ref}<\RunOneTwoUniformPriorAlphaTwoThirdsLimit$, and for $\alpha=3$, $\Omega_{\rm ref} < \RunOneTwoUniformPriorAlphaThreeLimit$. Finally, when we marginalize over the power law index $\alpha$, we obtain the upper limit $\Omega_{\rm ref}<\ngrRunOneTwoPolTUniformPriorLimitT$. The prior for $\alpha$ is described in the Supplementary Material.

  $\alpha$                    $\hat{\Omega}_{\rm ref}$ (O2)                                        $\hat{\Omega}_{\rm ref}$ (O1)                        O2 Sensitive band
  ---------- ---------------------------------------------------------------- ------------------------------------------------------------------------ -------------------
  0            $(\IsoAlphaZeroPtEst \pm \IsoAlphaZeroSigma) \times 10^{-8}$     $(\PrevIsoAlphaZeroPtEst \pm \PrevIsoAlphaZeroSigma) \times 10^{-8}$   
  2/3           $(\IsoAlphaCBCPtEst \pm \IsoAlphaCBCSigma) \times 10^{-8}$       $(\PrevIsoAlphaCBCPtEst \pm \PrevIsoAlphaCBCSigma) \times 10^{-8}$    
  3           $(\IsoAlphaThreePtEst \pm \IsoAlphaThreeSigma) \times 10^{-9}$   $(\PrevIsoAlphaThreePtEst \pm \PrevIsoAlphaThreeSigma) \times 10^{-9}$  

  : Point estimates and $1\sigma$ uncertainties for $\Omega_{\rm ref}$ in O2, for different power law models, alongside the same quantities measured in O1 [@stoch_O1]. We also show the minimum contiguous frequency band containing 99% of the sensitivity. For each power law, the maximum of the frequency band is within 5% of the value found in O1. The value of the Hubble constant used in this paper is different than what was used in the O1 analysis [@stoch_O1] ($68 \,\text{km s$^{-1}$ Mpc$^{-1}$}$), which has led to some differences in the numerical values of the point estimates and error bars that we report for O1.[]{data-label="tab:ul"}

  ---------- --------------------------------------------- ------------------------------------ ------------------------------------------------ ---------------------------------------------
  $\alpha$                       O1+O2                                      O1                                       O1+O2                                            O1
  0             $\RunOneTwoUniformPriorAlphaZeroLimit$            $\IsoAlphaZeroPrevUL$            $\RunOneTwoLogUniformPriorAlphaZeroLimit$        $\RunOneLogUniformPriorAlphaZeroLimit$
  