stochastic searches and discuss the implications of our current results for the detection of the stochastic background from compact binaries and cosmic strings. Finally, we present estimates of the correlated noise in the LIGO detectors due to magnetic Schumann resonances [@Schumann_theory], and discuss mitigation strategies that are being pursued for future observing runs.

*Method*— The isotropic  can be described in terms of the energy density per logarithmic frequency interval $$\Omega_{\rm GW}(f) = \frac{f}{\rho_c} \frac{{\rm d} \rho_{\rm GW}}{{\rm d} f},$$ where ${\rm d}\rho_{\rm GW}$ is the energy density in gravitational waves in the frequency interval from $f$ to $f+{\rm d}f$, and $\rho_c = 3H_0^2 c^2/(8\pi G)$ is the critical energy density required for a spatially flat universe. Throughout this work we will use the value of the Hubble constant measured by the *Planck* satellite, $H_0 = \HubbleConstant$ [@Planck_2015].

We use the optimal search for a stationary, Gaussian, unpolarized, and isotropic , which is the cross-correlation search [@christensen92; @Allen_Romano_1999; @RomanoCornish; @Christensen_2018] (however, see [^1]). For two detectors, we define a cross-correlation statistic $\hat{C}(f)$ in every frequency bin $$\label{eq:def_cross_corr}
\hat C(f) = \frac{2}{T}  \frac{{\rm Re}[\tilde{s}_1^\star(f) \tilde{s}_2(f)]}{\gamma_T(f) S_0(f)},$$ where $\tilde{s}_i(f)$ is the Fourier transform of the strain time series in detector $i=\{1,2\}$, $T$ is 