Class. Quantum Grav. 40 175009 (2023); Corrigendum: Class. Quantum
Grav. 40 249503 (2023) We construct a numerical relativity code based on the
Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation ...for the gravitational
quadratic $f(R)$ Starobinsky model. By removing the assumption that the
determinant of the conformal 3-metric is unity, we first generalize the BSSN
formulation for general $f(R)$ gravity theories in the metric formalism to
accommodate arbitrary coordinates for the first time. We then describe the
implementation of this formalism to the paradigmatic Starobinsky model. We
apply the implementation to three scenarios: the Schwarzschild black hole
solution, flat space with non-trivial gauge dynamics, and a massless
Klein-Gordon scalar field. In each case, long-term stability and second-order
convergence is demonstrated. The case of the massless Klein-Gordon scalar field
is used to exercise the additional terms and variables resulting from the
$f(R)$ contributions. For this model, we show for the first time that
additional damped oscillations arise in the subcritical regime as the system
approaches a stable configuration.
The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it ...was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the full system follows as a consequence in many cases. We demonstrate this explicitly in specific examples, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite general conditions. Consequently the characteristic initial (boundary) value problem of GR in these gauges is rendered ill-posed in the simplest norms one would like to employ. The possibility of finding good alternative norms, in which well-posedness is achieved, is discussed. So motivated, we present numerical convergence tests with an implementation of full GR which demonstrate the effect of weak hyperbolicity in practice.
We construct a numerical relativity code based on the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation for the gravitational quadratic \(f(R)\) Starobinsky model. By removing the assumption that ...the determinant of the conformal 3-metric is unity, we first generalize the BSSN formulation for general \(f(R)\) gravity theories in the metric formalism to accommodate arbitrary coordinates for the first time. We then describe the implementation of this formalism to the paradigmatic Starobinsky model. We apply the implementation to three scenarios: the Schwarzschild black hole solution, flat space with non-trivial gauge dynamics, and a massless Klein-Gordon scalar field. In each case, long-term stability and second-order convergence is demonstrated. The case of the massless Klein-Gordon scalar field is used to exercise the additional terms and variables resulting from the \(f(R)\) contributions. For this model, we show for the first time that additional damped oscillations arise in the subcritical regime as the system approaches a stable configuration.
A primary goal of numerical relativity is to provide estimates of the wave strain, \(h\), from strong gravitational wave sources, to be used in detector templates. The simulations, however, typically ...measure waves in terms of the Weyl curvature component, \(\psi_4\). Assuming Bondi gauge, transforming to the strain \(h\) reduces to integration of \(\psi_4\) twice in time. Integrations performed in either the time or frequency domain, however, lead to secular non-linear drifts in the resulting strain \(h\). These non-linear drifts are not explained by the two unknown integration constants which can at most result in linear drifts. We identify a number of fundamental difficulties which can arise from integrating finite length, discretely sampled and noisy data streams. These issues are an artifact of post-processing data. They are independent of the characteristics of the original simulation, such as gauge or numerical method used. We suggest, however, a simple procedure for integrating numerical waveforms in the frequency domain, which is effective at strongly reducing spurious secular non-linear drifts in the resulting strain.
We present an improved numerical relativity (NR) calibration of the new effective-one-body (EOB) model for coalescing non precessing spinning black hole binaries recently introduced by Damour and ...Nagar Physical Review D 90, 044018 (2014). We do so by comparing the EOB predictions to both the phasing and the energetics provided by two independent sets of NR data covering mass ratios \(1\leq q \leq 9.989\) and dimensionless spin range \(-0.95\leq \chi\leq +0.994\). One set of data is a subset of the Simulating eXtreme Spacetimes (SXS) catalog of public waveforms; the other set consists of new simulations obtained with the Llama code plus Cauchy Characteristic Evolution. We present the first systematic computation of the gauge-invariant relation between the binding energy and the total angular momentum, \(E_{b}(j)\), for a large sample of, spin-aligned, SXS and Llama data. The dynamics of the EOB model presented here involves only two free functional parameters, one (\(a_6^c(\nu)\)) entering the non spinning sector, as a 5PN effective correction to the interaction potential, and one (\(c_3(\tilde{a}_1,\tilde{a}_2,\nu))\) in the spinning sector, as an effective next-to-next-to-next-to-leading order correction to the spin-orbit coupling. These parameters are determined (together with a third functional parameter \(\Delta t_{\rm NQC}(\chi)\) entering the waveform) by comparing the EOB phasing with the SXS phasing, the consistency of the energetics being checked afterwards. The quality of the analytical model for gravitational wave data analysis purposes is assessed by computing the EOB/NR faithfulness. Over the NR data sample and when varying the total mass between 20 and 200~\(M_\odot\) the EOB/NR unfaithfulness (integrated over the NR frequency range) is found to vary between \(99.493\%\) and \(99.984\%\) with a median value of \(99.944\%\).
In addition to the dominant oscillatory gravitational wave signals produced during binary inspirals, a non-oscillatory component arises from the nonlinear "memory" effect, sourced by the emitted ...gravitational radiation. The memory grows significantly during the late inspiral and merger, modifying the signal by an almost step-function profile, and making it difficult to model by approximate methods. We use numerical evolutions of binary black holes to evaluate the nonlinear memory during late-inspiral, merger and ringdown. We identify two main components of the signal: the monotonically growing portion corresponding to the memory, and an oscillatory part which sets in roughly at the time of merger and is due to the black hole ringdown. Counter-intuitively, the ringdown is most prominent for models with the lowest total spin. Thus, the case of maximally spinning black holes anti-aligned to the orbital angular momentum exhibits the highest signal-to-noise (SNR) for interferometric detectors. The largest memory offset, however, occurs for highly spinning black holes, with an estimated value of h^tot_20 \approx 0.24 in the maximally spinning case. These results are central to determining the detectability of nonlinear memory through pulsar timing array measurements.
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