which is uniform in $\Omega_{\rm ref}$ and one which is uniform in $\log{\Omega_{\rm ref}}$. We additionally marginalize over detector calibration uncertainties [@StochCalUncertainty]. In O2 we assume  and  amplitude uncertainties in Hanford and Livingston, respectively [@Cahillane:2017vkb; @Viets:2017yvy]. In O1, the calibration uncertainty for Hanford was 4.8% and for Livingston was 5.4% [@Cahillane:2017vkb]. Phase calibration uncertainty is negligible.

Figure \[fig:ULs\] shows the resulting posterior distribution in the $\Omega_{\rm ref}$ vs $\alpha$ plane, along with $68\%$ and $95\%$ credibility contours. Table \[tab:ul2\] lists the marginalized 95% credible upper limit on $\Omega_\mathrm{ref}$ (for both choices of amplitude prior), as well as the amplitude limits obtained when fixing $\alpha=0$, $2/3$, and $3$.

When adopting a uniform amplitude prior and fixing $\alpha=0$, we obtain an upper limit of $\Omega_{\rm ref}<\RunOneTwoUniformPriorAlphaZeroLimit$, improving the previous O1 result by a factor of . The $1\sigma$ error bar is $\IsoAlphaZeroSigma \times 10^{-8}$, a factor of 2.7 times smaller than the equivalent O1 uncertainty. This factor can be compared with the factor of  that would be expected based on increased observation time alone, indicating that the search has benefited from improvements in detector noise between O1 and O2. For the compact binary  model of $\alpha=2/3$, we place a limit of $\Omega_{\rm 