  Accreting black holes show characteristic reflection features in their X-ray
spectrum, including an iron K$\alpha$ line, resulting from hard X-ray continuum
photons illuminating the accretion disk. The reverberation lag resulting from
the path length difference between direct and reflected emission provides a
powerful tool to probe the innermost regions around both stellar-mass and
supermassive black holes. Here, we present for the first time a reverberation
mapping formalism that enables modeling of energy dependent time lags and
variability amplitude for a wide range of variability timescales, taking the
complete information of the cross-spectrum into account. We use a pivoting
power-law model to account for the spectral variability of the continuum that
dominates over the reverberation lags for longer time scale variability. We use
an analytic approximation to self-consistently account for the non-linear
effects caused by this continuum spectral variability, which have been ignored
by all previous reverberation studies. We find that ignoring these non-linear
effects can bias measurements of the reverberation lags, particularly at low
frequencies. Since our model is analytic, we are able to fit simultaneously for
a wide range of Fourier frequencies without prohibitive computational expense.
We also introduce a formalism of fitting to real and imaginary parts of our
cross-spectrum statistic, which naturally avoids some mistakes/inaccuracies
previously common in the literature. We perform proof-of-principle fits to
Rossi X-ray Timing Explorer data of Cygnus X-1.

High Energy Astrophysical Phenomena