ligo-comprehensive/comprehensive.tex

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% \fancyhead[C]{The Observation of Gravity waves at LIGO $\bullet$ May 2017 $\bullet$ Vol. XXI, No. 1} % Custom header text
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\section{Introduction}
\lettrine[nindent=0em,lines=3]{I}n 1916, Einstein predicted the existence of gravity waves as a consequence of his theory of general relativity. However, despite a century of experimental effort, direct observation eluded scientists until September 2015 when the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) observed a gravity wave resulting from a binary black hole merger.

The structure of this paper is as follows: In \sec{history} I will go through the history of gravitational waves beginning with their first theoretical predictions and going through early efforts at direct detection by Weber and others.  \sec{ligo} is a rather thorough description of the Advanced LIGO detector itself.  \sec{data} describes the data analysis strategies employed to find evidence of gravity waves in the data coming from the detector.  \sec{observations} describes the two published gravity wave observations made in the first observing run of Advanced LIGO.\@ \sec{implications} Briefly describes some of the implications of the observed events on cosmology. And finally \sec{conclusion} gives some closing remarks.

\section{The History of Gravity Waves}
\label{sec::history}
In this section I give an overview of some of the history of gravity waves from their tumultuous theoretical origins to early efforts at detection.

\subsection{The Existential Question}
In 1905 Henri Poincare published a paper entitled ``Sur la dynamique d' l'\`{e}lectron''\cite{poincare1905}. In the paper, Poincare described his theory of relativity which based the existence of gravity waves on analogy with the electro-magnetic waves produced by accelerating charges. However, it would take until 1916 for Einstein to publish his Theory of General Relativity\cite{einstein1916}. This theory, which was in many ways an extension of his Theory of Special Relativity, viewed gravity not as a force, a la Newton, but instead as a curvature in space-time brought about by the presence of mass and energy. Expressed mathematically, Einstein's field equations can be written in tensor form as \[R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu}= \frac{8\pi G}{c^4}T_{\mu\nu}\] where $R_{\mu\nu}$ is the Ricci curvature tensor, $R$ is the scalar curvature, $g_{\mu\nu}$ is the metric tensor, $\Lambda$ is the cosmological constant, $G$ is Newton's gravitational constant, and $T_\mu\nu$ is the stress-energy tensor.

Roughly speaking, the right-hand side of the equation is determined by a distribution of mass and energy and the left-hand side is the resulting space-time curvature. Unfortunately, although the equation appears relatively simple, both the Ricci tensor and the scalar curvature depend on the metric tensor in a complicated nonlinear manner. As a result, in only a small number of cases with readily exploitable symmetries have the field equations been solved analytically. In fact, it has taken great efforts in the field of Numerical Relativity to be able to obtain the necessary theoretical results for LIGO to know what the signature of various astronomic calamities (e.g.\ binary black-hole mergers)  would look like.

But back 1916, Einstein was still grappling with his nascent theory and its implications. For example, because there is no such thing as negative mass in General Relativity (in contrast to electro-magnetism where charge comes with positive \emph{and} negative signs), one cannot construct a gravitational dipole or resulting dipole radiation. By 1936, Einstein, together with his student Nathan Rosen, had arrived at the conclusion that gravity waves could not exist in the theory. Indeed, they submitted a paper to the \textit{Physical Review} stating as much. The editor forwarded the paper to the referee, a Howard Percy Robertson, who pointed out several flaws in the paper. Einstein, apparently unfamiliar with the peer review process, responded to the criticism.

\begin{displayquote}
\small{\textit{%
  July 27, 1936 \\[.3cm]
  Dear Sir. \\[0.1cm]
  ``We (Mr.\ Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed. I see no reason to address the\--in any case erroneous\--comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.''\\[0.2cm]
  Respectfully\\[0.2cm]
  Einstein\\[0.2cm]
  P.S.\ Mr.\ Rosen, who has left for the Soviet Union, has authorized me to represent him in the matter.
}}
\end{displayquote}

With the departure of Rosen, another young physicist named Leopold Infeld became the new assistant to Einstein. Infeld befriended Robertson (the referee who criticized the original paper) and together they confirmed the error in the original Einstein-Rosen submission. Infeld proceeded to point out the error to Einstein who was then obligated to submit a letter to the editor of the \textit{Journal of the Franklin Society} where he had eventually submitted the paper after its rejection by \textit{Physical Review}. The revised paper contained the following conclusion.
\begin{displayquote}
  \small{%
``Rigorous solution for Gravitational cylindrical waves is provided. For convenience of the reader the theory of gravitational waves and their production, known in principle, is presented in the first part of this article. After finding relationships that cast doubt on the existence of gravitational fields rigorous wavelike solutions, we have thoroughly investigated the case of cylindrical gravitational waves. As a result, there are strict solutions and the problem is reduced to conventional cylindrical waves in Euclidean space.''
  }
\end{displayquote}

In the end Einstein came to understand that gravitational waves were a real part of General Relativity. However, it remained to be seen how such waves could be seen in experiment.

\subsection{Physical Interpretations}
Unfortunately for the experimentalist, the coordinate systems commonly employed for calculations in the realm of General Relativity were chosen for reasons of mathematical simplicity, not for easily extracting observables or making comparisons with experiment. In 1956, Felix A.\ E.\ Pirani addressed this problem in his paper ``On the physical significance of the Riemann tensor''\cite{pirani1956}. In his paper, Pirani deduced the effects of space-time curvature to an observer in   Crucially, Pirani showed that particles are oscillated by the passing of a gravity wave.

It is worth taking a slight pause in the story here to describe in greater detail the physical nature of gravity waves. First of all, these waves travel at the speed of light, which could seen as either a direct consequence of General Relativity, or a result of a massless graviton. The stretching and squeezing of space is always and only transverse to the direction of propagation and is invariant under a 180\si{\degree} rotation. The waves come in two polarizations, the $+$ polarization, which affects free particles as depicted in \fig{plus_polarization}, and the $\times$ polarization which is rotated 45\si{\degree} with respect to the $+$ polarization. These waves can be red shifted and gravitationally lensed, just like electro-magnetic waves. However, unlike electro-magnetic waves, they are only negligibly dispersed by interactions with matter\cite{thorne2002}.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.35\textwidth]{figures/plus_polarization.png}
  \caption{The effect of a $+$ polarized gravitational wave propagating into the page on a ring of free particles. Initially, the particles form a circle (a), but as the wave passes into the page, space is stretched horizontally, and squeezed vertically (b). A half-period later, however the situation is reversed and space is squeezed horizontally and stretched vertically (d).}
\label{fig::plus_polarization}
\end{figure}


In the year following Pirani's paper, the seminal Chapel Hill conference was held on the campus of the University of North Carolina, and among the many notable attendees were Richard Feynman, Julian Schwinger, and John Wheeler. The conference has organized by the Institute of Field Physics (IOFP), under the patronage of eccentric millionaire Roger W.\ Babson. Of the many topics covered at the six day conference, one of the most hotly debated was the question of whether gravity waves were able to carry energy. To address this question, Feynman, in characteristic style, anonymously proposed a simple thought experiment known as the ``sticky bead'' experiment. It goes as follows.

Consider a rod threaded through two rings as depicted in \fig{sticky_bead}. The rings are allowed to slide along the rod, but there is some small friction between the rings and the rod. As a gravity wave traverses the experiment, space will get periodically stretched and compressed along the axis of the rod, meaning the proper-distance between the rings oscillates. On the other hand, the atomic restoring forces between the atoms in the rod will keep its length fixed. Consequently, the rings will rub against the rod, heating it. This implies that the gravitational wave is doing work on the system, and must therefore carry energy.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.35\textwidth]{figures/sticky_bead.png}
  \caption{Sketch of the ``sticky bead'' experiment.\cite{CC2016}.}
\label{fig::sticky_bead}
\end{figure}

Also at the Chapel Hill conference was an engineer from the University of Maryland named Joseph Weber. He was fascinated by the phenomena of gravitational waves. So much so, that we went on to design the first experiment to directly detect them.

\subsection{Early Experimental Efforts}
In the years following the Chapel Hill conference Weber designed a ground based ``antenna'' which could detect the presence of gravity waves. He detailed his ideas in his 1960 paper ``Detection and Generation of Gravity Waves''\cite{weber1960}, and By 1966 had built a detector and published evidence of it's performance\cite{weber1966}. 

Weber's experimental setup consisted of a large aluminum cylinder, 66\cm{} in diameter and 153\cm{} in length\cite{weber1972}. The cylinder was suspended by steel wires from a vibration isolating support. Piezoelectric sensors were mounted around the diameter of the cylinder to detect vibrations, such as those induced by passing gravitational waves.

Weber actually built two of these detectors: one at the University of Maryland, and another nearly 1000 \km{} away at Argonne National Laboratory outside Chicago, with the idea being to use them to cross-check each other and eliminate false positives from local noise sources.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/weber_with_bar.png}
  \caption{Weber working on his detector}
\label{fig::weber_with_bar}
\end{figure}

Amazingly, when Weber turned on his detectors, they picked up about one coincidence a day. He claimed this as evidence for the discovery of gravity waves. He went further to claim that many of the signals originated near the center of our galaxy and estimated from his measurements that our galaxy is radiating $\approx1000$ solar masses per year of energy in the form of gravity waves. This ran afoul of estimates from cosmologists who calculated an upper limit of 200 solar masses per year. Any larger, and the necessary mass to hold the galaxy together would have radiated away long ago.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/bar_coincidence.png}
  \caption{An example of a detector coincidence seen by Weber published in the May 1972 issue of \textit{Popular Science}}
\label{fig::bar_coincidence}
\end{figure}

A hallmark of all good science is repeatability. As such, efforts were undertaken by others to build similar resonating bar experiments to attempt to reproduce Weber's results. By the middle of the 1970s several additional experiments were running that incorporated improvements over Weber's original design such as cooling the cylinders to reduce thermal noise. Sadly, none of these improved detectors were able to reproduce Weber's results. This inability to confirm Weber's results, combined with the unresolved disagreement with astronomic observations convinced most members of the community that Weber's original observations were spurious.

With efforts at direct observation of gravity waves stymied for the moment, indirect observations would have to do. This came in the form of the observation of orbital decay in a binary pulsar system by Taylor and Hulse\cite{taylor1979}. They used a 305\si{\meter} diameter radio telescope to observe the electro-magnetic emissions of the pulsar over time and deduce changes in the relative distance between the earth and the pulsar over a period of several years. They then fit these measurements to a model to find the orbital period of the binary pulsar.

Their results (including measurements made after their original publication) are shown in \fig{orbit_decay}. As the pulsars orbit each other, they emit quadrupole gravitational radiation which removes kinetic energy from the system. As a result the orbital period decreases with time. This is precisely what was observed by Taylor et al.\ and their observations matched the predictions of General Relativity remarkably well, and excluded other theories of gravitation that predicted gravitational dipole radiation.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/orbit_decay.png}
  \caption{The Fuckin Bullshit}
\label{fig::orbit_decay}
\end{figure}

\subsection{First Generation Interferometers}
Given the lack of reproducible results from the Weber bar experiments, experimentalists began investigating completely different types of detectors, the most promising of which was the laser interferometer. The earliest use of an interferometer to detect gravity waves was made by a former student of Weber named Robert Forward. Forward's detector consisted of 8.5\si{\meter} arms and through 150 hours of observation found no coincidences with the Weber bar detectors simultaneously in operation\cite{forward1978}.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/forward_schematic.png}
  \caption{A schematic of Robert Forward's early interferometer. Note the multiple paths the laser takes through each leg of the detector and the two photo-detectors, used to mitigate electronic noise.}
\label{fig::forward_schematic}
\end{figure}

The idea was not dead, however, and by the mid 1990s there were several large collaborations working on constructing long baseline interferometers. These included the GEO600 experiment in Germany, Virgo in Italy, and, of most relevance here, LIGO in the United States.

The inception of what would eventually become LIGO happened in the summer of 1975 when Rainer Weiss, an experimentalist at MIT, and Kip Thorne, a theorist from Caltech, met at a conference hosted by NASA to explore the applications of space-based research to cosmology and relativity. As Thorne did not have a hotel room, he shared one with Weiss who recalls that night,

\begin{displayquote}
``We made a huge map on a piece of paper of all the different areas in gravity. Where was there a future? Or what was the future, or the thing to do?''\cite{janna2016}
\end{displayquote}

Inspired by his conversation with Weiss, Thorne returned to Caltech and brought in an experienced experimentalist, Ronald Drever, to endeavour to develop interferometer technology at Caltech to detect gravity waves. For several years Weiss and Drever competed from opposite coasts to create better and better interferometers. Eventually both groups came to the conclusion that a discovery strength interferometer would have to be constructed at such a scale that attracting funding for separate experiments would be nearly impossible. Therefore, with some prodding by the NSF, the Caltech and MIT groups joined forces and formed the ``Laser Interferometer Gravitational-Wave Observatory'' (LIGO). 

Unfortunately, the triumvirate of Thorne, Weiss, and Drever proved unable to effectively manage the project so in 1986 the NSF instead appointed Rochus E. Vogt as the single project manager\cite{russel1992}. Despite this reshuffling of leadership, progress was slow and by 1994 Drever had left the project and Vogt was replaced with Barry Clark Barish\footnote{Born in Omaha, NE}, an high-energy experimentalist who had experience working in large collaborations. Under Barish's leadership, the original plan for LIGO was reworked into a two-stage deployment. The first stage, named Initial LIGO, or iLIGO, would include the construction of two laboratories, one in Hanford, Washington, and the other in Livingston, Louisiana. They would be built with current generation interferometers that would serve as a proof-of-concept and development platform for the second stage, known as Advanced LIGO.\@ Advanced LIGO would use the same facilities as iLIGO, but replace the interferometers with next-generation designs.

Barish also suggested splitting the experiment into two separate entities. The first would be responsible for the administration and operation of the laboratory facilities, and the other, the ``Ligo Scientific Collaboration'', would be in charge of scientific and technological research, as well as forging alliances with other collaborations, most notably GEO600 and Virgo.

Initial LIGO begin construction in 1995 and finished in 1997, however it still took until 2002 to begin taking scientific data, whereupon it operated in months long runs across eight years before ceasing operation in 2010, having not yet observed gravity waves. This was not unexpected as the intention had always been to use Initial LIGO as a research and development platform to design Advanced LIGO, and the odds of actually making an observation with Initial LIGO were small.

\section{The Advanced LIGO Detector}
\label{sec::ligo}
After The shutdown of Initial LIGO, efforts promptly began to install the upgraded systems of Advanced LIGO, and by February 2015\cite{davide2015}, the experiments began taking ``engineering mode'' data to commission the new systems. And finally, in September of that same year began taking scientific data.

\subsection{Principle of Operation}
LIGO is at its heart a Michelson style interferometer. This type of interferometer is displayed schematically in \fig{interferometer_schematic}. The interferometer works by first producing a coherent light source, typically a laser, and then splitting the source into two beams that take different paths. The beams are then recombined and made to interfere with each other. The resulting interference pattern can then be used to infer the difference in the path lengths taken by the two lasers.

If the two path lengths are exactly the same, the lasers will combine constructively and stimulate the sensitive element (e.g.\ a photo-diode) in the detector. However, as the path lengths diverge, the two beams will begin to interfere destructively and the resulting signal seen by the detector will decrease in amplitude until it disappears entirely when the path length differ by a half-wavelength.

In the case of LIGO, the nominal operating point is destructive interference at the detector, at the so-called ``dark fringe''. As a gravity wave passes through the detector, it causes one leg of the detector to lengthen and the other to contract. This differential change in length appears as a departure from perfect destructive interference, and the larger the amplitude of the wave, the larger the signal seen by the photo-detector.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/interferometer_schematic.png}
  \caption{Schematic representation of a Michelson interferometer. Image credit: Wikipedia}
\label{fig::interferometer_schematic}
\end{figure}

Unfortunately a simple Michelson interferometer would not be sensitive enough to detect the tiny deformations from gravity waves. The limitation comes from the Michelson interferometer requiring a differential deformation on the order of $\lambda$ to shift from fully constructive to fully destructive interference. A typical strain for the types of waves LIGO is searching for is $h\approx 10^{-21}$ which gives a corresponding deformation of $4\si{\kilo\meter}*10^{-21}=4*10^{-9}\si{\nano\meter}$ which is about twelve orders of magnitude smaller than $\lambda$. With such a large disparity, the resulting phase change would be too tiny to create a measurable deviation from the dark-fringe. Clearly, a way to get much larger phase shifts for an amount of deformation is needed.

Enter the Fabry-P\`{e}rot Interferometer, or more specifically as one end is almost a perfect mirror, the Gires-Tournois Interferometer\cite{gires1964}. In addition to the optical elements of the Michelson interferometer, this device adds an additional optical element in each leg between the beam splitter and the outer mirror. The cavity formed by the new element and the outer mirror is tuned to be resonant with the incident laser. The Advanced LIGO setup is shown in \fig{aligo_optics}.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.49\textwidth]{figures/aligo_optics.png}
  \caption{The Advanced LIGO optical configuration.\cite{ALIGO}.}
\label{fig::aligo_optics}
\end{figure}

The Fabry-P\`{e}rot resonance chamber in each leg is formed at one end by the end test mass (ETM) and at the other by the input test mass (ITM). The surface of the ETM is a nearly perfect mirror (transmission of 5 ppm), and the ITM has a reflection coefficient close to unity ($R=98.6\%$). This configuration provides an output signal whose phase is highly dependent on changes to the quantity \[\delta\equiv \frac{4\pi}{\lambda}nt\cos{\theta_t}.\] In the case of LIGO, $n$, the index of refraction in the chamber, is 1 since the laser is propagating through vacuum, $\lambda$, the wavelength of the laser is 1064 \si{\nano\meter}, and $\theta_t$, the angle of refraction is zero as the laser is normally incident. Assuming these variables are well controlled, $\delta$ will depend only on $t$, the length of the resonant cavity.


As a function of $\delta$, the (complex) reflectivity is \[r=-\frac{r_1 - e^{-i\delta}}{1-r_1e^{-i\delta}}\] where $r_1$ is the internal reflectivity of the incident surface. If we assume that there are no  losses in the resonator, $r_1$ is real and $|r|=1$. However, $r$ will still have a phase shift given by \[\tan\left(\frac{\Phi}{2}\right)=-\frac{1+\sqrt{R}}{1-\sqrt{R}}\tan\left(\frac{\delta}{2}\right).\] Where $R=|r_1|^2$. In reality, there are losses in the dielectric coatings of both the ETM and the ITM, but great care has been taken to make these losses small.

If we now rewrite $t$ in terms of the gravitational strain, $h$ \[t=t_0 + \Delta t=t_0(1+h),\] we get \[\Phi=-2\tan^{-1}\left(\frac{1 + \sqrt{R}}{1-\sqrt{R}}\tan{\frac{\delta}{2}}\right); \delta=4\pi\left(\frac{4\si{\kilo\meter}}{1064\si{\nano\meter}}\right)h\] To illustrate how this improves the sensitivity of the detector for small values of $h$, \fig{gt_etalon} shows the phase shift as a function of strain for different internal reflectivities. Note that the $R=0$ case corresponds to the normal Michelson interferometer where the ITM is transparent to the returning laser. As $R$ tends towards unity, a more and more non-linear response appears due to increasing interference of the beam with itself after multiple traversals of the interferometer leg. This means that much better sensitivity to small $h$ can be achieved by pushing $R$ towards 1.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/gt_etalon.png}
  \caption{Relations between $\Phi$ and $h$ for various values of internal reflectivity $r$.}
\label{fig::gt_etalon}
\end{figure}






\subsection{}

\section{Signal Extraction}
\label{sec::data}

\section{Observations}
\label{sec::observations}

\subsection{\GWA}

\subsection{\GWB}

\section{Cosmological Implications}
\label{sec::implications}

\section{Conclusions}
\label{sec::conclusion}

\begin{appendices}
  \section{App 1}
  \section{App 2}
\end{appendices}

\footnotesize
\bibliography{references}

\end{document}

% \subsection{From GR to Gravity Waves}
% Ever since the initial prediction of gravity waves

% For the purposes of describing the detection of gravity waves, I will start by introducing the \emph{stress-energy tensor} $T^{\alpha\beta}$ which satisfies the relation \[T^{\alpha\beta}\Sigma_\beta = P^\alpha.\] Where $P^\alpha$ is the total 4-momentum that flows through the a small 3-volume $\Sigma_\beta$. Now the question is: how does one interpret the components of $T^{\alpha\beta}$? Consider first the $P^0$ equation. This can be expanded as \[P^0 = T^{00} \Sigma_0 + T^{0i} \Sigma_i, \quad i\in\left\{1,2,3\right\}.\]

% If this is unclear, consider the 3-dimensional case where $\Sigma_i$ is a surface. This surface can equivalently be expressed as \[\Sigma_i=A\hat{n}.\] Where A is the area of the surface and $\hat{n}$ is the unit-normal to the surface.

% Where \[\Sigma_\beta\] is a 3-volume vector in 4-dimensional lorentz space. If this is unclear, consider that a 2-volume in 3-dimensional space, i.e. a rectangle, can be combined with a vector to form a 3-volume. Similarly, a 3-volume in 4-dimensional space can be combined with another vector(tensor) to create a scalar(tensor with reduced rank).

% $P^\alpha$ is the 4-momentum that flows through the spatial

% \begin{figure}[b]
%   \centering
%   \includegraphics[width=0.40\textwidth]{figures/interferometer_fringe_formation.png}
%   \caption{Image credit: Wikipedia}
% \label{fig::interferometer_fringe_formation}
% \end{figure}