Fast and accurate inference on gravitational waves from precessing compact binaries

Inferring astrophysical information from gravitational waves emitted by compact binaries is one of the key science goals of gravitational-wave astronomy. In order to reach the full scientific potential of gravitational-wave experiments, we require techniques to mitigate the cost of Bayesian inference, especially as gravitational-wave signal models and analyses become increasingly sophisticated and detailed. Reduced-order models (ROMs) of gravitational waveforms can significantly reduce the computational cost of inference by removing redundant computations. In this paper, we construct the first reduced-order models of gravitational-wave signals that include the effects of spin precession, inspiral, merger, and ringdown in compact object binaries and that are valid for component masses describing binary neutron star, binary black hole, and mixed binary systems. This work utilizes the waveform model known as “IMRPhenomPv2.” Our ROM enables the use of a fast reduced-order quadrature (ROQ) integration rule which allows us to approximate Bayesian probability density functions at a greatly reduced computational cost. We find that the ROQ rule can be used to speed-up inference by factors as high as 300 without introducing systematic bias. This corresponds to a reduction in computational time from around half a year to half a day for the longest duration and lowest mass signals. The ROM and ROQ rules are available with the main inference library of the LIGO Scientific Collaboration, LALInference.

Figure 1

The $J$-aligned source frame of a precessing binary. IMRPhenomPv2 uses a single precessing spin approximation to describe the inspiral, merger and ringdown and is described by the parameter vector $\overrightarrow{\lambda }=(m_1,m_2,{\chi }_1,{\chi }_2,{\chi }_p,{\theta }_J,{\alpha }_0)$, with $\widehat{N}$ in the x-z plane. Here, ${\chi }_1$ and ${\chi }_2$ are the spin components that lie parallel to $\overrightarrow{L}$ on the heavier (${\chi }_1$) and lighter (${\chi }_2$) compact object; and the perpendicular spin parameter ${\chi }_p$ is the spin component that lies in the orbital plane and is associated with the heavier body $m_1$.

Figure 2

An example of the basis enrichment strategy applied to case D from Table 1 (the plus- and cross-polarizations only). Top: A sequence of histograms showing the distribution of the reduced-basis approximation error for $\approx 10$ million out-of-sample model evaluations. Notice a continual lowering of the maximum approximation error with each iteration. Bottom: The final distribution of greedy points in a three-dimensional subspace. This set is a mixture of the initial structured grid used in the zeroth iteration and random points identified through the enrichment process.

Figure 3

Illustration of how to translate basis results to new regions of mass and frequency. By using the scaling described in the text, the basis for case A (blue unhatched region) maps on to case ${\mathrm{A}}^{\prime }$ (blue hatched region). Similarly, the unhatched and hatched green regions, respectively, correspond to cases B and ${\mathrm{B}}^{\prime }$ in Table 1. These primed regions form a starting point for building a 10 Hz basis without extra computational effort.

Figure 4

Greedy error, defined as the maximum approximation error over the training set using the first $N_{\mathtt{L}}$ basis, versus basis number $N_{\mathtt{L}}$. The error profile is fairly similar across all cases. Since the ROQ speed-up is (almost) proportional to $N_{\mathtt{L}}$, further speed-up can be achieved at the expense of accuracy. Throughout the paper we select the first $N_{\mathtt{L}}$ basis satisfying a conservative $5\times {10}^{-12}$ greedy error tolerance.

Figure 5

Histogram of selected ROQ nodes and a representative waveform for case A (top) and case F (bottom). Evidently the selected frequency points cluster at small values. This is intuitively expected because lower frequency intervals contain a greater number of waveform cycles, a feature which is automatically detected by the empirical interpolation method. Histograms of those cases not shown are qualitatively similar, being a mixture of these two boundary cases.

Figure 6

Projection (RB) and empirical interpolation (EIM) errors (generically x-axis labeled as “Waveform error”) for $\approx 15$ million randomly drawn waveforms. Each subfigure reports on the errors for an approximation defined by the six cases listed in Table 1. The validations are performed using the same adaptive frequency sampling strategy as was used to find the basis (cf. Sec. 3d).

Figure 7

Empirical interpolation errors (generically x-axis labeled as “Waveform error”) for $\approx 15$ million randomly drawn waveforms. Each subfigure reports on the errors for an approximation defined by the six cases listed in Table 1. The validations are performed using an adaptive frequency sampling strategy and for three representative antenna pattern configurations.

Figure 8

Empirical interpolant approximation errors (the plus- and cross-polarizations only) when using an adaptive (solid green line) and uniform (dashed blue line) frequency sampling. The adaptive sampling is used during the ROQ building procedure. To compute log-likelihoods with our ROQ rule we upsample to a uniform frequency grid, and so this error (which constitutes the last in a series of approximations of the underlying model) is the most relevant for ROQ-accelerated inference studies. Results are shown for case E only; other cases are qualitatively similar. Maximum upsampled EIM errors of $6\times {10}^{-9}$, $1\times {10}^{-7}$, $1\times {10}^{-5}$, $7\times {10}^{-8}$, $4\times {10}^{-7}$ and $1\times {10}^{-9}$ were computed for cases A–F, respectively.

Figure 9

Comparing the recovered posterior PDFs for the black hole masses measured in their source frame (left) and the two dominant spin parameters (right), c.f. [44]. The parameter ${\chi }_{\text{eff}}=(m_1{\chi }_1+m_2{\chi }_2)/(m_1+m_2)$ is known as the “effective” spin and is a mass-weighted combination of the two spin components parallel to the orbital angular momentum. The dashed black lines mark the 90% credible interval for both the ROQ and Full likelihoods, which are the same within the statistical sampling uncertainty of $\sim 1\%$. The posteriors do not peak exactly at the injected values due to the presence of detector noise.

Figure 10

Theoretical parameter estimation speed-up (using the ROQ) for cases A–F in Table 1. The speed-up is calculated from the ratio $L/(N_{\mathtt{L}}+N_{\mathtt{Q}})$, where $L=(f_{\max }-f_{\min })/\mathrm{\Delta }f$ is the number of quadrature points in the Full likelihood, $N_{\mathtt{L}}$ is the size of the linear basis and $N_{\mathtt{Q}}$ is the size of the quadratic basis. The sum $N_{\mathtt{L}}+N_{\mathtt{Q}}$ is the number of points in the ROQ likelihood (11). The plot is annotated with the time (in hours) to compute $2\times 10^7$ ROQ (Full) likelihood evaluations, roughly the number of evaluations required for a typical PE analysis [5]. Our tests were performed using an Intel Xeon CPU with a 2.70 GHz clock speed.

Figure 11

Amplitudes of ${\tilde{h}}_{+}(f)$ for selected points in the ${\chi }_{\mathrm{p}}\approx 0$ cluster shown in Fig. 12. The parameter values for configurations (a)–(d) are given in Table 2. Abrupt, sharp features are clearly visible in the IMRPhenomPv2 amplitudes (blue solid lines), but are absent in the SEOBNRv3 amplitudes (red dashed lines). These features are difficult to capture with the reduced-basis method without sacrificing the sparsity and/or accuracy of the basis.

Figure 12

Top: When applied to the full seven-dimensional parameter space, the greedy algorithm identifies a “feature cluster” where the model exhibits fast changing (potentially nonsmooth) behavior. The cluster directly below the dashed blue line arises for values ${\chi }_p\approx 0$ large antialigned spin ${\chi }_1$ for unequal mass ratios. Figure 11 shows a few waveforms from this region. Bottom: Values of $\kappa$, the angle between the orbital angular momentum L and the total spin $S=S_1+S_2$, as a function of the symmetric mass ratio $\eta$ for the same cyan (${\chi }_p\approx 0$) cluster as shown in the top panel. We observe a clear clustering between 175° and 180°.