Detecting a stochastic gravitational wave signal

This note discusses trying to detect a generic gravitational wave with an unknown waveform emitted from a particular sky position in data from two separate gravitational wave detectors. We define two slightly different approaches to this problem.

The signal

First we will define the gravitational wave signal at one timestamp, \(i\), observed in one detector, \(L\). We envisage two possible methods for this analysis with slightly different model definitions. The first uses

\begin{equation} \label{eq:signal1}h_{i}^{L}=h_{0}\left({A_{+}}_{i}{F_{+}}_{i}^{L}(\psi_{i})+{A_{\times}}_{i}{F_{\times}}_{i}^{L}(\psi_{i})\right),\\ \end{equation}

where \({A_{+}}_{i}\) and \({A_{\times}}_{i}\) are the signal amplitudes scale factors (which could be positive or negative) in the plus and cross polarisations at timestamp \(i\) (which would be the same for different detectors), \({F_{+}}_{i}^{L}(\psi_{i})\) and \({F_{\times}}_{i}^{L}(\psi_{i})\) are the detector’s antenna response to the plus and cross polarisations for a given polarisation angle \(\psi_{i}\)11Note that \(\psi\) could change between data points, so this is also indexed for the current timestamp., and \(h_{0}\) is an overall underlying gravitational wave amplitude. The second uses

\begin{equation} \label{eq:signal2}h_{i}^{L}={A_{+}}_{i}{F_{+}}_{i}^{L}(\psi_{i})+{A_{\times}}_{i}{F_{\times}}_{i}^{L}(\psi_{i}),\\ \end{equation}

where, in this case, the \({A_{+}}_{i}\) and \({A_{\times}}_{i}\) are the actual signal amplitudes in the plus and cross polarisations at timestamp \(i\).