X-ray time lags and non-linear 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 that the linear rms-flux relation is present in eight observations of NGC 5408 X-1 and also in three observations of NGC 6946 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 5408 X-1; our results show that this is a persistent property of the variability of NGC 5408 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 NGC 5408 X-1, with the soft band (< 1 keV) delayed with respect to the hard band (> 1 keV) by up to similar to 10 s (similar to 0.2 rad) at frequencies above similar to 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 similar to 1 ks at frequencies of similar to 0.1 mHz. Our energy-resolved analysis shows that 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.