power spectrumαk4which doesn’t agree with the shape of the observed power spectrum at any scale. Magnetic fields generate curl modes of the velocity field which increase both the quadrupole and hexadecapole of the redshift space power spectrum. For curl modes, the hexadecapole dominates over quadrupole. So the presence of curl modes could be indicated by an anomalously large hexadecapole, which has not yet been computed from observation.

An Inversion Method for Measuring β in Large‐Redshift Surveys

A precision method for determining the value of β = Ω/b, where b is the galaxy bias parameter, is presented. In contrast to other existing techniques that focus on estimating this quantity by measuring distortions in the redshift-space galaxy-galaxy correlation function or power spectrum, this method removes the distortions by reconstructing the real-space density field and determining the value of β that results in a symmetric signal. To remove the distortions, the method modifies the amplitudes of a Fourier plane-wave expansion of the survey data parameterized by β. This technique is not dependent on the small-angle/plane-parallel approximation and can make full use of large-redshift survey data. It has been tested using simulations with four different cosmologies and returns the value of β to ±0.031, more than a factor of 2 improvement over existing techniques.

The 2dF QSO Redshift Survey – IV. The QSO power spectrum from the 10k catalogue

We present a power spectrum analysis of the 10k catalogue from the 2dF QSO Redshift Survey. Although the Survey currently has a patchy angular selection function, we use the Virgo Consortium's Hubble Volume simulation to demonstrate that we are able to make a useful first measurement of the power spectrum over a wide range of scales. We compare the redshift space power spectra of QSOs with those measured for galaxies and Abell clusters at low redshift and find that they show similar shapes in their overlap range, , with . The amplitude of the QSO power spectrum at is almost comparable to that of galaxies at the present day if and (the Λ cosmology), and a factor of ≈3 lower if (the EdS cosmology) is assumed. The amplitude of the QSO power spectrum is a factor of ≈10 lower than that measured for Abell clusters at the present day. At larger scales, the QSO power spectra continue to rise robustly to ≈400h−1Mpc, implying more power at large scales than in the APM galaxy power spectrum measured by Baugh & Efstathiou. We split the QSO sample into two redshift bins and find little evolution in the amplitude of the power spectrum, consistent with the result for the QSO correlation function. In models with this represents evidence for a QSO mass bias that evolves as a function of time. We compare the QSO power spectra with cold dark matter (CDM) models to obtain a constraint on the shape parameter, Γ. For two choices of cosmology , and , , we find that the best fitting model has . In addition, we have shown that a power spectrum analysis of the Hubble Volume ΛCDM mock QSO catalogues with as input produces a result that is statistically consistent with the data. The analysis of the mock catalogues also indicates that the above results for Γ are unlikely to be dominated by systematic effects, owing to the current catalogue window. We conclude that the form of the QSO power spectrum shows large scale power significantly in excess of the standard CDM prediction, similar to that seen in local galaxy surveys at intermediate scales.

Cluster correlations in redshift space

We test an analytic model for the two-point correlations of galaxy clusters in redshift space using the Hubble volume N-body simulations. The correlation function of clusters shows no enhancement along the line of sight, owing to the lack of any virialized structures in the cluster distribution. However, the distortion of the clustering pattern arising from coherent bulk motions is clearly visible. The distribution of cluster peculiar motions is well described by a Gaussian, except in the extreme high-velocity tails. The simulations produce a small but significant number of clusters with large peculiar motions. The form of the redshift-space power spectrum is strongly influenced by errors in measured cluster redshifts in extant surveys. When these errors are taken into account, the model reproduces the power spectrum recovered from the simulation to an accuracy of 15 per cent or better over a decade in wavenumber. We compare our analytic predictions with the power spectrum measured from the APM cluster redshift survey. The cluster power spectrum constrains the amplitude of density fluctuations, as measured by the linear rms variance in spheres of radius 8 h - Mpc, denoted by σ 8 . When combined with the constraints on σ 8 and the density parameter Ω derived from the local abundance of clusters, we find a best-fitting cold dark matter model with σ 8 1.25 and Ω 0.2, for a power spectrum shape that matches that measured for galaxies. However, for the best-fitting value of Ω and given the value of Hubble's constant from recent measurements, the assumed shape of the power spectrum is incompatible with the most readily motivated predictions from the cold dark matter paradigm.

The power spectrum of flux-limited X-ray galaxy cluster surveys

We compute the redshift space power spectrum of two X-ray cluster samples: the X-ray Brightest Abell Cluster Sample (XBACS) and the Brightest Cluster Sample (BCS) using the method developed by Feldman, Kaiser & Peacock. The power spectrums derived for these samples are in agreement with determinations of other optical and X-ray cluster samples. For XBACS we find the largest power spectrum amplitude expected given the high richness of this sample ($R \ge 2$). In the range $0.05 \uk < k < 0.4 \uk$ the power spectrum shows a power law behavior $P(k)\propto k^{n}$ with an index $n\simeq-1.2$. In a similar range $0.04 \uk < k < 0.3 \uk $ BCS power spectrum has a smaller amplitude with index $n\simeq-1.0$. We do not find significant evidence for a peak at $k \simeq 0.05 \uk$ suggesting that claims such of feature detections in some cluster samples could relay on artificial inhomogeneities of the data. We compare our results with power spectrum predictions derived by Moscardini et al. within current cosmological models (LCDM and OCDM). For XBACS we find that both models underestimate the amplitude of the power spectrum but for BCS there is reasonably good agreement at $k\gsim 0.03 \uk$ for both models.

Biased Galaxy Formation and Measurements of β

Measurements of the cosmological density parameter Omega using techniques that exploit the gravity-induced motions of galaxies constrain, in linear perturbation theory, the degenerate parameter combination beta = Omega^{0.6}/b, where the linear bias parameter b is the ratio of the fluctuation amplitudes of the galaxy and mass distributions. However, the relation between the galaxy and mass density fields depends on the complex physics of galaxy formation, and it can in general be non-linear, stochastic, and perhaps non-local. The one-parameter linear bias model is almost certainly oversimplified, which leads to the obvious question: What is the quantity beta that is actually measured by different techniques? To address this question, we estimate beta from galaxy distributions that are constructed by applying a variety of locally biased galaxy formation models to cosmological N-body simulations. We compare the values of beta estimated using three different techniques: a density-density comparison similar to the POTENT analysis, a velocity-velocity comparison similar to the VELMOD analysis, and an anisotropy analysis of the redshift-space power spectrum. In most cases, we find that beta estimated using all three methods is similar to the asymptotic value of Omega^{0.6}/b_{sigma}(R) at large R, where b_{sigma}(R) is the ratio of rms galaxy fluctuations to rms mass fluctuations on scale R. Thus, something close to the conventional interpretation of beta continues to hold even for complex bias models. Moreover, we find that beta estimates made using these three methods should, in principle, agree with each other. It is thus unlikely that non-linear or scale-dependent bias is responsible for the discrepancies that exist among current measurements of beta from different techniques.

The ROSAT-ESO Flux-Limited X-Ray (REFLEX) galaxy cluster survey

We present a measure of the power spectrum on scales from 15 to 800h 1 Mpc using the ROSAT- ESO Flux-Limited X-Ray (REFLEX) galaxy cluster catalogue. The REFLEX survey provides a sample of the 452 X-ray brightest southern clusters of galaxies with the nominal flux limit S =3 : 01 0 12 erg s 1 cm 2 for the ROSAT energy band (0:1 2:4) keV. Several tests are performed showing no signicant incompletenesses of the REFLEX clusters with X-ray luminosities brighter than 10 43 erg s 1 up to scales of about 800h 1 Mpc. They also indicate that cosmic variance might be more important than previous studies suggest. We regard this as a warning not to draw general cosmological conclusions from cluster samples with a size smaller than REFLEX. Power spectra, P (k), of comoving cluster number densities are estimated for flux- and volume-limited subsamples. The most important result is the detection of a broad maximum within the comoving wavenum- ber range 0:022 k 0:030h Mpc 1 . The data suggest an increase of the power spectral amplitude with X-ray luminosity. Compared to optically selected cluster samples the REFLEX P (k) is flatter for wavenumbers k 0:05h Mpc 1 thus shifting the maximum of P (k) to larger scales. The smooth maximum is not consis- tent with the narrow peak detected at k =0 :05h Mpc 1 using the Abell/ACO richness0 data. In the range 0:02 k 0:4h Mpc 1 general agreement is found between the slope of the REFLEX P (k) and those obtained with optically selected galaxies. A semi-analytic description of the biased nonlinear power spectrum in redshift space gives the best agreement for low-density Cold Dark Matter models with or without a cosmological constant.

The Power Spectrum of Rich Clusters on Near‐Gigaparsec Scales

Recently, there have been numerous analyses of the redshift space power spectrum of rich clusters of galaxies. Some of these analyses indicate a bump in the Abell/ACO cluster power spectrum around k= 0.05hMpc^-1. Such a feature in the power spectrum excludes most standard formation models and indicates possible periodicity in the distribution of large-scale structure. However, the data used in detecting this peak include clusters with estimated redshifts and/or clusters outside of Abell's (1958) statistical sample, i.e. R=0 clusters. Here, we present estimations of the redshift-space power spectrum for a newly expanded sample of 637 R >= 1 Abell/ACO clusters which has a constant number density to z = 0.10 in the Southern Hemisphere and a nearly constant number density to z = 0.14 in the Northern Hemisphere. The volume sampled, \~10^8h^{-3}Mpc^{3}, is large enough to accurately calculate the power per mode to scales approaching 10^3h^{-1}Mpc. We find the shape of the power spectrum is is a power-law on scales 0.02 <= k <= 0.10hMpc^{-1}, with enhanced power over less rare clusters, such as APM clusters. The slope of this power-law is n= -1.4. The power spectrum is essentially featureless, although we do see a dip near $k = 0.04h$Mpc$^{-1}$ which cannot be considered statistically significant based on this data alone. We do not detect a narrow peak at $k \sim 0.05h$Mpc$^{-1}$ and there is no evidence for a turn-over in the power spectrum as has been previously reported. We compare the shape of the Abell/ACO rich cluster power spectrum to various linear models.

The Power Spectrum of the CfA/SSRS UZC Galaxy Redshift Survey

The combined CfA2/SSRS redshift catalog has recently been improved and made public. We compute the redshift-space power spectrum of this catalog using the regions at -2.5 < delta < 50 north (8h < alpha < 17h) and south (20h < alpha < 3h, b < -13) of the Galactic plane, where it is 98% complete down to Zwicky magnitude 15.5 and contains 13,681 galaxies. Our analysis uses Heavens-Taylor mode expansion, Karhunen-Loeve data compression and the Fisher matrix technique to compute quadratic band power estimates. This allows an exact calculation of window functions, including the integral constraint, in addition to the production of a power spectrum with uncorrelated error bars. Our results with this larger data set agree well with previous studies. We analyze 101Mpc/h and 130Mpc/h volume-limited subsets in addition to the full magnitude-limited sample, and our results are well fit by, e.g., Lambda-CDM models with bias b=1.2, 1.4 and 1.4, respectively. We estimate the effect of extinction using the Schlegel, Finkbeiner & Davis dust map. Our results are exclusively for the redshift space galaxy power spectrum, so they can only be compared with theoretical predictions if appropriate corrections are made for biasing and redshift space distortions.

Gravitational Clustering from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $\chi ^{2}$ \end{document} Initial Conditions

We consider gravitational clustering from primordial non-Gaussian fluctuations provided by a χ2 model, as motivated by some models of inflation. The emphasis is on signatures that can be used to constrain this type of model from large-scale structure galaxy surveys. Non-Gaussian initial conditions provide additional nonlinear couplings otherwise forbidden by symmetry that cause nonlinear gravitational corrections to become important at larger scales than in the Gaussian case. In fact, the lack of hierarchical scaling in the initial conditions is partially restored by gravitational evolution at scales k > 0.1 h Mpc-1. However, the bispectrum shows much larger amplitude and residual scale dependence not present in evolution from Gaussian initial conditions, which can be used to test this model against observations. We include the effects of biasing and redshift distortions essential to compare this model with galaxy redshift surveys. We also discuss the effects of primordial non-Gaussianity on the redshift-space power spectrum and show that it changes the shape of the quadrupole to monopole ratio through nonlinear corrections to infall velocities.

The Power Spectrum of Clusters of Galaxies and the Press‐Schechter Approximation

We examine the power spectrum of clusters in the Press-Schechter (PS) theory and in N-body simulations to see how the power spectrum of clusters is related to the power spectrum of matter density fluctuations in the Universe. An analytic model for the power spectrum of clusters for their given number density is presented, both for real space and redshift space. We test this model against results from N-body simulations and find that the agreement between the analytic theory and the numerical results is good for wavelengths $\lambda >60h^{-1}$ Mpc. On smaller scales non-linear processes that are not considered in the linear PS approximation influence the result. We also use our analytic model to study the redshift-space power spectrum of clusters in cold dark matter models with a cosmological constant ($\Lambda$CDM) and with a scale-invariant Harrison-Zel'dovich initial spectrum of density fluctuations. We find that power spectra of clusters in these models are not consistent with the observed power spectra of the APM and Abell-ACO clusters. One possible explanation for the observed power spectra of clusters is an inflationary scenario with a scalar field with the potential that has a localized steplike feature. We use the PS theory to examine the power spectrum of clusters in this model.

The power spectrum of the Point Source Catalogue redshift survey

We measure the redshift-space power spectrum P(k) for the recently completed IRAS Point Source Catalogue (PSC) redshift survey, which contains 14500 galaxies over 84% of the sky with 60 micron flux � 0.6 Jansky. Comparison with simulations shows that our estimated errors on P(k) are realistic, and that systematic errors due to the finite survey volume are small for wavenumbers k > � 0.03hMpc 1 . At large scales our power spectrum is intermediate between those of the earlier QDOT and 1.2 Jansky surveys, but with considerably smaller error bars; it falls slightly more steeply to smaller scales. We have fitted families of CDM-like models using the Peacock-Dodds formula for non-linear evolution; the results are somewhat sensitive to the assumed small-scale

Constraining cosmological models with cluster power spectra

Using extensive N-body simulations we estimate redshift space power spectra of clusters of galaxies for different cosmological models (SCDM, TCDM, CHDM, ΛCDM, OCDM, BSI, τCDM) and compare the results with observational data for Abell–ACO clusters. Our mock samples of galaxy clusters have the same geometry and selection functions as the observational sample which contains 417 clusters of galaxies in a double cone of galactic latitude |b|>30° up to a depth of 240h−1Mpc. The power spectrum has been estimated for wave numbers k in the range 0.03≲k≲0.2hMpc−1. For k>kmax≃0.05hMpc−1 the power spectrum of the Abell–ACO clusters has a power-law shape, P(k)∝kn, with n≈−1.9, while it changes sharply to a positive slope at k<kmax. By comparison with the mock catalogues SCDM, TCDM (n=0.9), and also OCDM with Ω0=0.35 are rejected. Better agreement with observation can be found for the ΛCDM model with Ω0=0.35 and h=0.7 and the CHDM model with two degenerate neutrinos and ΩHDM=0.2 as well as for a CDM model with broken scale invariance (BSI) and the τCDM model. As for the peak in the Abell–ACO cluster power spectrum, we find that it does not represent a very unusual finding within the set of mock samples extracted from our simulations.

The Bispectrum as a Signature of Gravitational Instability in Redshift Space

The bispectrum provides a characteristic signature of gravitational instability that can be used to probe the Gaussianity of the initial conditions and the bias of the galaxy distribution. We study how this signature is affected by redshift distortions using perturbation theory and high-resolution numerical simulations. We obtain perturbative results for the multipole expansion of the redshift-space bispectrum that provide a natural way to break the degeneracy between bias and Ω present in measurements of the redshift-space power spectrum. We propose a phenomenological model that incorporates the perturbative results and also describes the bispectrum in the transition to the nonlinear regime. We stress the importance of nonlinear effects and show that inaccurate treatment of these can lead to significant discrepancies in the determination of bias from galaxy redshift surveys. At small scales we find that the bispectrum monopole exhibits a strong configuration dependence that reflects the velocity dispersion of clusters. Therefore the hierarchical model for the three-point function does not hold in redshift space.

Recovery of the Shape of the Mass Power Spectrum from the Lyα Forest

We propose a method for recovering the shape of the mass power spectrum on large scales from the transmission fluctuations of the Lyα forest that directly takes into account redshift-space distortions. The procedure, in discretized form, involves the inversion of a triangular matrix that projects the mass power spectrum in three-dimensional real space to the transmission power spectrum in one-dimensional redshift space. We illustrate the method by performing a linear calculation relating the two. A method that does not take into account redshift-space anisotropy tends to underestimate the steepness of the mass power spectrum in the case of linear distortions. The issue of the effective bias factor for the linear distortion kernel is discussed.

The power spectrum of rich clusters of galaxies on large spatial scales

We present an analysis of the redshift-space power spectrum, $P(k)$, of rich clusters of galaxies based on an automated cluster catalogue selected from the APM Galaxy Survey. We find that $P(k)$ can be approximated by a power law, $P(k)\proptok^{n}$, with $n\approx-1.6$ over the wavenumber range $0.04\hr<k<0.1\hr$. Over this range of wavenumbers, the APM cluster power spectrum has the same shape as the power spectra measured for optical and IRAS galaxies. This is consistent with a simple linear bias model in which different tracers have the same power spectrum as that of the mass distribution but shifted in amplitude by a constant biasing factor. On larger scales, the power spectrum of APM clusters flattens and appears to turn over on a scale $k \sim 0.03\hmpcrev$. We compare the power spectra estimated from simulated APM cluster catalogues to those estimated directly from cubical N-body simulation volumes and find that the APM cluster survey should give reliable estimates of the true power spectrum at wavenumbers $k \simgt 0.02\hmpcrev$. These results suggest that the observed turn-over in the power spectrum may be a real feature of the cluster distribution and that we have detected the transition to a near scale-invariant power spectrum implied by observations of anisotropies in the cosmic microwave background radiation. The scale of the turn-over in the cluster power spectrum is in good agreement with the scale of the turn-over observed in the power spectrum of APM galaxies.

Estimating Ω from Galaxy Redshifts: Linear Flow Distortions and Nonlinear Clustering

We propose a method to determine the cosmic mass density Ω from redshift-space distortions induced by large-scale flows in the presence of nonlinear clustering. Nonlinear structures in redshift space, such as fingers of God, can contaminate distortions from linear flows on scales as large as several times the small-scale pairwise velocity dispersion σv. Following Peacock & Dodds, we work in the Fourier domain and propose a model to describe the anisotropy in the redshift-space power spectrum; tests with high-resolution numerical data demonstrate that the model is robust for both mass and biased galaxy halos on translinear scales and above. On the basis of this model, we propose an estimator of the linear growth parameter β = Ω0.6/b, where b measures bias, derived from sampling functions that are tuned to eliminate distortions from nonlinear clustering. The measure is tested on the numerical data and found to recover the true value of β to within ~10%. An analysis of IRAS 1.2 Jy galaxies yields β=0.8+0.4-0.3 at a scale of 1000 km s-1, which is close to optimal given the shot noise and finite size of the survey. This measurement is consistent with dynamical estimates of β derived from both real-space and redshift-space information. The importance of the method presented here is that nonlinear clustering effects are removed to enable linear correlation anisotropy measurements on scales approaching the translinear regime. We discuss implications for analyses of forthcoming optical redshift surveys in which the dispersion is more than a factor of 2 greater than in the IRAS data.