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The stochastic dynamics of a grain embedded within a turbulent fluid subject to strong gravitational fields can be formulated as a random walk on a Riemannian manifold. Such curvature-weighted walks provide a framework to model the intrinsic variability of accretion onto compact objects. By solving the relevant Fokker-Planck equation on a black hole background, we find the counterintuitive result that the escape probability of a grain is actually higher compared to flat space. This is a consequence of the stretching of radial cells near the event horizon: there is a greater spatial volume for the particle to wander through before being captured. By simulating a large number of grain trajectories, initially distributed on concentric shells with a density profile set by the thin-disc structure equations, we also study particle fluxes through the horizon. Shallower spectral indices emerge at low frequencies relative to flat space, primarily due to time dilation, and steeper ones at high frequencies. We find that Schwarzschild-weighted spectra broadly match observations of low-frequency X-ray variability from systems like Cygnus X-1 in their hard state, suggesting that geometric drifts may be important in describing stochastic accretion processes.

We demonstrate that the properties of eccentric gravitational-wave (GW) signals enhance the detectability of GW phase shifts caused by environmental effects (EEs). The signal-to-noise ratio (SNR) of EEs can be boosted by up to <inline-formula> <mml:math><mml:msubsup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>max</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math> </inline-formula> with respect to corresponding circular signals, where <inline-formula> <mml:math><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub></mml:math> </inline-formula> is the highest modeled eccentric GW harmonic and n is the frequency scaling of the GW dephasing prescription associated to the EE. We investigate the impact on a population level, adopting plausible eccentricity distributions for binary sources observed by LIGO─Virgo─Kagra (A+ and A# sensitivities), as well as Cosmic Explorer (CE) and the Einstein Telescope (ET). For sources in the high-eccentricity tail of a distribution (e ≳ 0.2 at 10 Hz), phase shifts can systematically be up to <inline-formula> <mml:math><mml:msubsup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>max</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math> </inline-formula> times smaller than in a corresponding circular signal and still be detectable. For typical EEs, such as Roemer delays and gas drag, this effect amounts to SNR enhancements that range from 10<SUP>2</SUP> up to 10<SUP>5</SUP>. For CE and ET, our analysis shows that EEs will be an ubiquitous feature in the eccentric tail of merging binaries, regardless of the specific details of the formation channel. Additionally, we find that the joint analysis of eccentricity and phase shift is already plausible in current catalogs if a fraction of binaries merge in active galactic nucleus migration traps.

The $Λ$ cold dark matter model successfully describes a wide range of cosmological observations. However, the persistent discrepancy between the value of the Hubble constant inferred from cosmic microwave background measurements within this model and that obtained from local distance-ladder determinations points to a significant inconsistency. This short review examines theoretical responses across the cosmological inference chain, from the gravitational field equations to the pre-recombination sound horizon and the late-time distance-redshift relation. We focus primarily on modified gravity, and briefly discuss early- and late-time mechanisms that can alter the acoustic ruler, distance measures, structure growth, or gravitational response. Current proposals can reduce the nominal tension, but often at the cost of correlated shifts in cosmic microwave background spectra, standard-ruler distances, lensing, structure growth, or calibrator information. Further progress requires unified likelihoods and multiprobe tests linking all key observables under the same model assumptions.