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.a Undergraduate Student,

If you thought that science was certain -well, that is just an error on your part.

Gravitational waves (GW) from eccentric binaries have intricate signals encoding important features about the location, creation and evolution of the sources. Eccentricity shortens the merger time, making the emitted GW statistically predominant in the observed data once detectors will reach the required sensitivity. We present a novel implementation of fully analytical GW templates from eccentric binary black hole (BBH) mergers within the Wolfram Mathematica software. We increase the accuracy by identifying and minimizing the possible source of errors. We start with an overview of the physics involved in eccentric mergers, then assemble the strain for the inspiral by employing up to six post-Newtonian (PN) corrections. We complete the eccentric inspiral with the quasi-circular Backwards one Body (BOB) merger model in frequency, amplitude and phase, then we build the hybrid GW strain for the whole evolution of the binary. For low eccentricity, we reach coincidence in the overlap, with no ambiguity in the time interval, a remarkable improvement from the usual matching techniques. For high-eccentricity, we compensate for the implicit quasi-circular assumption of the BOB approach, by introducing a small rescaling in amplitude. Our streamlined implementation is relevant for the new field of GW astronomy and is straightforward to understand, use and extend, offering researchers in the field a valuable open resource tool.

It is important to doubt and that the doubt is not a fearful thing, but a thing of great value.

Merging black holes produce the loudest signal in the detectors. However, this is the most difficult signal to accurately predict with analytical techniques. Only computer simulations can account for the nonlinear physics during the collision, but they are inherently complex, costly, and affected by numerical errors. In order to bypass this problem, two analytical models for the merger have been developed: the Implicit Rotating Source (IRS) and the newer Backwards one Body (BoB). In this work, we assess the performance of the BoB model by comparing it with the older IRS model and with the numerical data, identifying its strengths and weaknesses. Our main finding reveals discrepancies in amplitude, but overall excellent accord in frequency. The BoB model is comparable with the IRS and NR simulations, having the added advantage that it depends only indirectly on numerical data, it accounts for spin, and it offers a seamless fit with the analytical formalisms for the inspiral. By independently evaluating and testing those models, we bring evidence of their reproducibility, thus upholding high scientific standards, and make available our implementation, enabling readers to evaluate our results themselves.