# 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

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

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

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

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

## First Method

\label{sec:method1}

Here we will examine the details of the first method, which uses the signal model defined in Equation \ref{eq:signal1}.

Now, if we had one data point for detector $$X$$, $$d_{i}^{X}$$, and assuming the noise in the detector is Gaussian with zero mean and standard deviation of $$\sigma_{i}^{X}$$, then the likelihood for the data given the model is

$$\label{eq:singlelikelihood}p(d_{i}^{X}|h_{0},{A_{+}}_{i},{A_{\times}}_{i},\psi,I)=\frac{1}{\sqrt{2\pi}\sigma_{i}^{X}}\exp{\left(-\frac{(d_{i}^{X}-h_{i}^{X})^{2}}{2{\sigma_{i}^{X}}^{2}}\right)}.\\$$

We now add another detector, $$Y$$, with data point $$d_{i}^{Y}$$, where the $$i$$ timestamp index in detector $$Y$$ is actually indexing a time that is shifted with respect to that in detector $$X$$ based on the time delay between detectors for the known signal position. So, e.g. $$t_{i}^{Y}=t_{i}^{X}+\Delta t_{i}(\alpha,\delta)$$. This gives a joint likelihood of the data for the two detectors of

$$\label{eq:jointlikelihood}p(\{d_{i}^{X},d_{i}^{Y}\}|h_{0},{A_{+}}_{i},{A_{\times}}_{i},\psi_{i},I)=\frac{1}{2\pi\sigma_{i}^{X}\sigma_{i}^{Y}}\exp{\left(-\sum_{L=X,Y}\frac{(d_{i}^{L}-h_{i}^{L})^{2}}{2{\sigma_{i}^{L}}^{2}}\right)}.\\$$

We would like to get a posterior probability distribution on $$h_{0}$$ alone (and in fact we also want the evidence marginalised over $$h_{0}$$ too). If we assume that the $$A$$ scale factors and $$\psi$$ change on the timescale of individual data points then we want to marginalise over them for each point, e.g.

\begin{align} p(\{d_{i}^{X},d_{i}^{Y}\}|h_{0},I)= & \int_{{A_{+}}_{i}}\int_{{A_{\times}}_{i}}\int_{0}^{\pi}p(\{d_{i}^{X},d_{i}^{Y}\}|h_{0},{A_{+}}_{i},{A_{\times}}_{i},\psi_{i},I)\times\notag \\ & \label{eq:h0likelihood} p({A_{+}}_{i}|I)p({A_{\times}}_{i}|I)p(\psi_{i}|I){\rm d}{A_{+}}_{i}{\rm d}{A_{\times}}_{i}{\rm d}\psi_{i},\\ \end{align}

where $$p({A_{+}}_{i}|I)$$ and $$p({A_{\times}}_{i}|I)$$ are the priors on the scale factors and $$p(\psi_{i}|I)$$ is the prior on $$\psi_{i}$$.

To get the joint likelihood over all the data we can just use the product of Equation \ref{eq:h0likelihood}, such that we have