Welcome talk

• Ruth Durrer

FLRW foundations

• Chris Clarkson

Spatial curvature, backreaction and voids in our cosmic neighbourhood

• Pierre Mourier (University of Canterbury)

The late Universe features nonlinear deviations from strict homogeneity and isotropy as matter structures develop. These local inhomogeneities may have a non-negligible dynamical impact on the cosmological expansion. Such backreaction effects from the presence of structures also include the growth of spatial curvature over large regions. They can be described in a general-relativistic picture, by explicitly coarse-graining the local inhomogeneous fields using a spatial averaging scheme. I will present our estimates of the contributions to the averaged energy budget, including the relativistic backreaction effects, over a range of scales around our Galaxy, up to a 300 Mpc/*h* comoving distance (*z* ~ 0.1). We used the CosmicFlows-4++ reconstruction of the peculiar velocity and density contrast fields within this range, and a mapping between Newtonian and GR frameworks, to evaluate the corresponding local density, expansion rate, shear, and scalar intrinsic spatial curvature fields. I will discuss the significant (*O*(10%)) regional deviations that we observe with respect to the imposed flat $\Lambda$CDM background, which extend all the way to the edge of the surveyed volume. The scale-dependent variations in all averaged fields point to a vast underdense shell roughly compatible with the proposed Local Hole, surrounding further nested wall and void structures at a more local level. I will highlight in particular the important role played by spatial curvature in the obtained regional energy balance and in its variations with the averaging scale.

Covariant Cosmography with exact solutions of GR

• Maharshi SARMA

Precision cosmology suggests that late-time inhomogeneities may no longer be treated as small corrections to the FLRW paradigm. Moreover, recent observational hints of axisymmetric anisotropies in the local expansion rate further motivate analyses beyond linear perturbation theory. Motivated by these issues, the framework called "Covariant Cosmography'' is adopted which describes the anisotropies in the local cosmic expansion rate within a fully relativistic framework and in a model-independent manner, free from a priori symmetry constraints on the underlying gravitational field and without resorting to linear perturbation theory approximations. In this work, we address the intriguing evidence of an axisymmetric pattern in the angular anisotropies of the local cosmic expansion rate within the Covariant Cosmographic (CC) framework by considering the cosmological set-up of an off-center observer in a spherically symmetric Lemaître-Tolman-Bondi (LTB) spacetime. By estimating the amplitudes of the covariant cosmographic parameters—including the Hubble, deceleration, curvature, and jerk parameters, we compare it with the observed value. Moreover, we test luminosity-distance reconstructions with the CC parameters within this model. Finally, we compare the LTB relativistic distance for small inhomogeneities with the corresponding result derived from linear perturbation theory (LPT) in the standard cosmological model.

Euclid summary and survey outlook

• Stefano Camera

Integrated effects in Large Scale Structure

• Jessie Hammond

Measuring primordial non-Gaussianity (PNG) is one of the goals of current and future galaxy surveys. Since general relativistic effects are degenerate with any signal measured from PNG, a fully relativistic, non-Newtonian approach is required. In previous studies, integrated line-of-sight effects, such as lensing convergence, Shapiro time delay and integrated Sachs-Wolfe, have largely been ignored in power spectrum analysis. In this talk, I will show that it is important to consider integrated effects to model the power spectrum accurately. In current galaxy surveys, such as Euclid, we show that by neglecting these contributions in our power spectrum analyses, we bias our PNG measurements on the order of 3$\sigma$. This effect becomes increasingly important at higher redshift and for future galaxy surveys like MegaMapper, making it crucial to include integrated effects in current and future power spectrum analyses.

Detecting Relativistic Effects by Multi-tracing a Single Galaxy Population

• Federico Montano (University of Turin)

Our well-established gravity model, the theory of general relativity, has been extensively tested in strong-field regimes. However, no observational test has yet confirmed its validity on cosmological scales, where the universe’s dark components are necessary to fit LSS data. The detection of large-scale relativistic effects via galaxy power spectrum measurements would provide such an unprecedented confirmation. Amongst these relativistic contributions, the Doppler term acts as an imaginary correction in the relation between the galaxy density contrast and that of matter, and mostly affects the large scales usually plagued by cosmic variance. Also, it is sample-dependent, thus different galaxy populations display different contributions in their power spectra. In the search for the optimal galaxy samples to achieve a detection of the relativistic term, we can split a galaxy population according to luminosity, and then perform a multi-tracer analysis with auto-correlations of the sub-samples and their cross-power spectrum. While forecasts claim relativistic Doppler dipole as detectable with data from ongoing galaxy surveys, analysis pipelines able to extract the signal from source catalogues need to be set up. In fact, they cannot help considering wide-angle and survey window effects, as well as being carefully tested against simulations that include relativistic corrections. I will be talking about current efforts in this sense, showing ongoing tests of the faint-bright multi-tracer approach with mock catalogues, which are crucial to pave the way towards a measurement on real data.

Large-Scale Structure and Primordial Non-Gaussianity with SPHEREx

• Henry Grasshorn Gebhardt

SPHEREx is in the process of surveying the full sky every six months in 102 spectro-photometric bands from 0.75 to 5 micrometers. This will allow SPHEREx to measure redshifts for about 500 million galaxies out to about z < 4. The unprecedentedly large volume is forecast to allow measurement of local non-Gaussianity with statistical uncertainty below order unity. For the main part of the talk I will introduce SPHEREx, and it's status after one year of observation. In the second part of the talk I will go into some of the analysis techniques we are developing for SPHEREx, especially the spherical-Fourier-Bessel (SFB) power spectrum that incorporates all wide- and first-order GR effects and does not need an effective redshift approximation, as well as our new in-progress technique to calculate the window convolution matrix, wide-angle effects, and integral constraint for the galaxy power spectrum and bispectrum. I will also highlight some of the systematics challenges associated with the measurement.

Future prospects for measuring fNL using redshift-weighted angular power spectra

• William Luke Matthewson (Korea Astronomy and Space science Institute (KASI))

The angular power spectrum of redshift fluctuations is a novel observable that involves a specific kind of weighting of the galaxy number counts, meant to recover complementary information, using the redshift distribution along the line of sight. This has been shown to be generally effective in improving constraining power when combined with the conventional angular power spectrum, and was recently used with Quaia data to provide the tightest constraint on $f_{\rm NL}$ from large scale structure 2-point statistics, second only to the DESI DR1 3D power spectrum analysis. Using data from DESI, there is an opportunity to use this technique to further improve these constraints. I will briefly review the theoretical framework of angular redshift fluctuations, discuss results from forecasts, in anticipation of future work using DESI, and compare to the current results.

The impact of our peculiar motion on primordial non-Gaussianity measurements using the LIGER4GAL framework

• Bartolomeo Bottazzi Baldi (University of Padua)

Current and forthcoming galaxy surveys will map the observable Universe with unprecedented depth, sky coverage, and precision. These maps are affected by relativistic redshift-space distortions (RSDs), which become increasingly relevant on ultra-large scales. Accurate modelling of these relativistic RSDs is essential to avoid systematic biases in key cosmological measurements, such as primordial non-Gaussianity (PNG). To address this, we introduce an updated implementation of the LIGER method, LIGER4GAL, which incorporates all linear-order relativistic RSDs directly at the tracer level of high-resolution N-body simulations. We demonstrate that LIGER4GAL improves upon previous iterations of the LIGER method by reproducing the expected non-linear clustering while maintaining accuracy for relativistic RSDs on large scales. We use the updated code to generate a DESI-like sample of luminous red galaxies from the Huge MultiDark Planck simulation. By measuring the power spectrum multipoles of this sample with and without the imprint of relativistic RSDs, we assess the impact of relativistic effects on measurements of the local PNG signal (fnl). We find that the omission of the “finger-of-the-observer” (sourced by the peculiar velocity of the observer) effect in the power spectrum modelling can bias measurements of fnl by more than 1 (0.25) σfnl in 40% (80%) of the possible realizations of the universe if scales down to kmin = 0.0015 h/Mpc are included.

Hard Mode: The Galaxy Angular Bispectrum with Relativistic Effects and Primordial Non-Gaussianity

• Thomas Montandon (Laboratoire Univers et Particules de Montpellier)

The bispectrum of galaxy number counts is a key probe of large-scale structure, offering sensitivity to nonlinear gravitational evolution, galaxy bias, and primordial non-Gaussianity (PNG). In this talk, I present the first full-sky computation of the angular bispectrum in second-order perturbation theory without the Limber approximation, formulated for finite redshift bins via tomographic spherical harmonics. The framework incorporates all Newtonian contributions (density, redshift-space distortions, quadratic terms), non-integrated relativistic projection effects, and dynamical general relativistic and radiation corrections. Notably, radiation effects dominate the relativistic signal in the squeezed limit at $z \sim 2$, exceeding projection effects by over an order of magnitude. I further extend this framework to include galaxy bias and the three standard PNG shapes (local, equilateral, orthogonal), demonstrating how unmodeled relativistic effects can contaminate $f_{\rm NL}$ estimates and discussing the second-order scale-dependent biases. Applications to forecasts for the Square Kilometre Array are discussed. The corresponding code, \texttt{ang\_bispec}, is publicly available.

Detecting the relativistic dipole in galaxy clustering with DESI

• Jade Piat

Understanding the accelerated expansion of the Universe remains one of the key challenges in cosmology. Leading explanations which do not rely on a cosmological constant are dark energy and modifications of General Relativity, both of which require robust tests on large scales. The Dark Energy Spectroscopic Instrument (DESI) provides unprecedented precision in measuring galaxy clustering from spectroscopic data, enabling the detection of relativistic effects beyond the standard redshift-space distortions. In particular, relativistic effects generate a dipole in the cross-power spectrum of two galaxy populations. We analyse the detectability of this dipole using mock galaxy catalogues produced from post-processed Newtonian simulations that mimic the DESI Bright Galaxy Survey, splitting galaxies into bright and faint populations, while validating the modelling against a relativistic simulation mock. Our results show that the detection significance improves with less bright sources and that the measured distortions are well described by linear theory predictions.

Our inhomogeneous universe as an axially symmetric Szekeres spacetime

• Marie-Noëlle Célérier (Observatoire de Paris-PSL)

In the current era of precision cosmology, it is well-known that the present standard $\Lambda$CDM model, despite its successes, is an insufficient average of our actual Universe. Given the recorded tensions between theory and observations, it appears that a more precise representation of the local universe is mandatory. Since General Relativity (GR) is the best theory currently available to describe gravitation (verified up to some $10^{-15}$ precision in the weak and strong regimes), the new model should agree with its specifications. The Szekeres GR solution possesses all the nice properties allowing it to represent an inhomogeneous small scale universe becoming homogeneous (FLRW) at some large transition scale. Now, a number of observations leads us to conclude that the distribution of matter and its expansion are anisotropic roughly in the direction of the CMB dipole. While most of the community has been reluctant up to now to use the general Szekeres model, due to its complexity, its axially symmetric version, more simple to deal with, might be more appealing. Therefore, taking advantage of both its compatibility with observations and its relative simplicity, we show how it can be used as a new standard cosmological model.

Quantifying Anisotropies in the Local Expansion Rate with Cosmic-Flows 4

• Jessica Santiago

The Cosmological Principle predicts that, on sufficiently large scales, cosmic expansion should appear statistically isotropic to all observers. Testing the limitations of this prediction in the local Universe is, therefore, a first step into properly considering the impacts of local large-scale anisotropies in data analysis and interpretation. In this work, we use the Cosmicflows-4 distance compilation to reconstruct spatial fluctuations in the local expansion rate (up to $z=0.1$), obtaining its low $\ell$ multipoles, with special focus on the dipole and quadrupole of the expansion. The analysis is done in a completely model-independent way, allowing for a direct test of statistical isotropy without assuming a fiducial cosmological model. While the multipole amplitudes remain broadly consistent with $\Lambda$CDM expectations, the dipole and quadrupole exhibit a persistent coherence in direction across different redshift shells, with a probability below 0.001% in standard cosmological realizations. In this talk I will discuss the details and consequences of such findings.

Detection of Gpc-scale dipoles in gravitational potentials

• Yan-Chuan Cai (University of Edinburgh)

I will present the detection of Gpc-scale dipoles in gravitational potentials in the large-scale structure of the Universe. Large-scale transverse peculiar velocities are used as signposts and correlated with CMB temperature maps and gravitational lensing maps, revealing dipoles on angular scales of tens of degrees. I will also present constraints on general relativity derived from these new observables.

Cosmology with strong lensing images

• Julien Larena

Massive galaxies, acting as gravitational lenses, can form greatly distorted images of background galaxies in so called strong lensing events. When background galaxy and lens are almost aligned along the line of sight, the image take the shape of an Einstein ring whose shape encodes basic properties of the lens. However, since lenses are not isolated objects in an otherwise perfectly homogeneous universe, these Einstein rings are sligthly altered by further weak lensing along the line of sight coming from the large scale matter distribution in the universe in front and behind the lens. This coupling between strong and weak lensing, dubbed LOS shear, offers a novel opportunity to probe cosmic shear, in a regime that is complementary to regular cosmic shear. We will present a new 6x2pt framework incorporating this LOS shear with standard probes and assess its detectability and potential for cosmology with the Euclid survey.

The impact of evolving dark energy on the Weyl potential measured from the Dark Energy Survey Year 3 data

• Benedetta Rosatello (University of Geneva)

A direct, model-independent probe of the validity of theories beyond General Relativity and the concordance LCDM model is provided by the Weyl potential, the sum of the temporal and spatial distortions of the spacetime geometry. Its measurement, obtained by combining galaxy clustering and galaxy-galaxy lensing data from the Dark Energy Survey (DES-Y3), is in tension with the LCDM prediction at low redshift. Recently, the novel baryon acoustic oscillation measurements from DESI, combined with luminosity distance measurements from type Ia supernovae and Cosmic Microwave Background measurements from Planck, indicated a preference for an evolving dark energy model with an equation of state crossing w=-1. In this talk, I will discuss how the background evolution in evolving dark energy models impacts the growth of the Weyl potential. I will show that evolving dark energy models that cross the phantom divide can naturally reduce the Weyl potential at intermediate redshift, providing a better agreement with DES measurements. These models, however, do not fully capture the measured redshift evolution of the Weyl potential. Further data from Euclid and LSST are then necessary to determine if an evolving dark energy background is enough to explain the low values of the Weyl potential at intermediate redshift, or if the evolution of the perturbations itself should be modified by changing the theory of gravity or by including additional interactions in the dark matter sector.

Lensing and relativistic observables

• Jose Fonseca

Statistical Field Theory for Weak Gravitational Lensing

• ZHENG ZHANG

In conventional weak-lensing analyses, the lensing projection is treated as a direct statistical remapping: the convergence power spectrum inherits from the matter power spectrum, the convergence bispectrum from the matter bispectrum, and so on. In this work we develop a first-principles theoretical framework that goes beyond this approximate projection scheme. By formulating the dynamics of null geodesic congruences as a path integral, we track how the nonlinear Sachs evolution defines, and distorts, the mapping between the statistical hierarchy of the driving field and that of the observable lensing field. Within this framework we show, concretely, three effects. First, nonlinear Sachs evolution generates non-Gaussianity in the lensing observables even when the driving fields are exactly Gaussian. Second, non-Gaussianity in the driving fields leaks into the two-point statistics of the lensing observables, a contamination that is invisible to any purely linear projection. Third, we can disentangle, along the line of sight, the contributions from white-noise-type uncorrelated small-scale fluctuations and from correlated stochastic driving components. Through the mapping to the redshift space, we have also tracked the effects of redshift modifications such as ISW. We emphasise that these effects originate in the nonlinear dynamics of the null geodesic congruence itself, rather than in the mode coupling induced by structure formation. They are therefore not captured by N-body-based studies.

Probing the Cosmological Principle with CMB lensing and cosmic shear

• James Adam (University of the Western Cape)

The Cosmological Principle, which assumes both homogeneity and isotropy on large scales, is a cornerstone of the standard model of cosmology and shapes how we view the Universe and our place within it. It is imperative, then, to devise multiple observational tests which can identify and quantify possible violations of this foundational principle. One possible method of probing large-scale anisotropies involves the use of weak gravitational lensing. Previously, we demonstrated how cross-correlations between $E$- and $B$-modes in cosmic shear can reveal signatures of anisotropy at late times. In this talk, I will present a complementary approach: using cross-correlations between Cosmic Microwave Background (CMB) lensing convergence and cosmic shear $B$-modes measured from galaxy shapes. Our analysis shows that this probe can reach percent-level sensitivity to anisotropy with data from a Euclid-like survey, with most of its constraining power deriving from large angular scales ($\ell\lesssim 200$).

Relativistic effects in large-scale structure

• Pritha Paul

Relativistic Effects on nonlinear scales in ΛCDM

• Marco Bruni (University of Portsmouth)

In this talk I will give an overview of work on non-perturbative nonlinear structure formation in ΛCDM in the context of General Relativity (GR). Using a "post-Friedmann" approximation, I will show how, at leading order, the gravito-magnetic vector potential (AKA frame-dragging potential) is generated by the momentum density on nonlinear scales, and can be extracted from ILLUSTRIS Newtonian N-body simulations, as well as from N-body simulations with GRAMSES, an approximate GR code. I will then present full-GR simulations of a toy-model “cosmic web” of over-densities, voids and filaments with the Einstein Toolkit fluid code, showing how the first shell-crossing at peaks of over-densities is very well predicted by the simple "top hat" model, while in the formation of the cosmic web a role is played by gravito-magnetism, especially around filaments. In the last part of the talk I will illustrate some work in progress, some aimed at extending the work on frame-dragging, some aimed at understanding the role of relativistic effects during collapse and past the first shall crossing and virialization.

Non-linear Relativistic Effects on the Redshift Drift in the LambdaCDM Universe and Beyond

• Pierre Béchaz (University of Pisa & INFN, Pisa)

With high-precision data about to be delivered by large-scale surveys, the development of higher-order perturbative descriptions of cosmological observables is becoming increasingly important. Among the others, the redshift drift, being sensitive to local variations in the Hubble factor, paves the way for direct tests of the cosmic acceleration history. More in general, it enables real-time cosmology, which aims to determine the redshift evolution of observables measured on the past light-cone of an observer. The Geodesic Light-Cone coordinates provide a natural framework for this purpose, as they are specifically adapted to the observer’s past light-cone and allow for a non-perturbative description of light propagation in an inhomogeneous Universe. In this talk, I will first review how these coordinates are defined. I will then describe a cosmological perturbation theory constructed directly on the observed past light-cone up to second order, using a fully gauge-invariant approach. This framework enables the consistent computation of higher-order corrections to cosmological observables in General Relativity as well as in modified gravity and dark energy scenarios. I will focus on non-linear relativistic effects on the redshift drift, by combining analytical and numerical results and discussing about their validity also for cosmological models beyond LambdaCDM. In particular, I will demonstrate that, unlike widely studied observables such as the galaxy number counts, the bispectrum of the redshift drift is inherently more sensitive to non-linear effects than its two-point statistics. [Based on JCAP 03 (2026) 075 and arXiv:2604.26690]

Numerical reconstruction of gravitational lensing observables from null geodesic integration in perturbed FLRW spacetimes

• Laurent MAGRI--STELLA (Laboratoire d'Annecy de Physique des Particules)

Are the standard approximations used in gravitational lensing still adequate in the era of precision cosmology? If not, how can we move beyond them? As cosmological tensions sharpen, a new generation of surveys such as Euclid and LSST is set to deliver an unprecedented volume of data, with billions of galaxies analyzed through increasingly sophisticated pipelines, including machine-learning techniques. Despite the technical progress, the theoretical framework used to interpret weak lensing observations still relies largely on approximations that have remained essentially unchanged for several decades. In this talk, we revisit the relativistic foundations of gravitational lensing and explore a framework that moves beyond these standard assumptions and towards the precision needs of the next generation of cosmological surveys. In particular, we introduce EXCALIBUR, a numerical approach being developed and designed to model light propagation in ΛCDM cosmological settings through the direct integration of null geodesics in first-order perturbed FLRW spacetimes. This method enables the extraction of fully relativistic lightcones for a given cosmological background and provides a unified description of weak and strong lensing within a single pipeline. It relies on evaluating the optical tidal matrix along photon trajectories using local computations of the Riemann tensor, from which the full Jacobi mapping is reconstructed without relying on perturbative expansions. This framework gives direct access to lensing observables such as convergence and shear, while naturally extending to higher-order distortions beyond the standard weak-lensing formalism. We will discuss the numerical implementation, the physical setup and interpretation of the method, as well as its implications for current and upcoming cosmological surveys.

The magnification bias in w0wa-nu-CDM from N-body simulations and relativistic ray-tracing

• Fabien Castillo (LUX, Observatoire de Paris)

The new generation of galaxy surveys will reach an unprecedented precision, allowing to test cosmology and gravity theories from the relativistic effects on the clustering of matter in the universe. In particular, the magnification bias, the change of observed galaxy density caused by lensing magnification, can be used as a cosmological probe. In this work, we investigate the impact of the magnification bias within the w0wa-nu CDM cosmological framework using a suite of high-resolution N-body simulations. We explore the parameter space using a Latin Hypercube Sampling (LHS), and simulate 32 distinct cosmological models (with a planned extension to at least 64 models) where the different cosmological parameters are varied. Building on the work of Breton et al., 2019 and Rasera et al., 2022, we perform relativistic ray-tracing for each cosmology, to generate lightcone catalogs that include the relativistic effects on the matter distribution, and measure the full set of relativistic 3x2pt correlation functions (galaxy clustering, galaxy–galaxy lensing and cosmic shear). We can then compare the observables computed in the relativistic catalog to the same observables computed assuming the Born approximation, to isolate the effect of the magnification bias on these observables. We then construct an emulator based on Gaussian Processes, trained on the measured correlation functions, to extrapolate the cosmological dependence of the amplitude of the magnification bias. Our emulator provides fast and accurate predictions of the relativistic corrections on the 3x2 pts correlation functions, and can be directly used to perform MCMC from correlation functions measured by observations. We show that the magnification bias induces measurable deviations in 3x2pt statistics, and that the relativistic correction to the galaxy-lensing correlation function is particularly sensitive to the dark energy equation of state. In future work, we plan to extend this analysis to additional observables, such as Minkowski functionals, and to broaden the cosmological framework to include models of modified gravity.

Marked statistics across the cosmic web: Environmental dependent clustering in modified gravity simulations

• Joaquin Armijo

We study environment-dependent clustering using the marked correlation function applied to Hu-Sawicki $f(R)$ modified gravity simulations. This gravity theory enriches the structure formation by enhancing gravity in a scale-dependent form. By employing a multi-scale cosmic structure finder algorithm, we define the cosmic environments divided in: nodes, filaments, walls and voids. We find a stronger impact of modified gravity in nodes and filament, which together dominate the information content by more than a factor of four relative to other environments. Combining environmental information further enhances the expected signal-to-noise ratio for CMASS- and DESI-like mock samples, particularly in configurations including filaments. Overall, marked correlation functions that incorporate environmental structure increase the information content by about a factor of two compared to standard density-based marks applied to the full galaxy sample. These results demonstrate the importance of environmental information, especially from filaments, in improving the constraining power of galaxy clustering tests of modified gravity.

General Relativistic Radiative Transfer Corrections to Cosmic UVB Fluctuations

• Sankalan Bhattacharyya

Current and anticipated large-scale structure surveys are mapping the Lyman-alpha forest on growing cosmological scales, enabling measurements of the Baryon Acoustic Oscillation (BAO) peak and motivating a careful accounting of theoretical systematics at sub-percent precision. Among these systematics, the inhomogeneous Ultra-Violet Background (UVB) has received growing attention, yet its large-scale fluctuations are rarely treated within a relativistic framework. We present a linear-theory description of large-scale UVB fluctuations, connecting ionising source phenomenology directly to large-scale structure observables. Our approach incorporates general relativistic radiative transfer corrections — of the same order as those identified in existing analysis of the relativistic Lyman-Alpha forest (Iršič et al., 2016) — and allows us to quantify their impact on clustering signatures in the cosmic ionising radiation field. The inclusion of GR radiative transfer corrections introduces sub-percent-level corrections to the UVB contribution to the clustering of the Lyman-alpha forest at ultra-large scales.

Hermes - Towards an Ultimate High-Performance Algorithm for Cosmic Statistics of Large Data Sets

• Tengpeng Xu (Sun Yat-sen University)

Driven by the need for fast and efficient algorithms in cosmic statistics to handle the massive data volumes from ongoing and forthcoming galaxy surveys, we present the first public release of Hermes (HypER-speed MultiREsolution cosmic Statistics) — an open-source, parallelized, and GPU-accelerated Python package. At its core, Hermes implements the MRA-CS (MultiResolution Analysis for Cosmic Statistics) framework. The talk will provide a concise overview of the mathematical foundations and core algorithms behind Hermes, demonstrate its performance through numerical tests, and highlight its broad potential for extracting cosmological information from next-generation galaxy surveys.