Model waveform accuracy standards for gravitational wave data analysis

Model waveforms are used in gravitational wave data analysis to detect and then to measure the properties of a source by matching the model waveforms to the signal from a detector. This paper derives accuracy standards for model waveforms which are sufficient to ensure that these data analysis applications are capable of extracting the full scientific content of the data, but without demanding excessive accuracy that would place undue burdens on the model waveform simulation community. These accuracy standards are intended primarily for broadband model waveforms produced by numerical simulations, but the standards are quite general and apply equally to such waveforms produced by analytical or hybrid analytical-numerical methods.

Figure 1

Amplitude $A_h$ of the frequency-domain gravitational waveform for an equal-mass nonspinning binary black-hole system. Constants representing the total mass, $M$, and the distance, $r$, to the binary system have been introduced to make the graphed quantities dimensionless, independent of the mass of the source, and (asymptotically) independent of the observer’s position; $G$ is Newton’s constant and $c$ is the speed of light.

Figure 2

Phase ${\Phi }_h$ (measured in radians) of the frequency-domain gravitational waveform for an equal-mass nonspinning binary black-hole system. The constant $M$, representing the total mass of the binary system, has been introduced to express the frequency in dimensionless units, and to make the graphed quantities independent of the mass of the source; $G$ is Newton’s constant and $c$ is the speed of light.

Figure 3

The solid line illustrates the model waveform submanifold, with particular members of a discrete template bank $h_b$ and $h_{b^{\prime }}$. The model waveform $h_{\overline{m}}$ has the maximum mismatch ${ϵ}_{\mathrm{MM}}$ with the template bank waveform $h_b$. The closest exact waveform $h_e$ has the mismatch ${ϵ}_{\mathrm{FF}}$ with $h_{\overline{m}}$ and the total effective mismatch ${ϵ}_{\mathrm{EFF}}={ϵ}_{\mathrm{MM}}+{ϵ}_{\mathrm{FF}}$ with $h_b$. The model waveform $h_m$ having the same physical parameters as $h_e$ may differ from $h_{\overline{m}}$ due to modeling errors, and its mismatch ${ϵ}_{\max }$ will always exceed ${ϵ}_{\mathrm{FF}}$: ${ϵ}_{\max }\ge {ϵ}_{\mathrm{FF}}$.

Figure 4

Solid curves illustrate $\overline{C}$ and $C$, as defined in Eqs. (22, 27), as functions of the total mass for nonspinning equal-mass binary black-hole waveforms and the Initial LIGO noise spectrum. Dashed curves give the same quantities computed with the Advanced LIGO noise curve.

Figure 5

Curves illustrate $\overline{C}$ and $C$, as defined in Eqs. (22, 27), as functions of the total mass for nonspinning equal-mass binary black-hole waveforms using the Barack and Cutler [36] approximate LISA noise spectrum.