Waveform accuracy and systematic uncertainties in current gravitational wave observations

The post-Newtonian formalism plays an integral role in the models used to extract information from gravitational wave data, but models that incorporate this formalism are inherently approximations. Disagreement between an approximate model and nature will produce mismodeling biases in the parameters inferred from data, introducing systematic error. We here carry out a proof-of-principle study of such systematic error by considering signals produced by quasicircular, inspiraling black hole binaries through an injection and recovery campaign. In particular, we study how unknown, but calibrated, higher-order post-Newtonian corrections to the gravitational wave phase impact systematic error in recovered parameters. As a first study, we produce injected data of nonspinning binaries as detected by a current, second-generation network of ground-based observatories and recover them with models of varying PN order in the phase. We find that the truncation of higher-order ($>3.5$) post-Newtonian corrections to the phase can produce significant systematic error even at signal-to-noise ratios of current detector networks. We propose a method to mitigate systematic error by marginalizing over our ignorance in the waveform through the inclusion of higher-order post-Newtonian coefficients as new model parameters. We show that this method can reduce systematic error greatly at the cost of increasing statistical error.

Figure 1

Schematic diagram to illustrate the relationship between systematic error ${\delta }_{\mathrm{sys}}$ and statistical error ${\delta }_{\mathrm{stat}}$. The gray curve in each panel depict a one-dimensional, inferred posterior probability distribution on a hypothetical parameter obtained by analyzing a synthetic noise-free data set and recovering it with some model that has some amount of inherent inaccuracy with respect to the true signal. The injected (true) value of the parameter is indicated with a red line, while the recovered value, taken in this case from the maximum posterior point, is indicated with a blue line. In Case A, the systematic error is smaller than the statistical error and the value of the injected parameter lies within the credible intervals of the recovered best-fit value of the parameter. In Case B, however, the systematic error is dominant and the injected value lies outside of the credible interval.

Figure 2

Absolute values of series functions ${\varphi }_i={\varphi }_i(f,\mathit{\theta })$ of the IMRPhenomD inspiral phase, as given in Eqs. (2)–(6), plotted as a function of velocity $v=(\pi Mf{)}^{1/3}$ for the systems presented in Table 1. Observe that the terms in the phase have a hierarchy, with the lowest PN order terms larger than the higher PN order terms. We do not plot the terms that contain $\log (\pi Mf)$ that occur at 2.5PN and 3PN of the classic PN expansion.

Figure 3

Partial corner plot produced after a Bayesian parameter estimation study, using each of the recovery models detailed in Table 2 to infer parameters from the unequal mass (left) and equal mass (right) SNR 80 injected signals, described in Table 1. The corner plots show the marginalized one-dimensional posterior probability distributions and the 90% credible region contours of the two-dimensional posterior probability distributions on chirp mass $\mathcal{M}$, mass ratio $q$, and (geocentric) reference time $t_c$. The injected “true” values are shown as black dots and vertical lines. Observe that the bias is very small for the “5.5PN” model in both cases, because the recovery model is the same as the injection model. The bias is also suppressed for the “5PN” model in the unequal mass case where the removed “5.5PN” term is very small. However, in the equal mass case where the “5.5PN” term has a larger contribution, the bias from the “5PN” model is significant. The bias of the “4.5PN” model is very large in both cases, because of the alternating structure of the phenomenological coefficients. The bias becomes smaller again, but is still not negligible, for the 3.5PN and “4PN” models. This suggests that the phenomenological coefficients of the IMRPhenomD model are important for parameter estimation and their inaccurate determination could lead to systematic bias.

Figure 4

Partial corner plot produced using the “4PN” recovery model detailed in Table 2 to estimate parameters from each of the unequal mass-injected signals described in Table 1. We depict the marginalized one-dimensional posterior probability distributions and 90% credible region contours of the two-dimensional posterior probability distributions on chirp mass $\mathcal{M}$ and mass ratio $q$. The injection values are indicated in black. Observe that, as expected, as the SNR increases, the width of the posterior probability distribution shrinks, reducing the support of the posterior on the injected values. This figure indicates clearly that models once thought to be accurate enough at small SNR can rapidly become insufficiently accurate at higher SNRs.

Figure 5

Ratio of bias to statistical error in the chirp mass (top) and mass ratio (bottom) for various recovery models (Table 2) and injected parameter (Table 1), including the unequal mass cases (left) and the equal mass cases (right). The black dots indicate the configurations of the parameter estimation analyses we performed, which we join through a nearest-neighbor linear interpolation. When the recovery model matches the injection model (i.e. in the “5.5PN” case), the bias is due entirely to sampling, and it is thus distinguished by plotting it with dashed lines. Observe that for the 3.5PN, “4PN”, and “4.5PN” recovery models in the unequal mass case and 3.5PN, “4PN”, “4.5PN’,’ and “5PN” recovery models in the equal mass case, where the bias is dominated by systematic error, the bias grows with SNR relative to the statistical error. Observe that this trend is violated for the mass ratio in the equal-mass case due to prior boundary effects. Observe also that systematic bias grows as “PN” terms are removed, until the “4.5PN” term is removed. Most significantly, note the instances where $\mathrm{\Delta }\theta /{\delta }_{\mathrm{stat}}\theta >1$. In these cases the recovery model was not accurate enough to reliably recover the injection parameters.

Figure 6

Mismatch, as defined in Eq. (8), between the injected signal $h_{“5.5\mathrm{PN}”}=h_{“5.5\mathrm{PN}”}({\mathit{\theta }}_{\mathrm{inj}})$ and signals produced by injecting the recovery models with the injection parameters $h_{\mathrm{x}}=h_{\mathrm{x}}({\mathit{\theta }}_{\mathrm{inj}})$. We consider the $\mathrm{SNR}=80$ unequal and equal mass injection detailed in Table 1. Observe that the relative magnitudes of the mismatch values are comparable to the relative magnitudes of the biases in chirp mass and mass ratio depicted in the corner plots in Fig. 3.

Figure 7

Partial corner plot using the “5.5PN” and “4.5PN” recovery models, as well as the enhanced model with a “4.5PN” order base and the two priors (IG and SWU). All partial corner plots are marginalized over the parameters not shown in the figure. In particular, the enhanced “4.5PN” model are marginalized over the higher PN order terms introduced. The upper-left and lower-right panels contain the one-dimensional PDFs for chirp mass $\mathcal{M}$ and mass ratio $q$, respectively, and the lower-left panel depicts the 90% credible region contours, with the injection values are indicated in black. Observe that the enhanced models remove the systematic bias introduced by the plain “4.5PN” model, at the cost of enlarging the width of the posterior probability distribution when the range of ${\overline{\sigma }}_{3,4}$ is assumed unknown (the SWU prior).