2/3         $\RunOneTwoUniformPriorAlphaTwoThirdsLimit$          $\IsoAlphaCBCPrevUL$          $\RunOneTwoLogUniformPriorAlphaTwoThirdsLimit$   $\RunOneLogUniformPriorAlphaTwoThirdsLimit$
  3             $\RunOneTwoUniformPriorAlphaThreeLimit$           $\IsoAlphaThreePrevUL$           $\RunOneTwoLogUniformPriorAlphaThreeLimit$       $\RunOneLogUniformPriorAlphaThreeLimit$
  Marg.          $\ngrRunOneTwoPolTUniformPriorLimitT$      $\ngrRunOnePolTUniformPriorLimitT$      $\ngrRunOneTwoPolTLogUniformPriorLimitT$         $\ngrRunOnePolTLogUniformPriorLimitT$
  ---------- --------------------------------------------- ------------------------------------ ------------------------------------------------ ---------------------------------------------

![Posterior distribution for the amplitude $\Omega_{\rm ref}$ and slope $\alpha$ of the , using a prior which is uniform in the logarithm of $\Omega_{\rm ref}$, along with contours with 68% and 95% confidence level, using combined O1 and O2 data. There is a small region of increased posterior probability centered around $\log \Omega_{\rm ref}=-8$ and $\alpha=2$. This is not statistically significant, and similar-size bumps have appeared in simulations of Gaussian noise. An analogous plot with a prior uniform in $\Omega_{\rm ref}$ can be found in the Supplementary Material.[]{data-label="fig:ULs"}](Fig2.pdf){width="49.00000%"}

  Polarization                 Uniform prior                            Log-uniform prior
  -------------- ----------------------------------------- --------------------------------------------
  Tensor          $\ngrRunOneTwoPolTVSUniformPriorLimitT$   $\ngrRunOneTwoPolTVSLogUniformPriorLimitT$
  Vector          $\ngrRunOneTwoPolTVSUniformPriorLimitV$   $\ngrRunOneTwoPolTVSLogUniformPriorLimitV$
  Scalar          $\ngrRunOneTwoPolTVSUniformPriorLimitS$   $\ngrRunOneTwoPolTVSLogUniformPriorLimitS$

  : Upper limits on different polarizations. To obtain the upper limits, we assume a log uniform and a uniform prior on the amplitude $\Omega_{\rm ref}$ for each polarization, using combined O1 and O2 data. We assume the presence of a tensor, vector, and scalar backgrounds, then marginalize over the spectral indices and two amplitudes for the three different polarization modes, as described in the main text.[]{data-label="tab:nongrUL"}

*Implications for compact binary background*— In Figure \[fig:cbc\] we 