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X-ray time lags and nonlinear variability in the ultraluminous X-ray sources NGC 5408 X-1 and NGC 6946 X-1

We present our analysis of the X-ray variability of two ultraluminous X-ray sources (ULXs) based on multiple XMM--Newton observations. We show the linear rms-flux relation is present in eight observations of NGC5408 X-1 and also in three observations of NGC6946 X-1, but data from other ULXs are generally not sufficient to constrain any rms-flux relation. The presence of this relation was previously reported in only two observations of NGC 5408X-1; our results show this is a persistent property of the variability of NGC5408 X-1 and extends to at least one other variable ULX. We speculate this is a ubiquitous property of ULX variability, as it is for X-ray variability in other luminous accreting sources. We also recover the time delay between hard and soft bands in NGC5408 X-1, with the soft band (<1 keV) delayed with respect to the hard band (>1 keV) by up to ~10 s (~0.2 rad) at frequencies above ~few mHz. For the first time, we extend the lag analysis to lower frequencies and find some evidence for a reversal of the lag, a hard lag of ~1 ks at frequencies of ~0.1 mHz. Our energy-resolved analysis shows the time delays are energy dependent. We argue that the lag is unlikely to be a result of reflection from an accretion disc (`reverberation') based on the lack of reflection features in the spectra, and the large size of the reflector inferred from the magnitude of the lag. We also argue that associating the soft lag with a quasi-periodic oscillation (QPO) in these ULXs - and drawing an analogy between soft lags in ULXs and soft lags seen in some low-frequency QPOs of Galactic X-ray binaries - is premature.