the segment duration used to compute the Fourier transform, and $S_{0}(f)$ is the spectral shape for an $\OGW={\rm const}$ background given by $$S_0(f) = \frac{3 H_0^2}{10\pi^2f^3}.$$ The quantity $\gamma_T(f)$ is the normalized overlap reduction function for tensor (T) polarizations [@christensen92], which encodes the geometry of the detectors and acts as a transfer function between strain cross-power and $\OGW(f)$. Equation has been normalized so that the expectation value of $\hat C(f)$ is equal to the energy density in each frequency bin $$\langle \hat C(f) \rangle =\OGW(f).$$ In the limit where the gravitational-wave strain amplitude is small compared to instrumental noise, the variance of $\hat C(f)$ is approximately given by $$\label{eq:def_sigma}
\sigma^2(f) \approx \frac{1}{2T \Delta f}  \frac{P_1(f) P_2(f)}{\gamma_T^2(f) S_0^2(f)},$$ where $P_{1,2}(f)$ are the one-sided noise power spectral densities of the two detectors and $\Delta f$ is the frequency resolution, which we take to be $1/32$ Hz.

An optimal estimator can be constructed for a model of any spectral shape by taking a weighted combination of the cross-correlation statistics across different frequency bins $f_k$ $$\begin{aligned}
\hat \Omega_{\rm ref} &=& \frac{\sum_k w(f_k)^{-1} \hat C(f_k)\sigma^{-2}(f_k)}{\sum_k w(f_k)^{-2}\sigma^{-2}(f_k)}, \nonumber \\
\sigma_\Omega^{-2} &=& \sum_k w(f_k)^{-2} \sigma^{-2}(f_k),\end{aligned}$$ where the optimal weights for spectral shape $\OGW(f)$ are given by $$w(f) 