Mismodeling in gravitational-wave astronomy: The trouble with templates

Waveform templates are a powerful tool for extracting and characterizing gravitational wave signals, acting as highly restrictive priors on the signal morphologies that allow us to extract weak events buried deep in the instrumental noise. The templates map the waveform shapes to physical parameters, thus allowing us to produce posterior probability distributions for these parameters. However, there are attendant dangers in using highly restrictive signal priors. If strong field gravity is not accurately described by general relativity (GR), then using GR templates may result in fundamental bias in the recovered parameters, or even worse, a complete failure to detect signals. Here we study such dangers, concentrating on three distinct possibilities. First, we show that there exist modified theories compatible with all existing observations that would fail to be detected by the LIGO/Virgo network using searches based on GR templates, but which would be detected using a one parameter post-Einsteinian extension. Second, we study modified theories that produce departures from GR that turn on suddenly at a critical frequency, producing waveforms that do not directly fit into the simplest parametrized post-Einsteinian (ppE) scheme. We show that even the simplest ppE templates are still capable of picking up these strange signals and diagnosing a departure from GR. Third, we study whether using inspiral-only ppE waveforms for signals that include merger and ringdown can lead to problems in misidentifying a GR departure. We present a simple technique that allows us to self-consistently identify the inspiral portion of the signal, and thus remove these potential biases, allowing GR tests to be performed on higher mass signals that merge within the detector band. We close by studying a parametrized waveform model that may allow us to test GR using the full inspiral-merger-ringdown signal.

Figure 1

Bounds that can be placed on the ppE strength parameter, $\beta$, for various values of the ppE exponent, $b$, using GW measurements [29] from a binary with component masses of $6M_{\odot }$ and $12M_{\odot }$, and measurements from binary pulsars [65]. The regions above the pulsar line are already ruled out by experiment. Reference [29] shows that this bound is robust over a range of mass ratios.

Figure 2

Percentage of signals missed [$100\times (1-{\mathrm{FF}}^3)$] when using a GR (dashed/blue line) and a one-parameter ppE template with $b=-1$ (solid/red line) and the injected, non-GR signal [of the form in Eq. (7), with constituent masses $1.4M_{\odot }$ and $3.73M_{\odot }$] as a function of ${\beta }_{-1}$. ($b=-1$ corresponds to a 2PN correction to the GR waveform.) The same quantity is plotted for spinning GR and ppE templates. The percentage of signals missed approaches 100% for values of ${\beta }_{-1}$ that are fully consistent with existing experimental bounds.

Figure 3

Recovered value of ${\chi }_1$ from a spinning, GR template used to match a nonspinning, non-GR signal of the form in Eq. (7). Once ${\beta }_{-1}\gtrsim 5$, ${\chi }_1$ is limited by the prior range.

Figure 4

Recovered value of $\mathcal{M}$ from both a GR (dashed/blue line) and a one-parameter ppE template with $b=-1$ (solid/red line) used to fit a non-GR signal of the form in Eq. (7), plotted as a function of ${\beta }_{-1}$. The true value of $\mathcal{M}$ is $\mathcal{M}=1.94M_{\odot }$. The constituent masses were $1.4M_{\odot }$ and $3.73M_{\odot }$. Using GR templates to fit non-GR signals leads to large biases in the recovered parameters.

Figure 5

The BFs between a nonspinning, GR template and a nonspinning, ppE template with $b=-2$.The injection was a GR system of two neutron stars with equal and aligned spin. The BF favors the non-GR model for realistic values of dimensionless spin parameter $\chi$, indicating the need to use spinning waveforms to recover these types of signals.

Figure 6

BF between GR and ppE models for injections of the form in Eq. (14), for varying values of $f^{*}$. The dashed (blue) line is for a standard and the simplest, inspiral ppE template, and the solid (red) line is one that has been modified with a Heaviside function. Both types of templates show the same general behavior, and both are successful at detecting deviations from GR for certain ranges of $f^{*}$. The injected signal had constituent masses $m_1/m_2=1.42/2.0M_{\odot }$.

Figure 7

Posterior distributions for $\beta$, recovered using standard ppE templates. The injected signal was of the form in Eq. (14), with $\beta =1e-06$, and $m_1/m_2=1.42/2.0M_{\odot }$. If there is little weight in the posterior at $\beta =0$, the signal is detectable as non-GR. In the top left panel, $f_{\min }$ is low, and $\beta$ is recovered at the correct value. In the bottom right panel, $f_{\min }$ is very high, and the GR model is clearly favored. In the bottom right panel, the signal is clearly non-GR, but the recovered value for $\beta$ is incorrect. Finally, in the top right panel, two peaks in the posterior are clearly visible—one mode near the correct value of $\beta$, and one at the incorrect, negative value. Because the chain swaps between the two peaks, there is significant weight at $\beta =0$, and this signal is not detectable as non-GR. For all injections, the constituent masses were $m_1/m_2=1.42/2.0M_{\odot }$.

Figure 8

The correlation between $\beta$ and $\mathcal{M}$, generated from a signal of the form in Eq. (14), with $f^{*}=47.5$. The two separate maxima in the likelihood are clearly visible, as well as the strong correlation between these two parameters. The injected chirp mass was $\mathcal{M}=1.463M_{\odot }$, with constituent masses $m_1/m_2=1.42/2.0M_{\odot }$.

Figure 9

Fractional uncertainty in the recovered value of $f^{*}$, for different injected values of $f^{*}$. The uncertainty is inversely proportional to the BF in favor of the non-GR model—i.e. when the BF indicates a clear departure from GR, the $f^{*}$ parameter is recovered with high accuracy. Again, the injected signals had masses $m_1/m_2=1.42/2.0M_{\odot }$.

Figure 10

BFs between GR and ppE templates. The injected signals were GR, PhenomC waveforms, and they were recovered using inspiral-only ppE waveforms. The dashed (blue) line shows the BFs calculated by using the frequency at the light ring as the cutoff frequency for the waveforms. The solid (red) line shows BFs calculated by using $f_{\mathrm{ISCO}}$ as the cutoff frequency. A BF larger than 1 indicates a preference for the non-GR model.

Figure 11

(upper panel) The bias in total mass, $M$, recovered when using an inspiral-only, GR signal to fit an IMR, GR signal. The dashed (blue) line was calculated using the light ring to determine cutoff frequency, and the solid (red) line used the ISCO. The error bars show the 1-$\sigma$ limits for the recovered values. For high-mass systems, the bias nears 10%. For lower-mass systems, the recovered mass is very close to the injected value. Each injected signal had SNR $\sim 25$. (lower panel) The bias in chirp mass, $\mathcal{M}$, recovered for the same systems injected in the upper panel, with the cutoff frequency determined using the ISCO. This plot illustrates that the recovered value of $\mathcal{M}$ is not strongly affected by the presence of merger and ringdown in the signal.

Figure 12

The posterior distributions for $M$ (left panels) and $\beta$ (right panels) for a $30M_{\odot }$ system, from Analyses I, II, and III. The bias in recovery of $M$ is removed in Analyses II and III, as is the model preference for ppE over GR. The injected value is shown by the vertical line in each panel.

Figure 13

BFs between ppE and GR templates, from Analysis I (solid/red), using $f_{\mathrm{ISCO}}$ as the cutoff frequency, and from Analysis II (dashed/blue), using $r=10M$ to calculate the cutoff frequency. All signals were GR signals. In Analyses II and III, the model selection process always favors GR.

Figure 14

Time-domain waveforms generated using the parametrization in Eq. (16), for an SNR 30 signal with total mass $M=50M_{\odot }$. Top left: GR waveform. Top right: $f_{\text{stretch}}=0.1$: the merger portion of the waveform is compressed, but the frequency at which merger begins and the structure of the ringdown are unaffected. Bottom left: $f_{\text{shift}}=-80\mathrm{Hz}$: the beginning of merger is shifted to a lower frequency by 80 Hz, but the duration of merger and the ringdown structure are unaffected. Bottom right: $\kappa =0.01$: merger is unaffected, but ringdown is changed such that the decay is much slower than in GR.

Figure 15

(upper panel) Uncertainty in the recovered value of $f_{\text{shift}}$ (solid/red) and $10\times$ the uncertainty in the recovered value of $\beta$ (dashed/blue), for different injected values of $M$. The uncertainty in $f_{\text{shift}}$ decreases as the total mass increases and the merger-ringdown portion of the waveform becomes more important. The uncertainty in $\beta$ increases due to correlations between the two parameters. All injected waveforms were GR signals. (lower panel) The injected value of $f_{\mathrm{IM}}$ (the transition between merger and ringdown) for the different injected systems. The error bars on this value are the uncertainty in $f_{\text{shift}}$, as this parameter moves the transition between merger and ringdown.

Figure 16

Correlation between the ppE phase parameter $\beta$ and the parameter $f_{\text{shift}}$ that controls the start of the merger phase. This correlation is only present for systems with large $M$, and thus leads to an increase in the uncertainty in the recovered value of $\beta$ for these systems.

Figure 17

Posterior distributions for the parameters $\kappa$ and $f_{\text{stretch}}$ for various values of $M$. As the total mass increases, the parameters go from being completely unconstrained to well measured by the data. All injected signals were GR signals. The vertical line in each panel indicates the injected, GR value for that parameter.

Figure 18

Correlation between the parameter $\tau$, which affects the ringdown phase, and the parameter $f_{\text{stretch}}$, which controls the length of the merger phase.