Reverberation Reveals the Truncated Disc in the Hard State of GX 339-4

The nature and geometry of the hard state in black hole binaries is controversial. The broadband continuum spectrum and fast variability properties can be explained in a model where the inner disc evaporates into a geometrically thick, hot flow. However these models are challenged by the persistent detection of an extremely broad iron line, which requires that the disc extends down to the last stable orbit of a high spin black hole. This line width can be considerably reduced if the Comptonisation continuum is multi-component rather than single temperature, but such models are highly degenerate. Here we show a specific model of a radially stratified continuum coupled to a model of propagating fluctuations, fit to some of the best hard state data from GX 339-4. This full spectral-timing model can fit the time averaged spectrum, the power spectra in different energy bands, and the frequency dependent lags between these bands. For the first time we also include disc reverberation and show that this same spectral-timing model successfully \textit{predicts} the lag-energy spectra on all timescales. This gives a more robust method to determine the inner radius of the disc, which is of order $20~R_g$, i.e. significantly truncated. This opens up the way to use the fast variability spectral-timing data to trace the source geometry of black hole binaries in all states.