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Self-interacting scalar field distributions around Schwarzschild black holes

Alejandro Aguilar-Nieto, Víctor Jaramillo, Juan Barranco, Argelia Bernal, Juan Carlos Degollado, and Darío Núñez
Phys. Rev. D 107, 044070 – Published 27 February 2023
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  • INTRODUCTION
  • SETUP
  • RESONANT STATES
  • TIME DOMAIN DESCRIPTION
  • CONCLUDING REMARKS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Long-lived configurations of massive scalar fields around black holes may form if the coupling between the mass of the scalar field and the mass of the black hole is very small. In this work we analyze the effect of self-interaction in the distribution of the long-lived cloud surrounding a static black hole. We consider both attractive and repulsive self-interactions. By solving numerically the Klein-Gordon equation on a fixed background in the frequency domain, we find that the spatial distribution of quasistationary states may be larger as compared to the noninteracting case. We performed a time evolution to determine the effect of the self-interaction on the lifetime of the configurations our findings indicate that the contribution of the self-interaction is subdominant.

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    • Received 29 November 2022
    • Accepted 31 January 2023

    DOI:https://doi.org/10.1103/PhysRevD.107.044070

    © 2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Dark matterFluids & classical fields in curved spacetimeGeneral relativityGeneral relativity equations & solutions
    Gravitation, Cosmology & Astrophysics

    Authors & Affiliations

    Alejandro Aguilar-Nieto1, Víctor Jaramillo1, Juan Barranco2, Argelia Bernal2, Juan Carlos Degollado3, and Darío Núñez1

    • 1Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543, México D.F. 04510, Mexico
    • 2Departamento de Física, División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, León 37150, Mexico
    • 3Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251, Cuernavaca, Morelos, Mexico

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    Issue

    Vol. 107, Iss. 4 — 15 February 2023

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    Images

    • Figure 1
      Figure 1

      Sample of solutions of Eq. (12) with an exponential decay at infinity. In the region close the horizon (shaded area, r*→−∞) solutions behave according to Eq. (13), the maximal amplitude in this region sets the value of A. The value in the far region set by the C also fixes the value of vmax. Thus, each solution can be characterized by the amplitudes A and vmax.

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    • Figure 2
      Figure 2

      Amplitude near the horizon A, and vmax, for solutions of Eq. (12). Each point of the curve characterizes an asymptotically decaying solution with a behavior close the horizon given by Eq. (13). Without specifying the boundary conditions the spectra in ω is a continuum. The weak self-interacting regime corresponds to the extreme left in both panels, where the behavior is almost linear.

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    • Figure 3
      Figure 3

      Color map of the ratio A/vmax for Mμ=0.14. The intensity of the color indicates the value of the ratio. The white bands, in which the value of vmax is much larger than A, correspond to resonant states. The left panel is for η=−1 and right panel is for η=+1. The weakly self-interacting regime correspond to the lower part in which vmax→0. For larger values of vmax solutions can not be found for η=+1. The asterisk symbols are the configurations of Figs. 6 and 7 with frequency ω=0.13837.

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    • Figure 4
      Figure 4

      Ratio A/vmax as a function of ω for different values of vmax and Mμ=0.14 The local minima on each curve denote resonant solutions. The case with λ=0 is also shown as a reference. In the weak regime the curves are almost indistinguishable from the noninteracting case.

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    • Figure 5
      Figure 5

      Fundamental frequencies of resonant states, as a function of vmax for several cases of Mμ for both, positive and negative self-interaction. For η=+1 (dashed line), as vmax grows, there are no resonant states.

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    • Figure 6
      Figure 6

      The left panel shows the profile of the scalar field and the right panel the corresponding radial energy density for a fixed value of Mω=0.13837 with different values of vmax. The case λ=0 is included for comparison. Both sets have been normalized for a better visualization.

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    • Figure 7
      Figure 7

      Same as Fig. 6 with η=1.

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    • Figure 8
      Figure 8

      Scalar-field profile for the first resonant state for different values of vmax for η=−1, left panel, and η=+1, right panel.

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    • Figure 9
      Figure 9

      Radial energy density for the first resonant states for different values of vmax. Left panel correspond to η=−1 whereas right panel corresponds to η=+1. For η=+1 the characteristic size of the boson cloud in the strong self-interacting regime is much larger than the size of the cloud in the weak self-interacting regime.

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    • Figure 10
      Figure 10

      Mass of the scalar field as defined by Eq. (9). In order to avoid the divergence produced by the oscillatory behavior at the horizon, the integration was performed from r=2M+ε to r=rmax. In practice we have taken ε such that |U(2M+ε)|=ξmax(|U|), with ξ=10−3. Other values of ξ where tested with similar results. rmax corresponds to the last point of the numerical grid.

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    • Figure 11
      Figure 11

      Effective potential for the fundamental states for different values of vmax in tortoise coordinates. The depth of the potential well increases as vmax grows for η=−1 and decreases for η=+1.

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    • Figure 12
      Figure 12

      The initial scalar field profiles correspond to regularized resonant states for configurations with Mμ=0.14. Initially only the real part is different from zero. States with frequencies ω=0.13837 and ω=0.1371 are in the weak self-interaction regime with η=+1 and η=−1, while states with ω=0.13930 and ω=0.1300 are in the strong self-interaction regime with η=+1 and η=−1, respectively.

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    • Figure 13
      Figure 13

      Energy of the scalar-field states Eq. (21) for Mμ=0.14. The vertical axis is presented in log scale and the properties of the configurations are summarized in Table 2. After an initial decay, the energy decays exponentially in time. The rate of decay s, is shown in the last column of Table 2.

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