Reliability of complete gravitational waveform models for compact binary coalescences

Accurate knowledge of the gravitational-wave (GW) signal from inspiraling compact binaries is essential to detect these signatures in the data from GW interferometers. With recent advances in post-Newtonian (PN) theory and numerical relativity (NR) it has become possible to construct inspiral-merger-ringdown waveforms by combining both descriptions into one complete hybrid signal. While addressing the reliability of such waveforms in different points of the physical parameter space, previous studies have identified the PN contribution as the dominant source of error, which can be reduced by incorporating longer NR simulations. In this paper we overcome the two outstanding issues that make it difficult to determine the minimum simulation length necessary to produce suitably accurate hybrids for GW astronomy applications: (1) the relevant criteria for a GW search is the mismatch between the true waveform and a set of model waveforms, optimized over all waveforms in the model, but for discrete hybrids this optimization was not yet possible. (2) these calculations typically require that numerical waveforms already exist, while we develop an algorithm to estimate hybrid mismatch errors without numerical data, which enables us to estimate the necessary NR waveform length before performing the simulation. Our procedure relies on combining supposedly equivalent PN models at highest available order with common data in the NR regime, and their difference serves as a measure of the uncertainty assumed in each waveform. Contrary to some earlier studies, we estimate that $\sim 10$ NR orbits before merger should allow for the construction of waveform families that are accurate enough for detection in a broad range of parameters, only excluding highly spinning, unequal-mass systems. Nonspinning systems, even with high mass-ratio ($q\gtrsim 20$) are well modeled for astrophysically reasonable component masses. In addition, the parameter bias is only of the order of 1% for total mass and symmetric mass-ratio and less than 0.1 for the dimensionless spin magnitude. We take the view that similar NR waveform lengths will remain the state of the art in the advanced detector era, and begin to assess the limits of the science that can be done with them.

Figure 1

Hybrid mismatches in the equal-mass, nonspinning case. Black solid lines are mismatches of actual $\mathrm{\text{Taylor}}\mathrm{T}1/\mathrm{T}4+\mathrm{NR}$ hybrids, whereas the dashed (red online) lines are our estimates obtained without directly using any NR data (NR amplitude taken from a phenomenological model). The gray dotted lines describe the PN mismatch contribution derived in Eq. (9) that does not include a possible dephasing of the NR part. The matching frequencies for each set are from bottom to top $M{\omega }_m=0.04$, 0.06, 0.08.

Figure 2

Contour plots of the mismatch (in %) between different fictitious hybrids, as a function of the symmetric mass-ratio $\eta$ and equal aligned spins with dimensionless magnitude $\chi$. Left panel: PN part either defined by the TaylorT1 or TaylorT4 approximant. Right panel: Comparison of TaylorT1 and TaylorF2 in the PN part.

Figure 3

The ambiguity function (16) between two phenomenological waveforms [41], where $h_2$ is fixed with $\eta =0.2$, $\chi =0.3$ and the total mass as indicated in the plots. The parameters of $h_1$ are varied individually while the others are kept constant at the values of $h_2$, respectively.

Figure 4

The mass-optimized mismatch between equal-mass, nonspinning $\mathrm{\text{Taylor}}\mathrm{T}1/\mathrm{\text{Taylor}}\mathrm{T}4+\mathrm{NR}$ hybrids (black solid lines) compared to our NR-free estimate (dashed lines). The matching frequencies are $M{\omega }_m\in \{0.04,0.06,0.08\}$ from bottom to top. The gray lines show the results of nonoptimized mismatches for comparison, see also Fig. 1. The inset illustrates the relative bias in the total mass (matching frequencies in reverse order).

Figure 5

Mismatch between TaylorT1- and TaylorF2-based waveforms for mass ratio $4:1$, $\chi =0.5$ and matching frequency $M{\omega }_m=0.06$ (left panel). The mismatch is not optimized (top solid line), mass-optimized (dashed line) or optimized with respect to all physical parameters of the TaylorF2-based model (lowest dotted line). The bias in the parameters are provided on the right panel.

Figure 6

The maximum of the optimized mismatch (in %) for hybrids constructed either with TaylorT1 (target signal) or TaylorF2 (template signal) and a matching frequency of $M{\omega }_m=0.06$.

Figure 7

The fully optimized mismatch ${\mathcal{M}}_{\mathrm{FF}}$ as a function of the matching frequency $M{\omega }_m$ for equal-mass systems (solid lines) and mass-ratio $4:1$ (dashed lines). The considered spins in each case are $\chi \in \{0,0.2,0.4,0.6,0.8\}$ from bottom to top.

Figure 8

The fully optimized mismatch of nonspinning target signals employing either the TaylorT1 or TaylorT4 approximant with model waveforms constructed with TaylorF2 inspirals. The assumed matching frequency is always $M{\omega }_m=0.06$. The bias in parameters is $|\Delta M|/M\lesssim 0.6\%$ (0.16%), $|\Delta \eta |/\eta \lesssim 1.0\%$ (0.3%), $|\Delta \chi |<0.04$ (0.017), where the values in brackets indicate the restriction to $q\le 4$.