Here we study the rapid X-ray variability (using XMM-Newton observations) of three neutron-star low-mass X-ray binaries (1RXS J180408.9-342058, EXO 1745-248, and IGR J18245-2452) during their recently proposed very hard spectral state (Parikh et al. 2017). All our systems exhibit a strong to very strong noise component in their power density spectra (rms amplitudes ranging from 34% to 102%) with very low characteristic frequencies (as low as 0.01 Hz). These properties are more extreme than what is commonly observed in the canonical hard state of neutron-star low-mass X-ray binaries observed at X-ray luminosities similar to those we observe from our sources. This suggests that indeed the very hard state is a distinct spectral-timing state from the hard state, although we argue that the variability behaviour of IGR J18245-2452 is very extreme and possibly this source was in a very unusual state. We also compare our results with the rapid X-ray variability of the accreting millisecond X-ray pulsars IGR J00291+5934 and Swift J0911.9-6452 (also using XMM-Newton data) for which previously similar variability phenomena were observed. Although their energy spectra (as observed using the Swift X-ray telescope) were not necessarily as hard (i.e., for Swift J0911.9-6452) as for our other three sources, we conclude that likely both sources were also in very similar state during their XMM-Newton observations. This suggest that different sources that are found in this new state might exhibit different spectral hardness and one has to study both the spectral as well as rapid variability to identify this unusual state.
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