Application of a hierarchical MCMC follow-up to Advanced LIGO continuous gravitational-wave candidates

We present the first application of a hierarchical Markov Chain Monte Carlo (MCMC) follow-up on continuous gravitational-wave candidates from real-data searches. The follow-up uses an MCMC sampler to draw parameter-space points from the posterior distribution, constructed using the matched-filter as a log-likelihood. As outliers are narrowed down, coherence time increases, imposing more restrictive phase-evolution templates. We introduce a novel Bayes factor to compare results from different stages: The signal hypothesis is derived from first principles, while the noise hypothesis uses extreme value theory to derive a background model. The effectiveness of our proposal is evaluated on fake Gaussian data and applied to a set of 30 outliers produced by different continuous wave searches on O2 Advanced LIGO data. The results of our analysis suggest all but five outliers are inconsistent with an astrophysical origin under the standard continuous wave signal model. We successfully ascribe four of the surviving outliers to instrumental artifacts and a strong hardware injection present in the data. The behavior of the fifth outlier suggests an instrumental origin as well, but we could not relate it to any known instrumental cause.

Figure 1

Distribution of the maximum $2\mathcal{F}$ value of a template bank obtained from its evaluation at $N_o=600$ different off-sourced right ascensions, excluding 90° around the sky position of the outlier of interest. The template bank corresponds to MCMC samples from the fully coherent stage follow-up of a simulated signal in Gaussian noise. The solid line represents the fit of a Gumbel distribution.

Figure 2

Illustration of different regimes in which an outlier could be located. Shaded regions represent probability distributions associated to the indicated hypothesis. Dashed vertical lines refer to the enumerated labels in the text.

Figure 3

Bayes factor in terms of the discrepancy of an outlier with respect to the noise and signal hypothesis as described in Eq. (27). Numerical values are computed using ${\sigma }_{\mathrm{N}}=3$ and ${\sigma }_{\mathrm{S}}=30$, consistent with Table 1. This representation will be referred to as the $({\xi }_{\mathrm{N}},{\xi }_{\mathrm{S}})$ plane.

Figure 4

Flowchart ilustrating the computation of $\ln {\mathcal{B}}_{\mathrm{S}/\mathrm{N}}^{*}$ for a CW outlier.

Figure 5

Distribution of $|{\xi }_{\mathrm{S}}|$ for the complete set of detected injections using different semicoherent stages, namely $N_{\mathrm{seg}}=500$ and $N_{\mathrm{seg}}=5$, as the reference to compute ${\mu }_{\mathrm{S}}$ and ${\sigma }_{\mathrm{S}}$.

Figure 6

$({\xi }_{\mathrm{S}},{\xi }_{\mathrm{N}})$ plane for the complete set of detected injections using $N_{\mathrm{seg}}=5$ as the reference stage to compute ${\mu }_{\mathrm{S}}$ and ${\sigma }_{\mathrm{S}}$. The horizontal axis represents the discrepancy with respect to the noise hypothesis, while the vertical axis represents the discrepancy with respect to the signal hypothesis.

Figure 7

$\ln {\mathcal{B}}_{\mathrm{S}/\mathrm{N}}^{*}$ computed by applying the multi-stage MCMC follow up on the three sets of software injections. Reference values were computed with respect to the $N_{\mathrm{seg}}=5$ stage and the Gaussian approximation was used to compute the signal contribution. Outliers marked by a star did not display an ensemble-level volume shrinkage, as explained in the text.

Figure 8

Estimation of 90% detection-probability threshold on $\ln {\mathcal{B}}_{\mathrm{S}/\mathrm{N}}^{*}$ for different sensitivity depths beyond the injections shown in Fig. 7. Each dot represents an empirical estimate of the 90% detection-probability threshold using 100 simulated signals at a fixed depth value. Error bars correspond to the bootstrap standard deviation using 200 resamples of 50 samples each. The solid line and the associated envelopes represent a linear fit using scipy.optimize.curve_fit [107] with 1, 2, and 3 sigma uncertainties.

Figure 9

Posterior volume shrinkage of a successfully detected software injection. Parameter-space volumes are estimated by taking the product of parameter-wise 90% central credible regions as explained in the text. The vertical axis represents the posterior volume as a fraction of the initial prior volume. The slope of the log-log plot is an approximation to the inverse of the volume shrinkage $1/{\gamma }^{(j+1)}$, where the volume shrinkage ${\gamma }^{(j+1)}$ was defined in Eq. (12).

Figure 10

Distribution of (${\log }_{10}$) posterior volume shrinkage rates of the detected injections. The prominent peak is within the order of magnitude of the expected value according to Eq. (12). The presence of negative shrinkage rates for a few injections is discussed in the text.

Figure 11

$({\xi }_{\mathrm{N}},{\xi }_{\mathrm{S}})$ plane associated to the outliers found in O2 data by the specified searches.

Figure 12

$\ln {\mathcal{B}}_{\mathrm{S}/\mathrm{N}}^{*}$ values obtained after the hierarchical MCMC follow-up of O2 outliers. Relative outlier positions in this figure are consistent with Fig. 11. Outliers enclosed by circles and diamonds score a $\ln {\mathcal{B}}_{\mathrm{S}/\mathrm{N}}^{*}$ value above 30. The outlier enclosed by a square returns a negative value of $\ln {\mathcal{B}}_{\mathrm{S}/\mathrm{N}}^{*}$ and is displayed as white due to the logarithmic color scale.

Figure 13

Segment-wise $2\widehat{\mathcal{F}}$ accumulation of the loudest template associated to the outlier J1831-0952@15.4012 Hz throughout the observing run using 500 coherent segments. The solid orange line shows the results using the LIGO Hanford detector (H1) only, while the dashed blue line shows the results using the LIGO Livingston detector (L1). The upper panel shows the $2\widehat{\mathcal{F}}$ accumulation throughout the duration of the run. The lower panel shows the segment-wise $2\widehat{\mathcal{F}}$ values per frequency bin. Frequency values are computed evaluating the frequency-evolution template at the starting time of each segment.

Figure 14

Maximum Fourier power over $N=200$ samples of zero-mean unit-variance zero-padded Gaussian noise. In each panel, the stair-case line represents a histogram over ${10}^6$ repeated trials of ${\max }_{\mathcal{N}}{\tilde{\rho }}_p$. The dashed line is the best fit of Eq. (16) on the effective number of templates ${\mathcal{N}}^{\prime }$, and the solid line is the best fit of a Gumbel distribution on the location and scale parameters. Zero-padding is indicated by $p$, where $p=1$ represents no zero-padding, as explained in the text.

Figure 15

Equivalent figure to Fig. 14 using the alternative normalization of Fourier power ${\widehat{\rho }}_p$. In this case, increasing the number of correlated templates narrows the resulting distribution with respect to Eq. (16).