Massive Scalar Field Perturbations of Black Holes Surrounded by Dark Matter

Abstract: We consider scalar field perturbations in the background of black holes immersed in perfect fluid dark matter (PFDM). We find, by using the sixth order Wentzel-Kramers-Brillouin (WKB) approximation, that the longest-lived modes are the ones with higher angular number, for a scalar field mass smaller than a critical value, known as anomalous decay rate of the quasinormal modes, while that beyond this critical value the behaviour is inverted. Moreover, we show that it is possible to recover the real part of the quasinormal frequencies (QNFs), the imaginary part of the QNFs, and the critical scalar field mass, of the Schwarzschild background for different values of the PFDM intensity parameter $k$, respectively. For values of $k$ smaller than these values, the mentioned quantities are greater than the Schwarzschild background. However, beyond of these values of $k$, these quantities are smaller than the Schwarzschild background.

• The paper introduces a sixth-order WKB approximation to compute quasinormal modes in black holes influenced by PFDM.
• Results reveal that dark matter modifies QNM frequencies and decay rates, identifying a critical threshold for scalar mass behavior.
• The study correlates PFDM intensity with photon sphere modes, offering new insights for refining gravitational wave astronomy.

Overview of "Massive Scalar Field Perturbations of Black Holes Surrounded by Dark Matter" (2310.00857)

This paper examines the perturbative dynamics of massive scalar fields around black holes enveloped in perfect fluid dark matter (PFDM). It employs the sixth-order Wentzel-Kramers-Brillouin (WKB) approximation to uncover quasinormal modes (QNMs), detailing the anomalous decay rates contingent upon the scalar field mass relative to a critical threshold. This exploration is crucial for understanding dark matter's influence on black hole metrics, pertinent to gravitational wave astronomy.

Theory Setup

The investigation revisits Einstein's field equations, addressing scenarios where black holes are encircled by perfect fluid dark matter. The metric solution applied is spherically symmetric, informed by prior studies that modeled dark matter with negative pressure as a scalar field [Kiselev:2002dx]. The scalar field is characterized by a Lagrangian promoting additivity and linearity, effectively modeling dark matter's gravitational influence within an isotropically pressured framework.

Noteworthy is the incorporation of a phantom field hypothesis, which posits dark matter as devoid of electromagnetic interactions while exerting tangible gravitational effects [Li:2012zx]. The parameter $k$ quantifies PFDM intensity, influencing the characteristic black hole spacetime described by the modified Schwarzschild metric function.

Scalar Field Perturbations

The paper explores the dynamics induced by massive scalar fields within the PFDM-modified spacetime. It demonstrates that PFDM's presence deviates QNM spectra compared to those of an isolated Schwarzschild black hole. Enhanced QNM frequencies and decay rates reflect the interplay between the scalar perturbations and PFDM.

Calculations leverage the Klein-Gordon equation, expressed in tortoise coordinates, yielding a Schrödinger-like equation that posits an effective potential sensitive to variations in PFDM intensity and scalar field mass. The paper identifies critical scalar masses beyond which perturbative behaviors become anomalous, characterized by a reversal in decay rate hierarchy among QNM modes.

Figure 1: The behaviour of the event horizon radius $r_h$ as a function of the PFDM intensity parameter $k$. Black line for $M=0.5$, blue line for $M=1.0$, and red line for $M=3.0.</p></p> <h2 class='paper-heading' id='photon-sphere-modes'>Photon Sphere Modes</h2> <p>Employing a WKB approximation facilitates analytical insights into QNM phenomena, helping to discern <a href="https://www.emergentmind.com/topics/photon-sphere-modes" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">photon sphere modes</a>&#39; peculiarities. This method highlights the effects of PFDM as the real and imaginary parts of QNMs become distinct at certain PFDM intensities. Specifically, the spectral extrema are observed at $k_0 \approx 0.81$$, a pivotal point reflecting maximal oscillatory frequencies and subsequently, an insight into shadow radius phenomena.</p> <p>Critical scalar mass behavior aligns with the theoretical forecast, peaking at a distinct PFDM intensity and reflecting correlated changes in quasinormal mode decay. The peculiarities of the longest-lived modes, considering scalar mass and PFDM intensity, further enrich the theoretical underpinnings explored. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2310-00857/ReQNFsN0k.png" alt="Figure 2" title="" class="markdown-image" loading="lazy"></p> <p><img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2310-00857/ImQNFsN0k.png" alt="Figure 2" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 2: The behaviour of Re($$\omega$) (left panel), and Im($\omega$) (right panel) for the fundamental mode ($n=0$) as a function of the PFDM intensity parameter$k$with$M=1$, and$\ell=20. Black line for massless scalar field ($m=0$), and blue line for massive scalar field ($m=1.0).

Implications and Prospects

The paper invites further exploration into PFDM's gravitational imprints, postulating potential extrapolations to environments with additional astrophysical complexity like charged black holes—an area ripe for future inquiry. It bridges modern observational discrepancies in gravitational wave studies with classical concepts, enabling a refined understanding of environmental effects on black hole metrics.

Prospective developments might involve simulations that incorporate dark matter influences with heightened fidelity, improving both theoretical forecasts and observational strategies. The recognition and correlation of QNM characteristics with astrophysical milieu expand our grasp on cosmic phenomena, particularly concerning gravitational wave emissions and black hole astrophysics.

Conclusion

The research presented advances the comprehension of black holes in an enriched astrophysical context, outlining substantial interactions between massive scalar fields and surrounding dark matter fields. Its insights contribute to refining gravitational wave astronomy methodologies and engage with broader theoretical inquiries into cosmic structure and behaviors.