Parameter estimation method that directly compares gravitational wave observations to numerical relativity

J. Lange et al.

Phys. Rev. D 96, 104041 – Published 22 November 2017

Abstract

We present and assess a Bayesian method to interpret gravitational wave signals from binary black holes. Our method directly compares gravitational wave data to numerical relativity (NR) simulations. In this study, we present a detailed investigation of the systematic and statistical parameter estimation errors of this method. This procedure bypasses approximations used in semianalytical models for compact binary coalescence. In this work, we use the full posterior parameter distribution for only generic nonprecessing binaries, drawing inferences away from the set of NR simulations used, via interpolation of a single scalar quantity (the marginalized log likelihood, $\ln L$ ) evaluated by comparing data to nonprecessing binary black hole simulations. We also compare the data to generic simulations, and discuss the effectiveness of this procedure for generic sources. We specifically assess the impact of higher order modes, repeating our interpretation with both $l \leq 2$ as well as $l \leq 3$ harmonic modes. Using the $l \leq 3$ higher modes, we gain more information from the signal and can better constrain the parameters of the gravitational wave signal. We assess and quantify several sources of systematic error that our procedure could introduce, including simulation resolution and duration; most are negligible. We show through examples that our method can recover the parameters for equal mass, zero spin, GW150914-like, and unequal mass, precessing spin sources. Our study of this new parameter estimation method demonstrates that we can quantify and understand the systematic and statistical error. This method allows us to use higher order modes from numerical relativity simulations to better constrain the black hole binary parameters.

17 More

Received 26 May 2017

DOI:https://doi.org/10.1103/PhysRevD.96.104041

Physics Subject Headings (PhySH)

Gravitational waves

Numerical relativity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Click to Expand

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 96, Iss. 10 — 15 November 2017

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
NR template bank: An illustration of all the simulations used in this study in the two-dimensional space of $1 / q$ and $χ_{eff}$ [Eq. (2)]. Combined with our interpolation methods, the wide range of mass ratios and spins represented in this illustration allows us to reproduce binary parameters for much of the parameter space.
Reuse & Permissions
Figure 2
Example of $\ln L_{marg} (M)$ : comparing a simulation to itself: Left panel: Blue and yellow points (with error bars) show results of evaluating $\ln L_{marg} (M)$ with RIT-1a as a source compared to itself. The yellow color represents the points used for the fit. The shaded region is derived by fitting a quadratic to these data via least squares [Eq. (25)], providing a mean and confidence interval (shown). The reference source has total mass $M = 70 M_{⊙}$ and an inclination $ı = 0.785$ ; all calculations are carried out using $f_{\min} = 30 Hz$ . This curve will be duplicated as a black curve in the bottom left-panel of Fig. 3 and Fig. 4. Right panel: Nominal one-dimensional posterior distributions [Eq. (22)] derived from the fit to left. This figure shows five examples, randomly drawn from the fit coefficient distribution derived by least squares, drawn to exemplify the propagated systematic uncertainties due to Monte Carlo integration error. For studies similar to this one (i.e., high-mass investigations where direct comparison to numerical relativity is most appropriate), this figure suggests that Monte Carlo error is much smaller than the posterior width (i.e., has little relevance given the substantial statistical uncertainty introduced by the limited number of GW cycles available for comparison from short NR simulations).
Reuse & Permissions
Figure 3
Example 1—assessing differences between two NR simulations with different parameters: Two representations of the different predictions of RIT-1a and RIT-2, which are aligned spin binaries with mass ratios $q = 1.22$ and $q = 2.0$ respectively, illustrating how dramatic differences propagate into our diagnostics. Top-left panel: The strain along a line of sight inclined at $ι = 0.785$ and evaluated for a total mass $M = 70 M_{⊙}$ , with RIT-1a in black and RIT-2 in red. Top-right panel: The mismatch between synthetic data and candidate templates as a function of the template’s mass. In both cases, the RIT-1a simulation is used as the template [i.e., as $h$ in Eq. (16)]. For the black curve, RIT-1a for a $70 M_{⊙}$ binary is also used as the source (i.e., $h_{0} = h_{RIT - 1 a}$ ). For the red curve, the source is RIT-2 set at $M = 70 M_{⊙}$ , while RIT-1a has a changing mass. Bottom-left panel: Points show the marginalized likelihood versus total mass calculated by applying the same template simulation (RIT-1a) to two different sources: RIT-1a in black and RIT-2 in red. Each source has fixed mass $M = 70 M_{⊙}$ and inclination $ı = 0.785$ ; as in Fig. 2, we evaluate $L$ using a low-frequency cutoff $f_{\min} = 30 Hz$ . For context, red and black solid curves show a corresponding quadratic least-squares fit to these data. Bottom-right panel: The corresponding one-dimensional posteriors $p_{c} (M)$ [Eq. (22)]. Both bottom panels illustrate how an ill-suited simulation with large mismatch (i.e., the red curve) correlates with a drastic shift in parameters (here, total mass) relative to the true best-fit solution (here, the black curve) [see Eq. (24)]. Also, the ill-matched simulation cannot recover all the information available to the true solution, so the peak $\ln L_{marg}$ for the red curve is substantially lower ( $≃ 20$ ) than the peak of the black curve.
Reuse & Permissions
Figure 4
Example 2—assessing differences in SEOB and NR waveforms that have the same parameters: This figure shows how subtle differences between a NR solution and an approximation to GR [here, effective-one-body (EOB)] can propagate into mismatch and parameter estimation. These two companion figures follow the pattern of Fig. 3. Top-left panel: The black and blue curves show the (2,2) mode evaluated from RIT-a and SEOBNRv2, respectively, for a source with identical parameters. Source parameters and strain results for the black curve are identical to Fig. 3. In this illustration, the two waveforms have been time and phase aligned at their maximum value. Top right-panel: Following the top-right panel of Fig. 3, this figure shows the match between the two waveforms on the top left with the corresponding template from RIT-1a. Bottom-left: The marginalized likelihood $\ln L_{marg}$ for the two waveforms shown above, evaluated using both RIT-1a and SEOBNRv2 as templates: NR source compared to the same NR template in black, the SEOBNRv2 source to a SEOBNRv2 template in red, the SEOBNRv2 source to a NR (RIT-1a) template in blue, and the NR (RIT-1a) source to a SEOBNRv2 template in cyan. Bottom-right: The one-dimensional posterior distributions $p_{c} (M)$ derived from the quadratic fits shown in the bottom-left. Both bottom panels show a clear change along the total mass for SEOBNRv2 sources. The NR/NR comparison has the highest $\ln L_{marg}$ with a corresponding total mass $\sim 70 M_{⊙}$ . The NR/SEOBNRv2 template curve correctly finds the total mass $\sim 70 M_{⊙}$ ; however, the $\ln L_{marg}$ is orders of magnitudes different than the null example. The differences between NR simulations and the SEOBNRv2 model are significant for parameter estimation.
Reuse & Permissions
Figure 5
Example 3—quantifying the impact of the low-frequency cutoff: Using analytic SEOBNRv2 templates with user-specified starting frequency and length, this figure quantifies the impact of our choice of low-frequency cutoff on parameter estimation. Left panel: Plot of $\ln L_{marg}$ versus total mass evaluated using SEOBNRv2 templates with different starting frequencies with $f_{\min} = 10 Hz$ (brown), $f_{\min} = 20 Hz$ (green), $f_{\min} = 30 Hz$ (red), and $f_{\min} = 40 Hz$ (magenta). In all cases, the source signal is also SEOBNRv2 using the same parameters as RIT-1a, but starting frequency $f_{\min} = 5 Hz$ . Right panel: The one-dimensional posteriors $p_{c} (M)$ [Eq. (22)] implied by the results to the left. As you increase the low-frequency cutoff, the $\ln L_{marg}$ decreases significantly, and both the posterior and $\ln L_{marg}$ are wider and offset from the true parameters.
Reuse & Permissions
Figure 6
Monte Carlo error revisited: Repeating the fitting process multiple times: This figure shows several repeated, independent end-to-end calculations of $\ln L_{marg}$ (left panel) and $p_{c} (M)$ (right panel), shown in different colors. The calculation performed is identical to the calculation described for Fig. 2. This figure demonstrates that we understand and have control over our Monte Carlo errors.
Reuse & Permissions
Figure 7
Mismatch between waveforms at different extraction radii using different NR groups and extraction techniques: Both panels show the mismatch between the radiation extracted from RIT-1a (left panel) and SXS-0233 (right panel) as a function of the extraction radius $r$ . All calculations are performed using the same configurations as Figs. 3 and 4: a total mass of $70 M_{⊙}$ and an inclination $ι = 0.785$ . In both panels, the green, blue, and red colors represent different choices of low-frequency cutoff: $f_{\min} = 20$ , 30, 40 Hz respectively. For context and motivated by Eq. (21), the dashed line denotes the mismatch threshold implied by $ρ = 25$ [i.e., $\log_{10} (1 / 2 5^{2})$ ]. Left panel: Mismatch calculations comparing a waveform perturbatively extracted at $r = 190 M$ with a waveform that is perturbatively extracted at other extraction radii [see Eq. (3)]. Right panel: Circles correspond to results using a reference waveform extracted at $r = 545 M$ via perturbative extraction from their $ψ_{4}$ data; triangles denote calculations using a reference waveform evaluated using the strain provided by SXS (i.e., using a polynomial extrapolation with $N = 2$ ). In both cases, the reference waveform is compared to a waveform constructed via perturbative extraction using $ψ_{4}$ data at the specified radius.
Reuse & Permissions
Figure 8
Propagating systematic error from finite extraction radius into posterior distributions: This figure shows how small systematic errors from finite NR extraction radius propagate into parameter estimation posterior distributions, by concrete example. Left panel: A plot of $\ln L_{marg}$ versus total mass. In all cases, the source is RIT-1a at $r = 190 M$ ; the templates are also RIT-1a, using different extraction radii as templates. Here, magenta is $r = 141.71 M$ , orange is $r = 162.34 M$ , and black is $r = 190 M$ . We focus our search on only the last few extraction radii to avoid clutter. The error is relatively small but bigger than what our match study naively suggests (i.e., changes in $\ln L$ of order $10^{- 4} ρ^{2} / 2 ≃ 2 \times 10^{- 2}$ , though this result only applies to the change in the peak value, which indeed changes by less than that amount). Right panel: One-dimensional posterior distributions $p_{c} (M)$ of each individual fit derived from the three plots [see Eq. (22)]. Even though there are small differences, these PDFs are virtually identical.
Reuse & Permissions
Figure 9
Single runs of ILE with RIT simulations with changing resolution and their corresponding PDFs: The left panel consists of $\ln L$ vs total mass curves with different numerical resolution. Here we use RIT-1a as the source and compare it to simulations with the same parameters at different resolutions, specifically RIT-1b and RIT-1c. The results were evaluated with $f_{\min} = 30 Hz$ at a total mass $M = 70 M_{⊙}$ with an inclination $ı = 0.785$ . Here black is n120, purple is n110, and blue is n100. Even though the error is clearly minuscule, we convert the fits to PDFs for completeness. The right panel shows the PDFs for the three different resolutions [see Eq. (22)]. It is clear that these are all the same PDFs, and the error introduced by different resolutions is irrelevant.
Reuse & Permissions
Figure 10
Single runs of ILE with SXS simulations with changing resolution and their corresponding PDFs: The left panel consists of $\ln L$ vs total mass curves with different numerical resolution. Here we use SXS-305a as the source and compare it to simulations with the same parameters at different resolutions, specifically SXS-305b and SXS-305c. The results were evaluated with $f_{\min} = 30 Hz$ at a total mass $M = 70 M_{⊙}$ with an inclination $ı = 0.785$ . Here black is Res6, purple is Res5, and blue is Res4. Even though the error is clearly minuscule, we convert the fits to PDFs for completeness. The right panel shows the PDFs for the three different resolutions [see Eq. (22)]. It is clear that these are all the same PDFs, and the error introduced by different resolutions is irrelevant.
Reuse & Permissions
Figure 11
Single runs of ILE with GT simulations with changing resolution and their corresponding PDFs: The left panel consists of $\ln L$ vs total mass curves with different numerical resolution. Here we use GT-1a as the source and compare it to simulations with the same parameters at different resolutions, specifically GT-1b and GT-1c. The results were evaluated with $f_{\min} = 30 Hz$ at a total mass $M = 70 M_{⊙}$ with an inclination $ı = 0.785$ . Here black is M240, purple is M200, and blue is M160. Even though the error is clearly minuscule, we convert the fits to PDFs for completeness. The right panel shows the PDFs for the three different resolutions [see Eq. (22)]. It is clear that these are all the same PDFs, and the error introduced by different resolutions is irrelevant.
Reuse & Permissions
Figure 12
Assessing the impact of low-frequency cutoff 2: Consistent cutoff choices: Revisiting the investigations shown in Fig. 5, this figure uses (in the top panels) comparisons of a NR simulation to itself as a method to isolate the impact of $f_{\min}$ (the low-frequency cutoff appearing in the likelihood). The bottom panels repeat a comparable analysis using SEOB. Top left panel: Plots different $\ln L_{marg}$ vs total mass curves with different $f_{\min}$ . Here we compared a RIT-4 source with a duration of 5.0 Hz source compared to itself at different $f_{\min}$ values. Specifically brown has a $f_{\min} = 10$ , green has a $f_{\min} = 20$ , red has a $f_{\min} = 30$ , and magenta has a $f_{\min} = 40$ . These results are similar to the SEOBNRv2 case in Fig. 5. As the cutoff increases, our $\ln L$ curve becomes wider, and the peak value $\ln L_{marg}$ is lower. Top right panel: One-dimensional posteriors $p_{c} (M)$ [Eq. (22)]. This figure qualitatively resembles Fig. 5; however, unlike the previous analysis, while the posterior is wider (i.e., less informative), no significant bias is introduced by the low-frequency cutoff. Bottom left panel: Similar to prior figures, a plot of $\ln L_{marg} (M)$ , evaluated using SEOBNRv2. In this comparison, the SEOBNRv2 source with a certain duration was compared to a SEOBNRv2 template with the same $f_{\min}$ . Specifically brown has a $f_{\min} = 10$ , green has a $f_{\min} = 20$ , red has a $f_{\min} = 30$ , and magenta has a $f_{\min} = 40$ . As the cutoff increases, our $\ln L$ curve becomes wider. Bottom right panel: The corresponding PDFs to the fits [see Eq. (22)]. We again see similarities between this case and Fig. 5 minus the shift in total mass with increasing $f_{\min}$ .
Reuse & Permissions
Figure 13
Parameter recovery for zero spin equal mass binary I: Each point represents a NR simulation and a particular total mass compared against a SXS-1 source. The left panel shows $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ and $χ_{eff}$ defined in Eq. (2), and the right panel shows $χ_{eff}$ vs $M$ . The gray points represent points that fall between $\ln L_{marg} = 130$ and $\ln L_{marg} = 127$ . The black points represent points that fall in $\ln L_{marg} > 130$ , i.e. templates that best match the source. The peak value with this run was $\ln L_{marg} = 134$ . These intervals were determined using the inverse $χ^{2}$ distribution [see Eq. (10)]. The rest of the colors represent all the points $\ln L_{marg} < 127$ with the red representing the highest in the region. The green contour is the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The big red dot represents the true parameters of the source. We are able to recover the two-dimensional posterior distribution that is consistent with the distributions with $\ln L_{marg} > 130$ (black points).
Reuse & Permissions
Figure 14
Parameter recovery for zero spin equal mass binary II: The left panel shows the $\ln L_{marg}$ as a function of $χ_{1 z}$ and $χ_{2 z}$ . The rainbow, gray, and black points represent the same intervals as in Fig. 13. The green contour also represents the same CI as Fig. 13. The right panel shows the one-dimensional posterior distribution for $1 / q$ . This one-dimensional posterior was derived from the quadratic fit of to $\ln L_{marg}$ for nonprecessing systems only. Here we show results for six inclinations: $ı = 0.0$ (black), $ı = 0.5$ (red), $ı = 0.785$ (blue), $ı = 1.0$ (green), $ı = 1.5$ (gray), $ı = 2.35$ (orange). We see that the results from all the inclinations are the same; i.e. no more information can be obtained with higher order modes.
Reuse & Permissions
Figure 15
Parameter recovery for an aligned, GW150914-like unequal mass binary I: Each point represents a NR simulation and a particular total mass compared against a SXS-0233 source. The left panel shows $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ , and the right panel shows $χ_{eff}$ vs M with $χ_{eff}$ defined in Eq. (2). The gray points represent points that fall between $\ln L_{marg} = 167$ and $\ln L_{marg} = 165$ . The black points represent points that fall in $\ln L_{marg} > 167$ , i.e. templates that best match the source. The rest of the colors represent all the points $\ln L_{marg} < 165$ with the red representing the highest in the region. The green contours are the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The dash line is the CI for $l \leq 3$ , and the solid line is the CI for $l \leq 2$ . The big red dot represents the true parameters of the source. We are able to better constrain the posterior by using higher modes for this system.
Reuse & Permissions
Figure 16
Parameter recovery for an aligned, GW150914-like unequal binary II: The left panel shows the $\ln L_{marg}$ as a function of $χ_{1 z}$ and $χ_{2 z}$ . The colored, gray, and black points represent the same intervals as in Fig. 15. The green contours also represent the same CI as Fig. 13. The big red dot represents the true parameters of the source. The right panel shows the one-dimensional posterior distribution for $1 / q$ . This one-dimensional posterior was derived from the quadratic fit of to $\ln L_{marg}$ for nonprecessing systems only. Here we show results for six inclinations all represented by the same colors as the zero spin case; see Fig. 14. In this case, we see significant differences between the curves implying that higher order modes could be important for accurate analysis of this source.
Reuse & Permissions
Figure 17
Parameter recovery for a precessing, short, unequal mass binary I: Each point represents a NR simulation and a particular total mass compared against a SXS-0234v2 source with $l \leq 2$ modes. The left panel shows the $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ and $χ_{eff}$ defined in Eq. (2), and the right panel shows the $χ_{eff}$ vs $M$ . The gray points represent points that fall between $\ln L_{marg} = 165$ and $\ln L_{marg} = 163$ . The black points represent points that fall in $\ln L_{marg} > 165$ , i.e. templates that best match the source. The rest of the colors represent all the points $\ln L_{marg} < 163$ with the red representing the highest in the region. The green contour is the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The big red dot represents the true parameters of the source. We are able to recover the two-dimensional posterior distribution that is consistent with the distributions with $\ln L_{marg} > 165$ (black points).
Reuse & Permissions
Figure 18
Parameter recovery for a precessing, short, unequal mass binary II: The left panel shows the $\ln L_{marg}$ as a function of $χ_{1 z}$ and $χ_{2 z}$ . The gray, black, and other color points represent the same intervals as in Fig. 17. The green contour represents the same contour as in Fig. 17. The big red dot represents the true parameters of the source. The green contour is consistent with the black point distribution. The right panel shows the one-dimensional posterior distribution for $1 / q$ . This one-dimensional posterior was derived from the quadratic fit of $\ln L_{marg}$ for nonprecessing systems only. Here we show results for the same six inclinations all represented by the same colors as the zero spin case; see Fig. 14. In this case, we see significant differences between the curves implying that higher order modes could be important for accurate analysis of this source. We also see large discrepancies between the $ı = 1.5$ distribution and the other inclinations. See Figs. 19 and 20 for further analyses.
Reuse & Permissions
Figure 19
Proof of parameter recovery for a precessing, short, unequal mass binary: Here is $\ln L_{marg} (M)$ of a single ILE null run comparing SXS-0234v2 with itself (black) and the $\ln L_{marg} (M)$ for the full end-to-end run with SXS-0234v2 as its source (gray). The left panel represents runs with a source with $ı = 0.0$ , and the right panel represent runs with a source with $ı = 1.5$ . The gray points only include the nonprecessing templates. If we take the difference between the $\ln L_{marg}$ from the whole end-to-end run and the $\ln L_{marg}$ from the null run, we get a $Δ \ln L \sim 0.97$ for $ı = 0.0$ and $Δ \ln L \sim 1.8$ for $ı = 1.5$ . Even if we were to include the best template in our end-to-end runs (which is itself), we only get a slight increase in the $\ln L_{marg}$ for the face-on inclination. However, the edge-on case change seems significant; see Fig. 20 for an investigation focusing on the peak values.
Reuse & Permissions
Figure 20
Discrepancy in $\ln L_{marg} (M)$ for $ı = 1.5$ : This is a plot of multiple $\ln L_{marg} (M)$ comparing SXS-0234v2 with itself at different inclinations. Here $ı = 0.0$ is black, $ı = 0.5$ is red, $ı = 0.785$ is blue, $ı = 1.0$ is green, $ı = 1.5$ is gray, and $ı = 2.35$ is orange. The edge-on case is clearly different than the rest of the inclinations; more needs to be done to discover the origin of this discrepancy; however, this could be due to many significant higher modes.
Reuse & Permissions
Figure 21
Parameter recovery for a zero spin, $q = 2$ binary: Each point represents a NR simulation and a particular total mass compared against a RIT-5 source. The top-left panel shows the $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ and $χ_{eff}$ defined in Eq. (2), the top-right panel shows the $χ_{eff}$ vs $M$ , and the bottom panel shows the $χ_{1}$ vs $χ_{2}$ . The gray points represent points that fall between $\ln L_{marg} = 166$ and $\ln L_{marg} = 164$ . The black points represent points that fall in $\ln L_{marg} > 167$ , i.e. templates that best match the source. The rest of the colors represent all the points $\ln L_{marg} < 164$ with the red representing the highest in the region. The green contour is the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The dashed line is the CI for $l \leq 3$ , and the solid line is the CI for $l \leq 2$ . The big red dot represents the true parameters of the source. We are able to recover the two-dimensional posterior distribution that is consistent with the distributions with $\ln L_{marg} > 167$ (black points).
Reuse & Permissions
Figure 22
Parameter recovery for a high, antialigned spin $q = 1.31$ binary: Each point represents a NR simulation and a particular total mass compared against a SXS-high-antispin source. The top-left panel shows the $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ and $χ_{eff}$ defined in Eq. (2), the top-right panel shows the $χ_{eff}$ vs $M$ , and the bottom panel shows the $χ_{1}$ vs $χ_{2}$ . The gray points represent points that fall between $\ln L_{marg} = 167$ and $\ln L_{marg} = 164$ . The black points represent points that fall in $\ln L_{marg} > 167$ , i.e. templates that best match the source. The rest of the colors represent all the points $\ln L_{marg} < 164$ with the red representing the highest in the region. The green contour is the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The dashed line is the CI for $l \leq 3$ , and the solid line is the CI for $l \leq 2$ . The big red dot represents the true parameters of the source. We are able to recover the two-dimensional posterior distribution that is consistent with the distributions with $\ln L_{marg} > 167$ (black points).
Reuse & Permissions
Figure 23
Parameter recovery for a $χ_{eff} = 0.4$ spin $q = 1.19$ binary: Each point represents a NR simulation and a particular total mass compared against a SXS- $χ_{eff} 0.4$ source. The top-left panel shows the $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ and $χ_{eff}$ defined in Eq. (2), the top-right panel shows the $χ_{eff}$ vs $M$ , and the bottom panel shows the $χ_{1}$ vs $χ_{2}$ . The gray points represent points that fall between $\ln L_{marg} = 167$ and $\ln L_{marg} = 164$ . The black points represent points that fall in $\ln L_{marg} > 167$ , i.e. templates that best match the source. The rest of the colors represent all the points $\ln L_{marg} < 164$ with the red representing the highest in the region. The green contour is the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The dashed line is the CI for $l \leq 3$ , and the solid line is the CI for $l \leq 2$ . The big red dot represents the true parameters of the source. We are able to recover the two-dimensional posterior distribution that is consistent with the distributions with $\ln L_{marg} > 167$ (black points).
Reuse & Permissions
Figure 24
Parameter recovery for a $χ_{1} = χ_{2} = - 0.8$ spin $q = 2.0$ binary: Each point represents a NR simulation and a particular total mass compared against a RIT-2 source. The top-left panel shows the $χ_{eff}$ vs $1 / q$ with $q = m_{1} / m_{2}$ and $χ_{eff}$ defined in Eq. (2), the top-right panel shows the $χ_{eff}$ vs $M$ , and the bottom panel shows the $χ_{1}$ vs $χ_{2}$ . The gray points represent points that fall between $\ln L_{marg} = 165$ and $\ln L_{marg} = 162$ . The black points represent points that fall in $\ln L_{marg} > 165$ , i.e. templates that best match the source. The rest of the colors represent all the points $\ln L_{marg} < 162$ with the red representing the highest in the region. The green contour is the 90% CI derived using the quadratic fit to $\ln L_{marg}$ for nonprecessing systems only. The dashed line is the CI for $l \leq 3$ , and the solid line is the CI for $l \leq 2$ . The big red dot represents the true parameters of the source. We are able to recover the two-dimensional posterior distribution that is consistent with the distributions with $\ln L_{marg} > 165$ (black points).
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology