Effects Of Redshift-space Distortions Research Articles

Peculiar Velocity Reconstruction from Simulations and Observations Using Deep Learning Algorithms

In this paper, we introduce a U-Net model of deep learning algorithms for reconstructions of the 3D peculiar velocity field, which simplifies the reconstruction process with enhanced precision. We test the adaptability of the U-Net model with simulation data under more realistic conditions, including the redshift space distortion effect and halo mass threshold. Our results show that the U-Net model outperforms the analytical method that runs under ideal conditions, with a 16% improvement in precision, 13% in residuals, 18% in correlation coefficient, and 27% in average coherence. The deep learning algorithm exhibits exceptional capacities to capture velocity features in nonlinear regions and substantially improve reconstruction precision in boundary regions. We then apply the U-Net model trained under Sloan Digital Sky Survey (SDSS) observational conditions to the SDSS Data Release 7 data for observational 3D peculiar velocity reconstructions.

Taming redshift-space distortion effects in the EFTofLSS and its application to data

Former analyses of the BOSS data using the Effective Field Theory of Large-Scale Structure (EFTofLSS) have measured that the largest counterterms are the redshift-space distortion ones. This allows us to adjust the power-counting rules of the theory, and to explicitly identify that the leading next-order terms have a specific dependence on the cosine of the angle between the line-of-sight and the wavenumber of the observable, μ. Such a specific μ-dependence allows us to construct a linear combination of the data multipoles, P̸, where these contributions are effectively projected out, so that EFTofLSS predictions for P̸ have a much smaller theoretical error and so a much higher k-reach. The remaining data are organized in wedges in μ space, have a μ-dependent k-reach because they are not equally affected by the leading next-order contributions, and therefore can have a higher k-reach than the multipoles. Furthermore, by explicitly including the highest next-order terms, we define a `one-loop+' procedure, where the wedges have even higher k-reach. We study the effectiveness of these two procedures on several sets of simulations and on the BOSS data. The resulting analysis has identical computational cost as the multipole-based one, but leads to an improvement on the determination of some of the cosmological parameters that ranges from 10% to 100%, depending on the survey properties.

Euclid preparation. XXXIV. The effect of linear redshift-space distortions in photometric galaxy clustering and its cross-correlation with cosmic shear

The cosmological surveys that are planned for the current decade will provide us with unparalleled observations of the distribution of galaxies on cosmic scales, by means of which we can probe the underlying large-scale structure (LSS) of the Universe. This will allow us to test the concordance cosmological model and its extensions. However, precision pushes us to high levels of accuracy in the theoretical modelling of the LSS observables so that no biases are introduced into the estimation of the cosmological parameters. In particular, effects such as redshift-space distortions (RSD) can become relevant in the computation of harmonic-space power spectra even for the clustering of the photometrically selected galaxies, as has previously been shown in literature. In this work, we investigate the contribution of linear RSD, as formulated in the Limber approximation by a previous work, in forecast cosmological analyses with the photometric galaxy sample of the survey. We aim to assess their impact and to quantify the bias on the measurement of cosmological parameters that would be caused if this effect were neglected. We performed this task by producing mock power spectra for photometric galaxy clustering and weak lensing as is expected to be obtained from the survey. We then used a Markov chain Monte Carlo approach to obtain the posterior distributions of cosmological parameters from these simulated observations. When the linear RSD is neglected, significant biases are caused when galaxy correlations are used alone and when they are combined with cosmic shear in the so-called 3times 2pt approach. These biases can be equivalent to as much as $5\ when an underlying Lambda CDM cosmology is assumed When the cosmological model is extended to include the equation-of-state parameters of dark energy, the extension parameters can be shifted by more than $1\ sigma$.

Guess the cheese flavour by the size of its holes: a cosmological test using the abundance of popcorn voids

ABSTRACT We present a new definition of cosmic void and a publicly available code with the algorithm that implements it. Underdense regions are defined as free-form objects, called popcorn voids, made from the union of spheres of maximum volume with a given joint integrated underdensity contrast. The method is inspired by the excursion-set theory and consequently no rescaling processing is needed, the removal of overlapping voids and objects with sizes below the shot noise threshold is inherent in the algorithm. The abundance of popcorn voids in the matter field can be fitted using the excursion-set theory provided the relationship between the linear density contrast of the barrier and the threshold used in void identification is modified relative to the spherical evolution model. We also analysed the abundance of voids in biased tracer samples in redshift space. We show how the void abundance can be used to measure the geometric distortions due to the assumed fiducial cosmology, in a test similar to an Alcock–Paczyński test. Using the formalism derived from previous works, we show how to correct the abundance of popcorn voids for redshift-space distortion effects. Using this treatment, in combination with the excursion-set theory, we demonstrate the feasibility of void abundance measurements as cosmological probes. We obtain unbiased estimates of the target parameters, albeit with large degeneracies in the parameter space. Therefore, we conclude that the proposed test in combination with other cosmological probes has potential to improve current cosmological parameter constraints.

Detection of Cosmological 21 cm Emission with the Canadian Hydrogen Intensity Mapping Experiment

We present a detection of 21 cm emission from large-scale structure (LSS) between redshift 0.78 and 1.43 made with the Canadian Hydrogen Intensity Mapping Experiment. Radio observations acquired over 102 nights are used to construct maps that are foreground filtered and stacked on the angular and spectral locations of luminous red galaxies (LRGs), emission-line galaxies (ELGs), and quasars (QSOs) from the eBOSS clustering catalogs. We find decisive evidence for a detection when stacking on all three tracers of LSS, with the logarithm of the Bayes factor equal to 18.9 (LRG), 10.8 (ELG), and 56.3 (QSO). An alternative frequentist interpretation, based on the likelihood ratio test, yields a detection significance of 7.1σ (LRG), 5.7σ (ELG), and 11.1σ (QSO). These are the first 21 cm intensity mapping measurements made with an interferometer. We constrain the effective clustering amplitude of neutral hydrogen (H i), defined as , where ΩH i is the cosmic abundance of H i, b H i is the linear bias of H i, and 〈f μ 2〉 = 0.552 encodes the effect of redshift-space distortions at linear order. We find for LRGs (z = 0.84), for ELGs (z = 0.96), and for QSOs (z = 1.20), with constraints limited by modeling uncertainties at nonlinear scales. We are also sensitive to bias in the spectroscopic redshifts of each tracer, and we find a nonzero bias Δ v = − 66 ± 20 km s−1 for the QSOs. We split the QSO catalog into three redshift bins and have a decisive detection in each, with the upper bin at z = 1.30 producing the highest-redshift 21 cm intensity mapping measurement thus far.

A local measurement of the growth rate from peculiar velocities and galaxy clustering correlations in the 6dF Galaxy Survey

ABSTRACT Galaxy peculiar velocities provide an integral source of cosmological information that can be harnessed to measure the growth rate of large-scale structure and constrain possible extensions to General Relativity. In this work, we present a method for extracting the information contained within galaxy peculiar velocities through an ensemble of direct peculiar velocity and galaxy clustering correlation statistics, including the effects of redshift space distortions, using data from the 6-degree Field Galaxy Survey. Our method compares the auto- and cross-correlation function multipoles of these observables, with respect to the local line of sight, with the predictions of cosmological models. We find that the uncertainty in our measurement is improved when combining these two sources of information in comparison to fitting to either peculiar velocity or clustering information separately. When combining velocity and density statistics in the range $27 \lt s \lt 123 \, h^{-1}$ Mpc we obtain a value for the local growth rate of fσ8 = 0.358 ± 0.075 and for the linear redshift distortion parameter β = 0.298 ± 0.065, recovering both with 20.9 per cent and 21.8 per cent accuracy, respectively. We conclude this work by comparing our measurement with other recent local measurements of the growth rate, spanning different data sets and methodologies. We find that our results are in broad agreement with those in the literature and are fully consistent with ΛCDM cosmology. Our methods can be readily scaled to analyse upcoming large galaxy surveys and achieve accurate tests of the cosmological model.

The BINGO project

Context.BINGO (Baryon Acoustic Oscillations from Integrated Neutral Gas Observations) is a radio telescope designed to survey from 980 MHz to 1260 MHz, observe the neutral hydrogen (H I) 21 cm line, and detect the baryon acoustic oscillation signal with the intensity mapping technique. Here we present our method for generating mock maps of the 21 cm intensity mapping signal that cover the BINGO frequency range and related test results.Aims.We would like to employN-body simulations to generate mock 21 cm intensity maps for BINGO and study the information contained in 21 cm intensity mapping observations about structure formation, H Idistribution and H Imass-halo mass relation.Methods.We fit an H Imass-halo mass relation from the ELUCID semianalytical galaxy catalog and applied it to the Horizen Run 4 halo catalog to generate the 21 cm mock map, which is called HOD. We also applied the abundance-matching method and matched the Horizen Run 4 galaxy catalog with the H Imass function measured from ALFALFA, to generate the 21 cm mock map, which is called HAM.Results.We studied the angular power spectrum of the mock maps and the corresponding pixel histogram. The comparison of two different mock map generation methods (HOD and HAM) is presented. We provide the fitting formula of ΩHi, H Ibias, and the lognormal fitting parameter of the maps, which can be used to generate similar maps. We discuss the possibility of measuring ΩHiand H Ibias by comparing the angular power spectrum of the mock maps and the theoretical calculation. We also discuss the redshift space distortion effect, the nonlinear effect, and the bin size effect in the mock map.Conclusions.By comparing the angular power spectrum measured from two different types of mock maps and the theoretical calculation, we find that the theoretical calculation can only fit the mock result at large scales. At small scales, neither the linear calculation nor the halofit nonlinear calculation can provide an accurate fitting, which reflects our poor understanding of the nonlinear distribution of H Iand its scale-dependent bias. We have found that the bias is highly sensitive to the method of populating H Iin halos, which also means that we can place constraints on the H Idistribution in halos by observing 21 cm intensity mapping. We also illustrate that only with thin frequency bins (such as 2 MHz), we can discriminate the Finger-of-God effect. All of our investigations using mocks provide useful guidance for our expectation of BINGO experiments and other 21 cm intensity mapping experiments.

Cosmology with the moving lens effect

Velocity fields can be reconstructed at cosmological scales from their influence on the correlation between the cosmic microwave background and large-scale structure. Effects that induce such correlations include the kinetic Sunyaev Zel'dovich (kSZ) effect and the moving-lens effect, both of which will be measured to high precision with upcoming cosmology experiments. Galaxy measurements also provide a window into measuring velocities from the effect of redshift-space distortions (RSDs). The information that can be accessed from the kSZ or RSDs, however, is limited by astrophysical uncertainties and systematic effects, which may significantly reduce our ability to constrain cosmological parameters such as $f\sigma_8$. In this paper, we show how the large-scale transverse-velocity field, which can be reconstructed from measurements of the moving-lens effect, can be used to measure $f\sigma_8$ to high precision.

The detectability of strong 21 centimetre forest absorbers from the diffuse intergalactic medium in late reionisation models

Abstract A late end to reionisation at redshift z ≃ 5.3 is consistent with observed spatial variations in the Lyα forest transmission and the deficit of Lyα emitting galaxies around extended Lyα absorption troughs at z = 5.5. In this model, large islands of neutral hydrogen should persist in the diffuse intergalactic medium (IGM) until z ≃ 6. We use a novel, hybrid approach that combines high resolution cosmological hydrodynamical simulations with radiative transfer to predict the incidence of strong $21\rm \, cm$ forest absorbers with optical depths τ21 > 10−2 from the diffuse IGM in these late reionisation models. We include the effect of redshift space distortions on the simulated $21\rm \, cm$ forest spectra, and treat the highly uncertain heating of the pre-reionisation IGM by soft X-rays as a free parameter. For a model with only modest IGM pre-heating, such that average gas kinetic temperatures in the diffuse IGM remain below $T_{\rm K}\simeq 10^{2} \rm \, K$, we find that strong $21\rm \, cm$ forest absorption lines should persist until z = 6. For a sample of ∼10 sufficiently radio loud background sources, a null-detection of $21\rm \, cm$ forest absorbers at z ≃ 6 with SKA1-low or possibly LOFAR should provide an informative lower limit on the still largely unconstrained soft X-ray background at high redshift and the temperature of the pre-reionisation IGM.

Nonlinear redshift-space distortions in the harmonic-space galaxy power spectrum

Future high spectroscopic resolution galaxy surveys will observe galaxies with nearly full-sky footprints. Modeling the galaxy clustering for these surveys, therefore, must include the wide-angle effect with narrow redshift binning. In particular, when the redshift-bin size is comparable to the typical peculiar velocity field, the nonlinear redshift-space distortion (RSD) effect becomes important. A naive projection of the Fourier-space RSD model to spherical harmonic space leads to diverging expressions. In this paper we present a general formalism of projecting the higher-order RSD terms into spherical harmonic space. We show that the nonlinear RSD effect, including the fingers-of-God (FoG), can be entirely attributed to a modification of the radial window function. We find that while linear RSD enhances the harmonic-space power spectrum, unlike the three-dimensional case, the enhancement decreases on small angular-scales. The fingers-of-God suppress the angular power spectrum on all transverse scales if the bin size is smaller than $\Delta r \lesssim \pi \sigma_u$; for example, the radial bin sizes corresponding to a spectral resolution $R=\lambda/\Delta \lambda$ of a few hundred satisfy the condition. We also provide the flat-sky approximation which reproduces the full calculation to sub-percent accuracy.

The completed SDSS-IV extended baryon oscillation spectroscopic survey: geometry and growth from the anisotropic void–galaxy correlation function in the luminous red galaxy sample

ABSTRACT We present an analysis of the anisotropic redshift-space void–galaxy correlation in configuration space using the Sloan Digital Sky Survey extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 luminous red galaxy (LRG) sample. This sample consists of LRGs between redshifts 0.6 and 1.0, combined with the high redshift z > 0.6 tail of the Baryon Oscillation Spectroscopic Survey Data Release 12 CMASS sample. We use a reconstruction method to undo redshift-space distortion (RSD) effects from the galaxy field before applying a watershed void-finding algorithm to remove bias from the void selection. We then perform a joint fit to the multipole moments of the correlation function for the growth rate fσ8 and the geometrical distance ratio DM/DH, finding $f\sigma _8(z_\rm {eff})=0.356\pm 0.079$ and $D_M/D_H(z_\rm {eff})=0.868\pm 0.017$ at the effective redshift $z_\rm {eff}=0.69$ of the sample. The posterior parameter degeneracies are orthogonal to those from galaxy clustering analyses applied to the same data, and the constraint achieved on DM/DH is significantly tighter. In combination with the consensus galaxy BAO and full-shape analyses of the same sample, we obtain fσ8 = 0.447 ± 0.039, DM/rd = 17.48 ± 0.23, and DH/rd = 20.10 ± 0.34. These values are in good agreement with the ΛCDM model predictions and represent reductions in the uncertainties of $13{{\ \rm per\ cent}}$, $23{{\ \rm per\ cent}}$, and $28{{\ \rm per\ cent}}$, respectively, compared to the combined results from galaxy clustering, or an overall reduction of 55 per cent in the allowed volume of parameter 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.

Perturbation theory approach to predict the covariance matrices of the galaxy power spectrum and bispectrum in redshift space

ABSTRACT In this paper, we predict the covariance matrices of both the power spectrum and the bispectrum, including full non-Gaussian contributions, redshift space distortions, linear bias effects, and shot-noise corrections, using perturbation theory (PT). To quantify the redshift-space distortion effect, we focus mainly on the monopole and quadrupole components of both the power and bispectra. We, for the first time, compute the 5- and 6-point spectra to predict the cross-covariance between the power and bispectra, and the autocovariance of the bispectrum in redshift space. We test the validity of our calculations by comparing them with the covariance matrices measured from the MultiDark-Patchy mock catalogues that are designed to reproduce the galaxy clustering measured from the Baryon Oscillation Spectroscopic Survey Data Release 12. We argue that the simple, leading-order PT works because the shot-noise corrections for the Patchy mocks are more dominant than other higher order terms we ignore. In the meantime, we confirm some discrepancies in the comparison, especially of the cross-covariance. We discuss potential sources of such discrepancies. We also show that our PT model reproduces well the cumulative signal-to-noise ratio of the power spectrum and the bispectrum as a function of maximum wavenumber, implying that our PT model captures successfully essential contributions to the covariance matrices.

Cosmological angular trispectra and non-Gaussian covariance

Angular cosmological correlators are infamously difficult to compute due to the highly oscillatory nature of the projection integrals. Motivated by recent development on analytic approaches to cosmological perturbation theory, in this paper we present an efficient method for computing cosmological four-point correlations in angular space, generalizing previous works on lower-point functions. This builds on the FFTLog algorithm that approximates the matter power spectrum as a sum over power-law functions, which makes certain momentum integrals analytically solvable. The computational complexity is drastically reduced for correlators in a “separable” form—we define a suitable notion of separability for cosmological trispectra, and derive formulas for angular correlators of different separability classes. As an application of our formalism, we compute the angular galaxy trispectrum at tree level, with and without primordial non-Gaussianity. This includes effects of redshift space distortion and bias parameters up to cubic order. We also compute the non-Gaussian covariance of the angular matter power spectrum due to the connected four-point function, beyond the Limber approximation. We demonstrate that, in contrast to the standard lore, the Limber approximation can fail for the non-Gaussian covariance computation even for large multipoles.

Cosmological parameters from the BOSS galaxy power spectrum

We present cosmological parameter measurements from the publicly available Baryon Oscillation Spectroscopic Survey (BOSS) data on anisotropic galaxy clustering in Fourier space. Compared to previous studies, our analysis has two main novel features. First, we use a complete perturbation theory model that properly takes into account the non-linear effects of dark matter clustering, short-scale physics, galaxy bias, redshift-space distortions, and large-scale bulk flows. Second, we employ a Markov-Chain Monte-Carlo technique and consistently reevaluate the full power spectrum likelihood as we scan over different cosmologies. Our baseline analysis assumes minimal ΛCDM, varies the neutrino masses within a reasonably tight range, fixes the primordial power spectrum tilt, and uses the big bang nucleosynthesis prior on the physical baryon density ωb. In this setup, we find the following late-Universe parameters: Hubble constant H0=(67.9± 1.1) km s−1Mpc−1, matter density fraction Ωm=0.295± 0.010, and the mass fluctuation amplitude σ8=0.721± 0.043. These parameters were measured directly from the BOSS data and independently of the Planck cosmic microwave background observations. Scanning over the power spectrum tilt or relaxing the other priors do not significantly alter our main conclusions. Finally, we discuss the information content of the BOSS power spectrum and show that it is dominated by the location of the baryon acoustic oscillations and the power spectrum shape. We argue that the contribution of the Alcock-Paczynski effect is marginal in ΛCDM, but becomes important for non-minimal cosmological models.

Anisotropies of galaxy ellipticity correlations in real and redshift space: angular dependence in linear tidal alignment model

ABSTRACT Investigating intrinsic alignments (IAs) of galaxy shapes is important not only to constrain cosmological parameters unbiasedly from gravitational lensing but also to extract cosmological information complimentary to galaxy clustering analysis. We derive simple and useful formulas for the various IA statistics, including the intrinsic ellipticity–ellipticity correlation, the gravitational shear–intrinsic ellipticity correlation, and the velocity–intrinsic ellipticity correlation functions. The angular dependence of each statistic is explicitly given, namely the angle between the line-of-sight direction and the separation vector of two points. It thus allows us to analyse anisotropies of baryon acoustic oscillations encoded in the IA statistics, and we can extract the maximum cosmological information using the Alcock–Paczynski and redshift-space distortion effects. We also provide these formulas for the intrinsic ellipticities decomposed into E and B modes.

Cosmological Constraints from the Redshift Dependence of the Alcock–Paczynski Effect: Fourier Space Analysis

Abstract The tomographic Alcock–Paczynski (AP) method utilizes the redshift evolution of the AP distortion to place constraints on cosmological parameters. In previous works, it was performed via the anisotropic two-point correlation function statistic. In this work we consider the feasibility of conducting the analysis in the Fourier domain. We use the integrated galaxy power spectrum as a function of direction, , to quantify the magnitude of anisotropy in the large-scale structure clustering, and use its redshift variation to do the AP test. The method is tested on the large, high-resolution Big-MultiDark Planck simulation at redshifts z = 0–1. Testing the redshift evolution of in the true cosmology and cosmologies deviating from the truth with δΩ m = 0.1, δw = 0.3, we find that the redshift evolution of the AP distortion overwhelms the redshift space distortions effects by a factor of ∼1.7–3.6. The method works well throughout the range of k ∈ (0.2, 1.8) h Mpc−1. We tune the halo mass within the range 2 × 1013–1014 M ⊙, and find that the change of halo bias results in ≲5% change in , which is less significant compared with the cosmological effect. Our work shows that it is feasible to conduct the tomographic AP analysis in the Fourier space.

The full-sky angular bispectrum in redshift space

We compute the redshift-dependent angular bispectrum of galaxy number counts at tree-level, including nonlinear clustering bias and estimating numerically for the first time the effect of redshift space distortions (RSD). We show that for narrow redshift bins the amplitude of nonlinear RSD is comparable with the matter density perturbations. While our numerical results only include terms relevant on sub-horizon scales, the formalism can readily be extended to the full tree-level bispectrum. Our approach does not rely on the flat-sky approximation and it can be easily generalized to different sources by including the appropriate bias expansion. We test the accuracy of Limber approximation for different z-bins. We highlight the subtle but relevant differences in the angular bispectrum of galaxy number counts with respect to CMB, due to the different scale dependence of perturbations. Our formalism can also be directly applied to the angular HI intensity mapping bispectrum.

Can the homogeneity scale be used as a standard ruler?

The Universe on comoving scales larger than 100 Mpc/h is assumed to be statistically homogeneous, with the transition scale where we have $1\%$ deviation from homogeneity being known as the homogeneity scale $R_H$. The latter was recently proposed in Ntelis et al. (2018) [arXiv:1810.09362] to be a standard ruler. In this paper we perform a comparison between the baryon acoustic oscillations and the homogeneity scales $R_H$, assuming a spatially flat universe, linear cosmological perturbation theory and the cosmological constant cold dark matter $\Lambda$CDM model. By direct theoretical calculations, we demonstrate that the answer to the question of whether the homogeneity scale can be used as a standard ruler, is clearly negative due to the non-monotonicity of the homogeneity scale $R_H$ with the matter density parameter $\Omega_{m0}$ of the $\Lambda$CDM model. Furthermore, we also consider the effect of redshift space distortions and we find that they do not break the degeneracies but instead only change the value of $R_H$ by a few percent. Finally, we also confirm our findings with the help of an N-Body simulation.

Large Scale Structure in Redshift Space

The mapping of dark matter clustering from real space to redshift space introduces the anisotropic property to the measured density power spectrum in redshift space, known as the redshift space distortion effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to indefinite cross correlations between the density and velocity fields, and the Finger–of–God effect due to the randomness of the peculiar velocity field. Whilst the full higher order polynomials remain unknown, the other systematics can be controlled consistently within the same order truncation in the expansion of the mapping formula, as shown in this paper. The systematic due to the unknown non–linear density and velocity fields is removed by separately measuring all terms in the expansion directly using simulations. The uncertainty caused by the velocity randomness is controlled by splitting the FoG term into two pieces, 1) the “one–point” FoG term being independent of the separation vector between two different points, and 2) the “correlated” FoG term appearing as an indefinite polynomials which is expanded in the same order as all other perturbative polynomials. We introduce the recent progress on understanding this mapping, and the application for the future analysis to reveal the nature of cosmic acceleration.