The power spectra of black hole binaries have been well studied for decades, giving a very detailed phenomenological picture of the variability properties and their correlation with the energy spectrum (spectral state) of the source. Here we take the truncated disc/hot inner flow picture which can describe the spectral changes, and show that propagating mass accretion rate fluctuations in the hot flow can match the broad-band power spectral properties seen in black hole binaries, i.e. give an approximately band-limited noise between a low- and a high-frequency break. The low-frequency break marks the viscous time-scale at the outer edge of the hot inner flow, which is the inner edge of the truncated disc. The fluctuations in mass accretion rate propagate towards the central object in a finite time meaning the high-frequency break is more complex than simply the viscous time-scale at the inner edge of the hot flow because fluctuations on time-scales shorter than the propagation time are incoherent. The model also predicts the Lense–Thirring precession time-scale of the hot flow, as this is set by the combination of inner and outer radius of the flow, together with its surface density which is self-consistently calculated from the propagating fluctuations. We show that this naturally gives the observed relation between the low-frequency break and quasi-periodic oscillation (QPO) frequency as the outer radius of the flow moves inwards, and that this model predicts many of the observed QPO properties such as correlation of coherence with frequency, and of the recently discovered correlation of frequency with flux on short time-scales.
We fit this total model of the variability to a sequence of five observed power spectra from the bright black hole binary XTE J1550−564 as the source transitioned from a low/hard to very high state. This is the first time that a power spectrum from a black hole binary has been fit with a physical model for the variability. The data are well fit if the inner radius of the flow remains constant, while the outer radius sweeps inwards from ∼75 to 12 gravitational radii. This range of radii is the same range as required by models of the energy spectral evolution, giving the first self-consistent description of the evolution of both the spectrum and variability of black hole binaries.