Markov Chain Monte Carlo (MCMC) and Bayesian Inference for Gravitational Waves

Abstract

The Laser Interferometer Space Antenna (LISA) is a space borne gravitational wave detec- tor set to launch in 2034, with the objective of detecting and studying the Gravitational Waves (GWs) of our universe. So far, ground-based detectors such as the Laser Interferometer Gravitational-Wave Observatory (LIGO) have been successful in detecting GWs, but the limitations of ground based detectors is what makes LISA so special. With three separate space crafts, each 2.5 million kilometers apart, the detector is expected to mea- sure gravitational radiation within the frequency regime of 0.1 mHz to 1 Hz. With LISA, astronomers will be able to determine the type, mass, as well as the energy released by GW sources. From the data measured with LISA, scientists will be able to challenge Einstein’s theory of General Relativity through the lens of gravity, and perhaps uncover vital information about our universe’s past, present and future.

As LISA follows an Earth trailing orbit around the Sun, it is expected to detect GWs emitted by numerous sources simultaneously. This results in one single signal that contains the information of all detectable gravitational sources in the universe. In order to study a single source, the development of a mathematical computer model is required to extract the desired information. This thesis implements Bayesian inference with a stochastic sampling algorithm known as Markov Chain Monte Carlo (MCMC) to tackle the multi-dimensional problem and statistically recover the parameters of the GW.

In this thesis, we focus on the ecliptic coordinates of the source, which are just two out of the seven parameters of a GW. We found that MCMC was successful in the localisation of the source from a simulated gravitational wave strain. The ecliptic coordinates were recovered with a standard deviation of less than 1 degree. Expanding the program, we were also able to test the effectiveness of MCMC in the presence of multiple waves, and how their amplitude and frequencies a↵ect the algorithms ability to recover the true position. Lastly, we conclude this paper with an alternative suggestion for extracting the parameters using Multinest, as well as comment on additional research.

Popular Abstract

Gravitation Waves (GWs), first predicted by Albert Einstein in the early 20th century, are a warping of spacetime invisible to the naked eye. Back in September of 2016, the most sensitive GW detector on Earth called LIGO - the Laser Interferometer Gravitational Wave Observatory – made the first ever direct observation of a GW. The GW was generated by the merging of two massive black holes about 1.4 billion light years away. More such GWs exist and travel through the universe from all directions and distances. In fact, whenever an object has mass, it has the potential of warping spacetime with its gravitational attractiveness. When an object is massive enough, such as a black hole or neutron star, a change in its velocity can create a large enough ripple in space-time to be detectable. The most common formation of GWs is when two massive gravitationally bound stellar objects in a binary system spiral inward and merge.

LIGO uses laser beams and light sensors stretched four kilometers apart to detect GWs, but even then, the detector is not sensitive enough to detect effectively. The European Space Agency (ESA) decided to build a GW detector sensitive enough to detect hundreds, if not thousands, of GWs with sensors stretched 2.5 million kilometers apart. How can such a detector exist on Earth? Well, it can’t. The answer lies in the detector’s name. The Laser Interferometer Space Antenna (LISA) is a GW detector stationed in space. It revolves around the Sun, trailing behind Earth, meant to collect GW signals for up to 5 years starting in 2034. There is a lot of work to do before then to make sure all the data collected by LISA is analyzable. One issue is determining where the GW originated and providing a coordinate system to pinpoint the direction of the source in space. Our thesis explores the effectiveness of using a Markov Chain Monte Carlo (MCMC), with the implementation of Baye’s Theorem from statistics, in a computer algorithm to locate GW sources.

A good way of understanding how MCMC works is by imagining a robot that makes random jumps around a hill with the goal of finding the peak. Each time the robot lands in a spot higher than its previous record height, it stores the location and continues this pattern until it eventually reaches the peak. In this case, the peak is the coordinates of the GW source. Implementing Baye’s Theorem allows us to utilize our prior knowledge of GWs to weed out any physically meaningless data. This allows the algorithm to be more ecient as it restricts the “robot’s” area of investigation.

Our project explores the basics of this method. The results obtained showed us now MCMC is able to recover the position of a GW source from it’s strain signal to an accuracy of ±0.6 degrees. The algorithm is also able to ‘tune’ to different frequencies of GWs to extract only the desired relevant information. Multiple sources of the same frequency were also detectable, albeit more research needs to be done about problems where source positions overlap, causing confusion in the results. Additionally, the algorithm can also be modified to search and recover all 7 parameters of a GW at the same time. By studying these GWs in greater detail, astronomers will be able to probe the universe from an angle we were unable to in the past. Physicists will be able to use LISA to challenge theories such as General Relativity, by studying the gravity of the early universe.

• author: Andersson, Christine
• supervisor: David Hobbs
• organization: Lund Observatory - Undergoing reorganization
• course: ASTK02 20211
• year: 2021
• type: M2 - Bachelor Degree
• subject: Physics and Astronomy
• keywords: Gravitational Waves, LISA, LISA mission, Bayesian Inference, Markov Chain Monte Carlo, MCMC, Bayes’ Theorem, stochastic sampling, Metropolis Hastings, histograms
• publication/series: Lund Observatory Examensarbeten
• report number: 2021-EXA179
• language: English
• id: 9053298
• date added to LUP: 2021-06-16 12:35:42
• date last changed: 2021-06-16 12:35:42