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Calibrating the correlation between black hole mass and X-ray variability amplitude: X-ray only black hole mass estimates for active galactic nuclei and ultra-luminous X-ray sources

A calibration is made for the correlation between the X-ray Variability Amplitude (XVA) and Black Hole (BH) mass. The correlation for 21 reverberation-mapped Active Galactic Nuclei (AGN) appears very tight, with an intrinsic dispersion of 0.20 dex. The intrinsic dispersion of 0.27 dex can be obtained if BH masses are estimated from the stellar velocity dispersions. We further test the uncertainties of mass estimates from XVAs for objects which have been observed multiple times with good enough data quality. The results show that the XVAs derived from multiple observations change by a factor of 3. This means that BH mass uncertainty from a single observation is slightly worse than either reverberation-mapping or stellar velocity dispersion measurements; however BH mass estimates with X-ray data only can be more accurate if the mean XVA value from more observations is used. Applying this relation, the BH mass of RE J1034+396 is found to be $4^{+3}_{-2} \times 10^6$ $M_{\odot}$. The high end of the mass range follows the relationship between the 2$f_0$ frequencies of high-frequency QPO and the BH masses derived from the Galactic X-ray binaries. We also calculate the high-frequency constant $C= 2.37 M_\odot$ Hz$^{-1}$ from 21 reverberation-mapped AGN. As suggested by Gierliński et al., $M_{\rm BH}=C/C_{\rm M}$, where $C_{\rm M}$ is the high-frequency variability derived from XVA. Given the similar shape of power-law dominated X-ray spectra in ULXs and AGN, this can be applied to BH mass estimates of ULXs. We discuss the observed QPO frequencies and BH mass estimates in the Ultra-Luminous X-ray source M82 X-1 and NGC 5408 X-1 and favor ULXs as intermediate mass BH systems (abridged).