4.5 Nonlinear mode coupling

The PITT code has been used to model the nonlinear generation of waveforms by scattering off a Schwarzschild black hole [255Jump To The Next Citation Point256Jump To The Next Citation Point]. The physical setup is similar to the perturbative study in Section 4.4. A radially compact pulse is prescribed on an early time outgoing null hypersurface − 𝒥 and Schwarzschild null data is given on the interior white hole horizon ℋ −, which is causally unaffected by the pulse. The input pulse is standardized to (ℓ = 2, m = 0) and (ℓ = 2, m = 2) quadrupole modes with amplitude A. The outgoing null hypersurfaces extend to future null infinity + ℐ on a compactified numerical grid. Consequently, there is no need for an artificial outer boundary. The evolution code then provides the news function at + ℐ, in the coordinates of an observer in an inertial frame at infinity, thus avoiding any gauge ambiguity in the waveform. This provides a simple setting for how the nonlinearities generated by high amplitudes affect the waveform.

The study reveals several features of qualitative importance:

  1. The mode coupling amplitudes consistently scale as powers An of the input amplitude A corresponding to the nonlinear order of the terms in the evolution equations which produce the mode. This allows much economy in producing a waveform catalog: Given the order n associated with a given mode generation, the response to any input amplitude A can be obtained from the response to a single reference amplitude.
  2. The frequency response has similar behavior but in a less consistent way. The dominant frequencies produced by mode coupling are in the approximate range of the quasinormal frequency of the input mode and the expected sums and difference frequencies generated by the order of nonlinearity.
  3. Large phase shifts, ranging up 15% in a half cycle relative to the linearized waveform, are exhibited in the news function obtained by the superposition of all output modes, i.e. in the waveform of observational significance. These phase shifts, which are important for design of signal extraction templates, arise in an erratic way from superposing modes with different oscillation frequencies. This furnishes a strong argument for going beyond the linearized approximation in designing a waveform catalog for signal extraction.
  4. Besides the nonlinear generation of harmonic modes absent in the initial data, there is also a stronger than linear generation of gravitational wave output. This provides a potential mechanism for enhancing the strength of the gravitational radiation produced during, say, the merger phase of a binary inspiral above the strength predicted in linearized theory.
  5. In the non-axisymmetric m = 2 case, there is also considerable generation of radiation in polarization states not present in the linearized approximation. In the simulations, input amplitudes in the range A = 0.1 to A = 0.36 lead to nonlinear generation of the ⊕ polarization mode which is of the same order of magnitude as the ⊗ mode (which would be the sole polarization in the linearized regime). As a result, significant nonlinear amplification and phase shifting of the waveform would be observed by a gravitational wave detector, depending on its orientation.

These effects arise from the nonlinear modification of the Schwarzschild geometry identified by Papadopoulos in his prior work on axisymmetric mode coupling [183], reported in Section 3.3.2. Although Papadopoulos studied nonlinear mode generation produced by an outgoing pulse, as opposed to the case of an ingoing pulse studied in [255Jump To The Next Citation Point256Jump To The Next Citation Point], the same nonlinear factors were in play and gave rise to several common features. In both cases, the major effects arise in the region near r = 3M. Analogs of Features 1, 2, 3, and 4 above are all apparent in Papadopoulos’s work. At the finite difference level, both codes respect the reflection symmetry inherent in Einstein’s equations and exhibit the corresponding selection rules arising from parity considerations. In the axisymmetric case considered by Papadopoulos, this forbids the nonlinear generation of a ⊕ mode from a ⊗ mode, as described in Feature 5 above.

The evolution along ingoing null hypersurfaces in the axisymmetric work of Papadopoulos has complementary numerical features with the evolution along outgoing null hypersurfaces in the 3D work. The grid based upon ingoing null hypersurfaces avoids the difficulty in resolving effects close to r = 2M encountered with the grid based upon outgoing null hypersurfaces. The outgoing code would require AMR in order to resolve the quasinormal ringdown for as many cycles as achieved by Papadopoulos. However, the outgoing code avoids the late time caustic formation noted in Papadopoulos’ work, as well as the complications of gauge ambiguity and backscattering introduced by a finite outer boundary. One attractive option would be to combine the best features of these approaches by matching an interior evolution based upon ingoing null hypersurfaces to an exterior evolution based upon outgoing null hypersurfaces, as implemented in [164Jump To The Next Citation Point] for spherically symmetric Einstein–Klein–Gordon waves.

The waveform of relevance to gravitational wave astronomy is the superposition of modes with different frequency compositions and angular dependence. Although this waveform results from a complicated nonlinear processing of the input signal, which varies with choice of observation angle, the response of the individual modes to an input signal of arbitrary amplitude can be obtained by scaling the response to an input of standard reference amplitude. This offers an economical approach to preparing a waveform catalog.

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