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

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 ≤ 2 as well as l ≤ 3 harmonic modes. Using the l ≤ 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.

Targeted numerical simulations of binary black holes for GW170104

“…IV, we directly compare our simulations to GW170104. These comparisons provide both a scalar measure of how well each simulation agrees with the data (a marginalized likelihood), as well as the best-fitting reconstructed waveform in each instrument [14,15]. Using our reconstructed waveforms, we graphically demonstrate that our simulations agree with each other and the data, with simulation differences far smaller than the residual noise in each instrument.…”

“…Catalogs of the simulated waveforms are publicly available [19][20][21] for its use for BBH parameter estimation [15], as well as for determining how the individual masses and spins of the orbiting binary relate to the properties of the final remnant black hole produced after merger. This relationship [22] can be used as a consistency check for the observations of the inspiral and, independently, the merger-ringdown signals as tests of general relativity [3,23,24].…”

“…For any proposed coalescing binary, characterized by its outgoing radiation as a function of all directions, we can compute a single quantity to assess its potential similarity to GW170104, accounting for all possible ways of orienting the source and placing it in the Universe: the marginalized likelihood (ln L marg ) [15,102,103]. To provide a sense of scale, the distribution of ln L marg over the posterior distribution including all intrinsic parameters is roughly universal [103], approximately distributed as Table II. ln L marg;max − χ 2 =2 where χ 2 has d degrees of freedom (d.o.f.)…”

“…Fortunately, nature has provided us with a natural benchmark with which to assess the efficacy of our calculations and the significance of any discrepancies: observations of BH-BH mergers. We use standard techniques [14,15] to directly compare GW170104 to our simulations. For context, we also compare these observations to the corresponding predictions of approximate and phenomenological models that purport to describe the same event.…”

“…This is due to the fact that NR waveforms include physics omitted by current approximate models, notably higher order modes and accurate precession effects. A full description of this methodology, including detailed tests of systematic errors and parameter estimation improvements, can be found in [15]. This paper is organized as follows.…”