$\tilde {\cal C}^{aa}$ constitute the discrete representation of the power spectrum of the noise $S_n(f)$.

Gaussian noise can be described by the distribution, $$\label{ndis}
p_0 ({\bf n}) = M_n{\exp\left[
    -\frac{1}{2}\sum\limits_{a,b=0}^{N-1}{[{\cal C}^{-1}]_{ab} n^a n^{b}}\right]},$$ where $M_n$ is a normalization constant given by, $$M_n = { \left[\ (2\pi)^N \det\left[{\cal C}^{ab}\right]\ \right]^{-1/2}}.$$ Equivalently in the Fourier domain this can be written as, $$\begin{aligned}
p_0({\bf n}) &=& M_n {\exp\left[
    -\frac{1}{2}\sum\limits_{a,b=0}^{N-1}{[\tilde {\cal C}^{-1}]_{ab} \tilde n^a 
\tilde n^{b*}}\right]}\nonumber\\
&=& M_n{\exp\left[
    -\frac{1}{2}\sum\limits_{a=0}^{N-1}{{\tilde n^a 
\tilde n^{a*}}/{\tilde {\cal C}^{aa}}}\right]},\end{aligned}$$ where in the last step we have used the diagonal property of the matrix $\tilde {\cal C}^{ab}$ which implies that $[\tilde {\cal C}^{-1}]_{aa} = 1/\tilde {\cal C}^{aa}$.

In the presence of the signal ${\bf s} (\bbox{\check\mu})$ the above probability density function (pdf) gets modified but in a very simple way since we have assumed that the noise is additive. We have, $$p_1 ({\bf x}) = p_0 ({\bf x} - {\bf s}(\bbox{\check\mu})),$$ where $p_1 ({\bf x})$ is the pdf of ${\bf x}$ when the signal ${\bf s} (\bbox{\check\mu})$ is present in the data stream.

In this paper we shall assume the noise to have a power spectrum consistent with the initial LIGO instrument. We use the fit to the noise curve 