Large-scale Structure Surveys Research Articles

Forecast constraints on the baryonic feedback effect from the future kinetic Sunyaev-Zel'dovich effect detection

The baryonic feedback effect is an important systematic error in the weak lensing (WL) analysis. It contributes partly to the S 8 tension in the literature. With the next generations of large scale structure (LSS) and CMB experiments, the high signal-to-noise kinetic Sunyaev-Zel'dovich (kSZ) effect detection can tightly constrain the baryon distribution in and around dark matter halos, and quantify the baryonic effect in the weak lensing statistics. In this work, we apply the Fisher matrix technique to predict the future kSZ constraints on 3 kSZ-sensitive Baryon Correction Model (BCM) parameters. Our calculations show that, in combination with next generation LSS surveys, the 3rd generation CMB experiments such as AdvACT and Simon Observatory can constrain the matter power spectrum damping S(k) to the precision of σ S (k) < 0.8%√(37.8 Gpc3h-3-3/V) at k ≲ 10h/Mpc, where V is the overlapped survey volume between the future LSS and CMB surveys. For the 4th generation CMB surveys such as CMB-S4 and CMB-HD, the constraint will be enhanced to σ S (k) < 0.4%√(37.8 Gpc3h-3/V). If extra-observations, e.g. X-ray detection and thermal SZ observation, can effectively fix the gas density profile slope parameter δ, the constraint on S(k) will be further boosted to σ S (k) < 0.3%√(37.8 Gpc3 h -3/V) and σ S (k) < 0.1%√(37.8 Gpc3 h -3/V) for the 3rd and 4th generation CMB surveys.

Sensitivity of Cosmological Parameter Estimation to Nonlinear Prescription from Cosmic Shear

Several ongoing and upcoming large-scale structure surveys aim to explore the nonlinear regime of structure formation with high precision. Making reliable cosmological inferences from these observations necessitates precise theoretical modeling of the mildly nonlinear regime. In this work we explore how the choice of nonlinear prescription would impact parameter estimation from cosmic shear measurements for a Euclid-like survey. Specifically, we employ two different nonlinear prescriptions of halofit and the Effective Field Theory of the Large Scale Structure and compare their measurements for the three different cosmological scenarios of ΛCDM, wCDM, and (w 0, w a ) CDM. We also investigate the impact of different nonlinear cutoff schemes on parameter estimation. We find that the predicted errors on most parameters shrink considerably as smaller scales are included in the analysis, with the amount depending on the nonlinear prescription and the cutoff scheme used. We use predictions from the halofit model to analyze the mock data from DarkSky N-body simulations and quantify the parameter bias introduced in the measurements due to the choice of nonlinear prescription. We observe that σ 8 and n s have the largest measurement bias induced by inaccuracies of the halofit model.

Simulating emission line galaxies for the next generation of large-scale structure surveys

ABSTRACT We investigate emission line galaxies across cosmic time by combining the modified L-Galaxies semi-analytical galaxy formation model with the JiuTian cosmological simulation. We improve the tidal disruption model of satellite galaxies in L-Galaxies to address the time dependence problem. We utilize the public code cloudy to compute emission line ratios for a grid of H ii region models. The emission line models assume the same initial mass function as that used to generate the spectral energy distribution of semi-analytical galaxies, ensuring a coherent treatment for modelling the full galaxy spectrum. By incorporating these emission line ratios with galaxy properties, we reproduce observed luminosity functions for H α, H β, [O ii], and [O iii] in the local Universe and at high redshifts. We also find good agreement between model predictions and observations for autocorrelation and cross-correlation functions of [O ii]-selected galaxies, as well as their luminosity dependence. The bias of emission line galaxies depends on both luminosity and redshift. At lower redshifts, it remains constant with increasing luminosity up to around $\sim 10^{42.5}\, {\rm erg\, s^{-1}}$ and then rises steeply for higher luminosities. The transition luminosity increases with redshift and becomes insignificant above z = 1.5. Generally, galaxy bias shows an increasing trend with redshift. However, for luminous galaxies, the bias is higher at low redshifts, as the strong luminosity dependence observed at low redshifts diminishes at higher redshifts. We provide a fitting formula for the bias of emission line galaxies as a function of luminosity and redshift, which can be utilized for large-scale structure studies with future galaxy surveys.

A new test of gravity – I. Introduction to the method

ABSTRACT We introduce a new scheme based on the marked correlation function to probe gravity using the large-scale structure of the Universe. We illustrate our approach by applying it to simulations of the metric-variation f(R) modified gravity theory and general relativity (GR). The modifications to the equations in f(R) gravity lead to changes in the environment of large-scale structures that could, in principle, be used to distinguish this model from GR. Applying the Monte Carlo Markov Chain algorithm, we use the observed number density and two-point clustering to fix the halo occupation distribution (HOD) model parameters and build mock galaxy catalogues from both simulations. To generate a mark for galaxies when computing the marked correlation function we estimate the local density using a Voronoi tessellation. Our approach allows us to isolate the contribution to the uncertainty in the predicted marked correlation function that arises from the range of viable HOD model parameters, in addition to the sample variance error for a single set of HOD parameters. This is critical for assessing the discriminatory power of the method. In a companion paper, we apply our new scheme to a current large-scale structure survey.

Primordial non-Gaussianities with weak lensing: information on non-linear scales in the Ulagam full-sky simulations

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.

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

Cosmological gravity on all scales. Part III. Non-linear matter power spectrum in phenomenological modified gravity

Model-independent tests of gravity with cosmology are important when testing extensions to the standard cosmological model. To maximise the impact of these tests one requires predictions for the matter power spectrum on non-linear scales. In this work we validate the ReACT approach to the non-linear matter power spectrum against a suite of phenomenological modified gravity N-body simulations with a time-varying gravitational constant, covering a wider range of parameter space than previously examined. This vanilla application of ReACT has limited range and precision due to the different concentration-mass relation c(M) that occurs when gravity is modified. We extend this approach with a fitting function for a modified concentration-mass relation, allowing for accurate (1%) computation of the matter power spectrum up k = 2 h Mpc-1 across a substantial range of parameter space. This fitting function allows precision model-independent tests of modified gravity to be carried out using the data from upcoming large scale structure surveys.

Hybrid bias and displacement emulators for field-level modelling of galaxy clustering in real and redshift space

ABSTRACT Recently, hybrid bias expansions have emerged as a powerful approach to modelling the way in which galaxies are distributed in the Universe. Similarly, field-level emulators have recently become possible, thanks to advances in machine learning and N-body simulations. In this paper, we explore whether both techniques can be combined to provide a field-level model for the clustering of galaxies in real and redshift space. Specifically, here we will demonstrate that field-level emulators are able to accurately predict all the operators of a second-order hybrid bias expansion. The precision achieved in real and redshift space is similar to that obtained for the non-linear matter power spectrum. This translates to roughly 1–2 per cent precision for the power spectrum of a BOSS (Baryon Oscillation Spectroscopic Survey) and a Euclid-like galaxy sample up to $k\sim 0.6\ h\, {\rm Mpc}^{-1}$. Remarkably, this combined approach also delivers precise predictions for field-level galaxy statistics. Despite all these promising results, we detect several areas where further improvements are required. Therefore, this work serves as a road map for the developments required for a more complete exploitation of upcoming large-scale structure surveys.

Accurate Fourier-space statistics for line intensity mapping: Cartesian grid sampling without aliased power

ABSTRACT Estimators for n-point clustering statistics in Fourier-space demand that modern surveys of large-scale structure be transformed to Cartesian coordinates to perform Fast Fourier Transforms (FFTs). In this work, we explore this transformation in the context of pixelized line intensity maps (LIM), highlighting potential biasing effects on power-spectrum measurements. Current analyses often avoid a complete resampling of the data by approximating survey geometry as rectangular in Cartesian space, an increasingly inaccurate assumption for modern wide-sky surveys. Our simulations of a $20\, {\times }\, 20\, \text{deg}^2$ 21 cm LIM survey at $0.34\, {\lt }\, z\, {\lt }\, 0.54$ show this assumption biases power-spectrum measurements by ${\gt }\, 20~{{\ \rm per\ cent}}$ across all scales. We therefore present a more robust framework for regridding the voxel intensities on to a 3D FFT field by coordinate transforming large numbers of Monte-Carlo sampling particles. Whilst this unbiases power-spectrum measurements on large scales, smaller scale discrepancies remain, caused by structure smoothing and aliasing from separations unresolved by the grid. To correct these effects, we introduce modelling techniques, higher order particle assignments, and interlaced FFT grids to suppress the aliased power. Using a piecewise cubic spline (PCS) particle assignment and an interlaced FFT field, we achieve sub-per cent accuracy up to 80 per cent of the Nyquist frequency for our 21 cm LIM simulations. We find a more subtle hierarchical improvement in results for higher order assignment schemes, relative to the gains made for galaxy surveys, which we attribute to the extra complexity in LIM from additional discretizing steps. python code accompanying this paper is available at github.com/stevecunnington/gridimp.

Taming assembly bias for primordial non-Gaussianity

Primordial non-Gaussianity of the local type induces a strong scale-dependent bias on the clustering of halos in the late-time Universe. This signature is particularly promising to provide constraints on the non-Gaussianity parameter f NL from galaxy surveys, as the bias amplitude grows with scale and becomes important on large, linear scales. However, there is a well-known degeneracy between the real prize, the f NL parameter, and the (non-Gaussian) assembly bias i.e., the halo formation history-dependent contribution to the amplitude of the signal, which could seriously compromise the ability of large-scale structure surveys to constrain f NL. We show how the assembly bias can be modeled and constrained, thus almost completely recovering the power of galaxy surveys to competitively constrain primordial non-Gaussianity. In particular, studying hydrodynamical simulations, we find that a proxy for the halo properties that determine assembly bias can be constructed from photometric properties of galaxies. Using a prior on the assembly bias guided by this proxy degrades the statistical errors on f NL only mildly compared to an ideal case where the assembly bias is perfectly known. The systematic error on f NL that the proxy induces can be safely kept under control.

Cosmological parameters derived from the final Planck data release (PR4)

We present cosmological parameter constraints using maps from the last Planck data release (PR4). In particular, we detail an upgraded version of the cosmic microwave background likelihood, HiLLiPoP, that is based on angular power spectra and relies on a physical modeling of the foreground residuals in the spectral domain. This new version of the likelihood retains a larger sky fraction (up to 75%) and uses an extended multipole range. Using this likelihood, along with low-ℓ measurements from LoLLiPoP, we derived constraints on ΛCDM parameters that are in good agreement with previous Planck 2018 results, but with smaller uncertainties by 10% to 20%. We demonstrate that the foregrounds can be accurately described in the spectral domain, with a negligible impact on ΛCDM parameters. We also derived constraints on single-parameter extensions to ΛCDM, including AL, ΩK, Neff, and ∑mν. Noteworthy results from this updated analysis include a lensing amplitude value of AL = 1.039 ± 0.052, which is more closely aligned with theoretical expectations within the ΛCDM framework. Additionally, our curvature measurement, ΩK = −0.012 ± 0.010, is now fully consistent with a flat universe and our measurement of S8 is closer to the measurements derived from large-scale structure surveys (at the 1.5σ level). We also added constraints from PR4 lensing, making this combination the most tightly constrained data set currently available from Planck. Additionally, we explored the addition of baryon acoustic oscillation data, which tightens the limits on some particular extensions to the standard cosmology.

Patchy screening of the CMB from dark photons

We study anisotropic (patchy) screening induced by the resonant conversion of cosmic microwave background (CMB) photons into dark-sector massive vector bosons (dark photons) as they cross non-linear large scale structure (LSS). Resonant conversion takes place through the kinetic mixing of the photon with the dark photon, one of the simplest low energy extensions to the Standard Model. In the early Universe, resonant conversion can occur when the photon plasma mass, obtained as the photon propagates through the ionized interstellar and intergalactic media, matches the dark photon mass. After the epoch of reionization, resonant conversion occurs mainly in the ionized gas that occupies virialized dark matter halos, for a range of dark photon masses between 10-13 eV ≲ m A' ≲ 10-11 eV. This leads to new CMB anisotropies that are correlated with LSS, which we refer to as patchy dark screening, in analogy with anisotropies from Thomson screening. Its unique frequency dependence allows it to be distinguished from the blackbody CMB. In this paper, we use a halo model approach to predict the imprint of dark screening on the CMB temperature and polarization anisotropies, as well as their correlation with LSS. We then examine the two- and three-point correlation functions of the dark-screened CMB, as well as correlation functions between CMB and LSS observables, to project the sensitivity of future measurements to the kinetic mixing parameter and dark photon mass. We demonstrate that an analysis with existing CMB data can improve upon current constraints on the kinetic mixing parameter by two orders of magnitude with the two-point correlation functions, while data from upcoming CMB experiments and LSS surveys can further improve the reach by another order of magnitude with two- and three-point correlation functions.

The Three Hundred project: Contrasting clusters optical and IR properties in hydrodynamical and dark matter only simulations

Cluster number count is a major cosmological probe for the next generation of cosmological large scale-structure surveys like the one expected from the Euclid satellite mission. Cosmological constraints will be mainly limited by the understanding of the selection function (SF), which characterize the probability of detecting a cluster of a given mass and redshift. The SF can be estimated by injecting realistic simulated clusters into the survey and re-applying the detection procedure. In this paper, we use the galaxy clusters from The Three Hundred project to study this effect with their member galaxies. We further study possible resolution effects by comparing low and high resolution simulations. Finally, we present the density profiles of the member galaxies and discuss their evolution with cluster mass and redshift.

Propagating photo-z uncertainties: a functional derivative approach

ABSTRACT Photometric redshifts are a key ingredient in the analysis and interpretation of large-scale structure (LSS) surveys. The accuracy and precision of these redshift estimates are directly linked to the constraining power of photometric surveys. It is hence necessary to define precision and accuracy requirements for the redshift calibration to not infer biased results in the final analysis. For weak gravitational lensing of the LSS, the photometry culminates in the estimation of the source-redshift distribution (SRD) in each of the tomographic bins used in the analysis. The focus has been on shifts of the mean of the SRDs and how well the calibration must be able to recover those. Since the estimated SRDs are usually given as a normalized histogram with corresponding errors, it would be advantageous to propagate these uncertainties accordingly to see whether the requirements of the given survey are indeed fulfilled. Here, we propose the use of functional derivatives to calculate the sensitivity of the final observables, for examples, the lensing angular power spectrum, with respect to the SRD at a specific redshift. This allows the propagation of arbitrarily shaped small perturbations to the SRD, without having to run the whole analysis pipeline for each realization again. We apply our method to a survey with Euclid-like specifications and demonstrate it with SRDs of the KV450 data set, recovering previous results. Lastly, we note that the moments of the SRD of order larger than two will probably not be relevant when propagating redshift uncertainties in cosmic shear analysis.

Parity-violating trispectrum from Chern-Simons gravity

A potential source for parity violation in the Universe is inflation. The simplest inflationary models have two fields: the inflaton and graviton, and the lowest-order parity-violating coupling between them is dynamical Chern-Simons (dCS) gravity with a decay constant f. Here, we show that dCS imprints a parity-violating signal in primordial scalar perturbations. Specifically, we find that, after dCS amplifies one graviton helicity due to a tachyonic instability, the graviton-mediated correlation between two pairs of scalars develops a parity-odd component. This correlation, the primordial scalar trispectrum, is then transferred to the corresponding curvature correlator and thus is imprinted in both large-scale structure (LSS) and the cosmic microwave background (CMB). We find that the parity-odd piece has roughly the same amplitude as its parity-even counterpart, scaled linearly by the degree of gravitational circular polarization Πcirc ∼ √ε[h 2/(M Pl f)] ≤ 1, with ε the slow-roll parameter, H the inflationary Hubble scale, and the upper bound saturated for purely circularly-polarized gravitons. We also find that, in the collapsed limit, the ratio of the two trispectra contains direct information about the graviton's spin. In models beyond standard inflationary dCS, e.g. those with multiple scalar fields or superluminal scalar sound speed, there can be a large enhancement factor F ≳ 106 to the trispectrum. We find that an LSS survey that contains N modes linear modes would place an nσ constraint on Πcirc r of ∼ 0.04 (n/3)(106/F)(106/N modes)1/2 from the parity-odd galaxy trispectrum, for tensor-to-scalar ratio r. We also forecast for several spectroscopic and 21-cm surveys. This constraint implies that, for high-scale single-field inflation parameters, LSS can probe very large dCS decay constants f ≲ 4 × 109 GeV(3/n)(F/106)(N modes/106)1/2. Our result is the first example of a massless particle yielding a parity-odd scalar trispectrum through spin-exchange.

FLAMINGO: calibrating large cosmological hydrodynamical simulations with machine learning.

To fully take advantage of the data provided by large-scale structure surveys, we need to quantify the potential impact of baryonic effects, such as feedback from active galactic nuclei (AGN) and star formation, on cosmological observables. In simulations, feedback processes originate on scales that remain unresolved. Therefore, they need to be sourced via subgrid models that contain free parameters. We use machine learning to calibrate the AGN and stellar feedback models for the FLAMINGO (Fullhydro Large-scale structure simulations with All-sky Mapping for the Interpretation of Next Generation Observations) cosmological hydrodynamical simulations. Using Gaussian process emulators trained on Latin hypercubes of 32 smaller volume simulations, we model how the galaxy stellar mass function (SMF) and cluster gas fractions change as a function of the subgrid parameters. The emulators are then fit to observational data, allowing for the inclusion of potential observational biases. We apply our method to the three different FLAMINGO resolutions, spanning a factor of 64 in particle mass, recovering the observed relations within the respective resolved mass ranges. We also use the emulators, which link changes in subgrid parameters to changes in observables, to find models that skirt or exceed the observationally allowed range for cluster gas fractions and the SMF. Our method enables us to define model variations in terms of the data that they are calibrated to rather than the values of specific subgrid parameters. This approach is useful, because subgrid parameters are typically not directly linked to particular observables, and predictions for a specific observable are influenced by multiple subgrid parameters.

Sweeping Horndeski canvas: new growth-rate parameterization for modified-gravity theories

We propose and numerically validate a new fitting formula that is sufficiently accurate to model the growth of structure in Horndeski theories of modified gravity for upcoming Stage IV and V large-scale structure surveys. Based on an analysis of more than 18,000 Horndeski models and adopting the popular parameterization of the growth rate f(z) = Ω M (z) γ , we generalize the constant growth index γ to a two-parameter redshift-dependent quantity, γ(z), that more accurately fits these models. We demonstrate that the functional form γ(z) = γ 0 + γ 0 z 2/(1 + z) improves the median χ 2 of the fit to viable Horndeski models by a factor of ∼ 40 relative to that of a constant γ, and is sufficient to obtain unbiased results even for precise measurements expected in Stage IV and V surveys. Finally, we constrain the parameters of the new fitting formula using current cosmological data.

The FLAMINGO project: cosmological hydrodynamical simulations for large-scale structure and galaxy cluster surveys

ABSTRACT We introduce the Virgo Consortium’s FLAMINGO suite of hydrodynamical simulations for cosmology and galaxy cluster physics. To ensure the simulations are sufficiently realistic for studies of large-scale structure, the subgrid prescriptions for stellar and AGN feedback are calibrated to the observed low-redshift galaxy stellar mass function and cluster gas fractions. The calibration is performed using machine learning, separately for each of FLAMINGO’s three resolutions. This approach enables specification of the model by the observables to which they are calibrated. The calibration accounts for a number of potential observational biases and for random errors in the observed stellar masses. The two most demanding simulations have box sizes of 1.0 and 2.8 Gpc on a side and baryonic particle masses of 1 × 108 and $1\times 10^9\, \text{M}_\odot$, respectively. For the latter resolution, the suite includes 12 model variations in a 1 Gpc box. There are 8 variations at fixed cosmology, including shifts in the stellar mass function and/or the cluster gas fractions to which we calibrate, and two alternative implementations of AGN feedback (thermal or jets). The remaining 4 variations use the unmodified calibration data but different cosmologies, including different neutrino masses. The 2.8 Gpc simulation follows 3 × 1011 particles, making it the largest ever hydrodynamical simulation run to z = 0. Light-cone output is produced on-the-fly for up to 8 different observers. We investigate numerical convergence, show that the simulations reproduce the calibration data, and compare with a number of galaxy, cluster, and large-scale structure observations, finding very good agreement with the data for converged predictions. Finally, by comparing hydrodynamical and ‘dark-matter-only’ simulations, we confirm that baryonic effects can suppress the halo mass function and the matter power spectrum by up to ≈20 per cent.

The LSST AGN Data Challenge: Selection Methods

Development of the Rubin Observatory Legacy Survey of Space and Time (LSST) includes a series of Data Challenges (DCs) arranged by various LSST Scientific Collaborations that are taking place during the project's preoperational phase. The AGN Science Collaboration Data Challenge (AGNSC-DC) is a partial prototype of the expected LSST data on active galactic nuclei (AGNs), aimed at validating machine learning approaches for AGN selection and characterization in large surveys like LSST. The AGNSC-DC took place in 2021, focusing on accuracy, robustness, and scalability. The training and the blinded data sets were constructed to mimic the future LSST release catalogs using the data from the Sloan Digital Sky Survey Stripe 82 region and the XMM-Newton Large Scale Structure Survey region. Data features were divided into astrometry, photometry, color, morphology, redshift, and class label with the addition of variability features and images. We present the results of four submitted solutions to DCs using both classical and machine learning methods. We systematically test the performance of supervised models (support vector machine, random forest, extreme gradient boosting, artificial neural network, convolutional neural network) and unsupervised ones (deep embedding clustering) when applied to the problem of classifying/clustering sources as stars, galaxies, or AGNs. We obtained classification accuracy of 97.5% for supervised models and clustering accuracy of 96.0% for unsupervised ones and 95.0% with a classic approach for a blinded data set. We find that variability features significantly improve the accuracy of the trained models, and correlation analysis among different bands enables a fast and inexpensive first-order selection of quasar candidates.

Isotropic N-point basis functions and their properties

Isotropic functions of positions , i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch–Gordan coefficients. An orthonormal basis of such functions provides a formalism suitable for analyzing isotropic distributions such as those that arise in cosmology, for instance in the clustering of galaxies as revealed by large-scale structure surveys. The algebraic properties of the basis functions are conveniently expressed in terms of 6-j and 9-j symbols. The calculation of relations among the basis functions is facilitated by ‘Yutsis’ diagrams for the addition and recoupling of angular momenta.