Effect of Solar System Models on Pulsar Timing Experiments Cooper Nicolaysen May 27, 2021 An undergraduate thesis advised by Xavier Siemens and NANOGrav submitted to the Department of Physics, Oregon State University in partial fulfillment of the requirements for the degree BSc in Physics. Submitted on May 27, 2021 Abstract Gravitational wave (GW) astronomy is a key ingredient in confirming Einstein's theory of General Relativity and showing how the universe sends ripples through space- time, distorting distances between two points. The confirmation of high-frequency GWs observed by The Laser Interferometer Gravitational-Wave Observatory (LIGO) in 2015 was a breakthrough in our understanding of general relativity and was the start of a new field of observational astronomy. Pulsar timing arrays are currently a promising experiment for the detection of low-frequency gravitational waves, with the potential to detect a stochastic gravitational wave background. Recently, an international group of pulsar timing array researchers discovered an inconsistency in the difference between the expected and actual time of arrivals (timing residuals) in a local pulsar J1909. As they switched between solar system models (DE414 and DE436) there were higher timing residuals in the newer model, DE436, which is worrisome given that residuals should decrease as we increase precision. We propose these residuals are most likely affected by the difference in Roemer delay, a light travel time distortion due to earth- sun orbit radii changing, between the two models. We show that the Roemer delay difference between two solar system models can mimic a stochastic gravitational wave background around 30% of the time confirming this occurs in nature. 1 1 Acknowledgments Throughout my college experience and the last years, I have received some of the greatest supervision and support I could ever ask for. I would like to thank my advisor, Xavier Siemens, for allowing me into his group and giving me a role to play in cosmological research. Thank you to him and our team, for welcoming me in with open arms. A special thanks to Jacob Taylor for his supervision and Nima Laal for the use of Tempo2. I then thank NANOGrav, whom without Xavi I'd never joined, for the incredible ex- perience with astrophysicists around the world. Additionally, without Parkes pulsar timing array there would be no question for me to answer, so their aim to advance knowledge is quite notable and commendable. I come to thank both Oregon State's College of Science and College of Mathematics, for the incredible education delivered to me over the past 4 years. Finally, without the initial help from STEM Leaders my freshman year I do not know where I'd be professionally. Therefore, a big warm thank you goes to Kevin Ahern for helping me address many personal and professional concerns, Sophie Pierszalowski for the incredible research outreaching advice, and Stephanie Ramos for many professional life lessons. All of your help lead me to this publication; thank you. 2 Contents 1 Acknowledgments 2 2 Introduction 5 2.1 Background and Motivation . .5 2.2 Pulsars . .6 2.3 Gravitational Waves and Timing Residuals . .8 3 Theory 11 3.1 General Relativity . 11 3.1.1 Gravitational Wave Timing Residuals . 12 3.2 Pulsars Timing Arrays . 13 3.2.1 Timing Residuals . 13 3.2.2 Noise . 13 3.2.3 Cross Correlation . 14 3.3 Roemer Delay . 15 3.3.1 Effective Roemer Delay . 15 4 Methods 16 4.1 Roemer Delay . 16 4.2 Pulsars . 16 4.3 Injecting Gravitational Waves . 16 4.4 Root-Mean Square . 17 4.5 Histogram Analysis . 17 5 Results and Discussion 18 5.1 Pulsar Residuals . 18 5.2 Histograms . 19 6 Conclusion 22 7 Bibliography 23 Appendix A Perturbation Tensor 25 3 List of Figures 1 Binary system producing gravitational waves . .5 2 Gravitational wave frequencies vs characteristic strain . .6 3 Lighthouse & pulsar comparison . .7 4 Hellings and Downs Curve . .8 5 Earth-Sun-pulsar system . .9 6 White vs red noise . 14 7 3 pulsar's timing residuals . 18 8 2 pulsars residuals with Roemer delay interlaced . 19 9 Gravitational wave background caused timing residuals . 20 10 Timing residuals due to a GWB with Roemer delay . 20 11 Timing residuals comparison . 21 4 2 Introduction 2.1 Background and Motivation Einstein's "General Theory of Relativity" was a revolutionary theory published in 1915 which unified space with time and energy with momentum, showing precisely how massive objects cause spacetime to bend and, in return, how spacetime tells masses to move. These bends are felt as an acceleration, like gravity on Earth, as mass bends spacetime inward. In 2015, 100 years after Einstein published his paper, a distant binary system in which two black holes are orbiting around each other while spinning faster and faster was observed using gravitational waves (GW) (see Fig. 1 [2].) When these waves reached Earth all distances were distorted slightly and detected by by the large laser interferometer observatory LIGO. These ripples vary in magnitude dependent on the size of the system that produces them. Figure 1: An illustration of a binary orbiting system disrupting spacetime and sending out gravitational waves [3]. The gravitational wave spectrum spans frequencies between 10−16 Hz to 104 Hz. High- frequency (10 Hz to 10 kHz) GWs are measurable at ground-based interferometers like LIGO and medium-range waves (10−6 Hz to 1 Hz) are theorized to be detectable with the Laser Interferometer Space Antenna (LISA) launching by 2034. As the frequency of GWs decreases, the larger our detector needs to be. Although we've never detected gravitational waves near the nanohertz (10−9 Hz) range, pulsar timing arrays (PTAs) are a potential tool for this task. Pulsars have a near-perfect rotational period and span the entire galaxy. In this project, I will analyze the effects of how gravitational waves can interact with uncertainties in the solar system to minimize the observed pulsar timing residuals. 5 Figure 2: Plot of gravitational wave frequencies vs their characteristic strain with various detector sensitivities. LIGO can measure longer wavelengths while PTAs can measure the lower frequency waves. Adapted from Ref. [7]. 2.2 Pulsars Pulsars are formed when a massive star collapses during a supernova and creates a neutron star, a much smaller and denser object which begins to spin rapidly along one direc- tion. Due to the large change in radius from the star's initial to the final state (about 10km, around 50,000 times smaller than the Sun) the moment of inertia is drastically decreased causing a large increase in angular velocity. Exactly like a ballerina who tucks her arms in before speeding up. Along the magnetic axis of the pulsar, which does not have to be aligned with the rotational axis, a beam of radiation is emitted. If the beam of radiation coming from one of the magnetic poles crosses the line of sight between Earth and the pulsar, the pulsar is observed as a point source of periodic bursts of radio waves (as if we set our FM radio to a pulsars' frequency). As seen in Figure 3, a lighthouse and a pulsar are very similar objects as they both sweep past our field of view with some form of light-like radiation. 6 Figure 3: On the left is a lighthouse with electromagnetic beams (light waves) rotating across our field of view, seen periodically. On the right is a pulsar with electromagnetic beams (radio waves) rotating both along its magnetic and angular axis [10]. These pulses arrive at our radio telescopes so steadily that an array of pulsars spread across our galaxy can be used to detect gravitational waves by measuring variations in the arrival time of their pulses. The expected arrival time subtracted from the actual arrival time gives the timing residual, which may contain gravitational waves. Due to the gravitational waves being homogeneous (the same strength wherever you stand) the effect of GWs on a pair of pulsars only depends on their angular separation and the residuals. Therefore, if we know the expected arrival times of pulses from a pulsar we can cross-correlate across a whole PTA to detect the overall effect of the GW. This is best illustrated through the Hellings and Downs curve in Fig. 4 and is the key signature of a stochastic gravitational wave background [1]. 7 Figure 4: Graph corresponding to the TOAs cross-correlation between pulsars separated by some angle γIJ where I and J indicate pulsars [6]. 2.3 Gravitational Waves and Timing Residuals Gravitational waves are measured by the distortion in the time of arrivals (TOAs) of pulses from the pulsar system; where the distortion is quantified by a red or blue shift [5]. Due to the gravitational ripples passing through the pulsar-Earth line of sight, the electromagnetic radiation from the pulse increases or decreases in wavelength, causing the light to shift between red and blue periodically. When we calculate timing residuals for a pulsar we're integrating the redshift over the whole temporal and physical area of observation. When observing a pulse signal there will often be errors (noise) due to measurement techniques, objects orbiting the pulsar, intrinsic properties of the system, or GW effects. When referring to a GW, the noise produced is given the label red noise as the wave is stronger in the low- frequency part of the spectrum whereas the high-frequency bins are weaker. It is useful to characterize the timing residual time-series with the root-mean-square (RMS), and measure the effect of GWs using this quantity. To calculate results the timing residuals we need an intertial coordinate system which is chosen to be the solar system's center of mass.