of any spectral shape.

We perform several tests that the cross-correlation spectrum is consistent with uncorrelated Gaussian noise. The $\chi^2$ per degree of freedom for the observed spectrum is 0.94. The loudest individual frequency bin is 51.53 Hz, with a signal-to-noise ratio $C(f)/\sigma(f)$ of 4.2. With a total of 46,227 (un-notched) frequency bins, there is a 71% probability that random Gaussian noise would yield an equally loud bin.

In Table  \[tab:ul\], we list the broadband point estimates and 1$\sigma$ uncertainties obtained from the O2 data when assuming power laws with $\alpha=0$, $2/3$, and $3$. Given the uncertainties, uncorrelated Gaussian noise would produce point estimates at least this large with probability 30%, 22%, and 21%, respectively. We conclude there is not sufficient evidence to claim detection of the .

*Upper limits on isotropic stochastic background*— Since we do not find evidence for the , we place upper limits on the amplitude $\Omega_\mathrm{ref}$. We use the parameter estimation framework described in [@StochPE; @TestingGR_stoch; @stoch_nongr_O1], applied to the cross-correlation spectrum obtained by combining the results from O1 given in [@stoch_O1], with those from O2 which are described above (please see the Supplementary Material for more details). We present results assuming two priors, one 