Distribution function approach to redshift space distortions

Massive spectroscopic surveys will measure the redshift space distortion\n(RSD) induced by galaxy peculiar velocity to unprecedented accuracy. We develop\na new method to improve the RSD modeling and to carry out robust reconstruction\nof the 3D large scale peculiar velocity through galaxy redshift surveys, in\nlight of RSD. (1) We propose a mathematically unique and physically motivated\ndecomposition of peculiar velocity into three eigen-components of different\norigins, different scale dependences and different impacts on RSD. (2) This\ndecomposition has the potential to simplify and improve the RSD modeling. We\nderive a new formula for the redshift space power spectrum. Under the velocity\ndecomposition scheme, all high order Gaussian corrections and non-Gaussian\ncorrection of order $\\delta^3$ can be taken into account without introducing\nextra model uncertainties. We find that the leading order contribution $\\propto\n(1+f\\tilde{W}(k)u^2)^2$ differs from the Kaiser formula by a function\n$\\tilde{W}(k)\\leq 1$. We hence identify a significant systematic error causing\nunderestimation of the structure growth parameter $f$ by as much as $O(10%)$\neven at relatively large scale $k=0.1h/$Mpc. The velocity decomposition also\nreveals the three origins of the finger of God (FOG) effect. (3) The velocity\ndecomposition clarifies issues in peculiar velocity reconstruction through 3D\ngalaxy distribution. We discuss two possible ways to carry out the velocity\nreconstruction. Both use the otherwise troublesome RSD in velocity\nreconstruction as a valuable source of information. Both have the advantage to\nrender the reconstruction of a stochastic 3D field into the reconstruction of a\ndeterministic window function $W^s(k,u)$ of limited degrees of\nfreedom...[abridged]\n

The highly redshifted 21 cm line of neutral hydrogen has become recognized as\na unique probe of cosmology from relatively low redshifts (z ~ 1) up through\nthe Epoch of Reionization (z ~ 8) and even beyond. To date, most work has\nfocused on recovering the spherically averaged power spectrum of the 21 cm\nsignal, since this approach maximizes the signal-to-noise in the initial\nmeasurement. However, like galaxy surveys, the 21 cm signal is affected by\nredshift space distortions, and is inherently anisotropic between the\nline-of-sight and transverse directions. A measurement of this anisotropy can\nyield unique cosmological information, potentially even isolating the matter\npower spectrum from astrophysical effects. However, in interferometric\nmeasurements, foregrounds also have an anisotropic footprint between the\nline-of-sight and transverse directions: the so-called foreground "wedge".\nAlthough foreground subtraction techniques are actively being developed, a\n"foreground avoidance" approach of simply ignoring contaminated modes has\narguably proven most successful to date. In this work, we analyze the effect of\nthis foreground anisotropy in recovering the redshift space distortion\nsignature in 21 cm measurements at both high and intermediate redshifts. We\nfind the foreground wedge corrupts nearly all of the redshift space signal for\neven the largest proposed EoR experiments (HERA and the SKA), making\ncosmological information unrecoverable without foreground subtraction. The\nsituation is somewhat improved at lower redshifts, where the redshift-dependent\nmapping from observed coordinates to cosmological coordinates significantly\nreduces the size of the wedge. Using only foreground avoidance, we find that a\nlarge experiment like CHIME can place non-trivial constraints on cosmological\nparameters.\n

We present an emulator that accurately predicts the power spectrum of galaxies in redshift space as a function of cosmological parameters. Our emulator is based on a second-order Lagrangian bias expansion that is displaced to Eulerian space using cosmological N-body simulations. Redshift space distortions are then imprinted using the non-linear velocity field of simulated particles and haloes. We build the emulator using a forward neural network trained with the simulations of the BACCO project, which covers an eight-dimensional parameter space including massive neutrinos and dynamical dark energy. We show that our emulator provides unbiased cosmological constraints from the monopole, quadrupole, and hexadecapole of a mock galaxy catalogue that mimics the BOSS-CMASS sample down to non-linear scales ($k\sim 0.6{h\, {\rm Mpc}^{-1}}$). This work opens up the possibility of robustly extracting cosmological information from small scales using observations of the large-scale structure of the universe.

We have quantified the degree of nonlinearity and stochasticity of the clustering of biased objects using cosmological $N$-body simulations. Adopting the peaks and the halos as representative biasing models, we focused on the two-point correlation of the biased objects, dark matter and their cross-correlation. Especially, we took account of the effect of redshift-space distortion and attempted to clarify the scale dependence and the time dependence by analyzing the biasing factor and the cross-correlation factor. On small scales, stochasticity and nonlinearity become appreciable and strongly object-dependent, especially in redshift space due to the pair-wise velocity dispersion of the biased objects. Nevertheless, an approximation of deterministic linear biasing, $\delta_\mathrm{obj}\simeq b_\mathrm{obj} \delta_\mathrm{mass}$, works reasonably well even in the quasi-linear regime, $r \gtrsim 10\,h^{-1} \textrm{Mpc}$, and linear redshift-space distortion explains the clustering amplitudes in redshift space in this regime.

The peculiar velocities of galaxies distort the clustering pattern anisotropically in redshift space. This effect on the statistics of isodensity contours is examined by linear theory. The statistics considered in this paper are the three- and two-dimensional genus of isodensity contours, the area of isodensity contours, the length of isodensity contours in the 2-dimensional slice and the level crossing statistics on the line. We find that all these statistics in redshift space as functions of density threshold of contours have the same shape as in real space. The redshift space distortion affects only amplitudes of these statistics. The three-dimensional genus and the area hardly suffer from the redshift space distortion for $0 \leq \Omega b^{-5/3} \leq 1$, where $b$ is a linear bias parameter. The other statistics are defined in one- or two-dimensional slices of the sample volume and depend on the direction of these slices relative to the line of sight. This dependence on direction of these statistics provides a way to determine the density parameter of the universe.

Redshift space distortions are an important probe of the growth of large-scale structure and for constraining cosmological parameters in general. As galaxy redshift surveys approach per cent level precision in their observations of the two point clustering statistics, it is timely to review what effects baryons and associated processes such as feedback may have on small-scale clustering in redshift space. Contrary to previous studies in the literature, we show using the large-volume Bahamas hydrodynamic simulations that the effect of baryons can be as much as 1 per cent in the $k \sim 0.1\, h\,$Mpc$^{-1}$ range for the monopole and 5 per cent for quadrupole, and that this could rise to as much as 10 per cent at $k \sim 10 \, h\,$Mpc$^{-1}$ in both measurements. For the halo power spectra, this difference can be as much 3-4 per cent in the monopole on scales of $0.05 \lt k \lt 0.3 \, h\,$Mpc$^{-1}$ for 10$^{13}\, h^{-1}$ M$_{\odot }$ haloes. We find that these deviations can be mitigated to the sub- per cent level in the both the monopole and quadrupole up to $k\sim 0.3\, h$ Mpc$^{-1}$ if the baryon corrected halo masses are used to calculate the redshift space power spectra. Finally, we use the cosmo-OWLS simulation suite to explore the changes in the redshift space power spectra with different feedback prescriptions, finding that there is a maximum of 15–20 per cent difference between the redshift space monopole and quadrupole with and without baryons at $k \sim 1\!-\!2\, h\,$Mpc$^{-1}$ within these models.

We present a public code to generate a mock galaxy catalog in redshift space assuming a log-normal probability density function (PDF) of galaxy and matter density fields. We draw galaxies by Poisson-sampling the log-normal field, and calculate the velocity field from the linearised continuity equation of matter fields, assuming zero vorticity. This procedure yields a PDF of the pairwise velocity fields that is qualitatively similar to that of N-body simulations. We check fidelity of the catalog, showing that the measured two-point correlation function and power spectrum in real space agree with the input precisely. We find that a linear bias relation in the power spectrum does not guarantee a linear bias relation in the density contrasts, leading to a cross-correlation coefficient of matter and galaxies deviating from unity on small scales. We also find that linearising the Jacobian of the real-to-redshift space mapping provides a poor model for the two-point statistics in redshift space. That is, non-linear redshift-space distortion is dominated by non-linearity in the Jacobian. The power spectrum in redshift space shows a damping on small scales that is qualitatively similar to that of the well-known Fingers-of-God (FoG) effect due to random velocities, except that the log-normal mock does not include random velocities. This damping is a consequence of non-linearity in the Jacobian, and thus attributing the damping of the power spectrum solely to FoG, as commonly done in the literature, is misleading.

Redshift space distortions within voids provide a unique method to test for environmental dependence of the growth rate of structures in low density regions, where effects of modified gravity theories might be important. We derive a linear theory model for the redshift space void-galaxy correlation that is valid at all pair separations, including deep within the void, and use this to obtain expressions for the monopole $\xi^s_0$ and quadrupole $\xi^s_2$ contributions. Our derivation highlights terms that have previously been neglected but are important within the void interior. As a result our model differs from previous works and predicts new physical effects, including a change in the sign of the quadrupole term within the void radius. We show how the model can be generalised to include a velocity dispersion. We compare our model predictions to measurements of the correlation function using mock void and galaxy catalogues modelled after the BOSS CMASS galaxy sample using the Big MultiDark $N$-body simulation, and show that the linear model with dispersion provides an excellent fit to the data at all scales, $0\leq s\leq120\;h^{-1}$Mpc. While the RSD model matches simulations, the linear bias approximation does not hold within voids, and care is needed in fitting for the growth rate. We show that fits to the redshift space correlation recover the growth rate $f(z=0.52)$ to a precision of $2.7\%$ using the simulation volume of $(2.5\;h^{-1}\mathrm{Gpc})^3$.

The observed density field in redshift space is directly affected by the radial motions generated from mass fluctuations outside the volumeoccupied by a given galaxy redshift survey. These motions introduce redshift space anisotropies which are more pronounced at larger distances from thesurvey's center, thus offering clues to the nature of mass fluctuations on super-survey scales. Furthermore, we note that all estimates of the growthfactor derived from redshift space distortions are based on relations which explicitly assume that the velocity field is generated by mass fluctuationsinside the survey volume. This may cause uncertainties in these estimates which are on the order of a few percent.

The anisotropies in the galaxy two-point correlation function measured from redshift survey exhibits deviations from the predictions of the linear theory of redshift space distortion on scales as large as 20 h -1 Mpc where we expect linear theory to hold in real space. Any attempt at analysing the anisotropies in the redshift correlation function and determining the linear distortion parameter β requires these deviations to be correctly modelled and taken into account. These deviations are usually attributed to galaxy random motions and these are incorporated in the analysis through a phenomenological model in which the linear redshift correlation is convolved with the random pairwise velocity distribution function along the line of sight. We show that a substantial fraction of the deviations arise from non-linear effects in the mapping from real to redshift space caused by the coherent flows. Models that incorporate this effect provide as good a fit to N-body results as the phenomenological model which only includes the effect of random motions. We find that the pairwise velocity dispersions predicted by all of the models that we have considered are in excess of the values determined directly from the N-body simulations. This indicates a shortcoming in our understanding of the statistical properties of peculiar velocities and their relation to redshift distortion.

Forward modeling the galaxy density within the Effective Field Theory of Large Scale Structure (EFT of LSS) enables field-level analyses that are robust to theoretical uncertainties. At the same time, they can maximize the constraining power from galaxy clustering on the scales amenable to perturbation theory. In order to apply the method to galaxy surveys, the forward model must account for the full observational complexity of the data. In this context, a major challenge is the inclusion of redshift space distortions (RSDs) from the peculiar motion of galaxies. Here, we present improvements in the efficiency and accuracy of the RSD modeling in the perturbative LEFTfield forward model. We perform a detailed quantification of the perturbative and numerical error for the prediction of momentum, velocity and the redshift-space matter density. Further, we test the recovery of cosmological parameters at the field level, namely the growth rate f, from simulated halos in redshift space. For a rigorous test and to scan through a wide range of analysis choices, we fix the linear (initial) density field to the known ground truth but marginalize over all unknown bias coefficients and noise amplitudes. With a third-order model for gravity and bias, our results yield < 1 % statistical and < 1.5 % systematic error. The computational cost of the redshift-space forward model is only ∼ 1.5 times of the rest frame equivalent, enabling future field-level inference that simultaneously targets cosmological parameters and the initial matter distribution.

We study the statistical properties of voids in the distribution of mass, dark-matter haloes and galaxies (BJ 1011.5 h−1 M⊙, similar to the median halo mass, Mmed= 1011.3 h−1 M⊙. We also find that the number density of galaxy- and halo-defined voids can be up to two orders of magnitude higher than mass-defined voids for large void radii, however, we observe that this difference is reduced to about half an order of magnitude when the positions are considered in redshift space. As expected, there are outflow velocities that show their maximum at larger void-centric distances for larger voids. We find a linear relation for the maximum outflow velocity, vmax=v0rvoid. The void-centric distance where this maximum occurs follows a suitable power-law fit of the form log(d)=(rvoid/A)B. At sufficiently large distances, we find mild infall motions on to the subdense regions. The galaxy velocity field around galaxy-defined voids is consistent with the results of haloes with masses above the median, showing milder outflows than the mass around mass-defined voids. We find that a similar analysis in redshift space would make both outflows and infalls appear with a lower amplitude. We also find that the velocity dispersion of galaxies and haloes is larger in the direction parallel to the void walls by ≃10–20 per cent. Given that voids are by definition subdense regions, the cross-correlation function between galaxy-defined voids and galaxies is close to ξ=−1 out to separations comparable to the void size, and at larger separations the correlation function level increases, approaching the values of the auto-correlation function of galaxies. The cross-correlation amplitude of mass-defined voids versus mass has a more gentle behaviour remaining negative at larger distances. The cross- to auto-correlation function ratio as a function of the distance normalized to the void radius shows a small scatter around a relation that depends only on the object used to define the voids (galaxies or haloes for instance). The distortion pattern observed in ξ(σ, π) is that of an elongation along the line of sight that extends out to large separations. Positive ξ contours evidence finger-of-god motions at the void walls. Elongations along the line of sight are roughly comparable between galaxy-, halo- and mass-defined voids.

Redshift-space distortions (RSDs) present a significant challenge in building models for the three-point correlation function (3PCF). We compare two possible lines of attack: the streaming model and standard perturbation theory (SPT). The two approaches differ in their treatment of the non-linear mapping from real to redshift space: SPT expands this mapping perturbatively, while the streaming model retains its non-linear form but relies on simplifying assumptions about the probability density function (PDF) of line-of-sight velocity differences between pairs or triplets of tracers. To assess the quality of the predictions and the validity of the assumptions of these models, we measure the monopole of the matter 3PCF and the first two moments of the pair- and triplewise velocity PDF from a suite of N-body simulations. We also evaluate the large-scale limit of the streaming model and determine under which conditions it aligns to SPT. On scales larger than 10 h -1 Mpc, we find that the streaming model for the 3PCF monopole is dominated by the first two velocity moments, making the exact shape of the PDF irrelevant. This model can match the accuracy of a Stage-IV galaxy survey, if the velocity moments are measured directly from the simulations. However, replacing the measurements with perturbative expressions to leading order generates large errors already on scales of 60–70 h -1 Mpc. This is the primary drawback of the streaming model. On the other hand, the SPT model for the 3PCF cannot account for the significant velocity dispersion that is present at all scales, and consequently provides predictions with limited accuracy. We demonstrate that this issue can be approximately addressed by isolating the large-scale limit of the dispersion, which leads to typical Fingers-of-God damping functions. Overall, the SPT model with a damping function provides the best compromise in terms of accuracy and computing time.

We investigate the generalized cubic covariant Galileon model, a kinetically driven dark energy model within the Horndeski class of theories. The model extends the cubic covariant Galileon by including power laws of the field derivatives in the K-essence and cubic terms which still allow for tracker solutions. We study the shape of the viable parameter space by enforcing stability conditions which include the absence of ghost, gradient and tachyon instabilities and the avoidance of strong coupling at early time. We study here the relevant effects of the modifications induced by the model on some cosmological observables such as the cosmic microwave background (CMB), the lensing potential auto-correlation and the matter power spectrum. For this goal, we perform parameter estimation using data of CMB temperature and polarization, baryonic acoustic oscillations (BAO), redshift-space distortions (RSD), supernovae type Ia (SNIa) and Cepheids. Data analysis with CMB alone finds that the today's Hubble parameter $H_0$ is consistent with its determination from Cepheids at $1\sigma$, resolving the famous tension of the cosmological standard models. The joint analysis of CMB, BAO, RSD and SNIa sets a lower bound for the sum of neutrino masses which is $\Sigma m_\nu >0.11$ eV at 1$\sigma$, in addition to the usual upper limit. The model selection analysis based on the effective $\chi_\text{eff}^2$ and Deviance Information Criterion is not able to clearly identify the statistically favored model between $\Lambda$CDM and the generalized cubic covariant Galileon, from which we conclude that the latter model deserves further studies.

Modern galaxy surveys demand extensive survey volumes and resolutions surpassing current dark matter-only simulations' capabilities. To address this, many methods employ effective bias models on the dark matter field to approximate object counts on a grid. However, realistic catalogs necessitate specific coordinates and velocities for a comprehensive understanding of the Universe.In this research, we explore sub-grid modeling to create accurate catalogs, beginning with coarse grid number counts at resolutions of approximately 5.5 h -1 Mpc per side. These resolutions strike a balance between modeling nonlinear damping of baryon acoustic oscillations and facilitating large-volume simulations. Augmented Lagrangian Perturbation Theory (ALPT) is utilized to model the dark matter field and motions, replicating the clustering of a halo catalog derived from a massive simulation at z = 1.1.Our approach involves four key stages: Tracer Assignment: Allocating dark matter particles to tracers based on grid cell counts, generating additional particles to address discrepancies. Attractor Identification: Defining attractors based on particle cosmic web environments, acting as gravitational focal points. Tracer Collapse: Guiding tracers towards attractors, simulating structure collapse. Redshift Space Distortions: Introducing redshift space distortions to simulated catalogs using ALPT and a random dispersion term. Results demonstrate accurate reproduction of monopoles and quadrupoles up to wave numbers of approximately k = 0.6 h Mpc-1. This method holds significant promise for galaxy surveys like DESI, EUCLID, and LSST, enhancing our understanding of the cosmos across scales.