Clustering effects on the Dark Siren determination of $H_0$: A simulation study

Abstract

Gravitational waves (GWs) offer an alternative way to measure the Hubble parameter. The optimal technique, the ``bright siren'' approach, requires the identification of an electromagnetic counterpart. However, a significant fraction of gravitational waves signals will not have counterparts. Such events can still constrain the Hubble parameter $H_0$ via statistical methods, exploiting galaxy information from the GWs sky localisation volume. In this work, we investigate the power of this method using high-resolution, cosmological simulations that include realistic clustering. We find that clustering leads to increased convergence of the $H_0$ posteriors, with clear recovery of the input value as early as $N_{\rm gw}=40$ events, compared to uniform catalogues, where the posterior remains largely unconstrained, even with $N_{\rm gw}=100$ events. In addition, we quantify the role of catalogue incompleteness. We show that catalogues with completeness levels as low as $f=25\%$ can be competitive with fully complete catalogues, confirming the impact of clustering. Completeness levels of $f=50\%$ perform statistically similar to complete catalogues with as few as $N_{\rm gw}=40$ events. This indicates the need to focus on improving gravitational waves detection capabilities, rather than obtaining more complete galaxy catalogues. Finally, we investigate additional properties of the method by taking into consideration physical weights, different observational errors, potential biases from the $H_0$ priors, a variety of detectors' horizon distances, and different methods of catalogue completion and statistical analysis.

Figures (26)

• Figure 1: DM halo mass function for three of the large particle number LEGACY boxes that are used in the main analysis of this work. (Green solid) $L=100$ Mpc/h, (Orange solid) $L\simeq700$ Mpc/h zoom-in simulation, and the $L=1600$ Mpc/h box (Blue solid). Their mass functions are shown in the same order from left to right (as expected, only the largest box can form the more massive haloes). The grey dashed line provides the fit based on Tinker_et_al_2008.
• Figure 2: For each simulation we define an "observable spherical cell" in $3D$. A $2D$ plane projection is shown here, with $[r_{\rm min}=10\ {\rm Mpc}, r_{\rm max} = L_{\rm box}/2]$, as described in the main text.
• Figure 3: Graphical summary of our analysis: a) We cut an "observational sphere" from the test simulation box. A GW event is modeled as a 3D cone (red dots); b) the observer (black dot) is put at the centre of the "observational" sphere and the galaxies in the cone are assigned weights (geometrical or physical); c) each event produces an $H_0$ posterior; d) this procedure is repeated for multiple events and their $H_0$ posterior pdfs are combined. The $L_{\rm box}=100$ Mpc/h box, with fewer galaxies, has been chosen for visualisation purposes.
• Figure 4: Example of cone construction and of geometric weights: We show one example of cone construction and the weights assigned to each galaxy inside the cone. The black, dashed lines show the coordinates of the cone centre, the cyan star shows the position of the true GWs source, while the other dots show the galaxies that belong to this cone. Left: We show the angular position of the galaxies ($\phi, \theta$), together with the geometric weights they are assigned with respect their position to the cone centre. Centre: Same cone, but from the side, with the x-axis, showing the luminosity distance of the galaxies. The weights here correspond to the combination of angular and distance weights $w=w_{\rm \theta} \cdot w_{L}$ (see section \ref{['sec:Geometric_physical_weights']}). We also plot the distance likelihood, eq. (\ref{['eq:distance_lhd']}) - blue, solid line. Note that the maximum distance of a galaxy inside the cone is bigger than the $95\%$ range of the likelihood, and this is due to our consideration of the $H_0$ prior (recall section \ref{['sec:Cone_construction']}), i.e. for larger $H_0$ distances with the same redshift would be smaller, and as a result will be within the limits of our cone. Right: 3D configuration of the cone. The black dot corresponds to the centre of the box, i.e. the observer. The $L_{\rm box}=100$ Mpc/h box, with fewer galaxies, has been chosen for visualisation purposes.
• Figure 5: The most important distances in our cone construction. See discussion in section \ref{['sec:Cone_construction']} for details.