On the importance of source population models for gravitational-wave cosmology

Knowledge of the shape of the mass spectrum of compact objects can be used to help break the degeneracy between the mass and redshift of the gravitational wave (GW) sources and thus can be used to infer cosmological parameters in the absence of redshift measurements obtained from electromagnetic observations. In this paper, we study extensively different aspects of this approach, including its computational limits and achievable accuracy. Focusing on ground-based detectors with current and future sensitivities, we first perform the analysis of an extensive set of simulated data using a hierarchical Bayesian scheme that jointly fits the source population and cosmological parameters. We consider a population model (power-law plus Gaussian) which exhibits characteristic scales (extremes of the mass spectrum, presence of an accumulation point modeled by a Gaussian peak) that allow an indirect estimate of the source redshift. Our analysis of this catalog highlights and quantifies the tight interplay between source population and cosmological parameters, as well as the influence of initial assumptions (whether formulated on the source or cosmological parameters). We then validate our results by an “end-to-end” analysis using simulated GW $h(t)$ data and posterior samples generated from Bayesian samplers used for GW parameter estimation, thus mirroring the analysis chain used for observational data for the first time in literature. Our results then lead us to re-examine the estimation of $H_0$ obtained with GWTC-1 in Abbott et al. [LIGO Scientific, Virgo Collaborations, Astrophys. J. 909, 218 (2021)], and we show explicitly how population assumptions impact the final $H_0$ result. Together, our results underline the importance of inferring source population and cosmological parameters simultaneously (and not separately as is often assumed). The only exception, as we discuss, is if an electromagnetic counterpart was to be observed for all the BBH events; then, the population assumptions have less impact on the estimation of cosmological parameters.

Figure 1

Simulated population of 1024 observed events, showing the mass distributions (in the detector and source frames) and redshift distribution.

Figure 2

Posterior probability density distributions on the different population parameters (PLG mass model) as more and more GW detections are analyzed (horizontal axis). The horizontal black dashed line indicates the true parameters of the population. The blue posteriors are obtained by fixing ${\mathrm{\Omega }}_{m,0}=0.308$, while the orange posteriors are marginalized over the estimation of ${\mathrm{\Omega }}_{m,0}$.

Figure 3

(Left) Accuracy at the 90% CL on the population parameters (see levels) when we combine more and more GW detections. For this case, we fix ${\mathrm{\Omega }}_{m,0}=0.308$ to the injected value. (Right) Same but varying ${\mathrm{\Omega }}_{m,0}$ in the range $\in [0.1,0.5]$.

Figure 4

Posterior distribution on $H_0$ and ${\mathrm{\Omega }}_{m,0}$ for 1024 BBH events detected with LIGO and Virgo at current sensitivities. The blue lines show the true parameters.

Figure 5

Cumulative distributions of the population source-frame masses built by stacking the events posterior samples of the detected binaries (top, primary mass; bottom, secondary mass) inferred from the simulated population and assuming different values for $H_0$. The position of the Gaussian peak and the maximum BH mass for BH are indicated by red and green areas, respectively, see Sec. 4b for an interpretation of this plot. The original prior has not been removed.

Figure 6

Posterior distribution on the $H_0$, $m_{\max }$, and ${\mu }_g$ for 64 BBH events detected with LIGO and Virgo at current sensitivities. The blue lines show the true parameters. The contours indicate the $1\sigma$ and $2\sigma$ confidence level intervals.

Figure 7

Posterior probability density distributions for the population parameters and $H_0$ using posterior from 200 events generated with full parameter estimation. The blue lines indicate the population injected values. Levels indicate the 68% and 90% confidence intervals.

Figure 8

Posterior distribution for $H_0$ obtained by fixing $m_{\max }$ and ${\mu }_g$ in a range around their true values $m_{\max }=85M_{\odot }$ and ${\mu }_g=40M_{\odot }$. The black dashed line indicates the true value of $H_0$.

Figure 9

(Top) Posterior predictive check labeled as “expected” for the two source-frame masses and redshift fitting an underlying PLG population with the correct mass model in comparison with 64 observed events, labeled as “detected”. The detected distributions match the posterior predictive checks. (Bottom) Same but fitting a subcomplete PL model to the underlying PLG model. The model struggles to fit the lower end part of the mass distribution and the excess of BHs around $40–50M_{\odot }$.

Figure 10

Posterior distribution on $H_0$ using the six GWTC-1 events with $\mathrm{SNR}>12$ and the GLADE and DES galaxy catalogs. The plot compares the results obtained in [25] with the new results of this paper, see discussion in Sec. 6.

Figure 11

Hubble constant posterior generated from 64 of our synthetic population of BBHs (Sec. 3a) fixing different population models and providing the redshift of the GW source (assumed from an EM counterpart). An incorrect choice of one of the population parameters (see legend) does not affect significantly the $H_0$ estimation. The vertical dashed line indicates the injected value. Note that for this plot the posterior samples are generated taking into account the correlations between masses and luminosity distance [point (ii) above is dropped].