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%\title{CH.1 Intro: GWs and LIGO}
\date{}

\begin{document}
\chapter{Gravitational waves}
\label{ch: Ch.1}
The approach given in this thesis to introduce gravitational waves starts from the concept of gravity: gravity is not a force, but it is a property of the spacetime. This statement has been proved by General Relativity.\\
In this chapter we will briefly go through the reason why gravitational waves are an astrophysical phenomenon and how it is possible to detect them.\\
The purpose of the writer is to offer a contribution to the improvement of techniques of gravitational waves detection, extremely sophisticated in practice as well as elegant and straightforward in theory.

\section{Space, time and gravitation}
%RELATIVITA' GENERALE E EEs
%In the first years of XVIII century, the assumption that the fundamental laws of physics were invariant for all inertial observers was undisputed; however, they find hard applications in the electromagnetic theory: this led Einstein to develop the theory of relativity. He found that the Maxwell's equations imply that the electromagnetic fiends propagate in space as waves at the speed of light and that they are not invariant for all inertial observers in the Newtonian dynamics, but they are invariant in a frame were the space and time coordinates are linked together to be a unified system. Thus, for inertial observers moving with different relative velocities, space and time form a four-dimensional structure, called \textit{spacetime}.\\


%\noindent
The replacement of the Newtonian concepts of absolute space and absolute time with the merged concept of \textit{spacetime} introduced the Special Relativity\footnote{It is not in the aims of this work to demostrate the Theory of Relativity: a suggested detailed resource about the important transition between classical to relativistic physics is \cite{wei}.}. In this theory the light speed is assumed as preserved everywhere and every time for all interactions. However, gravitation seems not following this rule: the gravitational attraction seems to be instantaneous, or to propagate with \textit{infinite} speed.\\
%As well as the Newtonian space and time concepts have been modified to adapt to electromagnetic propagation
The Special Relativity then needs to be modified to adapt to gravity, taking into account its features: wherever there is matter, there is a gravitational effect, which is always present and acts as a radial inverse-square law. When trying to apply Newtonian gravitation to Special Relativity, the problem to solve becomes non-linear \cite{nar}; the key feature used by Einstein to adapt Special relativity to the presence of gravitation, is to consider that the permanence of gravitational effects makes gravity to be an intrinsic feature of space and time. Einstein's intuition marks the transition to the General Relativity: gravity is not anymore a force but an effect on the \textit{geometry} of the spacetime.\\

\noindent
General Relativity is then built on a non-Euclidean geometry able to explain the presence of gravity without considering it as a force: in such a geometry, matter under no force should move in straight like paths with uniform speeds.\\
This is possible if the spacetime is \textit{curved}.

\paragraph{Physics in a curved spacetime}
In General Relativity, the way to describe the physics in the curved spacetime modifies the usual Cartesian coordinate system used to measure the distance $s$ in space and time for two observers in an inertial frame:

\begin{equation}
\centering
ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2
\end{equation}

\noindent
with the following:

\begin{equation}
\centering
ds^2 = \Sigma_{i,k = 0}^{3}  g_{ik}dx^{i}dx^{k}
\label{a}
\end{equation}

\noindent
where the coordinates are now called $x^i$, with $i$=1,2,3 representing the three space coordinates and $i$=0 is the time coordinate\footnote{From now on we will omit the $\Sigma$ symbol, thanks to the summation convention of indeces: \textit{whenever an index appears as subscribt and superscript in the same expression, it is summed over all values}.} . The new entry is the coefficient $g_{ik}$, which is a series of functions of $x^i$. Eq. \ref{a} describes the geometry of the spacetime: its properties depend on $g_{ik}$, which transforms as a covariant tensor\footnote{A covariant vector is a quantity that transforms as $A'_{k} = \frac{\partial x^i}{\partial x^{'k}} B_i $. A tensor T is defined as the 4 $\times$ 4 product of two vectors and a covariant tensor transforms as $T'_{ik} = \frac{\partial x^m}{\partial x^{'i}}\frac{\partial  x^n}{\partial x^{'k}} T_{mn}$.} and it is called \textit{metric tensor} (a simple demonstration of the tensorial property of $g_{ik}$ can be found in \cite{nar}).\\

\noindent
%All physical interactions and how they behave in presence of gravitation can be described in this geometry in terms of vectors and tensors and the effect of gravitation enters in a condition of \textit{curved spacetime}:
The defined metric explains how to measure distances, while in order to define parallel vectors in a curved spacetime (and build equations of motion in such conditions), we need functions $\Gamma$ of space and time that can describe variations of components of a vector $B_i$ during a displacement $\delta x^k$ along the curve:

\begin{equation}
\centering
\delta B_i = \Gamma^{l}_{ik} B_l \delta x^k
\end{equation}

\noindent
that are determined once the metric tensor is known \cite{nar}:

\begin{equation}
\centering
\Gamma^{l}_{ik} = \frac{1}{2} g^{im} \left(\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{lm}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m}\right).
\end{equation}
\\
\noindent
The spacetime curvature is instead described by the Riemann tensor:

\begin{equation}
\centering
R_{ikn}^{m} \equiv \frac{\partial \Gamma^{m}_{ik}}{\partial x^n} - \frac{\partial \Gamma^{m}_{in}}{\partial x^n} + \Gamma^{l}_{ik}\Gamma^{m}_{ln} - \Gamma^{l}_{in}\Gamma^{m}_{lk} = 0.
\end{equation}
\\
\noindent
Spacetime is \textit{flat} if the Riemann tensor is null everywhere, otherwise it is \textit{curved}.\\
In this frame, the energy-momentum tensor of matter $T_{ik}$ is defined from the action principle and a variation of the metric over a volume V:

\begin{equation}
\centering
\delta \Sigma cm_a \int ds_a = \frac{1}{2c} \int_{V} \delta g_{ik} T_{ik} d^{4}x
\end{equation}
\\
\noindent
where m is the mass of a particle a.\\
Now that we have the elements describing geometry, displacements, the curvature of the spacetime and the energy distribution, physical interactions can be built on them. In particular, we are interested in the gravitational fields.

\paragraph{The Einstein's equations}
%NON DERIVARLE NEL DETTAGLIO, SPIEGA IL METODO EURISTICO
%SPIEGA LE APPROSSIMAZIONI PER MANIPOLARLE
The dynamical equations derivated by Einstein consider the energy-momentum tensor $T_{ik}$ acting as a source of gravity  propagating in the spacetime as a wave, having the form of series of wave equations for the metric $g_{ik}$ in presence of curvature\footnote{A heuristic derivation of the Eintein's equations can be found in \cite{nar}, while a derivation based on the non-linear effect of the gravitational fields can be found in \cite{wei}.}:

\begin{equation}
\centering
G_{ik} = R_{ik} - \frac{1}{2}g_{ik}R = \frac{8\pi G}{c^4}T_{ik}.
\label{EE}
\end{equation}
\\
\noindent
The Einstein's equations expressed in the form of eq. \ref{EE} are valid in the weak field (or Newtonian) approximation:
\begin{enumerate}
\item The motion of particles is non-relativistic.
\item The gravitational fields are weak: $g_{ik} = \eta_{ik} + h_{ik}$, where the term $\eta_{ik}$ = (-,+,+,+) expresses the flat-space metric and the perturbation is $\mid h_{ik}\mid$ $\ll$ 1.
\item The fields change slowly with time, so time derivatives are negligible with respect to space derivatives.
\end{enumerate}

\noindent
This approximation allows to handle the field equations in a linearized condition.

\paragraph{Gravitational-wave radiation}
%RISOLUZIONE DELLE EEs PER TROVARE L'EQ D'ONDA DELLE GWs\\
%SOLUZIONI DELL'EQ D'ONDA: COME SI PROPAGANO LE GW\\
The wave equations in the weak field approximation and under linearized conditions describe the gravitational radiation:

\begin{equation}
\centering
\square \bar{h}_{ik} = -\frac{16\pi G}{c^4}T_{ik},
\label{GW}
\end{equation}
\\
\noindent
where $\square = \eta_{ik}\partial ^{i} \partial ^{k}$ and $\bar{h}_{ik} = h_{ik} - \frac{1}{2}\eta_{ik}h$.\\
Physically, eqs. \ref{GW} say that bodies acting as sources of gravitational radiation move in a flat spacetime along a trajectory defined by their mutual influence. The gravitational field propagating in waves is known as \textit{Gravitational Waves}.\\
Solutions of the wave equations will tell how the gravitational waves propagate, how they interact with other bodies, the energy carried by the waves and the physical objects that could be gravitational waves sources.\\
If we consider what happens far from the source, i.e. when gravitational waves propagate, we need $T_{ik}$ = 0: in this way we just need to solve $\square \bar{h}_{ik}$ = 0. This gives a matrix-type form of the tensor when the wave propagates along the direction z\footnote{Full computations to derive the solutions of the wave equations for inspiriling masses can be found in \cite{mag}}:

\begin{equation}
\centering
h_{ab}^{TT}(t,z) =
\begin{bmatrix}
h_+ & h_\times & 0\\
h_\times & -h_+ & 0\\
0 & 0 & 0
\end{bmatrix}_{ab}\cos \left[\omega \left(t-\frac{z}{c}\right)\right].
\end{equation}
\\
\noindent
The TT index indicates a Transverse-Traceless condition of the solution. The two amplitudes of the gravitational wave depend on the mass of two objects at distance $r$ from each other \cite{mag}:

\begin{equation}
\centering
h_+ = \frac{4}{r} \left(\frac{GM_c}{c^2}\right)^{5/3} \left(\frac{\pi f_{gw}}{c}\right)^{2/3} \frac{1+\cos ^2 \theta}{2} \cos (2\pi f_{gw}t + 2\phi);
\end{equation}

\begin{equation}
\centering
h_\times = \frac{4}{r} \left(\frac{GM_c}{c^2}\right)^{5/3} \left(\frac{\pi f_{gw}}{c}\right)^{2/3} \cos \theta \sin (2\pi f_{gw}t + 2\phi)
\end{equation}

\noindent
where 0 $< \theta <$ $\pi/2$ and $f_{gw} = 2 \omega/(2\pi)$ is the frequency of the gravitational wave emitted. The mass of the two-system source of $m_1$ and $m_2$ each is defined as \textit{chirp mass}:

\begin{equation}
\centering
M_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}.
\end{equation}\\
\noindent
Typically, the amplitude of a gravitational wave is of the order of $h \sim 10^{-21} $ $1/\sqrt{Hz}$, very small: masses able to deform the fabric of the spacetime and generate gravitational waves are of the order of more than the solar mass $M_{\odot}$, so they need to be looked for in the Universe.
%servono masse molto grandi, quindi vanno cercate nell'Universo: possibili candidati

\paragraph{Sources of gravitational waves}
Fig. \ref{spec} summarizes the possible objects that can be gravitational waves sources, their frequency emission and what kind of instrument can detect them. The terrestrial interferometric detectors are the most involved at present times, but the efforts of the scientific community are going towards the development of new detectors both ground- and space-based in order to widen the frequency window of observation.

\begin{figure}[h!]
\centering
\includegraphics[scale=1.6]{images/spectrum}
\caption{Spectrum of emission of sources of gravitational waves (adapted from https://lisa.nasa.gov).}
\label{spec}
\end{figure}

\noindent
The best modelled sources are binary systems, typically Neutron stars (NS), White Dwarfs (WD) and Black Holes (BH), orbiting each other. Fig. \ref{binary} shows the main phases of the evolution of the systems, emitting gravitational waves at different frequencies, depending on the phase.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/bin.png}
\caption{The three phases of a BH-BH binary system emitting gravitational waves (amplitude vs time). \textbf{Inspiral phase}: the orbits shrink, velocity increases and frequency of the waves emitted increases as $f_{gw} = 2f_{orbital}$. \textbf{Merging phase}: the objects merge and the signal is maximum. \textbf{Ring-down phase}: a new BH is formed and the signal emitted decreases in frequency as a damped sinusoid.}
\label{binary}
\end{figure}

\noindent
Gravitational waves from binary systems can provide several information about the equation of state of Neutron stars, masses and spin of Black holes, test of General Relativity theory.\\
Currently, the ground-based observatories are tuned to detect binary systems sources: interferometers are the instruments that have been able to detect gravitational waves from binary systems.

\section{Interferometric detectors}
%COSA SONO GLI INTERFEROMETRI\\
%INTERAZIONE COL DETECTOR E LAVORO A POTENZA ZERO (DARK FRINGE)\\
The interaction of gravitation waves with two objects moving along the x axis produces effects on their distance $d = x_2 - x_1$:

\begin{equation}
\centering
s \simeq d \left(1 + \frac{1}{2}h_+ \cos \left[\omega \left(t-\frac{z}{c}\right)\right]\right).
\end{equation}
\\
\noindent
So the effect of the gravitational waves can be observed on the distance of the masses involved. A method is to measure the time it takes for light to travel from one mass to the other: this is the basic principle of the \textit{interferometer}.

\begin{figure}[h!]
\centering
\includegraphics[scale=1.2]{images/itf.png}
\caption{Basic features of an interferometer.}
\label{itf}
\end{figure}

\noindent
As shown in Fig. \ref{itf}, an interferometer is an instrument where a laser beam of wavelength $\lambda$ is split into two beams which propagate in two perpendicular arms of the same length. At the end of each arm, a mirror reflects the beam back to be recombined with the other one. The recomposed beam is then deviated to a power sensor.\\
If we consider the length of arms oriented to the x and y directions to be $L_x = L_y = L$, the power measured depends on the difference of path length travelled by the two beams:

\begin{equation}
\centering
P_{out} = E^{2}_{0} \sin^2 [k(L_x - L_y)] 
\end{equation}

\noindent
where $E^{2}_{0}$ is the amplitude of the electric field generated by the laser source and k = $2\pi / \lambda$.\\
We know that the effect of a gravitational wave is to modify the distance of two masses: in the case of the interferometer the path length difference in the arms is proportional to the gravitational wave amplitude $h$ \cite{mag}:

\begin{equation}
\centering
\Delta L = \frac{1}{2}hL
\end{equation}

\noindent
and so the key feature of this detector is that the beam coming from the recombination of the two ones that travelled in the arms brings a phase difference:

\begin{equation}
\centering
\Delta \phi = \frac{4\pi h L}{\lambda}
\end{equation}

\noindent
which results in a variation of the power measured:

\begin{equation}
\centering
P_{out} = E^{2}_{0} \sin^2 [k(L_x - L_y) + \Delta \phi].
\end{equation}

\noindent
The amplitude of a gravitational wave is typically very small and corresponds to a variation of the arm length of the order of $\Delta L \sim 10^{-18}$ m. This means that, if we want to measure a considerable phase shift, the sensitivity of the instrument depends on the length of the arms.

\paragraph{Fabry-Perot cavities}
A useful method to increase the length of the arms is to make the laser beam travel back and forth into a cavity delimited by two mirrors, called \textit{Fabry-Perot cavity}: here, the optical path length will be longer, due to the multiple reflections, and the field amplitude will increase due to constructive interference. This process returns a longer arm length, proportionally to the quality factor of the cavity, which depends on the reflection coefficients of the two mirrors and it is called \textit{Finesse} (F):

\begin{equation}
\centering
L_{F} \propto L \frac{2 F}{\pi}
\end{equation}

\noindent
which gives a phase shift:

\begin{equation}
\centering
\Delta \phi_{F} = \frac{8hFL}{\lambda}.
\end{equation}

\noindent
The higher is F, the higher is the effective length of the cavity and higher is the measureble phase shift.\\

\noindent
The first detection of gravitational waves happened on the 14th September 2015 and confirmed the Theory General Relativity, opening a new window on the Universe: among others, black holes have been observed thanks to their emission of gravitational waves, confirming the existence of these object, still mostly unknown \cite{first}. The detector responsible of the new discovery is based in the USA and it is one of the terrestrial interferometers currently in use for gravitational waves detection.

%\section{LIGO}
%The ambition of this work is to give a contribution to the improvement of one of the interferometric detectors in use at present time, based in the USA: the Advanced Laser Interferometric Gravitational-wave Observatory (aLIGO).\\
%The configuration of aLIGO is shown in Fig. \ref{aligo}: it is a Michelson interferometer provided with Fabry-Perot cavities, power and signal recycling cavities and 4 km-long arms. The light source is a solid-state Nd:YAG laser of wavelength $\lambda$= 1064 nm, injected at a power between 5 - 125 W.\\ The instrument design is extremely intricate in its details: this thesis will provide technical information useful for the understanding of the work made on specific sections of LIGO.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.9]{images/aligo.png}
%\caption{Advanced LIGO configuration as proposed in \cite{ligo}.}
%\label{aligo}
%\end{figure}
%
%\noindent
%The fused silica mirrors at the end of each arm, called End Test Masses (ETM), are 34 cm $\times$ 20 cm in size and 40 kg in weight. A photodiode (PD) detects the power at the output. The optic able to split the injected beam into two parts along the arms is called Beam Splitter (BS) and it is placed at 45$^{\circ}$ between the arms. There are two LIGOs in the USA, one in Hanford (WA) and one in Livingston (LA): some of the work that will be presented in the next chapters has been physically done in Hanford, in remote collaboration with Livingston team.
%
%\paragraph{LIGO sensitivity and noise sources}
%The performance of LIGO in terms of how far in the Universe it can detect gravitational waves and from which sources depends on the sensitivity: this in turn depends on the quality of the technologies involved and on the limitation given by nature.
%Fig. \ref{sens} shows the sensitivity of LIGO during the first observation run with the main noises shown.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.7]{images/ligosens.png}
%\caption{Advanced LIGO sensitivity during the first observation run (O1) \cite{abb}. The sensitivity curve tells us that we can observe an event emitting gravitational waves of a given amplitude at a given frequency in an average observation time of 1 s. Since every source emits waves at a certain frequency and amplitude, lowering the curve means opening the viewing on currently hidden sources.}
%\label{sens}
%\end{figure}
%
%\noindent
%Advanced LIGO can be tuned to adjust the frequency band of detection: for each operational mode and detection frequency band there is a gravitational wave source candidate, typically mergers of neutron stars (NS-NS) and black holes (BH-BH).\\
%Noise sources make LIGO blind in some frequency windows: technological limitations can be in principle overcome thanks to improvements in science, and this is what this present work is aiming to offer. The most important noise sources for LIGO are shown in the noise budget for LIGO Hanford (LHO) in Fig. \ref{lho}.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=1]{images/LHO.png}
%\caption{Noise budget of LIGO Hanford Obsevatory \cite{mar}.}
%\label{lho}
%\end{figure}
%
%\noindent
%Noises can be of fundamental, technical and environmental origin. Fundamental noises come from first principles, and they determine the ultimate design sensitivity of the instrument. They include thermal and quantum noise, and cannot be reduced without a major instrument upgrade. Quantum noises include shot noise of the sensors, causing power fluctuations, and radiation pressure forces, causing a physical displacement of the test masses. Thermal noise arises from the suspensions and the optical coatings and dominates in the 5-100 Hz frequency range.\\
%Technical noises arise from electronics, control loops, charging noise and other effects that can be reduced once identified and carefully studied.\\
%Environmental noises include seismic motion, acoustic and magnetic noises. This thesis focuses on the improvement of the seismic isolation system, which noises affect the inertial sensors placed on the suspension benches.
%
%\subsection{LIGO seismic isolation system}
%Every optic needs to be stable with respect to seismic motion, because movements in the mirrors will cause unwanted displacement of the laser beam on the optical surface, resulting in noise during the laser journey into the cavities and then at the output. The main mirrors (test masses and beam splitter) are suspended from a stabilized bench and every suspension chain is placed in vacuum chambers called \textit{Basic Symmetric Chamber} (BSC). The auxiliary optics are placed on optical benches enclosed in the \textit{Horizontal Access Module} (HAM) chambers.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=1]{images/chambers.png}
%\caption{Schematic view of the vacuum chambers enclosing the optics \cite{mat}. There are 5 BSCs and 6 HAMs, for a total of 11 vacuum chambers for each LIGO. Each chamber provides a mixture of passive-active isolation from seismic motion, using pendulums, inertial sensors and hydraulic systems.}
%\end{figure}
%
%\noindent
%The HAMs provide five levels of isolation, among which there is the Internal Seismic Isolation platform (HAM-ISI), where the auxiliary optics are placed, giving both passive and active isolation. A detailed drawing in Fig. \ref{ham} shows the design of a HAM chamber. The control system of the ISI 
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=1]{images/HAM.png}
%\caption{Schematic (a) and CAD model (b) of a HAM chamber \cite{mat}. Suspensions of auxiliary optics provide levels of passive isolation above 10 Hz. The ISI platforms where the suspensions live are optical tables actively isolated via low noise inertial sensors at low frequency ($\sim 0.1 Hz$). The hydraulic attenuators of the \textit{Hydraulic External Pre-Isolator} (HEPI) and the geophones gives isolation from ground motion.}
%\label{ham}
%\end{figure}
%
%\noindent
%The BSCs have a similar design as the HAMs, but they have two stages of ISI to support the suspensions isolating the test masses (Fig. \ref{bsc}).
%
%\begin{figure}
%\centering
%\includegraphics[scale=1]{images/BSC.png}
%\caption{Schematic (a) and CAD model (b) of a BSC chamber \cite{mat}. The active isolation is similar to the one exposed for HAM chambers. The two ISIs provide two stages of isolation while and the suspensions  are design to be quadruple pendulums, for a total of seven levels of isolation.}
%\label{bsc}
%\end{figure}
%
%\paragraph{Stabilizing the ISI}
%Part of the work presented in this thesis focussed on the improvement of the performances of the active isolation system of the ISIs of both BSC and HAM chambers.\\
%Active isolation implies a sensing system of the noise to reduce and a control system to compensate the disturbance. Each platform includes relative position sensors, inertial sensors and actuators, working in all degrees of freedom.\\
%The control loop of a generic ISI stage on the X degree of freedom is  simplified in the block diagram in Fig. \ref{control}. The platform motion is the sum of the input disturbance and the contribution from the control signal and it is measured by relative position and inertial sensors; then this motion is low- and high-passed via filters suitably built to fit the requirements and tuned to obtain the best performances combining the best results of both filters: this technique is called \textit{blending}, and the frequency where the relative and the inertial sensors contribute at their best is called \textit{blend frequency}. The result of this blend is called \textit{super sensor}. The output of the super sensor feeds the feedback loop, where the actuators close the loop \footnote{A general overview of control loops theory is exposed in Appendix B}.\\
%The sensor correction loop takes the ground motion signal from an inertial instrument, filtering it before adding it to the relative sensor signal. This filter is needed because the sum of the motions from the ground inertial and the relative sensors can in principle provide a measurement of the absolute motion of the platform. However, the ground sensors are affected by low frequency noise and need to be suitably filtered. 
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.7]{images/control.png}
%\caption{Control loop of a generic HAM-ISI platform. Similar block diagrams can be applied for BSC-ISI platforms, including relative position sensors between the two stages of ISIs. \textbf{Green:} there is an inertial sensor measuring the ground motion along the x axis (GNDx), a Capacitive Position Sensor (CPS) measuring relative motions between the platform and the ground. Rotational sensors take care of tilt motion and GS13 are seismometers measuring seismic motion. Tilt and GS13 sensors are both placed on the platform. \textbf{Blue:} the Sensor Correction (SC) filter is typically a Finite Impulse Response (FIR) designed to provide required magnitude and phase match at 100 mHz (where isolation is needed). High- and low-pass filters (LP and HP) manipulate the signals from the low and high frequency sensors and are blended to form the super sensor, which output is sent to the control loop in \textbf{pink}. The overall corrected signal is then sent to the plant (\textbf{yellow}), which represents the processing phase for platform motion actuation.}
%\label{control}
%\end{figure}
%
%\subsection{Length Sensing and Control}




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\bibitem{nar} J. V. Narlikar \textit{An introduction to Relativity}, Cambridge University Press, 2011

\bibitem{mag} M. Maggiore \textit{Gravitational waves - Vol. 1: Theory and Experiments}, Oxford University Press, 2013

\bibitem{first} B. P. Abbott, \textit{Observation of Gravitational Waves from a Binary Black Hole Merger}, Phys. Rev. Lett. 116, 061102, 2016 

\bibitem{ligo} Advanced LIGO Systems Group, \textit{Advanced LIGO Systems Design}, DCC document T010075-v3, 2017

\bibitem{abb} B. P. Abbott et al, \textit{GW150914: The Advanced LIGO Detectors in the Era of First Discoveries}, Phys. Rev. Lett. 116, 131102, 2016

\bibitem{mar} D. Martynov et al., \textit{The Sensitivity of the Advanced LIGO Detectors at the
Beginning of Gravitational Wave Astronomy}, ...

\bibitem{mat} F. Matichard et al, \textit{Seismic isolation of Advanced LIGO: Review of strategy, instrumentation and performance}, Class. Quantum Grav. 32 185003, 2015

\end{thebibliography}

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