**Title: **Where are the supermassive black holes measured by PTAs?

**Authors: **Gabriela Sato-Polito, Matias Zaldarriaga, Eliot Quataert

**First Author Institution: **School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, USA

**Status:** Published in Physical Review D [closed access]

Cosmology and astrophysics have accumulated several “tensions” in recent years – or inconsistencies between different observational results. Some of these are the discrepancy in measuring the Universe’s expansion rate, the discrepancy in the measurement of the distribution of matter in the Universe, and the more-massive-than-expected early galaxies seen in JWST… But in the past year, another discrepancy in our measurements of the universe has appeared.

**The Unexpectedly-Loud Gravitational Wave Background**

As supermassive black hole (SMBH) binaries orbit each other, they emit gravitational waves (GWs). We expect there to be many SMBH binaries emitting GWs from all directions, creating a background hum that permeates the universe. This ‘gravitational wave background’ (GWB) cannot be sourced to any individual SMBH binary (unless they are loud enough), and is quite like the waves on the surface of the ocean. The amplitude of the GWB tells us about the number of SMBH binaries that are contributing to the GWB. For example, since SMBH binaries are the product of galaxy mergers, that will tell us about the rate of galaxy mergers, and from that we can learn more about the evolutionary history of our universe.

Last year, various pulsar timing array (PTA) experiments, such as NANOGrav, EPTA & InPTA, and PPTA, published evidence for a nanohertz-frequency GWB and a measured amplitude. Astrophysicists noticed an inconsistency between the amplitude of the GWB as measured by PTAs, and the expected amplitude given the number of SMBHs that we have directly observed in our local universe. This discrepancy can be seen in Figure 1.

*Figure 1: The **characteristic strain** (or ‘amplitude’) of the GWB at a frequency of 1/yr for various models of the population of supermassive black holes (red and blue points) compared to the measurements by NANOGrav, EPTA+InPTA, and PPTA. The right-hand axis shows the ratio of characteristic strain from these models/measurements relative to NANOGrav’s measurements, showing that the NANOGrav measurement is about 2x greater than the theory expects. Fig. 4 from today’s paper.*

So, why the discrepancy? The amplitude measured by PTAs may be inaccurate, and as they collect more data and improve their measurements and noise analysis, the amplitude could be revised in the future.

Alternatively, our model of the number of SMBHs could be inaccurate, which would affect the predicted amplitude of the GWB. Today’s paper sets out to investigate what aspects of the model could be changed to reconcile the expected GWB amplitude determined from these models with the measurements made by PTAs.

**Method 1: Modifying the Galaxy Scaling Relations**

One method of estimating the masses of black holes is to use the “M- relation,” an empirical relationship between the mass of a supermassive black hole (M) at the center of a galaxy, and the velocity dispersion of the stars () in the galaxy bulge. By observing the redshift in the light from the galactic bulge of a galaxy, which allows us to determine the velocity dispersion of the stars in the bulge, we can estimate the mass of the black hole from this empirical relation which was determined from optical observations. While there is a strong direct relationship between mass and velocity dispersion, there is a bit of scatter, which is parameterised by a variable . Increasing allows the mass of the black hole to be larger for a given measurement of .

Figure 2 shows some of the problems that we have to deal with here. The blue curve shows the values of . and required for the number density of black holes in the Universe to be consistent with the NANOGrav amplitude measurement. The point labelled x_{fid} is the best-fit value for . and from direct observations of supermassive black holes and their host galaxies. The red curve shows the values of . and are required to explain the current measurement of the mass density of black holes. *These curves are wildly inconsistent*. The red and blue lines intersect – x_{1} is the point at which they intersect at their respective central values, while x_{2} is the point at which they intersect at the 90% confidence level. Notice that x_{2 }has a similar to x_{fid} but a greater ., therefore increasing the intrinsic scatter . to allow for higher mass black holes would increase the expected amplitude. However, there is a catch – this choice of and . is inconsistent with the M- relation. Go figure.

*Figure 2: The values of and required to explain the measured amplitude of the GWB from PTA measurements (blue) and black hole mass density from direct observations of supermassive black hole host galaxies(blue). x _{1} and x_{2} are the two curves’ central and 90% confidence level intersections, while x_{fid} are the best-fit parameters from direct observations.*

Therefore, the paper’s authors introduce a new M- relation – one that increases the number of supermassive black holes at higher masses. To achieve a GWB amplitude consistent with the PTA measurements, we need about 10 black holes with masses of around within 100 Mpc of us. This prediction could be tested with future optical observatories.

**Method 2: Increase the number of mergers**

Another way to boost the GWB amplitude in population models is to increase the number of mergers that the black holes experience, which would increase the number of black holes that were there before merging (requiring the initial population to start with lower masses). Hence, this increases the number of GWs contributing towards the background, increasing the amplitude. Unfortunately, the authors find (via a lovely mathematical result), that the upper limit to the boost in amplitude of the GWB is a factor of 1.67. From Figure 1, we need it to be 2, so this is not enough to resolve the amplitude problem.

Therefore, we have three options to resolve the observational discrepancies – (1) there is a population of around 10 black holes with masses of within 100 Mpc of us that we haven’t observed with optical observatories yet; (2) our measurement of the GWB amplitude is inaccurate, or; (3) the GWB is not astrophysical, and could have a cosmological origin. Watch this space.

*Edited by Abbé Whitford*

*Featured image credit: AUI/NRAO, NAOJ, and Science/Nicole Rager Fuller*

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