Universe

Nested sampling is widely used in astrophysics and cosmology to reliably infer model parameters and compare models within a Bayesian framework. To address models with many parameters, Markov Chain Monte Carlo (MCMC) random walks are used within nested sampling to advance a live point population. Diagnostics for nested sampling inference are essential to verify that trustworthy astrophysical conclusions can be drawn. We develop a diagnostic to identify poorly mixed random walks that do not meet the requirements of nested sampling. The distance from the start to the end of the random walk, the jump distance, is compared to the typical neighbour distance between live points, to obtain a relative jump distance (RJD). A robust estimate of a typical neighbour Mahalanobis distance is obtained with the MLFriends algorithm. In mock and real-world inference applications such as inferring the distance to GW170817, we observe the relative jump distance. We propose the geometric mean <RJD> and fraction of RJD > 1 as new summary diagnostics. To evaluate the sensitivity of the new diagnostics, problematic runs are identified by comparison with reruns using much longer MCMC chains. Trustworthy inference runs are consistently associated with <RJD> above 1.1, while problematic runs can fall below these thresholds. The RJD is more sensitive than previous tests based on the live point insertion order. The RJD diagnostic is proposed as a widely applicable diagnostic to verify inference with nested sampling. It is implemented in the UltraNest package in version 4.1.

In the original publication [...]

This article is the second part of a review devoted to spacetime singularities in classical general relativity. While Primer I developed the mathematical framework for defining, classifying, and proving the existence of singularities, the present work examines their concrete realization in spatially homogeneous cosmological models. After reviewing the geometrical foundations of homogeneity and the Bianchi classification, we derive the field equations for orthogonal and tilted perfect-fluid cosmologies and use the Kasner and Bianchi I solutions as elementary prototypes of anisotropic singular behavior. Raychaudhuri focusing provides general conditions under which the homogeneous Cauchy development reaches a finite-proper-time boundary. We then distinguish matter singularities, characterized by divergent density and Ricci curvature, from more subtle non-scalar or ‘whimper’ singularities associated with Cauchy horizons and extreme fluid tilt. The classical Ellis–King analysis is reconsidered in the light of later developments, including rigorous curvature-blow-up results for Bianchi VIII and IX models, kinematic singularities in future geodesically complete tilted spacetimes, and the recent reappearance of whimper behavior in dipole cosmology. The resulting picture shows that even spatially homogeneous cosmologies exhibit a remarkably rich hierarchy of singular boundaries. The detailed dynamical approach to these boundaries—Hamiltonian cosmology, Kasner transitions, Mixmaster dynamics, and the BKL conjecture—is reserved for Primer III.