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Representation of the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} and \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} approximation models, according to the legend. The blue line corresponds to the comoving distance in the standard \lcdm Universe; the yellow dotted-dashed line represents the matter-only approximation from Eq.~\ref{eq:matter-only} with $z_p=3$ and the two orange dashed lines the redshift bins approach in Eq.~\ref{eq:linear_fit} for two redshift bins with $z_p=1.5$ and $z_p=3$ (black stars). Green points correspond to the MBHBs from a random realisation of Q3d with the corresponding errors on redshift and comoving distance, accounting also for lensing and peculiar velocities errors as described in Sec.~\ref{sec:cat_construction}. For low-redshift events, the errors are smaller than the size of the dot.

Representation of the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} and \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} models. The blue, solid line corresponds to the comoving distance in the standard \lcdm Universe; the yellow, dotted-dashed line represents the matter-only approximation from Eq.~\ref{eq:matter-only} with $z_p=3$; the two red, dashed lines denote the redshift bins approach in Eq.~\ref{eq:linear_fit} for two redshift bins with $z_p=1.5$ and $z_p=3$ (black stars) \textcolor{\colorref}{and the vertical dotted-dashed grey lines visualise the boundaries of the corresponding redshift bins}. The green points show the MBHBs from a random Universe realisation, \textcolor{\colorref}{based on the astrophysical model Q3d, assuming 10 yrs of data (and therefore correspondingly of LISA mission duration),} with the respective \textcolor{\colorref}{1$\sigma$} errors bars on the redshift and comoving distance, accounting also for lensing and peculiar velocities errors (as described in Sec.~\ref{sec:cat_construction}). For the low-redshift events, the errors are smaller than the size of the dot.

Scatter plot of the luminosity distance uncertainty at 1$\sigma$ from LISA parameter estimation, as a function of redshift. The blue points correspond to all the MBHBs we have simulated within the Q3d model, i.e.~the 90 years of data. \textcolor{\colorref}{We convolve the luminosity distance uncertainty from the GW detection with the lensing and peculiar velocity errors as described in Sec.~\ref{sec:cat_construction}, to get the luminosity distance uncertainties adopted in the cosmological analysis}. The green solid line represents the errors from peculiar velocities as in Eq.~\ref{eq:pv_error}. The red (yellow) dashed line corresponds to the lensing error as in Eq.~\ref{eq:lensing_Q3} without (with) the delensing correction. \textcolor{\colorref}{The grey dotted-dashed line corresponds to the arbitrary cut-off we impose on the $1\sigma$ error on $d_L$ in order to choose which systems ro rerun, assuming their sky position is known, i.e. $\sigma_{68,d_L} > 0.5 \, \rm \sigma_{\rm delens}$ or $0.5\sigma_{\rm pv}$. The grey points above the grey dotted-dashed correspond to the subset of systems for which we rerun the parameter estimation assuming perfect localisation, leading to the corresponding blue points (connected with a vertical thin dashed grey line)}. For points below the grey dotted-dashed line, the error on $d_L$ is already dominated by lensing or peculiar velocities.

Ratio of the luminosity distance error fixing the sky position in the parameter inference over the one obtained in the full analysis, i.e. including as parameters the two sky position coordinates. Both errors are at 90\% level. Colors and line styles correspond to the three astrophysical models, as reported in the legend.

Example of JS divergence computed between the posterior distribution of $h(z_p=3)$ and an uniform prior between $[0.1,50]$ versus the inferred median value of $h(z_p=3)$. Each point corresponds to a single realisation for each astrophysical model, according to legend. The horizontal grey dotted-dashed line corresponds to the arbitrary cut-off of 0.5 on the value of the JS divergence (more details in the text). The blacked dashed line represents the true value of $h(z_p=3)\sim3.06$, according to $\Lambda$CDM. Uninformative realisations show inferred median values at $\sim25$, corresponding to the midpoint of the prior range, and JS divergence closer to 0, i.e. the posterior is similar to the prior.

\textcolor{\colorref}{The JS divergence as a function of the median of the posterior on the parameter $h(z_p)$, for all the EMcps catalogues simulated for the three MBHB formation scenarios, and taking as example the redshift bin centred on $z_p=3$. The JS divergence is computed between the posterior distribution of $h(z_p=3)$ and its uniform prior, which is extended to the interval $[0.1,50]$ to distinguish the \emph{uninformative} realisations. Each point corresponds to a single realisation for each astrophysical model, according to the legend. The horizontal grey dotted-dashed line corresponds to the arbitrary cut-off of 0.5 which we impose on the value of the JS divergence to select the informative realisations (more details in the text). The black dashed line represents the \lcdm true value, $h(z_p=3)= 3.06$. Uninformative realisations show inferred median values far from the \lcdm one and, specifically, decaying towards $ h(z_p=3)\sim25$, corresponding to the midpoint of the prior range. Furthermore, the JS divergence decays towards 0, meaning that the posterior is similar to the prior. Informative realisations, on the other hand, cluster around the \lcdm value, and have $\mathrm{JS}\gtrsim 0.5$.}

Cumulative distribution of events with p-value smaller or equal to the abscissa. Colors and line styles according to legend. The dotted lines represent the 90\% uncertainty region expected from the finite number of realisations. The dashed black line represents the expected value. Grey dashed line: same as the orange one for Pop3 but including all the realisations. It is clear that selecting only the informative realisations provide results consistent with the expected values.

Cumulative distribution of events with p-value smaller or equal to the abscissa. The solid, dashed and dot-dashed lines represent the \emph{informative} realisations only, for the different MBHB formation scenarios according to the legend. The dotted lines represent the 90\% uncertainty region expected from the finite number of realisations. The dashed black line represents the expected value for unbiased results. Grey dashed line: same as the orange one for Pop3, but including all the realisations, even the uninformative ones (according to the JS criterion (see the meain text). It is clear that selecting only the informative realisations provide results consistent with the expected values.

Cumulative distributions over realisations of the relative uncertainties for $h$ (left panel) and $\Omega_m$ (right panel) in the \hyperref[itm:lcdm]{\color{mycolormodel}$(h, \Omega_m)$} model, namely $\Lambda$CDM. Solid (dashed) lines correspond to 10 (4) yrs of observations with LISA. Colors represent different astrophysical models as described in the legend and the grey area represents uncertainties larger than $100\%$. We expect relative errors on $h$ of $\lesssim 5\%$ in 4 yrs and $\lesssim 2\%$ in 10 yrs for at least $50\%$ of the realisations. The relative errors on $\Omega_m$ descend below $\lesssim 10\%$ for at least $50\%$ of the realisations only in the 10 yrs LISA scenario.

Corner plot for $(h, \Omega_m)$ for the \lcdm model in 4 yrs of LISA mission, for the average, best and worst realisations of the Q3d MBHBs formation scenario, i.e.~those corresponding to the median, 5th and 95th percentile of the distribution pf the realisations (more details in the text). Colors according to legend.

Relative errors at 90\% for $h(z)$ (upper panels) and $d_L(z)$ (lower panels) as a function of redshift, derived using~\cref{eq:hubble_rate_h0_omegam} and \cref{eq:dl} from the posterior samples of the \hyperref[itm:lcdm]{\color{mycolormodel}$(h, \Omega_m)$} model, shown in \cref{fig:H0_and_omegam}. Error bars correspond to the $90\%$ uncertainty coming from the distribution over realisations. The three colors correspond to different astrophysical models, according to the legend. To avoid null values in the lower panels, the first point for $d_L$ is at $z=0.1$.

Same as Fig.~\ref{fig:H0_and_omegam} but with $h, \Omega_m$ and $\omega_0$ from the \hyperref[itm:cpl]{\color{mycolormodel}$(h, \Omega_m, \omega_0, \omega_a)$} model. $\omega_a$ is unconstrained and not reported. The addition of two parameters worsen the estimates on $(h, \Omega_m)$.

Same as Fig.~\ref{fig:H0_and_omegam} but for $\omega_0$ and $\Xi_0$ for the \hyperref[itm:belgacem19]{\color{mycolormodel}$(h, \Omega_m, \omega_0, \Xi_0)$} model. Note that we report the absolute $1\sigma$ uncertainties for comparison with Tab.2 in \cite{Belgacem19}. The uncertainties on $h$ and $\Omega_m$ coincide with the CMB priors. We can constrain $\Xi_0$ to $<10\%$ using only MBHBs bright sirens.

Same as Fig.~\ref{fig:H0_and_omegam} but for $h$ and $\beta$ for the \hyperref[itm:lcdm_beta]{\color{mycolormodel}$(h, \Omega_m, \beta)$} model. The parameter $\beta$ models a possible deviation from zero in the matter equation of state, i.e. $\omega_m = \beta$. We forecast a relative accuracy on $\beta$ of $<20\%$ with 4 yrs of LISA observations and $\lesssim 10\%$ with 10 yrs for at least 50\% of the realisations.

Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} approximation. Left panels: $z_p=2$. Right panels: $z_p=3$. In 10yr of observation, we expect constraints on $h(z_p=2)$ at $3-5\%$ and on $h(z_p=3)$ at $\lesssim 10\%$.

Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} approximation at different pivot redshift, as specified in the x-axis labels. \textcolor{\colorref}{Colors represent different astrophysical models as reported in the legend. The vertical dashed black line represents the model's accuracy. In 10yrs of observation, we expect constraints on $h(z_p=2)$ at $3-5\%$. At higher redshift, estimates get slightly worse but we still have relative errors on $h(z_p)$ of $10\%$ at $z_p=7$.}

Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} approach at different pivot redshifts, according to the x-axis labels. Colors represent different astrophysical models as reported in the legend. The cumulative distributions reach the fraction of \emph{informative} realisations (see Sec.~\ref{subsec:bins}): for example at $z_p=3.5$, only $\sim75\%$ of the Q3d and Q3nd realisations provide useful constraints.

Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} approach at different pivot redshift, as specified in the x-axis labels. Colors represent different astrophysical models as reported in the legend. \textcolor{\colorref}{The vertical dashed black line represents the model's accuracy.} The cumulative distributions reach the fraction of \emph{informative} realisations (see Sec.~\ref{subsec:bins}): for example at $z_p=3.5$, only $\sim75\%$ of the Q3d and Q3nd realisations provide useful constraints.

Relative errors at 90\% for $d_L(z)$ (upper panels) and $h(z)$ (lower panels) as a function of redshift from the \hyperref[itm:splines]{\color{mycolormodel}spline interpolation} model. Error bars correspond to the $90\%$ uncertainty on the relative errors from the distribution of realisations. Light blue boxes highlight the uncertainties at the knot redshifts. Splines can constrain the luminosity distance to $<10\%$ in 4 yrs at $1

Median relative uncertainty on $h$ as a function of the number of realisations. Errors bars represent the 90 percentile. For all astrophysical models, 100 realisations are sufficient to construct a statistically representative sample.

Upper left panel: ratio between the true value of the luminosity distance and the median value from the $d_L$ posterior distributions. Upper right panel: ratio between the $1\sigma$ uncertainty from fisher analysis and from $d_L$ posterior distribution. Lower left panel: skewness of the $d_L$ posteriors. Lower right panel: same as the left one but for the kurtosis. Aquamarine (crimson) lines correspond to the distribution for the entire MBHBs catalogues (the subset of EMcps). This plot is only for the Q3d model. Overall, the $d_L$ posterior distributions for the EMcps can be considered as Gaussian distributions.

Scatter plot of JS divergence computed between the posterior distribution of $h(z_p=3)$ and different priors (as specified in the legend) versus the inferred median value of $h(z_p=3)$. The dashed black line represents the value of $h(z_p=3)\sim3.06$, according to $\Lambda$CDM.

Scatter plot of JS divergence computed between the posterior distribution of $h(z_p=3)$ and different priors (as specified in the legend) versus the inferred median value of $h(z_p=3)$. The dashed black line represents the value of $h(z_p=3)\sim3.06$, according to $\Lambda$CDM.

Comparison between JS divergence values from different priors, as stated in the legend. On the x-axis we report the JS divergence adopted in the main text, i.e. with uniform prior in [0.1,50].

Comparison between JS divergence values for $h(z_p=3)$ for different priors, as stated in the legend. On the x-axis we report the JS divergence adopted in the main text, i.e. with uniform prior in [0.1,50].

Number of informative realisations as a function of the JS divergence for the three astrophysical models, as in the legend. The JS divergence is computed for $h(z_p=3)$ with the prior adopted in the main text.

Number of informative realisations as a function of the JS divergence for the three astrophysical models, as in the legend. The JS divergence is computed for $h(z_p=3)$ with the prior adopted in the main text.

Comparison between the posterior distributions on $h(z_p)$ and the corresponding priors, according to legend. Here we select a particular event realisation of Q3d \am{for future note: I selected the realisation 1}

\textcolor{\colorref}{Difference between the luminosity distance from spline interpolation $d_L^{spline}$ and the $\Lambda$CDM luminosity distance $d_L^{ \Lambda CDM}$, normalised over $d_L^{ \Lambda CDM}$ and for different choices of knot redshifts as reported in the legends. Upper and lower panels report the same quantity: we split the results and use solid and dashed lines only to facilitate the visual comparison. At the values of the knots, the spline is forced to pass through $\Lambda$CDM values and the difference goes to zero.}

PP-plot for $h$ and $\Omega_m$

Uncertainty on $h$ for all the realisations in the analysis for $(h, \Omega_m)$. Error bars correspond to 90th confidence interval. \am{add that the x-axis is irrelevant and the order is irrelevant} \am{the point is that this plot give the information about the fact the median is consistent while Fig.~\ref{fig:H0_and_omegam} tells us information only about the error}