Work Description

Last updated: 7 July, 2023

Title: The Impact of Black Hole Scaling Relation Assumptions on the Mass Density of Black Holes

Authors: Cayenne Matt, Kayhan Gültekin, Joseph Simon

Abstract: We examine the effect of supermassive black hole (SMBH) mass scaling relation choice on the inferred SMBH mass population since redshift z ∼ 3. To make robust predictions for the gravitational wave background (GWB) we must have a solid understanding of the underlying SMBH demographics. Using the SDSS and 3D HST+CANDELS surveys for 0 < z < 3 we evaluate the inferred SMBH masses from two SMBH-galaxy scaling relations: MBH–Mbulge and MBH–sigma. Our SMBH mass functions come directly from stellar mass measurements for MBH–Mbulge, and indirectly from stellar mass and galaxy radius measurements along with the galaxy mass fundamental plane for MBH–sigma. We find that there is a substantial difference in predictions especially for z > 1, and this difference increases out to z = 3. In particular we find that using velocity dispersion predicts a greater number of SMBHs with masses greater than 10^9Msun . The GWB that pulsar timing arrays find evidence for is higher in amplitude than expected from GWB predictions which rely on high redshift extrapolations of local SMBH mass-galaxy scaling relations. The difference in SMBH demographics resulting from different scaling relations may be the origin for the mismatch between the signal amplitude and predictions. Generally, our results suggest that a deeper understanding of the potential redshift evolution of these relations is needed if we are to draw significant insight from their predictions at z > 1.

Contact: Cayenne Matt, cayenne@umich.edu

Methodology: The galaxies in this data with redshifts less than 0.5 were queried from the Sloan Digital Sky Survey DR7 data base. The stellar masses and half-light radii for these low redshift galaxies were determined and published by Chang et al. (2015) and Simard et al. (2011) respectively and we use their data. The remaining galaxies are from the 3D+HST CANDELS survey, their stellar masses and half-light radii were determined and published by Leja et al. (2019) and van der Wel et al. (2014) respectively and we use their data. The data in MasterFile.csv contain the published values for the above parameters along with several other parameters included in their tables. From this file, number densities were calculated and fit using PyMC sampling, the output of which is provided in the rest of the files.

All data are .csv files with delimiter="|"

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The file titled "MasterFile.csv" is the data that were used to create the number density functions. The data were first published by several other authors whose results we used have been compiled here, the original papers are

Data in MasterFile.csv compiled from others' work:
Chang et al. (2015) DOI: 10.1088/0067-0049/219/1/8
Simard et al. (2011) DOI: 10.1088/0067-0049/196/1/11
van der Wel et al. (2014) DOI: 10.1088/0004-637X/788/1/28
Leja et al. (2019) DOI: 10.3847/1538-4357/ab7e27

The column descriptions for this file:

ID : Identifier in Field using Filter
Fld : Field name (COSMOS = C, GOODS-South = G or UDS = U)
F : [HJY] Filter (H, J or Y)
ra : Right ascension in decimal degrees (J2000)
dec : Declination in decimal degrees (J2000)
Early0_Late1 : Index indicating if a galaxy is considered early/quiescent (0) or late/star forming (1)
z_1 : Photometric redshift
logstellar_mass_median : Median logarithmic stellar mass [Msol]
logstellar_mass_errup : Upper bound on logarithmic stellar mass uncertainty [Msol]
logstellar_mass_errdn : Lower bound on logarithmic stellar mass uncertainty [Msol]
r : Half-light semi-major axis [arcsec]
r_err : Uncertainty in r [arcsec]
rtrue : Half-light semi-major axis [kpc]
veldisp : velocity dispersion [km/s]
veldisperr : Uncertainty in velocity dispersion [km/s]
vmax : (SDSS Galaxies only, not used in this work) Maximum volume for stellar mass [Mpc^3]

"-999.0" is used to indicate a parameter that does not have a measured/calculated value for a given galaxy. For example, velocity dispersion measurements only exist for SDSS galaxies in this sample, so all entries in veldisp and veldisperr for 3D HST+CANDELS galaxies will read "-999.0".

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Original PyMC fitting outputs (all other files):

The eight fits reported in the paper and presented in these files are:
1. Stellar mass functions for all galaxies (SMF_data.zip -> SMFitsAllTypes)
2. Velocity dispersion functions for all galaxies (VDF_data.zip -> SigFitsAllTypes)
3. Black hole mass functions for all galaxies using the M-M relation (BHMF_data.zip -> MMFitsAllTypes)
4. Black hole mass functions for star forming galaxies using the M-M relation (BHMF_data.zip -> MMFitsLate)
5. Black hole mass functions for quiescent galaxies using the M-M relation (BHMF_data.zip -> MMFitsEarly)
6. Black hole mass functions for all galaxies using the M-sigma relation (BHMF_data.zip -> MSigFitsAllTypes)
7. Black hole mass functions for star forming galaxies using the M-sigma relation (BHMF_data.zip -> MSigFitsLate)
8. Black hole mass functions for quiescent galaxies using the M-sigma relation (BHMF_data.zip -> MSigFitsEarly)

Within each of the 8 functions, we divided the redshifts into 7 bins:
1. 0.5 < z < 0.8
2. 0.8 < z < 1.1
3. 1.1 < z < 1.4
4. 1.4 < z < 1.8
5. 1.8 < z < 2.2
6. 2.2 < z < 2.6
7. 2.6 < z < 3.0

Fits to each redshift bin used four chains with 10,000 steps, this process was repeated 100 times.

At the beginning of each of the 100 iterations, the data to fit were created with some random scatter informed by the local intrinsic scatter of the scaling relations and error in the measurement values. Because of this repeating the iterations 100 times is more informative of the scatter in the fits than increasing the number of steps or chains.

File names all follow the format [Fit Type]Fits[Galaxy Type][Redshift Range].csv where

Fit Type:
"Sig" -> Velocity dispersion
"SM" -> stellar mass
"MSig" -> SMBH mass (M-sigma)
"MM" -> SMBH mass (M-M)

Galaxy Type:
"All Types" -> All galaxies
"Early" -> Quiescent
"Late" -> Star-forming

Redshift Range:
0508 - > 0.5 < z < 0.8
0811 - > 0.8 < z < 1.1
1114 - > 1.1 < z < 1.4
1418 - > 1.4 < z < 1.8
1822 - > 1.8 < z < 2.2
2226 - > 2.2 < z < 2.6
2630 - > 2.6 < z < 3.0

Column Descriptions:

chain_num : index of PyMC chain, ranges from 0-3. Repeating sequence of 0, 1, 2, 3 (40,000 rows) correspond to each of the 100 fits
logphi1 : The base 10 logarithm of the phi_* variable in the schechter function
logphi2 : (Stellar mass functions only)
logmstar : base 10 logarithm of the charateristic mass (this column is "logsigstar" for velocity dispersion functions)
alpha1 : The alpha variable in the schechter function
alpha2 : (Stellar mass functions only)
logscatter : the base 10 logarithm of the gaussian scatter in the fits returned by PyMC

In python, the above parameters could be used to produce a Schechter:
schechter_function = (((10**logphi1) * np.log(10) * 10**((xarray - logmstar) * (alpha1 + 1)) * np.exp(-10**(xarray - logmstar))))

where xarray would be a range of (log10) masses / velocity dispersions.

The chain data structure is as follows:

Three zipped folders each corresponding to stellar mass fits, velocity dispersion fits, or supermassive black hole mass fits.

SMF_data.zip and VDF_data.zip each contain seven files (one for every bin in redshift), each file has all 100 x 40,000 chain steps

BHMF_data.zip is the same as SMF_data.zip and VDF_data.zip except star forming and quiescent fits are included. In this folder are the fits to SMBH masses derived from both stellar mass and velocity dispersion for a total of 42 files.