Head-on Collisions of Boson Stars with Bowen-York Type Initial Data

Technical Summary: Head-on Collisions of Boson Stars with Bowen-York Type Initial Data

Problem Statement
The construction of valid initial data for numerical relativity simulations involving boson stars (BSs) remains a challenge. While the Bowen-York approach is the standard for black hole binaries (BBHs), offering analytic solutions to the momentum constraint and a natural link to post-Newtonian approximations, its application to BSs has been limited. Existing methods for BSs often rely on superposing boosted stationary solutions, which are approximate and violate constraint equations, or require solving complex constraint equations via conformal thin-sandwich decompositions. The authors address the need for a constraint-satisfying initial data method for BSs that retains the simplicity of the Bowen-York formalism, specifically for head-on collisions between two BSs (BBS) and between a BS and a black hole (BHBS).

Methodology
The study employs the Conformal-Transverse-Traceless (CTT) decomposition within the BSSN formulation using the moving puncture gauge. The core innovation is adapting the Bowen-York solution for the momentum constraint to include matter sources specific to boson stars.

1. Initial Data Construction:

   • Momentum Constraint: The authors derive a solution for the momentum constraint $\tilde{\nabla}_j \tilde{A}^{ij} = 8\pi \tilde{S}^i$ for a spherically symmetric source with linear momentum, following Bowen's extended solution for perfect fluids. For BSs, the source term $\tilde{S}^i$ is derived from the complex scalar field $\Phi$ and its conjugate momentum $\Pi$.
   • Scalar Field Structure: The scalar field is assumed to have the form $\tilde{\Phi} = \phi(r) e^{i(\omega t - k_j x^j)}$. This allows the momentum density to be expressed as $\tilde{S}^i = \sigma P^i$, where $\sigma$ is a source function derived from the stationary BS profile and $P^i$ is the linear momentum.
   • Superposition: For binary systems, the authors superpose the scalar fields ($\tilde{\Phi} = \tilde{\Phi}_+ + \tilde{\Phi}_-$) and the extrinsic curvature solutions ($\tilde{A}_{ij} = \tilde{A}_{ij}^+ + \tilde{A}_{ij}^-$). The Hamiltonian constraint is then solved numerically to obtain the conformal factor $\psi$, ensuring the initial data satisfies all constraints.

2. Simulation Setup:

   • Code: Simulations were performed using the Maya code (a local version of the Einstein Toolkit) based on the BSSN formulation.
   • Parameters: The study focuses on mini-boson stars with a solitonic potential (limit $\sigma_0 \to \infty$). Simulations cover equal-mass BBS collisions and equal-mass BHBS collisions.
   • Variations: The central amplitude $\phi_*$ (ranging from 0.02 to 0.04) and the initial linear momentum $P/M_*$ (0.1, 0.2, 0.3) are varied. Initial separations are set to $d = 80 \mu^{-1}$.
   • Comparisons: Results are compared against equivalent BBH collisions and previous studies on BBS and BHBS dynamics.

Key Contributions and Results

• Validation of the Method: The method was first tested on a single BS with linear momentum. The authors observed that the assumption of conformal flatness and spherical symmetry induces oscillations in the star's structure (specifically in the scalar field amplitude $\phi$) during evolution. These oscillations correspond to normal modes of the BS, confirming that the perturbations are intrinsic to the initial data construction rather than numerical noise.
• BBS Head-on Collisions:
  • Outcomes: For high compactness ($\phi_* = 0.03, 0.04$), all collisions resulted in black hole formation, evidenced by quasi-normal mode (QNM) ringing and apparent horizon detection. For lower compactness ($\phi_* = 0.02$), low momentum ($P/M_* = 0.1$) led to black hole formation, while higher momenta ($0.2, 0.3$) resulted in highly distorted, oscillating boson stars that avoided collapse due to high kinetic energy.
  • Gravitational Wave (GW) Emission: The study confirms that BBS systems emit more energy in gravitational waves than equivalent BBH systems. The authors attribute this to the collision of less compact objects producing a more perturbed remnant.
• BHBS Head-on Collisions:
  • Outcomes: All BHBS collisions resulted in black hole formation with standard QNM ringing.
  • Gravitational Wave Emission: In contrast to BBS, BHBS systems radiate less energy than their equivalent BBH counterparts. The authors suggest this is because the deformation of the black hole in a BHBS encounter is less significant compared to the deformation of a boson star in a BBS or BBH collision.
• Quantitative Analysis: The study provides detailed tables of ADM energy, radiated energy percentages, remnant black hole masses, and QNM frequencies/decay timescales. The QNM frequencies and decay times for the resulting black holes generally align with Schwarzschild values, though deviations occur when significant scalar field remnants persist near the horizon (e.g., in the $\phi_* = 0.02, P/M_* = 0.1$ case).

Significance and Claims
The paper claims that the proposed Bowen-York inspired initial data method is effective and preserves the simplicity required to incorporate information from the early, post-Newtonian inspiral phase. The authors demonstrate that this approach yields results consistent with previous studies regarding the outcomes of collisions and the efficiency of gravitational wave emission.

Specifically, the study corroborates the finding that:

1. BBS binaries are more efficient GW emitters than equivalent BBH binaries.
2. BHBS binaries are less efficient GW emitters than equivalent BBH binaries.

The authors conclude that this method provides a robust foundation for future studies, specifically mentioning plans to apply this prescription to inspiraling compact object binaries involving spinning black holes and neutron stars, thereby bridging the gap between post-Newtonian approximations and full numerical relativity simulations for astrophysically relevant scenarios.