added GWDA tutorial

tuturials/gw_tutorials.tex

@@ -168,12 +168,12 @@ A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equ
 \begin{itemize}
 	\item In the class we have derived 
 	\begin{equation}\label{hij_quad}
-		\left[h_{ij}^{TT}\right]_{quad} = \frac{1}{r}\frac{2G}{c^4}\Lambda_{ij,kl}(\hat{n})\ddot{M}^{kl}\left(t-\frac{r}{c}\right)
+		\left[h_{ij}^\mathrm{TT}\right]_\mathrm{quad} = \frac{1}{r}\frac{2G}{c^4}\Lambda_{ij,kl}(\hat{\mathbf{n}}) ~ \ddot{M}^{kl}\left(t-\frac{r}{c}\right)
 	\end{equation}
-    where $$M^{ij} = \frac{1}{c^2}\int \rho\left(t,\vec{x}\right) x^i x^j d^3\vec{x}$$
+    where $$M^{ij} = \frac{1}{c^2}\int \rho\left(t,\mathbf{x}\right) x^i x^j d^3 \mathbf{x}$$
 \end{itemize}
 \begin{enumerate}
-	\item\label{p1-5} Consider GW travelling in a generic direction $\hat{n}$ where $n_i = \left(sin\theta cos\phi,sin\theta sin\phi, cos\theta\right)$. Compute the angular distribution for the quadrupole radiation (i.e $h_{+}(\theta,\phi)$ and $h_{\times}(\theta,\phi)$)  for given ${M_{ij}}$ 
+	\item\label{p1-5} Consider GW travelling in a direction $\hat{\mathbf{n}}$ where $n_i = \left(\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta\right)$. Compute the angular distribution for the quadrupole radiation (i.e $h_{+}(\theta,\phi)$ and $h_{\times}(\theta,\phi)$)  for given ${M_{ij}}$ 
 	\item Consider a binary system of mass $m_1$ and $m_2$, and assume circular motion in relative coordinate (neglect back radiation on motion due to GW). Compute $h_+$ and $h_{\times}$ (use result of \ref{p1-5}), and radiated power. Plot the angular distribution of the radiated power. 
 	\item Consider a mass $m$ is performing simple harmonic motion along $z$ axis. Compute $h_{+}$ and $h_{\times}$ and power radiated. Plot the angular distribution of emitted power. 
 	\item Discuss the physical differences in the two radiation pattern obtained in Problem 2 and 3. Discuss the symmetry of the radiation pattern for both cases.
@@ -182,7 +182,7 @@ A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equ
 \begin{enumerate}
 	\item link to mathematica notebook \href{http://gitlab.icts.res.in/ajith/gwcourse2023/tree/master/tuturials}{http://gitlab.icts.res.in/ajith/gwcourse2023/tree/master/tuturials}
 \end{enumerate}
-\subsection{Tutorial 7}
+\subsection{Tutorial 7: Astrophysics of GW sources}
 \begin{enumerate}
 	\item Assume that all the spin down of pulsar is due to GW emissions, compute the strain amplitude of all pulsars in the \href{https://www.atnf.csiro.au/research/pulsar/psrcat/}{ATNF} catalog, using the $f$ and $\dot{f}$ values from the catalog.
 	\item Plot the amplitude of the GW signals as a function of the Fourier frequency for the following continuous sources of GWs.
@@ -196,6 +196,30 @@ A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equ
 	\end{itemize}
 \end{enumerate}
 
+\subsection{Tutorial 8: GW data analysis}
+
+In the case a known signal $h(t)$ buried in stationary Gaussian noise, the optimal technique for 
+signal extraction is the \emph{matched filtering}, which is a noise-weighted correlation of the data 
+with a signal {template}. If the noise is white, this is just a simple cross correlation.  The correlation function between two real valued time series $x(t)$ and $\hat h(t)$ is:
+\begin{equation}
+\label{eq:corrFn}
+R(\tau) = \int_{-\infty}^{\infty}x(t)\,\hat{h}(t+\tau)\,dt. 
+\end{equation}
+Above, $\hat{h(t)} := h(t) / ||h||$, where the norm $||h||$
+of the template is defined by $||h||^2 = \int \, |h(t)|^2/\sigma^2 \, dt,$ where 
+$\sigma^2$ is the variance of the noise. 
+%The optimal signal-to-noise ratio (SNR) is obtained when the template exactly matches with the signal. i.e., $\mathrm{SNR}_\mathrm{opt} = ||h||$. If the SNR is greater than a predetermined threshold (which corresponds to an acceptably small false alarm probability), a detection can be claimed. 
+
+\subsubsection{Problems}
+\begin{enumerate}
+\item You are given a time-series data $d(t)$. This contains a simulated gravitational-wave signal from a black hole binary buried in zero-mean, Gaussian white noise $n(t)$ with standard deviation $\sigma = 10^{-21}$. i.e., $d(t) = n(t) + h(t)$, where $h(t)$ is the leading order PN waveform from a binary with (unknown masses) $m_1$ and $m_2$. Detect the location of the signal in the data and values of $m_1$ and $m_2$ by maximising the correlation of the waveform templates with the data.: 
+\begin{equation}
+R_\mathrm{max} = \mathrm{max}~_{m_1, m_2,\tau}~ R(\tau),
+\end{equation}
+We have the prior information that the parameters of the signal are in the following range: $26 M_\odot < m_1, m_2 < 33 M_\odot$. 
+\item Compute the posterior distribution of the parameters $m_1$ and $m_2$ from the data assuming uniform prior in $m_1$ and $m_2$. 
+\end{enumerate}
+
 
 \bibliography{Lab}
 \end{document}