# Expectations for Horizon-Scale Supermassive Black Hole Population Studies with the ngEHT

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## Abstract

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## 1. Introduction

- the angular size of the SMBH shadow ($\theta $);
- the total horizon-scale flux density emitted by the source (${S}_{\nu}$); and
- the optical depth of the emitting material.

## 2. Measurable Proxies for Quantities of Interest

#### 2.1. Proxy for SMBH Shadows

#### 2.2. Proxy for SMBH Masses

#### 2.3. Proxy for SMBH Spins

## 3. Synthetic Data Generation and Fitting Procedure

`ngehtsim`4 package, which expands on the synthetic data generating functionality of the

`ehtim`library [30,31]. We assume the observations are carried out at an observing frequency of 230 GHz and with 8 GHz of bandwidth using the “full” ngEHT Phase 1 array configuration from [32], which consists of the 2022 EHT array plus the OVRO 10.4 m dish, the Haystack 37 m dish, and three 6.1 m dishes located in Baja California (Mexico), Las Campanas Observatory (Chile), and the Canary Islands (Spain). We use historical weather data to determine appropriate system equivalent flux densities at each site following a procedure similar to that in Raymond et al. [33]. To emulate fringe-finding signal-to-noise ratio (SNR) thresholds, we flag any visibilities from baselines that contain a station not participating in at least one other baseline that achieves an SNR of 5 in a 10-s integration time. We add complex station gain corruptions at the level of 10% in amplitude and uniformly sampled within $[0,2\pi ]$ in phase for all stations on every 300-s time interval, to emulate scans, and we assume that the data have been calibrated to remove polarimetric leakage effects.

`ngEHTforecast`5 package. This approach does not explicitly carry out fits of the model to the data; instead, it assumes that a “good” fit to the data has already been achieved, and it then provides an estimate of the uncertainty in each of the fitted parameters via a second-order expansion of the logarithmic probability density around the best-fit location. We compute parameter precision estimates assuming that the fits have been carried out using complex visibilities as the input data products, with broad priors on the station gain amplitudes and phases at every scan.

## 4. Results: The Expected Number of Measurable SMBH Masses, Spins, and Shadows

- Our condition for whether a SMBH has a measurable mass is that the fractional uncertainty in the measurement of the ring diameter d must be at the level of 20% or lower (i.e., it is measured with a statistical significance $\gtrsim 5\sigma $). Values of $(\theta ,{S}_{\nu})$ for which this condition is satisfied fall to the upper right of the red dashed curve in Figure 3.
- Our condition for whether a SMBH has a measurable spin is that the uncertainty in the measurement of all spin-relevant parameters (as determined by Qiu et al. [26]; see also Section 2.3) must be at the level of 20% or lower. Specifically, we require the fractional uncertainty in $|{\alpha}_{1}|$, $|{\beta}_{1}|$, and $|{\beta}_{2}|$ and the uncertainty in $\mathrm{arg}\left({\beta}_{1}\right)$ and $\mathrm{arg}\left({\beta}_{2}\right)$ to all be less than 0.2 (i.e., 20%). Values of $(\theta ,{S}_{\nu})$ for which this condition is satisfied fall to the upper right of the green dashed curve in Figure 3.
- Our condition for whether a SMBH has a measurable shadow is that the fractional width $W/d$ deviates from unity with an uncertainty of 20% or smaller; i.e., we require that $W<d$ with a statistical significance $\gtrsim 5\sigma $. Values of $(\theta ,{S}_{\nu})$ for which this condition is satisfied fall to the upper right of the blue dashed curve in Figure 3.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notes

1 | The procedure Pesce et al. [19] used to determine the number of observable SMBHs involves integrating the supermassive black hole mass function (BHMF) to determine how many objects have shadow diameters larger than $\theta $, while also using a semi-analytic spectral energy distribution model and adopting an empirically motivated prescription for the SMBH Eddington ratio distribution function to restrict the objects under consideration to those that have flux densities greater than ${S}_{\nu}$ and accretion flows that are optically thin. The distribution of sources used in this paper assumes an observing frequency of 230 GHz and a BHMF determined using the stellar mass function from Behroozi et al. [20] scaled according to the relation determined by Kormendy and Ho [21] (i.e., the “upper BHMF” from Pesce et al. [19]). |

2 | |

3 | |

4 | https://github.com/Smithsonian/ngehtsim, accessed on 5 November 2022. |

5 | https://github.com/aeb/ngEHTforecast, accessed on 5 November 2022. |

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**Figure 1.**Example polarized source model used for generating the synthetic data described in Section 3. The grayscale image shows the Stokes I emission, while the colored ticks mark the EVPA of the linear polarization structure. The length of each tick is proportional to the intensity of the linear polarization (i.e., $\left|P\right|$), while the color of each tick reflects the fractional polarization (i.e., $\left|P\right|/I$).

**Figure 2.**Estimated sky density of SMBHs with measurable masses (top), spins (middle), and shadows (bottom), as a function of right ascension and declination. These estimates have been determined according to the criteria outlined in Section 4, and they assume an underlying distribution of observable SMBHs from Pesce et al. [19]. The stochastic variations seen from pixel to pixel are primarily the result of sampling noise. The location of M87* is marked with a red star.

**Figure 3.**Approximate number density of SMBHs that are expected to satisfy different thresholds of measurability, assuming an observing frequency of 230 GHz. The background colorscale and contours mark the number density (per unit solid angle) of SMBHs that have flux densities greater than ${S}_{\nu}$ and shadow diameters larger than $\theta $, as a function of ${S}_{\nu}$ and $\theta $ and assuming that sources are distributed isotropically on the sky [19]. The solid contours start with the thick contour indicating a count of 1 and then increase by factors of 10 towards the lower left, while the dashed contours each decrease by a factor of ten towards the upper right. The overplotted colored dashed contours indicate where various parameters of interest could be measurable for different combinations of $(\theta ,{S}_{\nu})$, assuming observations appropriate for the “full” ngEHT Phase 1 array observing at a declination of 10 degrees (i.e., averaged over right ascension). The red dashed contour marks the lower boundary of the region in which black hole mass can be measured, the green dashed contour marks the lower boundary of the region in which black hole spin can be measured, and the blue dashed contour marks the lower boundary of the region in which black hole shadow can be measured.

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## Share and Cite

**MDPI and ACS Style**

Pesce, D.W.; Palumbo, D.C.M.; Ricarte, A.; Broderick, A.E.; Johnson, M.D.; Nagar, N.M.; Natarajan, P.; Gómez, J.L. Expectations for Horizon-Scale Supermassive Black Hole Population Studies with the ngEHT. *Galaxies* **2022**, *10*, 109.
https://doi.org/10.3390/galaxies10060109

**AMA Style**

Pesce DW, Palumbo DCM, Ricarte A, Broderick AE, Johnson MD, Nagar NM, Natarajan P, Gómez JL. Expectations for Horizon-Scale Supermassive Black Hole Population Studies with the ngEHT. *Galaxies*. 2022; 10(6):109.
https://doi.org/10.3390/galaxies10060109

**Chicago/Turabian Style**

Pesce, Dominic W., Daniel C. M. Palumbo, Angelo Ricarte, Avery E. Broderick, Michael D. Johnson, Neil M. Nagar, Priyamvada Natarajan, and José L. Gómez. 2022. "Expectations for Horizon-Scale Supermassive Black Hole Population Studies with the ngEHT" *Galaxies* 10, no. 6: 109.
https://doi.org/10.3390/galaxies10060109