Date of Award

August 2021

Degree Type


Degree Name

Doctor of Philosophy



First Advisor

Jolien Creighton

Committee Members

Patrick Brady, David Kaplan, Dawn Erb, Daniel Agterberg


Cosmology, General Relativity, Gravitational Waves, Pulsars, Supermassive Black Holes


The ability to detect gravitational waves now gives scientists and astronomers a new way in which they can study the universe. So far, the scientific collaboration LIGO has been successful in detecting binary black hole and binary neutron star mergers. These types of sources produce gravitational waves with frequencies of the order hertz to millihertz. But there do exist other theoretical sources which would produce gravitational waves in different parts of the frequency spectrum. Of these are the theoretical mergers of supermassive black hole binaries (SMBHBs), which could occur upon the merging of two galaxies with supermassive black holes at their cores. Sources like these would produce gravitational waves generally around the nanohertz regime, and the current main effort for detecting and measuring these waves comes from pulsar timing experiments. Detection of gravitational waves in these experiments would come as small fluctuations in the otherwise extremely regular period of pulsars over a long period of time (months to decades).

There are numerous goals for this dissertation. The first is to re-present much of the fundamental physics and mathematics concepts behind the calculations in this paper. While there exist many reference sources in the literature, we simply try to offer a fully self-contained explanation of the fundamentals of this research which we hope the reader will find helpful. It is often a challenge when jumping into a new field of study to quickly learn and understand the fundamentals (like the derivations of various formulae and the assumptions behind the models), so if this dissertation can help future readers to connect the dots between the blanks not filled in by other literature sources, then this goal will be accomplished. The pedantic approach to this dissertation is also helpful since much of the initial work for this dissertation was theoretical development of the mathematical models used in pulsar timing.

The next goal broadly speaking has been to combine the efforts of two previous studies by Deng & Finn (2011) and Corbin & Cornish (2010) to further develop the mathematics behind the currently used pulsar timing models for detecting gravitational waves with pulsar timing experiments. Previous timing residual models have first been derived assuming that the pulsar timing array receives plane-waves coming from distant sources (with the notable exception of Deng & Finn). Then these models can either treat the SMBHB as a monochromatic gravitational wave source, or model the frequency evolution of the gravitational waves over the thousands to tens of thousands of years it takes light to travel from the pulsar to the Earth. Our research began by first generalizing these models by removing the plane-wave assumption. In Chapter 3 we classify four regimes of interest (Figure 3.5), governed by the main assumptions made when deriving each regime. Of these four regimes the plane-wave models are well established in previous literature. We add a new regime which we label "Fresnel," as we will show it becomes important for significant Fresnel numbers describing the curvature of wavefronts.

With these mathematical models developed, in Chapter 6 we present the first main study investigated which was to forecast the ability of future pulsar timing experiments to probe and measure these Fresnel effects. Here we show the constraints needed on the pulsar timing experiments themselves (largely explained by the discussion in Chapter 5), and the types of precision measurements which could theoretically be achieved.

Then we generalize our models to a cosmologically expanding universe in Chapter 7. We show that in the fully general Fresnel frequency evolution regime, the Hubble constant enters the model and can now be measured directly. In this chapter we investigate what we will need of future experiments in order to obtain a measurement of this parameter. This offers future pulsar timing experiments the unique possibility of being able to procure a purely gravitational wave-based measurement of the Hubble constant.

Finally, Chapter 8 shows the initial steps taken to extend this work in the future, specifically for Doppler tracking experiments. The main goal of the inclusion of this final section, which was not the primary focus of this dissertation, is to point out how the mathematics and models derived in Chapters 2 and 3 can be applied and extended more generally.