Leonard E Parker Center for Gravitation, Cosmology and Astrophysics

*Abstract The first direct detection of gravitational waves (GWs) was observed on Sep 14, 2015. We report the properties of GW150914 derived from a coherent analysis of the data from the two LIGO detectors surrounding this event, based on the most accurate modeling of the coalescence signal as predicted by general relativity. We show that GW150914 originates from a BBH with components of masses 36 M☉ +5−4 M☉ and 29 M☉ +/−4 M☉ at redshift 0.09+0.03-0.04 assuming a standard Cosmology. The binary merges to form a BH of mass 62 M☉ +/−4 M☉ and spin 0.67+0.05−0.07 but we cannot place meaningful limits on the precession effects. About 3 M☉ of energy was radiating through the propagation of GWs. The final BH is more massive than any other found in the stellar-mass range. These results herald the beginning of GW astronomy and provide the first observational insights into the physics of BBHs.*

In general relativity, two bodies in orbit slowly spiral together due to the loss of energy and momentum through gravitational radiation. This is in contrast to Newtonian gravity where bodies can follow closed orbits. Radiation reaction is efficient in circularizing orbits before the signal enters the sensitivity band of the instruments. The gravitational wave observed for GW150914 comprises of order of 10 cycles during the binary BHs' inspiral phase from where it enters LIGO’s sensitive band at 20 Hz, followed by the merger which results in a single perturbed BH, and the ringdown during which the final BH settles down in its final state by radiating GWs at constant frequency with amplitude damped over a few cycles.

The properties of the binary affect the phase and amplitude evolution of the signal, leaving fingerprints that can be exploited to measure the source parameters. The binary BHs are fully modeled by 8 intrinsic parameters: the masses m1,2 and spins S1,2 (magnitude and 2-dimensional orientation) of the individual BHs, and 9 extrinsic parameters: the location (luminosity distance D_L, right ascension α and declination δ); orientation (the binary’s orbital inclination ι and polarization ψ); time tc and phase φc of coalescence, and the eccentricity (two parameters) of the system.

The gravitational radiation is described by two independent and time-dependent polarizations, h+ and h×. Each instrument of the LIGO detectors measures a strain that can be decomposed into the two polarizations, depending on the source location in the sky and the polarization of the waves. During the inspiral, the phase evolution can be computed using post-Newtonian (PN) theory, which is a perturbative expansion in powers of the orbital velocity v/c for weak gravitational fields. For GW150914, v/c is in the range ≈ 0.2–0.5 in the LIGO sensitivity band. As the BHs get closer to each other and their velocities increase, the accuracy of the PN expansion degrades, and eventually full solutions of Einstein’s equations are needed for an accurate description of the strong-field merger stage of the binary.

Full information about the properties of the source is provided by the probability density functions (PDFs) of the unknown parameters, given the two data-streams from the instruments. The posterior PDFs are computed through a straight forward application of Bayes’ theorem. The computations are addressed by using a suite of Bayesian parameter-estimation and model-selection algorithms tailored to this problem. From the posterior PDFs, shown in Figures 1–5 for selected parameters, we then construct credible intervals for the parameters of the binary system, reported in Table I.

TABLE I from the article. Summary of the parameters that characterise GW150914. |

One application of parameter estimation is the direct measurement of luminosity distance of a source from GW observations only. Whereas in traditional E&M observations we can measure the redshift of a galaxy from its spectra but its distance can only be indicated assuming some model of cosmology.

Throughout this work we have considered a model for the binary evolution in the LIGO sensitivity band that assumes a circular orbit. Preliminary investigations suggest that eccentricities of e < ∼ 0.1 at 10 Hz would not introduce measurable deviations from a circular-orbit signal; however, even larger eccentricities may have negligible effects on the recovered source parameters. At this time, the lack of a model that consistently accounts for the presence of spins and eccentricity throughout the full coalescence prevents us from placing more stringent constraints. We plan to report improved limits in the future.