Multi-messenger Approaches to Supermassive Black Hole Binary Detection and Parameter Estimation: Implications for Nanohertz Gravitational Wave Searches with Pulsar Timing Arrays

We first construct a simulated PTA using properties of the pulsars from the NANOGrav 11 yr data set ² (Arzoumanian et al. 2018b). Of the 45 pulsars that are timed over an 11 yr period, 34 have time spans of more than three years and were used for NANOGrav's suite of 11 yr GW analyses (Arzoumanian et al. 2018a, 2020b, 2021; Aggarwal et al. 2019, 2020). We additionally remove PSR B1937+21 from the simulated PTA, whose timing residual is known to be characterized by a large amount of red noise above the white noise level (e.g., Lam et al. 2017; Arzoumanian et al. 2018b). Our simulated PTA thus consists of 33 pulsars that retain the observing cadence and sky locations (Figure 1, left panel) of the 11 yr pulsars.

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To simulate their timing residuals, we used the libstempo software package ³ and modeled them as a combination of white noise, red noise, and residuals induced by a CW signal. The white noise component represents the nominal TOA measurement uncertainties (EFAC = 1), and the red noise component encompasses various effects such as spin noise intrinsic to the pulsar and interstellar medium propagation effects and is modeled as a power law. The power-law model is parameterized by an amplitude A_rn and a spectral index γ_rn, which have been measured separately for each pulsar.

To inject a CW signal using the functionality in libstempo, the following signal model parameters are required: {$\phi ,\theta ,{d}_{{\rm{L}}},{ \mathcal M },{f}_{\mathrm{gw}},i,{{\rm{\Phi }}}_{0},\psi$}.

• 1.  
  The parameters ϕ and θ are the azimuthal angle and polar angle, respectively, and are conventionally used in PTA GW analyses to represent the CW source's position on the sky. In observational astronomy, they correspond to R.A. α and decl. δ, respectively: ϕ = α, θ = π/2 − δ. We inject CW sources at two representative sky locations (Figure 1, left panel): a sensitive sky location where a large number of pulsars are being monitored nearby (R.A. 18^h, decl. −15°), and a less sensitive sky location where very few of them are nearby (R.A. 10^h, decl. −15°).
• 2.  
  The parameter d_L is the luminosity distance of the CW source. Assuming a cosmology, this can be obtained observationally by measuring the redshift z of the host galaxy of the SMBHB, or the AGN for an active SMBHB. We injected mock CW sources in the very nearby universe at z < 0.1 to mimic the detectable signals that are already present in the respective simulated data sets.
• 3.  
  The parameter f_gw is the GW frequency and is related to the orbital frequency of the SMBHB: f_gw = 2f_orb (assuming a circular orbit). We inject CW signals at two different frequencies (Figure 1; right panel): f_gw = 8 nHz, which is approximately the most sensitive frequency in the 11 yr data set; and f_gw = 80 nHz, which is motivated by the observational search for AGN periodicity: in time-series data with a baseline of a few years (which is typical for current time-domain surveys), at least a few cycles are required before claiming a periodic detection, hence making the EM search more sensitive to periods of approximately months. Furthermore, the stochastic GW background (GWB), on which the CW signal would be superimposed, has a canonical power-law spectral shape and thus has a lower amplitude at high GW frequencies, potentially making the search for a CW source emitting at a high GW frequency more fruitful.
• 4.  
  ${ \mathcal M }$ is the (redshifted) "chirp mass" of the SMBHB and is a combination of the component masses: ${ \mathcal M }\equiv {\left({m}_{1}{m}_{2}\right)}^{3/5}/{\left({m}_{1}+{m}_{2}\right)}^{1/5}$. Since our injected CW sources are in the very local universe, we make the approximation that the rest-frame chirp mass ${{ \mathcal M }}_{{\rm{r}}}\approx { \mathcal M }$. The amplitude of the GW signal is then related to d_L, f_gw, and ${ \mathcal M }$:

  $$\begin{eqnarray}&&{h}_{0}=\displaystyle \frac{2{{ \mathcal M }}^{5/3}{\left(\pi {f}_{\mathrm{gw}}\right)}^{2/3}}{{d}_{{\rm{L}}}}.\end{eqnarray} \tag{ 1 }$$

  In our analyses, we parameterize the signal strength in terms of the total mass M_tot = m₁ + m₂ and the mass ratio q ≡ m₂/m₁ ≤ 1 rather than the chirp mass because those parameters are more closely tied to EM observations. Observationally, the total mass of the system can be measured via standard methods such as single-epoch virial black hole mass estimation (assuming the method is still valid for close binary systems), which suffers from an ∼0.3 dex systematic uncertainty (e.g., Shen 2013 and references therein). In this study, we assign astrophysically reasonable black hole masses to the injected CW sources: $\mathrm{log}{ \mathcal M }/{{ \mathcal M }}_{\odot }=9.4\mbox{--}9.5$. The mass ratio of a binary system cannot always be directly determined from the light curve of a periodically varying SMBHB (see Section 4 for further discussion), and we keep the mass ratio fixed at q = 0.8 in all injections for simplicity.
• 5.  
  The parameter i is the inclination of the binary orbit and could be observationally constrained, including through light-curve fitting. We keep the inclination fixed at i = π/2 (which corresponds to an edge-on binary) for all injections for simplicity. We note that, while an edge-on or near edge-on orientation is optimal for EM observers in the cases of relativistic beaming and binary self-lensing, it is the least optimal for GW detection because it induces the smallest pulsar timing residuals.
• 6.  
  The parameters Φ₀ and ψ are the initial orbital phase and the GW polarization angle in units of radians, respectively. As they cannot be measured from EM observations, we keep them fixed at Φ₀ = 2.0 and ψ = 2.0 in all injections for simplicity.

The parameter values for each injected CW signal are summarized in Table 1. Additionally, we have included the so-called "pulsar term" in the injected signal, which is the signal induced at the pulsar (whereas "Earth term" is the signal induced at the Earth). In contrast to the alternative approach that uses a signal model that only contain the Earth term, our analysis is sensitive to the evolution, or "chirping," of the signal, and hence the chirp mass, as GWs pass through the pulsar and the Earth.

Table 1. Injected CW Parameters

Parameter	Simulation
	0	A	B	C	D
R.A. (hr)	(18)	18	18	10	10
Decl. (deg)	(−15)	−15	−15	−15	−15
fgw (nHz)	(8)	8	80	8	80
$\mathrm{log}{ \mathcal M }$ (M⊙)	(9.4)	9.4	9.4	9.5	9.5
z	(0.06)	0.06	0.08	0.02	0.05
log h0	⋯	−14.4	−13.9	−13.7	−13.5

Note. For all simulations: q = 0.8, i = π/2, Φ₀ = 2.0, ψ = 2.0. Additionally, S-0 does not contain an injected CW signal, but the search was performed using the parameters of S-A (shown in parentheses).

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For each simulation (Simulations A-D), we first calculate the GW strain h₀ from the binary parameters (Figure 1, right panel; Table 1). We have generated two realizations for each simulation, denoted S-X1 and S-X2, in order to study the potential effect of pulsar white and red noise realizations on the CW search. For each realization, we compute the ${ \mathcal F }$-statistic, which is the log-likelihood ratio maximized over the so-called extrinsic parameters of the source and is used as a detection statistic for GW searches (Jaranowski et al. 1998; Cornish & Porter 2007; Babak & Sesana 2012). In the context of this work, we evaluate the pulsar-term ${ \mathcal F }$-statistic (${{ \mathcal F }}_{{\rm{p}}}$, Ellis et al. 2012) at the injected GW frequency. We additionally compute the corresponding SNR, since $2{{ \mathcal F }}_{{\rm{p}}}$ follows a χ² distribution with 2N degrees of freedom, where N is the number of the pulsars: SNR = $\sqrt{2{{ \mathcal F }}_{{\rm{p}}}-2N}$. We show the ${{ \mathcal F }}_{{\rm{p}}}$ and SNR of each realization of S-A–S-D in Table 2. The different mock data sets containing the same injected CW signal differ slightly in ${{ \mathcal F }}_{{\rm{p}}}$ and SNR, as expected given their different noise realizations.

Table 2.  ${{ \mathcal F }}_{{\rm{p}}}$ Values and SNRs of different Realizations of S-A–S-D

	${{ \mathcal F }}_{{\rm{p}}}$	SNR
S-A1	55	6.6
S-A2	45	4.9
S-B1	40	3.7
S-B2	44	4.7
S-C1	42	4.2
S-C2	43	4.5
S-D1	30	<1
S-D2	32	<1

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As we see in Tables 1 and 2, Simulation A has the highest ${{ \mathcal F }}_{{\rm{p}}}$ statistic or SNR despite having the lowest nominal GW amplitude, which can be expected as it is emitting at a GW frequency favorable to the PTA. Conversely, Simulation D has the largest GW amplitude but has the lowest detection statistic or SNR, due to its GW frequency to which the NANOGrav PTA is less sensitive. Overall, Simulations A-D represent CW signals of a range of parameters and are capable of serving as test cases in this proof-of-concept study to investigate any effects of sky location and GW frequency on CW searches. Their comparable SNRs further permit reasonable comparisons between the mock CW search results. Moreover, their modest SNRs are suitable proxies for the first CW signals in some future PTA data set, which are expected to be weak.

We note that, in real PTA data, any CWs detectable as discrete signals would be superimposed on the stochastic GWB, which is the superposition of GW signals from a cosmological population of SMBHBs, and it is probable that multiple CW signals may be detectable simultaneously above the GWB. However, since the 11 yr NANOGrav analysis does not show evidence for the GWB (Arzoumanian et al. 2018a), which is expected to emerge in a data set spanning at least 15 yr (Taylor et al. 2016; Pol et al. 2021), we do not inject a GWB in our simulated PTA data sets. Additionally, the joint search for one or more CW signals in addition to a GWB (Bécsy & Cornish 2020) requires techniques such as a trans-dimensional Reversible Jump MCMC sampler (e.g., Green 1995) to explore a parameter space whose dimensionality is not fixed, which is beyond the scope of this work. For these reasons, we only inject one CW source in a given simulated PTA data set.

We further note that we have based our simulated PTA on the NANOGrav 11 yr data set, instead of the more recent 12.5 yr data set (Alam et al. 2021). Since a red noise process common among the 12.5 yr pulsars is present in this data set, but whose origin cannot yet be determined (Arzoumanian et al. 2020a), we do not utilize this data set in our analysis. Also, an 11 yr like simulated data set permits reasonable comparisons with NANOGrav results from the latest CW analysis, which was performed with the 11 yr data set (Aggarwal et al. 2019). Furthermore, we argue that the main results from this proof-of-concept work should not strongly depend on the length and size of the particular data set, since the SNRs of the injected CW signals are relative to the sensitivity of the data set. That said, extending this work to a later data set, either actual or projected, would be of interest to future work.

Finally, we generate two additional mock data sets with no injected CW signals (S-0). In Sections 2.2 and 2.3, we will treat S-0 in the same manner as S-A when performing uninformed and targeted searches (Table 1) and will compare and contrast the search results. This is motivated by the high false-positive rate of SMBHB searches (see, e.g., the footnote in Section 1 for a brief discussion), and searching for a CW signal in S-0 thus represents the case where a CW search is performed using the information of an EM candidate which turns out to be a false positive.

For each injection, we first perform a "blind" search, where the MCMC sampler is free to explore the prior ranges of all parameters (Table 3). This procedure thus mimics an unguided and uninformed search for a CW signal in PTA data. We perform these mock searches using the NANOGrav GW detection software package enterprise. ⁴ As we largely adopt the same data analysis methods as the previous CW analyses (modified for a simulated PTA data set), we refer the reader to Ellis et al. (2012) or Aggarwal et al. (2019) for a detailed description of the CW signal model and search methods. To explore the parameter space, we use PTMCMCSampler ⁵ (Ellis & van Haasteren 2017), with a geometrically spaced temperature ladder for parallel tempering (e.g., Earl & Deem 2005). Each chain is then sampled for (1–2) × 10⁶ iterations and individually inspected for convergence using the Geweke diagnostic (Geweke 1992).

Table 3. Priors in Uninformed Searches

Parameter	Prior	Range	Symbol in Table 5
ϕ	Uniform	[0, 2π]	⨯
$\cos \theta$	Uniform	[−1.0, 1.0]	⨯
$\mathrm{log}{d}_{{\rm{L}}}$	Uniform	[−2.0, 4.0]	⨯
$\mathrm{log}{M}_{\mathrm{tot}}$	Uniform	[6.0, 12.0]	⨯
$\mathrm{log}{f}_{\mathrm{gw}}$	Uniform	[−9.0, −7.0]	⨯
q	Uniform	[0.0001,1.0]	⋯
Φ0	Uniform	[0, π]	⋯
ψ	Uniform	[0, π]	⋯
$\cos i$	Uniform	[−1.0, 1.0]	⋯

Note. q, $\cos i$, Φ₀, and ψ are always searched over their respective priors and are hence omitted in Table 5.

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To study the effects of applying EM-informed priors on the detection and parameter estimation of a CW signal, we perform three separate searches for the same injected signal in a progressive scheme:

• 1.  
  ra_dec_z: We search for the signal at its injected sky location and luminosity distance. This mimics the search for a CW signal in the PTA data set which is prompted by the detection (or observation) of its EM counterpart (or candidate). In this scenario, we assume the SMBHB's location and redshift can be measured exactly, which is a reasonable assumption for EM observations of a source.
• 2.  
  ra_dec_z_dmtot: We search for the signal at its injected sky location, luminosity distance, and in addition, we take into account a 0.3 dex systematic uncertainty on the black hole mass measurement. In practice, this is implemented through adopting a normal prior for $\mathrm{log}{M}_{\mathrm{tot}}$ which is centered at the injected value. We note that, in practice, this scenario almost always contains the previous case; in other words, if an EM counterpart can be identified at all, its black hole mass can usually be estimated, and therefore in practice, ra_dec_z is never a stand-alone case for a targeted search. However, here we study both cases in order to isolate the effects of knowing the black hole mass on the targeted search.
• 3.  
  ra_dec_z_dmtot_dfgw: This represents the case where the SMBHB is detected electromagnetically as a periodically varying AGN, where the observed period is interpreted as the imprint of the binary orbital period. In addition to the priors above, we take into account the uncertainty of the GW frequency, which is motivated by factors including measurement uncertainties and being agnostic about how the observed variability period translates to the binary orbital period (which may have a binary model and/or mass-ratio dependence, see Section 4 for details). In practice, we apply a uniform prior to $\mathrm{log}{f}_{\mathrm{gw}}$, while allowing a factor of two uncertainty around the injected value, i.e., a total of a factor of four uncertainty on f _gw.

The priors applied in targeted searches are listed in Table 4, and we summarize the uninformed search and all targeted searches in Table 5.

Table 4. Priors in Targeted Searches

Parameter	Prior	Symbol in Table 5
ϕ	Fixed	✓
$\cos \theta$	Fixed	✓
$\mathrm{log}{d}_{{\rm{L}}}$	Fixed	✓
$\mathrm{log}{M}_{\mathrm{tot}}$	Normal	0.3 dex
$\mathrm{log}{f}_{\mathrm{gw}}$	Uniform	×4

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Table 5. Names of Uninformed and Targeted Searches in Figures 3–10

Parameter	Name
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	⨯	✓	✓	✓
$\cos \theta$	⨯	✓	✓	✓
$\mathrm{log}{d}_{{\rm{L}}}$	⨯	✓	✓	✓
$\mathrm{log}{M}_{\mathrm{tot}}$	⨯	⨯	0.3 dex	0.3 dex
$\mathrm{log}{f}_{\mathrm{gw}}$	⨯	⨯	⨯	×4

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We first investigate the effects of a targeted search on the detectability of the CW signal. To this aim, we compute the Bayes factor for the presence of a CW signal. We use the Savage–Dickey formula (Dickey 1971), which can be used to compute the Bayes factor ${{ \mathcal B }}_{10}$ for two nested models ${{ \mathcal H }}_{1}$ and ${{ \mathcal H }}_{0}$. In this case, ${{ \mathcal H }}_{0}$ is a model with pulsar noise only, and ${{ \mathcal H }}_{1}$ is a model with noise plus a CW signal, which is equivalent to noise only for h₀ = 0. For more details, see Appendix A.

In Figure 2, we show the evolution of the Bayes factor from an uninformed search to targeted searches. In uninformed, all Bayes factors ${{ \mathcal B }}_{10}\lt 1$, indicating that no evidence for the CW signal can be found. This is expected from the low SNRs of the signals. In ra_dec_z, ${{ \mathcal B }}_{10}$ has increased slightly, but stays below the nominal threshold of ${{ \mathcal B }}_{10}=1$, or barely above it. Thus, knowing the source's sky location does not appear to significantly improve the detectability of the source. However, by additionally knowing the the black hole mass (ra_dec_dmtot), the increases in ${{ \mathcal B }}_{10}$ are more significant (except for S-C1); in particular, ${{ \mathcal B }}_{10}$ in S-A and S-B have increased by at least a factor of a few. In ra_dec_dmtot_dfgw, where the frequency is also known, ${{ \mathcal B }}_{10}$ increases even further in some cases. S-A and S-B have the larger final ${{ \mathcal B }}_{10}$ values (∼5–10) among the four injections. While a Bayes factor of ∼10 is shy of being able to claim a confident detection, it is noteworthy that evidence for the (weak) CW signal can strengthen by at least a factor of a few via the EM observation of the source.

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Additionally, the overall trend described above appears to hold for both realizations, albeit with mild variation between realizations. The variation is likely the result of the MCMC analysis or the noise realization, rather than a simple correlation with SNR of the source. For example, S-A2 contains the same CW signal as S-A1 and has a slighter lower SNR, but has the highest final ${{ \mathcal B }}_{10}$. Similarly, S-C1 and S-C2 have almost identical SNRs, but an overall increase in ${{ \mathcal B }}_{10}$ is seen in S-C2, while a slight decrease is observed in S-C1.

Despite the individual variations, the evolution trends of S-A and S-B appear to be separated from those of S-C and S-D (last panel in Figure 2). This is likely due to the fact that S-A and S-B are located at a sensitive sky location, which is favorable for CW searches originally (see, e.g., Aggarwal et al. 2019) and benefits from EM information even further. However, we note that the Savage–Dickey approximation starts to break down for stronger detections, and therefore we advise the reader to take the ${{ \mathcal B }}_{10}$ values (and their error bars) of S-B2 with a grain of salt.

Furthermore, comparing the trends of S-A versus S-B, the latter appears to be more detectable than the former: both realizations, S-B1 and S-B2, have higher final ${{ \mathcal B }}_{10}$ than S-A. This is despite the fact that S-B has a quieter signal than S-A. We speculate that this is because S-B is emitting at a higher GW frequency than S-A, so that the CW signal is observed for more cycles over the same data length. While higher frequencies are conventionally considered "less sensitive" frequencies in terms of the GW amplitude, our results suggest that sources emitting at high frequencies could be as detectable as (or even more detectable than) those emitting at "sensitive" frequencies. This is encouraging for the possibility of strong synergies with EM observations (see Section 4 for further discussion). However, we are unable to observe the same dependence on frequency for S-C versus S-D, as their improvements in ${{ \mathcal B }}_{10}$ are much weaker.

We caution that these trends ought to be confirmed with a larger number of realizations and parameters than we have included here, especially given the aforementioned variation between different noise realizations. Unfortunately, it is prohibitively expensive to perform MCMC analysis on a statistically meaningful number of CW searches, for which a more robust and efficient detection pipeline is still needed. Therefore, while our current analyses suggest a dependence of CW detectability on sky location and GW frequency, at present we are not able to fully investigate the effect of different noise realizations on this trend.

We then proceed to investigate whether the above trend applies to S-0, which does not contain an injected CW signal. As we show in the bottom left panel in Figure 2, both realizations have slightly increased Bayes factors as a result of targeted searches. Therefore, applying (incorrect) EM priors appears to also risk increasing the false alarm probability. Additionally, the final Bayes factor in ra_dec_z_dmtot_dfgw is ∼2, which is comparable to that of S-C or S-D. Thus in actual searches, it would be difficult to distinguish a false-positive detection from a low SNR source, if the Bayes factor and its increase are interpreted at face value.

However, it is important to note that the final Bayes factor is significantly below that of S-A, whose source parameters were used in the search for a signal in S-0. Therefore, we argue that the trend of increased ${{ \mathcal B }}_{10}$ of true CW signals (and S-A in particular) cannot simply be the result of possibly increased false alarm probability due to the EM-informed search. Instead, this indicates that a false-positive detection is less likely to be mistaken for a true CW source if it is located at a favorable sky location.

More importantly, this emphasizes that the Bayes factor is not a "one size fits all" number to quantify the evidence for a signal, nor is there an absolute threshold between detection and non-detection. In actual CW searches (and especially targeted searches), the resulting Bayes factor should be interpreted in the appropriate context, for example by performing a false alarm analysis similar to this one. Finally, the "detection" of a CW signal is only part of the story. EM observations carry with them the parameters of the possible binary, which should be used for cross validation with GW searches in order to further reduce possible false alarms. In the next section, we will investigate the ability of the PTA to estimate, and independently verify, binary parameters in targeted searches.

In this section, we focus on the ability of the PTA to estimate binary parameters, and the effects of an EM-informed targeted search on parameter estimation or limits at various sky locations and GW frequencies. We discuss the results separately for each injected signal (S-A to S-D), which are the posteriors we obtained for the parameters {M_tot, q, f_gw, i, Φ₀, ψ} from the MCMC analysis. The posterior distributions are shown in Figures 3–10. Note that the x-axis plotting ranges and histogram bin sizes for M_tot and f_gw vary between simulations S-A–S-D for presentation and are kept fixed between realizations (e.g., S-A1 and S-A2) for visual comparison.

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To quantify the improvements in parameter estimation, we compute the width of the 90% percent of the posterior distribution (90% confidence interval, CI), which directly measures the precision of the parameter estimation. We also compute the Kullback–Leibler (KL) divergence (Kullback & Leibler 1951), which quantifies the difference between two probability distributions, and is given by

$$\begin{eqnarray}&&{D}_{\mathrm{KL}}=\sum _{x}p(x)\mathrm{ln}\left[\displaystyle \frac{p(x)}{q(x)}\right],\end{eqnarray} \tag{ 2 }$$

where here p(x) and q(x) are the posterior and prior probability distributions of the parameter x. Therefore the KL divergence measures the difference between the prior used for a given parameter in a given search and the resultant posterior. A small value of D_KL indicates that the posterior is indistinguishable from the prior, whereas a larger value suggests that the parameter estimation is improved relative to the prior (recall that the parameters have different EM priors in the targeted searches). In Tables 6 and 7, we show the 90% CI and D_KL for each parameter from each search. A more detailed discussion of the KL divergence can be found in Appendix B. We note that we calculate the KL divergence for each parameter in terms of the marginalized prior and posterior distributions, instead of a single N-dimensional KL divergence for all parameters. We also note that here we compute the KL divergence using the summation form (as opposed to integral form), because the posterior distributions cannot be fit to simple, known functions. Therefore, we choose the discrete form, to which the histograms of the priors and posteriors can be directly applied.

Table 6. 90% Confidence Interval and Kullback–Leibler Divergence in Uninformed and Targeted Searches (S-X1)

Simulation A1 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.524/0.021	⋯	⋯	⋯
$\cos \theta$	1.801/0.014	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.253/0.087	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	4.480/0.185	3.867/0.575	0.504/0.313	0.460/0.366
$\mathrm{log}{f}_{\mathrm{gw}}$	1.776/0.135	1.666/0.616	0.574/2.162	0.103/1.555
q	0.902/0.002	0.904/0.003	0.855/0.018	0.850/0.028
Φ0	2.827/0.005	2.819/0.010	2.822/0.047	2.767/0.049
ψ	2.847/0.004	2.741/0.046	2.460/0.279	2.171/0.343
$\cos i$	1.794/0.011	1.817/0.006	1.769/0.014	1.785/0.017
Simulation B1 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.810/0.025	⋯	⋯	⋯
$\cos \theta$	1.825/0.015	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.249/0.038	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	4.498/0.141	3.810/1.100	0.468/0.549	0.378/0.791
$\mathrm{log}{f}_{\mathrm{gw}}$	1.822/0.175	1.796/1.352	1.644/2.741	0.014/3.188
q	0.909/0.005	0.887/0.038	0.823/0.056	0.869/0.035
Φ0	2.856/0.002	2.617/0.098	2.401/0.173	2.158/0.289
ψ	2.803/0.005	2.607/0.180	1.903/0.477	1.168/0.754
$\cos i$	1.801/0.005	1.579/0.126	1.267/0.253	0.980/0.456
Simulation C1 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.646/0.004	⋯	⋯	⋯
$\cos \theta$	1.791/0.003	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.220/0.051	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	3.801/0.340	3.512/0.393	0.797/0.318	0.741/0.411
$\mathrm{log}{f}_{\mathrm{gw}}$	1.804/0.012	1.796/0.015	1.794/0.428	0.486/0.225
q	0.907/0.002	0.904/0.002	0.909/0.048	0.910/0.077
Φ0	2.829/0.001	2.838/0.001	2.797/0.007	2.786/0.007
ψ	2.812/0.003	2.833/0.003	2.829/0.020	2.700/0.107
$\cos i$	1.795/0.005	1.774/0.005	1.652/0.056	1.597/0.106
Simulation D1 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.821/0.014	⋯	⋯	⋯
$\cos \theta$	1.800/0.008	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.384/0.042	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	4.080/0.267	3.742/0.301	0.841/0.120	0.609/0.317
$\mathrm{log}{f}_{\mathrm{gw}}$	1.835/0.047	1.815/0.030	1.824/0.355	0.462/0.765
q	0.910/0.002	0.905/0.001	0.905/0.011	0.900/0.006
Φ0	2.821/0.001	2.817/0.001	2.841/0.001	2.806/0.009
ψ	2.815/0.004	2.820/0.002	2.802/0.011	2.709/0.097
$\cos i$	1.792/0.004	1.784/0.003	1.753/0.015	1.624/0.121

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Table 7. Same as Table 6, but for S-X2

Simulation A2 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.559/0.016	⋯	⋯	⋯
$\cos \theta$	1.793/0.006	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.287/0.048	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	4.229/0.227	3.557/1.031	0.565/0.262	0.541/0.320
$\mathrm{log}{f}_{\mathrm{gw}}$	1.780/0.047	1.348/0.726	0.581/1.656	0.290/1.256
q	0.903/0.002	0.878/0.007	0.866/0.022	0.886/0.011
Φ0	2.838/0.001	2.767/0.009	2.824/0.034	2.779/0.024
ψ	2.850/0.006	2.786/0.039	2.605/0.140	1.927/0.347
$\cos i$	1.808/0.004	1.807/0.013	1.790/0.010	1.762/0.013
Simulation B2 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.563/0.005	⋯	⋯	⋯
$\cos \theta$	1.797/0.004	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.289/0.044	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	4.033/0.312	0.330/2.911	0.193/1.399	0.264/1.129
$\mathrm{log}{f}_{\mathrm{gw}}$	1.817/0.023	0.010/3.894	0.010/3.910	0.010/3.915
q	0.906/0.003	0.860/0.086	0.776/0.116	0.826/0.061
Φ0	2.817/0.001	1.996/0.332	2.142/0.318	1.901/0.368
ψ	2.826/0.003	0.978/1.007	1.115/0.927	0.968/0.992
$\cos i$	1.797/0.005	1.175/0.368	1.189/0.354	1.173/0.378
Simulation C2 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.591/0.012	⋯	⋯	⋯
$\cos \theta$	1.788/0.005	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.173/0.061	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	3.955/0.295	3.686/0.436	0.661/0.288	0.619/0.376
$\mathrm{log}{f}_{\mathrm{gw}}$	1.754/0.061	1.737/0.189	1.275/1.029	0.406/0.533
q	0.905/0.002	0.903/0.004	0.897/0.009	0.892/0.015
Φ0	2.835/0.003	2.794/0.010	2.859/0.014	2.862/0.012
ψ	2.854/0.004	2.860/0.034	2.929/0.110	3.001/0.200
$\cos i$	1.789/0.008	1.744/0.039	1.570/0.184	1.478/0.209
Simulation D2 (90%CI/DKL)
	uninformed	ra_dec_z	ra_dec_z_dmtot	ra_dec_z_dmtot_dfgw
ϕ	5.797/0.038	⋯	⋯	⋯
$\cos \theta$	1.822/0.033	⋯	⋯	⋯
$\mathrm{log}{d}_{{\rm{L}}}$	5.057/0.110	⋯	⋯	⋯
$\mathrm{log}{M}_{\mathrm{tot}}$	4.103/0.282	4.406/0.204	0.927/0.085	0.567/0.379
$\mathrm{log}{f}_{\mathrm{gw}}$	1.796/0.147	1.853/0.069	1.822/0.377	0.494/0.674
q	0.902/0.003	0.901/0.002	0.888/0.008	0.893/0.011
Φ0	2.849/0.008	2.812/0.002	2.818/0.005	2.853/0.006
ψ	2.856/0.018	2.837/0.022	2.780/0.020	2.256/0.142
$\cos i$	1.781/0.021	1.782/0.007	1.781/0.029	1.754/0.017

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Simulation A (Figures 3 and 4): S-A is located at a sensitive sky location and a sensitive GW frequency. In uninformed, evidence for a CW signal at the injected f_gw is very weak, with all parameters essentially unconstrained (blue histograms). Searching for the source at its injected sky location and distance (ra_dec_z) assists the algorithm in locating the correct M_tot (which manifests as a small peak in the posterior distribution, see the orange histogram in the top-left panel). It has also improved the estimation of f_gw (orange histogram in top center), even though both parameters are being searched over their respective full prior ranges. Applying prior information on M_tot in ra_dec_z_dmtot slightly improves the actual estimation of the parameter (green histogram in top left), but improves the estimation of f_gw more significantly (green histogram in top center); see their respective D_KL values in Tables 6 and 7. The improvement on f_gw is worth noting, since the parameter is still being searched over the wide prior range at this stage. Including the additional prior on f_gw in ra_dec_z_dmtot_dfgw does not further improve the estimation of M_tot or f_gw significantly, as the green and purple histograms in the top-left and top-center panels, respectively, essentially converge and the D_KL value largely plateaus. Note that the estimation of f_gw greatly exceed the level of accuracy provided by the EM prior, which is indicated by the large final KL divergence (recall that the prior on f_gw is log-uniform with a factor of four uncertainty on f _gw on the linear scale). However, the mass ratio q remains unconstrained in all searches (top right).

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Additionally, we observe an overall improvement in the estimation of ψ (bottom center), especially in ra_dec_z_dmtot and ra_dec_z_dmtot_dfgw (green and purple histograms), even though we did not apply any priors on the parameter. It is therefore noteworthy that EM-informed priors on a subset of parameters have improved the estimation of the other parameters for which we have no EM information. However, Φ₀ and $\cos i$ remain essentially unconstrained.

Simulation B (Figures 5 and 6): S-B is located at the same sensitive sky location as S-A, but emitting at a higher GW frequency. All parameters are again unconstrained in uninformed (blue histograms). In the targeted searches, we observe a similar trend of enhanced parameter estimation for all parameters. Furthermore, for each parameter, the posterior distributions are in good agreement with each other and peak near the injected value (orange, green, and purple histograms), except for q, on which there are only lower limits. We note that these improvements are achieved despite the fact that the source is quieter than S-A (in terms of the ${{ \mathcal F }}_{p}$ statistic or SNR). A comparison of their KL divergence values shows that parameters in S-B are overall better constrained than S-A, especially the f_gw parameter, which has the highest KL divergence. This echos the better detectability of S-B compared to S-A (Section 3.1). Again this is likely due to the higher frequency of the S-B signal, which appears to be beneficial not only for detecting the signal, but the parameter estimation as well.

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Simulation C (Figures 7–8) and Simulation D (Figures 9–10): S-C and S-D are the counterparts of S-A and S-B, respectively, at a less sensitive sky location. While the parameter estimation shares similar overall trends as S-A and S-B, we also observe some differences, which are mainly (1) the improvements are less pronounced overall and (2) S-C1 is the only simulation where the EM prior on M_tot allows the algorithm to place a lower limit on q (green and purple histograms in the top-right panel of Figure 7).

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