Inferring Mbh-Mbulge Evolution from the Gravitational Wave Background

TL;DR

This work investigates whether the gravitational wave background (GWB) amplitude observed by pulsar timing arrays can be reconciled with theory by allowing evolution in the black hole–bulge mass relation or by altering the galaxy stellar mass function (GSMF). Using a semi-analytic holodeck framework, the authors synthesize SMBH populations by convolving the GSMF with a redshift-dependent $M_ ext{BH}$–$M_ ext{bulge}$ amplitude, then apply merger and hardening physics to predict the GWB spectrum and fit it to PTA data, while simultaneously evaluating consistency with electromagnetic constraints via $D_{KL}$ and a GSMF boost metric $\Xi$. They explore eight model variants with varying GSMF priors and whether the amplitude evolves as $\alpha(z)=\alpha_0(1+z)^{\alpha_z}$, finding a positive evolution (median $\alpha_z$ around 0.8–1.0, with $\alpha_z=1.04\pm0.5$ in models with strong evidence) generally provides a better description of the GWB, though substantial degeneracy exists with the GSMF; the results favor a SMBH-first growth scenario, where SMBHs grow relatively faster than their hosts at high redshift. The study highlights the importance of robust binary hardening modeling and the potential of GW data to complement EM-based SMBH–galaxy coevolution constraints, while cautioning that GSMF uncertainties can mimic BH-growth signals and that future PTA data will be crucial to break these degeneracies.

Abstract

We test the impact of an evolving supermassive black hole (SMBH) mass scaling relation (Mbh-Mbulge) on the predictions for the gravitational wave background (GWB). The observed GWB amplitude is 2-3 times higher than predicted by astrophysically informed models which suggests the need to revise the assumptions in those models. We compare a semi-analytic model's ability to reproduce the observed GWB spectrum with a static versus evolving-amplitude Mbh-Mbulge relation. We additionally consider the influence of the choice of galaxy stellar mass function on the modeled GWB spectra. Our models are able to reproduce the GWB amplitude with either a large number density of massive galaxies or a positively evolving Mbh-Mbulge amplitude (i.e., the Mbh / Mbulge ratio was higher in the past). If we assume that the Mbh-Mbulge amplitude does not evolve, our models require a galaxy stellar mass function that implies an undetected population of massive galaxies (Mstellar > 10^11 Msun at z > 1). When the Mbh-Mbulge amplitude is allowed to evolve, we can model the GWB spectrum with all fiducial values and an Mbh-Mbulge amplitude that evolves as alpha(z) = alpha_0 (1 + z)^(1.04 +/- 0.5).

Figures (14)

• Figure 1: Here we show the four posterior distributions for $\alpha_z$ from our models. The upper red and lower blue horizontal lines indicate the 68% confidence region for Le03ev and Le11ev. The gray dashed line shows our uniform prior. These are functionally identical to that of LM11ev and Le00ev respectively. In each distribution 87.2% - 99.3% of values are positive, indicating a moderate to strong positive evolution in the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude. The posterior distributions for our models fall into two categories: (i) A wide range of $\alpha_z$ values with a significant (greater than 10%) fraction of the distribution falling below $\alpha_z = 0$ and (ii) A very narrow, nearly symmetrical, distribution of $\alpha_z$ values with only a negligible (under 5%) fraction of the distribution sitting below $\alpha_z = 0$. This first category corresponds to models that sampled all 11 GSMF parameters (Le11ev and LM11ev), these distributions, while largely positive, are consistent with no significant redshift evolution of the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ relation. The latter category, however, show strong evidence for a positive $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude evolution. The distributions from this second category had some or all of the GSMF parameters fixed (Le03ev and Le00ev) in the prior set up. With fewer degrees of freedom these models converged to higher values of $\alpha_z$ to a higher degree of confidence suggesting a better constraint for the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude evolution than the larger models. This is indicative of the degeneracy between the GSMF and $M_\mathrm{BH}$--$M_\mathrm{bulge}$ parameters in our models.
• Figure 2: The posterior distributions for each of the evolving (red solid) and non-evolving (blue dashed) models. The priors are represented by the gray histograms in each panel and the gray vertical lines indicate the fiducial values. Each row represents a different evolving / non-evolving $M_\mathrm{BH}$--$M_\mathrm{bulge}$ model pair---top: Le11ne and Le11ev, middle: Le03ne and Le03ev, bottom: Le00ne and Le00ev. In the case where a parameter was fixed, there is no histogram and only the vertical line is shown. In each panel we show the Kullback-Leibler divergence, $D_{KL}$, between each model posterior and the prior. The top, blue, number is for the Le_ne models and the bottom, red, number is for Le_ev models. For each parameter, $D_{KL}$ is equal to or lower for models which allow for $M_\mathrm{BH}$--$M_\mathrm{bulge}$ evolution, indicating that the posterior distributions are in equal or better agreement with the priors when compared to the fixed $\alpha_z=0$ counterparts. All three posterior distributions for $\alpha_z$ (right-most column) demonstrate a preference for positive values. The $\alpha_z$ posterior is relatively broad for the top row (Le11ev), but when the evolutionary GSMF parameters are fixed (Le03ev, middle row) the distribution shifts towards higher values and also narrows. When only the local GSMF parameters were sampled, but $\alpha_z$ was fixed to 0 (Le03ne), the posterior distributions deviated from the priors, not only for GSMF parameters (e.g, $M_{c, 0}$, $M_{c, 1}$, and $M_{c, 2}$), but also for $\alpha_0$. When $\alpha_z$ was allowed to vary, the posteriors recover the prior GSMF parameters. Additionally, when any number of the GSMF parameters are fixed (bottom two rows, Le03ev and Le00ev), the $\alpha_z =0$ models return high posterior values for the local $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude, $\alpha_0$. This is indicative of the degeneracy between increasing galaxy number density and increasing the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude (either locally and/or at high-$z$). We also see little change between $\alpha_0$ and $\alpha_z$ posteriors in the middle and lower rows suggesting that fixing the three local GSMF parameters (Le00ev) does not have a further affect over fixing only the evolutionary parameters. Overall, the models that allow the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ relation to evolve are otherwise in better agreement with observational constraints for the GSMF and local $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude and have posterior distributions for $\alpha_z$ that are between 87 and 99% positive.
• Figure 3: The GWB spectra associated with the best-fit parameters from our 8 models fit to the first five frequency bins. The blue dashed lines indicate models with $\alpha_z$ fixed to 0 and red solid lines indicate models which allowed for $M_\mathrm{BH}$--$M_\mathrm{bulge}$ evolution. In all four panels, the spectrum from the evolving $M_\mathrm{BH}$--$M_\mathrm{bulge}$ models are in good agreement with the data though we note that the likelihood value for Le00ev is the lowest of the 8 models (discussed further in section \ref{['sec:res_gsmf_choice']} and appendix \ref{['sec:app_comp']}). There are some differences between the slope of the high-frequency end, but this portion of the spectrum is poorly constrained and it is not possible to distinguish between goodness of fit in this regime at this time. When all GSMF parameters are sampled (left- and right-hand panels, Le11ne and LM11ne), the spectra are nearly identical between the evolving and non-evolving models. The models with some/all GSMF parameters fixed (Le03ne and Le00ne) show more significant differences between the evolving and non-evolving models. For these models, those that allowed $\alpha_z$ to vary are consistently in better agreement with the data than the fixed $\alpha_z = 0$ counterparts indicating that these evolving models are a better description of the GWB than their non-evolving counterparts.
• Figure 4: Here we plot the difference between the posterior GSMF from both the evolving (Le11ev, red solid) and non-evolving (Le11ne, blue dashed) $M_\mathrm{BH}$--$M_\mathrm{bulge}$ model and the observed GSMF from Leja_2020. The light gray region represents the 1$\sigma$ error in the GSMF from Leja_2020, we show the equivalent error range for Le11ne in blue and for Le11ev in red. We compare the models only within the redshift range used in Leja_2020$0.2 < z < 3$. We see that, when the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ relation is not allowed to evolve, the GSMF shows higher number densities of high-mass galaxies with an increasing discrepancy as redshift increases. This difference is highest for galaxies with $M_\star > 10^{11} M_\odot$ though by $z = 3$ the posterior GSMF from Le11ne is inconsistent with the observed GSMF at all masses $M_\star > 10^{9} M_\odot$. When the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude is allowed to evolve, however, the posterior GSMF is consistent within the uncertainties of the observed GSMF though with a minor positive offset. This difference is evidence that our best-fit models with an evolving $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude are in better agreement with observational constraints for galaxy number density than the non-evolving models.
• Figure 5: Top left: The prior GSMF from Leja_2020. Top middle: The posterior GSMF for Le11ne. Top right: The posterior GSMF for Le11ev. Bottom: Same as top row, but for models based on our strongly evolving version of the Liepold_2024 GSMF. We see that, in both middle plots, the models that have $\alpha_z = 0$ require a large number density of galaxies relative to their respective fiducial model. Despite the increased local number density, the posterior GSMF from LM11ne returns a greater number density for $z = 3$ compared to the Leja_2020 GSMF. The prior for this model had a similar number density to that of Leja_2020 in this redshift range, by design. The number densities for $2 \lesssim z \lesssim 3$ are similar between the two posterior models with $\alpha_z = 0$. This means that, to reproduce the GWB without an evolving $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude, we need a significantly higher number density of massive galaxies across out to $z \sim 3$. For both models that allowed $\alpha_z$ to vary (right-most panels) the posterior GSMFs are only negligibly different from the priors and the corresponding median posterior values for $\alpha_z$ are similar (see Figure \ref{['fig:alphaz']} and Table \ref{['tab:alphaz']}) implying that the locally increased number density for LM11ev does not sufficiently increase the GWB amplitude without additionally increasing the high-$z$ GSMF number density or the $M_\mathrm{BH}$--$M_\mathrm{bulge}$ amplitude.