A Complete Analytic Gravitational Wave Model for Undergraduates

Abstract

Gravitational waves are produced by orbiting massive binary objects, such as black holes and neutron stars, and propagate as ripples in the very fabric of spacetime. As the waves carry off orbital energy, the two bodies spiral into each other and eventually merge. They are described by Einstein's equations of General Relativity. For the early phase of the orbit, called the inspiral, Einstein equations can be linearized and solved through analytical approximations, while for the late phase, near the merger, we need to solve the fully nonlinear Einstein's equations on supercomputers. In order to recover the gravitational wave for the entire evolution of the binary, a match is required between the inspiral and the merger waveforms. Our objectives are to establish a streamlined matching method, that will allow an analytical calculation of the complete gravitational waveform, while developing a gravitational wave modeling tutorial for undergraduate physics students. We use post-Newtonian (PN) theory for the inspiral phase, which offers an excellent training ground for students, and rely on Mathematica for our calculations, a tool easily accessible to undergraduates. For the merger phase we bypass Einstein's equations by using a simple analytic toy model named the Implicit Rotating Source (IRS). After building the inspiral and merger waveforms, we construct our matching method and validate it by comparing our results with the waveforms for the first detection, GW150914, available as open-source. Several future projects can be developed based from this project: building complete waveforms for all the detected signals, extending the post-Newtonian model to take into account non-zero eccentricity, employing and testing a more realistic analytic model for the merger, building a separate model for the ringdown, and optimizing the matching technique.