The origin of long soft lags and the nature of the hard-intermediate state in black hole binaries

Fast variability of the X-ray corona in black hole binaries can produce a soft lag by reverberation, where the reprocessed thermalized disc photons lag behind the illuminating hard X-rays. This lag is small, and systematically decreases with increasing mass accretion rate towards the hard-soft transition, consistent with a decreasing truncation radius between the thin disc and X-ray hot inner flow. However, the soft lag suddenly increases dramatically just before the spectrum becomes disc dominated (hard-intermediate state). Interpreting this as reverberation requires that the X-ray source distance from the disc increases dramatically, potentially consistent with switching to X-rays produced in the radio jet. However, this change in lag behaviour occurs without any clear change in hard X-ray spectrum, and before the plasmoid ejection event that might produce such a source (soft-intermediate state). Instead, we show how the soft lag can be interpreted in the context of propagation lags from mass accretion rate fluctuations. These normally produce hard lags, as the model has radial stratification, with fluctuations from larger radii modulating the harder spectra produced at smaller radii. However, all that is required to switch the sign is that the hottest Comptonized emission has seed photons that allow it to extend down in energy below the softer emission from the slower variable turbulent region from the inner edge of the disc. Our model connects the timing change to the spectral change, and gives a smooth transition of the X-ray source properties from the bright hard state to the disc-dominated states.