= \frac{\OGW(f_{\rm ref})}{\OGW(f)}.
\label{eq:optimal-w}$$ The broadband estimators are normalized so that $\langle \hat \Omega_{\rm ref}\rangle =\OGW(f_{\rm ref})$. By appropriate choices of the weights $w(f)$, one may construct an optimal search for  with arbitrary spectral shapes, or for  with scalar and vector polarizations.

Many models of the  can be approximated as a power laws [@Allen_Romano_1999; @StochPE], $$\Omega_{\rm GW}(f) = \Omega_{\rm ref} \left(\frac{f}{f_{\rm ref}}\right)^\alpha,$$ with a spectral index $\alpha$ and an amplitude $\Omega_\mathrm{ref}$ at a reference frequency $f_\mathrm{ref}$. As in the search in Advanced LIGO’s first observing run (O1) [@stoch_O1], we will take $f_{\rm ref}=\fref$, which is a convenient choice in the most sensitive part of the frequency band. While we will seek to generically constrain both $\Omega_\mathrm{ref}$ and $\alpha$ from the data, we will also investigate several specific spectral indices predicted for different gravitational-wave sources. In the frequency band probed by Advanced LIGO, the  from compact binaries is well approximated by a power law with $\alpha= 2/3$ [@Regimbau_2011]. Slow roll inflation and cosmic string models can be described with $\alpha = 0$ [@stoch_cosmo_review_2018]. Finally, following previous analyses [@stoch_O1], we use $\alpha=3$ as an approximate value to stand in for a variety of astrophysical models with positive slopes, such as unresolved supernovae 