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encore : An Estimator for Galaxy N -Point Correlation Functions
Abstract
We present a new algorithm for efficiently computing the N-point correlation functions (NPCFs) of a 3D density field for arbitrary N. This can be applied both to a discrete spectroscopic galaxy survey and a continuous field. By expanding the statistics in a separable basis of isotropic functions built from spherical harmonics, the NPCFs can be estimated by counting pairs of particles in space, leading to an algorithm with complexity for Ng particles, or when using a Fast Fourier Transform with NFFT grid-points. In practice, the rate-limiting step for N > 3 will often be the summation of the histogrammed spherical harmonic coefficients, particularly if the number of radial and angular bins is large. In this case, the algorithm scales linearly with Ng. The approach is implemented in the encore code, which can compute the 3PCF, 4PCF, 5PCF, and 6PCF of a BOSS-like galaxy survey in ∼ 100 CPU-hours, including the corrections necessary for non-uniform survey geometries. We discuss the implementation in depth, along with its GPU acceleration, and provide practical demonstration on realistic galaxy catalogs. Our approach can be straightforwardly applied to current and future datasets to unlock the potential of constraining cosmology from the higher-point functions.
... 1 Efficient methods for computing this exist (e.g. [17,18]; using mathematical tools developed in [19]), which have facilitated parity measurements from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) galaxy catalogue containing O(10 6 ) galaxies [20,21]. ...
... Given the datasets, 4PCFs are obtained using the encore code [17,61]. This computes the following quantity (with a complexity quadratic in the number of galaxies): ...
... On each of the data and 2 Six MultiDark-Patchy mocks were corrupted in data transfer and are thus dropped from the analysis. 3 In practice, we use the data and random catalogues to create a Landy-Szalay-type estimator [62] to remove the window function, as described in [17,61]. 4 Though the encore GPU implementation described in [17] is considerably faster, we do not use it here due to the far larger number of CPUs available on a typical cluster. ...
Recent works have uncovered an excess signal in the parity-odd four-point correlation function measured from the Baryon Oscillation Spectroscopic Survey (BOSS) galaxy catalogue. If physical in origin, this could indicate new parity-breaking processes in inflation. At heart, these studies compare the observed four-point correlator with the distribution obtained from parity-conserving mock galaxy surveys; if the simulations underestimate the covariance of the data, noise fluctuations may be misinterpreted as a signal. To test this, we reanalyse the BOSS CMASS parity-odd dataset with the noise distribution model using the newly developed GLAM-Uchuu suite of mocks. These comprise full N-body simulations that follow the evolution of 20003 dark matter particles and represent a significant upgrade compared with the formerly used MultiDark-Patchy mocks, which were based on an alternative (non N-body) gravity solver. We find no significant evidence for parity-violation (with a baseline detection significance of 1.0σ), suggesting that the former signal (2.9σ with our data cuts) could be caused by an underestimation of the covariance in MultiDark-Patchy. The significant differences between results obtained with the two sets of BOSS-calibrated galaxy catalogues (whose covariances differ at the 10−20% level) showcase the heightened sensitivity of beyond-two-point analyses to nonlinear effects and indicate that previous constraints may suffer from large systematic uncertainties.
This article is part of the discussion meeting issue ‘Challenging the standard cosmological model’.
... In [1], these isotropic basis functions were developed, and their importance and other related previous work is more fully outlined there. The idea of using them to parameterize N-Point Correlation Functions, quantifying the spatial clustering of the distribution of galaxies, was presented in [2], with a detection of the even-parity, "connected" (due to nonlinear gravitational evolution) 4-Point Correlation Function (4PCF) in [3]. The idea of using the decomposition of the 4PCF into even and odd-parity 3-argument basis functions to search for parity violation first presented in [4], and the covariance matrix required for this search computed in the isotropic basis in [5]. ...
... We then turn to expanding the Dirac Delta function in this basis ( §6), and subsequently present connections of the generating function to spherical Bessel function overlap integrals ( §7). In §8 we suggest one possible application of the Cartesian forms of the isotropic functions, outlining a new algorithm to compute the N-Point Correlation Function (NPCF) of a given density field; this approach may offer a speed-up over the spherical-harmonic-based encore code [2] when run on a CPU. §9 concludes. ...
... We now convert Y * ℓm (p) = (−1) m Y ℓ−m (p) so that our dΩ p integral is over all conjugated spherical harmonics, in which case it is then equal to what it would be had they been all unconjugated (this equality follows from the fact that the result must be real). 2 We then may rewrite this integral as ...
Recently isotropic basis functions of N unit vector arguments were presented; these are of significant use in measuring the N-point correlation functions (NPCFs) of galaxy clustering. Here we develop the generating function for these basis functions—i.e. that function which, expanded in a power series, has as its angular part the isotropic functions. We show that this can be developed using basic properties of the plane wave. A main use of the generating function is as an efficient route to obtaining the Cartesian basis expressions for the isotropic functions. We show that the methods here enable computing difficult overlap integrals of multiple spherical Bessel functions, and we also give related expansions of the Dirac Delta function into the isotropic basis. Finally, we outline how the Cartesian expressions for the isotropic basis functions might be used to enable a faster NPCF algorithm on the CPU.
... we can express this function in terms of two triangle sides and the cosine of the angle between them and subsequently expand the angular dependence in a Legendre series. This results in the following estimator for the L th multipole coefficient: For our measurements, we employ this multipole estimator as implemented in the public code Encore 4 [82], which evaluates Eq. (3.1) by comparing the dark-matter particle distribution to a corresponding random catalogue that models the (constant) mean density. Measuring the multipole coefficients for the entire set of particles is still computationally expensive, which is why we randomly downsample the original distributions from 512 3 to 128 3 particles, while we use a random catalogue that is 10 times denser. ...
... Starting from N r = 5N p random particles, we increase their number in two steps up to 32 times the particles contained in the data catalogue. We find some disagreement in the measurements Figure 13: Quadrupole of the isotropic 3PCF estimated by ENCORE [82] for a fixed bin of s 12 ∈ (62, 69) h −1 Mpc. We show the mean and standard deviation taken measuring the 3PCF from 30 realisations of the Quijote simulation. ...
... We do not consider isosceles configurations, s12 = s23 as Encore cannot handle them. As noted in[82], a more sophisticated estimator would have to be implemented to deal with overlapping bins. ...
Redshift-space distortions 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 , 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 . This is the main drawback of the streaming model. Conversely, 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 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 deal in terms of accuracy and computing time.
... To generalize to a realistic survey geometry, we use the edge correction procedure described in section 2 of [1] or section II.C of [2]. This edge correction is implemented in the ENCORE software [35], which we use to compute E in our pipeline. ...
... The input to E is a galaxy field δ g (x) defined by the usual "data minus randoms" prescription [35][36][37]: ...
... To compute E, we run the public ENCORE software [35] adapting the shell script available in the GitHub repository. 12 The strategy is to reduce the runtime by splitting the randoms into 32 equal files (after first randomizing their order to ensure that each random subset covers the full area). ...
Recent studies have found evidence for parity violation in the BOSS spectroscopic galaxy survey, with statistical significance as high as . These analyses assess the significance of the parity-odd four-point correlation function (4PCF) with a statistic called . This statistic is biased if the parity-even eight-point correlation function (8PCF) of the data differs from the mock catalogs. We construct new statistics , that separate the parity violation signal from the 8PCF bias term, allowing them to be jointly constrained. Applying these statistics to BOSS, we find that the parity violation signal ranges from 0 to depending on analysis choices, whereas the 8PCF bias term is . We conclude that there is no compelling evidence for parity violation in BOSS. Our new statistics can be used to search for parity violation in future surveys, such as DESI, without 8PCF biases.
... In [1], these isotropic basis functions were developed, and their importance and other related previous work is more fully outlined there. The idea of using them to parameterize N-Point Correlation Functions, quantifying the spatial clustering of the distribution of galaxies, was presented in [2], with a detection of the even-parity, "connected" (due to nonlinear gravitational evolution) 4-Point Correlation Function (4PCF) in [3]. The idea of using the decomposition of the 4PCF into even and odd-parity 3-argument basis functions to search for parity violation first presented in [4], and the covariance matrix required for this search computed in the isotropic basis in [5]. ...
... We then turn to expanding the Dirac Delta function in this basis ( §6), and subsequently present connections of the generating function to spherical Bessel function overlap integrals ( §7). In §8 we suggest one possible application of the Cartesian forms of the isotropic functions, outlining a new algorithm to compute the N-Point Correlation Function (NPCF) of a given density field; this approach may offer a speed-up over the spherical-harmonic-based encore code [2] when run on a CPU. §9 concludes. ...
... We now convert Y * ℓm (p) = (−1) m Y ℓ−m (p) so that our dΩ p integral is over all conjugated spherical harmonics, in which case it is then equal to what it would be had they been all unconjugated (this equality follows from the fact that the result must be real). 2 We then may rewrite this integral as ...
Recently isotropic basis functions of N unit vector arguments were presented; these are of significant use in measuring the N-Point Correlation Functions (NPCFs) of galaxy clustering. Here we develop the generating function for these basis functions -- i.e. that function which, expanded in a power series, has as its angular part the isotropic functions. We show that this can be developed using basic properties of the plane wave. A main use of the generating function is as an efficient route to obtaining the Cartesian basis expressions for the isotropic functions. We show that the methods here enable computing difficult overlap integrals of multiple spherical Bessel functions, and we also give related expansions of the Dirac Delta function into the isotropic basis. Finally, we outline how the Cartesian expressions for the isotropic basis functions might be used to enable a faster NPCF algorithm on the CPU.
... This may be done about many primaries so that the spheres of galaxies around each out to the maximal radius to which * A footnote to the article title one wishes to measure the NPCF do not overlap. One could then simply run an NPCF algorithm such as [3] by flagging only the primaries desired as those to be used. This would be a suitable setup for testing the effects outlined above. ...
... Let us notice that the CMB bispectrum actually involves four points: the observer, and the three points on the CMB spherical shell. This geometry is exactly the same as that used in the spherical-harmonic decomposition NPCF algorithms [3,[8][9][10], but in that case, the "primary" galaxy is set to be at the origin and the "secondaries" are on spherical shells with different radii. In particular, the geometry for the CMB bispectrum is the same as that for the 4PCF around a given primary. ...
... The fundamental requirements of the algorithm of this work are simply that the NPCFs be estimated around a primary galaxy by decomposing the density field on spherical shells in spherical harmonics. Thus, the algorithm trivially extends to NPCFs for N > 4. We note this is useful because [3] indeed can measure the 5PCF and 6PCF in the isotropic basis of [7]. ...
Here we show how to produce a 3D density field with a given set of higher-order correlation functions. Our algorithm enables producing any desired two-point, three-point, and four-point functions, including odd-parity for the latter. We also outline how to generalize the algorithm to i) N-point correlations with , ii) dimensions other than 3, and iii) beyond scalar quantities. This algorithm should find use in verifying analysis pipelines for higher-order statistics in upcoming galaxy redshift surveys such as DESI, Euclid, Roman, and Spherex, as well as intensity mapping
... The factorizability of these functions is important to the speed-up of the 4PCF algorithm (Philcox et al. 2022 ); in practice, it enables us to compute the 4PCF as a sum o v er the spherical harmonic coefficients a i m i of the density field about a given primary galaxy at s . Each unit vector ˆ r i is associated with one total angular momentum i , with z-component m i . ...
... To construct a density fluctuation field from the discrete galaxy counts, and also to account for the surv e y geometry, we use a generalized Landy-Szalay estimator (Landy & Szalay 1993 ;Szapudi & Szalay 1998, see also Kerscher, Szapudi & Szalay 2000 as first outlined for the angular momentum basis in Slepian & Eisenstein ( 2015b ) and further developed in Philcox et al. ( 2021Philcox et al. ( , 2022. It is ...
... Doing so giv es optimally weighted estimates of each in the shot-noise limit, as discussed in Slepian & Eisenstein ( 2015b , section 4, equations (24-26) and surrounding text). Multiplying equation ( 7 ) through by R , expanding each side of the resulting relation in the isotropic basis, reducing a product of two isotropic basis functions to a sum o v er single ones, and finally taking an inverse to solve the linear system so obtained (Slepian & Eisenstein 2015b ;Philcox et al. 2022 ), we find the edge-corrected 4PCF estimator as ˆ ζ 1 2 3 ( r 1 , r 2 , r 3 ) ...
A tetrahedron is the simplest shape that cannot be rotated into its mirror image in three-dimension (3D). The 4-point correlation function (4PCF), which quantifies excess clustering of quartets of galaxies over random, is the lowest order statistic sensitive to parity violation. Each galaxy defines one vertex of the tetrahedron. Parity-odd modes of the 4PCF probe an imbalance between tetrahedra and their mirror images. We measure these modes from the largest currently available spectroscopic samples, the 280 067 luminous red galaxies (LRGs) of the Baryon Oscillation Spectroscopic Survey (BOSS) twelfth data release (DR12) LOWZ ( ) and the 803 112 LRGs of BOSS DR12 CMASS ( ). In LOWZ, we find 3.1σ evidence for a non-zero parity-odd 4PCF, and in CMASS we detect a parity-odd 4PCF at 7.1σ. Gravitational evolution alone does not produce this effect; parity-breaking in LSS, if cosmological in origin, must stem from the epoch of inflation. We have explored many sources of systematic error and found none that can produce a spurious parity-odd signal sufficient to explain our result. Underestimation of the noise could also lead to a spurious detection. Our reported significances presume that the mock catalogues used to calculate the covariance sufficiently capture the covariance of the true data. We have performed numerous tests to explore this issue. The odd-parity 4PCF opens a new avenue for probing new forces during the epoch of inflation with 3D large-scale structure; such exploration is timely given large upcoming spectroscopic samples such as Dark Energy Spectroscopic Instrument and Euclid.
... We notice that contrary to the precyclic results, is not clear that the amplitude of the multipoles start to decrease beyond some , actually, it seems to be similar for ≥ 2. In the literature there is not a formal proof that this happens even in the case of ΛCDM, and if convergence is not attained that would mean that we cannot reconstruct the whole 3PCF from its multipoles. However, this is not a significant obstacle to use them, since their advantage rely in that the computational complexity to get these statistics from the data is reduced drastically and the estimators to do so search directly for the multipoles and not for the whole 3PCF [17,79]. ...
... With this motivation in mind, we develop a theoretical framework for the three point correlation function (3PCF) of tracers in modified gravity, exemplified by two models with representative screening mechanisms: the Hu-Sawicki f (R) [15] and the nDGP [16] models. The final 3PCF result is expressed in a Legendre/Szapudi basis [81], whose coefficients, given by Eq. (79), help not only to visualize deviations from GR (see for example [96]) but also to directly calculate the signal using estimators that scale as the two-point estimators with the number of data points [17,79]. We use standard perturbation theory with Effective Field Theory ingredients and a consistent biasing model to build up the tree-level MG bispectrum in the multipole basis, which are then directly map to the coefficients of the 3PCF in the same basis. ...
... For homogeneous but non isotropic fields, one applies these methods to the direction-averaged statistical fields; see e.g.[17,20], and[79,80] for more general N -Point Correlation Functions. ...
The next generation of galaxy surveys will provide highly accurate measurements of the large-scale structure of the Universe, allowing for more stringent tests of gravity on cosmological scales. Higher order statistics are a valuable tool to study the non-Gaussianities in the matter field and to break degeneracies between modified gravity and other physical or nuisance parameters. However, understanding from first principles the behaviour of these correlations is essential to characterise deviations from General Relativity (GR), and the purpose of this work. This work uses contemporary ideas of Standard Perturbation Theory on biased tracers to characterize the three point correlation function (3PCF) at tree level for Modified Gravity models with a scale-dependent gravitational strength, and applies the theory to two specific models (f(R) and DGP) that are representative for Chameleon and Vainshtein screening mechanisms. Additionally, we use a multipole decomposition, which apart from speeding up the algorithm to extract the signal from data, also helps to visualize and characterize GR deviations.
... assuming homogeneity. For efficient measurement, it is useful to restrict to the isotropic component of the 4PCF (i.e. that averaged over rotations), and project the statistic into a basis of spherical harmonics, defined by [85,86]: ...
... involving spherical harmonics and the Wigner 3j symbol. The 4PCF multiplets ζ 1 2 3 (r 1 , r 2 , r 3 ) can be directly estimated from data (using the encore code [86]), 4 and are related to the full field via ...
... In all cases, we use the measured multiplets with i ≤ 4 (satisfying the triangle conditions) and ten radial bins in [20,160]h −1 Mpc, giving a total of 1288 elements in the data-vector. The 4PCF measurements and corresponding analysis pipeline has been made publicly available on GitHub 5 and further details of the dataset (including details of systematic weights and survey geometry correction) are presented in [54] (see also [55]), building on the results of [53,86]. ...
Could new physics break the mirror symmetry of the Universe? Utilizing recent measurements of the parity-odd four-point correlation function of BOSS galaxies, we probe the physics of inflation by placing constraints on the amplitude of a number of parity-violating models. Within canonical models of (single-field, slow-roll) inflation, no parity-asymmetry can occur; however, it has recently been shown that breaking of the standard assumptions can lead to parity violation within the Effective Field Theory of Inflation (EFTI). In particular, we consider the Ghost Condensate and Cosmological Collider scenarios - the former for the leading and subleading operators in the EFTI and the latter for different values of mass and speed of an exchanged spin-1 particle - for a total of 18 models. Each instance yields a definite prediction for the inflationary trispectrum, which we convert to a late-time galaxy correlator prediction (through a highly non-trivial calculation) and constrain using the observed data. We find no evidence for inflationary parity-violation (with each of the 18 models having significances below ), and place the first constraints on the relevant coupling strengths, at a level comparable with the theoretical perturbativity bounds. This is also the first time Cosmological Collider signatures have directly been searched for in observational data. We further show that possible secondary parity-violating signatures in galaxy clustering can be systematically described within the Effective Field Theory of Large-Scale Structure. We argue that these late-time contributions are subdominant compared to the primordial parity-odd signal for a vast region of parameter space. In summary, the results of this paper disfavor the notion that the recent hints of parity-violation observed in the distribution of galaxies are due to new physics.
... The simplest random fields are the so called Gaussian random fields, which present vanishing NPCF for N larger than three. However, higher than or equal to three NPCF have found successful applications in several scientific applications: molecular physics [24]; material science [25]; field theory [26]; diffusive systems [27,28]; quantum field theory [29]; computational physics [30,31]; cosmology [30][31][32]. These NPCFs try to quantify models of natural systems which most of them are built on manipulation of ingredients of the action principle (see [33] and references therein). ...
... The simplest random fields are the so called Gaussian random fields, which present vanishing NPCF for N larger than three. However, higher than or equal to three NPCF have found successful applications in several scientific applications: molecular physics [24]; material science [25]; field theory [26]; diffusive systems [27,28]; quantum field theory [29]; computational physics [30,31]; cosmology [30][31][32]. These NPCFs try to quantify models of natural systems which most of them are built on manipulation of ingredients of the action principle (see [33] and references therein). ...
... These mean that the observed-relative-to-the-matter N-point correlator in real space becomes 30) where r N = ( r 1 , . . . , r N −1 ), while in Fourier space it becomes ...
In this paper, we describe a mathematical formalism for a -dimensional manifold with N-correlators of types of objects, with cross correlations and contaminants. In particular, we build this formalism using simple notions of mathematical physics, field theory, topology, algebra, statistics n-correlators and Fourier transform. We discuss the applicability of this formalism in the context of cosmological scales, i.e. from astronomical scales to quantum scales, for which we give some intuitive examples.
... For many applications within cosmology, such as covariance matrix calculations, the orders of two of the sBFs within a triple-sBF integral each have a maximum value ≤ 10 (e.g. [31,34,36,37,54]). The maximum value of the order of the third sBF is set by the sum of the orders of the other two sBFs; this is because of the triangle inequalities that stem from Wigner 3-j symbols coupling the sBF orders. ...
... for ℓ > −(1/2). We first use equation (54) to write each sBF in equation (53) as a cylindrical Bessel function, then use equation (55) to rewrite each cylindrical Bessel function as an integral over q i , and change the order of integration so that the k integral will be performed first: ...
Spherical Bessel functions appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different weighting functions (which appear as toy models of a given physical observable, such as the galaxy power spectrum) is useful. Here we present a generalization of a recursion-based method for evaluating such integrals. It gives relatively simple closed-form results in terms of Legendre functions (for the exponentially-damped case) and Gamma, incomplete Gamma functions, and hypergeometric functions (for the Gaussian-damped case). We also present a new, non-recursive method to evaluate integrals of products of spherical Bessel functions with Gaussian damping in terms of incomplete Gamma functions and hypergeometric functions.
... The derivation below is largely follow [26,27,63,67,69,70]. Here, we show the derivation for the readers' convenience. ...
Parity-violating interactions are ubiquitous phenomena in particle physics. If they are significant during cosmic inflation, they can leave imprints on primordial perturbations and be observed in correlation functions of galaxy surveys. Importantly, parity-violating signals in the four-point correlation functions (4PCFs) cannot be generated by Einstein gravity in the late universe on large scales, making them unique and powerful probes of high-energy physics during inflation. However, the complex structure of the 4PCF poses challenges in diagnosing the underlying properties of parity-violating interactions from observational data. In this work, we introduce a general framework that provides a streamlined pipeline directly from a particle model in inflation to galaxy 4PCFs in position space. We demonstrate this framework with a series of toy models, effective-field-theory-like models, and full models featuring tree-level exchange-type processes with chemical-potential-induced parity violation. We further showed the detection sensitivity of these models from BOSS data and highlighted potential challenges in data interpretation and model prediction.
... While two-point statistics capture crucial information about most cosmological parameters, higher-order statistics offer valuable complementary insights, particularly when small-scale modes are incorporated into the analysis. Significant progress has been made in leveraging higher-order statistics, including the development of efficient estimators for observables (Philcox et al. 2021b) and the construction of theoretical models for covariance matrices in cosmological analyses (Philcox et al. 2021a). ...
In this work, we develop a theoretical model for the cross-power spectrum of the galaxy density field before and after standard baryonic acoustic oscillation (BAO) reconstruction. Using this model, we extract the redshift-space distortion (RSD) parameter from the cross-power spectrum. The model is validated against a suite of high-resolution N-body simulations, demonstrating its accuracy and robustness for cosmological analyses.
... Geometrically, it corresponds to a mirror reflection followed by a 180 • rotation. Recently, fast algorithms have been developed to measure the 4PCF [18,19]. The search proposed by [15] was carried out using the 4PCF of Sloan Digital Sky Survey (SDSS) Baryon Oscillation Spectroscopic Survey (BOSS) galaxies, finding 7.1σ evidence for parity violation in the CMASS sample of roughly 800, 000 Luminous Red Galaxies (LRGs) at 0.43 < z < 0.7 and roughly 3σ evidence in the LOWZ (280,000 galaxies) lowerredshift sample of LRGs (0.16 < z < 0.36) [20]; see also [21]. ...
Recent measurements of the galaxy 4-Point Correlation Function (4PCF) have seemingly detected non-zero parity-odd modes at high significance. Since gravity, the primary driver of galaxy formation and evolution is parity-even, any parity violation, if genuine, is likely to have been produced by some new parity-violating mechanism in the early Universe. Here we investigate an inflationary model with a Chern-Simons interaction between an axion and a U(1) gauge field, where the axion itself is the inflaton field. Evaluating the trispectrum (Fourier-space analog of the 4PCF) of the primordial curvature perturbations is an involved calculation with very high-dimensional loop integrals. We demonstrate how to simplify these integrals and perform all angular integrations analytically by reducing the integrals to convolutions and exploiting the Convolution Theorem. This leaves us with low-dimensional radial integrals that are much more amenable to efficient numerical evaluation. This paper is the first in a series in which we will use these results to compute the full late-time 4PCF for axion inflation, thence enabling constraints from upcoming 3D spectroscopic surveys such as Dark Energy Spectroscopic Instrument (DESI), Euclid, or Roman.
... [27][28][29]). For example, precise algorithms for performing such analyses were developed in [21], based on innovative methods [22,[30][31][32] for 4PCF. Other examples include the introduction of novel observables for studying parity violation in LSS analyses, such as Parity-Odd Power Spectra [33] in which the parity-odd trispectrum is compressed into simpler statistics computationally fast to construct 1 . ...
Recently, possible hints of parity violation have been observed in the connected galaxy four-point correlation function. Although the true origin of the signal from the analysis has been debated, should they have a physical origin, they might point to primordial non-Gaussianity and would be evidence of new physics. In this work, we examine the single-field slow-roll model of inflation within chiral scalar-tensor theories of modified gravity. These theories extend the Chern-Simons one by including parity-violating operators containing first and second derivatives of the non-minimally coupled scalar (inflaton) field. This model is capable of imprinting parity-violating signatures in late-time observables, such as the galaxy four-point correlation function. We perform an analysis of the graviton-mediated scalar trispectrum of the gauge-invariant curvature perturbation . We estimate that for a set of parameters of the theory it is possible to produce a signal-to-noise ratio for the parity-violating part of the trispectrum of order one without introducing modifications to the single-field slow-roll setup. Even if the signal found in the analysis turns out to be spurious or if no parity violation is ever detected in the galaxy four-point correlation function, our analysis can be used to constrain the free parameters of these theories.
... • For computing the 3pCF we use a Legendre polynomial decomposition described by Philcox et al. (2021), which uses the generic estimator of the 3pCF: ...
Feedback from active galactic nuclei (AGN) plays a critical role in shaping the matter distribution on scales comparable to and larger than individual galaxies. Upcoming surveys such as and LSST aim to precisely quantify the matter distribution on cosmological scales, making a detailed understanding of AGN feedback effects essential. Hydrodynamical simulations provide an informative framework for studying these effects, in particular by allowing us to vary the parameters that determine the strength of these feedback processes and, consequently, to predict their corresponding impact on the large-scale matter distribution. We use the EAGLE simulations to explore how changes in subgrid viscosity and AGN heating temperature affect the matter distribution, quantified via 2- and 3-point correlation functions, as well as higher order cumulants of the matter distribution. We find that varying viscosity has a small impact () on scales larger than Mpc, while changes to the AGN heating temperature lead to substantial differences, with up to variation in gas clustering on small scales ( Mpc). By examining the suppression of the power spectrum as a function of time, we identify the redshift range as a key epoch where AGN feedback begins to dominate in these simulations. The 3-point function provides complementary insight to the more familiar 2-point statistics, and shows more pronounced variations between models on the scale of individual haloes. On the other hand, we find that effects on even larger scales are largely comparable.
... The downside to directly calculating the bispectrum or trispectrum is that it can be computationally expensive, and can lead to a large increase ★ E-mail: jessica.cowell@physics.ox.ac.uk in the size of the resulting data vector (with the associated complexity of estimating its covariance matrix and theoretical predictions). However, significant progress has been made at accelerating these calculaions, however (Choustikov et al. 2023;Philcox et al. 2022b;Takahashi et al. 2020;Philcox & Flöss 2024). There is also no guarantee that this next-order approach leads to an optimal extraction of non-Gaussian information. ...
Marked power spectra provide a computationally efficient way to extract non-Gaussian information from the matter density field using the usual analysis tools developed for the power spectrum without the need for explicit calculation of higher-order correlators. In this work, we explore the optimal form of the mark function used for re-weighting the density field, to maximally constrain cosmology. We show that adding to the mark function or multiplying it by a constant leads to no additional information gain, which significantly reduces our search space for optimal marks. We quantify the information gain of this optimal function and compare it against mark functions previously proposed in the literature. We find that we can gain around times smaller errors in and times smaller errors in compared to using the traditional power spectrum alone, an improvement of compared to other proposed marks when applied to the same dataset.
... [31][32][33][34][35][36]. Thereafter, these have been extended to arbitrary N-point correlations [37][38][39]. ...
One of the main obstacles for the signal extraction of the three point correlation function using photometric surveys, such as the Rubin Observatory Legacy Survey of Space and Time (LSST), will be the prohibitive computation time required for dealing with a vast quantity of sources. Brute force algorithms, which naively scales as with the number of objects, can be further improved with tree methods but not enough to deal with large scale correlations of Rubin's data. However, a harmonic basis decomposition of these higher order statistics reduces the time dramatically, to scale as a two-point correlation function with the number of objects, so that the signal can be extracted in a reasonable amount of time. In this work, we aim to develop the framework to use these expansions within the Limber approximation for scalar (or spin-0) fields, such as galaxy counts, weak lensing convergence or aperture masses. We develop an estimator to extract the signal from catalogs and different phenomenological and theoretical models for its description. The latter includes halo model and standard perturbation theory, to which we add a simple effective field theory prescription based on the short range of non-locality of cosmic fields, significantly improving the agreement with simulated data. In parallel to the modeling of the signal, we develop a code that can efficiently calculate three points correlations of more than 200 million data points (a full sky simulation with Nside=4096) in 40 minutes on a single high-performance computing node, enabling a feasible analysis for the upcoming LSST data.
... Since, in most cases, we are measuring the orientation of close galaxy pairs relative to a distant tracer, this estimator can also be thought of as a squeezed three-point correlation function. Previous work has explored 3-point and higher-order correlations in spectroscopic data (Slepian & Eisenstein 2015;Philcox et al. 2022), including detecting evidence of the tidal field (Slepian et al. 2017) and investigating the squeezed 3-point function (Yuan et al. 2017). These describe correlations that arise from larger scales than multiplets, but are a similar framing of our estimator. ...
We explore correlations between the orientations of small galaxy groups, or "multiplets", and the large-scale gravitational tidal field. Using data from the Dark Energy Spectroscopic Instrument (DESI) Y1 survey, we detect the intrinsic alignment (IA) of multiplets to the galaxy-traced matter field out to separations of 100 Mpc/h. Unlike traditional IA measurements of individual galaxies, this estimator is not limited by imaging of galaxy shapes and allows for direct IA detection beyond redshift z = 1. Multiplet alignment is a form of higher-order clustering, for which the scale-dependence traces the underlying tidal field and amplitude is a result of small-scale (< 1 Mpc/h) dynamics. Within samples of bright galaxies (BGS), luminous red galaxies (LRG) and emission-line galaxies (ELG), we find similar scale-dependence regardless of intrinsic luminosity or colour. This is promising for measuring tidal alignment in galaxy samples that typically display no intrinsic alignment. DESI's LRG mock galaxy catalogues created from the AbacusSummit N-body simulations produce a similar alignment signal, though with a 33% lower amplitude at all scales. An analytic model using a non-linear power spectrum (NLA) only matches the signal down to 20 Mpc/h. Our detection demonstrates that galaxy clustering in the non-linear regime of structure formation preserves an interpretable memory of the large-scale tidal field. Multiplet alignment complements traditional two-point measurements by retaining directional information imprinted by tidal forces, and contains additional line-of-sight information compared to weak lensing. This is a more effective estimator than the alignment of individual galaxies in dense, blue, or faint galaxy samples.
... The possibility of parity violation in the galaxy distribution is made perhaps more interesting given a reported preference in Planck 2018 CMB polarization data for a nonzero value of the cosmic birefringence angle [5], which also hints at cosmological parity-violating physics [6] The possibility to seek parity violation in galaxy clustering was suggested briefly in Refs. [7,8], and precise algorithms to carry out such searches were developed in Ref. [9], capitalizing upon novel techniques [10][11][12][13] for the more general 4PCF. The implementation with BOSS data and evidence for parity breaking in Refs. ...
Recent searches for parity breaking in the galaxy four-point correlation function, as well as the prospects for greatly improved sensitivity to parity breaking in forthcoming surveys, motivate the search for physical mechanisms that could produce such a signal. Here we show that a parity-violating galaxy four-point correlation function may be induced by lensing by a chiral gravitational-wave background. We estimate the amplitude of a signal that would be detectable with a current galaxy survey, taking into account constraints to the primordial gravitational-wave-background amplitude. We find that this mechanism is unlikely to produce a signal large enough to be seen with a galaxy survey but note that it may come within reach with future 21cm observations.
... An alternative approach explored by [86,87] is a Fast Fourier Transform (FFT)-based calculation of the 3-point function with a complexity of O(N log 2 N ), which is the same as that of tree-based 2-point function estimators. We reproduce below the main aspects of the computational procedure for completeness and direct readers to [87, see their section 2.2] for a detailed derivation. ...
Primordial non-Gaussianities (PNGs) are signatures in the density field that encode particle physics processes from the inflationary epoch. Such signatures have been extensively studied using the Cosmic Microwave Background, through constraining their amplitudes, fX NL, with future improvements expected from large-scale structure surveys; specifically, the galaxy correlation functions. We show that weak lensing fields can be used to achieve competitive and complementary constraints. This is shown via the Ulagam suite of N-body simulations, a subset of which evolves primordial fields with four types of PNGs. We create full-sky lensing maps and estimate the Fisher information from three summary statistics measured on the maps: the moments, the cumulative distribution function, and the 3-point correlation function. We find that the year 10 sample from the Rubin Observatory Legacy Survey of Space and Time (LSST) can constrain PNGs to σ(f NL eq) ≈ 110, σ(f NL or, lss) ≈ 120, σ(f NL loc) ≈ 40. For the former two, this is better than or comparable to expected galaxy clustering-based constraints from the Dark Energy Spectroscopic Instrument (DESI). The PNG information in lensing fields is on non-linear scales and at low redshifts (z ≲ 1.25), with a clear origin in the evolution history of massive halos. The constraining power degrades by ∼60% under scale cuts of ≳ 20 Mpc, showing there is still significant information on scales mostly insensitive to small-scale systematic effects (e.g., baryons). We publicly release the Ulagam suite to enable more survey-focused analyses.
... For many applications within cosmology, such as covariance matrix calculations, the orders of two of the sBFs within a triple-sBF integral each have a maximum value ≤10 (e.g. [31,34,36,37,54]). The maximum value of the order of the third sBF is set by the sum of the orders of the other two sBFs; this is because of the triangle inequalities that stem from Wigner 3-j symbols coupling the sBF orders. ...
Spherical Bessel functions (sBFs) appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different weighting functions (which appear as toy models of a given physical observable, such as the galaxy power spectrum) is useful. Here, we present a generalization of a recursion-based method for evaluating such integrals. It gives relatively simple closed-form results in terms of Legendre functions (for the exponentially damped case) and Gamma, incomplete Gamma, and hypergeometric functions (for the Gaussian-damped case). We also present a new, non-recursive method to evaluate integrals of products of sBFs with Gaussian damping in terms of incomplete Gamma functions and hypergeometric functions.
... A first method relies on measuring the higher-order point functions, such as bispectrum and trispectrum. This methodology can be computationally expensive-especially when modeling the covariance matrices, even though later achievements (Philcox et al. 2022) showed that the analysis can be sped up-but is very promising (Gil-Marín et al. 2017;Philcox et al. 2020;Hahn & Villaescusa-Navarro 2021). Another approach involves the use of different summary statistics, such as the number density of voids and clusters (Pisani et al. 2019;Bayer et al. 2021;Kreisch et al. 2022); counts in cells (CIC; Bernardeau 1994;Sheth 1998;Uhlemann et al. 2020), which is the probability distribution function (pdf) of the mean matter density inside patches of a given size; minimum spanning tree (Naidoo et al. 2020(Naidoo et al. , 2022; scattering transforms (Valogiannis & Dvorkin 2022;Eickenberg et al. 2022); and many others. ...
Marked power spectra are two-point statistics of a marked field obtained by weighting each location with a function that depends on the local density around that point. We consider marked power spectra of the galaxy field in redshift space that up-weight low-density regions, and we perform a Fisher matrix analysis to assess the information content of this type of statistics using the Molino mock catalogs built on the Quijote simulations. We identify four different ways to up-weight the galaxy field, and we compare the Fisher information contained in their marked power spectra to that of the standard galaxy power spectrum, when considering the monopole and quadrupole of each statistic. Our results show that each of the four marked power spectra can tighten the standard power spectrum constraints on the cosmological parameters Ω m , Ω b , h , n s , and M ν by 15%–25% and on σ 8 by a factor of 2. The same analysis performed by combining the standard and four marked power spectra shows a substantial improvement compared to the power spectrum constraints that is equal to a factor of 6 for σ 8 and a factor of 2.5–3 for the other parameters. Our constraints may be conservative, since the galaxy number density in the Molino catalogs is much lower than the ones in future galaxy surveys, which will allow them to probe lower-density regions of the large-scale structure.
... Despite the numerous studies that have presented 3PCF measurements [248][249][250][251][252][253], a brute force measurement of the NPCF poses computational challenges. For N g galaxies, while a brute force approach has the complexity that scales as O(N N g ), an algorithm with the complexity of O(N 2 g ) was proposed for the 3PCF [254] and was extended to the arbitrary integer N ≥ 4 [255][256][257]. The algorithm relies on the radial-angular decomposition and can either be applied to discrete data points or gridded data. ...
The current standard cosmological model is constructed within the framework of general relativity with a cosmological constant Λ, which is often associated with dark energy, and phenomenologically explains the accelerated cosmic expansion. Understanding the nature of dark energy is one of the most appealing questions in achieving a self-consistent physical model at cosmological scales. Modification of general relativity could potentially provide a more natural and physical solution to the accelerated expansion. The growth of the cosmic structure is sensitive in constraining gravity models. In this paper, we aim to provide a concise introductory review of modified gravity models from an observational point of view. We will discuss various mainstream cosmological observables, and their potential advantages and limitations as probes of gravity models.
... Computing 4-point functions is much more challenging due to the high dimensionality of the integrals involved in the computation. Recently, some progress has been made on this problem, allowing to compute high-order statistics of this kind in an efficient way [117,118]. Similar techniques could be applied to lattice codes, and would allow computing the trispectrum of ζ from the simulation. ...
We present a nonlinear study of the inflationary epoch based on numerical lattice simulations. Lattice simulations are a well-known tool in primordial cosmology, and they have been extensively used to study the reheating epoch after inflation. We generalize this known machinery to the inflationary epoch. Being this the first simulation of the inflationary epoch much before the end of inflation, the first part of the thesis focuses on the minimal single-field model of inflation. We discuss the conceptual and technical ingredients needed to simulate inflation on a lattice. The simulation is used to reproduce the nearly scale-invariant spectrum of scalar perturbations, as well as the oscillations in the power spectrum caused by a step in the potential. In the second part, we focus on the more complicated axion-U(1) model of inflation and present the first lattice simulation of this model during the deep inflationary epoch. We use the simulation to discover new properties of primordial scalar perturbations from this model. In the linear regime of the theory, we find high-order non-Gaussianity (beyond trispectrum) to be key to describing the statistical properties of scalar perturbations. Conversely, we find perturbations to be nearly Gaussian in the nonlinear regime of the theory. This relaxes existing constraints from the overproduction of primordial black holes, allowing for a gravitational waves signal in the observable range of upcoming experiments such as LISA. Our results show that lattice simulations can be a powerful tool to study the inflationary epoch and its observational signatures.
Understanding the impact of neutrino masses on the evolution of Universe is a crucial aspect of modern cosmology. Due to their large free streaming lengths, neutrinos significantly influence the formation of cosmic structures at non-linear scales. To maximize the information yield from current and future galaxy surveys, it is essential to generate precise theoretical predictions of structure formation. One approach to achieve this is by running large sets of cosmological numerical simulations, which is a computationally intensive process. In this study, we propose a deep learning-based generative adversarial network (GAN) model to emulate the Universe for a variety of neutrino masses. Our model called GAN (for neutrino GAN) is able to generate 2D cosmic webs of the Universe for a number of neutrino masses ranging from 0.0 eV to 0.4 eV. The generated maps exhibit statistical independence, lack correlations with training data, and very closely resemble the distribution of matter in true maps. We assess the accuracy of our results both visually and through key statistics used in cosmology and computer vision analyses. Our results indicate that samples generated by GAN are accurate within a 5% error on power spectrum between k=0.01 to k=0.5 h/Mpc. Although this accuracy covers the mildly non-linear scales, consistent with other works and observations, achieving higher accuracy at fully non-linear scales requires more sophisticated models, such as diffusion models. Nevertheless, our work opens up new avenues for building emulators to generate fast and massive neutrino simulations, potentially revolutionizing cosmological predictions and analyses. This work serves as a proof-of-concept, paving the way for future extensions with higher-resolution 3D data and advanced generative models.
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.
One of the main obstacles for the signal extraction of the three point correlation function using photometric surveys, such as the Rubin Observatory Legacy Survey of Space and Time (LSST), will be the prohibitive computation time required for dealing with a vast quantity of sources. Brute force algorithms, which naively scales as 𝒪(N ³) with the number of objects, can be further improved with tree methods but not enough to deal with large scale correlations of Rubin's data. However, a harmonic basis decomposition of these higher order statistics reduces the time dramatically, to scale as a two-point correlation function with the number of objects, so that the signal can be extracted in a reasonable amount of time. In this work, we aim to develop the framework to use these expansions within the Limber approximation for scalar (or spin-0) fields, such as galaxy counts, weak lensing convergence or aperture masses. We develop an estimator to extract the signal from catalogs and different phenomenological and theoretical models for its description. The latter includes halo model and standard perturbation theory, to which we add a simple effective field theory prescription based on the short range of non-locality of cosmic fields, significantly improving the agreement with simulated data. In parallel to the modeling of the signal, we develop a code that can efficiently calculate three points correlations of more than 200 million data points (a full sky simulation with Nside=4096) in ∼40 minutes, or even less than 10 minutes using an approximation in the searching algorithm, on a single high-performance computing node, enabling a feasible analysis for the upcoming LSST data.
Marked power spectra provide a computationally efficient way to extract non-Gaussian information from the matter density field using the usual analysis tools developed for the power spectrum without the need for explicit calculation of higher-order correlators. In this work, we explore the optimal form of the mark function used for re-weighting the density field, to maximally constrain cosmology. We show that adding to the mark function or multiplying it by a constant leads to no additional information gain, which significantly reduces our search space for optimal marks. We quantify the information gain of this optimal function and compare it against mark functions previously proposed in the literature. We find that we can gain around times smaller errors in and times smaller errors in compared to using the traditional power spectrum alone, an improvement of compared to other proposed marks when applied to the same data set.
This paper presents a novel perspective on correlation functions in the clustering analysis of the large-scale structure of the universe. We begin with the recognition that pair counting in bins of radial separation is equivalent to evaluating counts-in-cells (CIC), which can be modelled using a filtered density field with a binning-window function. This insight leads to an in situ expression for the two-point correlation function (2PCF). Essentially, the core idea underlying our method is to introduce a window function to define the binning scheme, enabling pair-counting without binning. This approach develops an idea of generalised 2PCF, which extends beyond conventional discrete pair counting by accommodating non-sharp-edged window functions. In the context of multiresolution analysis, we can implement a fast algorithm to estimate the generalised 2PCF. To extend this framework to N-point correlation functions (NPCF) using current optimal edge-corrected estimators, we developed a binning scheme that is independent of the specific parameterisation of polyhedral configurations. In particular, we demonstrate a fast algorithm for the three-point correlation function (3PCF), where triplet counting is accomplished by assigning either a spherical tophat or a Gaussian filter to each vertex of triangles. Additionally, we derive analytical expressions for the 3PCF using a multipole expansion in Legendre polynomials, accounting for filtered field (binning) corrections. Our method provides an exact solution for quantifying binning effects in practical measurements and offers a high-speed algorithm, enabling high-order clustering analysis in extremely large datasets from ongoing and upcoming surveys such as Euclid, LSST, and DESI.
Feedback from active galactic nuclei (AGN) plays a critical role in shaping the matter distribution on scales comparable to and larger than individual galaxies. Upcoming surveys such as Euclid and LSST aim to precisely quantify the matter distribution on cosmological scales, making a detailed understanding of AGN feedback effects essential. Hydrodynamical simulations provide an informative framework for studying these effects, in particular by allowing us to vary the parameters that determine the strength of these feedback processes and, consequently, to predict their corresponding impact on the large-scale matter distribution. We use the EAGLE simulations to explore how changes in subgrid viscosity and AGN heating temperature affect the matter distribution, quantified via 2- and 3-point correlation functions, as well as higher order cumulants of the matter distribution. We find that varying viscosity has a small impact () on scales larger than 1h−1 Mpc, while changes to the AGN heating temperature lead to substantial differences, with up to 70% variation in gas clustering on small scales (≲ 1h−1 Mpc). By examining the suppression of the power spectrum as a function of time, we identify the redshift range z = 1.5 − 1 as a key epoch where AGN feedback begins to dominate in these simulations. The 3-point function provides complementary insight to the more familiar 2-point statistics, and shows more pronounced variations between models on the scale of individual haloes. On the other hand, we find that effects on even larger scales are largely comparable.
We explore correlations between the orientations of small galaxy groups, or ‘multiplets’, and the large-scale gravitational tidal field. Using data from the Dark Energy Spectroscopic Instrument (DESI) Y1 survey, we detect the intrinsic alignment (IA) of multiplets to the galaxy-traced matter field out to separations of 100h−1Mpc. Unlike traditional IA measurements of individual galaxies, this estimator is not limited by imaging of galaxy shapes and allows for direct IA detection beyond redshift z = 1. Multiplet alignment is a form of higher-order clustering, for which the scale-dependence traces the underlying tidal field and amplitude is a result of small-scale (<1h−1Mpc) dynamics. Within samples of bright galaxies (BGS), luminous red galaxies (LRG) and emission-line galaxies (ELG), we find similar scale-dependence regardless of intrinsic luminosity or colour. This is promising for measuring tidal alignment in galaxy samples that typically display no intrinsic alignment. DESI’s LRG mock galaxy catalogues created from the AbacusSummit N-body simulations produce a similar alignment signal, though with a 33% lower amplitude at all scales. An analytic model using a non-linear power spectrum (NLA) only matches the signal down to 20h−1Mpc. Our detection demonstrates that galaxy clustering in the non-linear regime of structure formation preserves an interpretable memory of the large-scale tidal field. Multiplet alignment complements traditional two-point measurements by retaining directional information imprinted by tidal forces, and contains additional line-of-sight information compared to weak lensing. This is a more effective estimator than the alignment of individual galaxies in dense, blue, or faint galaxy samples.
Here we show how to produce a three-dimensional density field with a given set of higher order correlation functions. Our algorithm enables producing any desired two-, three-, and four-point functions, including odd parity for the last ones. We note that this algorithm produces the desired correlations around a set of ‘primary’ points, matched to how the spherical-harmonic-based algorithms ENCORE and CADENZA measure them. These ‘primary points’ must be used as those around which the correlation functions are measured. We also generalize the algorithm to (i) N-point correlations with , (ii) dimensions other than three, and (iii) beyond scalar quantities. This algorithm should find use in verifying analysis pipelines for higher order statistics in upcoming galaxy redshift surveys, such as Dark Energy Spectroscopic Instrument (DESI), Euclid, Roman, and Spherex, as well as intensity mapping. In particular, it may be helpful in searches for parity violation in the four-point correlation function of these samples, for which producing initial conditions for N-body simulations is both costly and highly model dependent at present, and so alternative methods, such as that developed here, are desirable.
This paper presents a novel perspective on correlation functions in the clustering analysis of the large-scale structure of the universe. We first recognise that pair counting in bins of radial separation is equivalent to evaluating counts-in-cells (CIC), which can be modelled using a filtered density field with a binning-window function. This insight leads to an in situ expression for the two-point correlation function (2PCF). Essentially, the core idea underlying our method is to introduce a window function to define the binning scheme, enabling pair-counting without binning. This approach develops a concept of generalised 2PCF, which extends beyond conventional discrete pair counting by accommodating non-sharp-edged window functions. To extend this framework to N-point correlation functions (NPCF) using current optimal edge-corrected estimators, we developed a binning scheme independent of the specific parameterisation of polyhedral configurations. In particular, we demonstrate a fast algorithm for the three-point correlation function (3PCF), where triplet counting is accomplished by assigning either a spherical tophat or a Gaussian filter to each vertex of triangles. Additionally, we derive analytical expressions for the 3PCF using a multipole expansion in Legendre polynomials, accounting for filtered field (binning) corrections. Numerical tests using several suites of N-body simulation samples show that our approach aligns remarkably well with the theoretical predictions. Our method provides an exact solution for quantifying binning effects in practical measurements and offers a high-speed algorithm, enabling high-order clustering analysis in extremely large datasets from ongoing and upcoming surveys such as Euclid, LSST, and DESI.
Recent studies have found evidence for parity violation in the BOSS spectroscopic galaxy survey, with statistical significance as high as 7σ. These analyses assess the significance of the parity-odd four-point correlation function (4PCF) with a statistic called X ². This statistic is biased if the parity-even eight-point correlation function (8PCF) of the data differs from the mock catalogs. We construct new statistics X ² ×, X ² null that separate the parity violation signal from the 8PCF bias term, allowing them to be jointly constrained. Applying these statistics to BOSS, we find that the parity violation signal ranges from 0 to 2.5σ depending on analysis choices, whereas the 8PCF bias term is ~ 6σ. We conclude that there is no compelling evidence for parity violation in BOSS. Our new statistics can be used to search for parity violation in future surveys, such as DESI, without 8PCF biases.
The density fields constructed by traditional mass assignment methods are susceptible to irritating discreteness, which hinders morphological measurements of cosmic large-scale structure (LSS) through Minkowski functionals (MFs). To alleviate this issue, fixed-kernel smoothing methods are commonly used in the literature, at the expense of losing substantial structural information. In this work, we propose to measure MFs with the Delaunay tessellation field estimation (DTFE) technique, with the goal of maximizing the extraction of morphological information from sparse tracers. We perform our analyses starting from matter fields and progressively extending to halo fields. At the matter-field level, we elucidate how discreteness affects morphological measurements of LSS. Then, by comparing with the traditional Gaussian smoothing scheme, we preliminarily showcase the advantages of DTFE for enhancing measurements of MFs from sparse tracers. At the halo-field level, we first numerically investigate various systematic effects on MFs of DTFE fields, which are induced by finite voxel sizes, halo number densities, halo weightings, and redshift space distortions (RSDs), respectively. Then, we explore the statistical power of MFs measured with DTFE for extracting the cosmological information encoded in RSDs. We find that MFs measured with DTFE exhibit improvements by ∼2 orders of magnitude in discriminative power for RSD effects and by a factor of ∼3–5 in constraining power on the structure growth rate over the MFs measured with Gaussian smoothing. These findings demonstrate the remarkable enhancements in statistical power of MFs achieved by DTFE, showing enormous application potentials for our method in extracting various key cosmological information from galaxy surveys.
Context. Third-order lensing statistics contain a wealth of cosmological information that is not captured by second-order statistics. However, the computational effort it takes to estimate such statistics in forthcoming stage IV surveys is prohibitively expensive.
Aims. We derive and validate an efficient estimation procedure for the three-point correlation function (3PCF) of polar fields such as weak lensing shear. We then use our approach to measure the shear 3PCF and the third-order aperture mass statistics on the KiDS-1000 survey.
Methods We constructed an efficient estimator for third-order shear statistics that builds on the multipole decomposition of the 3PCF. We then validated our estimator on mock ellipticity catalogs obtained from N -body simulations. Finally, we applied our estimator to the KiDS-1000 data and presented a measurement of the third-order aperture statistics in a tomographic setup.
Results. Our estimator provides a speedup of a factor of ∼100–1000 compared to the state-of-the-art estimation procedures. It is also able to provide accurate measurements for squeezed and folded triangle configurations without additional computational effort. We report a significant detection of tomographic third-order aperture mass statistics in the KiDS-1000 data (S/N = 6.69).
Conclusions. Our estimator will make it computationally feasible to measure third-order shear statistics in forthcoming stage IV surveys. Furthermore, it can be used to construct empirical covariance matrices for such statistics.
The density fields constructed by traditional mass assignment methods are susceptible to irritating discreteness, which hinders morphological measurements of cosmic large-scale structure (LSS) through Minkowski functionals (MFs). For alleviating this issue, fixed-kernel smoothing methods are commonly used in literatures, at the expense of losing substantial structural information. In this work, we propose to measure MFs with Delaunay tessellation field estimation (DTFE) technique, with the goal to maximize extractions of morphological information from sparse tracers. We perform our analyses starting from matter fields and progressively extending to halo fields. At matter field level, we elucidate how discreteness affects the morphological measurements of LSS. Then, by comparing with traditional Gaussian smoothing scheme, we preliminarily showcase the advantages of DTFE for enhancing measurements of MFs from sparse tracers. At halo field level, we first numerically investigate various systematic effects on MFs of DTFE fields, which are induced by finite voxel sizes, halo number densities, halo weightings, and redshift space distortions (RSDs), respectively. Then, we explore the statistical power of MFs measured with DTFE for extracting cosmological information encoded in RSDs. We find that MFs measured with DTFE exhibit improvements by 2 orders of magnitude in discriminative power for RSD effects and by a factor of 3-5 in constraining power on structure growth rate over the MFs measured with Gaussian smoothing. These findings demonstrate the remarkable enhancements in statistical power of MFs achieved by DTFE, showing enormous application potentials of our method in extracting various key cosmological information from galaxy surveys.
Recent measurements of the 4-point correlation functions (4PCF) from spectroscopic surveys provide evidence for parity violations in the large-scale structure of the Universe. If physical in origin, this could point to exotic physics during the epoch of inflation. However, searching for parity violations in the 4PCF signal relies on a large suite of simulations to perform a rank test, or an accurate model of the 4PCF covariance to claim a detection, and this approach is incapable of extracting parity information from the higher-order N-point functions. In this work we present an unsupervised method which overcomes these issues, before demonstrating the approach is capable of detecting parity violations in a few toy models using convolutional neural networks. This technique is complementary to the 4-point method and could be used to discover parity violations in several upcoming surveys including DESI, Euclid, and Roman.
Context. The new generation of galaxy surveys will provide unprecedented data that will allow us to test gravity deviations at cosmological scales at a much higher precision than could be achieved previously. A robust cosmological analysis of the large-scale structure demands exploiting the nonlinear information encoded in the cosmic web. Machine-learning techniques provide these tools, but no a priori assessment of the uncertainties.
Aims. We extract cosmological parameters from modified gravity (MG) simulations through deep neural networks that include uncertainty estimations.
Methods. We implemented Bayesian neural networks (BNNs) with an enriched approximate posterior distribution considering two cases: the first case with a single Bayesian last layer (BLL), and the other case with Bayesian layers at all levels (FullB). We trained both BNNs with real-space density fields and power spectra from a suite of 2000 dark matter-only particle-mesh N -body simulations including MG models relying on MG-PICOLA, covering 256 h ⁻¹ Mpc side cubical volumes with 128 ³ particles.
Results. BNNs excel in accurately predicting parameters for Ω m and σ 8 and their respective correlation with the MG parameter. Furthermore, we find that BNNs yield well-calibrated uncertainty estimates that overcome the over- and under-estimation issues in traditional neural networks. The MG parameter leads to a significant degeneracy, and σ 8 might be one possible explanation of the poor MG predictions. Ignoring MG, we obtain a deviation of the relative errors in Ω m and σ 8 by 30% at least. Moreover, we report consistent results from the density field and power spectrum analysis and comparable results between BLL and FullB experiments. This halved the computing time. This work contributes to preparing the path for extracting cosmological parameters from complete small cosmic volumes towards the highly nonlinear regime.
We show how the galaxy four-point correlation function can test for cosmological parity violation. The detection of cosmological parity violation would reflect previously unknown forces present at the earliest moments of the Universe. Recent developments both in rapidly evaluating galaxy N-point correlation functions and in determining the corresponding covariance matrices make the search for parity violation in the four-point correlation function possible in current and upcoming surveys such as those undertaken by Dark Energy Spectroscopic Instrument, the Euclid satellite, and the Vera C. Rubin Observatory. We estimate the limits on cosmic parity violation that could be set with these data.
The next generation of galaxy surveys will provide highly accurate measurements of the large-scale structure of the Universe, allowing for more stringent tests of gravity on cosmological scales. Higher-order statistics are a valuable tool to study the non-Gaussianities in the matter field and to break degeneracies between modified gravity and other physical or nuisance parameters. However, understanding from first principles the behavior of these correlations is essential to characterize deviations from General Relativity (GR), and the purpose of this work. This work uses contemporary ideas of standard perturbation theory on biased tracers to characterize the three-point correlation function at tree level for modified gravity models with a scale-dependent gravitational strength, and applies the theory to two specific models [f(R) and DGP] that are representative for Chameleon and Vainshtein screening mechanisms. Additionally, we use a multipole decomposition, which apart from speeding up the algorithm to extract the signal from data, also helps to visualize and characterize GR deviations.
Context. A novel high-performance exact pair-counting toolkit called fast correlation function calculator (FCFC) is presented.
Aims. With the rapid growth of modern cosmological datasets, the evaluation of correlation functions with observational and simulation catalogues has become a challenge. High-efficiency pair-counting codes are thus in great demand.
Methods. We introduce different data structures and algorithms that can be used for pair-counting problems, and perform comprehensive benchmarks to identify the most efficient algorithms for real-world cosmological applications. We then describe the three levels of parallelisms used by FCFC, SIMD, OpenMP, and MPI, and run extensive tests to investigate the scalabilities. Finally, we compare the efficiency of FCFC with alternative pair-counting codes.
Results. The data structures and histogram update algorithms implemented in FCFC are shown to outperform alternative methods. FCFC does not benefit greatly from SIMD because the bottleneck of our histogram update algorithm is mainly cache latency. Nevertheless, the efficiency of FCFC scales well with the numbers of OpenMP threads and MPI processes, even though speedups may be degraded with over a few thousand threads in total. FCFC is found to be faster than most (if not all) other public pair-counting codes for modern cosmological pair-counting applications.
We present a new python package sarabande for measuring 3 & 4 Point Correlation Functions (3/4 PCFs) in time using Fast Fourier Transforms (FFTs), with Ng the number of grid points used for the FFT. sarabande can measure both projected and full 3 and 4 PCFs on gridded 2D and 3D datasets. The general technique is to generate suitable angular basis functions on an underlying grid, radially bin these to create kernels, and convolve these kernels with the original gridded data to obtain expansion coefficients about every point simultaneously. These coefficients are then combined to give us the 3/4 PCF as expanded in our basis. We apply sarabande to simulations of the Interstellar Medium (ISM) to show the results and scaling of calculating both the full and projected 3/4 PCFs.
Could new physics break the mirror symmetry of the Universe? Utilizing recent measurements of the parity-odd four-point correlation function of BOSS galaxies, we probe the physics of inflation by placing constraints on the amplitude of a number of parity-violating models. Within canonical models of (single-field, slow-roll) inflation, no parity asymmetry can occur; however, it has recently been shown that breaking of the standard assumptions can lead to parity violation within the effective field theory of inflation (EFTI). In particular, we consider the ghost condensate and cosmological collider scenarios—the former for the leading and subleading operators in the EFTI and the latter for different values of mass and speed of an exchanged spin-1 particle—for a total of 18 models. Each instance yields a definite prediction for the inflationary trispectrum, which we convert to a late-time galaxy correlator prediction (through a highly nontrivial calculation) and constrain using the observed data. We find no evidence for inflationary parity violation (with each of the 18 models having significances below 2σ), and place the first constraints on the relevant coupling strengths, at a level comparable with the theoretical perturbativity bounds. This is also the first time cosmological collider signatures have directly been searched for in observational data. We further show that possible secondary parity-violating signatures in galaxy clustering can be systematically described within the effective field theory of large-scale structure. We argue that these late-time contributions are subdominant compared to the primordial parity-odd signal for a vast region of parameter space. In summary, the results of this paper disfavor the notion that the recent hints of parity violation observed in the distribution of galaxies are due to new physics.
We present a new package SARABANDE for measuring 3 & 4 Point Correlation Functions (3/4 PCFs) in time using Fast Fourier Transforms (FFTs), with the number of grid points used for the FFT. SARABANDE can measure both projected and full 3 and 4 PCFs on gridded 2D and 3D datasets. The general technique is to generate suitable angular basis functions on an underlying grid, radially bin these to create kernels, and convolve these kernels with the original gridded data to obtain expansion coefficients about every point simultaneously. These coefficients are then combined to give us the 3/4 PCF as expanded in our basis. We apply SARABANDE to simulations of the Interstellar Medium (ISM) to show the results and scaling of calculating both the full and projected 3/4 PCFs.
Parity-violating physics in the early universe can leave detectable traces in late-time observables. While vector- and tensor-type parity violation can be observed in the B-modes of the cosmic microwave background, scalar-type signatures are visible only in the four-point correlation function (4PCF) and beyond. This work presents a blind test for parity violation in the 4PCF of the baryon oscillation spectroscopic survey (BOSS) CMASS sample, considering galaxy separations in the range [20,160] h−1 Mpc. The parity-odd 4PCF contains no contributions from standard ΛCDM physics and can be efficiently measured using recently developed estimators. Data are analyzed using both a nonparametric rank test (comparing the BOSS 4PCFs to those of realistic simulations) and a compressed χ2 analysis, with the former avoiding the assumption of a Gaussian likelihood. These find similar results, with the rank test giving a detection probability of 99.6% (2.9σ). This provides significant evidence for parity violation, from either cosmological sources or systematics. We perform a number of systematic tests: although these do not reveal any observational artifacts, we cannot exclude the possibility that our detection is caused by the simulations not faithfully representing the statistical properties of the BOSS data. Our measurements can be used to constrain physical models of parity violation. As an example, we consider a coupling between the inflaton and a U(1) gauge field and place bounds on the latter’s energy density, which are several orders of magnitude stronger than those previously reported. Upcoming probes such as DESI and Euclid will reveal whether our detection of parity violation is due to new physics, and strengthen the bounds on a variety of models.
The 3-point correlation function (3PCF) is a powerful tool for the current era of high-data volume, high-precision cosmology. It goes beyond the Gaussian cosmological perturbations probed by the 2-point correlation function, including late-time non-Gaussianities, and encodes information about peculiar velocities, which distort observed positions of galaxies along the line of sight away from their true positions. To access this information, we must track the 3PCF’s dependence not only on each triangle’s shape, but also on its orientation with respect to the line of sight. Consequently, different choices for the line of sight will affect the measured 3PCF. Up to now, the line of sight has been taken as the direction to a single triplet member, but which triplet member is used impacts the 3PCF by ∼20 per cent of the statistical error for a BOSS-like survey. For DESI (5× more precise) this would translate to ∼100 per cent of the statistical error. We propose a new method that is fully symmetric between the triplet members, and uses either the average of the three galaxy position vectors, or the average of their unit vectors. We prove that these methods are equivalent to , where θ is the angle subtended at the observer by any triangle side. By harnessing the solid harmonic shift theorem, we show how these methods can be evaluated scaling as N2, with N the number of objects. We expect that they can be used to make a robust, systematics-free measurement of the anisotropic 3PCF of upcoming redshift surveys such as DESI.
Massive neutrinos suppress the growth of structure on small scales and leave an imprint on large-scale structure that can be measured to constrain their total mass, M ν . With standard analyses of two-point clustering statistics, M ν constraints are severely limited by parameter degeneracies. Ref. [1] demonstrated that the bispectrum, the next higher-order statistic, can break these degeneracies and dramatically improve constraints on M ν and other cosmological parameters. In this paper, we present the constraining power of the redshift-space galaxy bispectrum monopole , B g 0 . We construct the Molino suite of 75,000 mock galaxy catalogs from the Quijote N-body simulations using the halo occupation distribution (HOD) model, which provides a galaxy bias framework well-suited for simulation-based approaches. Using these mocks, we present Fisher matrix forecasts for {Ω m , Ω b , h , n s , σ 8 , M ν } and quantify, for the first time, the information content of the B g 0 down to nonlinear scales. For k max = 0.5 h/Mpc, B g 0 improves constraints on Ω m , Ω b , h , n s , σ 8 , and M ν by 2.8, 3.1, 3.8, 4.2, 4.2, and 4.6× over the power spectrum, after marginalizing over HOD parameters. Even with priors from Planck , B g 0 improves all of the cosmological constraints by ≳ 2×. In fact, for P g 0 + P g 2 and B g 0 out to k max = 0.5 h/Mpc with Planck priors, we achieve a 1σ M ν constraint of 0.048 eV, which is tighter than the current best cosmological constraint. While effects such as survey geometry and assembly bias will have an impact, these constraints are derived for (1 h ⁻¹ Gpc) ³ , a substantially smaller volume than upcoming surveys. Therefore, we conclude that the galaxy bispectrum will significantly improve cosmological constraints for upcoming galaxy surveys — especially for M ν .
We develop an analytical forward model based on perturbation theory to predict the redshift-space galaxy overdensity at the field level given a realization of the initial conditions. We find that the residual noise between the model and simulated galaxy density has a power spectrum that is white on large scales, with size comparable to the shot noise. In the mildly nonlinear regime, we see a k ² μ ² correction to the noise power spectrum, corresponding to larger noise along the line of sight and on smaller scales. The parametric form of this correction has been predicted on theoretical grounds before, and our simulations provide important confirmation of its presence. We have also modeled the galaxy velocity at the field-level and compared it against simulated galaxy velocities, finding that about 10% of the galaxies are responsible for half of the rms velocity residual for our simulated galaxy sample.
The effective field theory likelihood for the density field of biased tracers allows for cosmology inference from the clustering of galaxies that consistently uses all available information at a given order in perturbation theory. This paper presents results and implementation details on the real-space (as opposed to Fourier-space) formulation of the likelihood, which allows for the incorporation of survey window functions. The implementation further uses a Lagrangian forward model for biased tracers which automatically accounts for all relevant contributions up to any desired order. Unbiased inference of σ 8 is demonstrated at the 2% level for cutoff values Ł ≲ 0.14 h Mpc ⁻¹ for halo samples over a range of masses and redshifts. The inferred value shows the expected convergence to the ground truth in the low-cutoff limit. Apart from the possibility of including observational effects, this represents further substantial improvement over previous results based on the EFT likelihood.
We perform for the first time a joint analysis of the monopole and quadrupoles for power spectrum, bispectrum and integrated trispectrum (i-trispectrum) from the redshift space matter field in N-body simulations. With a full Markov Chain Monte Carlo exploration of the posterior distribution, we quantify the constraints on cosmological parameters for an object density of n p = 5 10-4 (h Mpc-1)3, redshift z = 0.5, and a covariance corresponding to a survey volume of V survey = 25 (h -1Gpc)3, a set up which is representative of forthcoming galaxy redshift surveys. We demonstrate the complementarity of the bispectrum and i-trispectrum in constraining key cosmological parameters. In particular, compared to the state-of-the-art power spectrum (monopole plus quadrupole) and bispectrum (monopole) analyses, we find 1D 68% credible regions smaller by a factor of (72%,78%,72%,47%,46%) for the parameters (f,σ8,f nl,α∥,α⊥) respectively. This work motivates the additional effort necessary to include the redshift-space anisotropic signal of higher-order statistics in the analysis and interpretation of ongoing and future galaxy surveys.
The shapes of galaxy N-point correlation functions can be used as standard rulers to constrain the distance–redshift relationship. The cosmological density fields traced by late-time galaxy formation are initially nearly Gaussian, and hence, all the cosmological information can be extracted from their two-point correlation function. Subsequent non-linear evolution under gravity, as well as halo and then galaxy formation, generates higher order correlation functions. Since the mapping of the initial to the final density field is, on large scales, invertible, it is often claimed that the information content of the initial field’s power spectrum is equal to that of all the higher order functions of the final, non-linear field. This claim implies that reconstruction of the initial density field from the non-linear field renders analysis of higher order correlation functions of the latter superfluous. We show that this claim is false when the N-point functions are used as standard rulers. Constraints available from joint analysis of the two and three-point correlation functions can, in some cases, exceed those offered by the initial power spectrum. We provide a mathematical justification for this claim and demonstrate it using a large suite of N-body simulations. In particular, we show that for the z = 0 real-space matter field in the limit of vanishing shot-noise, taking modes up to kmax = 0.2 h Mpc−1, using the bispectrum alone offers a factor of 2 reduction in the variance on the cosmic distance scale relative to that available from the linear power spectrum.
We study two key issues militating against the use of the anisotropic three-point correlation function (3PCF) for cosmological parameter inference: difficulties with its computational estimation and high-dimensionality. We show how high-dimensionality may be reduced significantly by multipole decompositions of all angular dependence. This allows deriving the full expression for the multipole moments of the anisotropic 3PCF and its covariance matrix in a basis where the dimensionality reduces from nine to two at each multipole in the plane-parallel limit. We use 2D FFTLog formalism to show how the multipole moments with double momentum integrals over the product of bispectrum and two highly oscillating spherical Bessel functions and its covariance with double momentum integrals over the product of three galaxy power spectra and a combination of four highly oscillating spherical Bessel functions may be computed optimally.
We present the cosmological implications from final measurements of clustering using galaxies, quasars, and Lyα forests from the completed Sloan Digital Sky Survey (SDSS) lineage of experiments in large-scale structure. These experiments, composed of data from SDSS, SDSS-II, BOSS, and eBOSS, offer independent measurements of baryon acoustic oscillation (BAO) measurements of angular-diameter distances and Hubble distances relative to the sound horizon, rd, from eight different samples and six measurements of the growth rate parameter, fσ8, from redshift-space distortions (RSD). This composite sample is the most constraining of its kind and allows us to perform a comprehensive assessment of the cosmological model after two decades of dedicated spectroscopic observation. We show that the BAO data alone are able to rule out dark-energy-free models at more than eight standard deviations in an extension to the flat, ΛCDM model that allows for curvature. When combined with Planck Cosmic Microwave Background (CMB) measurements of temperature and polarization, under the same model, the BAO data provide nearly an order of magnitude improvement on curvature constraints relative to primary CMB constraints alone. Independent of distance measurements, the SDSS RSD data complement weak lensing measurements from the Dark Energy Survey (DES) in demonstrating a preference for a flat ΛCDM cosmological model when combined with Planck measurements. The combined BAO and RSD measurements indicate σ8=0.85±0.03, implying a growth rate that is consistent with predictions from Planck temperature and polarization data and with General Relativity. When combining the results of SDSS BAO and RSD, Planck, Pantheon Type Ia supernovae (SNe Ia), and DES weak lensing and clustering measurements, all multiple-parameter extensions remain consistent with a ΛCDM model. Regardless of cosmological model, the precision on each of the three parameters, ΩΛ, H0, and σ8, remains at roughly 1%, showing changes of less than 0.6% in the central values between models. In a model that allows for free curvature and a time-evolving equation of state for dark energy, the combined samples produce a constraint Ωk=−0.0022±0.0022. The dark energy constraints lead to w0=−0.909±0.081 and wa=−0.49−0.30+0.35, corresponding to an equation of state of wp=−1.018±0.032 at a pivot redshift zp=0.29 and a Dark Energy Task Force Figure of Merit of 94. The inverse distance ladder measurement under this model yields H0=68.18±0.79 km s−1 Mpc−1, remaining in tension with several direct determination methods; the BAO data allow Hubble constant estimates that are robust against the assumption of the cosmological model. In addition, the BAO data allow estimates of H0 that are independent of the CMB data, with similar central values and precision under a ΛCDM model. Our most constraining combination of data gives the upper limit on the sum of neutrino masses at ∑mν<0.115 eV (95% confidence). Finally, we consider the improvements in cosmology constraints over the last decade by comparing our results to a sample representative of the period 2000–2010. We compute the relative gain across the five dimensions spanned by w, Ωk, ∑mν, H0, and σ8 and find that the SDSS BAO and RSD data reduce the total posterior volume by a factor of 40 relative to the previous generation. Adding again the Planck, DES, and Pantheon SN Ia samples leads to an overall contraction in the five-dimensional posterior volume of 3 orders of magnitude.
We present a Fisher information study of the statistical impact of galaxy bias and selection effects on the estimation of key cosmological parameters from galaxy redshift surveys; in particular, the angular diameter distance, Hubble parameter, and linear growth rate at a given redshift, the cold dark matter density, and the tilt and running of the primordial power spectrum. The line-of-sight-dependent selection contributions we include here are known to exist in real galaxy samples. We determine the maximum wavenumber included in the analysis by requiring that the next-order corrections to the galaxy power spectrum or bispectrum, treated here at next-to-leading and leading order, respectively, produce shifts of ≲ 0.25σ on each of the six cosmological parameters. With the galaxy power spectrum alone, selection effects can deteriorate the constraints severely, especially on the linear growth rate. Adding the galaxy bispectrum helps break parameter degeneracies significantly. We find that a joint power spectrum-bispectrum analysis of a Euclid-like survey can still measure the linear growth rate to 10% precision after complete marginalization over selection bias. We also discuss systematic parameter shifts arising from ignoring selection effects and/or other bias parameters, and emphasize that it is necessary to either control selection effects at the percent level or marginalize over them. We obtain similar results for the Roman Space Telescope and HETDEX.
Creating accurate and low-noise covariance matrices represents a formidable challenge in modern-day cosmology. We present a formalism to compress arbitrary observables into a small number of bins by projection into a model-specific subspace that minimizes the prior-averaged log-likelihood error. The lower dimensionality leads to a dramatic reduction in covariance matrix noise, significantly reducing the number of mocks that need to be computed. Given a theory model, a set of priors, and a simple model of the covariance, our method works by using singular value decompositions to construct a basis for the observable that is close to Euclidean; by restricting to the first few basis vectors, we can capture almost all the constraining power in a lower-dimensional subspace. Unlike conventional approaches, the method can be tailored for specific analyses and captures nonlinearities that are not present in the Fisher matrix, ensuring that the full likelihood can be reproduced. The procedure is validated with full-shape analyses of power spectra from Baryon Oscillation Spectroscopic Survey (BOSS) DR12 mock catalogs, showing that the 96-bin power spectra can be replaced by 12 subspace coefficients without biasing the output cosmology; this allows for accurate parameter inference using only ∼100 mocks. Such decompositions facilitate accurate testing of power spectrum covariances; for the largest BOSS data chunk, we find the following: (a) analytic covariances provide accurate models (with or without trispectrum terms); and (b) using the sample covariance from the MultiDark-Patchy mocks incurs a ∼0.5σ shift in Ωm, unless the subspace projection is applied. The method is easily extended to higher order statistics; the ∼2000-bin bispectrum can be compressed into only ∼10 coefficients, allowing for accurate analyses using few mocks and without having to increase the bin sizes.
In the forthcoming large volume galaxy surveys, the use of higher order statistics will prove important to obtain complementary information to the usual two point statistics. In particular, one of those higher order statistical tools is the Three Point Correlation Function (3PCF) over discrete data points, whose lowest variance estimators count triangle configurations with vertices mixing data and random catalogues. A popular choice is to use large density random catalogues, which reduces the shot noise but leads to a computational cost of one or two orders of magnitude more than the pure data histogram calculation. In this paper, we explore ideas to time reduce the random sampling without using random catalogues. We focus on the isotropic 3PCF case over periodic boxes. In a first approach, based on Hamilton's construction of his famous two point estimator, we use an ad-hoc two point correlation piece for the mixed random-data histograms. A second approach relies on operators constructed from a geometrical viewpoint, using two sides and one angle to define the three dependencies of the isotropic 3PCF . We map the last result to the three triangle side basis either numerically or analytically, and show that the latter method performs best when applied to synthetic data. In addition, we elaborate on relaxing the no boundary condition, discuss other low variance n-point estimators and present useful 3PCF visualization schemes.
We present updates on the cosmology inference using the effective field theory (EFT) likelihood presented previously in Schmidt et al., 2018, Elsner et al., 2019 \cite{paperI,paperII}. Specifically, we add a cutoff to the initial conditions that serve as starting point for the matter forward model. We show that this cutoff, which was not employed in any previous related work, is important to regularize loop integrals that otherwise involve small-scale, non-perturbative modes. We then present results on the inferred value of the linear power spectrum normalization σ8 from rest-frame halo catalogs using both second- and third-order bias expansions, imposing uniform priors on all bias parameters. Due to the perfect bias-σ8 degeneracy at linear order, constraints on σ8 rely entirely on nonlinear information. The results show the expected convergence behavior when lowering the cutoff in wavenumber, L. When including modes up to k ⩽ Λ = 0.1 h Mpc⁻¹ in the second-order case, σ8 is recovered to within ≲ 6% for a range of halo masses and redshifts. The systematic bias shrinks to 4% or less for the third-order bias expansion on the same range of scales. Together with additional evidence we provide, this shows that the residual mismatch in σ8 can be attributed to higher-order bias contributions. We conclude that the EFT likelihood is able to infer unbiased cosmological constraints, within expected theoretical systematic errors, from physical biased tracers on quasilinear scales.
We propose a new, likelihood-free approach to inferring the primordial matter power spectrum and cosmological parameters from arbitrarily complex forward models of galaxy surveys where all relevant statistics can be determined from numerical simulations, i.e. black boxes. Our approach, which we call simulator expansion for likelihood-free inference (selfi), builds upon approximate Bayesian computation using a novel effective likelihood, and upon the linearization of black-box models around an expansion point. Consequently, we obtain simple ‘filter equations’ for an effective posterior of the primordial power spectrum, and a straightforward scheme for cosmological parameter inference. We demonstrate that the workload is computationally tractable, fixed a priori, and perfectly parallel. As a proof of concept, we apply our framework to a realistic synthetic galaxy survey, with a data model accounting for physical structure formation and incomplete and noisy galaxy observations. In doing so, we show that the use of non-linear numerical models allows the galaxy power spectrum to be safely fitted up to at least kmax = 0.5 h Mpc−1, outperforming state-of-the-art backward-modelling techniques by a factor of ∼5 in the number of modes used. The result is an unbiased inference of the primordial matter power spectrum across the entire range of scales considered, including a high-fidelity reconstruction of baryon acoustic oscillations. It translates into an unbiased and robust inference of cosmological parameters. Our results pave the path towards easy applications of likelihood-free simulation-based inference in cosmology. We have made our code pyselfi and our data products publicly available at http://pyselfi.florent-leclercq.eu.
Upcoming galaxy redshift surveys promise to significantly improve current limits on primordial non-Gaussianity (PNG) through measurements of 2- and 3-point correlation functions in Fourier space. However, realizing the full potential of this dataset is contingent upon having both accurate theoretical models and optimized analysis methods. Focusing on the local model of PNG, parameterized by , we perform a Monte-Carlo Markov Chain analysis to confront perturbation theory predictions of the halo power spectrum and bispectrum in real space against a suite of N-body simulations. We model the halo bispectrum at tree-level, including all contributions linear and quadratic in , and the halo power spectrum at 1-loop, including tree-level terms up to quadratic order in and all loops induced by local PNG linear in . Keeping the cosmological parameters fixed, we examine the effect of informative priors on the linear non-Gaussian bias parameter on the statistical inference of . An entirely agnostic, conservative analysis of the combined power spectrum and bispectrum, in which all parameters are marginalized over, can improve the constraint on by more than a factor of 5 relative to the power spectrum-only measurement. Imposing a strong prior on , or assuming bias relations for both and (motivated by a universal mass function assumption), improves the constraints further by a factor of few. In this case, however, we find a significant systematic shift in the inferred value of if the same range of wavenumber is used. Likewise, a Poisson noise assumption can lead to significant systematics, and it is thus essential to leave all the stochastic amplitudes free.
As we move towards future galaxy surveys, the three-point statistics will be increasingly leveraged to enhance the constraining power of the data on cosmological parameters. An essential part of the three-point function estimation is performing triplet counts of synthetic data points in random catalogues. Since triplet counting algorithms scale at best as with the number of particles and the random catalogues are typically at least 50 times denser than the data; this tends to be by far the most time-consuming part of the measurements. Here, we present a simple method of computing the necessary triplet counts involving uniform random distributions through simple one-dimensional integrals. The method speeds up the computation of the three-point function by orders of magnitude, eliminating the need for random catalogues, with the simultaneous pair and triplet counting of the data points alone being sufficient.
This paper focuses on two aspects of the statistics of cosmological observables that are important for the next stages of precision cosmology. First, we note that the theory of reduced angular N-point spectra has only been developed in detail up to the trispectrum case and in a fashion that makes it difficult to go beyond. To fill this gap, here we present a constructive approach that provides a systematic description of reduced angular N-point spectra and their covariance matrices, for arbitrary N. Second, we focus on the common practice in the literature on cosmological observables, which consists in simply discarding a part of the expression, namely, the terms containing fields evaluated at the observer position. We point out that this is not justified beyond linear order in perturbation theory, as these terms contribute to all the multipoles of the corresponding spectra and with a magnitude that is of the same order as the rest of the nonlinear corrections. We consider the possibility that the reason for neglecting these terms is a conceptual discomfort when using ensemble averages, which originates in an apparent tension between the ergodic hypothesis and the privileged position of the observer on the light-cone. We clarify this subtle issue by performing a careful derivation of the relation between the theoretical statistical predictions and the observational estimators for all N. We conclude that there is no inconsistency whatsoever in ensemble-averaging fields at and near the observer position, thus clearing the way for consistent and robust high-precision calculations.
We forecast the benefits induced by adding the bispectrum anisotropic signal to the standard, two- and three-point, clustering statistics analysis. In particular, we forecast cosmological parameter constraints including the bispectrum higher multipoles terms together with the galaxy power spectrum (monopole plus quadrupole) and isotropic bispectrum (monopole) data vectors. To do so, an analytical covariance matrix model is presented. This template is carefully calibrated on well-known terms of a numerical covariance matrix estimated from a set of simulated galaxy catalogues. After testing the calibration using the power spectrum and isotropic bispectrum measurements from the same set of simulations, we extend the covariance modelling to the galaxy bispectrum higher multipoles. Using this covariance matrix we proceed to perform cosmological parameter inference using a suitably generated mock data vector. Including the bispectrum mutipoles up to the hexadecapole, yields 1-D 68% credible regions for the set of parameters (b1,b2,f,σ8,fNL,α⊥, α∥) tighter by a factor of 30% on average for kmax=0.09 h/Mpc, significantly reducing at the same time the degeneracies present in the posterior distribution.
The precision of the cosmological data allows us to accurately approximate the predictions for cosmological observables by Taylor expanding up to a low order the dependence on the cosmological parameters around a reference cosmology. By applying this observation to the redshift-space one-loop galaxy power spectrum of the Effective Field Theory of Large-Scale Structure, we analyze the BOSS DR12 data by scanning over all the parameters of ΛCDM cosmology with massive neutrinos. We impose several sets of priors, the widest of which is just a Big Bang Nucleosynthesis prior on the current fractional energy density of baryons, Ωb h², and a bound on the sum of neutrino masses to be less than 0.9 eV. In this case we measure the primordial amplitude of the power spectrum, As, the abundance of matter, Ωm, the Hubble parameter, H0, and the tilt of the primordial power spectrum, ns, to about 19%, 5.7%, 2.2% and 7.3% respectively, obtaining ln (10¹⁰As)=2.91± 0.19, Ωm=0.314± 0.018, H0=68.7± 1.5 km/(s Mpc) and ns=0.979± 0.071 at 68% confidence level. A public code is released with this preprint.
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.
We present cosmological constraints from a joint analysis of the pre- and post-reconstruction galaxy power spectrum multipoles from the final data release of the Baryon Oscillation Spectroscopic Survey (BOSS). Geometric constraints are obtained from the positions of BAO peaks in reconstructed spectra, which are analyzed in combination with the unreconstructed spectra in a full-shape (FS) likelihood using a joint covariance matrix, giving stronger parameter constraints than FS-only or BAO-only analyses. We introduce a new method for obtaining constraints from reconstructed spectra based on a correlated theoretical error, which is shown to be simple, robust, and applicable to any flavor of density-field reconstruction. Assuming ΛCDM with massive neutrinos, we analyze clustering data from two redshift bins zeff=0.38,0.61 and obtain 1.6% constraints on the Hubble constant H0, using only a single prior on the current baryon density ωb from Big Bang Nucleosynthesis (BBN) and no knowledge of the power spectrum slope ns. This gives H0 = 68.6±1.1 km s⁻¹Mpc⁻¹, with the inclusion of BAO data sharpening the measurement by 40%, representing one of the strongest current constraints on H0 independent of cosmic microwave background data, comparable with recent constraints using BAO data in combination with other data-sets. Restricting to the best-fit slope ns from Planck (but without additional priors on the spectral shape), we obtain a 1% H0 measurement of 67.8± 0.7 km s⁻¹Mpc⁻¹. Finally, we find strong constraints on the cosmological parameters from a joint analysis of the FS, BAO, and Planck data. This sets new bounds on the sum of neutrino masses ∑ mν < 0.14 eV (at 95% confidence) and the effective number of relativistic degrees of freedom Neff = 2.90+0.15−0.16, though contours are not appreciably narrowed by the inclusion of BAO data.
The Effective Field Theory of Large-Scale Structure is a formalism that allows us to predict the clustering of Cosmological Large-Scale Structure in the mildly non-linear regime in an accurate and reliable way. After validating our technique against several sets of numerical simulations, we perform the analysis for the cosmological parameters of the DR12 BOSS data. We assume ΛCDM, a fixed value of the baryon/dark-matter ratio, Ωb/Ωc, and of the tilt of the primordial power spectrum, ns, and no significant input from numerical simulations. By using the one-loop power spectrum multipoles, we measure the primordial amplitude of the power spectrum, As, the abundance of matter, Ωm, and the Hubble parameter, H0, {to about 13%, 3.2% and 3.2% respectively, obtaining ln (10¹⁰As)=2.72± 0.13, 0Ωm=0.309± 0.01, H0=68.5± 2.2 km/(s Mpc) at 68% confidence level. If we then add a CMB prior on the sound horizon, the error bar on H0 is reduced to 1.6%.} These results are a substantial qualitative and quantitative improvement with respect to former analyses, and suggest that the EFTofLSS is a powerful instrument to extract cosmological information from Large-Scale Structure.
We derive, using functional methods and the bias expansion, the conditional likelihood for observing a specific tracer field given an underlying matter field. This likelihood is necessary for Bayesian-inference methods. If we neglect all stochastic terms apart from the ones appearing in the auto two-point function of tracers, we recover the result of Schmidt et al., 2018 [1]. We then rigorously derive the corrections to this result, such as those coming from a non-Gaussian stochasticity (which include the stochastic corrections to the tracer bispectrum) and higher-derivative terms. We discuss how these corrections can affect current applications of Bayesian inference. We comment on possible extensions to our result, with a particular eye towards primordial non-Gaussianity. This work puts on solid theoretical grounds the effective-field-theory-(EFT-)based approach to Bayesian forward modeling.
We present a Markov-Chain Monte-Carlo (MCMC) forecast for the precision of neutrino mass and cosmological parameter measurements with a Euclid-like galaxy clustering survey. We use a complete perturbation theory model for the galaxy one-loop power spectrum and tree-level bispectrum, which includes bias, redshift space distortions, IR resummation for baryon acoustic oscillations and UV counterterms. The latter encapsulate various effects of short-scale dynamics which cannot be modeled within perturbation theory. Our MCMC procedure consistently computes the non-linear power spectra and bispectra as we scan over different cosmologies. The second ingredient of our approach is the theoretical error covariance which captures uncertainties due to higher-order non-linear corrections omitted in our model. Having specified characteristics of a Euclid-like spectroscopic survey, we generate and fit mock galaxy power spectrum and bispectrum likelihoods. Our results suggest that even under very agnostic assumptions about non-linearities and short-scale physics a future Euclid-like survey will be able to measure the sum of neutrino masses with a standard deviation of 28 meV . When combined with the Planck cosmic microwave background likelihood, this uncertainty decreases to 13 meV . Over-optimistically reducing the theoretical error on the bispectrum down to the two-loop level marginally tightens this bound to 11 meV . Moreover, we show that the future large-scale structure (LSS) spectroscopic data will greatly improve constraints on the other cosmological parameters, e.g. reaching a percent (per mille) error on the Hubble constant with LSS alone (LSS + Planck).
We present an efficient implementation of Wiener filtering of real-space linear field and optimal quadratic estimator of its power spectrum Band-powers. We first recast the field reconstruction into an optimization problem, which we solve using quasi-Newton optimization. We then recast the power spectrum estimation into the field marginalization problem, from which we obtain an expression that depends on the field reconstruction solution and a determinant term. We develop a novel simulation based method for the latter. We extend the simulations formalism to provide the covariance matrix for the power spectrum. We develop a flexible framework that can be used on a variety of cosmological fields and present results for a variety of test cases, using simulated examples of projected density fields, projected shear maps from galaxy lensing, and observed Cosmic Microwave Background (CMB) temperature anisotropies, with a wide range of map incompleteness and variable noise. For smaller cases where direct numerical inversion is possible, we show that our solution matches that created by direct Wiener Filtering at a fraction of the overall computation cost. Even more significant reduction of computational is achieved by this implementation of optimal quadratic estimator due to the fast evaluation of the Hessian matrix. This technique allows for accurate map and power spectrum reconstruction with complex masks and nontrivial noise properties.
In this paper, we test the perturbative halo bias model at the field level. The advantage of this approach is that any analysis can be done without sample variance if the same initial conditions are used in simulations and perturbation theory calculations. We write the bias expansion in terms of modified bias operators in Eulerian space, designed such that the large bulk flows are automatically resummed and not treated perturbatively. Using these operators, the bias model accurately matches the Eulerian density of halos in N-body simulations. The mean-square model error is close to the Poisson shot noise for a wide range of halo masses, and it is rather scale independent, with scale-dependent corrections becoming relevant at the nonlinear scale. In contrast, for the linear bias, the mean-square model error can be higher than the Poisson prediction by factors of up to a few on large scales, and it becomes scale dependent already in the linear regime. We show that by weighting simulated halos by their mass, the mean-square error of the model can be further reduced by up to an order of magnitude, or by a factor of 2 when including 60% mass scatter. We also test the standard Eulerian bias model using the nonlinear matter field measured from simulations and show that it leads to a larger and more scale-dependent model error than the bias expansion based on perturbation theory. These results may be of particular relevance for cosmological inference methods that use a likelihood of the biased tracer at the field level or for initial condition and baryon acoustic oscillation reconstruction that requires a precise estimate of the large-scale potential from the biased tracer density.
We present a novel method to compress galaxy clustering three-point statistics and apply it to redshift space galaxy bispectrum monopole measurements from BOSS DR12 CMASS data considering a k-space range of 0.03−0.12h/Mpc. The method consists in binning together bispectra evaluated at sets of wave-numbers forming closed triangles with similar geometrical properties: the area, the cosine of the largest angle and the ratio between the cosines of the remaining two angles. This enables us to increase the number of bispectrum measurements for example by a factor of 23 over the standard binning (from 116 to 2734 triangles used), which is otherwise limited by the number of mock catalogues available to estimate the covariance matrix needed to derive parameter constraints. The 68% credible intervals for the inferred parameters (b1,b2,f,σ8) are thus reduced by (−39%,−49%,−29%,−22%), respectively. We find very good agreement with the posteriors recently obtained by alternative maximal compression methods. This new method does not require the a-priori computation of the data-vector covariance matrix and has the potential to be directly applicable to other three-point statistics (e.g. glaxy clustering, weak gravitational lensing, 21 cm emission line) measured from future surveys such as DESI, Euclid, PFS and SKA.
The large scale structure (LSS) of the universe is generated by the linear density gaussian modes, which are evolved into the observed nonlinear LSS. Given a set of data the posterior surface of the modes is convex in the linear regime, leading to a unique global maximum (MAP), but this is no longer guaranteed in the nonlinear regime. In this paper we investigate the nature of posterior surface using the recently developed MAP reconstruction method, with a simplified but realistic N-body simulation as the forward model. The reconstruction method uses optimization with analytic gradients from back-propagation through the simulation. For low noise cases we recover the initial conditions well into the nonlinear regime (k~ 1 h/Mpc) nearly perfectly. We show that the large scale modes can be recovered more precisely than the linear expectation, which we argue is a consequence of nonlinear mode coupling. For noise levels achievable with current and planned LSS surveys the reconstruction cannot recover very small scales due to noise. We see some evidence of non-convexity, specially for smaller scales where the non-injective nature of the mappings: several very different initial conditions leading to the same near perfect final data reconstruction. We investigate the nature of these phenomena further using a 1-d toy gravity model, where many well separated local maximas are found to have identical data likelihood but differ in the prior. We also show that in 1-d the prior favors some solutions over the true solution, though no clear evidence of these in 3-d. Our main conclusion is that on very small scales and for a very low noise the posterior surface is multi-modal and the global maximum may be unreachable with standard methods, while for realistic noise levels in the context of the current and next generation LSS surveys MAP optimization method is likely to be nearly optimal.
We forecast constraints on primordial non-Gaussianity (PNG) and bias parameters from measurements of galaxy power spectrum and bispectrum in future radio continuum and optical surveys. In the galaxy bispectrum, we consider a comprehensive list of effects, including the bias expansion for non-Gaussian initial conditions up to second order, redshift space distortions, redshift uncertainties and theoretical errors. These effects are all combined in a single PNG forecast for the first time. Moreover, we improve the bispectrum modelling over previous forecasts, by accounting for trispectrum contributions. All effects have an impact on final predicted bounds, which varies with the type of survey. We find that the bispectrum can lead to improvements up to a factor over bounds based on the power spectrum alone, leading to significantly better constraints for local-type PNG, with respect to current limits from \textit{Planck}. Future radio and photometric surveys could obtain a measurement error of . In the case of equilateral PNG, galaxy bispectrum can improve upon present bounds only if significant improvements in the redshift determinations of future, large volume, photometric or radio surveys could be achieved. For orthogonal non-Gaussianity, expected constraints are generally comparable to current ones.
We present the first high significance detection () of the Baryon Acoustic Oscillations (BAO) feature in the galaxy bispectrum of the twelfth data release (DR12) of the Baryon Oscillation Spectroscopic Survey (BOSS) CMASS sample (). We measured the scale dilation parameter, , using the power spectrum, bispectrum, and both simultaneously for DR12, plus 2048 PATCHY mocks in the North and South Galactic Caps (NGC and SGC, respectively), and the volume weighted averages of those two samples (N+SGC). The fitting to the mocks validated our analysis pipeline, yielding values consistent with the mock cosmology. By fitting to the power spectrum and bispectrum separately, we tested the robustness of our results, finding consistent values from the NGC, SGC and N+SGC in all cases. We found Mpc, Mpc, and Mpc from the N+SGC power spectrum, bispectrum and simultaneous fitting, respectively.
We present the first application of the angle-dependent 3-Point Correlation Function (3PCF) to the density fields magnetohydrodynamic (MHD) turbulence simulations intended to model interstellar (ISM) turbulence. Previous work has demonstrated that the angle-averaged bispectrum, the 3PCF's Fourier-space analog, is sensitive to the sonic and Alfv\'enic Mach numbers of turbulence. Here we show that introducing angular information via multipole moments with respect to the triangle opening angle offers considerable additional discriminatory power on these parameters. We exploit a fast, order ( the number of grid cells used for a Fourier Transform) 3PCF algorithm to study a suite of MHD turbulence simulations with 10 different combinations of sonic and Alfv\'enic Mach numbers over a range from sub to super-sonic and sub to super-Alfv\'{e}nic. The 3PCF algorithm's speed for the first time enables full quantification of the time-variation of our signal: we study 9 timeslices for each condition, demonstrating that the 3PCF is sufficiently time-stable to be used as an ISM diagnostic. In future, applying this framework to 3-D dust maps will enable better treatment of dust as a cosmological foreground as well as reveal conditions in the ISM that shape star formation.
We present an algorithm enabling computation of the anisotropic redshift-space galaxy 3-point correlation function (3PCF) scaling as , with N the number of galaxies. Our previous work showed how to compute the isotropic 3PCF with this scaling by expanding the radially-binned density field around each galaxy in the survey into spherical harmonics and combining these coefficients to form multipole moments. The scaling occurred because this approach never explicitly required the relative angle between a galaxy pair about the primary galaxy. Here we generalize this work, demonstrating that in the presence of azimuthally-symmetric anisotropy produced by redshift-space distortions (RSD) the 3PCF can be described by two triangle side lengths, two independent total angular momenta, and a spin. This basis for the anisotropic 3PCF allows its computation with negligible additional work over the isotropic 3PCF. We also present the covariance matrix of the anisotropic 3PCF measured in this basis. Our algorithm tracks the full 5-D redshift-space 3PCF, uses an accurate line of sight to each triplet, is exact in angle, and easily handles edge correction. It will enable use of the anisotropic large-scale 3PCF as a probe of RSD in current and upcoming large-scale redshift surveys.
We explore two methods of compressing the redshift space galaxy power spectrum and bispectrum with respect to a chosen set of cosmological parameters. Both methods involve reducing the dimension of the original data-vector ( e.g. 1000 elements ) to the number of cosmological parameters considered ( e.g. seven ) using the Karhunen-Lo\`eve algorithm. In the first case, we run MCMC sampling on the compressed data-vector in order to recover the one-dimensional (1D) and two-dimensional (2D) posterior distributions. The second option, approximately 2000 times faster, works by orthogonalising the parameter space through diagonalisation of the Fisher information matrix before the compression, obtaining the posterior distributions without the need of MCMC sampling. Using these methods for future spectroscopic redshift surveys like DESI, EUCLID and PFS would drastically reduce the number of simulations needed to compute accurate covariance matrices with minimal loss of constraining power. We consider a redshift bin of a DESI-like experiment. Using the power spectrum combined with the bispectrum as a data-vector, both compression methods on average recover the 68% credible regions to within 0.5% and 3.2% of those resulting from standard MCMC sampling respectively. These confidence intervals are also smaller than the ones obtained using only the power spectrum by (81%, 80%, 82%) respectively for the bias parameter b_1, the growth rate f and the scalar amplitude parameter A_s.
We present a new method for calculating loops in cosmological perturbation theory. This method is based on approximating a CDM-like cosmology as a finite sum of complex power-law universes. The decomposition is naturally achieved using an FFTLog algorithm. For power-law cosmologies, all loop integrals are formally equivalent to loop integrals of massless quantum field theory. These integrals have analytic solutions in terms of generalized hypergeometric functions. We provide explicit formulae for the one-loop and the two-loop power spectrum and the one-loop bispectrum. A chief advantage of our approach is that the difficult part of the calculation is cosmology independent, need be done only once, and can be recycled for any relevant predictions. Evaluation of standard loop diagrams then boils down to a simple matrix multiplication. We demonstrate the promise of this method for applications to higher multiplicity/loop correlation functions.
One of the main unsolved problems of cosmology is how to maximize the extraction of information from nonlinear data. If the data are nonlinear the usual approach is to employ a sequence of statistics (N-point statistics, counting statistics of clusters, density peaks or voids etc.), along with the corresponding covariance matrices. However, this approach is computationally prohibitive and has not been shown to be exhaustive in terms of information content. Here we instead develop a Bayesian approach, expanding the likelihood around the maximum posterior of linear modes, which we solve for using optimization methods. By integrating out the modes using perturbative expansion of the likelihood we construct an initial power spectrum estimator, which for a fixed forward model contains all the cosmological information if the initial modes are gaussian distributed. We develop a method to construct the window and covariance matrix such that the estimator is explicitly unbiased and nearly optimal. We then generalize the method to include the forward model parameters, including cosmological and nuisance parameters, and primordial non-gaussianity. We apply the method in the simplified context of nonlinear structure formation, using either simplified 2-LPT dynamics or N-body simulations as the nonlinear mapping between linear and nonlinear density, and 2-LPT dynamics in the optimization steps used to reconstruct the initial density modes. We demonstrate that the method gives an unbiased estimator of the initial power spectrum, providing among other a near optimal reconstruction of linear baryonic acoustic oscillations.
We investigate the potential sources of theoretical systematics in the anisotropic Baryon Acoustic Oscillation (BAO) distance scale measurements from the clustering of galaxies in configuration space using the final Data Release (DR12) of the Baryon Oscillation Spectroscopic Survey (BOSS). We perform a detailed study of the impact on BAO measurements from choices in the methodology such as fiducial cosmology, clustering estimators, random catalogues, fitting templates, and covariance matrices. The theoretical systematic uncertainties in BAO parameters are found to be 0.002 in in the isotropic dilation and 0.003 in in the quadrupolar dilation . We also present BAO-only distance scale constraints from the anisotropic analysis of the correlation function. Our constraints on the angular diameter distance and the Hubble parameter H(z) including both statistical and theoretical systematic uncertainties are 1.5% and 2.8% at , 1.4% and 2.4% at , and 1.7% and 2.6% at . This paper is part of a set that analyses the final galaxy clustering dataset from BOSS. The measurements and likelihoods presented here are cross-checked with others BAO analysis in Alam et. al. 2016. The systematic error budget concerning the methodology on post-reconstruction BAO analysis presented here is used in Alam et. al. 2016 to produce the final cosmological constraints from BOSS.
The redshift-space bispectrum (three point statistics) of galaxies depends on the expansion rate, the growth rate, and geometry of the Universe, and hence can be used to measure key cosmological parameters. In a homogeneous Universe the bispectrum is a function of five variables and unlike its two point statistics counterpart -- the power spectrum, which is a function of only two variables -- is difficult to analyse unless the information is somehow reduced. The most commonly considered reduction schemes rely on computing angular integrals over possible orientations of the bispectrum triangle, thus reducing it to sets of function of only three variables describing the triangle shape. We use Fisher information formalism to study the information loss associated with this angular integration. Without any reduction, the bispectrum alone can deliver constraints on the growth rate parameter f that are better by a factor of 2.5 compared to the power spectrum, for a sample of luminous red galaxies expected from near future galaxy surveys at a redshift of . At lower redshifts the improvement could be up to a factor of 3. We find that most of the information is in the azimuthal averages of the first three even multipoles. This suggests that the bispectrum of every configuration can be reduced to just three numbers (instead of a 2D function) without significant loss of cosmologically relevant information.
We present estimators for the small-scale power spectrum and bispectrum in cosmological simulations. In combination with traditional methods, these allow spectra to be efficiently computed across a vast range of scales, requiring orders of magnitude less computation time than Fast Fourier Transform based approaches alone. These methods are applicable to any tracer; simulation particles, halos or galaxies, and take advantage of the simple geometry of the box and periodicity to remove almost all dependence on large random particle catalogs. By working in configuration-space, both power spectra and bispectra can be computed via a weighted sum of particle pairs up to some radius, which can be reduced at larger k, leading to algorithms with decreasing complexity on small scales. These do not suffer from aliasing or shot-noise, allowing spectra to be computed to arbitrarily large wavenumbers. The estimators are rigorously derived and tested against simulations, and their covariances discussed. The accompanying code, HIPSTER, has been publicly released, incorporating these algorithms. Such estimators will be of great use in the analysis of large sets of high-resolution simulations.
We establish a practical method for the joint analysis of anisotropic galaxy two- and three-point correlation functions (2PCF and 3PCF) on the basis of the decomposition formalism of the 3PCF using tri-polar spherical harmonics. We perform such an analysis with MultiDark Patchy mock catalogues to demonstrate and understand the benefit of the anisotropic 3PCF. We focus on scales above 80 h−1 Mpc, and use information from the shape and the baryon acoustic oscillation (BAO) signals of the 2PCF and 3PCF. We also apply density field reconstruction to increase the signal-noise ratio of BAO in the 2PCF measurement, but not in the 3PCF measurement. In particular, we study in detail the constraints on the angular diameter distance and the Hubble parameter. We build a model of the bispectrum or 3PCF that includes the nonlinear damping of the BAO signal in redshift space. We carefully account for various uncertainties in our analysis including theoretical models of the 3PCF, window function corrections, biases in estimated parameters from the fiducial values, the number of mock realizations to estimate the covariance matrix, and bin size. The joint analysis of the 2PCF and 3PCF monopole and quadrupole components shows a and improvement in Hubble parameter constraints before and after reconstruction of the 2PCF measurements, respectively, compared to the 2PCF analysis alone. This study clearly shows that the anisotropic 3PCF increases cosmological information from galaxy surveys and encourages further development of the modeling of the 3PCF on smaller scales than we consider.
We present configuration-space estimators for the auto- and cross-covariance of two- and three-point correlation functions (2PCF and 3PCF) in general survey geometries. These are derived in the Gaussian limit (setting higher order correlation functions to zero), but for arbitrary non-linear 2PCFs (which may be estimated from the survey itself), with a shot-noise rescaling parameter included to capture non-Gaussianity. We generalize previous approaches to include Legendre moments via a geometry-correction function calibrated from measured pair and triple counts. Making use of importance sampling and random particle catalogues, we can estimate model covariances in fractions of the time required to do so with mocks, obtaining estimates with negligible sampling noise in ∼10 (∼100) CPU-hours for the 2PCF (3PCF) autocovariance. We compare results to sample covariances from a suite of BOSS DR12 mocks and find the matrices to be in good agreement, assuming a shot-noise rescaling parameter of 1.03 (1.20) for the 2PCF (3PCF). To obtain strongest constraints on cosmological parameters, we must use multiple statistics in concert; having robust methods to measure their covariances at low computational cost is thus of great relevance to upcoming surveys.
We present a high-fidelity realization of the cosmological N-body simulation from the Schneider et al. code comparison project. The simulation was performed with our AbacusN-body code, which offers high-force accuracy, high performance, and minimal particle integration errors. The simulation consists of 20483 particles in a box for a particle mass of with spline softening. Abacus executed 1052 global time-steps to z = 0 in 107 h on one dual-Xeon, dual-GPU node, for a mean rate of 23 million particles per second per step. We find Abacus is in good agreement with Ramses and Pkdgrav3 and less so with Gadget3. We validate our choice of time-step by halving the step size and find sub-percent differences in the power spectrum and 2PCF at nearly all measured scales, with errors at . On large scales, Abacus reproduces linear theory better than 0.01 per cent. Simulation snapshots are available at http://nbody.rc.fas.harvard.edu/public/S2016.
We present a configuration-space model of the large-scale galaxy 3-point correlation function (3PCF) based on leading-order perturbation theory and including redshift-space distortions (RSD). This model should be useful in extracting distance-scale information from the 3PCF via the baryon acoustic oscillation method. We include the first redshift-space treatment of biasing by the baryon-dark matter relative velocity. Overall, on large scales the effect of RSD is primarily a renormalization of the 3PCF that is roughly independent of both physical scale and triangle opening angle; for our adopted Ωm and bias values, the rescaling is a factor of ∼1.8. We also present an efficient scheme for computing 3PCF predictions from our model, important for allowing fast exploration of the space of cosmological parameters in future analyses.
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.
We present a new class of estimators for computing small-scale power spectra and bispectra in configuration space via weighted pair and triple counts, with no explicit use of Fourier transforms. Particle counts are truncated at via a continuous window function, which has negligible effect on the measured power spectrum multipoles at small scales. This gives a power spectrum algorithm with complexity (or for the bispectrum), measuring N galaxies with number density n. Our estimators are corrected for the survey geometry and have neither self-count contributions nor discretization artefacts, making them ideal for high-k analysis. Unlike conventional Fourier-transform-based approaches, our algorithm becomes more efficient on small scales (since a smaller R0 may be used), thus we may efficiently estimate spectra across k-space by coupling this method with standard techniques. We demonstrate the utility of the publicly available power spectrum algorithm by applying it to BOSS DR12 simulations to compute the high-k power spectrum and its covariance. In addition, we derive a theoretical rescaled-Gaussian covariance matrix, which incorporates the survey geometry and is found to be in good agreement with that from mocks. Computing configuration- and Fourier-space statistics in the same manner allows us to consider joint analyses, which can place stronger bounds on cosmological parameters; to this end we also discuss the cross-covariance between the two-point correlation function and the small-scale power spectrum.
We explore data analysis techniques for signatures from heavy particle production during inflation. Heavy particules can be produced by time dependent masses and couplings, which are ubiquitous in string theory. These localized excitations induce curvature perturbations with nonzero correlation functions at all orders. In particular, Flauger et al. [J. Cosmol. Astropart. Phys. 10 (2017) 058] have shown that the signal to noise as a function of the order N of the correlation function can peak for N of order O(1) to O(100) for an interesting space of models. As previous non-Gaussianity analyses have focused on N={3,4}, in principle this provides an unexplored data analysis window with new discovery potential. We derive estimators for arbitrary N-point functions in this model and discuss their properties and covariances. To lowest order, the heavy particle production phenomenology reduces to a classical Poisson process, which can be implemented as a search for spherically symmetric profiles in the curvature perturbations. We explicitly show how to recover this result from the N-point functions and their estimators. Our focus in this paper is on method development, but we provide an initial data analysis using WMAP data, which illustrates the particularities of higher N-point function searches.
The two-point correlation function of the galaxy distribution is a key cosmological observable that allows us to constrain the dynamical and geometrical state of our Universe. To measure the correlation function we need to know both the galaxy positions and the expected galaxy density field. The expected field is commonly specified using a Monte-Carlo sampling of the volume covered by the survey and, to minimize additional sampling errors, this random catalog has to be much larger than the data catalog. Correlation function estimators compare data–data pair counts to data–random and random–random pair counts, where random–random pairs usually dominate the computational cost. Future redshift surveys will deliver spectroscopic catalogs of tens of millions of galaxies. Given the large number of random objects required to guarantee sub-percent accuracy, it is of paramount importance to improve the efficiency of the algorithm without degrading its precision. We show both analytically and numerically that splitting the random catalog into a number of subcatalogs of the same size as the data catalog when calculating random–random pairs and excluding pairs across different subcatalogs provides the optimal error at fixed computational cost. For a random catalog fifty times larger than the data catalog, this reduces the computation time by a factor of more than ten without affecting estimator variance or bias.
Nonlinearities in the gravitational evolution, galaxy bias, and redshift-space distortion drive the observed galaxy density fields away from the initial near-Gaussian states. Exploiting such a non-Gaussian galaxy density field requires measuring higher-order correlation functions, or, its Fourier counterpart, polyspectra. Here, we present an efficient parallel algorithm for estimating higher-order polyspectra. Based upon the Scoccimarro estimator, the estimator avoids direct sampling of polygons using the fast Fourier transform, and the parallelization overcomes the large memory requirement of the original estimator. In particular, we design the memory layout to minimize the inter-CPU communications, which excels in the code performance.
We present an algorithm for the fast computation of the general N -point spatial correlation functions of any discrete point set embedded within an Euclidean space of . Utilizing the concepts of kd-trees and graph databases, we describe how to count all possible N -tuples in binned configurations within a given length scale, e.g., all pairs of points or all triplets of points with side lengths < r MAX . Through benchmarking, we show the computational advantage of our new graph-based algorithm over more traditional methods. We show measurements of the three-point correlation function up to scales of ∼200 Mpc (beyond the baryon acoustic oscillation scale in physical units) using current Sloan Digital Sky Survey (SDSS) data. Finally, we present a preliminary exploration of the small-scale four-point correlation function of 568,776 SDSS Constant (stellar) Mass (CMASS) galaxies in the northern Galactic cap over the redshift range of 0.43 < z < 0.7. We present the publicly available code GRAMSCI (GRAph Made Statistics for Cosmological Information; bitbucket.org/csabiu/gramsci ), under a Gnu is Not Unix (GNU) General Public License.
To fully extract cosmological information from nonlinear galaxy distribution in redshift space, it is essential to include higher-order statistics beyond the two-point correlation function. In this paper, we propose a new decomposition formalism for computing the anisotropic bispectrum in redshift space and for measuring it from galaxy samples. Our formalism uses tri-polar spherical harmonic decomposition with zero total angular momentum to compress the 3D modes distribution in the redshift-space bispectrum. This approach preserves three fundamental properties of the Universe: statistical homogeneity, isotropy, and parity-symmetry, allowing us to efficiently separate the anisotropic signal induced by redshift-space distortions (RSDs) and the Alcock-Paczy\'{n}ski (AP) effect from the isotropic bispectrum. The relevant expansion coefficients in terms of the anisotropic signal are reduced to one multipole index L, and the modes are induced only by the RSD or AP effects. Our formalism has two advantages: (1) we can make use of Fast Fourier Transforms (FFTs) to measure the bispectrum; (2) it gives a simple expression to correct for the survey geometry, i.e., the survey window function. As a demonstration, we measure the decomposed bispectrum from the Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12, and, for the first time, present a detection of the anisotropic bispectrum in the L=2 mode.
The significance of the present IAU symposium, "The Large Scale Structure of the Universe", fortunately requires no elaboration by the editors. The quality of the wide range of observational and theoretical astrophysics contained in this volume speaks for itself. The published version of the proceedings contains all the contributions presented at the symposium with the exception of the introductory lecture by V. A. Ambartsumian. Contributed papers, short contributions and discussions have been included according to the recommendations of the IAU. Many people contributed to the success of the symposium. First of all, thanks are due to the USSR Academy of Sciences and to the Estonian Academy of Sciences for sponsoring this symposium in Tallinn. The efforts of Academician K. Rebane, President of the Estonian Academy of Sciences, are particularly appreciated. The astronomical hosts of the symposium were the members of the W. Struve Astrophysical Observatory of Tartu who made outstanding efforts to lavish participants with Estonian hospitality which was greatly appreciated and enjoyed by them and their guests. The members of the Scientific and Local Organising Committees are listed below and we thank all of them for their contributions which were central to the success of the symposium. In addition are listed members of the Technical Organising Committee who were responsible for all details of the organisation and whose vigilance ensured that all aspects of the symposium ran smoothly and efficiently. Their contributions are all gratefully acknowledged.
We propose an efficient way to test rotational invariance in the cosmological perturbations by use of galaxy correlation functions. In symmetry-breaking cases, the galaxy power spectrum can have extra angular dependence in addition to the usual one due to the redshift-space distortion, k^·n^. We confirm that, via the decomposition into not the usual Legendre basis Lℓ(k^·n^) but the bipolar spherical harmonic one {Yℓ(k^)⊗Yℓ′(n^)}LM, the symmetry-breaking signal can be completely distinguished from the usual isotropic one since the former yields nonvanishing L≥1 modes but the latter is confined to the L=0 one. As a demonstration, we analyze the signatures due to primordial-origin symmetry breakings such as the well-known quadrupolar-type and dipolar-type power asymmetries and find nonzero L=2 and 1 modes, respectively. Fisher matrix forecasts of their constraints indicate that the Planck-level sensitivity could be achieved by the SDSS or BOSS-CMASS data, and an order-of-magnitude improvement is expected in a near future survey as PFS or Euclid by virtue of an increase in accessible Fourier mode. Our methodology is model-independent and hence applicable to the searches for various types of statistically anisotropic fluctuations.
In many astrophysical settings covariance matrices of large datasets have to be determined empirically from a finite number of mock realisations. The resulting noise degrades inference and precludes it completely if there are fewer realisations than data points. This work applies a recently proposed non-linear shrinkage estimator of covariance to a realistic example from large-scale structure cosmology. After optimising its performance for the usage in likelihood expressions, the shrinkage estimator yields subdominant bias and variance comparable to that of the standard estimator with a factor less realisations. This is achieved without any prior information on the properties of the data or the structure of the covariance matrix, at negligible computational cost.