Periodic orbits and gravitational waveforms around a Schwarzschild black hole with a cloud of strings embedded in perfect fluid dark matter

Abstract: In this study, we explore the dynamics of particle orbits and their corresponding gravitational wave signatures in the vicinity of a Schwarzschild black hole (BH) surrounded by a cloud of strings and embedded in a perfect fluid dark matter medium. The model is characterized by two parameters: $a$, associated with the string cloud, and $α$, representing the dark matter distribution. We systematically analyze how the marginally bound orbit (MBO) and the innermost stable circular orbit (ISCO) depend on these parameters. Our findings reveal that while both the orbital radius and angular momentum increase with increasing $a$, they decrease as $α$ increases; notably, the energy exhibits the opposite trend, decreasing with $a$ and increasing with $α$. Furthermore, we examine periodic orbits indexed by rational numbers $q$ and the gravitational waveforms they generate. The results demonstrate that an increase in the string cloud parameter $a$ induces a significant phase delay in the waveform. Specifically, waveforms with lower values of $a$ oscillate over shorter time intervals, whereas those with higher values extend to longer time scales. These distinct features, including noticeable differences in amplitude, allow the waveforms to be clearly distinguished from those in a pure Schwarzschild spacetime.

• The paper derives how string-cloud parameter a and dark-matter parameter α oppositely modify marginally bound orbits, ISCO properties, and the allowed energy–angular-momentum region.
• Periodic-orbit analysis shows that the frequency ratio q increases with energy, decreases with angular momentum, and shifts systematically with a and α, producing distinctive orbit morphologies.
• Quadrupole EMRI waveforms for a (3,2,2) orbit display parameter-dependent phase delays and amplitude changes, suggesting sensitivity to environmental matter with future LISA and Taiji observations, although radiation reaction is not included.

Spacetime model and geodesic framework

The paper analyzes timelike geodesics in a static, spherically symmetric spacetime combining two environmental matter components: a Letelier cloud of strings and a perfect fluid dark matter (PFDM) halo. The metric function is

$$f(r) = 1 - a - \frac{2M}{r} + \frac{\alpha}{r}\ln\frac{r}{|\alpha|},$$

where $a$ characterizes the string cloud and $\alpha$ the PFDM distribution; both reduce to zero to recover Schwarzschild. Working on the equatorial plane, the conserved specific energy $E = f(r)\dot{t}$ and angular momentum $L = r^2\dot{\phi}$ follow from the Killing vectors $\partial_t$ and $\partial_\phi$, yielding the radial equation $\dot{r}^2 = E^2 - V_{\text{eff}}$ with $V_{\text{eff}} = f(r)(1 + L^2/r^2)$.

The bound-orbit parameter space is bounded by the marginally bound orbit (MBO), defined by $V_{\text{eff}} = 1$ and $a$0, and the innermost stable circular orbit (ISCO), obtained from $a$1. Closed-form expressions for $a$2, $a$3, and $a$4 are derived in terms of $a$5 and its derivatives at the ISCO radius.

Dependence of critical orbits on $a$6 and $a$7

The central structural result is that the two parameters act in opposition. Both $a$8 and $a$9 increase with $\alpha$0 and decrease with increasing $\alpha$1. The ISCO quantities exhibit the same pattern for radius and angular momentum, while the energy behaves inversely: $\alpha$2 decreases with $\alpha$3 but increases with $\alpha$4. The authors interpret this as a competition between the two matter fields: the string cloud weakens effective attraction at orbital scales, pushing marginal orbits outward and requiring higher angular momentum, whereas PFDM deepens the potential well, allowing stable orbits closer to the hole with lower angular momentum but higher energy.

Consistently, the allowed $\alpha$5 region for bound orbits shifts toward lower energy and higher angular momentum as $\alpha$6 grows (at fixed $\alpha$7), and toward higher energy and lower angular momentum as $\alpha$8 grows (at fixed $\alpha$9). This directly implies that accretion-disk observables tied to the ISCO—radiative efficiency and disk inner edge—carry separable imprints of the two parameters, since their effects on $E = f(r)\dot{t}$0 and $E = f(r)\dot{t}$1 have opposite signs.

Classification of periodic orbits

Following the Levin–Perez-Giz taxonomy, periodic orbits are indexed by the rational number

$E = f(r)\dot{t}$2

computed via the integral over the radial libration between periapsis $E = f(r)\dot{t}$3 and apoapsis $E = f(r)\dot{t}$4. Two scans are performed: varying $E = f(r)\dot{t}$5 at fixed $E = f(r)\dot{t}$6, and varying $E = f(r)\dot{t}$7 at fixed $E = f(r)\dot{t}$8. In both cases, $E = f(r)\dot{t}$9 increases monotonically with $L = r^2\dot{\phi}$0 (with a steep rise near the upper energy bound) and decreases gradually with $L = r^2\dot{\phi}$1. Increasing $L = r^2\dot{\phi}$2 shifts the $L = r^2\dot{\phi}$3 profiles to lower energies; increasing $L = r^2\dot{\phi}$4 shifts them to higher energies.

Tabulated values for representative orbits $L = r^2\dot{\phi}$5 quantify these trends. For example, at fixed $L = r^2\dot{\phi}$6 and fixed $L = r^2\dot{\phi}$7, the energy of the $L = r^2\dot{\phi}$8 orbit drops from $L = r^2\dot{\phi}$9 at $\partial_t$0 to $\partial_t$1 at $\partial_t$2, compared with $\partial_t$3 in pure Schwarzschild. At fixed $\partial_t$4, the corresponding angular momentum rises from $\partial_t$5 ($\partial_t$6) to $\partial_t$7 ($\partial_t$8), versus $\partial_t$9 in Schwarzschild. Orbit plots confirm the expected morphology: larger zoom number $\partial_\phi$0 produces richer radial structure, larger whirl number $\partial_\phi$1 produces more azimuthal revolutions between successive apoapses, and the modified spacetime visibly deforms trajectories relative to the Schwarzschild benchmark.

Gravitational waveforms from EMRI systems

Waveforms are computed using the quadrupole-order metric perturbation appropriate for an extreme mass-ratio inspiral (EMRI), with plus and cross polarizations expressed in terms of the orbital phase $\partial_\phi$2, pericenter longitude $\partial_\phi$3, and inclination $\partial_\phi$4. The fiducial system is $\partial_\phi$5, $\partial_\phi$6, $\partial_\phi$7, $\partial_\phi$8, $\partial_\phi$9 Mpc, with the secondary following a $\dot{r}^2 = E^2 - V_{\text{eff}}$0 periodic orbit.

The principal waveform result is that increasing $\dot{r}^2 = E^2 - V_{\text{eff}}$1 induces a significant phase delay: waveforms generated at smaller $\dot{r}^2 = E^2 - V_{\text{eff}}$2 complete their oscillations within shorter time intervals, while those at larger $\dot{r}^2 = E^2 - V_{\text{eff}}$3 extend to later times, relative to the pure Schwarzschild benchmark. Amplitude differences are also present. Because the phase shift accumulates coherently over many cycles, this feature is precisely the kind of observable to which space-based detectors such as LISA and Taiji are sensitive, supporting the authors' claim that the waveforms can be clearly discriminated from vacuum Schwarzschild signals and used to constrain $\dot{r}^2 = E^2 - V_{\text{eff}}$4 and $\dot{r}^2 = E^2 - V_{\text{eff}}$5.

Limitations and open questions

Several caveats qualify these results. First, the waveform calculation is quadrupole-order only; no self-force or adiabatic radiation-reaction evolution is included, so the waveforms represent snapshots along prescribed periodic geodesics rather than full inspiral signals. Second, the analysis is restricted to equatorial motion around a non-rotating hole; extension to Kerr backgrounds with inclined orbits remains unaddressed. Third, the physical normalization and astrophysical plausibility of the parameters $\dot{r}^2 = E^2 - V_{\text{eff}}$6 and $\dot{r}^2 = E^2 - V_{\text{eff}}$7 are not constrained by observational data in this work—the claimed distinguishability is demonstrated qualitatively through waveform comparison rather than through a Fisher-matrix or Bayesian parameter-estimation study quantifying measurability at realistic signal-to-noise ratios. Whether the degeneracy between $\dot{r}^2 = E^2 - V_{\text{eff}}$8 and $\dot{r}^2 = E^2 - V_{\text{eff}}$9 can be broken observationally, given their opposing effects on orbital radii but partially compensating effects on energies, is left open.

Conclusion

This paper establishes how a string cloud and a PFDM halo jointly reshape the geodesic structure of a Schwarzschild black hole. The key quantitative findings are the monotonic, oppositely signed dependences of the MBO and ISCO properties on $V_{\text{eff}} = f(r)(1 + L^2/r^2)$0 and $V_{\text{eff}} = f(r)(1 + L^2/r^2)$1, the systematic shifts of the periodic-orbit classification index $V_{\text{eff}} = f(r)(1 + L^2/r^2)$2, and the pronounced phase delay and amplitude modulation induced by $V_{\text{eff}} = f(r)(1 + L^2/r^2)$3 in EMRI waveforms from $V_{\text{eff}} = f(r)(1 + L^2/r^2)$4 orbits. These results identify periodic-orbit waveforms as a concrete channel through which future low-frequency gravitational-wave observations could constrain dark matter distributions and string-like matter near supermassive black holes, pending more complete waveform modeling that includes radiation reaction and detector noise treatment.