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 continuing 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 represents 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 an inertial frame. 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 be 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 \SI{180}{\degree} rotation around the axis of propagation. 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 \SI{45}{\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 he 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 \SI{305}{\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 in 1979) 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 decay of the orbital period of PSR1913 due to the emission of gravity waves}\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 for direct detection, 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 (\fig{forward_schematic}) consisted of \SI{8.5}{\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 decided that the thing to do at Caltech was to develop interferometers to detect gravity waves, and to accomplish this, he brought in an experienced experimentalist, Ronald Drever, to construct the detectors. 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 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, or at least the distance modulo the wavelength of the laser.

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.

It may be tempting to think that just as the space between the test masses (the splitter and the mirrors) gets stretched and squeezed with the passing of the gravity wave, so does the wavelength of the laser pulse leading to no observable change in the interference pattern. It turns out that this is not the case. Although the laser's wavelength \emph{does} get stretched and squeezed by the passing gravity wave, the effect is approximately ten-thousand times smaller than the effect on the test masses, well below the noise floor of LIGO.\@ This is due to the fact that the spatial extent of LIGO ($\approx \SI{4}{\kilo\meter}$) is much smaller than the wavelength of the gravity waves being searched for ($\approx \SI{3000}{\kilo\meter}$).

\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, defined as $h\equiv \frac{\Delta l}{l}$, for the types of waves LIGO is searching for is $h\approx 10^{-21}$ which gives a corresponding deformation of
$\SI{4}{\kilo\meter}*10^{-21}=\SI{4E-9}{\nano\meter}$ 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 \fp{} 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 \fp{} 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 \[\delta\equiv \frac{4\pi}{\lambda}nL\cos{\theta_L}.\] 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 \SI{1064}{\nano\meter}, and $\theta_L$, the angle of refraction is zero as the laser is normally incident. Assuming these variables are well controlled, $\delta$ will depend only on $L$, 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 ITM.\@ 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\footnote{The drastically increased circulation power in Advanced LIGO, however, causes appreciable heating of the optics, leading to thermal lensing. Compensating heaters are added to the lenses to mitigate this effect. The fractional losses are still small enough, however, to make the following calculations valid.}.

If we now rewrite $L$ in terms of the gravitational strain, $h$ \[L=L_0 + \Delta L=L_0(1+h),\] we get \[\Phi=-2\tan^{-1}\left(\frac{1 + \sqrt{R}}{1-\sqrt{R}}\tan{\frac{\delta}{2}}\right); \delta=4\pi\frac{\SI{4}{\kilo\meter}}{\SI{1064}{\nano\meter}}(1+h)\] 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. Of consequence to the following discussion is the quantity known as \emph{finesse}. It can be defined as \[\mathcal{F}_c\equiv \frac{2\pi}{-\ln\left(R_1R_2\right)},\] where $R_1$ and $R_2$ and the intensity reflectivities of the mirrors on either end of the \fp{} cavity. Qualitatively, as the product of the reflectivities approaches unity, the finesse becomes large. In practice the finesse is adjusted to balance the power stored in the cavity, and the sharpness of the transmission/reflection curves as a function of frequency.

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

\subsection{The Laser Source and Input Optics}
LIGO employs a multi-stage neodymium-doped yttrium aluminum garnet (Nd:YAG) laser that can supply up to \SI{180}{\watt} of power to the interferometer. The laser source together with initial stabilizing and cleaning optics are referred to as the pre-stabilized laser (\fig{prestabilized_laser}).


The laser cavity produces radiation with a nominal wavelength of \SI{1064}{\nano\meter}, and frequency power distribution as shown in \fig{mode_power}.

\begin{figure}[h]
  \centering

  \begin{subfigure}[b]{0.5\textwidth}
    \includegraphics[width=\textwidth]{figures/mode_power.png}
    \subcaption{Mode scan of the laser used at the Livingston Detector demonstrating that nearly \SI{95}{\percent} of the laser's energy is in the $\mathrm{TEM}_{00}$ mode. The horizontal scale is in units of free spectral range (FSR) which is defined as $\frac{c}{2L}$ where $L$ is the length of the resonant cavity.}\label{fig::mode_power}
  \end{subfigure}

  \begin{subfigure}[b]{0.5\textwidth}
    \includegraphics[width=\textwidth]{figures/laser_modes.png}
    \subcaption{Intensity distributions of transverse laser modes for a cylindrically symmetric beam. Image Credit: Wikipedia}\label{fig::laser_modes}
  \end{subfigure}
\end{figure}


\begin{figure*}
  \centering
  \includegraphics[width=0.80\textwidth]{figures/prestabilized_laser.png}
  \caption{Schematic of the pre-stabilized laser system.  AOM:\@ acousto-optic modulator, EOM:\@ electro-optic modulator, FI:\@ Faraday isolator, PD:\@ photodetector}\label{fig::prestabilized_laser}
\end{figure*}

To understand this figure, first note that the radiation produced by a laser cavity can be decomposed into transverse modes (see \fig{laser_modes}). The presence of higher-order transverse modes in the beam is a problem because, due to the Gouy phase shift, the frequencies of these modes are higher than the frequency of the $\mathrm{TEM}_{00}$ mode. Since the interferometer is extremely sensitive to changes in laser frequency, it is therefore crucial to remove these other modes as much as possible. To that end, the beam is passed through the pre-mode-cleaner (PMC). This is a bow-tie cavity that strips higher-order modes from the beam. For example, it reduces the amplitudes of the $\mathrm{TEM}_{01}/\mathrm{TEM}_{10}$ modes by a factor of 63. It also serves to provide low-pass filtering of RF intensity fluctuations.

After passing through the PMC, the beam goes through a RF modulator which adds low-amplitude phase modulation at \SI{9}{\mega\hertz}, \SI{45}{\mega\hertz}, and \SI{24}{\mega\hertz}.  The 9 and \SI{45}{\mega\hertz} modulations are used for global sensing of the interferometer, while the \SI{24}{\mega\hertz} modulation is used for input-mode-cleaner (IMC) sensing.  These side bands are used by the Pound-Drever-Hall\cite{drever1983} stabilization method.  For a thorough description of the technique see~\cite{black2001}.  Pound-Drever-Hall can be used for both laser frequency stabilization and \fp{} cavity length stabilization. In fact, LIGO uses variations of the technique for both.

To illustrate the Pound-Drever-Hall method for laser frequency stabilization, consider a laser shining into a \fp{} cavity with identical mirrors on either end. The laser is equipped with an actuator that can deform the lasing cavity to slightly alter the laser's frequency allowing for active correction of deviations from the ideal frequency. The transmission function of the cavity is given by \[T_e=\frac{T^2}{1+R^2-2R\cos\left(\frac{2L}{c}\left(\omega_0+\Delta \omega\right)\right)}\quad,\] where $R$ and $T$ are the reflection and transmission coefficients of the cavity mirrors, $L$ is the length of the cavity, $\omega_0$ is the ideal, resonant, frequency in the cavity, and $\Delta\omega$ is a deviation from that ideal.

Of more interest for Pound-Drever-Hall is the reflectivity (\fig{pdh_demo}) given simply by \[R_e=1-T_e.\] Now, a detector can be placed on the upstream side of the cavity to sense reflected light. If the detector sees anything, one can infer that the detector is off resonance, and the laser frequency has deviated from its ideal value\footnote{Note that this is in contrast with the LIGO arm cavities where the intensity of the reflected light is independent of laser frequency or arm length. In this case, the equation for $T_e$ does not apply since the two ends have differing reflectivities. The applicable error signal in this case is the light shining into the dark port.}. Unfortunately, the error signal is symmetric about the resonance so the actuator in the lasing cavity does not know which way to adjust the laser to correct the error. Pound-Drever-Hall solves this by introducing a small phase modulation to the laser to operate the cavity slightly off resonance.

The modulated error signal then sinusoidally traces the reflection curve in the area highlighted by the inset of \fig{pdh_demo}. If the frequency is too high, the error will trace up and down the curve on the right side of the minimum so the error signal will be in phase with the modulating signal implying that the product of the error and modulation signal will be positive. On the other hand, if the frequency is too low, the signals will be half a period out of phase so the product will be negative. Therefore, by simply observing the sign of the product of the error signal and the modulating signal, the control system knows whether to adjust the frequency up or down.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.49\textwidth]{figures/pdh_demo.png}
  \caption{The reflection coefficient of a \fp{} cavity as a function of $\Delta \omega$, i.e.\ deviations in laser frequency.}\label{fig::pdh_demo}
\end{figure}

In a similar way, if the laser frequency is assumed stable, but the length of the cavity is varying, the same scheme can be applied with the only substantive difference being the actuator now applies a force to one or both of the cavity mirrors instead of the lasing cavity. This is the method used to keep all of the LIGO optical resonant cavities on their working point.

After modulation, the beam passes through the final stage of the pre-stabilized laser system, the input-mode-cleaner. The input-mode-cleaner is used to further fix the laser's mode content, as well as stabilize the beam's position as it shines onto the power recycling mirror (PRM). Using the \SI{24}{\mega\hertz} modulation and the Pound-Drever-Hall locking technique, it is able to help stabilize the frequency of the laser to $<\SI{1E-3}{\hertz\per\sqrt\hertz}$ at \SI{100}{\hertz}.

Finally, before being passed to the core optics of the interferometer, the beam passes through a Faraday Isolator which prevents light that is returning from the core optics from creating parasitic effects in the input chain.

\subsection{Core Optics}

The output of the PSL is routed into the core interferometer optics. These consist of one ETM and one ITM for each leg. Adjacent to each ITM, there is also a compensation plate (CP) which used used as a reaction mass for actuating the ITM.\@ Similarly, there is a end reaction mass for actuating the ETM.\@

The actuation of these optics is accomplished by applying a circular thin gold plating on the reaction masses that can be made to hold an electro-static charge by application of a voltage. This charge will push or pull on the dielectric test mass. The plating is divided into quadrants to also allow for aiming the test mass. Initial LIGO actually affixed magnets to the test masses and applyied forces to the magnets via magnetic fields induced by coils mounted on the reaction masses. The electro-static method is prefered because mounting magnets on the test masses modifies their natural vibration modes, leading to additional thermal noise.

To round out the core optics, there is a 50/50 beamsplitter, four curved mirrors for signal (SR2, SR3) and power (PR2, PR3) recycling as well as the partially transmitting power recycling mirror (PRM) and signal recycling mirror (SRM). Parameters for all these core optics, including dimensions, mass, transmission rates, radii-of-curvature, and beam size on the optic, are shown in \tab{optics}.

\begin{table*}
  \small
  \centering
  \begin{tabular}{llllll}
    \toprule
Optic & Diameter\ $\times$ thickness (\si{\centi\meter}) & Mass (\si{\kilo\gram}) & Transmission & ROC (\si{\meter}) & Beam size (\si{\centi\meter}) \\
    \midrule
    ITM & 34   $\times$ 20  & 40  & 1.4\% (0.5\--2\%) & 1934  & 5.3 \\
    ETM & 34   $\times$ 20  & 40  & 5 ppm (1\--4\%)   & 2245  & 6.2 \\
    CP  & 34   $\times$ 10  & 20  & AR < 50 ppm       & flat  & 5.3 \\
    ERM & 34   $\times$ 13  & 26  & AR < 1000 ppm     & flat  & 6.2 \\
    BS  & 37   $\times$ 6   & 14  & 50\%              & flat  & 5.3 \\
    PR3 & 26.5 $\times$ 10  & 12  & <15 ppm           & 36.0  & 5.4 \\
    SR3 & 26.5 $\times$ 10  & 12  & <15 ppm           & 36.0  & 5.4 \\
    PR2 & 15   $\times$ 7.5 & 2.9 & 225 ppm (>90\%)   & -4.56 & 0.62 \\
    SR2 & 15   $\times$ 7.5 & 2.9 & <15 ppm           & -6.43 & 0.82 \\
    PRM & 15   $\times$ 7.5 & 2.9 & 3.0\%             & -11.0 & 0.22 \\
    SRM & 15   $\times$ 7.5 & 2.9 & 20\%              & -5.69 & 0.21 \\
    \bottomrule
  \end{tabular}
  \caption{Parameters for the core optics.\@ All transmission values are at \SI{1064}{\nano\meter}, except for those in parentheses which are for \SI{532}{\nano\meter}. AR:\@ anti-reflection}\label{tab::optics}
\end{table*}

One of the key improvements from Initial LIGO to Advanced LIGO was the addition of the signal recycling system. This can be seen in \fig{aligo_optics} as the elements SR2, SR3, and SRM.\@ This serves to reshape the response of the interferometer to be more sensitive to signals in the low-audio frequency band that would be come from binary neutron star or black hole coalescences\cite{meers1988}. The parameters of the signal recycling subsystem can be adjusted to enhance other frequency bands as well.

\subsection{Environmental Isolation}
A great disadvantage of terrestrial experiments is that they are subject to the whims of nature, be it turbulent weather, the rumblings of the earth, or, in the case of the Livingston experiment, particuarly heavy-footed alligators\footnote{not measurable}. In fact, the ground motion at the sites of the two LIGO detectors is measured to be \texttildelow\SI{E-6}{\m\per\sqrt\hertz} while the pertubations from gravity waves are \texttildelow\SI{E-18}{\m\per\sqrt\hertz}. This twelve order of magnitude difference emphasizes the importance, and the difficulty, of isolating the experiment from environmental vibrations.

The environmental isolation system is divided into two parts. The subsystem closest to the ground is referred to as the seismic isolation system, and the subsystem between that and the test masses is the suspension system.

The seismic isolation system, shown in some detail in \fig{seismic_iso}. It consists of first the Hydraulic External Pre-Isolator (HEPI) system that uses a blend of geophones and inductive position sensors together with hydraulic actuators to actively damp low frequency (\SI{0.1}{\hertz}-\SI{10}{\hertz}) vibrations. This supports the Internal Seismic Isolation (ISI) system. The ISI sits inside vacuum and consists of three stages (labeled in the figure as Stage 0, Stage 1, and Stage 2) are sequentially suspended and sprung from each other. The stages are instrumented with capacitive position sensors and controlled with electromagnetic force actuators. The stage 2 structure includes an optics table from which the optical elements are suspended.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.49\textwidth]{figures/seismic_iso.png}
  \caption{}\label{fig::seismic_iso}
\end{figure}

All of the in-vacuum core optics are mounted on elaborate multi-stage suspension systems of various designs based on noise requirements. \tab{suspensions} lists these noise requirements along with number of spring-based vertical isolation stages, pendulum stages, and suspension wire types. An example of all of these things put together, in this case for the ITM, is shown in \fig{itm_suspension}.


A pendelum suspension is a wonderful choice for passive filtering of environmental vibrations due to the property that above its resonance frequency, $\omega_0$, it supresses noise by a factor of $\omega_0^2/\omega^2$. And this property can be chained by hanging peldula from other pendula. For the test masses, there are four pendula stages, yielding filtering $\propto \omega^{-8}$ for $\omega$ greater than all the resonance frequencies of the system. Unfortunately, this only serves to damp horizontal motion, and, although horizontal noise in the test masses causes the most direct damage to signal quality, vertical and angular misalignment also contribute. To provide for isolation from vertical noise, 

\begin{figure*}
  \centering
  \includegraphics[width=0.7\textwidth]{figures/itm_suspension.png}
  \caption{Illustration of the suspension system for the ITM showing the three vertical isolation stages and four pendulum stages. The center image shows the so-called ``earthquake-stop'', also known as the catcher structure.}\label{fig::itm_suspension}
\end{figure*}

% At low frequencies a potentially dominant noise contribution comes from seismic vibrations. To isolate the core optics from this noise, they have all been mounted in elaborate multi-stage suspension systems. These systems are divided into two parts, the seismic isolation subsystem and the suspension subsystem.



\begin{table*}
  \small
  \centering
  \begin{tabular}{lllll}
    \toprule
                      & Vertical       & Pendulum  & Final stage  & Longitudinal noise  \\
    Optical component &  iso.\ stages  & stages    & fibre type   & requirement @ 10 Hz (\si{\meter\per\sqrt\hertz}) \\
    \midrule
    Test masses (ITM, ETM)             & 3 & 4 & Fused silica & $1\times 10^{-19}$ \\
    Beamsplitter (BS)                  & 2 & 3 & Steel wire   & $6\times 10^{-18}$ \\
    Recycling cavity optics            & 2 & 3 & Steel wire   & $1\times 10^{-17}$ \\
    Input mode cleaner (IMC) optics    & 2 & 3 & Steel wire   & $3\times 10^{-15}$ \\
    Output mode cleaner (OMC) assembly & 2 & 2 & Steel wire   & $1\times 10^{-13}$ \\
    ETM transmission monitor           & 2 & 2 & Steel wire   & $2\times 10^{-12}$ \\
    \bottomrule
  \end{tabular}
  \caption{Suspension parameters for the core optics.}\label{tab::suspensions}
\end{table*}

\section{Signal Extraction}
\label{sec::data}
The raw signals coming from the photo-diodes at the anti-symmetric port must be cleaned of all possible detector effects before being analyzed for the presence of gravity waves. This is done by a running the signal through a sophisticated response function that uses information from the monitor and control system to remove contributions from, for example, the actuation of the ETMs in response to previous misalignments or motion of the optics due to fluctuations in radiation pressure. With these effects removed, the signal is promptly
\subsection{Noise Sources}

\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}