An exact analytic treatment of propagating mass accretion rate fluctuations in X-ray binaries

Many statistical properties of the aperiodic variability observed in X-ray radiation from accreting compact objects can be naturally explained by the propagating fluctuations model. This considers variations in mass accretion rate to be stirred up throughout the accretion flow. Variations from the outer regions of the accretion flow will propagate towards the central object, modulating the variations from the inner regions and eventually modulating the radiation, giving rise to the observed linear rms-flux relation and also Fourier frequency-dependent time lags. Previous treatments of this model have relied on computationally intensive Monte Carlo simulations which can only yield an estimate of statistical properties such as the power spectrum. Here, we find exact and analytic expressions for the power spectrum and lag spectrum predicted by the same model. We use our calculation to fit the model of Ingram & Done to a power spectrum of XTE J1550-564. The result we present here will apply to any treatment of the propagating fluctuations model and thus provides a very powerful tool for future theoretical modelling.