Gaussian Streaming Model Research Articles

Challenges in constraining gravity with cosmic voids

We compare void size and clustering statistics for nDGP and $f(R)$ gravity models and general relativity (GR) using N-body simulations. We show how it is critical to consider the statistics derived from mock galaxy catalogs rather than the dark matter halos alone. Marked differences between the void size functions for GR and $f(R)$ models which present when voids are identified using dark matter halos are removed when voids are identified, more realistically, from mock galaxy tracers of the halos. The void radial velocities and velocity dispersions in the $f(R)$ and nDGP models are enhanced relative to GR in both halos and mock galaxy identified voids. Despite this, we find that the redshift space void quadrupole moments derived from the mock galaxy tracers are strikingly similar across the three gravity models. The Gaussian streaming model (GSM) is shown to accurately reconstruct ${\ensuremath{\xi}}_{2}$ in modified gravity models, and we employ the GSM, using a functional derivative approach, to analyze the insensitivity of ${\ensuremath{\xi}}_{2}$ to the gravity model. Assuming linear theory, we show the void quadrupole to be an unbiased estimator of the redshift space growth rate parameter $\ensuremath{\beta}=f/b$ in the modified gravity theories.

Redshift-space effects in voids and their impact on cosmological tests – II. The void-galaxy cross-correlation function

ABSTRACT This is the second part of a thorough investigation of the redshift-space effects that affect void properties and the impact they have on cosmological tests. Here, we focus on the void-galaxy cross-correlation function, specifically, on the projected versions that we developed in a previous work. The pillar of the analysis is the one-to-one relationship between real and redshift-space voids above the shot-noise level identified with a spherical void finder. Under this mapping, void properties are affected by three effects: (i) a systematic expansion as a consequence of the distortions induced by galaxy dynamics, (ii) the Alcock–Paczynski volume effect, which manifests as an overall expansion or contraction depending on the fiducial cosmology, and (iii) a systematic off-centring along the line of sight as a consequence of the distortions induced by void dynamics. We found that correlations are also affected by an additional source of distortions: the ellipticity of voids. This is the first time that distortions due to the off-centring and ellipticity effects are detected and quantified. With a simplified test, we verified that the Gaussian streaming model is still robust provided all these effects are taken into account, laying the foundations for improvements in current models in order to obtain unbiased cosmological constraints from spectroscopic surveys. Besides this practical importance, this analysis also encodes key information about the structure and dynamics of the Universe at the largest scales. Furthermore, some of the effects constitute cosmological probes by themselves, as is the case of the void ellipticity.

Redshift-space distortions with split densities

ABSTRACT Accurate modelling of redshift-space distortions (RSD) is challenging in the non-linear regime for two-point statistics e.g. the two-point correlation function (2PCF). We take a different perspective to split the galaxy density field according to the local density, and cross-correlate those densities with the entire galaxy field. Using mock galaxies, we demonstrate that combining a series of cross-correlation functions (CCFs) offers improvements over the 2PCF as follows: (1) The distribution of peculiar velocities in each split density is nearly Gaussian. This allows the Gaussian streaming model for RSD to perform accurately within the statistical errors of a ($1.5\, h^{-1}$ Gpc)3 volume for almost all scales and all split densities. (2) The probability distribution of the density contrast at small scales is non-Gaussian, but the CCFs of split densities capture the non-Gaussianity, leading to improved cosmological constraints over the 2PCF. We can obtain unbiased constraints on the growth parameter fσ12 at the per cent level, and Alcock–Paczynski (AP) parameters at the sub-per cent level with the minimal scale of $15\, h^{-1}{\rm Mpc}$. This is a ∼30 per cent and ∼6 times improvement over the 2PCF, respectively. The diverse and steep slopes of the CCFs at small scales are likely to be responsible for the improved constraints of AP parameters. (3) Baryon acoustic oscillations (BAO) are contained in all CCFs of split densities. Including BAO scales helps to break the degeneracy between the line-of-sight and transverse AP parameters, allowing independent constraints on them. We discuss and compare models for RSD around spherical densities.

Redshift-space distortions in Lagrangian perturbation theory

We present the one-loop 2-point function of biased tracers in redshift space computed with Lagrangian perturbation theory, including a full resummation of both long-wavelength (infrared) displacements and associated velocities. The resulting model accurately predicts the power spectrum and correlation function of halos and mock galaxies from two different sets of N-body simulations at the percent level for quasi-linear scales, including the damping of the baryon acoustic oscillation signal due to the bulk motions of galaxies. We compare this full resummation with other, approximate, techniques including the moment expansion and Gaussian streaming model. We discuss infrared resummation in detail and compare our Lagrangian formulation with the Eulerian theory augmented by an infrared resummation based on splitting the input power spectrum into “wiggle” and “no-wiggle” components. We show that our model is able to recover unbiased cosmological parameters in mock data encompassing a volume much larger than what will be available to future galaxy surveys. We demonstrate how to efficiently compute the resulting expressions numerically, making available a fast Python code capable of rapidly computing these statistics in both configuration and Fourier space.

A Lagrangian perturbation theory in the presence of massive neutrinos

We develop a Lagrangian Perturbation Theory (LPT) framework to study the clustering of cold dark matter (CDM) in cosmologies with massive neutrinos. We follow the trajectories of CDM particles with Lagrangian displacements fields up to third order in perturbation theory. Once the neutrinos become non-relativistic, their density fluctuations are modeled as being proportional to the CDM density fluctuations, with a scale-dependent proportionality factor. This yields a gravitational back-reaction that introduces additional scales to the linear growth function, which is accounted for in the higher order LPT kernels. Through non-linear mappings from Eulerian to Lagrangian frames, we ensure that our theory has a well behaved large scale behavior free of unwanted UV divergences, which are common when neutrino and CDM densities are not treated on an equal footing, and in resummation schemes that manifestly break Galilean invariance. We use our theory to construct correlation functions for both the underlying matter field, as well as for biased tracers using Convolution-LPT. Redshift-space distortions effects are modeled using the Gaussian Streaming Model. When comparing our analytical results to simulated data from the \ extsc{Quijote} simulation suite, we find good accuracy down to r=20 Mpc h−1 at redshift z=0.5, for the real space and redshift space monopole particle correlation functions with no free parameters. The same accuracy is reached for the redshift space quadrupole if we additionally consider an effective field theory parameter that shifts the pairwise velocity dispersion. For modeling the correlation functions of tracers we adopt a simple Lagrangian biasing scheme with only density and curvature operators, which we find sufficient to reach down to r=20 Mpc h−1 when comparing to simulated halos.

Hybrid Pℓ(k): general, unified, non-linear matter power spectrum in redshift space

Constraints on gravity and cosmology will greatly benefit from performing joint clustering and weak lensing analyses on large-scale structure data sets. Utilising non-linear information coming from small physical scales can greatly enhance these constraints. At the heart of these analyses is the matter power spectrum. Here we employ a simple method, dubbed “Hybrid Pℓ(k)”, based on the Gaussian Streaming Model (GSM), to calculate the quasi non-linear redshift space matter power spectrum multipoles. This employs a fully non-linear and theoretically general prescription for the matter power spectrum. We test this approach against comoving Lagrangian acceleration simulation measurements performed in GR, DGP and f(R) gravity and find that our method performs comparably or better to the dark matter TNS redshift space power spectrum model {for dark matter. When comparing the redshift space multipoles for halos, we find that the Gaussian approximation of the GSM with a linear bias and a free stochastic term, N, is competitive to the TNS model.} Our approach offers many avenues for improvement in accuracy as well as further unification under the halo model.

Towards a non-Gaussian model of redshift space distortions

ABSTRACT To understand the nature of the accelerated expansion of the Universe, we need to combine constraints on the expansion rate and growth of structure. The growth rate is usually extracted from 3D galaxy maps by exploiting the effects of peculiar motions on galaxy clustering. However, theoretical models of the probability distribution function (PDF) of galaxy pairwise peculiar velocities are not accurate enough on small scales to reduce the error on theoretical predictions to the level required to match the precision expected for measurements from future surveys. Here, we improve the modelling of the pairwise velocity distribution by using the Skew-T PDF, which has non-zero skewness and kurtosis. Our model accurately reproduces the redshift space multipoles (monopole, quadrupole, and hexadecapole) predicted by N-body simulations, above scales of about $10\, h^{-1}{\rm Mpc}$. We illustrate how a Taylor expansion of the streaming model can reveal the contributions of the different moments to the clustering multipoles, which are independent of the shape of the velocity PDF. The Taylor expansion explains why the Gaussian streaming model works well in predicting the first two redshift space multipoles, although the velocity PDF is non-Gaussian even on large scales. Indeed, any PDF with the correct first two moments would produce precise results for the monopole down to scales of about $10\, h^{-1}{\rm Mpc}$, and for the quadrupole down to about $30\, h^{-1}{\rm Mpc}$. An accurate model for the hexadecapole needs to include higher order moments.

An accurate perturbative approach to redshift space clustering of biased tracers in modified gravity

We extend the scale-dependent Gaussian Streaming Model (GSM) to produce analytical predictions for the anisotropic redshift-space correlation function for biased tracers in modified gravity models. Employing the Convolution Lagrangian Perturbation Theory (CLPT) re-summation scheme, with a local Lagrangian bias schema provided by the peak-background split formalism, we predict the necessary ingredients that enter the GSM, the real-space halo pairwise velocity and the pairwise velocity dispersion. We further consider effective field theory contributions to the pairwise velocity dispersion in order to model correctly its large scale behavior. We apply our method on two widely-considered modified gravity models, the chameleon-screened f(R) Hu-Sawicki model and the nDGP Vainshtein model and compare our predictions against state-of-the-art N-body simulations for these models. We demonstrate that the GSM approach to predict the monopole and the quadrupole of the redshift-space correlation function for halos, gives very good agreement with the simulation data, for a wide range of screening mechanisms, levels of screening and halo masses at z=0.5 and z=1. Our work shows the applicability of the GSM, for cosmologies beyond GR, demonstrating that it can serve as a powerful predictive tool for the next stage of cosmological surveys like DESI, Euclid, LSST and WFIRST.

The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey DR14 LRG sample: structure growth rate measurement from the anisotropic LRG correlation function in the redshift range 0.6 < z < 1.0

ABSTRACT We analyse the anisotropic clustering of the Sloan Digital Sky Survey-IV Extended Baryon Oscillation Spectroscopic Survey (eBOSS) Luminous Red Galaxy Data Release 14 (DR14) sample combined with Baryon Oscillation Spectroscopic Survey (BOSS) CMASS sample of galaxies in the redshift range 0.6 < z < 1.0, which consists of 80 118 galaxies from eBOSS and 46 439 galaxies from the BOSS-CMASS sample. The eBOSS-CMASS Luminous Red Galaxy sample has a sky coverage of 1844 deg2, with an effective volume of 0.9 Gpc3. The analysis was made in configuration space using a Legendre multipole expansion. The Redshift Space Distortion signal is modelled as a combination of the Convolution Lagrangian Perturbation Model and the Gaussian Streaming Model. We constrain the logarithmic growth of structure times the amplitude of dark matter density fluctuations, f(zeff)σ8(zeff) = 0.454 ± 0.134, and the Alcock-Paczynski dilation scales which constraints the angular diameter distance $D_A(z_{\mathrm{ eff}})=1466.5 \pm 133.2 (r_s/r_s^{\rm fid})$ and $H(z_{\rm eff})=105.8 \pm 15.7 (r_s^{\rm fid}/r_s) \mathrm{km\, s^{-1}\, Mpc^{-1}}$, where rs is the sound horizon at the end of the baryon drag epoch and $r_s^{\rm fid}$ is its value in the fiducial cosmology at an effective redshift zeff = 0.72. These results are in full agreement with the current Λ-Cold Dark Matter (Λ-CDM) cosmological model inferred from Planck measurements. This study is the first eBOSS LRG full-shape analysis i.e. including Redshift Space Distortions simultaneously with the Alcock-Paczynski effect and the Baryon Acoustic Oscillation scale.

Exploring redshift-space distortions in large-scale structure

We explore and compare different ways large-scale structure observables in redshift-space and real space can be connected. These include direct computation in Lagrangian space, moment expansions and two formulations of the streaming model. We derive for the first time a Fourier space version of the streaming model, which yields an algebraic relation between the real- and redshift-space power spectra which can be compared to earlier, phenomenological models. By considering the redshift-space 2-point function in both configuration and Fourier space, we show how to generalize the Gaussian streaming model to higher orders in a systematic and computationally tractable way. We present a closed-form solution to the Zeldovich power spectrum in redshift space and use this as a framework for exploring convergence properties of different expansion approaches. While we use the Zeldovich approximation to illustrate these results, much of the formalism and many of the relations we derive hold beyond perturbation theory, and could be used with ingredients measured from N-body simulations or in other areas requiring decomposition of Cartesian tensors times plane waves. We finish with a discussion of the redshift-space bispectrum, bias and stochasticity and terms in Lagrangian perturbation theory up to 1-loop order.

A perturbative approach to the redshift space correlation function: beyond the Standard Model

We extend our previous redshift space power spectrum code to the redshift space correlation function. Here we focus on the Gaussian Streaming Model (GSM). Again, the code accommodates a wide range of modified gravity and dark energy models. For the non-linear real space correlation function used in the GSM we use the Fourier transform of the RegPT 1-loop matter power spectrum. We compare predictions of the GSM for a Vainshtein screened and Chameleon screened model as well as GR. These predictions are compared to the Fourier transform of the Taruya, Nishimichi and Saito (TNS) redshift space power spectrum model which is fit to N-body data. We find very good agreement between the Fourier transform of the TNS model and the GSM predictions, with ⩽ 6% deviations in the first two correlation function multipoles for all models for redshift space separations in 50Mpch ⩽ s ⩽ 180Mpc/h. Excellent agreement is found in the differences between the modified gravity and GR multipole predictions for both approaches to the redshift space correlation function, highlighting their matched ability in picking up deviations from GR. We elucidate the timeliness of such non-standard templates at the dawn of stage-IV surveys and discuss necessary preparations and extensions needed for upcoming high quality data.

Gaussian streaming with the truncated Zel’dovich approximation

We calculate the dark matter halo correlation function in redshift space using the Gaussian streaming model (GSM). To determine the scale dependent functions entering the streaming model we use local Lagrangian bias together with Convolution Lagrangian perturbation theory (CLPT) which constitutes an approximation to the Post-Zel'dovich approximation. On the basis of N-body simulations we demonstrate that a smoothing of the initial conditions with the Lagrangian radius improves the Zel'dovich approximation and its ability to predict the displacement field of proto-halos. Based on this observation we implement a "truncated" CLPT by smoothing the initial power spectrum and investigate the dependence of the streaming model ingredients on the smoothing scale. We find that the real space correlation functions of halos and their mean pairwise velocity are optimised if the coarse graining scale is chosen to be 1 Mpc/h at z=0, while the pairwise velocity dispersion is optimised if the smoothing scale is chosen to be the Lagrangian size of the halo. We compare theoretical results for the halo correlation function in redshift space to measurements within the Horizon Run 2 N-body simulation halo catalog. We find that this simple two-filter smoothing procedure in the spirit of the truncated Zel'dovich approximation significantly improves the GSM+CLPT prediction of the redshift space halo correlation function over the whole mass range from large galaxy to galaxy cluster-sized halos.

The Gaussian streaming model and convolution Lagrangian effective field theory

We update the ingredients of the Gaussian streaming model (GSM) for the redshift-space clustering of biased tracers using the techniques of Lagrangian perturbation theory, effective field theory (EFT) and a generalized Lagrangian bias expansion. After relating the GSM to the cumulant expansion, we present new results for the real-space correlation function, mean pairwise velocity and pairwise velocity dispersion including counter terms from EFT and bias terms through third order in the linear density, its leading derivatives and its shear up to second order. We discuss the connection to the Gaussian peaks formalism. We compare the ingredients of the GSM to a suite of large N-body simulations, and show the performance of the theory on the low order multipoles of the redshift-space correlation function and power spectrum. We highlight the importance of a general biasing scheme, which we find to be as important as higher-order corrections due to non-linear evolution for the halos we consider on the scales of interest to us.

Improving the modelling of redshift-space distortions– II. A pairwise velocity model covering large and small scales

We develop a model for the redshift-space correlation function, valid for both dark matter particles and halos on scales $>5\,h^{-1}$Mpc. In its simplest formulation, the model requires the knowledge of the first three moments of the line-of-sight pairwise velocity distribution plus two well-defined dimensionless parameters. The model is obtained by extending the Gaussian-Gaussianity prescription for the velocity distribution, developed in a previous paper, to a more general concept allowing for local skewness, which is required to match simulations. We compare the model with the well known Gaussian streaming model and the more recent Edgeworth streaming model. Using N-body simulations as a reference, we show that our model gives a precise description of the redshift-space clustering over a wider range of scales. We do not discuss the theoretical prescription for the evaluation of the velocity moments, leaving this topic to further investigation.

Edgeworth streaming model for redshift space distortions

We derive the Edgeworth streaming model (ESM) for the redshift space correlation function starting from an arbitrary distribution function for biased tracers of dark matter by considering its two-point statistics and show that it reduces to the Gaussian streaming model (GSM) when neglecting non-Gaussianities. We test the accuracy of the GSM and ESM independent of perturbation theory using the Horizon Run 2 $N$-body halo catalog. While the monopole of the redshift space halo correlation function is well described by the GSM, higher multipoles improve upon including the leading order non-Gaussian correction in the ESM: the GSM quadrupole breaks down on scales below $30\text{ }\text{ }\mathrm{Mpc}/h$ whereas the ESM stays accurate to 2% within statistical errors down to $10\text{ }\text{ }\mathrm{Mpc}/h$. To predict the scale-dependent functions entering the streaming model we employ convolution Lagrangian perturbation theory (CLPT) based on the dust model and local Lagrangian bias. Since dark matter halos carry an intrinsic length scale given by their Lagrangian radius, we extend CLPT to the coarse-grained dust model and consider two different smoothing approaches operating in Eulerian and Lagrangian space, respectively. The coarse graining in Eulerian space features modified fluid dynamics different from dust while the coarse graining in Lagrangian space is performed in the initial conditions with subsequent single-streaming dust dynamics, implemented by smoothing the initial power spectrum in the spirit of the truncated Zel'dovich approximation. Finally, we compare the predictions of the different coarse-grained models for the streaming model ingredients to $N$-body measurements and comment on the proper choice of both the tracer distribution function and the smoothing scale. Since the perturbative methods we considered are not yet accurate enough on small scales, the GSM is sufficient when applied to perturbation theory.

Testing general relativity with growth rate measurement from Sloan Digital Sky Survey – III. Baryon Oscillations Spectroscopic Survey galaxies

The measured redshift ($z$) of an astronomical object is a combination of Hubble recession, gravitational redshift and peculiar velocity. The line of sight distance to a galaxy inferred from redshift is affected by the peculiar velocity component of galaxy redshift, which is observed as an anisotropy in the correlation function. This anisotropy allows us to measure the linear growth rate of matter ($f\sigma_8$). We measure the $f\sigma_8$ at $z=0.57$ using the CMASS sample from Data Release 11 of Sloan Digital Sky Survey III (SDSS III) Baryon Oscillations Spectroscopic Survey (BOSS). The galaxy sample consists of 690,826 massive galaxies in the redshift range 0.43-0.7 covering 8498 deg$^2$. Here we report the first simultaneous measurement of $f\sigma_8$ and background cosmological parameters using Convolution Lagrangian Perturbation Theory (CLPT) with Gaussian streaming model (GSRSD). We arrive at a constraint of $f\sigma_8=0.462\pm0.041$ (9\% accuracy) at effective redshift ($\bar{z}=0.57$) when we include Planck CMB likelihood while marginalizing over all other cosmological parameters. We also measure $b\sigma_8=1.19\pm0.03$, $H(z=0.57)=89.2\pm3.6$ km s$^{-1}$ Mpc$^{-1}$ and $D_A(z=0.57)=1401\pm23$ Mpc. Our analysis also improves the constraint on $\Omega_c h^2=0.1196\pm0.0009$ by a factor of 3 when compared to the Planck only measurement($\Omega_c h^2=0.1196 \pm 0.0031$). Our results are consistent with Planck $\Lambda$CDM-GR prediction and all other CMASS measurements, even though our theoretical models are fairly different. This consistency suggests that measurement of $f\sigma_8$ from Redshift space distortions at multiple redshifts will be a sensitive probe of the theory of gravity that is largely model independent, allowing us to place model-independent constraints on alternative models of gravity.

The Zel'dovich approximation

This year marks the 100th anniversary of the birth of Yakov Zel'dovich. Amongst his many legacies is the Zel'dovich approximation for the growth of large-scale structure, which remains one of the most successful and insightful analytic models of structure formation. We use the Zel'dovich approximation to compute the two-point function of the matter and biased tracers, and compare to the results of N-body simulations and other Lagrangian perturbation theories. We show that Lagrangian perturbation theories converge well and that the Zel'dovich approximation provides a good fit to the N-body results except for the quadrupole moment of the halo correlation function. We extend the calculation of halo bias to 3rd order and also consider non-local biasing schemes, none of which remove the discrepancy. We argue that a part of the discrepancy owes to an incorrect prediction of inter-halo velocity correlations. We use the Zel'dovich approximation to compute the ingredients of the Gaussian streaming model and show that this hybrid method provides a good fit to clustering of halos in redshift space down to scales of tens of Mpc.

An analytic model for redshift-space distortions

Understanding the formation and evolution of large-scale structure is a central problem in cosmology and enables precise tests of General Relativity on cosmological scales and constraints on dark energy. An essential ingredient is an accurate description of the pairwise velocities of biased tracers of the matter field. In this paper we compute the first and second moments of the pairwise velocity distribution by extending the Convolution Lagrangian Perturbation theory (CLPT) formalism of Carlson et al. (2012). Our predictions outperform standard perturbation theory calculations in many cases when compared to statistics measured in N-body simulations. We combine the CLPT predictions of real-space clustering and velocity statistics in the Gaussian streaming model of Reid & White (2011) to obtain predictions for the monopole and quadrupole correlation functions accurate to 2 and 4 per cent respectively down to <25Mpc/h for halos hosting the massive galaxies observed by SDSS-III BOSS. We also discuss contours of the 2D correlation function and clustering "wedges". We generalize the scheme to cross-correlation functions.