@@ -210,7 +210,6 @@ 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 \href{http://gitlab.icts.res.in/ajith/gwcourse2023/blob/master/tuturials/fake_gw_data.dat.gz}{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}
...
...
@@ -220,6 +219,15 @@ We have the prior information that the parameters of the signal are in the follo
\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}
\subsection{Term paper topics}
\begin{enumerate}
\item Survey the formation channels of binary black holes, broadly outlining the physical and astrophysical processes involved. What are the observational properties that can be used to identify the formation mechanisms? Can you suggest something new?
\item Survey the the scientific capabilities of multi-band GW observations. Can you think of some new science that can be done by combining observations using future ground and space-based GW observatories?
\item Overview the gravitational lensing of GWs (strong lensing, weak lensing and micro lensing). What kind of information on astrophysics, cosmology and fundamental physics can be learned from observations of lensed GWs. Can you suggest something new (perhaps taking motivation from the lensing studies of electromagnetic waves)?
\item Provide an overview of possible astrophysical sources of GWs (both traditional and new). What kind of a new detector is needed to observe some of the hitherto unobserved sources? Broadly outline the sensitivity and bandwidth required.
\item Describe some of the new methods proposed for GW detection, and compute their sensitivities from first principles. Can you improve their sensitivities or change their frequency band by means of changing design parameters?
\item A recent work has computed 4.5PN accurate gravitational wave flux and binding energy in the post-Newtonian approximation. Compute different post-Newtonian approximants using these inputs. Compute their mismatches. Does the computation of the higher order terms improve the agreement between post-Newtonian and numerical relativity waveforms?
