Discriminating between different scenarios for the formation and evolution of massive black holes with LISA

Electromagnetic observations have provided strong evidence for the existence of massive black holes in the center of galaxies, but their origin is still poorly known. Different scenarios for the formation and evolution of massive black holes lead to different predictions for their properties and merger rates. LISA observations of coalescing massive black hole binaries could be used to reverse engineer the problem and shed light on these mechanisms. In this paper, we introduce a pipeline based on hierarchical Bayesian inference to infer the mixing fraction between different theoretical models by comparing them to LISA observations of massive black hole mergers. By testing this pipeline against simulated LISA data, we show that it allows us to accurately infer the properties of the massive black hole population as long as our theoretical models provide a reliable description of the Universe. We also show that measurement errors, including both instrumental noise and weak lensing errors, have little impact on the inference.

Figure 1

Normalized population distribution for different values of the mixing fraction between the fiducial LS and HS models. We show the 68% and 90% confidence intervals. The (source-frame) chirp mass distribution is the most sensitive to $\alpha$. The redshift distributions of detectable events look much more similar, unlike the effective spin distributions, as discussed in the main text. (a) Without SNR threshold. (b) With an SNR threshold of 10.

Figure 2

Normalized population distributions predicted by our fiducial model and the SN-delays model, both in the LS and HS scenarios. We show the 68% and 90% confidence intervals. While the chirp mass distributions in the two models are quite similar, the redshift and effective spin distributions are not.

Figure 3

Posterior distribution on $\alpha$ for observation sets with an increasing number of observed events, generated using different values of the mixing fraction ${\alpha }_0$: ${\alpha }_0=0.2$ (left), ${\alpha }_0=0.5$ (middle) and ${\alpha }_0=0.8$ (right). The posteriors peak near the true value and become narrower as we increase the number of events.

Figure 4

Evolution of the shift and the error on $\alpha$ (90% confidence interval) with the number of observed events. Evolution of the shift and the error on $\alpha$ (90% confidence interval) with the number of observed events. We consider two sets of observations, with 50 events (crosses) and 200 events (dots). The color scale indicates the value of ${\alpha }_0$. As expected, they tend to decrease as we observe more events. The fact that the points are equally distributed on both sides of the ${\alpha }_{\max }={\alpha }_0$ line indicates that there is little systematic bias in our analysis.

Figure 5

Kullback-Leibler divergence between the PPD and the population distribution for different observation sets generated with different values of ${\alpha }_0$. On the left (right) panel the observation sets contain 50 (200) observed events. The smaller the KL divergence, the better our inference of the population distribution. Increasing the number of events tends to improve the inference, as expected. (a) 50 events. (b) 200 events.

Figure 6

Population distribution and PPD for four sets of observations generated with different values of ${\alpha }_0$. Each observation set contains 100 events. On the upper-left and lower-right panels we show the cases that yield the largest and smallest values of the KL divergence among the cases shown in Fig. 5. The other two panels show cases yielding mid-range values of the KL divergence. (a) Worst case. (b) Mid-range case. (c) Mid-range case. (d) Best case.

Figure 7

Comparison between the population distribution for the SN-delays model and the PPD for an observation set containing 20 events from the same catalogue. Different rows refer to the HS variant (${\alpha }_0=0$, top), a mixing fraction ${\alpha }_0=0.5$ between the HS and LS variants (middle), and the LS variant (${\alpha }_0=1$, bottom). Panels on the left refer to the detectable population; panels on the right, to the intrinsic population.

Figure 8

Error on $\alpha$ and KL divergence between the rescaled posterior distribution of $\alpha$ and the (flat) prior. As we include the different sources of error, $\mathrm{\Delta }\alpha$ tends to increase and $D_{KL}$ tends to decrease, reflecting a degradation in the measurement of $\alpha$. Note that these are the errors and KL divergences for the rescaled posterior, i.e., we artificially bring the number of detected events to 1, as detailed in the main text.

Figure 9

Population distribution and PPDs obtained in the no-noise and $\mathrm{detector}+\mathrm{lensing}$ noise scenarios for a representative case. The dataset contains 500 events (before applying the detectability criterion). Including measurement errors barely affects our ability to infer the population distribution.

Figure 10

Comparison between different KDE approximations to the population probability density function of ${\log }_{10}({\mathcal{M}}_{c,s})$, using different values of the bandwidth. If the bandwidth is too small the KDE is not smooth, and if it is too large we cannot resolve the features of the distribution. For the case shown here, a bandwidth of 0.08 is a good choice. This value was obtained by minimizing the integrated squared error, as described in the main text.

Figure 11

Comparison between the selection functions obtained using different numbers of points.

Figure 12

Evolution of the bias and the error on $\alpha$ using the selection function in blue in Fig. 11 (top) and the one in orange (bottom). We can clearly observe a systematic bias in the latter case due to misevaluation of the selection function. (a) We use $8\times {10}^5$ points to evaluate the selection function of the LS and HS variants. (b) We use $2\times {10}^3$ points to evaluate the selection function of the LS and HS variants.

Figure 13

Comparison between the population distributions obtained from numerical simulations and the KDE we build from it. We purposefully did not smooth the corner plot in order to reflect the real level of agreement between the two distributions. The top and bottom panels refers to the LS and HS variants, respectively. The “bumpy” histograms for the HS variant (in particular for the spin) highlight that we do not have enough points to build an accurate enough KDE for our purposes. However the two distributions are overall in good agreement, and therefore we expect that the approximation of using the KDE as our “true” fiducial astrophysical model should not sensibly affect our results.