Investigating supermassive black hole mergers using cosmological simulations

Title: High-redshift supermassive black hole mergers in simulations with dynamical friction modelling

Authors: Colin DeGraf, Nianyi Chen, Yueying Ni, Tiziana Di Matteo, Simeon Bird, Michael Tremmel, Rupert Croft

First Author’s Institution: Department of Physics, Truman State University, Kirksville

Status: Submitted to MNRAS, available on arXiv

Ground-based gravitational wave (GW) detectors like LIGO and Virgo have been detecting gravitational waves from stellar-mass black hole mergers for the past several years. Higher-mass mergers like between two supermassive black holes (SMBH) produce gravitational waves with longer wavelengths. These mergers are beyond the sensitivity of the current gravitational wave detectors but will be seen by the future space-based gravitational-wave detector Laser Interferometer Space Antenna (LISA).

Supermassive black holes are found at the center of galaxies and have a “special relationship” with their host galaxies. When two galaxies merge, the central BHs move toward the new galactic center, forming an SMBH binary and eventually merging. Since the gravitational waves from such SMBH mergers are expected to be seen in LISA, it is necessary to study their population and get a prediction of their merger rates and the GW signals. As cosmological simulations model BHs and galaxy evolution and span a wide range of redshift (effectively cosmic time), they are the perfect tool for probing the population of SMBH mergers.  In today’s paper, the authors study the Astrid simulation to characterize supermassive black hole mergers at high redshift (z > 2). 

Merging populations in the simulation

The authors use the Astrid cosmological simulation to model the merging black hole population. Most cosmological simulations only resolve higher-mass black holes (around 10^6 solar masses) and it misses a lot of LISA-detectable mergers. The Astrid simulation however can resolve lower-mass black hole mergers as it uses a lower-mass seed model ( down to 10^{4.5} solar masses). Astrid also implements dynamical friction that accounts for the effective loss of kinetic energy and momentum of the black holes through gravitational interaction with surrounding matter. This helps in accurately modeling the orbital dynamics of the inspiralling BHs to small scales.

A plot showing the black hole mass function for the Astrid and TNG300 simulations. The x-axis represents the mass of the black holes in a log scale and the y-axis shows the BHMF. The black hole seed mass is seen as the lowest mass for both simulations. The Astrid black hole mass function is okitted in red and TNG300 is plotted in blue for different redshift values. We see that the TNG300 mass functions peaks around 10^6 solar mass whearas the peak for Astrid is around 10^4 solar mass. Bothe Astrid and TNG mass functions agree for redshifts greater than 4.
Figure 1: Black hole mass functions (BHMF) for the simulations (Astrid in red and TNG300 in blue), which is defined as the number of single BH sources per co-moving volume. The x-axis represents the mass of the black holes in a log scale and the y-axis shows the BHMF. The black hole seed mass is seen as the lowest mass for both simulations. TNG300 uses a higher mass seed (around $\latex 10^6$) whereas Astrid can go to lower BH masses. We also see that they agree for higher masses for redshifts, z>4. Figure 1 in the paper.

To study how accounting for dynamical friction and a lower-mass seed affects the merging black hole population, the authors compare results from the Astrid simulation to another cosmological simulation: TNG300. Figure 1 shows the overall black hole populations in both cosmological simulations over cosmic time. We can see that the Astrid model (in red) spans a wide range of black hole masses across all redshift values whereas the TNG300 simulation has a large spike at around 10^6 solar mass which corresponds to the lowest seed mass used in the TNG simulation. Both the models broadly agree above the TNG seed mass (10^6) for higher redshifts but TNG produces more higher mass black holes at lower redshifts (later times). 

Expectations for LISA

A plot of the black hole merger rates for Astrid and TNG simulations. A solid red line for Astrid and a solid blue line for TNG300. The rates for Astrid are greater than TNG which is because of a larger mass range. When limiting the masses to be greater than the seed masses in TNG, the Astrid merger rate is smaller. The limited range Astrid merger rate is plotted in dotted-red style and the limited range TNG is plotted in blue-dotted style.
Figure 2: The rate of the black hole mergers is plotted as a function of the redshift for different cosmological simulations. The dotted lines correspond to the case after imposing the condition that the seed masses must be greater than the seed mass of TNG (M > 2 * M_{seed, TNG}). This removes the peak we see in Figure 1 for TNG300; hence, the blue dotted line appears different from the blue solid one. The total merger rate for Astrid (solid red) is much greater than all other simulations because of a larger mass range. However, for the limiting case, Astrid has a lower merger rate (red dotted lines). Source: Figure 2 in the paper.

The authors calculate the rate at which gravitational wave signals from SMBH reach LISA by integrating the total mergers in the simulation over all redshift values. Figure 2 shows the merger rate as a function of redshift for the Astrid and TNG simulations. Astrid simulation has a higher merger rate than all the other TNG simulations as it includes black holes much lower in mass than those in TNG. However, when limiting Astrid to black hole mergers massive enough to be resolved in TNG300 (red dotted lines), we see that Astrid has a lower merger rate than TNG300. This is because, in the Astrid simulation, the black holes take longer to inspiral towards the galactic center due to the inclusion of dynamical friction. Therefore the inclusion of low-mass seeds and a dynamical friction model significantly affects the BH merger rates in cosmological simulations.

The authors also compute the expected gravitational wave signals from the BH mergers in the simulation and plot the GW strain (the strain is effectively a measure of the GW magnitude) against their frequencies (see Figure 3). When we compare the strain-frequency plot of Astrid (Figure 3a) to that of TNG (Figure 3b), it is clear that the TNG simulations only cover the low-frequency high-strain regime. Therefore, including lower-mass seed black holes, which produce higher GW signal frequencies, is crucial to probe the full range of LISA detections. The authors verify this with Figure 3c where they limit the mergers in Astrid to the same mass range as TNG and we see the higher frequency portion of the plot missing.

A 4x4 plot of the gravitational wave strain versus the frequency.  The gravitational wave strain for all mergers in the simulation is plotted against frequency along with the sensitivity curve of LISA represented by black dotted lines. The strain is also color-coded by redshift. The plot on the top left shows the strain-frequency for Astrid mergers and the plot on top right is for TNG300. The plot for TNG300 shows that it is only limited to the low-frequency high-strain regime. The plot on the bottom left shows Astrid mergers but only limited to the masses greater than the seed mass of TNG. When compared to the plot on the bottom right for TNG in the same limiting case and redshift range, we see they agree with each other.
Figure 3:  Frequency-strain plot for gravitational wave signals emitted by the mergers in the simulations color-coded by the redshift. The black dotted line is the LISA sensitivity curve above which LISA can detect the mergers. (a): All GW signals in the Astrid simulation for z>2. (b): All GW signals in TNG300 simulation for z>0. Notice that TNG300 only covers the low-frequency high-strain regime. (c): GW signals in Astrid but limited by the seed mass condition (M_{BH} > M_{seed, TNG}). (d) GW signals in TNG but only for z>2 to compare with Astrid simulation. Figure 6 in the paper.

In conclusion, the authors use the cosmological simulation Astrid to illustrate the importance of using a wide range of black hole seed mass down to lower mass black holes to help in probing BH mergers visible over a wide range of LISA detections. Further, they demonstrate the need to account for dynamical friction in cosmological simulations as it accurately models the smaller scale dynamics of the inspiralling BHs. Thus, cosmological simulations like the Astrid and TNG can help us in predicting expected gravitational wave signals in future gravitational wave observations.

Astrobite edited by Megan Masterson

Featured image credit: Adapted from illustris, NASA and Wikimedia Commons

About Pranav Satheesh

I am a second year graduate student in physics at the University of Florida. My research focuses on studying supermassive binary and triple black hole dynamics using cosmological simulations. In my free time, I love drawing, watching movies, cooking, and playing board games with my friends.

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