Redshift Space Distortion Measurements Research Articles

The Gigaparsec WiggleZ simulations: characterizing scale-dependant bias and associated systematics in growth of structure measurements

We use the Gigaparsec WiggleZ (GiggleZ) simulations to characterise galaxy bias and its scale dependence for a range of redshifts and halo masses in a standard $\Lambda$LCDM cosmology. Assuming bias converges to a scale independent form at large scales, we develop a phenomenological model which fully expresses the mass and redshift dependence of bias and its scale dependence in real or redshift space. We then use this to illustrate how scale-dependent bias can systematically skew measurements of the growth-rate of cosmic structure obtained from redshift-space distortion measurements. When data is fit only to scales $k_{\rm max}{\le}0.1$ $[h^{-1} \rm{Mpc}]^{-1}$, we find that these effects are significant only for large biases ($b{\gtrsim}3$) at large redshifts ($z{\gtrsim}1$). However, when smaller scales are incorporated ($k_{\rm max}{\le}0.2$ $[h^{-1} \rm{Mpc}]^{-1}$) to increase measurement precision, the combination of reduced statistical uncertainty and increased scale dependent bias can result in highly significant systematics for most large halos across all redshifts. We identify several new interesting aspects of bias, including a significant large-scale bias boost for small halos at low-redshifts due to substructure effects ($\sim$20\% for Milky Way-like systems) and a nearly redshift-independent halo mass (corresponding to a redshift-space bias of ${\sim}1.5$) for which halo bias has little-or-no scale dependence on scales greater than $3$ $[h^{-1} {\rm Mpc}]$. This suggests an optimal strategy of targeting bias ${\sim}{1.5}$ systems for clustering studies which are dominated by systematic uncertainties in how observed halo (or galaxy) distributions map to their underlying mass distribution, such as cosmological measurements of neutrino masses. Code for generating our fitting formula is publicly available at http://gbpoole.github.io/Poole_2014a_code/ (Abridged)

The impact of foregrounds on redshift space distortion measurements with the highly redshifted 21-cm line

The highly redshifted 21-cm line of neutral hydrogen has become recognized as a unique probe of cosmology from relatively low redshifts (z ∼ 1) up through the Epoch of Reionization (EoR) (z ∼ 8) and even beyond. To date, most work has focused on recovering the spherically averaged power spectrum of the 21-cm signal, since this approach maximizes the signal to noise in the initial measurement. However, like galaxy surveys, the 21-cm signal is affected by redshift space distortions, and is inherently anisotropic between the line of sight and transverse directions. A measurement of this anisotropy can yield unique cosmological information, potentially even isolating the matter power spectrum from astrophysical effects. However, in interferometric measurements, foregrounds also have an anisotropic footprint between the line of sight and transverse directions: the so-called foreground ‘wedge’. Although foreground subtraction techniques are actively being developed, a ‘foreground avoidance’ approach of simply ignoring contaminated modes has arguably proven most successful to date. In this work, we analyse the effect of this foreground anisotropy in recovering the redshift space distortion signature in 21-cm measurements at both high and intermediate redshifts. We find the foreground wedge corrupts nearly all of the redshift space signal for even the largest proposed EoR experiments (Hydrogen Epoch of Reionization Array and the Square Kilometre Array), making cosmological information unrecoverable without foreground subtraction. The situation is somewhat improved at lower redshifts, where the redshift-dependent mapping from observed coordinates to cosmological coordinates significantly reduces the size of the wedge. Using only foreground avoidance, we find that a large experiment like Canadian Hydrogen Intensity Mapping Experiment can place non-trivial constraints on cosmological parameters.

FORECASTS ON THE DARK ENERGY AND PRIMORDIAL NON-GAUSSIANITY OBSERVATIONS WITH THE TIANLAI CYLINDER ARRAY

The Tianlai experiment is dedicated to the observation of large scale structures (LSS) by the 21 cm intensity mapping technique. In this paper we make forecasts on its capability at observing or constraining the dark energy parameters and the primordial non-Gaussianity. From the LSS data one can use the baryon acoustic oscillation (BAO) and the growth rate derived from the redshift space distortion (RSD) to measure the dark energy density and equation of state. The primordial non-Gaussianity can be constrained either by looking for scale-dependent bias in the power spectrum, or by using the bispectrum. Here we consider three cases: the Tianlai cylinder array pathfinder which is currently being built, an upgrade of the pathfinder array with more receiver units, and the full-scale Tianlai cylinder array. Using the full-scale Tianlai experiment, we expect $\sigma_{w_0} \sim 0.082$ and $\sigma_{w_a} \sim 0.21$ from the BAO and RSD measurements, $\sigma_{\rm f_{NL}}^{\rm local} \sim 14$ from the power spectrum measurements with scale-dependent bias, and $\sigma_{\rm f_{NL}}^{\rm local} \sim 22$ and $\sigma_{\rm f_{NL}}^{\rm equil} \sim 157$ from the bispectrum measurements.

Measuring growth index in a Universe with sterile neutrinos

Consistency tests for the general relativity (GR) can be performed by constraining the growth index γ using the measurements of redshift-space distortions (RSD) in conjunction with other observations. In previous studies, deviations from the GR expected value of γ≈0.55 at the 2–3σ level were found. In this work, we reconsider the measurement of γ in a Universe with sterile neutrinos. We constrain the sterile neutrino cosmological model using the RSD measurements combined with the cosmic microwave background data (Planck temperature data plus WMAP 9-yr polarization data), the baryon acoustic oscillation data, the Hubble constant direct measurement, the Planck Sunyaev–Zeldovich cluster counts data, and the galaxy shear data. We obtain the constraint result of the growth index, γ=0.584−0.048+0.047, well consistent with the GR expected value (the consistency is at the 0.6σ level). For the parameters of sterile neutrino, we obtain Neff=3.62−0.42+0.26 and mν,sterileeff=0.48−0.14+0.11 eV. We also consider the BICEP2 data and perform an analysis on the model with tensor modes. Similar fit results are obtained, showing that once light sterile neutrino is considered in the Universe, GR will become well consistent with the current observations.

The 6dF Galaxy Survey: cosmological constraints from the velocity power spectrum

We present scale-dependent measurements of the normalised growth rate of structure $f\sigma_{8}(k, z=0)$ using only the peculiar motions of galaxies. We use data from the 6-degree Field Galaxy Survey velocity sample (6dFGSv) together with a newly-compiled sample of low-redshift $(z < 0.07)$ type Ia supernovae. We constrain the growth rate in a series of $\Delta k \sim 0.03 h{\rm Mpc^{-1}}$ bins to $\sim35\%$ precision, including a measurement on scales $>300 h^{-1}{\rm Mpc}$, which represents one of the largest-scale growth rate measurement to date. We find no evidence for a scale dependence in the growth rate, or any statistically significant variation from the growth rate as predicted by the {\it Planck} cosmology. Bringing all the scales together, we determine the normalised growth rate at $z=0$ to $\sim15\%$ in a manner {\it independent} of galaxy bias and in excellent agreement with the constraint from the measurements of redshift-space distortions from 6dFGS. We pay particular attention to systematic errors. We point out that the intrinsic scatter present in Fundamental-Plane and Tully-Fisher relations is only Gaussian in logarithmic distance units; wrongly assuming it is Gaussian in linear (velocity) units can bias cosmological constraints. We also analytically marginalise over zero-point errors in distance indicators, validate the accuracy of all our constraints using numerical simulations, and demonstrate how to combine different (correlated) velocity surveys using a matrix `hyper-parameter' analysis. Current and forthcoming peculiar velocity surveys will allow us to understand in detail the growth of structure in the low-redshift universe, providing strong constraints on the nature of dark energy.

Nonlocal gravity and structure in the Universe

The observed acceleration of the Universe can be explained by modifying general relativity. One such attempt is the nonlocal model of Deser and Woodard. Here we fix the background cosmology using results from the Planck satellite and examine the predictions of nonlocal gravity for the evolution of structure in the universe, confronting the model with three tests: gravitational lensing, redshift space distortions, and the estimator of gravity $E_G$. Current data favor general relativity (GR) over nonlocal gravity: fixing primordial cosmology with the best fit parameters from Planck leads to weak lensing results favoring GR by 5.9 sigma; redshift space distortions measurements of the growth rate preferring GR by 7.8 sigma; and the single measurement of $E_G$ favoring GR, but by less than 1-sigma. The significance holds up even after the parameters are allowed to vary within Planck limits. The larger lesson is that a successful modified gravity model will likely have to suppress the growth of structure compared to general relativity.

Cross-correlating spectroscopic and photometric galaxy surveys

AbstractDoes photometric and spectroscopic survey benefit from overlapping areas? The photometric survey measures 2D Weak Lensing (WL) information from galaxy shape distortions. On the other hand, the higher redshift precision of an spectroscopic survey allows measurements of redshift space distortions (RSD) and baryonic accustic oscillations (BAO) from 3D galaxy counts.The two surveys are combined using 2D-correlations, using sufficiently narrow bins to capture the radial information. This poster present effects of RSD and intrinsic correlations between narrow redshift bins. In understanding how the effects affects cosmological constrains, we first define two stage-IV and then present forecast for various configurations. When surveys overlap, they benefit from additional cross-correlations and sample variance cancellations from overlapping volumes. For a combined dark energy and growth history figure of merit, the result increase 50% for overlapping surveys, corresponding to 30% larger area.

Geometric and dynamic distortions in anisotropic galaxy clustering

We examine the signature of dynamic (redshift-space) distortions and geometric distortions (including the Alcock-Paczynski effect) in the context of the galaxy power spectrum measured in upcoming galaxy redshift surveys. Information comes from both the baryon acoustic oscillation (BAO) feature and the broadband power spectrum shape. Accurate modeling is required to extract this information without systematically biasing the result. We consider an analytic model for the power spectrum of dark matter halos in redshift space, based on the distribution function expansion, and compare with halo clustering measured in N-body simulations. We forecast that the distribution function model is sufficiently accurate to allow the inclusion of broadband information on scales down to k ∼ 0.2h Mpc−1, with somewhat better accuracy for higher bias halos. Compared with a BAO-only analysis with reconstruction, including broadband shape information can improve unbiased constraints on distance measures H(z) and DA(z) by ∼ 30% and 20%, respectively, for a galaxy sample similar to the DESI luminous red galaxies. The gains in precision are larger in the absence of BAO reconstruction. Furthermore, including broadband shape information allows the measurement of structure growth, through redshift-space distortions. For the same galaxy sample, the distribution function model is able to constrain fσ8 to ∼ 2%, when simultaneously fitting for H(z) and DA(z). We discuss techniques to optimize the analysis of the power spectrum, including removing modes near the line-of-sight that are particularly challenging to model, and whether these approaches can improve parameter constraints. We find that such techniques are unlikely to significantly improve constraints on geometry, although they may allow higher precision measurements of redshift-space distortions.

The clustering of Galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: including covariance matrix errors

We present improved methodology for including covariance matrices in the error budget of Baryon Oscillation Spectroscopic Survey (BOSS) galaxy clustering measurements, revisiting Data Release 9 (DR9) analyses, and describing a method that is used in DR10/11 analyses presented in companion papers. The precise analysis method adopted is becoming increasingly important, due to the precision that BOSS can now reach: even using as many as 600 mock catalogues to estimate covariance of 2-point clustering measurements can still lead to an increase in the errors of ~20%, depending on how the cosmological parameters of interest are measured. In this paper we extend previous work on this contribution to the error budget, deriving formulae for errors measured by integrating over the likelihood, and to the distribution of recovered best-fit parameters fitting the simulations also used to estimate the covariance matrix. Both are situations that previous analyses of BOSS have considered. We apply the formulae derived to Baryon Acoustic Oscillation (BAO) and Redshift-Space Distortion (RSD) measurements from BOSS in our companion papers. To further aid these analyses, we consider the optimum number of bins to use for 2-point measurements using the monopole power spectrum or correlation function for BAO, and the monopole and quadrupole moments of the correlation function for anisotropic-BAO and RSD measurements.

Probing neutrinos from Planck and forthcoming galaxy redshift surveys

We investigate how much the constraints on the neutrino properties can be improvedby combining the CMB, the photometric and spectroscopic galaxyredshift surveys which include the CMB lensing, galaxy lensingtomography, galaxy clustering and redshift space distortion observables. We pay a particular attention to the constraint onthe neutrino mass in view of the forthcoming redshift surveys such asthe Euclid satellite and the LSST survey along with the Planck CMB lensing measurements.Combining the transverse mode information from the angular powerspectrum and the longitudinal mode information from the spectroscopic survey with the redshiftspace distortion measurements can determine the total neutrino masswith the projected error of \U0001d4aa(0.02) eV.Ouranalysis fixes the mass splittings among the neutrino species to beconsistent with the neutrino oscillation data, and we accordingly study thesensitivity of our parameter estimations on the minimal neutrino mass. Thecosmological measurement of the total neutrino mass can distinguishbetween the normal and inverted mass hierarchy scenarios if the minimalneutrino mass ≲0.005 eV with the predicted 1–σuncertainties taken into account.

Simulations of Galileon modified gravity: Clustering statistics in real and redshift space

We use N-body simulations to study the statistics of massive halos and redshift space distortions for theories with a standard \Lambda CDM expansion history and a galileon-type scalar field. The extra scalar field increases the gravitational force, leading to enhanced structure formation. We compare our measurements of the real space matter power spectrum and halo properties with fitting formula for estimating these quantities analytically. We find that a model for power spectrum, halo mass-function and halo bias, derived from \Lambda CDM simulations can fit the results from our simulations of modified gravity when \sigma_8 is appropriately adjusted. We also study the redshift space distortions in the two point correlation function measured from these simulations, finding a difference in the ratio of the redshift space to real space clustering amplitude relative to standard gravity on all scales. We find enhanced clustering on scales r>10 Mpc/h and increased damping of the correlation function for scales r<9 Mpc/h. The boost in the clustering on large scales due to the enhanced gravitational forces cannot be mimicked in a standard gravity model by simply changing \sigma_8. This result illustrates the usefulness of redshift space distortion measurements as a probe of modifications to General Relativity.

Lower Growth Rate from Recent Redshift Space Distortion Measurements than Expected from Planck

We perform a metastudy of recently published redshift space distortion (RSD) measurements of the cosmological growth rate, f(z)σ8(z). We analyze the latest results from the 6dFGS, BOSS, LRG, WiggleZ, and VIPERS galaxy redshift surveys, and compare the measurements to expectations from Planck. In this Letter we point out that the RSD measurements are consistently lower than the values expected from Planck, and the relative scatter between the RSD measurements is lower than expected. A full resolution of this issue may require a more robust treatment of nonlinear effects in RSD models, although the trend for a low σ8 agrees with recent constraints on σ8 and Ω(m) from Sunyaev-Zeldovich cluster counts identified in Planck.

Quintessence: a review

Quintessence is a canonical scalar field introduced to explain the late-time cosmic acceleration. The cosmological dynamics of quintessence is reviewed, paying particular attention to the evolution of the dark energy equation of state w. For the field potentials having tracking and thawing properties, the evolution of w can be known analytically in terms of a few model parameters. Using the analytic expression of w, we constrain quintessence models from the observations of supernovae type Ia, cosmic microwave background and baryon acoustic oscillations. The tracking freezing models are hardly distinguishable from the Λ-cold-dark-matter model, whereas in thawing models the today's field equation of state is constrained to be w0 < −0.7 (95% CL). We also derive an analytic formula for the growth rate of matter density perturbations in dynamical dark energy models, which allows a possibility of putting further bounds on w from the measurement of redshift-space distortions in the galaxy power spectrum. Finally, we review particle physics models of quintessence—such as those motivated by supersymmetric theories. The field potentials of thawing models based on a pseudo-Nambu–Goldstone boson or extended supergravity theories have a nice property that a tiny mass of quintessence can be protected against radiative corrections.

Principal component analysis of modified gravity using weak lensing and peculiar velocity measurements

We perform a principal component analysis to assess ability of futureobservations to measure departures from General Relativity inpredictions of the Poisson and anisotropy equations on linearscales. In particular, we focus on how the measurements ofredshift-space distortions (RSD) observed from spectroscopic galaxyredshift surveys will improve the constraints when combined withlensing tomographic surveys. Assuming a Euclid-like galaxy imaging andredshift survey, we find that adding the 3D information decreases thestatistical uncertainty by a factor between 3 and 7 compared to thecase when only observables from lensing tomographic surveys areused. We also find that the number of well-constrained modes increasesby a factor between 3 and 6. Our study indicates the importance ofjoint galaxy imaging and redshift surveys such as SuMIRe and Euclid togive more stringent tests of the ΛCDM model and to distinguishbetween various modified gravity and dark energy models.

Constraining cosmic expansion and gravity with galaxy redshift surveys

We analyze the science reach of a next generation galaxy redshift survey such as BigBOSS to fit simultaneously for time varying dark energy equation of state and time- and scale-dependent gravity. The simultaneous fit avoids potential bias from assuming ΛCDM expansion or general relativity and leads to only modest degradation in constraints. Galaxy bias, fit freely in redshift bins, is self calibrated by spectroscopic measurements of redshift space distortions and causes little impact. The combination of galaxy redshift, cosmic microwave background, and supernova distance data can deliver 5-10% constraints on 6 model independent modified gravity quantities.

Statistical and systematic errors in redshift-space distortion measurements from large surveys

We investigate the impact of statistical and systematic errors on measurements of linear redshift-space distortions (RSD) in future cosmological surveys, analyzing large catalogues of dark-matter halos from the BASICC simulation. These allow us to estimate the dependence of errors on typical survey properties, as volume, galaxy density and mass (i.e. bias factor) of the adopted tracer. We find that measures of the specific growth rate \beta=f/b using the Hamilton/Kaiser harmonic expansion of the redshift-space correlation function \xi(r_p,\pi) on scales larger than 3/h Mpc are typically under-estimated by up to 10% for galaxy sized halos. This is significantly larger than the corresponding statistical errors, which amount to a few percent, indicating the importance of non-linear improvements to the Kaiser model to obtain accurate measurements of the growth rate. We compare the statistical errors to predictions obtained with the Fisher information matrix, based on the usual FKP prescription for the errors on the power spectrum. We show that this produces parameter errors fairly similar to the standard deviations from the halo catalogues, but only if applied to strictly linear scales in Fourier space (k<0.2 h/Mpc). Finally, we present an accurate scaling formula describing the relative error on {\beta} as a function of the survey parameters, which closely matches the simulation results in all explored regimes. This provides a handy and plausibly more realistic alternative to the Fisher matrix approach, to quickly and accurately predict RSD statistical errors expected from future surveys.

The complementarity of redshift-space distortions and the integrated Sachs-Wolfe effect: a 3D spherical analysis

Assuming general relativity is correct on large scales, redshift-space distortions (RSDs) and the integrated Sachs–Wolfe effect (ISW) are both sensitive to the time derivative of the linear growth function. We investigate the extent to which these probes provide complementary or redundant information when they are combined to constrain the evolution of the linear velocity power spectrum, often quantified by the function f(z)σ8(z), where f is the logarithmic derivative of σ8 with respect to (1 +z). Using a spherical Fourier–Bessel (SFB) expansion for galaxy number counts and a spherical harmonic expansion for the cosmic microwave background (CMB) anisotropy, we compute the covariance matrices of the signals for a large galaxy redshift survey combined with a CMB survey like Planck. The SFB basis allows accurate ISW estimates by avoiding the plane-parallel approximation, and it retains RSD information that is otherwise lost when projecting angular clustering on to redshift shells. It also allows straightforward calculations of covariance with the CMB. We find that the correlation between the ISW and RSD signals is low since the probes are sensitive to different modes. For our default surveys, on large scales (k < 0.05 Mpc h−1), the ISW can improve constraints on fσ8 by more than 10 per cent compared to using RSDs alone. In the future, when precision RSD measurements are available on smaller scales, the cosmological constraints from ISW measurements will not be competitive; however, they will remain a useful consistency test for possible systematic contamination and alternative models of gravity.

Distribution function approach to redshift space distortions. Part II: N-body simulations

Measurement of redshift-space distortions (RSD) offers an attractive method to directly probe the cosmic growth history of density perturbations. A distribution function approach where RSD can be written as a sum over density weighted velocity moment correlators has recently been developed. In this paper we use results of N-body simulations to investigate the individual contributions and convergence of this expansion for dark matter. If the series is expanded as a function of powers of μ, cosine of the angle between the Fourier mode and line of sight, then there are a finite number of terms contributing at each order. We present these terms and investigate their contribution to the total as a function of wavevector k. For μ2 the correlation between density and momentum dominates on large scales. Higher order corrections, which act as a Finger-of-God (FoG) term, contribute 1% at k ∼ 0.015hMpc−1, 10% at k ∼ 0.05hMpc−1 at z = 0, while for k > 0.15hMpc−1 they dominate and make the total negative. These higher order terms are dominated by density-energy density correlations which contributes negatively to the power, while the contribution from vorticity part of momentum density auto-correlation adds to the total power, but is an order of magnitude lower. For μ4 term the dominant term on large scales is the scalar part of momentum density auto-correlation, while higher order terms dominate for k > 0.15hMpc−1. For μ6 and μ8 we find it has very little power for k < 0.15hMpc−1, shooting up by 2–3 orders of magnitude between k < 0.15hMpc−1 and k < 0.4hMpc−1. We also compare the expansion to the full 2-d Pss(k,μ), as well as to the monopole, quadrupole, and hexadecapole integrals of Pss(k,μ). For these statistics an infinite number of terms contribute and we find that the expansion achieves percent level accuracy for kμ < 0.15hMpc−1 at 6-th order, but breaks down on smaller scales because the series is no longer perturbative. We explore resummation of the terms into FoG kernels, which extend the convergence up to a factor of 2 in scale. We find that the FoG kernels are approximately Lorentzian with velocity dispersions around 600 km/s at z = 0.

Interpreting large-scale redshift-space distortion measurements

The simplest theory describing large-scale redshift-space distortions (RSD), based on linear theory and distant galaxies, depends on the growth of cosmological structure, suggesting that strong tests of General Relativity can be constructed from galaxy surveys. As data sets become larger and the expected constraints more precise, the extent to which the RSD follow the simple theory needs to be assessed in order that we do not introduce systematic errors into the tests by introducing inaccurate simplifying assumptions. We study the impact of the sample geometry, non-linear processes, and biases induced by our lack of understanding of the radial galaxy distribution on RSD measurements. Using LasDamas simulations of the Sloan Digital Sky Survey II (SDSS-II) Luminous Red Galaxy (LRG) data, these effects are shown to be important at the level of 20 per cent. Including them, we can accurately model the recovered clustering in these mock catalogues on scales 30 -- 200 Mpc/h. Applying this analysis to robustly measure parameters describing the growth history of the Universe from the SDSS-II data, gives $f(z=0.25)\sigma_8(z=0.25)=0.3512\pm0.0583$ and $f(z=0.37)\sigma_8(z=0.37)=0.4602\pm0.0378$ when no prior is imposed on the growth-rate, and the background geometry is assumed to follow a $\Lambda$CDM model with the WMAP + SNIa priors. The standard WMAP constrained $\Lambda$CDM model with General Relativity predicts $f(z=0.25)\sigma_8(z=0.25)=0.4260\pm0.0141$ and $f(z=0.37)\sigma_8(z=0.37)=0.4367\pm0.0136$, which is fully consistent with these measurements.

Redshift-space distortions

Comparing measurements of redshift-space distortions (RSDs) with geometrical observations of the expansion of the Universe offers tremendous potential for testing general relativity on very large scales. The basic linear theory of RSDs in the distant-observer limit has been known for 25 years and the effect has been conclusively observed in numerous galaxy surveys. The next generation of galaxy survey will observe many millions of galaxies over volumes of many tens of Gpc(3). They will provide RSD measurements of such exquisite precision that we will have to carefully analyse and correct for many systematic deviations from this simple picture in order to fully exploit the statistical precision obtained. We review RSD theory and show how ubiquitous RSDs actually are, and then consider a number of potential systematic effects, shamelessly highlighting recent work in which we have been involved. This review ends by looking ahead to the future surveys that will make the next generation of RSD measurements.