Linear Density Field Research Articles

Bootstrapping Lagrangian perturbation theory for the large scale structure

We develop a model-independent approach to Lagrangian perturbation theory for the large scale structure of the universe. We focus on the displacement field for dark matter particles, and derive its most general structure without assuming a specific form for the equations of motion, but implementing a set of general requirements based on symmetry principles and consistency with the perturbative approach. We present explicit results up to sixth order, and provide an algorithmic procedure for arbitrarily higher orders. The resulting displacement field is expressed as an expansion in operators built up from the linear density field, with time-dependent coefficients that can be obtained, in a specific model, by solving ordinary differential equations. The derived structure is general enough to cover a wide spectrum of models beyond ΛCDM, including modified gravity scenarios of the Horndeski type and models with multiple dark matter species. This work is a first step towards a complete model-independent Lagrangian forward model, to be employed in cosmological analyses with power spectrum and bispectrum, other summary statistics, and field-level inference.

Analysis of an iterative reconstruction method in comparison of the standard reconstruction method

ABSTRACT We present a detailed analysis of a new iterative density reconstruction algorithm. This algorithm uses a decreasing smoothing scale to better reconstruct the density field in Lagrangian space. We implement this algorithm to run on the quijote simulations, and extend it to (a) include a smoothing kernel that smoothly goes from anisotropic to isotropic, and (b) a variant that does not correct for redshift space distortions. We compare the performance of this algorithm with the standard reconstruction method. Our examinations of the methods include cross-correlation of the reconstructed density field with the linear density field, reconstructed two-point functions, and BAO parameter fitting. We also examine the impact of various parameters, such as smoothing scale, anisotropic smoothing, tracer type/bias, and the inclusion of second order perturbation theory. We find that the two reconstruction algorithms are comparable in most of the areas we examine. In particular, both algorithms give consistent fittings of BAO parameters. The fits are robust over a range of smoothing scales. We find the iterative algorithm is significantly better at removing redshift space distortions. The new algorithm will be a promising method to be employed in the ongoing and future large-scale structure surveys.

Estimating major merger rates and spin parameters ab initio via the clustering of critical events

Abstract We build a model to predict from first principles the properties of major mergers. We predict these from the coalescence of peaks and saddle points in the vicinity of a given larger peak, as one increases the smoothing scale in the initial linear density field as a proxy for cosmic time. To refine our results, we also ensure, using a suite of ∼400 power-law Gaussian random fields smoothed at ∼30 different scales, that the relevant peaks and saddles are topologically connected: they should belong to a persistent pair before coalescence. Our model allows us to (a) compute the probability distribution function of the satellite-merger separation in Lagrangian space: they peak at three times the smoothing scale; (b) predict the distribution of the number of mergers as a function of peak rarity: haloes typically undergo two major mergers (>1:10) per decade of mass growth; (c) recover that the typical spin brought by mergers: it is of the order of a few tens of percent.

Improving constraints on primordial non-Gaussianity using neural network based reconstruction

We study the use of U-Nets in reconstructing the linear dark matter density field and its consequences for constraining cosmological parameters, in particular primordial non-Gaussianity. Our network is able to reconstruct the initial conditions of redshift z = 0 density fields from N-body simulations with 90% accuracy out to k ≤ 0.4 h/Mpc, competitive with state-of-the-art reconstruction algorithms at a fraction of the computational cost. We study the information content of the reconstructed z = 0 density field with a Fisher analysis using the QUIJOTE simulation suite, including non-Gaussian initial conditions. Combining the pre- and post-reconstructed power spectrum and bispectrum data up to k max = 0.52 h/Mpc, we find significant improvements in all parameters. Most notably, we find a factor 3.65 (local), 3.54 (equilateral), and 2.90 (orthogonal) improvement on the marginalized errors of f NL as compared to only using the pre-reconstructed data. We show that these improvements can be attributed to a combination of reduced data covariance and parameter degeneracy. The results constitute an important step towards a more optimal inference of primordial non-Gaussianity from non-linear scales.

The large-scale velocity field from the Cosmicflows-4 data

ABSTRACT The reconstruction of the large-scale velocity field from the grouped Cosmicflows-4 (CF4) database is presented. The lognormal bias of the inferred distances and velocities data is corrected by the Bias Gaussianization correction scheme, and the linear density and velocity fields are reconstructed by means of the Wiener filter and constrained realizations (CRs) algorithm. These tools are tested against a suite of random and constrained Cosmicflows-3-like mock data. The CF4 data consist of three main subsamples – the 6dFGS and the SDSS data – and the ‘others’. The individual contributions of the subsamples have been studied. The quantitative analysis of the velocity field is done mostly by the mean overdensity (ΔL(R)) and the bulk velocity (Vbulk(R)) profiles of the velocity field out to $300\, {{h^{-1}\, {\rm Mpc}}}$. The Vbulk(R) and ΔL(R) profiles of the CF4 data without its 6dFGS component are consistent with the cosmic variance to within 1σ. The 6dFGS sample dominates the Vbulk (ΔL) profile beyond $\sim 120\, {{h^{-1}\, {\rm Mpc}}}$, and drives it to roughly a 3.4σ (−1.9σ) excess (deficiency) relative to the cosmic variance at $R\sim 250\ (190)\ \, {{h^{-1}\, {\rm Mpc}}}$. The excess in the amplitude of Vbulk is dominated by its Supergalactic X component, roughly in the direction of the Shapley Concentration. The amplitude and alignment of the inferred velocity field from the CF4 data are at $\sim (2{-}3)\, \sigma$ discrepancy with respect to the Lambda cold dark matter model. Namely, it is somewhat atypical but yet there is no compelling tension with the model.

Inner cusps of the first dark matter haloes: formation and survival in a cosmological context

ABSTRACT We use very high resolution cosmological zoom simulations to follow the early evolution of 12 first-generation haloes formed from gaussian initial conditions with scale-free power spectra truncated on small scales by a gaussian in wavenumber. Initial collapse occurs with a diverse range of sheet- or filament-like caustic morphologies, but in almost all cases it gives rise to a numerically converged density cusp with ρ = Ar−3/2 and total mass comparable to that of the corresponding peak in the initial linear density field. The constant A can be estimated to within about 10 per cent from the properties of this peak. This outcome agrees with earlier work on the first haloes in cold and warm dark matter universes. Within central cusps, the velocity dispersion is close to isotropic, and the equidensity surfaces tend to align with those of the main body of the halo at larger radii. As haloes grow, their cusps are often (but not always) overlaid with additional material at intermediate radii to produce profiles more similar to the Einasto or Navarro–Frenk–White forms typical of more massive haloes. Nevertheless, to the extent that we can resolve them, cusps survive at the smallest radii. Major mergers can disturb them, but the effect is quite weak in the cases that we study. The cusps extend down to the resolution limits of our simulations, which are typically a factor of several larger than the cores that would be produced by phase-space conservation if the initial power spectrum cutoff arises from free streaming.

Non-linear reconstruction of features in the primordial power spectrum from large-scale structure

ABSTRACT Potential features in the primordial power spectrum have been searched for in galaxy surveys in recent years since these features can assist in understanding the nature of inflation. The null detection to date suggests that any such features should be fairly weak, and next-generation galaxy surveys, with their unprecedented sizes and precisions, are in a position to place stronger constraints than before. However, even if such primordial features once existed in the early Universe, they would have been significantly damped in the non-linear regime at low redshift due to structure formation, which makes them difficult to be directly detected in real observations. A potential way to tackle this challenge for probing the features is to undo the cosmological evolution, i.e. using reconstruction to obtain an approximate linear density field. By employing a set of N-body simulations, here we show that a recently proposed non-linear reconstruction algorithm can effectively retrieve damped oscillatory features from halo catalogues and improve the accuracy of the measurement of feature parameters (assuming that such primordial features do exist). We do a Fisher analysis to forecast how non-linear reconstruction affects the constraining power, and find that it can lead to significantly more robust constraints on the feature amplitude for a DESI-like survey. Comparing non-linear reconstruction with other ways of improving constraints, such as increasing the survey volume and range of scales, this shows that it is possible to achieve what the latter do, but at a lower cost.

Accurate Baryon Acoustic Oscillations Reconstruction via Semidiscrete Optimal Transport.

Optimal transport theory has recently re-emerged as a vastly resourceful field of mathematics with elegant applications across physics and computer science. Harnessing methods from geometry processing, we report on the efficient implementation for a specific problem in cosmology-the reconstruction of the linear density field from low redshifts, in particular the recovery of the baryonic acoustic oscillation (BAO) scale. We demonstrate our algorithm's accuracy by retrieving the BAO scale in noiseless cosmological simulations that are dedicated to cancel cosmic variance; we find uncertainties to be reduced by a factor of 4.3 compared with performing no reconstruction, and a factor of 3.1 compared with standard reconstruction.

Hamiltonian Monte Carlo reconstruction from peculiar velocities

ABSTRACTThe problem of the reconstruction of the large-scale density and velocity fields from peculiar velocity surveys is addressed here within a Bayesian framework by means of Hamiltonian Monte Carlo (HMC) sampling. The HAmiltonian Monte carlo reconstruction of the Local EnvironmenT (hamlet) algorithm is designed to reconstruct the linear large-scale density and velocity fields in conjunction with the undoing of lognormal bias in the derived distances and velocities of peculiar velocity surveys, such as the Cosmicflows (CF) data. The hamlet code has been tested against CF mock catalogues consisting of up to 3 × 104 data points with mock errors akin to those of the Cosmicflows-3 (CF3) data, within the framework of the Lambda cold dark matter standard model of cosmology. The hamlet code outperforms previous applications of Gibbs sampling Markov chain Monte Carlo reconstruction from the CF3 data by two to four orders of magnitude in CPU time. The gain in performance is due to the inherent higher efficiency of the HMC algorithm and due to parallel computing on GPUs rather than CPUs. This gain will enable an increase in the reconstruction of the large-scale structure from the upcoming CF4 data and the setting of constrained initial conditions for cosmological high-resolution simulations.

Modeling iterative reconstruction and displacement field in the large scale structure

The next generation of galaxy surveys like the Dark Energy Spectroscopic Instrument (DESI) and Euclid will provide datasets orders of magnitude larger than anything available to date. Our ability to model nonlinear effects in late time matter perturbations will be a key to unlock the full potential of these datasets, and the area of initial condition reconstruction is attracting growing attention. Iterative reconstruction developed in Ref. [1] is a technique designed to reconstruct the displacement field from the observed galaxy distribution. The nonlinear displacement field and initial linear density field are highly correlated. Therefore, reconstructing the nonlinear displacement field enables us to extract the primordial cosmological information better than from the late time density field at the level of the two-point statistics. This paper will test to what extent the iterative reconstruction can recover the true displacement field and construct a perturbation theory model for the postreconstructed field. We model the iterative reconstruction process with Lagrangian perturbation theory~(LPT) up to third order for dark matter in real space and compare it with $N$-body simulations. We find that the simulated iterative reconstruction does not converge to the nonlinear displacement field, and the discrepancy mainly appears in the shift term, i.e., the term correlated directly with the linear density field. On the contrary, our 3LPT model predicts that the iterative reconstruction should converge to the nonlinear displacement field. We discuss the sources of discrepancy, including numerical noise/artifacts on small scales, and present an ad hoc phenomenological model that improves the agreement.

Non-universality of the mass function: dependence on the growth rate and power spectrum shape

ABSTRACT The abundance of dark matter haloes is one of the key probes of the growth of structure and expansion history of the Universe. Theoretical predictions for this quantity usually assume that, when expressed in a certain form, it depends only on the mass variance of the linear density field. However, cosmological simulations have revealed that this assumption breaks, leading to 10–20 per cent systematic effects. In this paper, we employ a specially designed suite of simulations to further investigate this problem. Specifically, we carry out cosmological N-body simulations where we systematically vary growth history at a fixed linear density field, or vary the power spectrum shape at a fixed growth history. We show that the halo mass function generically depends on these quantities, thus showing a clear signal of non-universality. Most of this effect can be traced back to the way in which the same linear fluctuation grows differently into the non-linear regime depending on details of its assembly history. With these results, we propose a parameterization with explicit dependence on the linear growth rate and power spectrum shape. Using an independent suite of simulations, we show that this fitting function accurately captures the mass function of haloes over cosmologies spanning a vast parameter space, including massive neutrinos and dynamical dark energy. Finally, we employ this tool to improve the accuracy of so-called cosmology-rescaling methods and show they can deliver 2 per cent accurate predictions for the halo mass function over the whole range of currently viable cosmologies.

Evaluating the origins of the secondary bias based on the correlation of halo properties with the linear density field

Using two sets of large N-body simulations, we studied the origins of the correlations between halo assembly time (zf), concentration (vmax/v200), and spin (λ) with the large-scale evolved density field at given halo mass, namely, the secondary bias. We find that the secondary bias is a secondary effect resulting from the correlations of halo properties with the linear density estimated at the same comoving scale. Using the linear density on different scales, we find two types of correlations. The internal correlation, which reflects the correlation of halo properties with the mean linear over-density, δL, within the halo Lagrangian radius, RL, is positive for both zf and vmax/v200, and negative for λ. The external correlation, which describes the correlation of halo properties with linear overdensity at R > RL for a given δL, shows trends that are contrary to the internal correlation. Both of the external and internal correlations depend only weakly on halo mass, indicating a similar origin for halos of different masses. Our findings offer a transparent perspective on the origins of the secondary bias, which can be largely explained by the competition between the external and internal correlations with the correlation of the linear density field on different scales. The combination of these two types of correlations has the potential to establish the complex halo-mass dependence of the secondary bias observed in the simulations.

Fully relativistic predictions in Horndeski gravity from standard Newtonian N-body simulations

The N-body gauge allows the introduction of relativistic effects in Newtonian cosmological simulations. Here we extend this framework to general Horndeski gravity theories, and investigate the relativistic effects that the scalar field introduces in the matter power spectrum at intermediate and large scales. In particular, we show that the kineticity function at these scales enhances the amplitude of the signal of contributions coming from the extra degree of freedom. Using the Quasi-Static Approximation (QSA), we separate modified gravity effects into two parts: one that only affects small-scale physics, and one that is due to relativistic effects. This allows our formalism to be readily implemented in modified gravity N-body codes in a straightforward manner, e.g., relativistic effects can be included as an additional linear density field in simulations. We identify the emergence of gravity acoustic oscillations (GAOs) in the matter power spectrum at large scales, k ∼ 10-3–10-2 Mpc-1. GAO features have a purely relativistic origin, coming from the dynamical nature of the scalar field. GAOs may be enhanced to detectable levels by the rapid evolution of the dark energy sound horizon in certain modified gravity models and can be seen as a new test of gravity at scales probed by future galaxy and intensity-mapping surveys.

On the primordial information available to galaxy redshift surveys

We investigate the amount of primordial information that can be reconstructed from spectroscopic galaxy surveys, as well as what sets the noise in reconstruction at low wavenumbers, by studying a simplified universe in which galaxies are the Zeldovich displaced Lagrangian peaks in the linear density field. For some of this study, we further take an intuitive linearized limit in which reconstruction is a convex problem but where the solution is also a solution to the full nonlinear problem, a limit that bounds the effectiveness of reconstruction. The linearized reconstruction results in similar cross correlation coefficients with the linear input field as our full nonlinear algorithm. The linearized reconstruction also produces similar cross correlation coefficients to those of reconstruction attempts on cosmological N-body simulations, which suggests that existing reconstruction algorithms are extracting most of the accessible information. Our approach helps explain why reconstruction algorithms accurately reproduce the initial conditions up to some characteristic wavenumber, at which point there is a quick transition to almost no correlation. This transition is set by the number of constraints on reconstruction (the number of galaxies in the survey) and not by where shot noise surpasses the clustering signal, as is traditionally thought. We further show that on linear scales a mode can be reconstructed with precision well below the shot noise expectation if the galaxy Lagrangian displacements can be sufficiently constrained. We provide idealized examples of nonlinear reconstruction where shot noise can be outperformed.

Nonperturbative halo clustering from cosmological density peaks

Associating the formation sites of haloes with the maxima of the smoothed linear density field, we present nonperturbative predictions for the Lagrangian and evolved halo correlation functions that are valid at all separations. In Lagrangian space, we find significant deviations from the perturbative bias calculation at small scales, in particular, a pronounced exclusion region where $\ensuremath{\xi}=\ensuremath{-}1$ for maxima of unequal height. Our predictions are in good agreement with the Lagrangian clustering of dark matter proto-haloes reconstructed from $N$-body simulations. Our predictions for the mean infall and velocity dispersion of haloes, which differ from the local bias expansion, show a similar level of agreement with simulations. Finally, we displace the initial density peaks according to the Zeldovich approximation in order to predict the late-time clustering of dark matter haloes. While we are able to reproduce the early evolution of this conserved set of tracers, our approximation fails at the collapse epoch ($z=0$) on nonlinear scales $r\ensuremath{\lesssim}10{h}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{Mpc}$, emphasizing the need for a nonperturbative treatment of the halo displacement field.

Constraining primordial non-Gaussianity with postreconstructed galaxy bispectrum in redshift space

Galaxy bispectrum is a promising probe of inflationary physics in the early Universe as a measure of primordial non-Gaussianity (PNG), whereas its signal-to-noise ratio is significantly affected by the mode coupling due to nonlinear gravitational growth. In this paper, we examine the standard reconstruction method of linear cosmic mass density fields from nonlinear galaxy density fields to decorrelate the covariance in redshift-space galaxy bispectra. In particular, we evaluate the covariance of the bispectrum for massive-galaxy-sized dark matter halos with reconstruction by using 4000 independent $N$-body simulations. Our results show that the bispectrum covariance for the postreconstructed field approaches the Gaussian prediction at scale of $k<0.2\text{ }\text{ }h\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. We also verify the leading-order PNG-induced bispectrum is not affected by details of the reconstruction with perturbative theory. We then demonstrate the constraining power of the postreconstructed bispectrum for PNG at redshift of approximately 0.5. Further, we perform a Fisher analysis to make a forecast of PNG constraints by galaxy bispectra including anisotropic signals. Assuming a massive galaxy sample in the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey, we find that the postreconstructed bispectrum can constrain the local, equilateral, and orthogonal types of PNG with $\mathrm{\ensuremath{\Delta}}{f}_{\mathrm{NL}}\ensuremath{\sim}13$, 90, and 42, respectively, improving the constraints with the prereconstructed bispectrum by a factor of 1.3--3.2. In conclusion, the reconstruction plays an essential role in constraining various types of PNG signatures with a level of $\mathrm{\ensuremath{\Delta}}{f}_{\mathrm{NL}}\ensuremath{\lesssim}1$ from the galaxy bispectrum based on upcoming galaxy surveys.

When do cosmic peaks, filaments or walls merge? A theory of critical events in a multi-scale landscape.

Abstract The merging rate of cosmic structures is computed, relying on the Ansatz that they can be predicted in the initial linear density field from the coalescence of critical points with increasing smoothing scale, used here as a proxy for cosmic time. Beyond the mergers of peaks with saddle points (a proxy for halo mergers), we consider the coalescence and nucleation of all sets of critical points, including wall-saddle to filament-saddle and wall-saddle to minima (a proxy for filament and void mergers respectively), as they impact the geometry of galactic infall, and in particular filament disconnection. Analytical predictions of the one-point statistics are validated against multiscale measurements in 2D and 3D realisations of Gaussian random fields (the corresponding code being available upon request) and compared qualitatively to cosmological N-body simulations at early times (z ≥ 10) and large scales (≥ 5Mpc/h). The rate of filament coalescence is compared to the merger rate of haloes and the two-point clustering of these events is computed, along with their cross-correlations with critical points. These correlations are qualitatively consistent with the preservation of the connectivity of dark matter haloes, and the impact of the large scale structures on assembly bias. The destruction rate of haloes and voids as a function of mass and redshift is quantified down to z = 0 for a ΛCDM cosmology. The one-point statistics in higher dimensions are also presented, together with consistency relations between critical point and critical event counts.

Application of the iterative reconstruction to simulated galaxy fields

We apply an iterative reconstruction method to galaxy mocks in redshift space obtained from $N$-body simulations. Comparing the two-point correlation functions for the reconstructed density field, we find that although the performance is limited by shot noise and galaxy bias compared to the matter field, the iterative method can still reconstruct the initial linear density field from the galaxy field better than the standard method both in real and in redshift space. Furthermore, the iterative method is able to reconstruct both the monopole and quadrupole more precisely, unlike the standard method. We see that as the number density of galaxies gets smaller, the performance of reconstruction gets worse due to the sparseness. However, the precision in the determination of bias ($\sim20\%$) hardly impacts on the reconstruction processes.

Comparing approximate methods for mock catalogues and covariance matrices – I. Correlation function

This paper is the first in a set that analyses the covariance matrices of clustering statistics obtained from several approximate methods for gravitational structure formation. We focus here on the covariance matrices of anisotropic two-point correlation function measurements. Our comparison includes seven approximate methods, which can be divided into three categories: predictive methods that follow the evolution of the linear density field deterministically (ICE-COLA, Peak Patch, and Pinocchio), methods that require a calibration with N-body simulations (Patchy and Halogen), and simpler recipes based on assumptions regarding the shape of the probability distribution function (PDF) of density fluctuations (log-normal and Gaussian density fields). We analyse the impact of using covariance estimates obtained from these approximate methods on cosmological analyses of galaxy clustering measurements, using as a reference the covariances inferred from a set of full N-body simulations. We find that all approximate methods can accurately recover the mean parameter values inferred using the N-body covariances. The obtained parameter uncertainties typically agree with the corresponding N-body results within 5% for our lower mass threshold, and 10% for our higher mass threshold. Furthermore, we find that the constraints for some methods can differ by up to 20% depending on whether the halo samples used to define the covariance matrices are defined by matching the mass, number density, or clustering amplitude of the parent N-body samples. The results of our configuration-space analysis indicate that most approximate methods provide similar results, with no single method clearly outperforming the others.

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.