How Mass Flows Through Accretion Discs: A Spectral-Timing Vision for the 2040s

Understanding how mass and angular momentum flow through accretion discs remains a fundamental unsolved problem in astrophysics. Accreting white dwarfs offer an ideal laboratory for addressing this question: their variability occurs on accessible timescales of seconds to minutes, and their optical spectra contain continuum and emission-line components that trace distinct disc regions. Broad-band timing studies have revealed time-lags similar to those observed in X-ray binaries and active galactic nuclei, suggesting propagating fluctuations and possible coupling to an inner hot flow. However, the blending of line and continuum light in broad filters prevents a physical interpretation of these signals. The 2040s will bring an unprecedented number of disc-accreting systems discovered by Rubin-LSST, space-based gravitational-wave observatories, and third-generation ground and space-based detectors. To extract disc physics from these sources, high-cadence optical spectral-timing, simultaneously resolving continuum and individual lines, is essential. Such measurements would directly map how variability propagates through discs, determine how the outer disc responds to changes in the inner flow, and test whether accretion physics is scale-invariant from white dwarfs to supermassive black holes. This white paper outlines the scientific motivation and observational capabilities required to realise this vision. It highlights the opportunity for ESO to enable a transformative new window on accretion physics in the coming decade.