Gravitational waveforms from periodic orbits around a charged black hole with scalar hair

TL;DR

The study analyzes geodesic motion and gravitational-wave signatures around charged black holes with scalar hair, focusing on marginally bound orbits, ISCOs, and periodic orbits indexed by (z,w,v). Using effective-potential methods and a numerical kludge for EMRI waveforms, it demonstrates that the scalar-hair parameter $r_B$ systematically shifts orbital boundaries inward, increases the angular momentum required for stable motion, and lowers orbit energies. Periodic orbits exhibit q-based classifications whose energy and angular-momentum characteristics shift with $r_B$, producing increasingly intricate zoom-whirl morphologies. The resulting EMRI waveforms show clear $r_B$-dependent amplitude and phase modifications, suggesting that future space-based detectors like LISA could probe deviations from classical black-hole spacetimes through EMRI observations.

Abstract

We investigate geodesic motion and gravitational-wave signatures of charged black holes with scalar hair. Using the effective potential approach, we analyze marginally bound orbits and innermost stable circular orbits, showing how their positions and energy thresholds are modified by the scalar hair parameter $r_B$. These results demonstrate scalar hair's role in altering the boundary of stable motion. We further explore periodic orbits characterized by rational frequency ratios, labeled by the index $(z,w,v)$, and quantify how scalar hair affects their orbital energy and angular momentum. Based on these orbital properties, we compute gravitational waveforms from extreme mass-ratio inspirals where a stellar-mass compact object orbits a supermassive charged black hole with scalar hair. Using the numerical kludge method, we generate waveforms that exhibit clear zoom-whirl patterns with morphology visibly affected by $r_B$. Our results show that scalar hair leaves distinguishable imprints on waveforms, suggesting future space-based detectors could probe deviations from classical black hole spacetimes through extreme mass-ratio inspirals observations.

Figures (9)

• Figure 1: Effective potential \ref{['eq2']} for test particle motion in charged black holes with scalar hair. (\ref{['Fig1a']}) Potential profiles for varying angular momentum at fixed $r_B$. (\ref{['Fig1b']}) Behavior for varying $r_B$ at constant angular momentum.
• Figure 2: MBO properties around charged black holes with scalar hair. (a) MBO radius versus $r_B$. (b) MBO angular momentum versus $r_B$. Radius decreases while angular momentum increases with $r_B$.
• Figure 3: ISCO properties around charged black holes with scalar hair. (a) ISCO radius versus $r_B$. (b) ISCO angular momentum versus $r_B$. (c) ISCO energy versus $r_B$. Radius and energy decrease while angular momentum increases with $r_B$.
• Figure 4: Allowed bound orbit parameter space around charged black holes with scalar hair, showing orbital angular momentum versus energy for different $r_B$ values.
• Figure 5: (a) Rational number $q$ versus periodic orbit energy for different $r_B$, with fixed $L=(L_{\text{MBO}}+L_{\text{ISCO}})/2$. (b) $q$ versus angular momentum for different $r_B$, with fixed $E=0.96$.