# Failure to Reproduce the Results of “A new transferable interatomic
potential for molecular dynamics simulations of borosilicate glasses”

François-Xavier Coudert<EMAIL_ADDRESS>https://www.coudert.name/ Chimie ParisTech, PSL Research University, CNRS,
Institut de Recherche de Chimie Paris, 75005 Paris, France

###### Abstract

We reproduced the simulations described in Wang et al [_J. Non-Cryst. Sol._ ,
498 (2018) 294–304] and found we could not obtain the results reported. The
root cause was identified to be incorrect atom masses in the original
simulation files. As a consequence, the potential does not reproduce the
experimental glass density — and presumably, other structural properties — and
should be considered with great caution.

Wang et al. reported in Ref. [1] the development of a new empirical
interatomic potential for borosilicate glasses. Based on a potential from
Guillot and Sator,[2] that classical force field is composed of two-body
Buckingham potential energy terms and Coulombic interactions:

$U_{ij}(r_{ij})=\frac{z_{i}z_{j}}{r_{ij}}+A_{ij}\exp\left(-\frac{r_{ij}}{\rho_{ij}}\right)-\frac{C_{ij}}{r_{ij}^{6}}$

The Wang potential used the parameters from Guillot and Sator, but since that
did not contain any interaction parameters for B cations, the authors
optimized the terms involving boron, i.e., the energy terms for B–O, B–B, and
B–Si interactions. The parameters for these terms were determined by fitting
on experimental data over the whole range of glass composition, from pure
silicate to pure borate. The properties fitted include the coordination number
of B cations (for the B–O parameters) and the evolution of density as a
function of glass composition (for the B–B parameters). The authors then
describe the structural properties of the glasses as obtained with their
model.

Figure 1: Glass density as a function of composition, represented as fraction
of SiO2 (over the total amount of SiO2 and B2O3) Red diamonds: experimental
data.[3] Black squares: results published in Ref. [1]. Blue circles: our
reproduction of the work. Lines are guides for the eye.

##  I Validity of the interatomic potential

While trying to reproduce the work published in Ref. [1], we could not
reproduce the results in the authors’ Figure 1, i.e., the evolution of density
as a function of glass composition. We present the points from the original
paper, and our reproduction, in Figure 1. While the overall trend is somewhat
similar, there is a systematic difference and we found densities that were
systematically higher than reported, with a difference of up to 0.07 g/cm3.
The quantitative agreement with experimental data, reported by the authors, is
not reproduced at all.

While an example LAMMPS[4] input file was provided in Supplementary
information, the original configuration (LAMMPS data file) was not published.
The author kindly shared starting configurations from the different systems,
and we discovered that the original files had errors in the atomic masses of
boron and silicon atoms. This impacted all simulations and results reported in
Wang et al.

Moreover, its appears that the “new transferable interatomic potential” was
itself optimized with the incorrect atomic masses. The potential parameters
for the B–B terms were optimized to reproduce the experimental densities, but
they took into account the wrong masses. Therefore, the potential should be
reoptimized in the future before it is further used, especially the B–B
parameters.

##  II Works impacted

The paper by Wang et al. is higly cited, with 117 citations on Google Scholar
to date. Having reviewing the citations, we think the following works are
relying on the invalid potential for all or part of their molecular
simulations:

  * •

Wang, M.; Smedskjaer, M. M.; Mauro, J. C.; Bauchy, M. “Modifier Clustering and
Avoidance Principle in Borosilicate Glasses: A Molecular Dynamics Study”. _J.
Chem. Phys._ , 2019, 150, 044502. The densities reported are the same as Ref.
[1], and it is safe to assume it uses input files with invalid atomic masses.

  * •

Kumar, R.; Jan, A.; Bauchy, M.; Krishnan, N. M. A. “Effect of Irradiation on
the Atomic Structure of Borosilicate Glasses”. _J. Am. Ceramic Soc._ , 2021,
104, 6194–6206. This paper has common co-authors with Ref. [1], and because it
does reproduce the same densities as Wang et al., it seems that it used
incorrect atomic masses as well. All results should be taken with great
caution.

  * •

Tuheen, M. I.; Deng, L.; Du, J. “A Comparative Study of the Effectiveness of
Empirical Potentials for Molecular Dynamics Simulations of Borosilicate
Glasses”. _J. Non-Cryst. Sol._ , 2021, 553, 120413. The authors’ conclusion is
that the potential “fails to provide the correct B coordination change for a
wide range of compositions”, which may be linked to the optimization issues.

  * •

Fortino, M.; Berselli, A.; Stone-Weiss, N.; Deng, L.; Goel, A.; Du, J.;
Pedone, A. “Assessment of Interatomic Parameters for the Reproduction of
Borosilicate Glass Structures via DFT-GIPAW Calculations”. _J. Am. Ceramic
Soc._ , 2019, 102, 7225–7243. In this paper comparing different interatomic
potentials, the authors used $(N,V,T)$ simulations at fixed experimental
densities.

  * •

Bisbrouck, N.; Micoulaut, M.; Delaye, J. M.; Gin, S.; Angeli, F.
“Structure–Property Relationship and Chemical Durability of Magnesium-
Containing Borosilicate Glasses with Insight from Topological Constraints”.
_npj Mater. Degrad._ , 2022, 6, 58. These simulations were performed at fixed
density, in the $(N,V,T)$ and $(N,V,E)$ ensembles.

  * •

Zhai, C.; Zhong, Y.; Liu, J.; Zhang, J.; Zhu, Y.; Wang, M.; Yeo, J.
“Customizing the Properties of Borosilicate Foam Glasses via Additions under
Low Sintering Temperatures with Insights from Molecular Dynamics Simulations”.
_J. Non-Cryst. Sol._ , 2022, 576, 121273. This work includes a detailed
discussion of the evolution of density and other physical properties as a
function of boron content of the glasses, and is probably significantly
impacted.

  * •

Lee, K.; Yang, Y.; Ding, L.; Ziebarth, B.; Mauro, J. C. “Effect of
Pressurization on the Fracture Toughness of Borosilicate Glasses”. _J. Am.
Ceramic Soc._ , 2021, 105, 2536–2545. The Wang force field was compared to
other potentials, and is found to lead to higher densities and different
trends in Young’s modulus. This appears to be linked to the parameterization
issue.

  * •

Ding, L.; Lee, K.; Zhao, T.; Yang, Y.; Bockowski, M.; Ziebarth, B.; Wang, Q.;
Ren, J.; Smedskjaer, M. M.; Mauro, J. C. “Atomic Structure of Hot Compressed
Borosilicate Glasses”. _J. Am. Ceramic Soc._ , 2020, 103, 6215–6225. For one
of the systems (Boro33), this study features $(N,P,T)$ simulations. We note
that the authors report that “A computed pressure […] of $-1.8$ GPa is found
in Boro33 glass (Bauchy’s potential). The pressure at room temperature density
is due to the choice of potential”. Moreover, because several authors are also
co-authors of Ref. [1], it is unclear whether the correct atomic masses were
used in the simulations.

  * •

Lee, K.-H.; Yang, Y.; Ziebarth, B.; Mannstadt, W.; Davis, M. J.; Mauro, J. C.
“Evaluation of Classical Interatomic Potentials for Molecular Dynamics
Simulations of Borosilicate Glasses”. _J. Non-Cryst. Sol._ , 2020, 528,
119736. The work features $(N,P,T)$ simulations and the densities reported
could be higly impacted by the boron parameterization. Because one author is
also a co-author of Ref. [1], it is unclear whether the correct atomic masses
were used in the simulations.

  * •

Lee, K.; Yang, Y.; Ding, L.; Ziebarth, B.; Davis, M. J.; Mauro, J. C. “Atomic-
scale Mechanisms of Densification in Cold-compressed Borosilicate Glasses”.
_J. Am. Ceramic Soc._ , 2021, 104, 2506–2520. The same remarks as the previous
work apply.

## Data availability

The simulation files used to reproduce the work (both with correct and
incorrect atomic masses) are freely available at
https://github.com/fxcoudert/citable-data

###### Acknowledgements.

Mengyi Wang is acknowledged for sharing original simulation input files.

## References

  * Wang  _et al._ [2018] M. Wang, N. Anoop Krishnan, B. Wang, M. M. Smedskjaer, J. C. Mauro, and M. Bauchy, A new transferable interatomic potential for molecular dynamics simulations of borosilicate glasses, Journal of Non-Crystalline Solids 498, 294 (2018).
  * Guillot and Sator [2007] B. Guillot and N. Sator, A computer simulation study of natural silicate melts. Part I: Low pressure properties, Geochimica et Cosmochimica Acta 71, 1249 (2007).
  * Smedskjaer  _et al._ [2011] M. M. Smedskjaer, J. C. Mauro, R. E. Youngman, C. L. Hogue, M. Potuzak, and Y. Yue, Topological principles of borosilicate glass chemistry, J. Phys. Chem. B 115, 12930 (2011).
  * Thompson  _et al._ [2022] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS – a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).


# The Supersonic Project: Lighting up the faint end of the JWST UV luminosity
function

Claire E. Williams Department of Physics and Astronomy, UCLA, Los Angeles, CA
90095  Mani L. Bhaumik Institute for Theoretical Physics, Department of
Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA
William Lake Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA
Smadar Naoz Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA
Blakesley Burkhart Department of Physics and Astronomy, Rutgers, The State
University of New Jersey, 136 Frelinghuysen Rd, Piscataway, NJ 08854, USA
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA
Tommaso Treu Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Federico Marinacci Department of Physics & Astronomy “Augusto Righi”,
University of Bologna, via Gobetti 93/2, 40129 Bologna, Italy
Yurina Nakazato Department of Physics, The University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-0033, Japan  Mark Vogelsberger Department of Physics and
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA
Naoki Yoshida Department of Physics, The University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-0033, Japan  Kavli Institute for the Physics and Mathematics
of the Universe (WPI), UT Institute for Advanced Study, The University of
Tokyo, Kashiwa, Chiba 277-8583, Japan  Research Center for the Early Universe,
School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan  Gen Chiaki Astronomical Institute, Tohoku University, 6-3,
Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan  Yeou S. Chiou Department of
Physics and Astronomy, UCLA, Los Angeles, CA 90095  Mani L. Bhaumik Institute
for Theoretical Physics, Department of Physics and Astronomy, UCLA, Los
Angeles, CA 90095, USA
Avi Chen Department of Physics and Astronomy, Rutgers, The State University of
New Jersey, 136 Frelinghuysen Rd, Piscataway, NJ 08854, USA

###### Abstract

The James Webb Space Telescope (JWST) is capable of probing extremely early
eras of our Universe when the supersonic relative motions between dark matter
and baryonic overdensities modulate structure formation ($z\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10$).
We study low-mass galaxy formation including this “stream velocity” using high
resolution AREPO hydrodynamics simulations, and present theoretical
predictions of the UV luminosity function (UVLF) and galaxy stellar mass
function (GSMF) down to extremely faint and low mass galaxies
($M_{UV}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}-15$, $10^{4}M_{\odot}\leq M_{*}\leq
10^{8}M_{\odot})$. We show that, although the stream velocity suppresses early
star formation overall, it induces a short period of rapid star formation in
some larger dwarfs, leading to an enhancement in the faint-end of the UVLF at
$z=12$. We demonstrate that JWST observations are close to this enhanced
regime, and propose that the UVLF may constitute an important probe of the
stream velocity at high redshift for JWST and future observatories.

High-redshift galaxies — Primordial galaxies — Luminosity function — James
Webb Space Telescope — Hydrodynamical simulations

††software: The authors would like to acknowledge the Astropy (Astropy
Collaboration et al., 2013, 2018, 2022), Web Plot Digitizer (Rohatgi, 2022),
yt (Turk et al., 2011), NetworkX (Hagberg et al., 2008), and Matplotlib
(Hunter, 2007) software packages, which were used in this work.

##  1 Introduction

The James Webb Space Telescope (JWST) has opened a new window on the first
galaxies in the Universe, allowing for new insights regarding their properties
and formation. These galaxies at “Cosmic Dawn” host the yet undetected first
generation of stars (Population III), reionize the Universe, and enrich their
surroundings with the first metals. The $\Lambda$CDM (cold dark matter +
cosmological constant) cosmological model predicts the formation of these
early sources within the first few hundred million years after the Big Bang
(e.g., Vogelsberger et al., 2020a). Observations of faint galaxies at high
redshift represent an opportunity to test galaxy formation models and the
$\Lambda$CDM paradigm itself.

Since JWST began science operations in 2022, NIRCam datasets have provided a
number of intriguing galaxy candidates at $z>9$ (e.g., Naidu et al., 2022;
Castellano et al., 2022, 2023; Adams et al., 2023; Atek et al., 2023;
Finkelstein et al., 2022; Yan et al., 2023; Donnan et al., 2023a, b; Morishita
& Stiavelli, 2023; Bradley et al., 2022; Bouwens et al., 2023a; Pérez-González
et al., 2023). Furthermore, JWST’s spectrographic instruments allow for
precision redshift determination, and multiple galaxies with $z>9$ have been
confirmed, with gravitationally lensed systems providing an additional
magnified glimpse at the early Universe (e.g., Roberts-Borsani et al., 2023;
Williams et al., 2023b; Atek et al., 2023; Arrabal Haro et al., 2023).
Moreover, robust measurements of the UV luminosity function (UVLF) are now
possible down to very faint magnitudes (e.g., Naidu et al., 2022; Leung et
al., 2023; Castellano et al., 2023; Finkelstein et al., 2023; Donnan et al.,
2023b; Harikane et al., 2023a, b) at redshifts $z\sim 12$ and beyond. These
observations open a window into the physics of galaxy formation at
increasingly small scales and high redshifts.

Low-mass (baryonic mass ($M_{b}$) $\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10^{9}M_{\odot}$)
galaxies represent a sensitive test of galaxy formation and cosmological
models due to their small gravitational potentials, which are sensitive to the
precise nature of dark matter (DM) and baryonic physics (e.g., Bullock &
Boylan-Kolchin, 2017; Sales et al., 2022). In order to compare the wealth of
JWST data to cosmological models, a precise description of the baryonic
physics that dictates the formation of these dwarf galaxies is necessary
(e.g., Vogelsberger et al., 2020b; Shen et al., 2020, 2023). Within the
standard $\Lambda$CDM model, these galaxies originate from small scale
($k\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}20$ Mpc-1) baryonic density fluctuations, whose growth is
dictated by the overwhelming gravitational potential of DM overdensities. The
latter is orders of magnitude larger than baryonic overdensities by the time
of recombination (e.g., Naoz & Barkana, 2005; Loeb & Furlanetto, 2013).

Tseliakhovich & Hirata (2010) showed that a second-order term in standard
linear perturbation theory, the relative velocity between dark matter and
baryons ($v_{\rm bc}$), is non-negligible at these scales, with an rms value
($\sigma_{\rm vbc}$) at $z=1100$ of $30$ km s-1, five times the speed of sound
at that time. This stream velocity was shown to delay the formation of Pop III
stars, suppress halo abundance, and raise the minimum halo mass that can
retain baryons and form stars (e.g., Stacy et al., 2011; Greif et al., 2011;
Schauer et al., 2017a, 2021; Naoz et al., 2012, 2013; Bovy & Dvorkin, 2013;
Asaba et al., 2016; Nebrin et al., 2023; Hegde & Furlanetto, 2023; Conaboy et
al., 2023). Notably, the stream velocity was recently estimated locally to be
$v_{\rm bc}=1.75^{+0.13}_{-0.28}\sigma_{\rm vbc}$ (Uysal & Hartwig, 2023).

This supersonic velocity generates a phase shift between the baryonic and dark
matter perturbations that introduces a physical offset between the center of
mass of collapsed baryon structures and their parent DM halo (e.g., Naoz &
Narayan, 2014; Popa et al., 2016; Chiou et al., 2018, 2019, 2021; Lake et al.,
2021; Nakazato et al., 2022; Lake et al., 2023a; Williams et al., 2023a; Lake
et al., 2023b). The physical offset between DM and baryonic components of
$\Lambda$CDM overdensities and other effects of the stream velocity have
important implications for the formation of the first stars and galaxies, with
potential consequences for JWST observation. In some cases for very low mass
halos ($M_{b}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10^{5}M_{\odot}$), the physical offset
is so large that baryons collapse outside of their parent DM halo, creating
DM-free Supersonically-Induced Gas Objects (SIGOs) which may be the
progenitors of some early globular clusters (e.g., Popa et al., 2016; Chiou et
al., 2018, 2019, 2021; Lake et al., 2021; Nakazato et al., 2022; Lake et al.,
2023a). Future JWST observations may reveal a population of star clusters at
high galactocentric distances descended from SIGOs at high redshift (Lake et
al., 2023b). For halos with $M_{b}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10^{9}M_{\odot}$,
the physical offset results in the formation of diffuse structures called Dark
Matter + Gas Halos Offset by Streaming (DM GHOSts) (e.g., Williams et al.,
2023a). Unlike typical dwarf galaxies, these structures are highly elongated,
more rotationally supported, and gas-deficient, as the stream velocity advects
a portion of the gas component out of the halo (e.g., Williams et al., 2023a;
Hirano et al., 2023).

The delay of star formation and the diffuse nature of the structures that form
in regions of streaming have implications for the populations of dwarf
galaxies that exist in the high redshift Universe. Here, we suggest that the
stream velocity may affect the faint end of the UVLF by suppressing star
formation in the smallest galaxies and boosting the star formation rate (SFR)
of larger dwarfs for a short period around $z\sim 12$. We estimate the galaxy
stellar mass function (GSMF), showing that the stream velocity suppresses the
low mass end. We suggest that JWST and future extremely large telescopes
(ELTs) have the opportunity to observe galaxies at extremely high redshift in
regions of supersonic streaming. These results imply that the stream velocity
will enhance the scatter in the UVLF at faint $M_{UV}$.

The structure of this paper is as follows: in § 2 we describe the AREPO
simulations used and their initial conditions including the stream velocity.
In § 3, we describe the UV luminosity function (§ 3.3) and galaxy stellar mass
function (§ 3.2) at high redshifts in regions of the universe with and without
streaming. Finally, in § 4 we interpret these results and the outlook for JWST
observations.

In this study we assume a $\Lambda$CDM cosmology, with
$\Omega_{\rm\Lambda}=0.73$, $\Omega_{\rm m}=0.27$, $\Omega_{\rm b}=0.044$,
$\sigma_{8}=1.7$, and $h=0.71$. Magnitudes are calculated using the AB
magnitude system; $m=-2.5\log_{10}(f_{\nu}/\text{ nJy})+31.40$ (Oke, 1974).

##  2 Methods

###  2.1 Cosmological Simulation Description

We perform a suite of numerical simulations in AREPO (Springel, 2010),
evolving a box (2.5 h-1 Mpc)3 from $z=200$ to $z=12$. To generate initial
conditions, we use a modified CMBFAST code (Seljak & Zaldarriaga, 1996). As in
Popa et al. (2016), we include the first-order correction of scale-dependent
temperature fluctuations on the initial conditions and their transfer
functions as in Naoz & Barkana (2005); Naoz & Narayan (2013). This allows for
accurate gas fractions in halos at high redshift (e.g., Naoz et al., 2009,
2011, 2013). The box contains 7683 DM particles ($m_{\rm
DM}=775h^{-1}M_{\odot}$) and 7683 Voronoi mesh gas cells, giving a gas cell
resolution of $m_{\rm gas}=200\,M_{\odot}$. See Lake et al. (2023b) for the
simulation suite details.

On scales smaller than a few Mpc, the relative velocity is coherent and can be
modeled as a bulk stream motion (e.g., Tseliakhovich & Hirata, 2010; Popa et
al., 2016) with an rms value of $\sigma_{\rm bc}$. We initialize our
simulations at $z=200$, when a 2$\sigma_{\rm bc}$ fluctuation in the stream
velocity corresponds to $11.8$ km s -1 (e.g., Popa et al., 2016). The stream
velocity is thus implemented as a uniform boost of 11.8 ( km
s${}^{-1})\hat{x}$ to all baryon particles. In order to compare the properties
of early galaxies in regions with and without a highly supersonic stream
velocity, we perform two simulation runs, one with a $0\sigma_{\rm bc}$
fluctuation (i.e., no stream velocity) and one with a $2\sigma_{\rm bc}$
fluctuation (a value similar to the observed local one; Uysal & Hartwig 2023).

In order to increase the statistical power of our simulation without affecting
the underlying cosmology, we choose $\sigma_{8}=1.7$ (these are the same
simulations as in Lake et al. 2023b). Thus, our box represents a strong
density peak where structure in the Universe forms early, such as in the Virgo
cluster (e.g., Naoz & Barkana, 2007). These results and their robust
statistics can thus be scaled to other regions of the Universe accordingly
(e.g., Naoz & Narayan, 2013; Park et al., 2020) (see Appendix).

###  2.2 Subgrid Physics - Cooling and Star Formation

An accurate prescription of molecular cooling in the pristine gas of the early
Universe is necessary to model the formation and collapse of gas-rich
structures at high redshift (e.g., Schauer et al., 2021; Nakazato et al.,
2022; Lake et al., 2023a; Williams et al., 2023a). Following Nakazato et al.
(2022); Lake et al. (2023a); Williams et al. (2023a), we include molecular
cooling in both runs through the GRACKLE chemistry and cooling library (Smith
et al., 2017; Chiaki & Wise, 2019). This prescription explicitly accounts for
nonequilibrium chemical reactions and radiative cooling, including 15
primordial species (e-, H, H+, He, He+, He++, H-, H2, H${}_{2}^{+}$, D, D+,
HD, HeH+, D-, and HD+).

Once gas condenses sufficiently, the first generation of stars forms. As
described in Lake et al. (2023b), when a gas cell exceeds the Jeans mass, it
collapses to form a star particle on the free-fall timescale. The star
formation rate (SFR) for a given object is computed as the difference in its
stellar mass between two snapshots, excluding any star particles that were
part of a cluster that merged into the halo. We do not include any stellar
feedback effects, although the feedback processes are vital to understand the
efficiency of star formation over cosmic time to the present day. The
resulting estimates of the UVLF can thus be considered an upper limit, given
that feedback effects will suppress the efficiency of star formation.

###  2.3 Identification of structures

Using a friends-of-friends (FOF) algorithm, we search for two object classes.
First, we identify DM-primary/gas-secondary objects, corresponding to DM
GHOSts. In this case, the FOF algorithm is run on DM particles first, with gas
cells and star particles linked at a secondary stage. We also identify gas-
primary objects by running the FOF algorithm on gas cells only. This allows us
to find gas clumps without an associated DM halo ((SIGOs), see e.g., Chiou et
al., 2018, 2019, 2021; Lake et al., 2021; Nakazato et al., 2022; Lake et al.,
2023a, b). In order to avoid non-physical numerical effects, we require DM-
primary objects to contain at least 300 particles and gas-primary objects to
have at least 100 cells (e.g., Naoz et al., 2011).

Figure 1: Comparison of the projected density of gas and stars in the region
of a highly star forming object in our simulations at $z=12$ between the
$2\sigma_{\rm bc}$ run (left) and the $0\sigma_{\rm bc}$ (right) run. A bar is
shown in each panel denoting a distance of 1 kpc. The galaxy at the center of
the left two panels formed stars rapidly between $z=15$ and $z=12$, with a
star formation rate of $\dot{M}=0.176\,M_{\odot}$ yr-1, the fifth highest in
our $2\sigma_{\rm bc}$ run. It has $M_{*}=10^{7.51}M_{\odot}$ at $z=12$. The
stellar mass history of this object in the interval between $z=15$ and $z=12$
is shown in Fig. 7 in the Appendix.

###  2.4 The UV luminosity

The rest-frame UV $1500$Å luminosity of high redshift galaxies observed in the
infrared by JWST  traces the star formation rate (SFR) assuming that they are
dominated by young, massive stars with short lifetimes (e.g., Kennicutt, 1998;
Stark et al., 2009; Kennicutt & Evans, 2012). We calculate the SFR of each
object in the simulation by identifying objects in snapshots at $z=13$ and
$z=12$, and calculating the difference in stellar mass between the two
snapshots. Given the SFR of galaxies in our simulations at $z=12$, we then
estimate their rest-frame UV continuum luminosity ($L_{UV,1500}$):

$L_{UV,1500}=\frac{\dot{M}_{*}}{{\cal K}_{UV,1500}}\ ,$ (1)

where $\dot{M}_{*}$ is the star formation rate in units of $M_{\odot}$ yr-1.
${\cal K}_{UV,1500}$ is a fiducial constant evaluated for continuous-mode star
formation with a Salpeter IMF. We adopt ${\cal K}_{UV,1500}=1.15\times
10^{-28}$ ( $M_{\odot}$ yr-1 / (ergs s-1 Hz-1) (following e.g., Madau &
Dickinson, 2014; Sun & Furlanetto, 2016).

This seems to be a reasonable assumption given that we trace the star
formation on a timescale of $\sim 30$ Myr, which is comparable to the lifetime
of the most massive stars. ${\cal K}_{UV,1500}$ depends on the IMF of the
stellar population, which remains uncertain at high redshift. Proposed high-
redshift and Pop III IMFs may shift the magnitude at which galaxies of a given
mass enter the observable magnitudes of JWST. In particular, several studies
propose that a top heavy IMF may be present at high redshift, Pop III
environments due to their low metallicity and/or high CMB temperature (e.g.,
Omukai et al., 2005; Hirano et al., 2014, 2015; Chon et al., 2022). Such an
IMF would lead to increased UV luminosity from nebular emission, increasing
our $L_{UV,1500}$ for a given $\dot{M}_{*}$. Thus, we adopt the standard value
of ${\cal K}_{UV,1500}$ both as a conservative estimate of detectability and
for ease of comparison with other works. This work assumes that the stream
velocity does not affect the high-redshift IMF, and further work is needed to
ascertain whether such a dependence exists. Under this assumption, the shape
of the UVLF should be consistent between the stream velocity and non-stream
velocity scenarios, although the overall height of both curves may vary given
assumptions about ${\cal K}_{UV,1500}$.

We must also correct our number counts for the fact that we have artificially
increased $\sigma_{8}$ to $1.7$ in order to gain more complete statistics when
calculating number counts per volume in this study. We calculate a conversion
factor from $\sigma_{8}=1.7$ to $\sigma_{8}=0.8$ as a function of mass using
the Sheth & Tormen (2002) halo mass function (see Appendix). In each magnitude
bin, we divide the number counts by the correction factor corresponding to the
average mass in each bin. This introduces an additional uncertainty in the
UVLF because of the range of halo masses represented at each magnitude.

##  3 Results

###  3.1 The Effect of Streaming on Star-Forming Regions

As expected, the small-scale structure is suppressed in the presence of stream
velocity (e.g., Tseliakhovich & Hirata, 2010; Fialkov et al., 2012; Bovy &
Dvorkin, 2013; Maio et al., 2011; O’Leary & McQuinn, 2012; Naoz et al., 2012;
Tseliakhovich et al., 2011; Tanaka et al., 2013; Tanaka & Li, 2014). Figure 1
shows an example of the projected density of a highly star-forming region with
(left panels) and without (right panels) streaming. Both cases show the gas
and star components separately (see labels). As depicted, small clumps of gas
and stars in the $v_{\rm bc}=0\sigma_{\rm bc}$ are less apparent in the
$v_{\rm bc}=2\sigma_{\rm bc}$ case. even denser) stars and gas in $v_{\rm
bc}=2\sigma_{\rm bc}$. Although Fig. 1 shows just one example region, visual
inspection of tens of highly star forming objects in the simulation boxes
reveals a similar pattern. We quantify the extent of these effects on star
formation below, as they have implications for the star formation rate
estimation and the luminosity of these objects.

###  3.2 Star formation and the Galaxy Stellar Mass Function

Figure 2: Galaxy stellar mass function with (pink lines below) and without
(blue lines above) stream velocity at $z=12$ (solid) and $z=15$ (dashed).

Given that the stream velocity is expected to delay and suppress small-scale
star formation at high redshift (e.g., Schauer et al., 2023; Hegde &
Furlanetto, 2023; Conaboy et al., 2023), and that JWST can already provide
constraints on the low-mass end of the GSMF at redshifts as high as $z\sim 8$
(e.g., Navarro-Carrera et al., 2023), we calculate the GSMF in each simulation
box to determine the expected contribution from the stream velocity.

Figure 3:  Stellar mass ($M_{*}$) vs total halo mass ($M_{h}$) for the two
runs at $z=12$ (top) and $z=15$ (bottom). The data are binned along the y-axis
and shown with error bars corresponding to one standard deviation. The
$0\sigma_{bc}$ run is given in blue (left points), and the $2\sigma_{bc}$ run
in pink (points to the right). Underneath the binned data, the actual
distributions are shown, with the $0\sigma_{bc}$ runs shown in grey and the
$2\sigma_{bc}$ runs shown in pink.  Figure 4: Average stellar mass in the most
massive 100 galaxies in the simulation between $z=15$ and $z=12$. This panel
highlights the difference between the large masses shown Fig. 3 as a function
of redshift. Best fit lines in give,
$\log_{10}\left({M_{*}}/{M_{\odot}}\right)=0.0035\log_{10}\left({t}/{\text{Myr}}\right)+6.31$
for the stream velocity run and
$\log_{10}\left({M_{*}}/{M_{\odot}}\right)=0.0031\log_{10}\left({t}/{\text{Myr}}\right)+6.47$
for the no stream velocity run.

The results are presented in Fig. 2, where we show the galaxy stellar mass
function in our simulation volume for the runs with (pink lines) and without
(blue lines) the stream velocity. The two redshifts examined ($z=12$ and
$z=15$), are displayed with solid and dashed lines, respectively. Above
$M_{*}\gtrsim 10^{6.5}M_{\odot}$, the two curves agree. However, at lower
masses, there are up to an order of magnitude more galaxies at each stellar
mass in the run without the stream velocity.

The suppression in the GSMF in the $2\sigma_{\rm bc}$ run can be attributed to
two effects caused by the stream velocity. First, the number density of all
halos is suppressed in regions of streaming, leading to a suppression of the
GSMF (see Fig. 6 in the appendix, and, e.g., Naoz et al., 2013; Lake et al.,
2021). Additionally, we show in Fig. 3 the stellar mass as a function of halo
mass for objects in each run. For a given stellar mass, the stars lie in a
much larger halo in regions of streaming than without streaming. Thus, larger
halos, which are less abundant according to standard halo mass function, host
galaxies of the same stellar mass for $M_{*}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10^{6.5}M_{\odot}$,
contributing fewer galaxies to the GSMF.

As demonstrated by Fig. 3, while low stellar mass objects in the case with
stream velocity have higher DM mass, the stream velocity and classical cases
converge at large masses. Furthermore, the mass at which the two converge
decreases with decreasing redshift. In other words, the blue ($0\sigma_{\rm
vbc}$), and the red ($2\sigma_{\rm vbc}$) points, lie on top of each other for
$M\geq 10^{7.0}M_{\odot}$ at $z=15$. However, at $z=12$, they converge at
$M\geq 10^{7.25}M_{\odot}$, implying a fast SFR for the high masses with
stream velocity case, between these two snapshots at $z=15$ and $z=12$.

This behavior is further illustrated in Fig. 4, where the time evolution of
the average stellar mass is shown for the most massive galaxies by stellar
mass. As depicted, the star formation rate (the slope of the Figure) in the
stream velocity case is larger. Thus, although the star formation is
suppressed for most galaxies in the stream velocity simulation run until
$z\sim 15$, the star formation rate at large masses with a large stream
velocity is heightened between $z=15$ and $z=12$.

###  3.3 The UV luminosity function

Figure 5: Top: UV Luminosity Function at $z=12$ with (pink) and without the
stream velocity (blue) compared to the theoretical models of Yung et al.
(2023), Mason et al. (2023), Dayal et al. (2014), and observational
constraints close to the redshift presented from Pérez-González et al. (2023);
Naidu et al. (2022); Finkelstein et al. (2023); Leung et al. (2023); Harikane
et al. (2023b); Donnan et al. (2023b); Bouwens et al. (2023a, b). The UVLF is
then averaged over bins with $\Delta M_{UV}=1$, with the shaded regions
showing the minimum and maximum value in that box. Bottom: Distributions of
galaxies by AB absolute UV magnitude with and without the stream velocity
between $z=13$ and $z=12$.

Given the above prediction that the star formation rate is higher in patches
of the Universe with high stream velocity at the redshifts probed by our
simulation, we expect that high redshift dwarfs will be brighter in UV
emission in those patches.

Thus, as a second probe of the population of dwarf galaxies resulting from a
streaming scenario versus the classical scenario, we calculate the rest frame
absolute UV magnitude $M_{UV}$ of our objects at $z=12$. This is done based on
the SFR according to the procedure described in § 2.

In the bottom panel of Fig. 5 we show the distribution of objects in our
simulation box binned by $M_{UV}$. For objects fainter than $M_{UV}\sim-12$,
the UV magnitudes follow the same trend as the stellar masses, with the stream
velocity function only $1/6$ of the no streaming case. However, a turnover
occurs above $M_{UV}\sim-12$, and the stream velocity counts are enhanced by a
factor of over an order of magnitude at $M_{UV}\sim-14$.

We show the UVLF in regions with and without streaming in the top panel of
Fig. 5. The trend in the distributions is reflected in the UVLF at the bright
end of our sample as a turnover where the UVLF in the $2\sigma_{bc}$ region is
greater than the no streaming function. The enhancement in the UVLF below
$M_{UV}\sim-12$ in the presence of the stream velocity can be counterintuitive
given the suppression in star formation and number density caused by the
stream velocity in most other contexts, including the GSMF. As evident in Fig.
3, there is no underlying overabundance of more massive galaxies by stellar
mass (which have higher SFR) due to the stream velocity–at these larger
masses, there are roughly equal abundances in both simulation runs. Instead,
several effects play a role in boosting the UV luminosity of these more
massive galaxies. Importantly, the UV luminosity in our model is proportional
to the SFR of galaxies via Eq. (1). Here, we discuss how the stream velocity
in this model leads to a boost in the star formation rate of dwarf galaxies
with slightly higher stellar mass ($M_{*}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{6.5}M_{\odot}$).

Firstly, in regions with highly supersonic streaming flow, the small-scale
structure of gaseous clumps is washed away (as shown by e.g., Tseliakhovich &
Hirata, 2010; Fialkov et al., 2012; Bovy & Dvorkin, 2013; Maio et al., 2011;
O’Leary & McQuinn, 2012; Naoz et al., 2012; Tseliakhovich et al., 2011; Tanaka
et al., 2013; Tanaka & Li, 2014). An example of this suppression is evident in
the small region of our simulation snapshot at $z=12$ depicted in Fig. 1,
where at least thirteen small gaseous clumps are present around the central
galaxy in the $0~{}\sigma_{bc}$ run while only five similar clumps are present
in the $2\sigma$ run. Thus, star formation without the stream velocity can
proceed in small minihalos of condensed gas, contributing to the behavior of
the low mass GSMF in Fig. 2. Small clumps of star formation corresponding to
these gaseous clumps are seen in the right-hand panel of Fig. 1. In regions
with the stream velocity, however, the same overall cosmic density of gas is
still present. Having been swept away from clumps that would allow it to form
small clusters, the diffuse gas streams back onto larger halos thanks to their
gravitational influence over the region as the stream velocity decays. Thus,
star formation is concentrated in larger dwarf galaxies rather than smaller
minihalos. This suppresses the extreme faint end of the UVLF while increasing
the luminosity of the larger dwarfs, producing the trend in Fig. 5.

Furthermore, previous studies such as Stacy et al. (2011); Greif et al.
(2011); Schauer et al. (2017b, 2023) found a delay in star formation in
regions of streaming. Schauer et al. (2023) found that by $z=8$, dwarf
galaxies in their simulations with the stream velocity had reached roughly the
same stellar mass as counterparts in regions without streaming. For example,
Figure 4 of that study depicts a dwarf galaxy that reaches $\sim
10^{7}M_{\odot}$ by the end of the simulation and grows at a rate of $\sim
0.031M_{\odot}$ yr-1 between $z=15$ and $z=12$, while during the same period
in the run with the stream velocity it grows at $\sim 0.038M_{\odot}$ yr-1.
The authors note a modest effect of higher star formation rates in their
streaming run overall prior to $z\approx 8$, at which point the difference in
mass between the runs becomes mostly stochastic. If stream velocity-induced
dwarfs indeed catch up to the stellar mass of the non-streaming population by
the epoch of reionization after a delay in the onset of star formation, then
they undergo a period of faster SFR than the non-streaming case.

The UVLF at $z=12$, which roughly tracks the instantaneous SFR at the redshift
of observation, would be catching these objects in a period where they
experience enhanced star formation thanks to the mechanism of concentrating
gas into larger halos described above. We note that the baryonic feedback
resulting from star formation episodes in these structures is not investigated
here. Our UVLF can be understood as an upper limit in both cases given the
likely reduction in star formation efficiency that would result from the
inclusion of feedback in the simulation. A heightened SFR may lead to
increased disruption due to feedback, however, as in Fig 3, dwarf galaxies
with the stream velocity sit in larger DM halos in streaming cases which may
prevent disruption. The stream velocity may thus play a role in the
“burstiness” of star formation at early times.

We overplot our results to several other theoretical models in the literature
from Dayal et al. (2014); Behroozi et al. (2019); Yung et al. (2023); Mason et
al. (2023) at $z=12$. This Figure, thus, demonstrates that the high-resolution
investigation presented here allows us to reach the extremely faint end of the
luminosity function. We note that none of these studies included the stream
velocity effect. In addition, we overplot some current observational JWST
constraints. Our box size does not include enough objects bright enough to
fall within the observable range of current JWST  surveys. However, JWST
studies focused on the faint-end of the UVLF, such as Pérez-González et al.
(2023) and Leung et al. (2023) are within a few magnitudes of our brightest
galaxies. Given a strong lensing scenario, where magnifications allow for
observations of objects that are an order of magnitude fainter than these
studies, a measurement of the UVLF that reaches the predictions here may soon
be possible.

##  4 Discussion and Conclusion

Our results imply that JWST observations of the faint end of the UVLF are
close to a regime sensitive to the stream velocity. In particular, we predict
that a high-stream velocity patch will contribute to a higher UVLF compared to
the UVLF predicted by classic structure formation. This work also confirms
that a strong suppression of galaxies with low stellar mass
($M_{*}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{6.5}\,M_{\odot}$) is present in the early Universe due to
the stream velocity.

Although this faint regime has not yet been detected, given that dwarf
galaxies may contribute significantly to the production of ionizing photons at
high redshift, the stream velocity effect may contribute to a spatial
modulation in the progression of the early phases of reionization.
Additionally, the suppression may persist to lower redshifts beyond the end of
our simulation, where it could be detected by JWST or future missions that
survey dwarf galaxies.

Given that the stream velocity coherence scale is on the order of a few tens
of Mpc, observational measurements of the UVLF drawn from cosmic volumes much
larger than the coherence scale will represent objects drawn from both low and
high values of the stream velocity. Thus, the individual curves shown in our
Fig. 5 will not be resolved except by observations below the coherence scale.
When comparing surveys selected from volumes larger than the coherence scale,
the stream velocity effect should contribute to an overabundance of UV bright
galaxies, scattering points above the standard $\Lambda$CDM curve. Our results
predict that observations from regions comparable and smaller than the
coherent length at a certain redshift would fall along a curve corresponding
to a single value of the stream velocity. The stream velocity varies on
spatial scales $>$ few Mpc and varies randomly in magnitude and direction in
space following a Maxwell-Boltzmann distribution (e.g., Fialkov, 2014). We
note that this study only investigates the effects with a single value of the
stream velocity as a proof-of-concept, and we leave detailed studies of
intermediate or larger values to future work.

We suggest that as data sets at high redshift grow in completeness with future
observations, it will be possible to undertake a statistical comparison of a
larger cosmological volume to determine the presence of the stream velocity.
Furthermore, since high-density peaks often correlate to large stream velocity
values (e.g., Tseliakhovich et al., 2011), we expect that deep field JWST
observations will likely probe volumes formed in the presence of the stream
velocity. For an ultra deep field survey such as NGDEEP (e.g., Bagley et al.,
2023) observing a highly magnified field ($\mu\sim 10-50$) that reaches
$M_{UV}\sim-14$, we expect an enhanced luminosity function at the faint end.

JWST has opened a window into the high redshift Universe and the earliest
stages of galaxy formation, allowing for cosmological tests at earlier times
than previously possible. While current interest often focuses on the observed
discrepancies at the bright end of the UVLF, we propose that the faint end of
the UVLF contains information about baryonic physics and the interaction with
dark matter at early times. The stream velocity is a robust prediction of
$\Lambda$CDM and is not present or decays rapidly in some alternative DM
models (e.g., Nadler et al., 2019; Maamari et al., 2021; Driskell et al.,
2022), thus, our methods represent a new test of $\Lambda$CDM structure
formation and may help constrain alternative models. Such observed faint-end
fluctuations in the UVLF on the coherence scale of the stream velocity would
constitute the first high-redshift evidence of the presence of the stream
velocity in the early Universe.

## Acknowledgements

The authors would like to thank Steve Furlanetto and Sahil Hegde for
constructive conversations and feedback. C.E.W., W.L., S.N., Y.S.C, B.B.,
F.M., and M.V. thank the support of NASA grant No. 80NSSC20K0500 and the
XSEDE/ACCESS AST180056 allocation, as well as, the UCLA cluster Hoffman2 for
computational resources.

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##  Appendix A Clustering correction factor

Figure 6: Top Panel: The number of halos greater than mass $M$ for the
standard value $\sigma_{8}=0.8$ (dark grey) and $\sigma_{8}=1.7$ (light grey)
used in the simulations presented here. The halo mass function from Sheth &
Tormen (2002) is used. Also shown are the numerical results from our
simulations at a selection of redshifts (pink and blue dashed lines). Bottom
Panel: The ratio between $n(>M)$ at $\sigma_{8}=0.8$ and $\sigma_{8}=1.7$
($f_{1.7}$) as a function of halo mass. This ratio corresponds to the
correction factor used to calibrate our UVLF.

In our simulations, the value of $\sigma_{8}$ is increased from the nominal
value of $\sigma_{8}=0.8$ to $\sigma_{8}=1.7$ in order to increase the
statistical power of our study without affecting the underlying cosmology.
However, this results in an overabundance of halos. In order to describe the
number density of objects in a typical region with $\sigma_{8}=0.8$, we apply
an analytically calculated correction based on the Sheth & Tormen (2002) halo
mass function. We calculate the Sheth & Tormen halo mass function with
$\sigma_{8}=1.7$ and $\sigma_{8}$ at $z=12$. Our simulation number counts with
$v_{bc}=0\sigma_{vbc}$ are consistent with the Sheth & Tormen value (see top
panel of Fig. 6 comparing the numerical number counts to the analytical
prediction). The correction factor is a function of mass. To correct the UVLF,
we calculate the average mass in each bin and divide by the correction factor
corresponding to the average mass. This introduces an uncertainty in our
predicted value due to the range of masses in each magnitude bin. The
correction factor is shown in Fig. 6 in the bottom panel.

##  Appendix B Example buildup of stellar mass

Figure 7:  Stellar mass of massive objects between $z=12$ and $z=15$ with the
stream velocity (pink) and the same object without the stream velocity (blue).
An example object (the same object shown in Fig. 1 is shown in the thick
dashed lines. In the simulation run with stream velocity (shown in pink), the
object grows from $10^{6.39}M_{\odot}$ to $10^{7.49}M_{\odot}$, whereas
without the stream velocity (blue), the object has already formed the bulk of
its stellar mass by that time.

Figure 7 shows the stellar mass as a function of redshift for an example
object in the simulation (solid dashed lines). In a period between $z=15$ and
$z=12$, the object in the run with the stream velocity builds up the majority
of its stellar mass, whereas its counterpart in the no stream velocity run
does not undergo significant star formation because it already accrued the
bulk of its stellar mass prior to that time. By $z=12$, the object with the
stream velocity has reached a similar mass to the object in the $0\sigma_{\rm
bc}$ run. In the background, similar tracks comparing objects with and without
the stream velocity are shown in faint solid lines.