Effect of Solar System Models on 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 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 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 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 ...... 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 ...... 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 ” was a revolutionary theory published in 1915 which unified space with time and energy with momentum, showing precisely how massive objects cause 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 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 and creates a , 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 over the whole temporal and physical area of observation. When observing a pulse 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. To calculate the latter we use Developmental Ephemerides (version model labeled by ”DE” followed by 3 numbers) created by NASA’s JPL for spacecraft navigation. The models consist of numerical values for position, velocity,

8 and acceleration of the Sun, eight major planets with Pluto, and the moon in rectangular coordinates [8]. A recent version, DE436, was created with more precise orbital data for Mercury, Mars, and Jupiter missions. Modern models are derived from DE430, a 1000 year- long model with the highest precision for lunar orbits. These models are a useful tool for calculating the solar system barycenter: the center of mass of the solar system as seen in Fig. 5. Every object in the solar system orbits around the barycenter, including the sun. Due to the elliptical orbits between planets about the barycenter, the positions and lengths between the two evolve with time.

Figure 5:n ˆ points to a local pulsar, ~r points from the SSB to Earth,

and difference in light travel time is seen with tb.

Now we’re equipped to talk about the light travel time between the Earth and the barycenter, otherwise known as the Roemer delay. Members of the Australian pulsar timing group (called the Parkes Pulsar Timing Array) noticed that the timing residuals of one pulsar, J1909, are smaller in DE414 (RMS ∼ 100 ns) than DE436 (RMS ∼ 170 ns), and have suggested that versions beyond DE414 might be incorrect as they make the timing residuals of J1909 increase. This presents a puzzle: if the excess RMS in DE436 is real then it must be that the difference between DE436 and DE414 cancels out with a real red noise process in J1909 due to DE436 being more accurate. Here I calculate how often this happens

9 if the red noise process is a gravitational wave. This turns out to happen about 30% of the time.

10 3 Theory

General relativity (GR) is used to calculate the arrival time of a pulsar pulse. From the GR calculation we will see the effect of gravitational waves on pulse arrival time, and the effect of the earth-SSB distance on pulse arrival times. Section 3.1 explains the GR calculation leading up to gravitational waves, then section 3.2 connects pulsar timing arrays and timing residuals, and section 3.3 defines the Roemer delay.

3.1 General Relativity

We start with Einstein’s field equations which directly relate the curvature of spacetime (left-hand side) to the mass and energy in that spacetime (right-hand side) are,

8πG G + Λg = T . (1) µν µν c4 µν

The subscripts µ and ν signifies a run over both space and time µ, ν = t, x1, x2, x3. Therefore, Einstein’s field equations are 16 equations as seen in Eq. 1. The Einstein equation contains three 4 x 4 tensors, the Einstein tensor Gµν, the metric tensor gµν, the stress-energy tensor Tµν, and a cosmological constant Λ. Each tensor is of the form

  gtt gtx1 gtx2 gtx3   gx1t gx1x1 gx1x2 gx1x3  gµν =   g g g g   x2t x2x1 x2x2 x2x3 

gx3t gx3x1 gx3x2 gx3x3 . Where every entry is a series of differential equations that can be solved. From here we can extract gravitational waves by assuming the metric tensor is of the form

gab(x) = ηab + hab(x). (2)

Where ηab is the Minkowski metric that details flat space and hab is a small perturbation. Eq. 9 describes the change in spacetime ”distance” measurements between two points. We then note the vacuum equation which says the Ricci curvature tensor is

Rab = 0, (3)

meaning

11 1 G = R − g R, (4) µν µν 2 µν simplifies to

1 G = g R, (5) µν 2 µν

where R is the scalar curvature given by the trace of Rµν. From this, we can derive the solution for the perturbation tensor as seen in [1],

  0 0 0 0   0 h+ hx 0 i2πf( z −t) hab =   e c . 0 h −h 0  x +  0 0 0 0

The plus (+) and cross (x) on hab describe the polarization directions of a gravitational

wave propagating in the z direction, where c is the . h+ bends the axis of the

x & y planes while hx distorts across x = ±y.

3.1.1 Gravitational Wave Timing Residuals

Now we want to analyze how gravitational waves distort spacetime between two points. We start with the redshift of the gravitational wave z adapted from Ref. 1,

1 pˆipˆj z(t, rˆ) = ∆h . (6) 2 1 +r ˆ · pˆ ij Where pˆi and pˆj indicate spatial coordinates from Earth to a pulsar,r ˆ is the propagation direction of a gravitational wave (ˆr = (sin θ cos φ, sin θ sin φ, cos θ) for spherical coordinates),

and ∆hij is the difference in the metric perturbation tensor described by changes between

when the pulse is emitted (tp) and received (te) seen in Eq. 19,

∆hij ≡ he(te) − hp(tp) (7)

We find the total red shift by integrating z(t, rˆ) over the 2-sphere, S2,

Z z(t) = drzˆ (t, rˆ). (8) S2 Yet, the timing residual is still not defined. We will have to integrate over the time domain to sum every bit of redshift.

12 Z t r(t) = dt0z(t0), (9) 0 which finally gives the total gravitational wave signal affecting our measurements.

3.2 Pulsars Timing Arrays

3.2.1 Timing Residuals

Pulsar timings have so far given us accurate measurements confirming the effects of general relativity as their measurements are very precise and consistent. The only things needed to analyze a pulsar system are to take the expected and recorded time of arrivals (TOAs) of their pulses and take the difference, giving us a timing residual

tresidual = tobserved − tmodel. (10)

These timing residuals can be caused due to fluctuations in interstellar medium that disperse the radio pulses or interference between us and the signal, measurement error, or that of a time-dependent gravitational wave propagating through our line of sight. As our model increases and theory improves the timing residuals decrease accordingly.

3.2.2 Noise

When dealing with stochastic processes, it’s important to talk about noise and the various colors associated with different frequency patterns observed in a signal. When we record an observation in the field of audio engineering, electronics, or gravitational wave astronomy, we are required to discuss the effect of the GW background as it is a source of noise in the data. For pulsar timing arrays we often encounter different variations of noise that affect our TOAs, such as generic or in some cases, red noise. White noise is named after white light which has a flat frequency spectrum, so there’s equal power in any band of the noise signal. This is the most generic type of noise as there’s equal power at all frequencies (high and low) being observed. We can describe the power density per unit of bandwidth (width between frequencies) as being proportional to

1 , (11) f β where β = 0 for white noise [12].

13 What if β 6= 0? There’s only one other color of noise needing analysis in this project: red noise. We observe stochastic gravitational waves as being more apparent in the lower frequency spectrum than higher frequencies, so we suggest β = 2 to show the power density 1 falls off like f 2 . We call this ”red noise” as shown in Fig. 6 where two frequency are plot against their intensity.

Figure 6: On the left is an example of white noise with a flat power spectrum and on the right is an example of red noise where lower frequencies are much more intense [11].

3.2.3 Cross Correlation

Knowing only the angle between a pair of Earth-pulsar baselines (lines from Earth to the pulsar) and the gravitational wave noise, we can find the expected (cross) correlation between the timing residuals due to an isotropic stochastic gravitational wave background. This matrix is given by the Hellings and Down (HD) equation

1 − cos γ 1 − cos γ 1 1 − cos γ 1 χ = IJ ln | IJ | − IJ + . (12) IJ 2 2 6 2 3 Where γIJ is the angle between the (I, J) pulsar pair in the array. This relation tells us how the GWB by is correlated among pulsars. If we know the residuals of two pulsars and we cross-correlate them, diving by their HD value yields the cross-correlation strength of a stochastic gravitational wave background.

14 3.3 Roemer Delay

When we have a solar system model we must pay attention to the small differences in orbits throughout to guarantee precision. Any differences in orbit could interfere with pulsar timings or other experiments as lengths change and cause noise. This is why we calculate the timing residual due to the Roemer delay seen in Fig. 5 given by

1 t = t − nˆ · ~r, (13) observed c to see the difference in light travel throughout the Earth’s orbital period. Heren ˆ is a vector pointing to a pulsar, ~r is the earth-barycenter length that is always changing, c is the speed of light, and tobserved is the time of arrival of a pulse [4]. We subtract the speed of light terms to remove any noise caused by this effect. Between solar system models like SSEs, these values may change, so it’s important to observe the change in Roemer delay as they add to the timing residuals.

3.3.1 Effective Roemer Delay

When working with a set of real data (xi, yi) it is important to find the true correlation between them. With this in mind, we want a transformation that will remove basis functions (1, x, x2, ..., xn, cos(x), sin(x)) from each data point. We construct a ”Design Matrix” M

  n M = 1 tn ... tn sin(tn) cos(tn) . (14) This is a matrix with N rows (one for each point) and more than 5 columns depending on the choice of functions to subtract. From this, we further express the R-matrix used to transform our data points, derived from Ref. [6]

R = I − M(M T M)−1M T . (15)

We can then multiply the set of Roemer delays by the R-matrix to transform the data. Now we have a set of transformed data points that take up a lot of space and vary like a wave. Therefore, We find the ”mean” of the results using quadratic mean or root mean square (RMS). This process gives a result that corresponds to the average power of a . For a set of values xi the RMS is given by

r 1 x = [x2 + x2 + ... + x2 ]. (16) RMS n 1 2 n

15 4 Methods

In this section I detail how we use Matlab, a mathematical processing software, to generate a stochastic gravitational wave background, a pulsar timing array, and any given solar system ephemeris models such as DE414 and DE436. The code used is run 200 times to get GWB realizations from over 30000 pulsars to supply large quantities of frequencies.

4.1 Roemer Delay

Solar system ephemerides are models created of our solar system by NASA to facilitate spacecraft navigation. They are excellent depictions of the orbits of all planets, the sun, and most other masses within it. This includes position, velocity, and acceleration, therefore giving us access to a vast amount of precise historical data to analyze. With the discrepancies in mind, we look at DE414 and DE436 to find the difference in Roemer delay between the two models. With the help of Tempo2, a pulsar timing software, we were able to easily calculate this change in RD. Now that we have the Roemer delay data, we recall from Ch 3.3.1 that we wish to remove a set of basis functions that are fit out as part of the timing procedure. This calls for the use of the R-matrix Eq. 6 derived from the design matrix Eq. 5 to remove polynomial and sinusoidal terms. We, therefore, set the design matrix equal to

  2 M = 1 tn tn sin(tn) cos(tn). (17) This fits out a constant, a linear term (frequency of the pulsar), a quadratic term (spin- down), and a yearly sinusoidal term (sky location) which can all affect the TOAs.

4.2 Pulsars

We then proceed by loading in 150 random local pulsars’ θ and φ velocities provided by NANOGrav; these will act as our pulsar timing array. For each pulsar in the array, we assign a little residual white noise data to simulate generic deviations in TOAs.

4.3 Injecting Gravitational Waves

With a PTA now in our environment we’re finally set to inject a simulated stochastic gravitational wave background. For this analysis, we simulate 150 local pulsars from 3.2 by

16 looping over all possible pulsar pairs and calculating the Hellings and Downs χ matrix from 2.2.3. To simulate a gravitational wave background1 we must create it artificially through levels of white and red noise. To do this we create a white noise signal and then scale it identically to how red noise drops off. Then for each pulsar in our collection, a white noise signal is added to the red noise to create a timing residual. At this point, we have a set of pulsars with timing residuals affected by a stochastic GWB and can cross-correlate them via the χ matrix. 2

4.4 Root-Mean Square

Now we have a set of timing residuals including white noise and red noise and a set of Roemer delays. Our goal is to see how the pulsar timing residuals look if we subtract the Roemer delay from them, so we create two independent sets σ and σgw. We define σ to be the root-mean-square of the RD subtracted from the timing residuals and σgw to be the root-mean-square of just the timing residuals. Both σ and σGW act as a wave due to their periodicity concerning position and time of orbits. These RMS values correspond to the average power of the residuals over that domain.

4.5 Histogram Analysis

Now that we have two sets of data with varying frequencies, we plot σ and σGW onto a histogram to get a total count of each residual for both systems. Our goal is to under- stand how often a Roemer delay subtracted timing residual could look like one caused by a stochastic GWB, so we take the mean of all σGW points to compare it to σ. With this value, we will count how many σ values are lower than the mean of σGW , which will lead us to understand how probable a Roemer delay is to cause distortions close enough to a GWB.

1Process derived from [9] 2A detailed analysis of the χ matrix lives in [Ref. 1]

17 5 Results and Discussion

5.1 Pulsar Residuals

We want to analyze the difference in timing residuals with and without the Roemer delay included, so we first look at Fig. 7 which shows the timing residual of three random pulsars over 15 years.

Figure 7: First three pulsar timing residuals due to an injected stochastic gravitational wave background discussed in Ch. 2.1.1.

As we see, the timing residuals behave almost periodically, often intersect, and share around the same range of values. Now in Fig. 8 we remove one random pulsar and add in the Roemer delay over the same period.

18 Figure 8: First 2 pulsar’s timing residuals with the Roemer Delay over 15 years.

This shows that the recorded Roemer delay looks close to that of the timing residuals, so we suspect that after the RMS analysis of σ and σGW the cancellation rate will be very high, meaning a GWB often looks like the difference in Roemer Delay between SSEs.

5.2 Histograms

After running the Matlab code 200 times, we come out with 30000 pulsars. We can separate the residuals ranging from near 0 nanoseconds to 800 nanoseconds and count each pulsar by putting them into frequency bins.

19 Figure 9: RMS of gravitational wave time residuals across 30000 pulsars injections.

Following this, we can subtract the Roemer delay from the timing residuals seen in Fig. 9. This will allow us to see how often the subtracted residuals end up less than the mean of the ordinary GWB red noise.

Figure 10: RMS of Roemer delay subtracted from gravitational wave timing residuals

We take the mean of the timing residuals from Fig. 9 and add a vertical line for that

20 value onto the histogram seen in Fig. 10, resulting in Fig. 11 below.

Figure 11: Plot of subtracted timing residuals with a mean gravitational wave signal at t = 253 ns.

After counting the total number of pulsars (P) less than the mean, we divide by the total number of pulsars to get a percentage. This result correlates to the cancellation rate of Roemer delay with the GWB as we desire, showing that this effect does occur in nature.

P < σ¯ 9857 gw = = 32.8%. (18) P 30000 We can consider this mystery solved because we see in Eq. 22 that 32.8% of the time the Roemer delay cancels out with an injected gravitational wave background. This is the result we wanted as we only care about if this event happens or not, so a result of almost half is incredible.

21 6 Conclusion

Einstein’s theory of General Relativity was once again confirmed via observation of grav- itational waves in 2015 by LIGO, a large interferometer based on the idea that these ripples in spacetime distort distances between mirrors that can be measured using electromagnetic radiation like light. As systems like binaries begin to orbit each other and accel- erate inward they distort spacetime with GWs. It was noticed that as a system becomes more massive the lower the frequency of its gravitational waves; as does the detectability of the wave. Thus, we suspect that binary systems produce incredibly low-frequency GWs. These have yet to be detected, but the process of analyzing pulsar tim- ing array data appears to be a very promising avenue. As pulsars radiate electromagnetic beams like a lighthouse with an extremely regular rotation we can observe the distortions and noise in our signal with timing residuals. The Parkes Pulsar Timing Array noticed that the timing residuals of one pulsar, J1909, are smaller in DE414 (RMS ∼ 100 ns) than DE436 (RMS ∼ 170 ns), and have suggested that versions beyond DE414 might be incorrect as they make the timing residuals of J1909 increase. This presents a puzzle as we know DE436 is more accurate, so our goal was to find the solution. We showed that the excess RMS in DE436 for J1909 is being canceled out with a real red noise process, a gravitational wave. We simulated a stochastic gravitational wave background and added that to 150 random pulsar timing residuals. We then applied the R-matrix to account for the timing model. This process was done 200 times to guarantee accuracy with 30000 pulsar realizations and shows us how often the average power of the Roemer delay included timing residuals look like that of a GWB. We saw in Ch. 4.2 that 32.8% of the time this cancellation occurs. We now know that there is no puzzle left to be solved as enough of the time these two wave-like objects often cancel each other out. With more developments in the field of PTAs and gravitational wave astronomy like this, the closer we get to understanding the cause and possible existence of a stochastic gravitational wave background.

22 7 Bibliography

[1] Chamberlin, Sydney J. et al. “Time-domain implementation of the optimal cross- correlation statistic for stochastic gravitational-wave background searches in pulsar timing data”. In: Physical Review D 91.4 (Feb. 2015). issn: 1550-2368. doi: 10.1103/ physrevd.91.044048. url: http://dx.doi.org/10.1103/PhysRevD.91.044048. [2] Chen, Zu-Cheng, Yuan, Chen, and Huang, Qing-Guo. Non-tensorial Gravitational Wave Background in NANOGrav 12.5-Year Data Set. 2021. arXiv: 2101.06869 [astro-ph.CO]. [3] Choi, Charles Q. With New Gravitational-Wave Detectors, More Cosmic Mysteries Will Be Solved. May 2019. url: https://www.space.com/gravitational-waves- future-discoveries.html. [4] Edwards, R. T., Hobbs, G. B., and Manchester, R. N. tempo2, a new pulsar timing package – II. The timing model and precision estimates. Oct. 2006. url: https:// academic.oup.com/mnras/article/372/4/1549/1186764.

[5] Howell, Elizabeth. What Are Redshift and Blueshift? Mar. 2018. url: https://www. space.com/25732-redshift-blueshift.html. [6] Jenet, Fredrick A. and Romano, Joseph D. Understanding the gravitational-wave Hellings and Downs curve for pulsar timing arrays in terms of and electromagnetic waves. 2015. arXiv: 1412.1142 [gr-qc]. [7] Moore, C J, Cole, R H, and Berry, C P L. “Gravitational-wave sensitivity curves”. In: Classical and 32.1 (Dec. 2014), p. 015014. issn: 1361-6382. doi: 10.1088/0264- 9381/32/1/015014. url: http://dx.doi.org/10.1088/0264- 9381/32/1/015014.

[8] Park, Ryan. JPL Planetary and Lunar Ephemerides. Dec. 2020. url: https://ssd. jpl.nasa.gov/?planet_eph_export. [9] Press, William H. et al. Numerical Recipes in C: The Art of Scientific Computing. USA: Cambridge University Press, 1988. isbn: 052135465X. [10] Saxton, B. Parts of a Pulsar. Dec. 2020. url: https://public.nrao.edu/gallery/ parts-of-a-pulsar/.

[11] Warrakk. . Apr. 2021. url: https : / / en . wikipedia . org / wiki / Colors_of_noise.

23 [12] Wisniewski, Joseph S. The Colors of Noise. Oct. 1996. url: https://web.archive. org/web/20110430151608/https://www.ptpart.co.uk/colors-of-noise.

24 A Perturbation Tensor

We define the plane wave expansion of the perturbation tensor to be of the form

Z ∞ Z X ˆ i2πf(t−Ω·~x ˆ A hij(t, ~x) = df dΩe )hA(f, Ω)eij (19) 2 A −∞ S Which is unneeded for this project but nice to list.

25