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Power spectral density $S_n$ with and without a SGWB. For comparison, we also plot illustrative signals including MBHBs for different total masses and redshifts, a GW150914-like event~\cite{LIGOScientific:2016aoc} and galactic and verification binaries. The latter are taken from Ref.~\cite{Kupfer:2023nqx,gitlab_verification_binaries}, and we only plot those with SNR$>5$. For MBHBs and GW150914, we only plot the + polarization of GWs for definiteness. The parameters for the SGWBs are reported in the main text and are chosen to saturate the CMB+BBN bound. All the noise curves also include galactic foregrounds.

[First column.] We plot the ${\rm SNR}$ distribution averaged over 100 realizations of our catalogs of MBHBs populations. We emphasize the point ${\rm SNR}=10$ with a vertical black line and report the number of events detected with ${\rm SNR}>10$ in the legend. [Second, third, and fourth columns.] For a specific realization of our populations, we plot the distribution of events as a function of redshift and the total mass of the binary. The color bar indicates the ${\rm SNR}$ of each event, and we emphasize events with ${\rm SNR}>10$ with a colored circle. We plot results for \texttt{PopIII}, \texttt{Q3delays}, and \qtn populations in the first, second, and third rows respectively. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively.

Relative 68\% CL errors on the total mass $M_z$ [top] and the luminosity distance $d_L$ [center] and absolute 68\% CL errors on the sky localization $\Omega$ [bottom] from our FIM analysis. We color the region where the percentage error is $>100\%$ with black. We only perform the Fisher analysis for the events that are detected with ${\rm SNR}>10$ in the absence of an SGWB. The errors are shown as function of the fiducial values in the scatter plots, and we also provide histograms to visualize their distribution. Our results are averaged over 15 random realizations of our populations. We plot results for \texttt{PopIII}, \texttt{Q3delays}, and \qtn populations in the first, second, and third rows respectively. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively.

Relative 68\% CL errors on the total mass $M_z$ [top] and the luminosity distance $d_L$ [center] and absolute 68\% CL errors on the sky localization $\Omega$ [bottom] from our FIM analysis. We color the region where the percentage error is $>100\%$ with black. We only perform the Fisher analysis for the events that are detected with ${\rm SNR}>10$ in the absence of an SGWB. The errors are shown as function of the fiducial values in the scatter plots, and we also provide histograms to visualize their distribution. Our results are averaged over 15 random realizations of our populations. We plot results for \texttt{PopIII}, \texttt{Q3delays}, and \qtn populations in the first, second, and third rows respectively. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively.

Results of simulated Bayesian parameter estimation for the MBHB 1 [left panels] and MBHB 2 [right panels] systems of Table~\ref{tab:MBHBparams}. Shown is the posterior distribution on intrinsic parameters, masses, and spins [upper panels] and on the two angles for sky localization, as well as luminosity distance [lower panels]. Contours are shown at 90\% CL. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively. All intrinsic and extrinsic parameters are varied and marginalized over in the analysis. We present them in separate corner plots only for convenience.

Results of simulated Bayesian parameter estimation for the MBHB 1 [left panels] and MBHB 2 [right panels] systems of Table~\ref{tab:MBHBparams}. Shown is the posterior distribution on intrinsic parameters, masses, and spins [upper panels] and on the two angles for sky localization, as well as luminosity distance [lower panels]. Contours are shown at 90\% CL. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively. All intrinsic and extrinsic parameters are varied and marginalized over in the analysis. We present them in separate corner plots only for convenience.

Results of simulated Bayesian parameter estimation for the MBHB 1 [left panels] and MBHB 2 [right panels] systems of Table~\ref{tab:MBHBparams}. Shown is the posterior distribution on intrinsic parameters, masses, and spins [upper panels] and on the two angles for sky localization, as well as luminosity distance [lower panels]. Contours are shown at 90\% CL. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively. All intrinsic and extrinsic parameters are varied and marginalized over in the analysis. We present them in separate corner plots only for convenience.

Results of simulated Bayesian parameter estimation for the MBHB 1 [left panels] and MBHB 2 [right panels] systems of Table~\ref{tab:MBHBparams}. Shown is the posterior distribution on intrinsic parameters, masses, and spins [upper panels] and on the two angles for sky localization, as well as luminosity distance [lower panels]. Contours are shown at 90\% CL. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively. All intrinsic and extrinsic parameters are varied and marginalized over in the analysis. We present them in separate corner plots only for convenience.

Results of simulated Bayesian parameter estimation for the MBHB 1 [left panels] and MBHB 2 [right panels] systems of Table~\ref{tab:MBHBparams}. Shown is the posterior distribution on intrinsic parameters, masses, and spins [upper panels] and on the two angles for sky localization, as well as luminosity distance [lower panels]. Contours are shown at 90\% CL. Blue, orange, green, and red colors represent our results without SGWB, for {\bf PL}, {\bf Bump High $f$}, and {\bf Bump Low $f$} respectively. All intrinsic and extrinsic parameters are varied and marginalized over in the analysis. We present them in separate corner plots only for convenience.

Scatter plot for our 3 populations showing the value of an approximate measure for sky multimodality, $\Delta \ln \mathcal{L}_{\rm reflected}$, computed in the absence of an SGWB (x-axes) versus its value computed in the presence of our 3 benchmark SGWBs (y-axes). Events on the left of the vertical dashed gray line are likely to feature multimodal sky localizations even in the absence of an SGWB. Events in the bottom right quadrants, i.e. to the right of the vertical and below the horizontal dashed gray lines, are likely to feature a multimodal sky localization only when the SGWB is included.

SNR histogram for a SOBHB system similar to GW150914, varying inclination, phase, sky position and polarization, for the different SGWB cases.

[Left] SNR histogram for Galactic Binaries from the LDC catalogue, for the different SGWB cases. We start from a reduced catalogue with purely instrumental $\mathrm{SNR}>5$, plotted in black. [Center] Percentage loss of detached and interacting binaries with ${\rm SNR}>5$ (with respect to the No SGWB case) as a function of frequency. [Right] Degradation of the SNR of the verification binaries. We only plot those that have ${\rm SNR}>5$ in the No SGWB case. In the left and right panels, dashed and solid lines indicate SNR $=5$ and $8$ respectively.

[Left] SNR histogram for Galactic Binaries from the LDC catalogue, for the different SGWB cases. We start from a reduced catalogue with purely instrumental $\mathrm{SNR}>5$, plotted in black. [Center] Percentage loss of detached and interacting binaries with ${\rm SNR}>5$ (with respect to the No SGWB case) as a function of frequency. [Right] Degradation of the SNR of the verification binaries. We only plot those that have ${\rm SNR}>5$ in the No SGWB case. In the left and right panels, dashed and solid lines indicate SNR $=5$ and $8$ respectively.

[Left] SNR histogram for Galactic Binaries from the LDC catalogue, for the different SGWB cases. We start from a reduced catalogue with purely instrumental $\mathrm{SNR}>5$, plotted in black. [Center] Percentage loss of detached and interacting binaries with ${\rm SNR}>5$ (with respect to the No SGWB case) as a function of frequency. [Right] Degradation of the SNR of the verification binaries. We only plot those that have ${\rm SNR}>5$ in the No SGWB case. In the left and right panels, dashed and solid lines indicate SNR $=5$ and $8$ respectively.