pith. sign in

arxiv: 2504.10235 · v1 · pith:WNGS5UF4new · submitted 2025-04-14 · 🌀 gr-qc

Eccentric mergers of binary Proca stars

Gabriele Palloni , Nicolas Sanchis-Gual , Jos\'e A. Font , Carlos Herdeiro , Eugen Radu This is my paper

Pith reviewed 2026-05-22 20:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Proca starsbinary mergerseccentric mergersgravitational wavesnumerical relativitybosonic starsblack hole formationodd modes
0
0 comments X

The pith

The relative phase between two merging Proca stars controls whether the outcome is prompt black-hole formation, a hypermassive remnant, or an unstable spinning m-bar=2 star, and can generate odd-mode gravitational waves absent from black-hp

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper runs numerical-relativity simulations of eccentric, equal-mass mergers of rotating m-bar=1 Proca stars while changing the initial orbital boost and the relative phase of the Proca fields. The phase turns out to dominate the post-merger fate: some alignments produce a black hole plus short-lived Proca remnant, others leave a hypermassive m-bar=1 star, and still others yield a spinning m-bar=2 star that is dynamically unstable. The same phase choices also excite odd multipoles such as the l=m=3 gravitational-wave mode that never appear in ordinary black-hole mergers. A reader cares because Proca stars are candidate models for ultralight bosonic dark matter, so these phase-driven differences could supply observable signatures that distinguish them from conventional compact objects.

Core claim

Systematic variation of the relative phase and orbital boost in eccentric equal-mass m-bar=1 Proca-star binaries shows that the phase exerts paramount control over the nonlinear evolution: prompt black-hole formation with a transient Proca remnant, formation of a hypermassive m-bar=1 Proca star, or emergence of a dynamically unstable spinning m-bar=2 Proca star can all occur. Under certain phase alignments the emitted gravitational waves contain significant odd modes (for example the l=m=3 mode) that are absent in black-hole mergers and may serve as unique identifiers of these objects.

What carries the argument

The relative phase between the Proca fields of the two stars, which sets the interference pattern that governs the nonlinear post-merger dynamics and the selection of gravitational-wave multipoles.

If this is right

  • Different relative phases produce qualitatively distinct remnants: black holes with transient Proca clouds, long-lived hypermassive m-bar=1 stars, or spinning m-bar=2 stars that are dynamically unstable.
  • Gravitational-wave signals from some mergers contain odd multipoles such as the l=m=3 mode that do not appear in black-hole mergers.
  • The internal phase structure of the Proca field can therefore act as an additional degree of freedom that shapes both the final object and the observable waveform.
  • These mergers enlarge the known phenomenology of bosonic-star collisions and the possible astrophysical roles of ultralight bosonic fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If ultralight bosonic fields constitute dark matter, phase-dependent merger outcomes could produce a subpopulation of compact objects whose stability and spin properties differ from those formed by ordinary stellar collapse.
  • Detection of odd-mode gravitational waves in a merger event without an electromagnetic counterpart might indicate a Proca-star origin even when the remnant itself is not directly resolved.
  • The same phase-sensitivity mechanism may operate in mergers of other self-interacting scalar or vector boson stars, suggesting a broader class of phase-controlled exotic compact-object collisions.

Load-bearing premise

The chosen numerical code and initial-data construction faithfully reproduce the nonlinear Proca-field evolution without dominant discretization or gauge artifacts when the relative phase is altered.

What would settle it

A set of otherwise identical simulations in which changing only the relative phase produces no measurable difference in remnant type or in the amplitude of the l=m=3 gravitational-wave mode.

Figures

Figures reproduced from arXiv: 2504.10235 by Carlos Herdeiro, Eugen Radu, Gabriele Palloni, Jos\'e A. Font, Nicolas Sanchis-Gual.

Figure 1
Figure 1. Figure 1: Difference with respect to the zero-dephase config [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Top-left panel: rΨ 22 4 mode for simulation 85A0 at extraction radius rext = 100. Top-right panel: rΨ 22 4 mode for simulation 85C0 at several extraction radii (rext = 100, 120 and 200). Bottom row: convergence study for simulation 85A0 (left) and 85C0 (right). The waveforms are artificially rescaled according to fourth-order convergence. −4 −2 0 2 4 −30 −20 −10 0 10 20 30 Star 1 Star 2 y/µ x/µ −15 −10 −5 … view at source ↗
Figure 4
Figure 4. Figure 4: Trajectories of PSs in binaries for simulations 83A0a (left) and 87C6 (right) in the equatorial plane [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: rΨ 22 4 mode for in-phase equal-mass mergers (left ω/µ = 0.83; right ω/µ = 0.87), varying the initial boost v/c = 0.015 (black), 0.02 (red) and 0.03 (blue). 0 0.3 0.6 0.9 1.2 1.5 1.8 0 500 1000 1500 2000 2500 3000 ∆ǫ = 0, v/c = 0.030 Mass tµ 1 × 10−10 1 × 10−9 1 × 10−8 1 × 10−7 1 × 10−6 1 × 10−5 0.0001 0.001 −100 400 900 1400 1900 2400 2900 ∆ǫ = 0, v/c = 0.030 Re( r Ψl= m=2 4 /µ) uµ (retarded time) [PITH_… view at source ↗
Figure 6
Figure 6. Figure 6: Top panel: Evolution of the integrated mass on the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left column: equatorial plane snapshots of the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the GW luminosity (top panel) and [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: l = m = 2 modes of rΨ lm 4 comparing the period￾icity of the relative phase (∆ϵ) for simulation 85C0 and 85C6 (top panel) and 85C3 and 85C9 (bottom panel) equal-mass merger (ω/µ = 0.8500) comparing for periodic relative phase (∆ϵ) for initial boost of 0.030. sion, [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left column: rΨ 22 4 waveforms for equal-mass mergers with ω/µ = 0.87 and initial boost v/c = 0.020 (models 87B0, 87B2, 87B5 and 87B6). Right column: rΨ 33 4 waveforms for the same set of models [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Contribution to the total GW luminosity of several [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of rΨ 22 4 for models 83B0 (in-phase on top panel) and 83B6 (in-phase on bottom panel), both with ω/µ = 0.83. difference between the binary components and a signa￾ture of exotic physics, since equal-mass BBH mergers do not produce such modes (at least without precession). Since eccentric mergers take longer to merge than head-on collisions, the overall radiated energy is also higher. In addition… view at source ↗
Figure 16
Figure 16. Figure 16: Formation of a hypermassive Proca star for model [PITH_FULL_IMAGE:figures/full_fig_p012_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Eccentric anti-phase models may yield more than one encounter. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p013_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: rΨ 33 4 GW modes for models 87C5 (red solid line) and 87C6 (black solid line). system can either undergo prompt collapse to a black hole with an accreting Proca cloud (not in equilibrium), as in the case of models 87C0 and 87C3, or experience a prolonged transient phase corresponding to a highly￾spinning “hypermassive” Proca star (model 87C5). This is a transient remnant which has an excess angular mo￾men… view at source ↗
Figure 19
Figure 19. Figure 19: l = m = 2 (top panel) and l = m = 3 (middle panel) modes of rΨ lm 4 and angular momentum and integrated Proca mass on spherical volume of radius equals to the radial extend of the grid (bottom panel) as a function of retarded time for simulation 87D6. ℓ = m = 3 mode in model 87C5 reaches an amplitude nearly comparable to that of the quadrupole ℓ = m = 2 mode for ∆ϵ = 5π/6. However, the black solid line in… view at source ↗
Figure 20
Figure 20. Figure 20: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p015_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: rΨ 22 4 waveform for model 87E6. complex amplitude of the Proca field reads |A|2 = Re(A) 2 + Im(A) 2 ∼ 4 cos2  (ω1 − ω2) 2 t + ∆ϵ  = = 2  1 + cos  (ω1 − ω2)t + ∆ϵ 2  . (16) The square of the amplitude of the Proca field is propor￾tional to the GW energy, EGW ∝ |A|2 . We can use this 1 × 10−18 1 × 10−16 1 × 10−14 1 × 10−12 1 × 10−10 1 × 10−8 1 × 10−6 1 × 10−4 −100 150 400 650 900 1150 1400 1650 Total… view at source ↗
Figure 23
Figure 23. Figure 23: Dependence of the GW energy on the relative phase for equal-mass Proca star mergers with [PITH_FULL_IMAGE:figures/full_fig_p016_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Absolute value of the Fourier transform of the [PITH_FULL_IMAGE:figures/full_fig_p017_24.png] view at source ↗
read the original abstract

We present a numerical relativity study of eccentric mergers of equal-mass rotating $\bar m=1$ Proca stars, focusing on their gravitational-wave (GW) emission. By systematically varying key binary parameters, such as the initial orbital boost, which determines the orbital angular momentum, and the relative phase between the stars, we examine how the internal phase structure of the Proca field influences the merger dynamics and the properties of the emitted GWs. Our simulations demonstrate that the relative phase has paramount impact on the post-merger evolution, resulting in prompt black hole formation accompanied by a transient Proca remnant, the formation of a hypermassive $\bar m=1$ Proca star or even the emergence of a dynamically-unstable spinning $\bar m=2$ Proca star. Under certain conditions, the GW signal exhibits significant odd-modes (e.g., the $\ell=m=3$ mode) that are absent in conventional black hole mergers, potentially serving as unique signatures of these exotic objects. Our findings offer new insights into the phenomenology of bosonic star mergers and the potential astrophysical role of ultralight bosonic fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper presents a numerical relativity study of eccentric mergers of equal-mass rotating m-bar=1 Proca stars. By varying the initial orbital boost (controlling orbital angular momentum) and the relative phase between the two stars, the authors investigate how the internal phase structure of the Proca vector field affects merger dynamics and gravitational-wave emission. The central results are that the relative phase has a dominant effect on post-merger evolution, producing outcomes ranging from prompt black-hole formation with a transient Proca remnant, to a hypermassive m-bar=1 Proca star, or a dynamically unstable spinning m-bar=2 Proca star; under some conditions the GW signal contains significant odd modes (e.g., the l=m=3 mode) absent from conventional black-hole mergers.

Significance. If the reported phase-dependent phenomenology is confirmed to be free of numerical artifacts, the work would be significant for the phenomenology of bosonic-star mergers. It identifies potential unique GW signatures (odd-parity modes) that could serve as observational discriminants for ultralight vector fields and adds concrete examples of how internal field structure influences the remnant and waveform in exotic compact-object mergers.

major comments (1)
  1. [Numerical Methods and Results sections] The manuscript provides no information on grid resolution, convergence tests, or error budgets for the family of simulations in which the relative phase is varied. Because the headline claim is that constructive/destructive interference in the massive vector field produces qualitatively different post-merger fates (prompt collapse, hypermassive m-bar=1 remnant, or unstable m-bar=2 star) and excites odd GW modes, the absence of these tests leaves open the possibility that the reported differences arise from phase-dependent numerical dissipation, constraint violation, or gauge drift rather than from the physics of the Proca field.
minor comments (1)
  1. [Abstract] The notation m-bar=1 and m-bar=2 is used without an explicit definition or reference to prior Proca-star literature in the abstract; a brief parenthetical clarification would aid readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for recognizing the potential significance of the phase-dependent phenomenology in Proca-star mergers. We address the single major comment below and will revise the manuscript accordingly to strengthen the numerical validation.

read point-by-point responses

  1. Referee: [Numerical Methods and Results sections] The manuscript provides no information on grid resolution, convergence tests, or error budgets for the family of simulations in which the relative phase is varied. Because the headline claim is that constructive/destructive interference in the massive vector field produces qualitatively different post-merger fates (prompt collapse, hypermassive m-bar=1 remnant, or unstable m-bar=2 star) and excites odd GW modes, the absence of these tests leaves open the possibility that the reported differences arise from phase-dependent numerical dissipation, constraint violation, or gauge drift rather than from the physics of the Proca field.

    Authors: We agree that explicit documentation of resolutions, convergence tests, and error estimates for the phase-variation suite is essential to rule out numerical artifacts. The current Numerical Methods section outlines the Einstein Toolkit infrastructure and Proca implementation but does not tabulate the specific grid parameters or present dedicated tests across relative phases. In the revised manuscript we will add a new subsection (Numerical Methods, §3.3) that (i) lists the finest-grid spacing, number of refinement levels, and domain sizes used for each relative-phase run, (ii) shows convergence of the L2 norm of the Hamiltonian constraint and of the extracted (ℓ,m)=(2,2) and (3,3) GW amplitudes for representative cases at three resolutions, and (iii) reports the estimated truncation error on remnant mass and spin. These tests will demonstrate that the qualitative distinctions in post-merger fate and the appearance of odd modes remain robust under refinement, thereby confirming that the reported behavior originates from the vector-field interference rather than from phase-dependent dissipation or gauge effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct numerical evolution

full rationale

The paper reports outcomes from numerical relativity simulations of eccentric mergers of equal-mass rotating m-bar=1 Proca stars. Central claims concern the impact of relative phase and orbital boost on post-merger fates (prompt black hole formation with transient remnant, hypermassive m-bar=1 star, or dynamically unstable spinning m-bar=2 star) and the excitation of odd-parity GW modes such as l=m=3. These are obtained by evolving the Einstein-Proca system under varied initial data rather than any closed analytical derivation, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations reduce the reported phenomenology to its inputs by construction; the work is self-contained against external benchmarks of the NR code and initial-data construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claims rest on the validity of the Einstein-Proca system as a model for ultralight vector fields and on the assumption that the chosen numerical setup faithfully represents the physical dynamics.

free parameters (2)
  • initial orbital boost
    Parameter varied to control orbital angular momentum and eccentricity; not fitted to data but chosen to explore different regimes.
  • relative phase
    Parameter varied to probe internal field alignment effects; not derived from first principles.
axioms (1)
  • domain assumption The dynamics of Proca stars are governed by the Einstein equations coupled to a massive vector field.
    Standard modeling assumption for these objects in general relativity.
invented entities (1)
  • m-bar=2 Proca star no independent evidence
    purpose: Dynamically unstable spinning configuration that can emerge from certain phase choices
    Outcome reported from the simulations; no independent observational handle given.

pith-pipeline@v0.9.0 · 5739 in / 1522 out tokens · 77815 ms · 2026-05-22T20:13:25.903837+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

89 extracted references · 89 canonical work pages · 10 internal anchors

  1. [1]

    5 0 1 2 3 4 5 6 ×10−16 ||H||2 µ 2 ||Mx||2 µ 2 × 100 ||My||2 µ 2 ||Mz||2 µ 2 × 10 ∆ǫ Figure 1. Difference with respect to the zero-dephase config- uration of the L2-norm of the Hamiltonian constraint (black) and of the momentum constraint ( x component in blue, y component in green and z component in red) for equal- mass Proca star mergers with ω/µ = 0 .83...

    work page

  2. [2]

    030 Mass tµ 1 × 10−10 1 × 10−9 1 × 10−8 1 × 10−7 1 × 10−6 1 × 10−5

    8 0 500 1000 1500 2000 2500 3000 ∆ǫ = 0, v/c = 0. 030 Mass tµ 1 × 10−10 1 × 10−9 1 × 10−8 1 × 10−7 1 × 10−6 1 × 10−5

    work page 2000

  3. [3]

    030 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 6

    001 −100 400 900 1400 1900 2400 2900 ∆ǫ = 0, v/c = 0. 030 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 6. Top panel: Evolution of the integrated mass on the full radial extend of the grid for model 87C0. Bottom panel: rΨ22 4 mode for the same model. The vertical dashed black line indicates the merger time (u µ ∼ 360). A. The role of the orbital angular mom...

    work page 1900

  4. [4]

    Evolution of rΨ22 4 for models 83A0a (black), 83A0b (red) and 83A0c (blue) with zero relative phase, showing that the waveforms perfectly overlap

    02 250 300 350 400 450 500 550 ǫ = 0 ǫ = π/ 4 ǫ = π/ 2 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 9. Evolution of rΨ22 4 for models 83A0a (black), 83A0b (red) and 83A0c (blue) with zero relative phase, showing that the waveforms perfectly overlap. gular momentum result in a more luminous event with more energy emitted through GW. B. The role of the compa...

    work page

  5. [5]

    06 300 350 400 450 500 550 600 ∆ǫ = π/ 2 ∆ǫ = 3π/ 2 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0. 06 −0. 03 0

    work page

  6. [6]

    06 300 350 400 450 500 550 600 ∆ǫ = 0 ∆ǫ = π Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 10. l = m = 2 modes of rΨlm 4 comparing the period- icity of the relative phase (∆ϵ) for simulation 85C0 and 85C6 (top panel) and 85C3 and 85C9 (bottom panel) equal-mass merger (ω/µ = 0.8500) comparing for periodic relative phase (∆ϵ) for initial boost of 0 .030. sion...

    work page

  7. [7]

    06 300 400 500 600 700 800 ∆ǫ = 0 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0. 02 −0. 01 0

    work page

  8. [8]

    02 300 400 500 600 700 800 ∆ǫ = 0 Re(rΨl=m=3 4 /µ ) uµ (retarded time) −0. 06 −0. 03 0

    work page

  9. [9]

    06 300 400 500 600 700 800 ∆ǫ = π/ 3 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0. 02 −0. 01 0

    work page

  10. [10]

    02 300 400 500 600 700 800 ∆ǫ = π/ 3 Re(rΨl=m=3 4 /µ ) uµ (retarded time) −0. 06 −0. 03 0

    work page

  11. [11]

    06 300 400 500 600 700 800 ∆ǫ = 5π/ 6 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0. 02 −0. 01 0

    work page

  12. [12]

    02 300 400 500 600 700 800 ∆ǫ = 5π/ 6 Re(rΨl=m=3 4 /µ ) uµ (retarded time) −0. 06 −0. 03 0

    work page

  13. [13]

    06 300 400 500 600 700 800 ∆ǫ = π Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0. 02 −0. 01 0

    work page

  14. [14]

    Left column: rΨ22 4 waveforms for equal-mass mergers with ω/µ = 0.87 and initial boost v/c = 0.020 (models 87B0, 87B2, 87B5 and 87B6)

    02 300 400 500 600 700 800 ∆ǫ = π Re(rΨl=m=3 4 /µ ) uµ (retarded time) Figure 11. Left column: rΨ22 4 waveforms for equal-mass mergers with ω/µ = 0.87 and initial boost v/c = 0.020 (models 87B0, 87B2, 87B5 and 87B6). Right column: rΨ33 4 waveforms for the same set of models. 11 1 × 10−18 1 × 10−16 1 × 10−14 1 × 10−12 1 × 10−10 1 × 10−8 1 × 10−6 1 × 10−4 −...

    work page

  15. [15]

    03 −0. 03 −0. 02 −0. 01 0

    work page

  16. [16]

    Evolution of rΨ22 4 for models 83B0 (in-phase on top panel) and 83B6 (in-phase on bottom panel), both with ω/µ = 0.83

    03 250 300 350 400 450 500 550 600 ∆ǫ = 0 Re(rΨl=m=2 4 /µ ) ∆ǫ = π Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 13. Evolution of rΨ22 4 for models 83B0 (in-phase on top panel) and 83B6 (in-phase on bottom panel), both with ω/µ = 0.83. difference between the binary components and a signa- ture of exotic physics, since equal-mass BBH mergers do not produce s...

    work page

  17. [17]

    01 −100 150 400 650 900 1150 1400 1650 ∆ǫ = 5π/ 6 Re(rΨl=m=2 4 /µ ) uµ (retarded time) 0

    work page

  18. [18]

    Formation of a hypermassive Proca star for model 87C5

    8 0 250 500 750 1000 1250 1500 1750 ∆ǫ = 5π/ 6 Mass tµ Figure 16. Formation of a hypermassive Proca star for model 87C5. Top panel: Evolution of rΨ22 4 . Bottom panel: Evo- lution of the integrated mass on the entire spherical volume of the computational grid. The dashed black line indicates the merger time (u µ ∼ 390), while the dashed red line corre- sp...

    work page

  19. [19]

    8300 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0

    08 −100 100 300 500 700 900 1100 ∆ǫ = π, ω/µ = 0. 8300 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −0. 03 −0. 02 −0. 01 0

    work page

  20. [20]

    8700 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 17

    03 −100 150 400 650 900 1150 1400 1650 1900 2150 ∆ǫ = π, ω/µ = 0. 8700 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 17. Eccentric anti-phase models may yield more than one encounter. Evolution of rΨ22 4 for models 83C6 (left) and 87C6 (right). See also Fig. 4 for an illustration of the trajectories. −0. 02 −0. 01 0

    work page 1900

  21. [21]

    hypermassive

    02 −100 150 400 650 900 1150 1400 1650 1900 2150 ∆ǫ = 5π/ 6 ∆ǫ = π Re(rΨl=m=3 4 /µ ) uµ (retarded time) Figure 18. rΨ33 4 GW modes for models 87C5 (red solid line) and 87C6 (black solid line). system can either undergo prompt collapse to a black hole with an accreting Proca cloud (not in equilibrium), as in the case of models 87C0 and 87C3, or experience ...

    work page 1900

  22. [22]

    040 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −1 −0

    5 −100 300 700 1100 1500 1900 2300 2700 ×10−2 ∆ǫ = π, v/c = 0. 040 Re(rΨl=m=2 4 /µ ) uµ (retarded time) −1 −0. 5 0

    work page 1900

  23. [23]

    040 Re(rΨl=m=3 4 /µ ) uµ (retarded time) 0 1.5 3 4.5 0 400 800 1200 1600 2000 2400 2800 Proca mass Proca angular momentum Mass tµ Figure 19

    5 1 −100 300 700 1100 1500 1900 2300 2700 ×10−2 ∆ǫ = π, v/c = 0. 040 Re(rΨl=m=3 4 /µ ) uµ (retarded time) 0 1.5 3 4.5 0 400 800 1200 1600 2000 2400 2800 Proca mass Proca angular momentum Mass tµ Figure 19. l = m = 2 (top panel) and l = m = 3 (middle panel) modes of rΨlm 4 and angular momentum and integrated Proca mass on spherical volume of radius equals ...

    work page 1900

  24. [24]

    050 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 21

    5 1 −100 400 900 1400 1900 2400 2900 3400 3900 ×10−3 ∆ǫ = π, v/c = 0. 050 Re(rΨl=m=2 4 /µ ) uµ (retarded time) Figure 21. rΨ22 4 waveform for model 87E6. complex amplitude of the Proca field reads |A|2 = Re(A)2 + Im(A)2 ∼ 4 cos2 (ω1 − ω2) 2 t + ∆ϵ = = 2 1 + cos (ω1 − ω2)t + ∆ϵ 2 . (16) The square of the amplitude of the Proca field is propor- tional to th...

    work page 1900

  25. [25]

    6 5 0 0 . 5 1 1 . 5 2 ×10−3×10−3×10−3 v/c = 0. 015, ω/µ = 0. 8700 EGW µ ∆ǫ/π 0 1 2 3 4 0 0 . 5 1 1 . 5 2 ×10−2 v/c = 0. 020, ω/µ = 0. 8700 EGW µ ∆ǫ/π 0 1 2 3 4 5 0 0 . 5 1 1 . 5 2 ×10−1 v/c = 0. 030, ω/µ = 0. 8700 EGW µ ∆ǫ/π Figure 23. Dependence of the GW energy on the relative phase for equal-mass Proca star mergers with ω/µ = 0 .8500 (top row) and 0.87...

    work page

  26. [26]

    Higher-order interactions Pairwise interactions Interaction network Active Inactive Pairwise activation Inactivation Inactivation μ μ β Higher-order activation β, βΔ β βΔ β FIG

    01 0 . 1 ∆ǫ = 0 ∆ǫ = 1π/ 6 ∆ǫ = π/ 3 ∆ǫ = π/ 2 ∆ǫ = 2π/ 3 ∆ǫ = 5π/ 6 ∆ǫ = π | ˜Ψl=m=2 4 | µf /µ Figure 24. Absolute value of the Fourier transform of the l = m = 2 mode of Ψ4 for equal-mass PSs orbital merger with ω/µ = 0.8300 and initial boost v/c = 0.020 as a function of the relative phase of the two stars. much longer and even experiences a recoil kick...

    work page doi:10.13039/501100011033 2022

  27. [27]

    B. P. Abbott, R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, LIGO Scientific Collaboration, and Virgo Collaboration, GWTC-1: A Gravitational-Wave Transient Catalog of Compact Bi- nary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Physical Review X 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  28. [28]

    GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run

    R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, A. Adams, C. Adams, LIGO Scientific Collab- oration, and Virgo Collaboration, GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo dur- ing the First Half of the Third Observing Run, Physical Review X 11, 021053 (2021), arXiv:2010.14527 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  29. [29]

    GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run

    R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, LIGO Scientific Collaboration, and the Virgo Collaboration, GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run, Phys. Rev. D 109, 022001 (2024), arXiv:2108.01045 [gr- qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  30. [30]

    GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run

    R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, V. C. Ligo Scientific Collaboration, and Kagra Collaboration, GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of 18 the Third Observing Run, Physical Review X 13, 041039 (2023), arXiv:2111.03606 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  31. [31]

    Abbott, R

    B. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Ac- ernese, K. Ackley, C. Adams, R. X. Adhikari, V. Adya, C. Affeldt, et al., Binary black hole population properties inferred from the first and second observing runs of ad- vanced ligo and advanced virgo, The Astrophysical Jour- nal Letters 882, L24 (2019)

    work page 2019

  32. [32]

    Abbott, T

    R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. X. Adhikari, V. B. Adya, C. Affeldt, D. Agarwal, et al. , Search for intermediate- mass black hole binaries in the third observing run of advanced ligo and advanced virgo, Astronomy & Astro- physics 659, A84 (2022)

    work page 2022

  33. [33]

    Abbott, T

    R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. Adhikari, V. Adya, C. Affeldt, D. Agar- wal, et al. , Population of merging compact binaries in- ferred using gravitational waves through gwtc-3, Physical Review X 13, 011048 (2023)

    work page 2023

  34. [34]

    Punturo, M

    M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun, F. Barone, B. Barr, M. Bar- suglia, M. Beker, N. Beveridge, S. Birindelli, S. Bose, L. Bosi, S. Braccini, C. Bradaschia, T. Bulik, E. Cal- loni, G. Cella, E. Chassande Mottin, S. Chelkowski, A. Chincarini, J. Clark, E. Coccia, C. Colacino, J. Co- las, A. Cumming, L. Cunningham, E. Cuo...

    work page 2010

  35. [35]

    A. Abac, R. Abramo, S. Albanesi, A. Albertini, A. Agapito, M. Agathos, C. Albertus, N. Andersson, T. Andrade, I. Andreoni, et al. , The science of the ein- stein telescope, arXiv preprint arXiv:2503.12263 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  36. [36]

    Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO

    D. Reitze, R. X. Adhikari, S. Ballmer, B. Barish, L. Bar- sotti, G. Billingsley, D. A. Brown, Y. Chen, D. Coyne, R. Eisenstein, et al. , Cosmic explorer: the us contribu- tion to gravitational-wave astronomy beyond ligo, arXiv preprint arXiv:1907.04833 (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1907

  37. [37]

    A Horizon Study for Cosmic Explorer: Science, Observatories, and Community

    M. Evans, R. X. Adhikari, C. Afle, S. W. Ballmer, S. Biscoveanu, S. Borhanian, D. A. Brown, Y. Chen, R. Eisenstein, A. Gruson, A. Gupta, E. D. Hall, R. Hux- ford, B. Kamai, R. Kashyap, J. S. Kissel, K. Kuns, P. Landry, A. Lenon, G. Lovelace, L. McCuller, K. K. Y. Ng, A. H. Nitz, J. Read, B. S. Sathyaprakash, D. H. Shoemaker, B. J. J. Slagmolen, J. R. Smit...

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  38. [38]

    Bezares and N

    M. Bezares and N. Sanchis-Gual, Exotic compact objects: a recent numerical-relativity perspective, arXiv preprint arXiv:2406.04901 (2024)

    work page arXiv 2024

  39. [39]

    Veltmaat, J

    J. Veltmaat, J. C. Niemeyer, and B. Schwabe, Formation and structure of ultralight bosonic dark matter halos, Physical Review D 98, 043509 (2018)

    work page 2018

  40. [40]

    E. G. Ferreira, Ultra-light dark matter, The Astronomy and Astrophysics Review 29, 1 (2021)

    work page 2021

  41. [41]

    Matos, L

    T. Matos, L. A. Ure˜ na-L´ opez, and J.-W. Lee, Short re- view of the main achievements of the scalar field, fuzzy, ultralight, wave, bec dark matter model, Frontiers in As- tronomy and Space Sciences 11, 1347518 (2024)

    work page 2024

  42. [42]

    Cardoso, ´O

    V. Cardoso, ´O. J. Dias, G. S. Hartnett, M. Middle- ton, P. Pani, and J. E. Santos, Constraining the mass of dark photons and axion-like particles through black-hole superradiance, Journal of Cosmology and Astroparticle Physics 2018 (03), 043

    work page 2018

  43. [43]

    W. E. East and F. Pretorius, Superradiant instability and backreaction of massive vector fields around kerr black holes, Physical review letters 119, 041101 (2017)

    work page 2017

  44. [44]

    C. A. R. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112, 221101 (2014), arXiv:1403.2757 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  45. [45]

    Kerr black holes with Proca hair

    C. Herdeiro, E. Radu, and H. R´ unarsson, Kerr black holes with Proca hair, Class. Quant. Grav. 33, 154001 (2016), arXiv:1603.02687 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  46. [46]

    Abbott et al

    R. Abbott et al. , All-sky search for gravitational wave emission from scalar boson clouds around spinning black holes in LIGO O3 data, Phys. Rev. D105, 102001 (2022), arXiv:2111.15507 [astro-ph.HE]

    work page arXiv 2022

  47. [47]

    F. E. Schunck and E. W. Mielke, General relativis- tic boson stars, Class. Quant. Grav. 20, R301 (2003), arXiv:0801.0307 [astro-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  48. [48]

    Brito, V

    R. Brito, V. Cardoso, C. A. Herdeiro, and E. Radu, Proca stars: Gravitating bose–einstein condensates of massive spin 1 particles, Physics Letters B 752, 291 (2016)

    work page 2016

  49. [49]

    J. C. Bustillo, N. Sanchis-Gual, A. Torres-Forn´ e, J. A. Font, A. Vajpeyi, R. Smith, C. Herdeiro, E. Radu, and S. H. W. Leong, Gw190521 as a merger of proca stars: A potential new vector boson of 8.7 ×10−13 eV, Phys. Rev. Lett. 126, 081101 (2021)

    work page 2021

  50. [50]

    Calder´ on Bustillo, N

    J. Calder´ on Bustillo, N. Sanchis-Gual, S. H. W. Leong, K. Chandra, A. Torres-Forn´ e, J. A. Font, C. Herdeiro, E. Radu, I. C. F. Wong, and T. G. F. Li, Search- ing for vector boson-star mergers within LIGO-Virgo intermediate-mass black-hole merger candidates, Phys. Rev. D 108, 123020 (2023), arXiv:2206.02551 [gr-qc]

    work page arXiv 2023

  51. [51]

    R. Luna, M. Llorens-Monteagudo, A. Lorenzo-Medina, J. C. Bustillo, N. Sanchis-Gual, A. Torres-Forn´ e, J. A. Font, C. A. Herdeiro, and E. Radu, Numerical relativity surrogate waveform models for exotic compact objects: The case of head-on mergers of equal-mass proca stars, Physical Review D 110, 024004 (2024)

    work page 2024

  52. [52]

    Evstafyeva, U

    T. Evstafyeva, U. Sperhake, I. M. Romero-Shaw, and 19 M. Agathos, Gravitational-wave data analysis with high- precision numerical relativity simulations of boson star mergers, Physical Review Letters 133, 131401 (2024)

    work page 2024

  53. [53]

    Palenzuela, P

    C. Palenzuela, P. Pani, M. Bezares, V. Cardoso, L. Lehner, and S. Liebling, Gravitational wave signatures of highly compact boson star binaries, Physical Review D 96, 104058 (2017)

    work page 2017

  54. [54]

    Sanchis-Gual, J

    N. Sanchis-Gual, J. C. Bustillo, C. Herdeiro, E. Radu, J. A. Font, S. H. W. Leong, and A. Torres-Forn´ e, Im- pact of the wavelike nature of proca stars on their gravitational-wave emission, Phys. Rev. D 106, 124011 (2022)

    work page 2022

  55. [55]

    B.-X. Ge, E. A. Lim, U. Sperhake, T. Evstafyeva, D. Cors, E. de Jong, R. Croft, and T. Helfer, Hair is complicated: Gravitational waves from stable and unsta- ble boson-star mergers, arXiv preprint arXiv:2410.23839 (2024)

    work page arXiv 2024

  56. [56]

    Palenzuela, L

    C. Palenzuela, L. Lehner, and S. L. Liebling, Orbital dy- namics of binary boson star systems, Phys. Rev. D 77, 044036 (2008)

    work page 2008

  57. [57]

    Bezares and C

    M. Bezares and C. Palenzuela, Gravitational waves from dark boson star binary mergers, Classical and Quantum Gravity 35, 234002 (2018)

    work page 2018

  58. [58]

    Sanchis-Gual, C

    N. Sanchis-Gual, C. Herdeiro, J. A. Font, E. Radu, and F. Di Giovanni, Head-on collisions and orbital mergers of proca stars, Phys. Rev. D 99, 024017 (2019)

    work page 2019

  59. [59]

    Jaramillo, N

    V. Jaramillo, N. Sanchis-Gual, J. Barranco, A. Bernal, J. C. Degollado, C. Herdeiro, M. Megevand, and D. N´ u˜ nez, Head-on collisions of l-boson stars, Physical Review D 105, 104057 (2022)

    work page 2022

  60. [60]

    Helfer, U

    T. Helfer, U. Sperhake, R. Croft, M. Radia, B.-X. Ge, and E. A. Lim, Malaise and remedy of binary boson-star initial data, Classical and Quantum Gravity 39, 074001 (2022)

    work page 2022

  61. [61]

    Croft, T

    R. Croft, T. Helfer, B.-X. Ge, M. Radia, T. Evstafyeva, E. A. Lim, U. Sperhake, and K. Clough, The gravita- tional afterglow of boson stars, Classical and Quantum Gravity 40, 065001 (2023)

    work page 2023

  62. [62]

    J. C. Aurrekoetxea, K. Clough, and E. A. Lim, Cttk: a new method to solve the initial data constraints in nu- merical relativity, Classical and Quantum Gravity 40, 075003 (2023)

    work page 2023

  63. [63]

    Siemonsen and W

    N. Siemonsen and W. E. East, Binary boson stars: Merger dynamics and formation of rotating rem- nant stars, Phys. Rev. D 107, 124018 (2023), arXiv:2302.06627 [gr-qc]

    work page arXiv 2023

  64. [64]

    Siemonsen and W

    N. Siemonsen and W. E. East, Generic initial data for binary boson stars, Phys. Rev. D 108, 124015 (2023), arXiv:2306.17265 [gr-qc]

    work page arXiv 2023

  65. [65]

    Atteneder, H

    F. Atteneder, H. R. R¨ uter, D. Cors, R. Rosca-Mead, D. Hilditch, and B. Br¨ ugmann, Boson star head-on colli- sions with constraint-violating and constraint-satisfying initial data, Physical Review D 109, 044058 (2024)

    work page 2024

  66. [66]

    J. C. Aurrekoetxea, S. E. Brady, L. Arest´ e-Sal´ o, J. Bam- ber, L. Chung-Jukko, K. Clough, E. de Jong, M. Elley, P. Figueras, T. Helfer, et al., Grtresna: An open-source code to solve the initial data constraints in numerical relativity, arXiv preprint arXiv:2501.13046 (2025)

    work page arXiv 2025

  67. [67]

    Sanchis-Gual, F

    N. Sanchis-Gual, F. Di Giovanni, M. Zilh˜ ao, C. Herdeiro, P. Cerd´ a-Dur´ an, J. A. Font, and E. Radu, Nonlinear dy- namics of spinning bosonic stars: Formation and stabil- ity, Phys. Rev. Lett. 123, 221101 (2019)

    work page 2019

  68. [68]

    Sanchis-Gual, M

    N. Sanchis-Gual, M. Zilh˜ ao, C. Herdeiro, F. Di Gio- vanni, J. A. Font, and E. Radu, Synchronized gravita- tional atoms from mergers of bosonic stars, Phys. Rev. D 102, 101504 (2020)

    work page 2020

  69. [69]

    Siemonsen and W

    N. Siemonsen and W. E. East, Stability of rotating scalar boson stars with nonlinear interactions, Physical Review D 103, 044022 (2021)

    work page 2021

  70. [70]

    Sanchis-Gual, F

    N. Sanchis-Gual, F. Di Giovanni, C. Herdeiro, E. Radu, and J. A. Font, Multifield, multifrequency bosonic stars and a stabilization mechanism, Physical Review Letters 126, 241105 (2021)

    work page 2021

  71. [71]

    Jaramillo, D

    V. Jaramillo, D. N´ u˜ nez, M. Ruiz, and M. Zilh˜ ao, Full 3d nonlinear dynamics of charged and magnetized boson stars, Physical Review D 111, 024070 (2025)

    work page 2025

  72. [72]

    Herdeiro, I

    C. Herdeiro, I. Perapechka, E. Radu, and Y. Shnir, Asymptotically flat spinning scalar, Dirac and Proca stars, Phys. Lett. B 797, 134845 (2019), arXiv:1906.05386 [gr-qc]

    work page arXiv 2019

  73. [73]

    N. M. Santos, C. L. Benone, L. C. B. Crispino, C. A. R. Herdeiro, and E. Radu, Black holes with synchronised Proca hair: linear clouds and fundamental non-linear so- lutions, JHEP 07, 010, arXiv:2004.09536 [gr-qc]

    work page arXiv 2004

  74. [74]

    P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light- ring stability for ultracompact objects, Phys. Rev. Lett. 119, 251102 (2017)

    work page 2017

  75. [75]

    P. V. P. Cunha, C. Herdeiro, E. Radu, and N. Sanchis- Gual, Exotic compact objects and the fate of the light- ring instability, Phys. Rev. Lett. 130, 061401 (2023)

    work page 2023

  76. [76]

    Di Giovanni, N

    F. Di Giovanni, N. Sanchis-Gual, P. Cerd´ a-Dur´ an, M. Zil- hao, C. Herdeiro, J. A. Font, and E. Radu, Dynamical bar-mode instability in spinning bosonic stars, Physical Review D 102, 124009 (2020)

    work page 2020

  77. [77]

    L¨ offler, J

    F. L¨ offler, J. Faber, E. Bentivegna, T. Bode, P. Di- ener, R. Haas, I. Hinder, B. C. Mundim, C. D. Ott, E. Schnetter, G. Allen, M. Campanelli, and P. Laguna, The einstein toolkit: a community computational infras- tructure for relativistic astrophysics, Classical and Quan- tum Gravity 29, 115001 (2012)

    work page 2012

  78. [78]

    Zilh˜ ao, H

    M. Zilh˜ ao, H. Witek, and V. Cardoso, Nonlinear interac- tions between black holes and proca fields, Classical and Quantum Gravity 32, 234003 (2015)

    work page 2015

  79. [79]

    Witek and M

    H. Witek and M. Zilh˜ ao, Canuda

    work page

  80. [80]

    Witek, M

    H. Witek, M. Zilhao, G. Bozzola, C.-H. Cheng, A. Dima, M. Elley, G. Ficarra, T. Ikeda, R. Luna, C. Richards, N. Sanchis-Gual, and H. Silva, Canuda: a public nu- merical relativity library to probe fundamental physics (2023)

    work page 2023

Showing first 80 references.