Rapid gravitational wave parameter estimation with a single spin: Systematic uncertainties in parameter estimation with the SpinTaylorF2 approximation

Reliable low-latency gravitational wave parameter estimation is essential to target limited electromagnetic follow-up facilities toward astrophysically interesting and electromagnetically relevant sources of gravitational waves. In this study, we examine the trade-off between speed and accuracy. Specifically, we estimate the astrophysical relevance of systematic errors in the posterior parameter distributions derived using a fast-but-approximate waveform model, SpinTaylorF2 (stf2), in parameter estimation with lalinference_mcmc. Though efficient, the stf2 approximation to compact binary inspiral employs approximate kinematics (e.g., a single spin) and an approximate waveform (e.g., frequency domain versus time domain). More broadly, using a large astrophysically motivated population of generic compact binary merger signals, we report on the effectualness and limitations of this single-spin approximation as a method to infer parameters of generic compact binary sources. For most low-mass compact binary sources, we find that the stf2 approximation estimates compact binary parameters with biases comparable to systematic uncertainties in the waveform. We illustrate by example the effect these systematic errors have on posterior probabilities most relevant to low-latency electromagnetic follow-up: whether the secondary has a mass consistent with a neutron star (NS); whether the masses, spins, and orbit are consistent with that neutron star’s tidal disruption; and whether the binary’s angular momentum axis is oriented along the line of sight.

Figure 1

SNR ratio distribution: Left panel: Ratio $r=({\overline{\rho }}_B)/({\overline{\rho }}_A)$ of the largest SNR recovered for STF2 to the SNR recovered for STT2, described in Sec. 3a. This distribution is significantly different from unity, with a magnitude consistent with the mismatch found in previous studies [17]. Right panel: Cumulative distribution of ${\mathcal{T}}_{\overline{\rho }}$ [Eq. (1)] divided by the square root of $N$; in these units, the horizontal axis measures the difference in mean in units of the standard deviation. The dashed curve is the theoretical cumulative $t$ distribution. This plot demonstrates that there is a statistically significant difference between the average recovered SNR ($\rho$) by the two different models stf2 and stf2.

Figure 2

Example of posterior distributions: Using one random example, this figure shows the one-dimensional distribution of chirp mass inferred from the stf2 (red line) and stf2 (blue line) distributions. Though the two methods give very similar results, the two distributions are slightly offset. Our study demonstrates these small offsets occur ubiquitously, at a statistically significant level.

Figure 3

Scaled difference in mean chirp mass: For the same data, we performed parameter estimation twice, reporting the cumulative distribution $t$-test scores in Eq. (1) divided by $\sqrt{N}$. In these units, the horizontal axis measures the difference in mean in units of the standard deviation. The dashed curve shows the expected result, a $t$ distribution; the purple and blue curves show the results when both models were stf2 with the same or unequal numbers of nonzero spins; the orange curve shows the results when comparing (double spin) stf2 to stf2; and the red curve shows a comparison between single-spin stt4 and double-spin stf2.

Figure 4

Results for additional parameters and PN approximants: Like Fig. 3, a cumulative plot of ${\mathcal{T}}_x$ for $x={\chi }_1,\eta ,{\theta }_{JN}$. The orange curve corresponds to comparisons between stt4 (single spin) and stf2 (double spin). The orange curve demonstrates that parameter estimation with post-Newtonian approximation schemes that adopt otherwise identical physics yield considerably different posterior distributions for intrinsic parameters. All approximations agree on geometric parameters like ${\theta }_{JN}$.

Figure 5

Simulation widths: Different approximations will predict posterior distributions of often significantly different widths. The width of the distribution is strongly impacted by non-Gaussian tails and is a less-robust measure of distribution similarity. These results suggest that the posteriors produced by F2 are more similar to the “true” distributions than the results from T4. (To emphasize the differences in distribution widths, we intentionally selected the parameter with the most Gaussian distribution: the chirp mass. This test is less well suited to posterior distributions with significant non-Gaussian tails and cutoffs, notably $\eta$ and ${\chi }_{1,2}$.)

Figure 6

Distribution of ${\varphi }_{12}$: Report on parameter measurement accuracy of ${\varphi }_{12}$ using two-spin stf2. (The parameter ${\varphi }_{12}$ does not exist in a single-spin model, including stf2.) Left panel: Cumulative distribution of ${\sigma }_{{\varphi }_{12}}$ [Eq. (3)]. For most cases, ${\varphi }_{12}$ is approximately uniformly distributed. Right panel: Scatterplot of ${\sigma }_{{\varphi }_{12}}$ versus $\rho$. Sources with high amplitude provide more information about all parameters, including ${\varphi }_{12}$. In some exceptional cases—with high amplitude, large spin, and fortuitous orientation—${\varphi }_{12}$ can thereby be tightly constrained.

Figure 7

Identifying a candidate NS companion: The posterior probability that the secondary is a neutron star versus the smaller object’s mass, as estimated using STT2 ($P_A$; blue dots) and STF2 ($P_B$; red dots).

Figure 8

Aligned with the line of sight: Distribution $P_A(\mathrm{beam})$ (red line) and $P_B(\mathrm{beam})$ for the probability of alignment between the line of sight and the total angular momentum direction, shown both for all elements of the astrophysical sample (yellow dots) and for the sources with neutron star companions (purple stars).

Figure 9

Tidal disruption probability: The posterior tidal disruption probability evaluated via Eq. (4) using parameters drawn from SpinTaylorF2 [$P_{F2}$] and using parameters drawn from SpinTaylorT2 [$P_{T2}$].