Large and complex X-ray time lags from black hole accretion discs with compact inner coronae

Black hole X-ray binaries in their hard and hard-intermediate states display hard and soft time lags between broad-band noise variations (high-energy emission lagging low-energy and vice versa), which could be used to constrain the geometry of the disc and Comptonizing corona in these systems. Comptonization and reverberation lag models, which are based on light-travel delays, can imply coronae that are very large (hundreds to thousands of gravitational radii, $R_{g}$) and in conflict with constraints from X-ray spectral modelling and polarimetry. Here, we show that the observed large and complex X-ray time lags can be explained by a model where fluctuations are generated in and propagate through the blackbody-emitting disc to a relatively compact ($\sim$10 $R_{g}$) inner corona. The model naturally explains why the disc variations lead coronal variations with a Fourier-frequency dependent lag at frequencies $\lt 1$ Hz, since longer variability time-scales originate from larger disc radii. The propagating fluctuations also modulate successively the coronal seed photons from the disc, heating of the corona via viscous dissipation and the resulting reverberation signal. The interplay of these different effects leads to the observed complex pattern of lag behaviour between disc and power-law emission and different power-law energy bands, the energy-dependence of power-spectral shape, and a strong dependence of spectral-timing properties on coronal geometry. The observed spectral-timing complexity is thus a natural consequence of the response of the disc-corona system to mass-accretion fluctuations propagating through the disc.