Abstract
Understanding the connections between galaxy stellar mass, star
formation rate, and dark matter halo mass represents a key goal
of the theory of galaxy formation. Cosmological simulations that
include hydrodynamics, physical treatments of star formation,
feedback from supernovae, and the radiative transfer of ionizing
photons can capture the processes relevant for establishing
these connections. The complexity of these physics can prove
difficult to disentangle and obfuscate how mass-dependent trends
in the galaxy population originate. Here, we train a machine
learning method called Explainable Boosting Machines (EBMs) to
infer how the stellar mass and star formation rate of nearly 6
million galaxies simulated by the Cosmic Reionization on
Computers (CROC) project depend on the physical properties of
halo mass, the peak circular velocity of the galaxy during its
formation history vpeak, cosmic environment, and redshift. The
resulting EBM models reveal the relative importance of these
properties in setting galaxy stellar mass and star formation
rate, with vpeak providing the most dominant contribution.
Environmental properties provide substantial improvements for
modeling the stellar mass and star formation rate in only ≲10%
of the simulated galaxies. We also provide alternative
formulations of EBM models that enable low-resolution
simulations, which cannot track the interior structure of dark
matter halos, to predict the stellar mass and star formation
rate of galaxies computed by high-resolution simulations with
detailed baryonic physics.
Highlights
-
\(S\!F\!R\) and \(M_\star\) primarily depend on
\(M_\textrm{vir}\) and \(v_\textrm{peak}\), followed by
redshift, environmental density, and environmental gas
temperature.
-
When including \(M_\textrm{vir}\) and
\(v_\textrm{peak}\) in the parameter set used to train
the EBM, the model recovers better than 97% of the
distribution of \(M_\star\) or \(S\!F\!R\) with virial
mass \(M_\textrm{vir}\) in the CROC simulations.
-
If the model fit excludes \(v_\textrm{peak}\), the
fraction of outliers in the CROC data relative to the
predicted model distribution increases to 7.6% for
\(S\!F\!R\) and 2.8% for \(M_\star\).
-
To ameliorate the degradation of the model performance
when excluding vpeak, we define a composite EBM model
comprised of a weighted sum of the base EBM model fit to
main trend of \(S\!F\!R\) and \(M_\star\) with the halo
properties and a second EBM model to fit the outliers
not represented in the base EBM. The weighting
coefficients are themselves determined by an EBM model
fit.
-
The Composite EBM model improves the performance to
\(\approx\) 95 - 98% accuracy in the distribution of
\(S\!F\!R\) or \(M_\star\) with virial mass, even when
excluding vpeak measurements from the training dataset.
Demonstration EBM Predicting Star Formation Rate
In this work, we released four models: two predicting the star
formation rate and stellar mass of a galaxy as a function of \(
M_{\text{vir}} \), \( z \), \( v_{\text{peak}} \), \( \rho_1 \), \(
T_1 \), and \( \Upsilon_{0.1} \), and two to predict star formation
rate and stellar mass as a function of \( M_{\text{vir}} \), \( z
\),q \( \rho_1 \), \( T_1 \), and \(\Upsilon_{0.1} \) (excluding \(
v_{\text{peak}} \)). Where the input pararmeters indicate the
following:
Intrinsic Properties of the Galaxy
-
\( \text{log}_{10}M_{\text{vir}}[M_{\odot}] \): Galaxy virial
mass
- \( z \): Redshift
-
\( \text{log}_{10}v_{\text{peak}}[\text{kms}^{-1}] \): Maximum
peak circular velocity
Extrinsic Properties of the Galaxy
-
\( \text{log}_{10}\rho_{1} \): Environmental density,
\(\rho_1 \equiv 1 + \Delta_1\), where \( \Delta_1 \) is the
dimensionless matter overdensity measured within 1 Mpc scales
-
\( \text{log}_{10}T_{1}[K] \): Environmental gas temperature
averaged on 1 Mpc scales
-
\( \text{log}_{10}\Upsilon_{0.1} \): The ratio of the virial
mass of the most massive neighbor within 100 kpc
(\( M_{\text{max},0.1} \)) to the virial mass of the galaxy.
Specifically, we define the ratio as the following:
\( \Upsilon_{0.1} \equiv 1 + M_{\text{max},0.1}/M_{\text{vir}} \)
The generic EBM is formulated as follows:
\( E[y|\mathbf{x}] = \beta + \sum_{i=0}^{n} f_i(\mathbf{x}_i) + \sum_{i=0,i \ne j}^{n} \sum_{j=0}^{n} f_{ij}(\mathbf{x}_i, \mathbf{x}_j) \)
In this case, the EBM models the expected star formation rate given
the galaxy features using a sum of a baseline \( \beta \),
univariate functions \( f_i \), and bivariate \( f_{ij} \)
functions. The univariate functions model the impact of each feature
on the expected star formation rate, and the bivariate functions
model how two features together impact the star formation rate.
Below, we provide a demonstration of the EBM trained to predict star
formation rate as a function of \( M_{\text{vir}} \), \( z \), \(
v_{\text{peak}} \), \( \rho_1 \), \( T_1 \), and \( \Upsilon_{0.1}
\). Insert values into the input boxes below and click calculate to
see what the EBM predicts for the star formation rate.
Univariate Functions \( f_i \)
Bivariate Functions \( f_{ij} \)
Acknowledgements
This work was supported by the NASA Theoretical and
Computational Astrophysics Network (TCAN) grant 80NSSC21K0271.
The authors acknowledge use of the lux supercomputer at UC Santa
Cruz, funded by NSF MRI grant AST 1828315. This manuscript has
been co-authored by Fermi Research Alliance, LLC under Contract
No. DEAC02-07CH11359 with the U.S. Department of Energy, Office
of Science, Office of High Energy Physics. CROC project relied
on resources of the Argonne Leadership Computing Facility, which
is a DOE Office of Science User Facility supported under
Contract DE-AC02-06CH11357. An award of computer time was
provided by the Innovative and Novel Computational Impact on
Theory and Experiment (INCITE) program. CROC project is also
part of the Blue Waters sustained-petascale computing project,
which is supported by the National Science Foundation (awards
OCI0725070 and ACI-1238993) and the state of Illinois. Blue
Waters is a joint effort of the University of Illinois at
Urbana-Champaign and its National Center for Supercomputing
Applications. This research used resources of the Oak Ridge
Leadership Computing Facility, which is a DOE Office of Science
User Facility supported under Contract DE-AC05- 00OR22725. We
have used resources from DOE INCITE award AST 175.
