Probing Hairy Kerr Black Holes through Quasi-Periodic Oscillations I: A study based on the kinematic models

TL;DR

This work investigates rotating hairy Kerr black holes generated by extended gravitational decoupling, characterized by deviation parameters $\alpha$ and $l$, and analyzes their horizon structure and geodesic epicyclic frequencies. By computing $f_{\phi}$, $f_{r}$, and $f_{\theta}$ in the hairy Kerr spacetime and applying two kinematic HFQPO models—Relativistic Precession (RPM) and Tidal Disruption (TDM)—the authors confront six BH sources with HFQPO data, using $\chi^2$ fitting and MCMC to constrain the hair parameters. The results show that for GRO J1655-40 and XTE J1859+226, RPM favors hairy Kerr with $l\sim0.1$, $\alpha\sim12$, while for several other sources the data are inconclusive, often preferring Kerr or yielding similar goodness for both models. Across sources, TDM generally provides weaker constraints on the hair parameters, and model comparison via AIC indicates (in many cases) no decisive preference between RPM and TDM. The study demonstrates how HFQPO observations can probe non-Kerr spacetimes and outlines a framework for integrating future, more precise data to distinguish between hairy and Kerr black holes in strong gravity regimes.

Abstract

Black hole (BH) solutions endowed with nontrivial scalar or matter fields - commonly known as hairy black holes - have attracted significant interest in recent years. They admit extra parameters beyond mass, charge, and angular momentum, leading to a richer phenomenology. Characterizing their allowed parameter space is therefore crucial, especially in light of the available data from electromagnetic and gravitational-wave observations. The rotating hairy black hole solutions studied here are inspired by the gravitational decoupling method and satisfy the Einstein field equations with a conserved energy-momentum tensor that respects the strong energy conditions (SECs). We explore in detail the horizon structure of such black holes and report for the first time certain unique features, not observed in Kerr BHs. We investigate the sensitivity of the hairy parameters on the fundamental frequencies associated with the motion of matter in the hairy Kerr spacetime, and compare them with the Kerr scenario. The theoretical models aimed to explain the observed high-frequency observations (HFQPOs) in BHs are associated with the fundamental frequencies. Hence, such a study enables us to constrain the parameter space of the hairy Kerr spacetime by comparing the model-dependent HFQPO frequencies with the available observations. By comparing the kinematic models of HFQPOs with the observations of six BH sources, we report the most favored parameter space for each of these BHs. Our analysis also provides a framework to discern the most suitable model for each of these sources. Interestingly, even with the present precision of the data, the Relativistic Precession Model seems to be less suitable compared to the Tidal Disruption Model for the sources GRO J1655-40 and XTE J1859+226. The implications are discussed.

Figures (21)

• Figure 3: In the figure above, we consider the case $l = e^{-2}$. Figure (a) shows the behavior of $\Delta(r)$ for several values of $\alpha$, including $\alpha < \alpha_{crit}$ ($\alpha = 5$), the critical case $\alpha = \alpha_{crit} = e^{2}$, and values $\alpha > \alpha_{crit}$ ($\alpha = 7.8,\, 8.15,\, 10$). Solid curves represent the non-rotating case $a = 0$, while dashed curves correspond to the extremal configurations $a = a_{max}$. The vertical dotted lines indicate $r = r_{SEC}$ for each choice of $\alpha$. Figure (b) displays the corresponding dependence of the maximum allowed spin $a_{max}$ on $\alpha$ within the interval $\alpha_{crit} \leq \alpha \leq \alpha_{max}$.
• Figure 4: In the figure above, we consider the case $l = 0.1 < e^{-2}$. Figure (a) shows the behavior of $\Delta(r)$ for several values of $\alpha$, including $\alpha < \alpha_{crit}$ ($\alpha = 10$), the critical case $\alpha = \alpha_{crit}$, and values $\alpha > \alpha_{crit}$ ($\alpha = 13,\, \alpha_c,\, 15$). Solid curves represent the non-rotating case $a = 0$, while dashed curves correspond to the extremal configurations $a = a_{max}$. The vertical dotted lines indicate $r = r_{SEC}$ for each choice of $\alpha$. Figure (b) displays the corresponding dependence of the maximum allowed spin $a_{max}$ on $\alpha$ within the interval $\alpha_{crit} \leq \alpha \leq \alpha_{max}$.
• Figure 5: The above figures show the variation of $f_\phi$ (a) with $r$ for $\alpha=0$ (Kerr scenario), (b) with $r$ for $l=0.1, \alpha=12$, and (c) with $r$ for $l=0.1, \alpha=14$ ,(d) with $\alpha$ for $l=0.1$, $r=5.5$, (e) with $r$ for $l=e^{-2}, \alpha=7.5$, (f) with $r$ for $l=e^{-2}, \alpha=8$, (g) with $\alpha$ for $l=e^{-2}, r=5.5$, (h) with $r$ for $l=0.3, \alpha=2$, (i) with $r$ for $l=0.3, \alpha=4$, (j) with $\alpha$ for $l=0.3, r=6$. In each case we have considered variation with respect to spins from $-a_{max}$ to $a_{max}$. The above figure is plotted for a $M=10 M_\odot$ BH.
• Figure 6: The above figures show the variation of $f_\phi$ (a) with $r$, for $\alpha=0.5, l=1$, (b) with $r$, for $\alpha=0.5, l=2$, (c) with $l$, for $\alpha=0.5$ at $r=6$, (d) with $r$, for $\alpha=2, l=0.3$, (e) with $r$, for $\alpha=2, l=0.6$, (f) with $l$, for $\alpha=2$, at $r=6$, (g) with $r$, for $\alpha=6, l=0.15$ , (h) with $r$, for $\alpha=6, l=0.2$, (i) with $r$, for $\alpha=6, l=0.3$, (j) with $r$, for $\alpha=12, l=0.1$, (k) with $r$, for $\alpha=12, l=e^{-2}$, (l) with $r$, for $\alpha=12, l=0.15$. In each case we have considered variation with respect to spins from $-a_{max}$ to $a_{max}$. The above figures are made for a $M=10 M_\odot$ BH.
• Figure 7: The above figures show the radial variation of $f_r$ for (a) $\alpha=0$ (Kerr scenario), (b) $l=0.1, \alpha=12$, (c) $l=0.1, \alpha=14$, (d) $l=0.1,\alpha=16$, (e) $l=e^{-2}, \alpha=7.5$, (f) $l=e^{-2}, \alpha=8$, (g) $l=e^{-2}, \alpha=10$, (h) $l=0.3, \alpha=2$, (i) $l=0.3, \alpha=4$, and (j) $l=0.3, \alpha=6$. In each subfigure, we have shown variation with respect to spins in the entire allowed range. The above plot is made for a $M=10 M_\odot$ BH.