Alan F. Heavens

The Euclid satellite will deliver a catalogue of optically selected galaxy clusters spanning from around 2000 deg ² in Data Release (DR) 1 to around 14 000 deg ² in DR3. In this work, we assess the validity of cluster clustering (CC) models for template-fitting, which complements the full-shape methodology by providing cosmological information from the anisotropy of the redshift-space two-point correlation function (2PCF). Both methods will be used to analyse the cluster 2PCF multipoles with Euclid . We examined the multipoles of the two-point redshift-space clustering of galaxy clusters simulated with the semi-analytic PINOCCHIO code using third-order Lagrangian perturbation theory, assuming a Euclid DR1-like footprint of 500 deg ² in the northern hemisphere and 1400 deg ² in the southern hemisphere. We estimated the first three even multipoles of the 2PCF and associated covariance matrix from 1000 DR1-like synthetic catalogues. We studied the impact of modelling the relevant non-linearities, halo bias, and photometric redshift uncertainties on the 2PCF. We applied three clustering models to the mock catalogues at 0 < z < 2 and with a virial mass of M _vir > 10 ¹⁴ h ⁻¹ M _⊙ under realistic and optimistic photometric redshift uncertainty scenarios. We formulated a set of permissive and conservative criteria that ought to be fulfilled by the multipole cut-off scales and validated them against 100 mock catalogues via an inference of the growth rate multiplied by the matter power spectrum normalisation parameter, fσ ₈ . We tested the dispersion, Scoccimarro, and Taruya–Nishimichi–Saito models. We find that the simplest of the three (i.e. the dispersion model) yields unbiased inferences on fσ ₈ from CC down to 10 h ⁻¹ Mpc in a DR1-like setting. All clustering models provide very similar goodness-of-fit metrics in the presence of DR1-like cluster redshift uncertainties.

The statistical analysis of cosmological data often assumes a Gaussian sampling distribution and relies on covariance matrices estimated from simulations. In this setting, the likelihood function of the data is not Gaussian but is instead a multivariate Student-t distribution, arising from marginalisation over an inverse-Wishart distribution for the true covariance matrix. This framework, introduced by Sellentin & Heavens (2016) and extended by Percival et al. ( 2022), provides a principled drop-in replacement to the Gaussian likelihood with Hartlap correction (Hartlap et al. 2007). The latter removes bias in the precision matrix; it is still widely used, despite failing to reproduce the heavy tails of the true distribution (thus yielding inaccurate probabilities, especially in the case of tensions between datasets). In practice, cosmological analyses frequently involve additional Gaussian error contributions, for example from instrumental noise, foregrounds, super-sample covariance, or emulator uncertainties. The resulting likelihood function is a convolution of the Sellentin-Heavens or Percival likelihoods with an extra Gaussian contribution, and does not have a simple expression. In this note, we derive an accurate approximation for the combined likelihood function, another multivariate Student-t distribution which inherits the heavy tails. The parameters of the Student-t distribution are determined by matching the covariance and multivariate kurtosis to those of the true distribution. We also include a slightly more expensive but fast sampling algorithm, based on the mixture representation of the Student-t distribution, which avoids the approximation altogether, but is not the drop-in replacement for the normal Gaussian or Hartlap likelihood function that the Student-t approximation in this paper provides.

In Hamiltonian Monte Carlo sampling, the shape of the potential and the choice of the momentum distribution jointly give rise to the Hamiltonian dynamics of the sampler. An efficient sampler propagates quickly in all regions of the parameter space, so that the chain has a low autocorrelation length and the sampler has a high acceptance rate, with the goal of optimising the number of near-independent samples for given computational cost. Standard Gaussian momentum distributions allow arbitrarily large velocities, which can lead to inefficient exploration in posteriors with ridges or funnel-like geometries. We investigate alternative momentum distributions based on relativistic and Student's t kinetic energies, which naturally limit particle velocities and may improve robustness. Using Almanac, a sampler for cosmological posterior distributions of sky maps and power spectra on the sphere, we test these alternatives in both low- and high-dimensional settings. We find that the choice of parameterization and momentum distribution can improve convergence and effective sample rate, though the achievable gains are generally modest and strongly problem-dependent, reaching up to an order of magnitude in favorable cases. Among the momentum distributions that we tested, those with moderately heavy tails achieved the best balance between efficiency and stability. These results highlight the importance of sampler design and encourage future work on adaptive and self-tuning strategies for kinetic energy parameter optimization in high-dimensional settings.

The Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) will produce unprecedented volumes of heterogeneous astronomical data (images, catalogs, and alerts) that challenge traditional analysis pipelines. The LSST Dark Energy Science Collaboration (DESC) aims to derive robust constraints on dark energy and dark matter from these data, requiring methods that are statistically powerful, scalable, and operationally reliable. Artificial intelligence and machine learning (AI/ML) are already embedded across DESC science workflows, from photometric redshifts and transient classification to weak lensing inference and cosmological simulations. Yet their utility for precision cosmology hinges on trustworthy uncertainty quantification, robustness to covariate shift and model misspecification, and reproducible integration within scientific pipelines. This white paper surveys the current landscape of AI/ML across DESC's primary cosmological probes and cross-cutting analyses, revealing that the same core methodologies and fundamental challenges recur across disparate science cases. Since progress on these cross-cutting challenges would benefit multiple probes simultaneously, we identify key methodological research priorities, including Bayesian inference at scale, physics-informed methods, validation frameworks, and active learning for discovery. With an eye on emerging techniques, we also explore the potential of the latest foundation model methodologies and LLM-driven agentic AI systems to reshape DESC workflows, provided their deployment is coupled with rigorous evaluation and governance. Finally, we discuss critical software, computing, data infrastructure, and human capital requirements for the successful deployment of these new methodologies, and consider associated risks and opportunities for broader coordination with external actors.

The current standard model of cosmology successfully describes a variety of measurements, but the nature of its main ingredients, dark matter and dark energy, remains unknown. Euclid is a medium-class mission in the Cosmic Vision 2015–2025 programme of the European Space Agency (ESA) that will provide high-resolution optical imaging, as well as near-infrared imaging and spectroscopy, over about 14 000 deg² of extragalactic sky. In addition to accurate weak lensing and clustering measurements that probe structure formation over half of the age of the Universe, its primary probes for cosmology, these exquisite data will enable a wide range of science. This paper provides a high-level overview of the mission, summarising the survey characteristics, the various data-processing steps, and data products. We also highlight the main science objectives and expected performance.

Cosmological inference relies on compressed forms of the raw data for analysis, with traditional methods exploiting physics knowledge to define summary statistics, such as power spectra, that are known to capture much of the information. An alternative approach is to ask a neural network to find a set of informative summary statistics from data, which can then be analysed either by likelihood- or simulation-based inference. This has the advantage that for non-Gaussian fields, they may capture more information than two-point statistics. However, a disadvantage is that the network almost certainly relearns that two-point statistics are informative. In this paper, we introduce a new hybrid method, which combines the best of both: we use our domain knowledge to define informative physics-based summary statistics, and explicitly ask the network to augment the set with extra statistics that capture information that is not already in the existing summaries. This yields a new, general loss formalism that reduces both the number of simulations and network size needed to extract useful non-Gaussian information from cosmological fields, and guarantees that the resulting summary statistics are at least as informative as the power spectrum. In combination, they can then act as powerful inputs to implicit inference of model parameters. We use a generalisation of Information Maximising Neural Networks (IMNNs) to obtain the extra summaries, and obtain parameter constraints from simulated tomographic weak gravitational lensing convergence maps. We study several dark matter simulation resolutions in low- and high-noise regimes. We show that i) the information-update formalism extracts at least 3× and up to 8× as much information as the angular power spectrum in all noise regimes, ii) the network summaries are highly complementary to existing 2-point summaries, and iii) our formalism allows for networks with extremely lightweight architectures to match much larger regression networks with far fewer simulations needed to obtain asymptotically optimal inference.

We develop extreme data compression for use in Bayesian model comparison via the MOPED algorithm, as well as more general score compression. We find that Bayes Factors from data compressed with the MOPED algorithm are identical to those from their uncompressed datasets when the models are linear and the errors Gaussian. In other nonlinear cases, whether nested or not, we find negligible differences in the Bayes Factors, and show this explicitly for the Pantheon-SH0ES supernova dataset. We also investigate the sampling properties of the Bayesian Evidence as a frequentist statistic, and find that extreme data compression reduces the sampling variance of the Evidence, but has no impact on the sampling distribution of Bayes Factors. Since model comparison can be a very computationally-intensive task, MOPED extreme data compression may present significant advantages in computational time.