# The resolved star formation law in NGC 7469 from JWST, ALMA and VLA

Maria Vittoria Zanchettin SISSA, Via Bonomea 265, I-34136 Trieste, Italy  INAF
Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34143 Trieste, Italy
Marcella Massardi INAF, Istituto di Radioastronomia, Italian ARC, Via Piero
Gobetti 101, I-40129 Bologna, Italy  Francesco Salvestrini INAF Osservatorio
Astronomico di Trieste, via G.B. Tiepolo 11, 34143 Trieste, Italy  Manuela
Bischetti Dipartimento di Fisica, Università di Trieste, Sezione di
Astronomia, Via G.B. Tiepolo 11, I-34131 Trieste, Italy  INAF Osservatorio
Astronomico di Trieste, via G.B. Tiepolo 11, 34143 Trieste, Italy  Chiara
Feruglio INAF Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34143
Trieste, Italy  IFPU - Institute for fundamental physics of the Universe, Via
Beirut 2, 34014 Trieste, Italy  Fabrizio Fiore INAF Osservatorio Astronomico
di Trieste, via G.B. Tiepolo 11, 34143 Trieste, Italy  IFPU - Institute for
fundamental physics of the Universe, Via Beirut 2, 34014 Trieste, Italy
Andrea Lapi SISSA, Via Bonomea 265, I-34136 Trieste, Italy  IFPU - Institute
for fundamental physics of the Universe, Via Beirut 2, 34014 Trieste, Italy
INAF, Istituto di Radioastronomia, Italian ARC, Via Piero Gobetti 101, I-40129
Bologna, Italy  INFN, Sezione di Trieste, via Valerio 2, Trieste I-34127,
Italy

###### Abstract

We investigate the star formation process within the central 3.3 kpc region of
the nearby luminous infrared Seyfert NGC 7469, probing scales ranging from 88
to 330 pc. We combine JWST/MIRI imaging with the F770W filter, with CO(2-1)
and the underlying 1.3 mm dust continuum data from ALMA, along with VLA radio
continuum observations at 22 GHz. NGC 7469 hosts a starburst ring which
dominates the overall star formation activity. We estimate a global star
formation rate SFR $\sim 11.5$ $\rm M_{\odot}~{}yr^{-1}$ from the radio at 22
GHz, and a cold molecular gas mass $\rm M(H2)\sim 6.4\times 10^{9}M_{\sun}$
from the CO(2-1) emission. We find that the 1.3 mm map shows a morphology
remarkably similar to those traced by the 22 GHz and the 7.7 µm polycyclic
aromatic hydrocarbon (PAH) emission observed with JWST. The three tracers
reproduce the morphology of the starburst ring with good agreement. We further
investigate the correlations between the PAHs, the star formation rate and the
cold molecular gas. We find a stronger correlation of the PAHs with the star
formation than with the CO, with steeper correlations within the starburst
ring ($n>2$) than in the outer region ($n<1$). We derive the correlation
between the star formation rate and the cold molecular gas mass surface
densities, the Kennicutt-Schmidt star formation law. Comparisons with other
galaxy populations, including starburst galaxies and active galactic nuclei,
highlighted that NGC 7469 exhibits an intermediate behavior to the Kennicutt-
Schmidt relations found for these galaxy populations.

AGN host galaxies (2017) — Molecular gas (1073) – Star formation (1569) —
Interstellar medium (847)

††facilities: JWST(MIRI), ALMA, VLA††software: astropy (Astropy Collaboration
et al., 2013, 2018), emcee (Foreman-Mackey et al., 2013), CASA (McMullin et
al., 2007), WebbPSF (Perrin et al., 2014)

##  1 Introduction

Schmidt (1959, 1963) proposed the existence of empirical scaling relations
between the properties of the interstellar gas and the star formation rate
(SFR) in the same environment. Later, Kennicutt (1998a, b) confirmed the
correlation between global measurements of galaxy gas mass and SFR in nearby
galaxies. Consequently, this correlation has become known as the Kennicutt-
Schmidt (K-S) star formation law.

Spatially unresolved studies that target molecular gas and SFR help to derive
global disk properties and are necessary to test the redshift dependence of
the gas–star formation relation (Genzel et al., 2010; Schruba et al., 2011).
However, to understand the physical mechanisms behind the star formation (SF)
process, spatially resolved measurements in nearby targets are necessary.
These observations allow us to study how the SF properties, such as the
timescales for gas consumption and the SF efficiency, vary across the galaxy
environments (Onodera et al., 2010; Bigiel et al., 2011; Leroy et al., 2013;
Ford et al., 2013; Kreckel et al., 2018; Dey et al., 2019; Pessa et al., 2022;
Sun et al., 2023). The tracers most commonly used to quantify the SFR are the
most commonly observed optical spectral lines (e.g. $\rm H_{\alpha}$ and [O
II]), the global radio continuum, mid- and far- infrared (MIR and FIR), and
ultraviolet (UV) emission. Such SFR indicators refer to various wavelengths
that trace stellar populations at different evolutionary stages (see Kennicutt
& Evans, 2012, for a review). Optical lines and UV light are emitted
predominantly by young stars and are sensitive to SF on $\sim$10 Myr and
$\sim$10–100 Myr timescales respectively (Hao et al., 2011; Murphy et al.,
2011). However, accurate SFR estimates from UV data are affected by large
uncertainties due to the dust extinction correction (Mahajan et al., 2019).
The thermal free-free and non-thermal synchrotron radio emission at low
frequency (1-10 GHz) are a measure of past formation of massive stars,
resulting in a SF time sensitivity of about 150-300 Myr (Klein & Emerson,
1981; Gioia et al., 1982; Tabatabaei et al., 2017; Arango-Toro et al., 2023).
The MIR (3–25 µm) emission of star forming galaxies is dominated by the
polycyclic aromatic hydrocarbon (PAH) features both at low and high redshift
(Puget & Leger, 1989; Genzel et al., 1998; Armus et al., 2007; Valiante et
al., 2007; Farrah et al., 2008; Veilleux et al., 2009; Riechers et al., 2014).
In particular PAHs has been proposed as a good indicator of SFR over
timescales of up to a few hundred million years (Peeters et al., 2004; Shipley
et al., 2016; Xie & Ho, 2019). Moreover a connection between PAHs and the
properties of the cold molecular gas, traced by the CO emission, has been
proposed by several studies (Cortzen et al., 2019; Alonso-Herrero et al.,
2020). Taking advantage of interferometric mm and cm, and JWST data of nearby
galaxies, it is possible to perform spatially resolved studies of the SF
process in galaxies, down to hundred-parsec scales and to investigate the
correlations between the PAHs, the SFR surface density ($\rm\Sigma_{SFR}$) and
the mass surface density of molecular gas ($\rm\Sigma_{H2}$), which is the
direct fuel of the formation of stars.

In this paper, we study the SF process in the inner ($\sim$3 kpc) region of
NGC 7469, a nearby (z = 0.01627) luminous infrared galaxy ($\rm
L_{IR,8-1000\rm\mu m}=10^{11.6}L_{\odot}$), and type-1.5 Seyfert (Landt et
al., 2008) with a bolometric AGN luminosity of $\rm L_{bol,AGN}=3.5\times
10^{44}$ erg s-1(Gruppioni et al., 2016). From VLA 8.4 GHz observations at 0.3
arcsec resolution, Pérez-Torres et al. (2009) estimated the nuclear radio
emission $\rm L_{8.4~{}GHz}\sim 10^{36.96}$ erg s-1 . NGC 7469 may host a
radio core–jet structure as resolved on 0.3 arcsec scale by the VLA (Lonsdale
et al., 2003; Alberdi et al., 2006; Orienti & Prieto, 2010), and no evidence
of a radio jet–ISM interaction has yet been reported (Xu & Wang, 2022). NGC
7469 is known to host a prominent starburst (SB) ring with an outer radius of
900 pc, which dominates the global SFR of $\sim 20-50$ $\rm
M_{\odot}~{}yr^{-1}$(Genzel et al., 1995; Soifer et al., 2003; Gruppioni et
al., 2016; Song et al., 2021). A bar-like feature connects the SB ring to a
circumnuclear disk (CND) located in the inner 300 pc (see Davies et al., 2004;
Izumi et al., 2015, 2020, and references therein). The molecular gas mass
derived from CO(1-0) within the inner $\sim$1.2 kpc region is $\rm
M(H2)=2.7\times 10^{9}M_{\odot}$ (Davies et al., 2004; Izumi et al., 2020). To
investigate the SF process inside and outside the SB ring of NGC 7469, we
combine James Webb Space Telescope (JWST) imaging with the F770W filter of the
Mid-InfraRed Instrument (MIRI), with CO(2-1) and the underlying dust continuum
(224.2-243.7 GHz) data obtained from the Atacama Large
Millimeter/submillimeter Array (ALMA), along with the Karl G. Jansky Very
Large Array (VLA) radio continuum imaging at 22 GHz, resolving physical scales
in the range 88 pc (MIRI) to 330 pc (VLA).

This Paper is organized as follows. In Section 2, we describe the observations
and data reduction. Section 3 provides details on the data analysis and
quantity derivations. Section 4 describes our results. The adopted luminosity
distance of the galaxy is 72.1 Mpc and the corresponding scale is 0.338
kpc/arcsec.

##  2 Observations and Data Reduction

We analyzed band 6 ALMA 12 m observations (program ID 2017.1.00078.S, PI Izumi
Takuma) of NGC 7469 of the CO(2-1) line (observed frequency 226.85 GHz) and
the underlying continuum (224.2-243.7 GHz). NGC 7469 was observed in September
2018 by using the 12m array for a total integration time of 1.48 h in the
configuration C43-4 with 44 antennas, proving an angular resolution of about
0.4 arcsec and a largest angular scale of 4.8 arcsec. Visibilities were
calibrated by the NRAO Observatory, using the standard pipeline and default
calibrators. Imaging was performed with the Common Astronomy Software
Applications (CASA; McMullin et al., 2007), version 5.1.1-5. We created a
continuum map by averaging the visibilities in the four spectral windows
(spws), excluding the spectral range covered by the CO(2-1) emission line. We
produced continuum-subtracted datacubes by fitting and subtracting a first-
order polynomial from the visibilities of the spw covering CO(2-1). Imaging
was performed with task tclean using the hogbom algorithm, a threshold of
three times the root mean square (rms) noise, a briggs weighting scheme with
robust parameter $b=$ 0.5 for the continuum map and a natural weighting scheme
for the CO(2-1) data cube in order to increase the map sensitivity. The
resulting synthesized beam for the continuum map is 0.492 arcsec $\times$
0.360 arcsec (corresponding to 164 pc $\times$ 120 pc), with an rms of 0.0121
mJy/beam. The CO(2-1) data cube has a beam size of 0.508 arcsec $\times$ 0.374
arcsec and a rms noise of 0.518 mJy/beam for a channel width of 5 km s-1. This
translates into a mass surface density sensitivity of 7.8 $\rm M_{\odot}$/beam
for a channel width of 5 km s-1, computed using equation 1. Figure 1 shows
(left panel) the millimetre continuum map and (right panel) the continuum-
subtracted velocity-integrated map of the CO(2-1) emission.

We analysed archival VLA observations of NGC 7469 in band K (project code
20A-158, PI Krista L. Smith). The observations were carried out in April 2020
for a total integration time of 58.9 h in the frequency range 18-26 GHz. We
used the standard calibrated visibilites provided by the VLA data archive
l111(https://science.nrao.edu/facilities/vla/archive and we used CASA 5.1.1-5
software to generate the map of K band continuum. We averaged the 62 spw and
we produced a clean map using the tclean task and a briggs weighting scheme
with robust parameter $b=$ 0.5. We used the hogbom cleaning algorithm with a
detection threshold of 3 times the rms sensitivity. The final continuum map
has an rms of 23.2 $\rm\mu Jy/beam$ and a clean beam of 1.029 arcsec $\times$
0.967 arcsec (i.e. 342 pc $\times$ 322 pc).

JWST/MIRI (Bouchet et al., 2015; Rieke et al., 2015) imaging of NGC 7469 was
obtained with the F770W (7.7 $\mathrm{\mu m}$) filter with both the BRIGHTSKY
and SUB128 subarray modes (program ID 1328, PI Armus Lee). In this paper we
focus the analysis on the SUB128 subarray data, where the nucleus remained
unsaturated, taken on 2022 July 2 with an exposure time of 48 s. The field of
view of 14.8 arcsec $\times$ 16.4 arcsec and the pixel size is 0.11 arcsec.
The calibrated JWST/MIRI 7.7 µm map has a sensitivity of 12 $\rm
MJy\,sr^{-1}\,pixel^{-1}$. We used the calibrated images provided by the STSCI
Observatory, downloaded through the MAST
portal222https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html.

Figure 1: Left panel: 1.3 mm continuum map. Labels from A to E indicate the
five clumps of emission. The position of A corresponds to the AGN position
(23:03:15.617 +08:52:26.00 J2000, Izumi et al., 2020). Contours are drawn at
(3, 5, 10, 15, 20, 40)$\rm\sigma$, $\rm\sigma=0.0121$ mJy/beam. The clean beam
(0.492 arcsec $\times$ 0.360 arcsec, PA=46 deg) is indicated by the gray
ellipse. Right panel: velocity integrated CO(2-1) (moment-0) map. White
contours are drawn at (3, 5, 10, 15, 20, 30)$\rm\sigma$, $\rm\sigma=0.56$
Jy/beam.km/s. The clean beam (0.508 $\times$ 0.374 arcsec, PA=45 deg) is
indicated by the gray ellipse. Regions with emission below 1$\rm\sigma$ are
blanked out.

Figure 2: Left panel: VLA 22 GHz radio continuum map (color scale), with
corresponding white contours drawn at (3, 5, 8, 10, 15)$\rm\sigma$,
$\rm\sigma=0.023$ mJy/beam. The clean beam (1.03 arcsec $\times$ 0.97 arcsec,
PA=-13 deg) is indicated by the gray ellipse. Right panel: JWST MIRI/F770W
image (color scale), with white contours drawn at (3, 5, 10, 15, 20, 30,
50)$\rm\sigma$, $\rm\sigma=27$ MJy/sr. The black contours in both panels show
the 1.3 mm continuum emission (same as in Figure 1, left panel).

Figure 3: Surface brightness at 7.7 $\rm\mu m$ from JWST/MIRI imaging versus $\rm\Sigma_{SFR}$ (left panel), and $\rm\Sigma_{H_{2}}$ (right panel), within the central 10 arcsec $\times$ 10 arcsec region of NGC 7469. The symbols are colored according to their distance from the galaxy center (color bar). The dashed blue (red) lines are the fits to data for $\rm r<r_{SB}$ ($\rm r>r_{SB}$), where $\rm r_{SB}\sim 900pc$ is the outer SB ring radius (Song et al., 2021).  Table 1: Derived parameters for the $\rm\Sigma_{SFR}$ \- $\rm SB_{PAH}$ and $\rm\Sigma_{H2}$ \- $\rm SB_{PAH}$ correlations. |  | $\rm\Sigma_{SFR}-\rm SB_{PAH}$ | $\rm\Sigma_{H2}-\rm SB_{PAH}$
---|---|---|---
radius | number data points | $\rho$ | $\tau$ | $n$ | $k$ | $\rho$ | $\tau$ | $n$ | $k$
(a) | (b) | (c) | (d) | (e) | (f) | (c) | (d) | (e) | (f)
$\rm r>0$ | 82 | 0.91 | 0.75 | 1.77 $\pm$ 0.08 | -2.56 $\pm$ 0.03 | 0.84 | 0.68 | 2.17 $\pm$ 0.11 | -15.48 $\pm$ 0.36
$\rm r<r_{SB}$ | 31 | 0.76 | 0.56 | 2.16 $\pm$ 0.20 | -2.78 $\pm$ 0.02 | 0.71 | 0.52 | 2.61 $\pm$ 0.25 | -10.25 $\pm$ 0.03
$\rm r>r_{SB}$ | 51 | 0.77 | 0.56 | 0.83 $\pm$ 0.10 | -2.36 $\pm$ 0.03 | 0.66 | 0.47 | 0.72 $\pm$ 0.09 | -4.39 $\pm$ 0.03

Figure 4: Spatially resolved star formation law within the central 10 arcsec
$\times$ 10 arcsec region of NGC 7469. Circles show the derived quantities in
each cell of the grid, coloured according to their distance from the AGN
(color bar). The dashed green lines are the linear fit to the data $\rm
log\Sigma_{SFR}=(1.02\pm 0.07)log\Sigma_{H2}+(-2.85\pm 0.02)$, derived by
using the python emcee library (Foreman-Mackey et al., 2013). Triangles show
the derived quantities for NGC 2992 (Zanchettin et al., 2023). The yellow
shaded area shows the best-fit correlation with 0.33 dex scatter found using
global measurements for (Kennicutt & De Los Reyes, 2021): $\rm
log\Sigma_{SFR}=(0.98\pm 0.07)log\Sigma_{H2}+(-2.27\pm 0.21)$. The purple
shaded area shows the range of values found by Casasola et al. (2015) for
local AGN host galaxies: $\rm log\Sigma_{SFR}=(1.14\pm
0.01)log\Sigma_{H2}+(-3.41\pm 0.02)$. Solid, dashed, and dotted black lines
represent gas depletion times $\tau_{dep}$ of 0.1, 0.5, and 1 Gyr,
respectively.

##  3 Analysis

Within CASA, we modeled and subtracted the central point source emission in
the VLA map to remove its contribution from the circumnuclear region. We used
the task os.system to model a point source at the AGN position (23:03:15.617
+08:52:26.00 J2000, Izumi et al., 2020) with a flux of 5.037 mJy at 22 GHz,
corresponding to 90% of the central beam flux. We performed several tests with
different point source flux normalization, and checked that with the assumed
fraction we avoid artificial over-subtracted regions. The final circumnuclear
region flux is corrected by 15 % and its value is consistent with the central
beam flux after point source subtraction. We used the uvsub task to subtract
the point source model from the visibility data, so obtaining the final
visibilities. We followed the cleaning procedure described in section 2 to
obtain the final map reported in Figure 2, left panel.

We used the WebbPSF333https://webbpsf.readthedocs.io tool version 1.2.1
(Perrin et al., 2014) to simulate the point-spread function (PSF) of the
JWST/MIRI F770W filter image, with a pixel size of 0.027 arcsec. We modeled
the radial PSF contribution by using circular sectors with a radius equal to
the grid cells size, see Section 3, and we derived the radial correction for
the F770W filter map. We corrected both the F770W filter fluxes extracted with
the grid and their associated errors for the PSF contribution according to
their distance from the center, by following standard error propagation (i.e.,
convolving the square of the error map with the square of the PSF error). Due
to the limitations in the current theoretical model for the JWST PSF, we do
not have a robust method for removing the PSF spikes (see Williams et al.,
2024, and references therein). Therefore we applied this conservative method
to subtract and model the PSF contribution. We excluded from the analysis the
central 1 arcsec ($\sim$330 pc) region where the PSF contribution is dominant.

We used the continuum-subtracted velocity-integrated CO(2-1) line intensity
map to derive the mass surface density of the cold molecular gas, VLA radio
map at 22 GHz to estimate the SFR surface density and JWST/MIRI F770W filter
map to estimate the PAH emission at 7.7 $\rm\mu m$. Since we analyzed one
single PAH feature at 7.7 $\rm\mu m$, we omit the 7.7 $\rm\mu m$ wording from
this point onward. We extract fluxes from a common grid applied to all maps.
First, we regridded all the maps at the VLA spatial resolution, i.e. the
coarser resolution of our data. The grid is centered at the AGN position and
has 1 arcsec wide (i.e. 0.330 kpc) squared cells, equal to regridded maps
resolution, and a total size of 10 arcsec $\times$ 10 arcsec (i.e. 3.3 kpc
$\times$ 3.3 kpc). Fluxes are extracted from each cell. If the mean flux in
the cell is below 3rms per pixel, we consider it as an upper limit. For each
grid cell, we derived the corresponding cold molecular gas mass ($\rm M_{H2}$)
surface density ($\rm\Sigma_{H2}$), SFR surface density ($\rm\Sigma_{SFR}$)
and PAH flux density ($\rm\Sigma_{PAH}$). The central cell (i.e. the one that
contains the AGN) is not used, since it is not possible to determine and
disentangle the AGN contribution to the measured flux with good accuracy.
Indeed, in this region, we expect both a significant contribution in the JWST
image arising from the dust-reprocessed AGN emission from the torus, and a
possible contribution to the VLA continuum due to the presence of a core jet-
like structure (Lonsdale et al., 2003; Alberdi et al., 2006). We note that
Armus et al. (2023) verified that in the total mid-infrared spectrum of NGC
7469 PAH emissions are significantly weaker at the nucleus, due to the AGN
heating effect. They reported that the SFR from PAH in the nuclear region, 0.3
arcsec ($\sim$100pc) diameter circular region, is less than 1% of the total
SFR in NGC 7469.

The $\rm\Sigma_{H2}$ in units of $\rm M_{\odot}\,pc^{-2}$ is derived from the
velocity-integrated CO(2-1) flux density as:

$\Sigma_{H2}=\alpha_{CO}\frac{S_{CO(2-1)}}{R_{21}}F_{21}cos~{}i$ (1)

where $F_{21}$ is the conversion factor from the flux density to the line
luminosity $L^{\prime}$ in K km s-1pc2 (see Solomon & Vanden Bout, 2005), i is
the galaxy inclination angle, and $\alpha_{\rm CO}$ is the CO-to-$\rm H_{2}$
conversion factor. We adopt a CO ladder correction factor of $R_{21}$ = 0.7
equal to the mean value found by Casasola et al. (2015), and $\alpha_{\rm
CO}=4.2^{+1.961}_{-1.341}\rm M_{\odot}(K\,km\,s^{-1}\,pc^{2})^{-1}$ derived
adopting equation 25 in Accurso et al. (2017), for a stellar mass of
$\log(M_{*}/M_{\odot})=10.5$ (Díaz-Santos et al., 2007). The $cos~{}i$ factor
is the correction for galaxy inclination, which we assume i=45 deg (Davies et
al., 2004).

Assuming that most of the radio 22 GHz continuum emission arises from SF, we
adopt the radio luminosity at 22 GHz to derive the SFR. First, we derived the
radio luminosity at 1.4 GHz by assuming an average radio spectrum steepness
$\alpha=-0.85$ for the entire galaxy. We obtained the $\alpha$ parameter by
fitting the radio Spectral Energy Distribution (SED) using the integrated
radio fluxes reported by NED444https://ned.ipac.caltech.edu. The $\alpha$
parameter adopted is consistent with the values used to describe SF regions in
normal galaxies and LIRGs (e.g. Murphy et al., 2012; Linden et al., 2019,
2020) and with the values derived by Song et al. (2021). However, we cannot
exclude local variations in the steepness of the radio spectrum that can
affect the final SFR values. Song et al. (2021) derived the spectral index
measured from 15 to 33 GHz, finding values in between $\sim$-0.7 and
$\sim$-0.9 in the NGC 7469 nuclear ring. These variations translate into
corrections to the SFR up to 10$\%$ which will be taken into account further
in the discussion. We then obtained the SFR adopting the correlation reported
by Shao et al. (2018) in section 3.2, and the corresponding star formation
surface density, $\rm\Sigma_{SFR}$, by correcting for galaxy inclination, i.e.
multiplying by the $cos~{}i$ factor as in equation 1.

The MIRI F770W allows us to sample the 7.7 $\rm\mu m$ PAH emission complex in
NGC7469. Although the band is dominated by the PAH emission feature, it may
also include continuum emission due to hot dust. Whitcomb et al. (2023) showed
that the Spitzer/IRS spectra of the star-forming regions in 75 nearby galaxies
(SINGS sample) may have up to 10% continuum contamination in the F770W filter.
Therefore, in a conservative approach, we assumed that 90% of the flux is
attributed to PAH (Chastenet et al., 2023).

##  4 Discussion and Conclusion

###  4.1 Morphology of the SB ring

We estimate the global SFR integrated over the central 3.3 kpc $\times$ 3.3
kpc into SFR = $11.5\pm 0.3$ $\rm M_{\odot}~{}yr^{-1}$, following the
prescription of Shao et al. (2018), see Section 3. This estimate does not
include the contribution from the central 330 pc $\times$ 330 pc cell. The
uncertainty accounts for a realistic estimate of the systematic scatter in the
1.4 GHz-based method ($\sim$0.3 dex according to Shao et al., 2018). Our SFR
estimate is consistent with the SFR = $21\pm 2.1$ $\rm M_{\odot}~{}yr^{-1}$ by
Song et al. (2021) from the 3-to-33 GHz radio continuum, and with the range of
SFR = 10-30 $\rm M_{\odot}~{}yr^{-1}$ reported by Lai et al. (2022) using
optical emission lines, namely the $\rm Pf\alpha$ recombination line, [Ne II]
and [Ne III]. We found that the majority, approximately 60%, of the SF occurs
in the SB ring, consistent with Song et al. (2021).

We computed the molecular gas mass over the central 3.3 kpc $\times$ 3.3 kpc
$\rm M(H2)=(6.4\pm 2.0)\times 10^{9}M_{\sun}$, where the error is dominated by
the uncertainty on $\alpha_{\rm CO}$ (0.165 dex, Accurso et al., 2017). This
estimate is consistent within 2 sigma with the values reported in the
literature using different CO transitions (Davies et al., 2004; Izumi et al.,
2020; Nguyen et al., 2021; Zhang & Ho, 2023). We note that these works focus
on the inner $\sim$1 kpc region of NGC 7469, i.e. the SB ring. We computed
that $\sim 50\%$ of the total cold molecular gas mass is contained in the SB
ring, $\rm M(H2)_{SB}=(3.3\pm 1.0)\times 10^{9}M_{\sun}$. The central
concentration of gas and SFR is in line with those observed in local LIRGs
(Díaz-Santos et al., 2010, 2011). Adopting our estimate of $\rm M(H2)$ and
SFR, we derived the characteristic gas depletion timescale, $\rm\tau_{dep}$ =
$\rm M(H_{2})/SFR$. We obtained $\rm\tau_{dep}=(0.56\pm 0.18)~{}Gyr$ for the
central 3.3 kpc $\times$ 3.3 kpc region, and a shorter $\rm\tau_{dep}=(0.47\pm
0.15)~{}Gyr$ for the SB ring. Considering the errors associated with the
values, the depletion times are consistent within one sigma.

Figure 1 shows the distribution of the 1.3 mm continuum and the integrated
CO(2-1) flux map (Moment 0 map). In the continuum map, we identify five
representative positions (A–E), that are detected as clumps with SNR$>15$.
Clumps from A to D are named as in Izumi et al. (2020), who first identified
the clumps using the ALMA 860 $\rm\mu m$ continuum map. In addition, we
identify clump E (23:03:15.60 +08:52:24.56, J2000) where we detect 1.3 mm
emission with a statistical significance of $\sim$18$\rm\sigma$. The 1.3 mm
central peak position, labeled as clump A, is consistent with the AGN position
from VLA 3.5 cm and ALMA 860 $\rm\mu m$ continuum emission (Condon et al.,
1991; Orienti & Prieto, 2010; Izumi et al., 2020). Due to the PSF smearing of
the central AGN emission in MIRI 7.7 $\rm\mu m$, we excluded from the
following analysis the central cell, which contains also the CND (Davies et
al., 2004; Izumi et al., 2015). Figure 2 shows the VLA 22 GHz emission map
(left panel) and the JWST/MIRI PAH emission map (right panel), both with 1.3
mm emission contours overlaid. The 1.3 continuum map traces the dust
distribution across the SB ring, which shows a morphology that is remarkably
similar to those traced by PAHs and radio continuum at 22 GHz. The three
tracers reproduce the morphology of the SB ring with good agreement. In
particular, clumps B, C, and D are detected with similar morphology and size
in the dust continuum, CO(2-1), 22 GHz, and the PAH emission. The overall
clumps distribution appears to be consistent also with the 860 $\rm\mu m$
continuum (Izumi et al., 2020), suggesting that the 1.3 mm emission is mainly
tracing thermal dust emission heated by young stars in the SB ring. While the
dusty clumps B, C, D, and E all show prominent counterparts also in the CO map
(Figure 2), the dusty clump E exhibits no evident counterpart in the radio or
mid-IR maps. Interestingly, the 22 GHz radio continuum map presents an
emission peak with a significance of 15$\rm\sigma$ (23:03:15.63 +08:52:24.98
J2000), offset by about 0.5 arcsec with respect to E along the north-east
direction (white and black contours in the left panel of Figure 3). We exclude
a contribution from the unresolved radio jet at this location, as a core jet-
like structure was only detected at the nucleus (Lonsdale et al., 2003;
Alberdi et al., 2006). Indeed, the local enhancement in the radio emission may
be attributed to a gas inflow associated to the SB ring and to the two-arm
spiral structure, suggested by the high velocity dispersion of the cold
molecular gas at this location (Nguyen et al., 2021). Clumps B and D are
located at the endpoints of a bar-like feature identified by Davies et al.
(2004); Fathi et al. (2015); Izumi et al. (2015). Although the bar structure
is detected at a significance level of 12$\rm\sigma$ in the CO(2-1) map, it
remains undetected down to 3$\rm\sigma$ in the radio continuum and in the 7.7
$\rm\mu m$ maps, despite the high angular resolution of JWST/MIRI.

###  4.2 PAH emission

In the following, we investigate the correlation between $\rm\Sigma_{SFR}$,
$\rm\Sigma_{H_{2}}$ and PAH surface brightness, $\rm SB_{PAH}$. Figure 3 shows
$\rm SB_{PAH}$ versus $\rm\Sigma_{SFR}$ (left panel), and $\rm\Sigma_{H_{2}}$
(right panel), within the region of NGC 7469 sampled by the grid. The PAH
surface brightness correlates both with the SFR and the cold molecular gas
surface densities. We derived the Pearson $\rho$ and the Kendall $\tau$ rank
correlation coefficients (Table 1), finding a higher correlation of $\rm
SB_{PAH}$ with $\rm\Sigma_{SFR}$ than with $\rm\Sigma_{H2}$, both for the SB
ring and for external regions. Although the signal-to-noise ratio (SNR) of the
radio map is lower than that of the CO map (Section 2), the correlation
between $\rm SB_{PAH}$ and $\rm\Sigma_{SFR}$ is stronger and narrower than
that with $\Sigma_{H2}$. This is likely due to PAH emission being more
affected by the ionizing radiation field of young stars rather than by the
availability of molecular gas (see Jensen et al., 2017; Belfiore et al., 2023,
and references therein).

We also observe a steepening of both relations moving towards the center of
the galaxy, as highlighted in Figure 3 by the color gradient of the symbols.
To quantify this steepening, we divided the data into two subsamples: an inner
region within the SB ring ($\rm r<r_{SB}$) and an outer region with $\rm
r>r_{SB}$, where $\rm r_{SB}=900~{}pc$. This value has been measured by (Song
et al., 2021) using the 3 to 30 GHz continuum and is consistent with our maps.
We fit a linear relation $y=nx+k$ to each subsample using Python package emcee
(Foreman-Mackey et al., 2013), and assuming a constant error on
$\rm\Sigma_{H2}$ and $\rm\Sigma_{SFR}$, equal to the rms noise of each
regridded map. The uncertainty associated to $\rm SB_{PAH}$ is obtained by
convolving the rms noise of the regridded MIRI map with the uncertainty due to
the PSF contribution (Section 3). Blue and red solid lines in Figure 3 show
the fits for $\rm r<r_{SB}$ and for $\rm r>r_{SB}$, respectively. For both
$\rm\Sigma_{SFR}$ and $\rm\Sigma_{H2}$, the SB ring shows a steeper slope with
respect to $\rm r>r_{SB}$ (Table 1). In particular, the slope at the SB ring
is $n=(2.16\pm 0.20)$ whereas at $\rm r>r_{SB}$ is $n=(0.83\pm 0.10)$. A
similar behavior is seen for the $\rm\Sigma_{H2}$ correlation. The steepening
of these relations may be attributed to an enhancement of recent SF activity
in the SB ring structure, as already observed using the 11.3µm PAH feature by
Jensen et al. (2017). This is consistent with the presence of a young
population of ionizing stars in the SB ring, as reported by Díaz-Santos et al.
(2007). Jensen et al. (2017) measured a negligible contribution of AGN-heated
dust to the continuum flux at 11µm above 200 pc. Therefore, we exclude that
this steepening is due to an increased contribution of AGN-heated dust to the
continuum emission measured in the JWST F770W filter, at the location of the
SB ring. Regarding a possible AGN contribution to PAH destruction, we do not
expect a significant effect given the modest luminosity of our target ($\rm
L_{bol,AGN}=3.5\times 10^{44}$ erg s-1 , Gruppioni et al., 2016).

Still local variations in gas properties, including composition, turbulence,
and metallicity, significantly influence the CO-to-H2 conversion factor (Leroy
et al., 2011; Narayanan et al., 2012; Papadopoulos et al., 2012; Bolatto et
al., 2013; Sandstrom et al., 2013; Teng et al., 2024). Oxygen abundances
compatible with solar metallicity were reported for NGC 7469 SB ring and outer
region by Cazzoli et al. (2020) and Zamora & Díaz (2023) using optical nebular
lines. However, metallicity determination is challenging at all spatial
scales, particularly in AGN-affected nuclear regions where it remains unknown.
Considering a CO-to-H2 conversion factor accounting for both solar-like
metallicity and gas excitation, as prescribed by Narayanan et al. (2012), we
found that the change in slope between SB ring and outer disk remains
appreciable. A significantly decreasing metallicity gradient toward the
nucleus could still mitigate the slope change in the
$\mathrm{SB_{PAH}-\Sigma_{H2}}$ relation. However, there are no evidence for
such trend. Additionally, several factors, including grain size distribution,
temperature, ionization field, metallicity, and dust amount can affect PAH
luminosity (e.g. Chastenet et al., 2023; Whitcomb et al., 2024). The 7.7 µm
complex is associated with larger, positively charged ions (Galliano et al.,
2008; Draine et al., 2021; Rigopoulou et al., 2021; Maragkoudakis et al.,
2022), and its emission can increase around young stars due to a stronger
radiation field. However, investigating PAH properties such as grain size
distribution and ionization fraction involves examining the ratios between
various MIR PAH features (Lai et al., 2023; Bohn et al., 2023). Finally,
spatially resolved studies in nearby galaxies have reported a steepening of
the radio spectral index with increasing distance from the center (e.g.
Paladino et al., 2009; Basu et al., 2015; Roy & Manna, 2021), with additional
local variations in regions of active SF (Westcott et al., 2018). We therefore
cannot exclude that the superposition of these effects could cause the
observed slope change in the $\rm logSB_{PAH}-log\Sigma_{SFR}$ relation.

###  4.3 The star formation law

Figure 4 shows the spatially resolved star formation law on a $\sim 300$ pc
scale for NGC 7469. The plot also includes NGC 2992 measurements obtained with
a similar method on $\sim 200$ pc scales (Zanchettin et al., 2023), and the
relations derived for compilation of local AGN (Casasola et al., 2015), and
ultraluminous/luminous infrared galaxies (U/LIRGs, $\rm
L_{IR}>10^{12}\,L_{\odot}$, and $\rm>10^{11}\,L_{\odot}$, respectively,
Kennicutt & De Los Reyes, 2021). The purple shaded area shows the K-S relation
for local AGN hosts drawn from the NUclei of GAlaxies (NUGA) survey,
characterized by SFR within $\rm-0.9<log~{}SFR/M_{\odot}yr^{-1}<0.8$ and cold
molecular gas masses in the range $\rm 8.5<logM(H_{2})/M_{\odot}yr^{-1}<9.6$
(see Casasola et al., 2015, and references therein). The yellow shaded are
shows the star formation law found for dusty starbursts U/LIRGs, with $\rm
1.1<log~{}SFR/M_{\odot}yr^{-1}<1.8$, and cold molecular gas masses $\rm
9.4<logM(H_{2})/M_{\odot}yr^{-1}<10.2$ (Kennicutt & De Los Reyes, 2021). We
adopted the same fitting method as described above, taking into account the
uncertainties both on $\rm\Sigma_{SFR}$ and on $\rm\Sigma_{H2}$, and the best
fit linear relation for all data points is $\rm log\Sigma_{SFR}=(1.02\pm
0.07)log\Sigma_{H2}+(-2.85\pm 0.02)$ (green solid line in Figure 4). The slope
of this relation is consistent with that obtained by fitting the SB ring
alone, while we find a flatter slope for the fit of the outer region
($n=0.5\pm 0.1$). As the latter fit is dominated by data points with low SNR,
we consider the global $\rm log\Sigma_{SFR}-log\Sigma_{H2}$ relation as best-
fit for NGC 7469.

The star formation relation derived for the NUGA AGN (purple shaded area)
exhibits a slightly steeper slope ($n=1.14\pm 0.01$) and a $\sim$0.6 dex
offset towards lower SFR surface densities with respect to NGC 7469. Kennicutt
& De Los Reyes (2021) found that starburst galaxies and U/LIRGs are fitted by
a linear relation with slope $n=(0.98\pm 0.07)$, consistent with the one
derived for NGC 7469 in this work, and a $\sim$0.6 dex offset in the zero-
point towards higher SFR surface densities (yellow shaded area). Considering
the associated errors, the slopes of the relations reported by Kennicutt & De
Los Reyes (2021) and Casasola et al. (2015) are roughly consistent within
three sigma. Figure 4 indicates that the individual regions of NGC 7469 occupy
an intermediate area of the $\rm log\Sigma_{SFR}-log\Sigma_{H2}$ plane,
corresponding to an intermediate star formation regime between local AGNs and
U/LIRGs. The observed scatter in the data points can reflect local variations
in the physical conditions of the ISM from which stars form, such as the
morphology and concentration of the gas, and any differences in gas
excitation. Additionally, local fluctuations in physical quantities affect the
conversion (e.g., CO-to-H2 conversion factor, radio steepness) from
observables to physical quantities. In Figure 4, we report the characteristic
gas depletion timescales: NGC 7469 data points show an average
$\rm\tau_{dep,mean}$ = ($0.66\pm 0.02$) Gyr, where the uncertainty is the
standard error of the mean. The measurements of the circumnuclear region of
NGC 2992 (black triangles, Zanchettin et al., 2023), derived with a spatial
resolution of 200 pc, populate regions of the star formation law chart
characterized by both relatively lower $\rm log\Sigma_{SFR}$ and $\rm
log\Sigma_{H2}$, with respect to NGC 7469. The difference observed between NGC
2992 and our target may reflect the fact that individual galaxies may show
different star formation laws, as highlighted by Casasola et al. (2022) in
their work on a set of nearby spiral
DustPedia555http://dustpedia.astro.noa.gr/ galaxies. However, NGC 2992 has an
average $\rm\tau_{dep,mean}$ = ($0.38\pm 0.07$) Gyr, consistent with that
measured in our target.

##  5 Acknowledgments

.

This work is based in part on observations made with the NASA/ESA/CSA James
Webb Space Telescope. The data were obtained from the Mikulski Archive for
Space Telescopes at the Space Telescope Science Institute, which is operated
by the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127 for JWST. These observations are associated with program
No. 1328. The JWST data used in this paper can be found in MAST:
http://dx.doi.org/10.17909/3x02-0q74 (catalog 10.17909/3x02-0q74). This paper
makes use of the following ALMA data: ADS/JAO.ALMA#2017.1.00078.S ALMA is a
partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. Based on observations No.
20A-158 with the National Radio Astronomy Observatory which is a facility of
the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc. We acknowledge financial support from PRIN MIUR
contract 2017PH3WAT, and PRIN MAIN STREAM INAF ”Black hole winds and the
baryon cycle”. We acknowledge support from the INAF Mini Grant GO/GTO ”SHORES:
Serendipitous H-ATLAS fields Observations of Radio Extragalactic Sources”.
M.B. acknowledges support from INAF project 1.05.12.04.01 - MINI-GRANTS di
RSN1 ”Mini-feedback” and FVG LR 2/2011 project D55-microgrants23 ”Hyper-gal”.

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# The Average Size of a Connected Vertex Set of a $k$-connected Graph

Andrew Vince  Department of Mathematics
University of Florida
Gainesville, FL 32611, USA<EMAIL_ADDRESS>
###### Abstract.

The topic is the average order $A(G)$ of a connected induced subgraph of a
graph $G$. This generalizes, to graphs in general, the average order of a
subtree of a tree. In 1984, Jamison proved that the average order, over all
trees of order $n$, is minimized by the path $P_{n}$, the average being
$A(P_{n})=(n+2)/3$. In 2018, Kroeker, Mol, and Oellermann conjectured that
$P_{n}$ minimizes the average order over all connected graphs $G$ \- a
conjecture that was recently proved. In this short note we show that this
lower bound can be improved if the connectivity of $G$ is known. If $G$ is
$k$-connected, then

$A(G)\geq\frac{n}{2}\Bigg{(}1-\frac{1}{2^{k}+1}\Bigg{)}.$

###### Key words and phrases:

graph, connectedness, average order

###### 2010 Mathematics Subject Classification:

05C30

This work was partially supported by a grant from the Simons Foundation
(322515 to Andrew Vince).

##  1\. Introduction

Although connectivity is a basic concept in graph theory, problems involving
the enumeration of the connected induced subgraphs of a given graph have only
recently received attention. The topic of this paper is the average order of a
connected induced subgraph of a graph. Let $G$ be a connected finite simple
graph with vertex set $V=\\{1,2,\dots,n\\}$, and let $U\subseteq V$. The set
$U$ is said to be a connected set if the subgraph of $G$ induced by $U$ is
connected. Denote the collection of all connected sets, excluding the
emptyset, by $\mathcal{C}=\mathcal{C}(G)$. The number of connected sets in $G$
will be denoted by $N(G)$. Let

$S(G)=\sum_{U\in\mathcal{C}}|U|$

be the sum of the sizes of the connected sets. Further, let

$A(G)=\frac{S(G)}{N(G)}\qquad\qquad\text{and}\qquad\qquad D(G)=\frac{A(G)}{n}$

denote, respectively, the average size of a connected set of $G$ and the
proportion of vertices in an average size connected set. The parameter $D(G)$
is referred to as the density of connected sets of vertices. The density
allows us to compare the average size of connected sets of graphs of different
orders. If, for example, $G$ is the complete graph $K_{n}$, then $A(K_{n})$ is
the average size of a subset of an $n$-element set, which is $n/2$ (counting
the empty set for simplicity), and the density is $1/2$.

Papers [2, 4, 5, 7, 8, 9, 12, 13] on the average size and density of connected
sets of trees appeared beginning with Jamison’s 1984 paper [4]. The invariant
$A(G)$, in this case, is the average order of a subtree of a tree. Concerning
lower bounds, Jamison proved that the average size of a subtree of a tree of
order $n$ is minimized by the path $P_{n}$. In particular $A(T)\geq(n+2)/3$
for all trees $T$ with equality only for $P_{n}$. Therefore $D(T)>1/3$ for all
trees $T$. Vince and Wang [9] proved that if $T$ is a tree all of whose non-
leaf vertices have degree at least three, then $\frac{1}{2}\leq
D(T)<\frac{3}{4},$ both bounds being best possible.

Although results are known for trees, little was known until recently for
graphs in general. Kroeker, Mol, and Oellermann conjectured in their 2018
paper [6] that the lower bound of Jamison for trees extends to graphs in
general.

######  Conjecture 1.

The $P_{n}$ minimizes the average size of a connected set over all connected
graphs.

In [1] Balodis, Mol, and Oellermann verified the conjecture for block graphs
of order $n$, i.e., for graphs each maximal $2$-connected component of which
is a complete graph. The conjecture was proved recently for connecterd graphs
in general in [11] and independently shortly thereafter in [3].

######  Theorem 1.

If $G$ is a connected graph of order $n$, then

$A(G)\geq\frac{n+2}{3},$

with equality if and only if $G$ is a path. In particular, $D(G)>1/3$ for all
connected graphs $G$.

In [10] it was conjectured that the lower bound of Vince and Wang for trees
extends to graphs in general.

######  Conjecture 2.

If $G$ is a conectred graph all of whose vertices have degree at least $3$,
then $D(G)\geq\frac{1}{2}$.

Kroeker, Mol, and Oellermann [6] verified Conjecture 2 for connected cographs.
A cograph can be defined recursively: the one vertex graph is a cograph and,
if $G$ and $H$ are cographs, then so is their disjoint union and their join.
(The join is obtained by joining by an edge each vertex of $G$ to each vertex
of $H$.) Complete bipartite graphs are examples of cographs. Conjecture 2
remains open, but in this note we prove a lower bound on $D(G)$ close to $1/2$
if $G$ is highly connected. More precisely:

######  Theorem 2.

If $G$ is $k$-connected, then

$D(G)\geq\frac{1}{2}\Bigg{(}1-\frac{1}{2^{k}+1}\Bigg{)}.$

##  2\. Proof of Theorem 2

If $i\in V$, let $N(G,i),S(G,i)$, and $A(G,i)$ denote the number of connected
sets in $G$ containing $i$, the sum of the sizes of all connected sets
containing $i$, and the average size of a connected set containing $i$,
respectively. The following result appears in [11, Corollary 3.2].

######  Theorem 3.

If $i\in V$ is any vertex of a connected graph $G$ of order $n$, then

$A(G,i)\geq\frac{n+1}{2}.$

######  Corollary 4.

If $G$ is a connected graph of order $n$, then

$\sum_{U\in\mathcal{C}}|U|^{2}\geq\Big{(}\frac{n+1}{2}\Big{)}S(G).$

###### Proof.

From Theorem 3 we have

$S(G,i)\geq\frac{n+1}{2}N(i)$

for all $i\in V$. Now count the number of pairs in the set
$\\{(i,U):U\in\mathcal{C},i\in U\subseteq V\\}$ in two ways to obtain

$S(G)=\sum_{U\in\mathcal{C}}|U|=\sum_{i\in V}N(G,i).$

Similarly

$\sum_{U\in\mathcal{C}}|U|^{2}=\sum_{U\in\mathcal{C}}\sum_{i:i\in
U}|U|=\sum_{i\in V}\sum_{U\in\mathcal{C}:i\in U}|U|=\sum_{i\in
V}S(G,i)\geq\frac{n+1}{2}\sum_{i\in V}N(G,i)=\Big{(}\frac{n+1}{2}\Big{)}S(G).$

∎

###### Proof of Theorem 2.

The proof is by induction on $k$. When $k=1$, the statement is
$D(G)\geq\frac{1}{3}$, which follows from Theorem 1 in the introduction.

Let

$a_{k}:=\frac{1}{2}\Bigg{(}1-\frac{1}{2^{k}+1}\Bigg{)}.$

A straightforward calculation shows that

$a_{k}=\frac{2a_{k-1}}{2a_{k-1}+1}.$

Assume that the statement of Theorem 2 holds for all $(k-1)$-connected graphs
and asume that $G$ is $k$-connected. Let
$\mathcal{C}^{\prime}=\\{U\in\mathcal{C}:U\neq V\\}$, and denote by
$N^{\prime}(G)$ and $S^{\prime}(G)$ the number of connected sets and the sum
of the sizes of the connected sets, respectively, not including $V$. For $i\in
V$, denote by $G_{i}$ the graph induced by the vertices $V\setminus\\{i\\}$.
Note that $G_{i}$ is $(k-1)$-connected for all $i\in V$ and therefore, by the
induction hypothesis, we have $S(G_{i})\geq a_{k-1}(n-1)N(G_{i})$ for all
$i\in V$. Now

$\displaystyle nS^{\prime}(G)-\sum_{U\in\mathcal{C}^{\prime}}|U|^{2}$
$\displaystyle=\sum_{U\in\mathcal{C}^{\prime}}|U|(n-|U|)=\sum_{i\in
V}S(G_{i})\geq a_{k-1}(n-1)\sum_{i\in V}N(G_{i})$
$\displaystyle=a_{k-1}(n-1)\sum_{U\in\mathcal{C}^{\prime}}(n-|U|)=a_{k-1}n(n-1)N^{\prime}(G)-a_{k-1}(n-1)S^{\prime}(G).$

This implies

$\big{(}n+a_{k-1}(n-1)\big{)}(S(G)-n)\geq
a_{k-1}n(n-1)(N(G)-1)+\sum_{U\in\mathcal{C}}|U|^{2}-n^{2},$

equivalently

$\big{(}n+a_{k-1}(n-1)\big{)}S(G)\geq
a_{k-1}n(n-1)N(G)+\sum_{U\in\mathcal{C}}|U|^{2}.$

By Corollary 4 this implies

$\Big{(}\frac{n-1}{2}+a_{k-1}(n-1)\Big{)}S(G)\geq a_{k-1}n(n-1)N(G),$

or

$D(G)=\frac{S(G)}{nN(G)}=\frac{a_{k-1}}{\frac{1}{2}+a_{k-1}}=\frac{2a_{k-1}}{2a_{k-1}+1}=a_{k}.$

∎

## References

  * [1] K. J. Balodis, L. Mol, L, and R. Oellermann, On the mean order of connected induced subgraphs of block graphs, Australasian J. Comb. 76 (2020) 128–148.
  * [2] J. Haslegrave, Extremal results on average subtree density of series-reduced trees, J. Combin. Theory Ser. B 107 (2014), 26–41.
  * [3] J. Haslegrave, The path minimises the average size of a connected set, arXiv:2103.16491.
  * [4] R. Jamison, On the average number of nodes in a subtree of a tree. J. Combin. Theory Ser. B 35 (1983) 207–223.
  * [5] R. Jamison, Monotonicity of the mean order of subtrees, J. Combin. Theory Ser. B 37 (1984), 70–78.
  * [6] M. E. Kroeker, L. Mol, and O. Oellermann, On the mean connected induced subgraph order of cographs, Australasian J. Combin.71 (2018) 161–183.
  * [7] L. Mol and O. Oellermann, Maximizing the mean subtree order, J. Graph Theory, doi.org/10.1002/jgt.22434, 2018.
  * [8] A. M. Stephens and O. Oellermann, The mean order of sub-$k$-trees of $k$-trees, J. Graph Theory 88 (2018), 61–79.
  * [9] A. Vince and H. Wang, The average order of a subtree of a tree, J CombinTheory Ser B 100 (2010) 161–170.
  * [10] A. Vince, The average size of a connected vertex set of a graph - explicit formulas and open problems, J. Graph Theory 97 (2020), 82–103. DOI: https://doi.org/10.1002/jgt.22643.
  * [11] A. Vince, A lower bound on the average size of a connected vertex set of a graph, arXiv:2103.15174.
  * [12] S. Wagner and H. Wang, On the local and global means of subtree orders, J. Graph Theory 81 (2016), 154–166.
  * [13] W. Yan and Y. Yeh, Enumeration of subtrees of trees, Theoret. Comput. Sci. 369 (2006), 256–268.


$\displaystyle|g_{t}|\mathbb{P}\left[\sign\left[\sum_{i=1}^{M}\sign(\tilde{g}_{i,j})\right]\not=\sign(g_{j})\right]\leq\frac{\sigma_{i}}{\sqrt{M}}.$
(4)

As Lemma 1 proved, in the existence of the malicious clients, we can get
(4)$\leq\frac{\sigma_{i}}{\sqrt{M}}\cdot\mathcal{O}(\sqrt{M/h})=\frac{\sigma_{i}}{\mathcal{O}(\sqrt{h})}$.
Therefore, by plugging the result into (3), we can get the convergence rate of
BRIEF is

$\displaystyle\mathbb{E}\left[\frac{1}{T}\sum_{t=0}^{T-1}\left\|g_{t}\right\|_{1}\right]^{2}\leq$
$\displaystyle\frac{1}{\sqrt{N}}\Bigg{[}\sqrt{\|\textbf{L}\|_{1}}\left(f_{0}-f_{*}+\frac{1}{2}\right)$
(5)
$\displaystyle+\frac{2}{\sqrt{M}}\|\bm{\sigma}\|_{1}\cdot\mathcal{O}(\sqrt{M/h})\Bigg{]}^{2}$
$\displaystyle=\frac{1}{\sqrt{N}}\Bigg{[}\sqrt{\|\textbf{L}\|_{1}}\left(f_{0}-f_{*}+\frac{1}{2}\right)$
$\displaystyle+\frac{2}{\mathcal{O}(\sqrt{h})}\|\bm{\sigma}\|_{1}\Bigg{]}^{2}.$

∎

##  Appendix F Proof of Theorem 1

###### Proof.

We denote $a=(a_{1},\cdots,a_{np})$ and $b=(b_{1},\cdots,b_{np})$ are two
vectors uploaded by malicious clients, where $np$ refers to the number of
model parameters:

$Pr(a_{i},b_{i}=\mathsf{sign})=\begin{cases}\frac{1}{2},&\mathsf{sign}=+1\\\
\frac{1}{2},&\mathsf{sign}=-1\end{cases},\ \forall i\in np.$

The cosine similarity can be computed as:

$\displaystyle COS\\_similarity$
$\displaystyle=\frac{a_{1}b_{1}+\cdots+a_{np}b_{np}}{\sqrt{a_{1}^{2}+\cdots+a_{np}^{2}}\cdot\sqrt{b_{1}^{2}+\cdots+b_{np}^{2}}}$
$\displaystyle=\frac{a_{1}b_{1}+\cdots+a_{np}b_{np}}{np}.$

Since $a_{i}$ and $b_{i}$ are relatively independent, we have:

$Pr(a_{i}\cdot
b_{i}=\mathsf{sign})=\begin{cases}\frac{1}{2},&\mathsf{sign}=+1\\\
\frac{1}{2},&\mathsf{sign}=-1\end{cases},\ \forall i\in np.$

According to the Law of large numbers,

$E(COS\\_similarity)\sim E(a_{i}b_{i})=0.$

This conclusion can be generalized to any two malicious clients, and malicious
clients have the same distance as a semi-honest client. Therefore, if we
calculate the cosine similarity vector of two malicious clients, then there
should be only two elements of difference. The Euclidean distance of these two
vectors is $\sqrt{2}$. ∎

##  Appendix G Other Experimental Results

In Figure 7 and 8, we present the comparison of testing accuracy among
baseline, semi-honest and malicious groups under targeted attacks and ASR
between the groups under untargeted attacks. We see that the results are
consistent with the analysis in Section 5.1: the untargeted attacks nearly
have no impact on the accuracy of the final model (only slightly decreasing
the speed of convergence). Since GA may directly upload noise, it may be
easily detected from the beginning of the training to the end, resulting in
the same convergence as the baseline. For LFA, Krum, Trim and AA Attacks,
BRIEF is also difficult to distinguish between semi-honest and malicious
clients at the beginning. Thus, the speed of convergence is slightly
decreased. The main difference is that: LeNet achieves around $78\%$ in
FMNIST, while ResNet-18 provides approx. $56\%$ accuracy in CIFAR-10.

We also provide comparisons with the state-of-the-art Byzantine-robust methods
in Figure 9 and 10 on FMNIST and CIFAR-10 respectively. Similar to Figure 5,
under malicious majority of untargeted attacks, the testing accuracies of
_FLTrust, BRIEF_ , and _weights-BRIEF_ are maintained at the same level of the
baseline. Under the targeted attacks, _FLTrust, BRIEF_ , and _weights-BRIEF_
can restrain ASR to about 0%-10%. For more detailed explanations, please refer
to Section 5.1.

(a) Gaussian Attack

(b) Label Flipping Attack

(c) Krum Attack

(d) Trim Attack

(e) Adaptive Attack

(f) Backdoor Attack

Figure 7: Comparison of testing accuracy among baseline, semi-honest and
malicious groups under targeted attacks ((a)-(e)) and ASR between the groups
under untargeted attack (f), where we use LeNet to train FMNIST by default
settings in Table II.

(a) Gaussian Attack

(b) Label Flipping Attack

(c) Krum Attack

(d) Trim Attack

(e) Adaptive Attack

(f) Backdoor Attack

(g) Edge-case Attack

Figure 8: Comparison of testing accuracy among baseline, semi-honest and
malicious groups under targeted attacks ((a)-(e)) and ASR between the groups
under untargeted attacks ((f)-(g)), where we use ResNet-18 to train CIFAR-10
by the default settings in Table II.

(a) Gaussian Attack

(b) Label Flipping Attack

(c) Krum Attack

(d) Trim Attack

(e) Adaptive Attack

(f) Backdoor Attack

Figure 9: Comparison with Byzantine-robust methods in FMNIST by $\xi=0.1-0.9$.

(a) Gaussian Attack

(b) Label Flipping Attack

(c) Krum Attack

(d) Trim Attack

(e) Adaptive Attack

(f) Backdoor Attack

(g) Edge-case Attack

Figure 10: Comparison with Byzantine-robust methods in CIFAR-10 by
$\xi=0.1-0.9$.

##  Appendix H Discussions

###  H.1 Dynamic Attacks

Recall that BRIEF can perform well in terms of testing accuracy under the
assumption that malicious clients consistently perform one type of attack
(e.g., GA) throughout the whole training period. It may also perform well if
we allow malicious clients to perform different attacks on the epochs, e.g.,
GA to the first 10 epochs and then Krum attacks to the remaining epochs. We
notice that all the attacks (we consider in this work) except GA may require
several epochs of training (as a buffer) to produce attack effects. But these
buffer epochs may bring an advantage to BRIEF’s clustering performance. This
is so because the clustering ability is enhanced with the increase of training
rounds. On the other hand, the TNR and TPR (of the clustering) may be
relatively low in these epochs. Some malicious clients can be clustered into a
semi-honest group, but this will not seriously affect the accuracy performance
of the model.

One may think malicious clients may be allowed to perform all the attacks in a
single epoch. But so far, it is unknown how to group those attacks together
friendly and meanwhile maximize their attack effects. In practice, the attacks
may deliver an update in different directions, and further, they may even
yield influence on each other. For example, GA may easily destroy the
convergence of BA. We say that it is an interesting open problem to consider
launching GA, LFA, Krum, Trim, and AA attacks together in an epoch to evaluate
the accuracy and ARS.

###  H.2 Varying Clients Subsampling Rate

We assert that BRIEF may perform well under different clients’ subsampling
rates. This benefits from the proposed _Model Segmentation_ that can resist
malicious-majority clients. Imagine that in the context of an honest majority
(e.g., 40 out of 100 are malicious), if the subsampling rate is set relatively
low (e.g., 10%), we may eventually have a malicious majority case in one
training epoch with a high probability. Our experimental results have
demonstrated that BRIEF can still achieve Byzantine-robustness under a low
subsampling rate.

###  H.3 Learning Rate and Local Epoch

They are subtle parameters that may decide the performance of FL training. A
low learning rate may slow down the speed of convergence, and on the other
hand, a high rate may cause the model not to converge and ruin the accuracy.
As for the local epoch, in the case of iid, if carefully increasing the number
of epochs, we may make the model converge fast. But under a large degree of
non-iid (e.g., q=0.5), the increase of epoch may lead to updates in different
directions, making the FedAvg algorithm invalid [40]. In this work, we set
these two parameters according to the recommendations given in [40, 6].
Exploring their impacts on training may be orthogonal to the main focus of
this work.

###  H.4 Privacy-preserving DBSCAN

This is one of the core parts we used to build BRIEF. It may be applied to
other real-world domains, e.g., anomaly detection and encrypted traffic
analytics, where data should be clustered securely. But we note that the
current BRIEF with optimization may not scale well in these applications. We
did the optimization for the sake of efficiency by using SignSGD and binary
secret sharing, which may not support precise floating-point arithmetic.

###  H.5 Test Datasets

We notice that the EA is not applicable for FMINST since the research work
[59] did not provide a backdoor dataset. We may leave this as an open problem.
Based on the performance of BRIEF on MNIST and CIFAR-10, we may claim that
even in FMNIST, the ASR of EA may stay low, which is consistent with our main
conclusion.

In the experiments, we conduct three image datasets (MNIST, FMNIST, and
CIFAR-10). We claim that BRIEF can be further used to capture the Byzantine-
robust and privacy-preserving features of the models trained on text and
speech datasets, where the text dataset for the next-word prediction task may
require Recurrent Neural Networks (RNNs). To train these datasets, we may also
use weights or gradients to update the models (in the context of FL). If the
models or datasets are poisoned (by malicious clients), malicious updates may
have differences in updates directions from benign ones. Then BRIEF may be
used to protect benign clients. We may use other types of datasets during
training, but this will not affect our conclusions on Byzantine robustness and
privacy preservation.

###  H.6 Boosting Accuracy

Intuitively, CIFAR-10 is more difficult to train than (F)MNIST. The SignSGD
[6] achieves 90% accuracy with centralized learning. The reasons for the
accuracy drop in FL are (1) The number of clients: when the number increases,
the amount of data allocated to a client becomes less, and the local model is
more prone to overfitting; (2) Data heterogeneity: a larger degree of non-iid
leads to less information the local model can learn, which harms accuracy; (3)
Malicious clients: malicious clients taking a part of the training data can be
filtered out by BRIEF. This is equivalent to discarding that part of the
training data. We say that some potential improvements in accuracy could be to
use another model, e.g. ViT [20], hyperparameters grid search for FL, etc.

###  H.7 Comparison in Overheads with _Private FLAME/FLGUARD_.

Intuitively, the systems that use clustering, like _Private FLAME/FLGUARD_
[48, 47], may suffer the efficiency problem without proper optimization. In
our design, MPC reflects on 3 main phases: computation of cosine similarity
matrix, Euclidean distance, and element-wise comparison. The 1 _st_ stage is
the most computationally expensive, and thus we perform optimization for it.
After the optimization, the servers avoid using “heavy” tools, like
homomorphic encryption and Beaver’s multiplication, to calculate cosine so
that communication and computational overheads are naturally reduced. BRIEF
has lighter complexity than _FLGUARD_.

We noticed _FLGUARD_ uses a 2-party semi-honest MPC scheme which is very
similar to the semi-honest situation in our design (Figure 6). But
theoretically, our optimized version is less computationally and
communicationally complex than _FLGUARD_. In BRIEF, cosine similarity is
calculated by doing XOR locally on servers, and the size of the matrix reduces
from the number of clients (c) * the number of model updates (n) to c*c after
calculation. Then this c*c matrix is used to calculate the pairwise-Euclidean
distance (where only multiplication is involved). In _FLGUARD_ , directly
calculating cosine similarity requires multiplication, division, and the
square root of the c*n matrix, which are relatively intensive, expensive
operations in MPC. Since _FLGUARD_ ’s code is not published, a
code/implementation-based comparison could be difficult.

We followed state-of-the-art Byzantine-robust FL [13]’s default setting on the
client number (number = 100). Figure 6 shows overheads (the runtime and
communication costs) on the server side are acceptable; in particular, only
0.4s and 16 MB are required with LeNet in the semi-honest case.

###  H.8 Defending Against Other Attacks

In the experiments, we considered various popular untargeted and targeted
attacks. We say that interested reader may use other attacks to test BRIEF and
the Byzantine-robustness may not be seriously affected. We take Distributed
Backdoor Attack (DBA) [63] for an example. DBA decomposes a global trigger
into several pieces distributed to local clients. But it has to yield
significant changes to some dimensions of updates to maintain the ASR of the
backdoor task. Since the cosine distance between malicious and benign updates
are distinguishable, BRIEF can still work well under DBA.

###  H.9 Limitations

_Using weights as updates._ To provide cost-effective secure computations, the
proposed BRIEF only implements the update method by SignSGD. If we use weights
as updates, the secret shares sent to the servers will be in floating point
format. In this case, we will have to downgrade the design to the unoptimized
BRIEF in Figure 6, which may cause a considerable amount of both communication
costs and runtime. An interesting future work could be to propose a more
lightweight (than the current design) and secure MPC framework for BRIEF.

_Performance under EA._ Although BRIEF does achieve good performance in terms
of accuracy and (to some extent) efficiency, under the EA, BRIEF’s ASR cannot
be eventually reduced to nearly 0%. In future work, we will improve the TNR
and TPR of the clustering algorithm so as to recognize subtle differences
between malicious and semi-honest clients.


# Probabilistic reconstruction of Dark Matter fields from biased tracers using
diffusion models

Core Francisco Park
Harvard University
17 Oxford St., Cambridge, MA 02138, USA
<EMAIL_ADDRESS>
&Victoria Ono
Harvard University
17 Oxford St., Cambridge, MA 02138, USA
<EMAIL_ADDRESS>
Nayantara Mudur
Harvard University
17 Oxford St., Cambridge, MA 02138, USA
<EMAIL_ADDRESS>
&Yueying Ni
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Cambridge, MA 02138, USA
<EMAIL_ADDRESS>
Carolina Cuesta-Lazaro
The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
<EMAIL_ADDRESS>

###### Abstract

Galaxies are biased tracers of the underlying cosmic web, which is dominated
by dark matter components that cannot be directly observed. The relationship
between dark matter density fields and galaxy distributions can be sensitive
to assumptions in cosmology and astrophysical processes embedded in the galaxy
formation models, that remain uncertain in many aspects. Based on state-of-
the-art galaxy formation simulation suites with varied cosmological parameters
and sub-grid astrophysics, we develop a diffusion generative model to predict
the unbiased posterior distribution of the underlying dark matter fields from
the given stellar mass fields, while being able to marginalize over the
uncertainties in cosmology and galaxy formation.

##  1 Introduction

The nature of dark matter (DM) remains one of the most enigmatic questions in
cosmology. Direct observation of dark matter remains elusive. N-body
simulations have in the past [1] been used to unveil the dark matter density
field from tracers such as galaxies. Over the last decade, significant
progress has been made in cosmological hydrodynamic simulations of galaxy
formation [e.g. 2, 3, 4]. These simulations study how galaxies evolve over
cosmic time and allow more accurate exploration of the relationship between
observed galaxy distributions and underlying dark matter fields. For example,
previous work of Hong et al. [5] developed a convolutional neural network
model based on the IllustrisTNG simulation, using the stellar density and
velocities to regress the otherwise unobservable dark matter density maps.
However, machine learning models trained solely on a single simulation can not
accommodate the substantial uncertainties inherent in the astrophysical
processes assumed by galaxy formation models, as well as the specific
cosmological parameters employed in that simulation. Moreover, it is important
to develop probabilitic models, such as [6], that can describe our
uncertainties in the way that galaxies connect to the underlying dark matter
distribution to answer questions such as what is the likelihood of a certain
dark matter halo mass associated with a given galaxy.

In this work, we develop a diffusion generative model trained based on CAMELS
simulation suites [7, 8] that contain more than 1000 state-of-the-art galaxy
formation simulations with varied cosmological parameters and sub-grid
astrophysics, to reconstruct the underlying DM fields from stellar fields. The
primary goal of the model is to capture the relationship between the stellar
fields and DM fields by marginalizing over the uncertainties in the modeling
of galaxy formation, and predict the unbiased posterior distribution of the DM
fields conditioned on the given stellar field,
$p(x_{\mathrm{DM}}|x_{\mathrm{stars}})$.

Figure 1: Top row: Input stellar field, corresponding true DM, and a sample DM
from the diffusion model. Bottom row: Posterior mean, posterior standard
deviation, and pixelwise $z$-score from 100 posterior samples.

##  2 Methodology

#### Model

We use the variational diffusion model developed by Kingma et al. [9] with a
denoising architecture similar to the U-Net [10] to model the posterior of DM
density fields given stellar fields. The conditional diffusion model generates
a target DM density field in $T=250$ refinement steps. It begins with a random
noise field $x_{\mathrm{DM}}^{T}\sim\mathcal{N}(0,1)$, and iteratively
denoises it according to a learned conditional probability
$p_{\theta}(x_{\mathrm{DM}}^{t-1}|x_{\mathrm{DM}}^{t},x_{\mathrm{stars}})$ to
ultimately generate a sample $x_{\mathrm{DM}}^{0}\sim
p(x_{\mathrm{DM}}|x_{\mathrm{stars}})$. The denoising model therefore takes as
input $\\{x_{\mathrm{DM}}^{t},x_{\mathrm{stars}},t\\}$ and estimates the noise
that was added to the image at that timestep. In the forward diffusion
process, we progressively add noise to an image by sampling from
$q(x_{\mathrm{DM}}^{t}|x_{\mathrm{DM}}^{0})=\mathcal{N}(\alpha_{t}x_{\mathrm{DM}}^{0},\sigma_{t}^{2}\mathbf{I})$,
where $\alpha_{t}$ and $\sigma^{2}_{t}$ are functions of $\gamma_{t}$, which
is assumed to be a linear function of time and whose free parameters are
learned during training. Noise is added to the sample in a variance-preserving
way, i.e. $\alpha_{t}^{2}=\text{sigmoid}(-\gamma(t))$. The loss function we
optimize is the variational lower bound of the marginal likelihood.

#### Dataset

For training, we use the Latin Hypercube set of the IllustrisTNG suite from
the 2D CAMELS Multifield Dataset [11] at $z=0$. This contains 1000 simulations
sampling a wide variation range in cosmological and astrophysical parameters,
reflecting the uncertainties of cosmology and the complex astrophysical
processes taking place in our current understanding of galaxy formation. This
training set includes 15 samples for each of the 1000 simulations, where each
simulation is uniquely varied by the parameter values of
$\Omega_{\text{m}},\sigma_{8}$ (cosmological),
$A_{\text{SN1}},A_{\text{AGN1}},A_{\text{SN2}}$, and $A_{\text{AGN2}}$
(astrophysical). The range of parameters is: $0.1\leq\Omega_{m}\leq
0.5,0.6\leq\sigma_{8}\leq 1.0,0.25\leq(A_{\rm SN1},A_{\rm AGN1})\leq 4.00$,
and $0.5\leq(A_{\rm SN2},A_{\rm AGN2})\leq 2.0$. The samples are 2D
projections of $5\,h^{-1}\text{Mpc}$ thickness from the 3D density fields of
stars and dark matter. We evaluate our model on the one-parameter (1P) set,
which contains 11 variations of a single parameter at a time. We take the log
of each field and standardize the dataset to have zero mean and unit variance.
We augment the dataset by adding random translations (with periodic boundary
conditions), flips and permutations of the input and output images. We keep
the image size to the original $256\times 256$ pixels.

#### Training

We use a U-Net [10] like architecture with 4 blocks of double convolution
followed by strided downsampling layers. We employ group normalization [12]
and residual connections [13] in each block, and use the AdamW optimizer [14]
with a learning rate of $1\times 10^{-4}$. We also initialize the learned
linear noise schedule with $\gamma(t)=18.3\>t-13.3$. We train the model using
the PyTorch Lightning framework [15] with a batch size of 12 for 60000
gradient steps.

##  3 Results

We first compare the DM fields sampled from the diffusion model for the CAMELS
simulations at the fiducial parameters. In the top row of Fig. 1, we show an
example stellar map together with the ground truth underlying DM map, and one
of the samples from the diffusion model. The second row shows the diffusion
model’s posterior mean and standard deviation from 100 samples, conditioned on
the same stellar field. The posterior standard deviation and the pixelwise
$z$-score, computed as $\frac{\text{Truth - Posterior Mean}}{\text{Posterior
Std}}$, allow us to assess whether the diffusion model’s uncertainty is
sufficient to account for deviations from the target field. The posterior
standard deviation is lowest at the positions of structures that were present
in the stellar field, while the $z$-score is higher for filaments that are
present in the target DM field but do not correspond to structures in the
stellar field. Note that the $z$-score is not symmetric since the overdensity
field is lower bounded by $-1$.

Fig. 2 shows a quantitative comparison between the summary statistics of the
true DM fields and those of the generated ones. We find good statistical
agreement between the true DM field and the sampled DM fields, both in density
histogram (the left panel) and in the clustering of the DM fields (the middle
panel).

The right panel compares the statistics of the cross-correlations.

The cross-correlations between the sampled and true DM fields are always
higher than $0.8$; the large-scale cross-correlation is as expected larger
than that on small scales, but it is overall very well reproduced by the
diffusion model samples.

Figure 2:  Summary statistics of the stellar and DM density field at the
fiducial parameters. Left panel: density histogram of stars (copper), true DM
(solid purple lines) and DM inferred by the diffusion model (light blue). Note
that we show the 10-90th percentiles of $100$ samples from the posterior
distribution. Middle panel: Power spectra for star, true DM and sampled DM
fields. Right panel: Cross correlations between i) stellar and DM density
fields, for both true and inferred DM fields (the brown line and shaded
region), and ii) between true and sampled DM fields (the blue shaded region).
Figure 3:  1st and 2nd panels: Illustrations of the DM maps generated from
stellar fields from two simulations of different SN wind strength (low
$A_{\text{SN1}}$ for the first panel and high $A_{\text{SN1}}$ for the second
panel). The red panels show the respective true DM maps for comparison. 3rd
panel: Histogram of the stellar density (top) and generated DM density
(bottom) for the simulations run with SN strengths of low, fiducial and high
values. The shaded regions show the 10-90th percentiles based on 100 generated
samples for each of the input stellar maps.

To demonstrate the model’s capability to marginalize over the range of
cosmological and astrophysical parameters we trained on, we test the model on
the 1P set of CAMELS which varies one parameter (from the fiducial value) at a
time for each simulation. Fig. 3 shows samples from the diffusion model when
conditioned on stellar fields from simulations with varying strengths of
supernovae (SN) feedback $A_{\rm SN1}$.

In these simulations, higher $A_{\rm SN1}$ (stronger SN feedback) leads to
significant regularization of star formation and results in a lower stellar
population compared to the scenario with lower $A_{\rm SN1}$ (weaker SN
feedback), as evident from the stellar image and density histogram. However,
within both scenarios of strong and weak SN feedback, the diffusion model is
able to predict the underlying DM field from the given stellar field that well
resembles the ground truth DM distribution both visually and on a statistical
basis. This exhibits the capability of the diffusion model to predict the DM
distribution from biased tracers of galaxies produced from different
astrophysical scenarios.

Figure 4:  Summary statistics of the density fields when varying $\Omega_{m}$,
$A_{\mathrm{SN1}}$ and $A_{\mathrm{AGN1}}$. The first row shows a comparison
of the density field PDFs. The second row shows the power spectrum ratio
between the sampled fields and the true DM distribution. The solid lines in
the top panel give the true DM distributions from simulations, while the
shaded regions correspond to 10-90th percentiles based on 100 generated DM
samples from the input stellar fields. Note that the dark matter projected
density distribution does not vary noticeably with the astrophysical
parameters $A_{\mathrm{SN1}}$ and $A_{\mathrm{AGN1}}$, and the two solid lines
overlap.

Fig. 4 shows a quantitative comparison between the generated DM density fields
and the true ones based on the 1P set simulations that separately vary the
parameters $\Omega_{m}$, $A_{\rm SN1}$ and $A_{\rm AGN1}$. The DM distribution
is clearly sensitive to the cosmological parameter $\Omega_{m}$. We can see
that the model is able to predict systematically lower (higher) DM
distribution for the low (large) $\Omega_{m}$ scenario, exhibiting the
potential to generalize the DM predictions across varying cosmological
assumptions. However, the model struggles to generalize for low $\Omega_{m}$
values, predicting the DM power with amplitude larger than expected, albeit
showing larger uncertainties in the samples. Similarly, the model also
overpredicts the DM power for the low $A_{\rm SN1}$ scenarios. Improving the
capacity of the model to generalize over the entire parameter space will be
the subject of future work.

By comparing the dark matter power spectrum, we find that the predicted and
observed DM fields exhibit a broad level of agreement, with a deviation of
approximately $20\%$, although the uncertainty in the diffusion model is large
enough for the true DM field to not be anomalous. The consistency of the power
spectrum with varied cosmological and astrophysical assumptions demonstrates
that the trained model can well capture the clustering properties of the
underlying DM fields based on the stellar fields, while marginalizing over the
cosmological and astrophysical parameters.

##  4 Conclusions and Outlook

We have presented a diffusion generative model that can sample the posterior
distribution of dark matter density fields conditioned on observable stellar
maps. Previous deep learning approaches used regression-based frameworks, but
we show that a generative model can be used to turn this mapping into a
probabilistic one and capture the inherent uncertainty of the model due to
lacking information in the stellar maps. The diffusion model can generate DM
density fields with summary statistics and visual inspections consistent to
those in the authentic simulations. This consistency persists even when
stellar maps are changed notably by variations in parameters controlling the
SN and AGN feedback, illustrating the model’s capacity to effectively
marginalize over astrophysical uncertainties. Our future work will extend the
current framework to 3D density fields for an application to galaxy surveys
and explore how additional information, such as galaxy velocities,
luminosities and colors, can be folded in to better constrain the DM density
field.

##  5 Code Availability

Our code is publicly available at https://github.com/cfpark00/vdm4cdm

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##  6 Supplementary Material

###  6.1 Statistical analysis of the sampled fields

We illustrate some more statistical properties of the sampled DM fields in
Figure 5. We show the variance of the power spectrum due to cosmic variance
and due to sampling the conditional posterior distribution described by the
diffusion model. The latter is only bigger at the smallest scales. The results
are within the expected level of variance from the ground truth DM fields. We
show that the GT DM power spectrum is statistically consistent with the mean
of the distribution of the sampled DM field’s power spectra, and the absolute
mean of the Z-score is also order of unity. Finally, we find that the pixels
with non-zero stellar mass form a distinct branch (shown in red) in the joint
distribution of the posterior mean and variance values. Thus, the variance is
the biggest in the pixel where there is non-zero but small underlying stellar
mass.

Figure 5: Statistics on 500 samples generated from an input stellar mass map
from the CV (Cosmic Variance) set. This set of simulations have the same
fiducial parameters but a different initial condition. Top left: Cosmic
Variance of GT Dark Matter fields and variance from Cosmic Variance and
diffusion seed of the generated samples. Top right: Hotelling’s T-test with a
diagonal covariance assumption shows that, for a single sample, the GT Dark
Matter’s power spectrum is consistent with the mean power spectrum of the
samples. Bottom left: The Z-score distribution of a single GT Dark Matter
field with respect to the per-pixel posterior mean and variance of generated
samples. Bottom right: Joint distribution of the posterior mean and variance
of the DM field for a single conditioning stellar mass field. Pixels
corresponding to a non-zero stellar mass fields are in red circles.

###  6.2 Illustrations of the diffusion process

We show the diffusion process generating DM fields conditioned on stellar mass
fields in Figure 6. Visually, we find that most of the structure forms between
steps 80 and 160 when discretizing $t=[0,1]$ into 250 steps and a learned
noise schedule of $\gamma(t)=18.347\>t-7.202$. We hope to investigate if there
could be a better noise scheduling for our task which can improve the results.

Figure 6: Input conditioning and sampled images at different steps of the
diffusion process. Visually, most of the structure are produced between steps
80 and 160.

In Figure 7, we show for 4 input stellar mass fields the evolution of the
power spectrum in the diffusion process. The diffusion process seems to
produce the large scale features first before completing the small scales.

Figure 7: The evolution of the powerspectrum during the diffusion process. We
show the powerspectrum of the DM field in different steps of the diffusion
process for 4 randomly chosen stellar mass inputs.

###  6.3 Compute Time

We used a single NVIDIA A100 40 GB GPU for training and evaluation. With a
batch size of 12, each training step took $\sim$ 0.2 s and sampling a batch of
images with 250 steps took 8.4 s.