Fractal temporal dynamics in black hole accretion and quasi-periodic oscillation scaling

Abstract Accretion processes around black holes play a central role in high-energy astrophysics by governing compact object growth and driving luminous emissions across the electromagnetic spectrum. In this work, we propose a theoretical framework in which the local flow of time near the event horizon acquires a fractal structure, potentially arising from quantum gravitational fluctuations at Planckian length scales. This fractal temporal behavior, characterized by a scaling relation $$t_f = t^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$0 &lt; \alpha \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , modifies the time evolution of surface density in viscous accretion disks. We derive a generalized diffusion equation incorporating the fractal-time exponent $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and perform numerical simulations across a wide range of black-hole masses ( $$5\,M_\odot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>5</mml:mn> <mml:mspace/> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:mrow> </mml:math> to $$10^9\,M_\odot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>9</mml:mn> </mml:msup> <mml:mspace/> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:mrow> </mml:math> ) and accretion rates (0.01– $$1.0\,\dot{M}_\textrm{Edd}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.0</mml:mn> <mml:mspace/> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mtext>Edd</mml:mtext> </mml:msub> </mml:mrow> </mml:math> ). Our results reveal enhanced modulation in the accretion luminosity and the emergence of quasi-periodic oscillations (QPOs) in the X-ray light curves. The predicted QPO frequencies, which range from 4– $$8\,\textrm{Hz}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mspace/> <mml:mtext>Hz</mml:mtext> </mml:mrow> </mml:math> for stellar-mass black holes, scale with the fractal exponent and follow $$\nu _\textrm{QPO}\propto M_\textrm{BH}^{-\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mtext>QPO</mml:mtext> </mml:msub> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mtext>BH</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> (classical limit $$\alpha =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), remaining within the detection capabilities of current instruments such as NICER, XMM-Newton, and NuSTAR. These findings indicate that microscopic spacetime fluctuations may leave macroscopic imprints on accretion dynamics, thereby providing an observational probe of fractal-time behavior near black holes. This work offers a testable connection between quantum gravitational microphysics and observable X-ray variability.