Revisiting the X-ray Variability Plane of AGNs: The Significant Role of the Photon Index

X-ray variability provides a powerful probe of the innermost regions of active galactic nuclei (AGNs), offering valuable insights into the accretion process and the structure of the corona. Previous studies have established a correlation between the X-ray variability timescale, black hole mass, and luminosity, forming the AGN X-ray variability plane. A possible link between the X-ray spectral photon index and X-ray variability was noted in early studies but has rarely been incorporated into subsequent analyses of the variability plane. Moreover, the limited sample sizes in earlier works have limited the robustness and universality of the X-ray variability plane. In this work, we compile a sample of 112 AGNs with 399 exposures from the 4XMM-DR14 catalog and constrain the correlations between X-ray variability timescale, black hole mass, luminosity, and photon index using the recently developed fitting method, BADDAT {(Baseline-Aware Dependence fitting for DAmping Timescales)}, which enables a robust exploration of an extended parameter space. Our analysis confirms the dependence of the rest-frame variability timescale ($\tau_{\rm rest}$) on black hole mass ($M_{\rm BH}$) and further incorporates the photon index ($\Gamma$) into the variability plane, yielding a best-fit relation of $\log (\tau_{\rm rest}/{\rm s}) = 1.22\log (M_{\rm BH}/M_\odot) - 0.24\Gamma - 3.53$, which is strongly favored over the model with $M_{\rm BH}$ alone. In contrast, the inclusion of luminosity does not produce a comparable improvement. The correlation with $\Gamma$ likely reflects the effects of Comptonization and the geometry of the corona.