Title: A novel approach to understanding the link between supermassive black holes and host galaxies
Authors: Gabriel Sasseville, Julie Hlavacek-Larrondo, Samantha C. Berek, Gwendolyn M. Eadie, Carter Lee Rhea, Aaron Springford, Mar Mezcua, Daryl Haggard
First Author’s Institution: Department of Physics, University of Montreal, Montreal, Canada
Status: Published in The Astrophysical Journal [OPEN Access]
Most galaxies are thought to host a supermassive black hole (SMBH) at their center, and these black holes play a crucial role in shaping how galaxies evolve. There is a strong connection found between the properties of a galaxy and its central black hole. One of the most well-known examples is the relationship between the black hole’s mass and the random motions of stars near the galaxy’s center, measured by a quantity called stellar velocity dispersion (denoted as σ). This connection, known as the M-σ relationship, has been studied for decades and is often used to estimate the mass of a galaxy’s central black hole.
Interestingly, some galaxies appear to lack a central black hole. In many cases, observations only provide an upper limit on the possible mass of any black hole that might be there. In today’s paper, the authors study the M-σ relationship with a new statistical approach to account for galaxies that might not have a central black hole and improve the calculation of this relationship.
Hurdle-up, the Bayesian way
The authors use a statistical method called the Bayesian hurdle model. To update our understanding, bayesian modeling combines prior beliefs or initial guesses with new evidence. The Bayesian hurdle model approach tackles two key questions: first, whether a galaxy is likely to host a central black hole (BH), and second, if it does, how the black hole’s mass relates to the galaxy’s velocity dispersion (σ). The model works in two stages. In the first step, called the “hurdle,” a logistic regression determines the probability that a galaxy has a central BH. If the galaxy clears this hurdle (i.e., is likely to host a BH), the second step uses linear regression to establish the relationship between the BH’s mass and the galaxy’s velocity dispersion, analyzed on a logarithmic scale.
The study examines a sample of 244 galaxies where either the central BH mass has been measured directly or an upper limit has been estimated. From the logistic regression step, the authors find that galaxies with σ > 126 km/s have a 99% probability of hosting a central BH. This result suggests that while massive galaxies are almost certain to have BHs, smaller dwarf galaxies are much less likely to host them.
A steeper correlation
The hurdle model reveals a relationship of \(M \propto \sigma^{5.8}\) between black hole mass and velocity dispersion, as shown in Figure 1 (solid black line). This result is steeper than the correlations found in previous studies. The difference likely arises from the authors’ inclusion of upper limits on black hole masses in their hurdle model analysis. The hurdle model also predicts several undermassive black holes in the range of \(10^1-10^5\) solar masses compared to the other linear model studies. This downward shift in the lower mass range compared to other models is seen in Figure 1. Also, the breakpoint between under-massive and over-massive black holes occurs at a lower mass than previously reported. This shift is due to the hurdle model’s handling of upper limits in the logistic regression step, which pulls the curve downward. Also, the turning point (see Figure 1) between under-massive and over-massive black holes occurs at a lower mass than previously reported. This shift is due to the hurdle model’s handling of upper limits in the logistic regression step, which pulls the curve downward.
The authors highlight that their model could benefit from better parameter estimates for the hurdle step, requiring further analysis of upper limits on black hole masses. They also recommend exploring alternative scaling relationships for calculating black hole masses. For example, the more well-established, BH mass – galaxy stellar mass relationship can be used. Combining stellar mass with the hurdle model’s insights on the M-σ relationship could provide an even more comprehensive understanding of black hole-galaxy connection. Understanding this connection is critical to unraveling how black holes and galaxies co-evolve.
Astrobite edited by Cole Meldorf
Featured image credit: Pranav Satheesh and Figure 1 from the paper
Thank you Pranav for your comments here. I want to be clear on a few things. First, the most important aspect of the Bayesian approach seems to be that it provides researchers with a statistical sample that is more valid (if we are to study the mass/dispersion link, we should not include galaxies that do not have MBH. The logistic regression appears to do this, but not sure “how”. Second, you write: “This result suggests that while massive galaxies are almost certain to have BHs, smaller dwarf galaxies are much less likely to host them” — I assume this inference is because smaller dwarf galaxies have too low of a stellar velocity dispersion such that a MBH is very unlikely. Is this a correct inference? Last, I assume the main point is that velocity dispersions are much easier to measure than finding a MBH, thus we can more quickly build a database of galaxies which we believe have a MBH (and establish its mass) simply by measuring the nearby stellar velocity dispersion. Correct?