Distinguishing between dark matter-black hole systems and naked singularities via quasi-periodic oscillations

Abstract Quasi-Periodic Oscillations (QPOs) are an important phenomenon commonly observed in the X-ray radiation of black holes (BH) and neutron stars, closely related to the dynamics of accretion disks around Compact Objects (CO) and general relativistic effects. The objective of this study is to use the QPO phenomenon to distinguish between Dark Matter (DM)-BH systems and Naked Singularities (NS), as well as to investigate the effects of different dark matter models (Cold Dark Matter, CDM, and Scalar Field Dark Matter, SFDM) on the accretion disk dynamics. By introducing a DM correction model within the framework of general relativity, we systematically investigate the differences in dragging effects, characteristic frequency distribution, and the Innermost Stable Circular Orbit (ISCO) radius between DM-BH systems and NS, while analyzing the potential coupling between QPO frequencies and DM distribution. The main results of this study are as follows: In BH cases, the radial precession frequency $$\nu _r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> is consistent with the High-Frequency QPO range only in the low-spin region ( $$a = 0.1 \sim 0.2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn> <mml:mo>∼</mml:mo> <mml:mn>0.2</mml:mn> </mml:mrow> </mml:math> ); the vertical precession frequency $$\nu _\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:math> matches the Low-Frequency QPO range around $$a = 0.5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math> , while the nodal precession frequency $$\nu _{\text {nod}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mtext>nod</mml:mtext> </mml:msub> </mml:math> agrees with the Low-Frequency QPO range around $$a = 0.7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.7</mml:mn> </mml:mrow> </mml:math> , and exhibits frequency reversal in the high-spin region ( $$0.7&lt; a &lt; 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.7</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>a</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ). NS cases, however, display markedly different characteristics: both radial and vertical precession frequencies show monotonic decrease with significant amplification, with $$\nu _r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> reaching kHz levels at $$a = 1 \sim 1.2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>∼</mml:mo> <mml:mn>1.2</mml:mn> </mml:mrow> </mml:math> , and $$\nu _\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:math> reaching 700+ Hz at $$a = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , both far exceeding the QPO observational range; while the nodal precession frequency $$\nu _{\text {nod}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mtext>nod</mml:mtext> </mml:msub> </mml:math> remains consistently in the reverse state but with values falling within the High-Frequency QPO observational range. CDM and SFDM modulate the accretion disk dynamics at the order of $$10^{-6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> </mml:math> .