Nonspherical Perturbations of 3D Boson Stars

Table of Contents

• Nonspherical Perturbations : scalar field pert = epsilon x Y_lm
  Zerilli and Psi4 waveforms and modefit with the 12 quasinormal modes determined by Yoshida, Eriguchi and Futamase, 1994
  • Scalar field pert = 0.01 Y_20
    Excellent agreement between all gravitational waveforms and quasinormal modefits
    We observed a spread of modes contributing (see table of coeficients of the modefit). It is noteworthy that the first quasinormal mode does not have a large contribution.
  • Scalar field pert = 0.01 Y_20 + 0.01 Y_22
    Excellent agreement between all gravitational waveforms and quasinormal modefits
    • The Zerilli l=2, m=0 signals for the Y_20 and the mixed Y_20/Y22 perturbation have the same size and frequency
    • The Psi4 signals for the Y_20 and the mixed Y_20/Y22 perturbation have the same frequency. The latter is larger in size because it incorporates more modes.
    • The Zerilli l=2, m=0 and l=2, m=2 signals have the same frequency
  • Mathematica notebook for modefit generation
    We use Mathematica's "Fit" function which performs a linear least square fit to determine the coefficient of each mode.
• Nonspherical Perturbations : "spherical pert" weighted on the z axis
  These perturbations change the mass of the star significantly: Zerilli calculation appears to break down due to scalar radiation. Psi4 waveform and modefit for:
  • Stable Star: 178 x 178 x 178 simulation
  • Critical Star
• Nonspherical Perturbations : numerical resolution perturbations on spherical stars
  Figure 1: Stable vs. Critical vs. Unstable
  • The Zerilli waveform shows that the more compact and dense the boson star configuration, the higher the frequency of the emitted waveform (frequency is highest for the unstable configuration, lowest for the stable one).
  For this perturbation Psi4 shows no definite signal. Figure 2: Axisymmetric vs. Non-Axisymmetric perturbations : critical star
  
  • Under Non-axiymmetric perturbation: non-trivial Zerilli waveforms for both l=2,m=0 and l=2,m=2 modes. These signals have the same frequency. The same behavior is observed for the non-spherical qusinormal modes.
  • Under Axiymmetric perturbation: non-trivial Zerilli waveform exists only for l=2, m=0 mode.
• IVP Solver
  • Convergence test - roughly second order convergence
  2D and 3D Plots of Conformal Factor for the 0.001 Y 20 perturbation
  • 2D xy and yz slices: XY slice shows spherical symmetry in the conformal factor, while the YZ slice shows the contours of a dumb bell shape
  • 3D plot shows that the conformal factor has the expected dumb bell shape characteristic to the Y 20 spherical harmonic

Nonspherical Perturbations : scalar field pert = epsilon x Y_lm

Scalar field pert = 0.01 Y_20

Scalar field pert = 0.01 Y_20 + 0.01 Y_22

Nonspherical Perturbations : "spherical pert" weighted on the z axis

Non-spherical Perturbations : numerical resolution perturbations on spherical stars

Figure 1 Model Grid Size Resolution Stable star 128x128x124 dx=0.5,dy=0.5,dz=0.516 Critical star 128x128x124 dx=0.4,dy=0.4,dz=0.416 Unstable star 128x128x124 dx=0.4,dy=0.4,dz=0.416	Figure 2 Perturbation Type Grid Size Resolution Axisymmetric 128x128x124 dx=0.4,dy=0.4,dz=0.416 Nonaxisymmetric 128x126x124 dx=0.4,dy=0.408,dz=0.416

Convergence of the IVP Solver

We consider a stable branch boson star with central field density 0.1414. The stable star is perturbed by taking a spherically perturbed configuration (of size which adds 7% to the mass of the star) and weighting the configuration preferentially in the z direction, creating a nonspherical perturbation. An approximate solution to the Hamiltonian constraint equation is obtained from the Initial Value Problem solver after providing the nonspherical configuration as input.