ligo-comprehensive/comprehensive.tex

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\newcommand{\GWA}{\textsc{GW150914}}
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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.

\section{The History of Gravity Waves}
\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
\subsection{Early Efforts at Detection}

\section{The Advanced LIGO Detector}

\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 coherant 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.

\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}

%To be more concrete, consider \fig{interferometer_fringe_formation}.

% In the case of LIGO, the nominal operating point is perfect alignment with destructive interference at the sensor. This means that absent of gravity waves and detector noise, the detector sees no laser light. Differential length changes between the two detecting arms, such as those resulting from gravity waves, appear as deviations from this perfect destructive interference.

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 peak-to-peak phase change would be too tiny to create a measurable signal. 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 coefficent 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 propogating 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 $R=r_1^2$. 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, \fig{gt_etalon} shows the phase shift as a function of strain for different internal reflectivities. The $R=0$ case corresponds to the Michelson case where the ITM is transparent to the returning laser. As $R$ increases a more and more non-linear effect becomes apparent. 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}





% \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}

\subsection{}

\section{Signal Extraction}


\section{Observations}

\subsection{\GWA}

\subsection{\GWB}

\section{Cosmological Implications}

\section{Conclusions}

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

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