# Observed Dust Surface Density Across Cosmic Times

Céline Péroux1,2, Annalisa De Cia3 and J. Christopher Howk4
1 European Southern Observatory, Karl-Schwarzschildstrasse 2, D-85748 Garching
bei München, Germany
2 Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de
Marseille) UMR 7326, 13388, Marseille, France
3 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290
Versoix, Switzerland
4 Department of Physics and Astronomy, University of Notre Dame, Notre Dame,
Indiana 46556, USA
E-mail<EMAIL_ADDRESS>
(Accepted 2023 April 20. Received 2023 April 20; in original form 2022
December 20)

###### Abstract

Our ability to interpret observations of galaxies and trace their stellar,
gas, and dust content over cosmic time critically relies on our understanding
of how the dust abundance and properties vary with environment. Here, we
compute the dust surface density across cosmic times to put novel constraints
on simulations of the build-up of dust. We provide observational estimates of
the dust surface density consistently measured through depletion methods
across a wide range of environments, going from the Milky Way up to z=5.5
galaxies. These conservative measurements provide complementary estimates to
extinction-based observations. In addition, we introduce the dust surface
density distribution function – in analogy with the cold gas column density
distribution functions. We fit a power law of the form: $\log f(\Sigma_{\rm
Dust})=-1.92\times\log\Sigma_{\rm Dust}-3.65$ which proves slightly steeper
than for neutral gas and metal absorbers. This observed relation, which can be
computed by simulations predicting resolved dust mass functions through 2D
projection, provides new constraints on modern dust models.

###### keywords:

galaxies: abundances – galaxies: evolution – galaxies: high-redshift –
Galaxies Magellanic Clouds – quasars: absorption lines – Interstellar Medium
(ISM), Nebulae – ISM: dust, extinction

††pubyear: 2020

##  1 Introduction

Dust grains absorb stellar light in the ultraviolet (UV)-optical wavelengths
and re-emit it in the far-infrared, which represents 30%–50% of the radiative
output of a galaxy (Roman-Duval et al., 2017). Therefore, our ability to
interpret observations of galaxies and trace their stellar, gas, and dust
content with redshift across the entire spectral range critically relies on
measurements of the dust abundance and properties vary with environment and
cosmic times. This in turn requires us to understand the processes responsible
for dust formation, destruction, and transport, as well as their associated
timescales.

A fraction of metals in the interstellar medium of galaxies in both the local
and high-redshift Universe resides in microscopic solid particles or dust
grains (Field, 1974; Savage & Sembach, 1996; Jenkins, 2009a; De Cia et al.,
2016). Interstellar dust has manifold impact on the physics and chemistry of
the interstellar medium (Zhukovska et al., 2016, 2018; Galliano, 2022) as well
as the intra-cluster medium (Shchekinov et al., 2022). Because dust locks some
elements away from the gas phase, it affects our measurements of the
metallicity of galaxies. One of the most important roles of interstellar
grains is that they facilitate the formation of molecular hydrogen, ${\rm
H}_{2}$, on their surfaces (Hollenbach & Salpeter, 1971). The ${\rm H}_{2}$
molecule is the main component of molecular clouds, which are the cradle of
star formation in most of the Universe (Klessen & Glover, 2016). Because dust
absorbs UV emission from young massive stars and re-emits it in the infrared,
the spectral energy distribution from dust is one of the primary indicators of
star formation (Calzetti et al., 2000).

Yet, in both local and high-redshift galaxies, the dust production rates in
evolved stars (Bladh & Höfner, 2012; Riebel et al., 2012) and supernova
remnants (Matsuura et al., 2011; Leśniewska & Michałowski, 2019; Slavin et
al., 2020) are largely insufficient compared to the dust destruction rates in
interstellar shocks (Jones et al., 1996) to explain the dust masses of
galaxies over cosmic times (Morgan & Edmunds, 2003; Boyer et al., 2012;
Rowlands et al., 2014; Zhukovska & Henning, 2013). This so-called dust budget
crisis poses an important challenge to our modelling of dust into a
cosmological context (Mattsson, 2021).

Observationally, the amount of dust in astrophysical objects has been
quantified with a number of different methods probing various dust signature.
Extinction refers to the amount of light dimming due to all the material lying
along the line of sight between the astrophysical object and the observer.
Extinction is therefore an integrated quantity which encapsulates absorption
and scattering away from the line-of-sight. The observational determination of
extinction requires backlights such as stars, Gamma-Ray Bursts, quasars, or
other objects with much smaller angular extent than a galaxy. The extinction
at a given wavelength results from a combination of the grain size
distribution (Mattsson, 2020), metallicity (Shivaei et al., 2020a) and the
optical properties of the grains (which is itself dependent on the chemical
composition of the grains). Therefore, the extinction scales with dust column
density or surface density. Similarly, infra-red emission has been used to
probe the dust content of galaxies (Chiang et al., 2021). In particular,
significant work has been put into characterizing these quantities in galaxies
beyond the Milky Way, assessing observational constraints both on the
integrated values (e.g., Rémy-Ruyer et al., 2014; De Vis et al., 2019) and
spatially-resolved values within galaxies (e.g., Vílchez et al., 2019). These
works have placed particular emphasis on the variation of the dust properties
with metallicity, stellar mass, star formation rate, and gas content of the
galactic environments, as these help shape our understanding of the factors
that drive the formation/destruction balance of dust. Reddening, expressed
through the colour excess, E(B–V), quantifies the differential extinction.

Attenuation represents the effect of dust on the light continuum from the
geometric mix of stars and dust in galaxies. It thus reflects the net effect
on light due to a combination of multiple effects, including extinction,
scattering back into the line-of-sight as well as contribution from unobscured
stars (Salim & Narayanan, 2020). Reddening is often expressed in terms of UV
continuum slope, $\beta$ (Shivaei et al., 2020b). The UV continuum slope
depends on the column density of dust along the line-of-sight to the observer
that is dimming the UV light of background objects. A fully consistent
comparison between depletion-estimated extinction and colour-based estimates
is found in Konstantopoulou et al. (2023).

The total far-infrared emission is a proxy for the total dust mass. At z$>$5,
that emission is successfully probed at mm wavelengths with facilities such as
ALMA. The dust temperature however is less well-constrained in these early
times (see Figure 1 of Bouwens et al., 2020), leading to a degeneracy in the
current estimates (Faisst et al., 2020; Sommovigo et al., 2020; Bakx et al.,
2021; Sommovigo et al., 2022; Chen et al., 2022a; Fudamoto et al., 2022; Viero
et al., 2022; Drew & Casey, 2022; Ferrara et al., 2022). Alternatively, the
dust mass is being derived from the mm-continuum of high-redshift galaxies and
the gas mass is then estimated assuming a dust-to-gas ratio – often taken to
be the value of the Milky Way (Scoville et al., 2017; Dunne et al., 2022, but
see Popping & Péroux (2022)). The ratio of dust emission in infrared to the
observed UV emission, known as the infrared excess (IRX), is a measure of the
UV dust attenuation. The attenuation/extinction curve/law is characterised by
its slope and normalisation (e.g. Calzetti et al., 2000). Two galaxies having
the same attenuation curves (i.e. same shape) might still differ in their
normalisation (i.e. column density of dust). The two quantities, $\beta$ and
IRX, are often related into one diagram (Meurer et al., 1999; Faisst et al.,
2017; Shivaei et al., 2020b). The relation is sensitive to a range of
interstellar properties including dust geometries, dust-to-gas ratios, dust
grain properties, and the spatial distribution of dust. The relation provides
a powerful empirical constraint on dust physical properties because it laid
the foundation for a straightforward correction of UV emission in galaxies
where infrared observations are lacking (especially at higher redshifts),
based on the easily observable UV slope (or colour).

Dust is made of some of the available metals produce by stars. In the
interstellar medium, a fraction of these elements is in the gas, and the rest
is locked up in dust. Indeed, most metals are under-abundant in the
interstellar gas of the Milky Way (Jenkins, 2009a), reflecting the amount of
dust in our galaxy. Dust depletion can be used to give hints on the dust
composition (Savage & Sembach, 1996; Jenkins, 2014; Dwek, 2016; Mattsson et
al., 2019; Roman-Duval et al., 2022a) and this often indicates that
significant amounts of Fe-rich dust should be present in the interstellar
medium. Dust measurements based on depletion estimated at UV and optical
wavelengths might suffer from a bias where the dustier systems would be
obscured at these wavelengths. In that sense, dust depletion provides
conservative measurements which can be seen as a lower limit on the total
amount of dust in a population. Depletion measurements provide a direct
estimate of the dust content (Dwek, 1998; Draine & Li, 2007; Galliano et al.,
2018). The depletion can be used to calculate the extinction through the gas
which is proportional to the column density of metals (see equation 1 of
Savaglio et al., 2003). In their Figure 9, Wiseman et al. (2017) offer a
comparison of the two measurements between depletion-estimated extinction and
colour-based estimates in a sample of Gamma-Ray Bursts, indicating significant
discrepancies highlighting the possible limitations described above (see also
De Cia et al., 2013; Zafar et al., 2014).

These multiple observational results have triggered a number of simulation
efforts. These works come into two main families: i) semi-analytical models
which use empirical relation to approximate some of the physical processes at
play; and ii) hydrodynamical cosmological simulations which include full
treatments of dust production and destruction. Contemporary models have
reached a new level of realism by including a large number of physical
processes (Draine, 2003). Dust shattering refers to the breaking of large
grains into small dust grains, due to high-velocity collisions. Conversely,
coagulation describes the processes of large grains being made out of smaller
entities because of low-velocity collisions. Therefore, both these processes
do not change the mass but the size distribution of dust grains. Sputtering
refers to dust destruction by shocks, including those produced by Supernovae
blasts. Astration reflects the process of dust being absorbed by stars. We
note that the same processes can both produce and destroy small/big grains, so
that the production sources and destruction sinks are complex processes to
simulate.

A number of efforts based on semi-analytical models have made predictions on
the dust mass of galaxies (Bekki, 2015a; Pantoni et al., 2019; Lapi et al.,
2020; Gjergo et al., 2020; Dayal et al., 2022) and associated dust mass
function (Popping et al., 2017; Triani et al., 2020; Vijayan et al., 2019). As
subset of these studies has provided information on the dust surface density
specifically (Bekki, 2013, 2015b; Osman et al., 2020; Gjergo et al., 2020). In
parallel, there have been a number of works introducing dust physical
processes within hydrodynamical cosmological models (Moseley et al., 2023).
Many report estimates of the dust mass density (Gioannini et al., 2017; Aoyama
et al., 2018; Lewis et al., 2023), while others report the dust mass function
(McKinnon et al., 2017; Graziani et al., 2020; Li et al., 2019; Hou et al.,
2019; Baes et al., 2020). A number of these efforts have made predictions on
dust surface densities in particular, as the ones presented here (McKinnon et
al., 2016; Trayford & Schaye, 2019).

The goal of this work is two-fold. On one hand, we provide observational
estimates of the dust surface density consistently measured through depletion
methods across a wide range of environments, going from the Milky Way up to
z=5.5 galaxies. While previous works have estimated the dust mass in local
galaxies (De Vis et al., 2019; Millard et al., 2020; De Looze et al., 2020;
Morselli et al., 2020; Casasola et al., 2020; Nanni et al., 2020; Galliano et
al., 2021), this study focuses on the dust column density. On the other hand,
we introduce the dust surface density distribution function – in analogy with
the gas ($N(\mbox{H$\;${{i}}})$ or $N({\rm H}_{2})$) column density
distribution functions (Péroux et al., 2003; Zwaan et al., 2005; Zwaan &
Prochaska, 2006; Klitsch et al., 2019; Péroux & Howk, 2020; Szakacs et al.,
2022). Spatially resolved simulations predicting the dust-mass function (Pozzi
et al., 2020; Millard et al., 2020) can predict the dust surface density
distribution function through 2D projection. Thus, the observed dust surface
density distribution function potentially offers new constraints on modern
dust models.

The manuscript is organized as follows: Section 2 presents the methods used in
this study. Section 3 details how dust surface density relates to the global
physical properties of galaxies. We summarize and conclude in Section 4. Here,
we adopt an H0 = 67.74 km s-1 Mpc-1, $\Omega_{M}$ = 0.3089, and
$\Omega_{\Lambda}$ = 0.6911 cosmology. We use the latest solar abundance
values from Asplund et al. (2021).

##  2 Differential Dust Depletion

###  2.1 Dust Depletion in our Galaxy

####  2.1.1 Methodology

Figure 1: Dust surface density as a function of cosmic times. In this figure
and the following, the dust column densities are derived from depletion
measurements performed in absorption at UV wavelengths. The shade of colours
from light to dark refers to the Milky Way, the Large and Small Magellanic
Clouds, Gamma-Ray Bursts and quasar absorbers at z$>$0\. We note that Gamma-
Ray Bursts galaxy hosts preferentially lie above the quasar absorbers. There
is also a clear trend of increasing dust surface density with cosmic time with
a large scatter at any given redshift, as expected from the building of dust
with time.

The depletion of metals is differential, with some elements showing a higher
affinity for incorporation into solid-phase grains than others, based on their
chemical properties (Pettini et al., 1997; Vladilo, 2002; Jenkins, 2009a). The
differential nature of elemental depletion has traditionally been used to
correct the observed abundance for unseen metals. Early works used lightly-
depleted elements to derive abundances (e.g., focusing on Zn or S or to a
lesser degree Si). More recent works have taken advantage of the patterns of
differential depletion to estimate the extent of the dust-depletion
correction. These efforts follow the spirit of Vladilo (1998) and Jenkins
(2009a), using the observed abundances to study the dust depletion beyond the
Milky Way (Jenkins & Wallerstein, 2017; Roman-Duval et al., 2022a).

Specifically, De Cia et al. (2016) developed a method to characterize dust
depletion, $\delta_{X}$, without assumption on the total gas+dust metallicity.
This is achieved through the study of relative abundances of several metals
with different nucleosynthetic and refractory properties, as follows. The
relative gas-phase abundance of the metals X and Y is written: $[X/Y]=\rm
log(N(X)/N(Y))-log(N(X)_{\odot}/N(Y)_{\odot})$. This approach enables to
calculate the depletion without assumptions on the total metallicity of the
gas, including metals locked onto dust grains (see also De Cia et al., 2021;
De Cia, 2018a). We derive the depletions of different elements as follows:

$\delta_{X}=A2_{X}+(B2_{X}\times[Zn/Fe]_{\rm fit}),$ (1)

where [Zn/Fe]fit traces the overall amount of dust depletion and is taken from
De Cia et al. (2021). This quantity is equivalent to the observed [Zn/Fe],
although it is based on the observations of all available metals. The
coefficients A2X and B2X are taken from Konstantopoulou et al. (2022).

The total dust-corrected metallicity is then computed as:

$[X/H]_{\rm total}=[X/H]_{\rm observed}-\delta_{X}$ (2)

where [X/H]total is the total metallicity including metals locked into dust
grains, [X/H]observed is the observed abundance of X in the gas phase, and
$\delta_{X}$, i.e. the logarithm of the fraction of X in the gas phase. Given
that each element has a different propensity to deplete onto dust grains, the
observations of multiple element ratios provide a measure of the interstellar
depletions for various elements X. Given estimates of $\delta_{X}$ and
observations of [X/H]observed, one can derive the quantity: [X/H]total.

####  2.1.2 Observational Results in the Milky Way

In the Milky Way, depletions have been studied for many heavy elements in
several hundreds of sightlines through the diffuse neutral medium (Field,
1974; Phillips et al., 1982; Jenkins, 2009b; De Cia et al., 2021). Here, we
make use of results from two works: while Jenkins (2009a) assumes solar
metallicity (and solar abundance pattern) for the Milky Way, De Cia et al.
(2021) report both depletion and metallicity measurements along different
line-of-sight. For consistency with other measurements and to avoid
complications related to ionisation corrections, we focus here on log N(H)
$\geq$ 20.3 measurements from Jenkins (2009a). We also note that the
methodology of Jenkins (2009a) follows the one presented for the Magellanic
Clouds next as a fixed global metallicity is assumed. These studies indicate
that in total about 50% of the metals in the Milky Way’s interstellar medium
are incorporated into grains (Draine, 2003). The dust-to-gas (${\rm DTG}$) and
dust-to-metal (${\rm DTM}$) mass ratios are calculated from these values of
the dust depletions (see Section 3).

###  2.2 Dust Depletion in the Magellanic Clouds

####  2.2.1 Methodology

The method that we use to estimate the amount of dust in the Magellanic Clouds
is slightly different from the one used to estimate the same quantity in the
Milky Way. The depletion for element X is expressed as follows:

$\delta_{X}=[X/H]_{\rm observed}-[X/H]_{\rm assumed\\_total}$ (3)

where $[X/H]_{\rm assumed\\_total}=\log(X/H)_{\rm
assumed\\_total}-\log(X/H)_{\odot}$ is the total abundance of element X (gas +
dust) which here is assumed to be equal to the abundance of element X in the
photospheres of young stars that have formed out of the interstellar medium
(as done for the Milky Way by Savage & Sembach, 1996; Jenkins, 2009a;
Tchernyshyov et al., 2015; Jenkins & Wallerstein, 2017; Roman-Duval et al.,
2019). These works compare the chemical abundances in neutral interstellar gas
based on UV spectroscopy to stellar abundances of OB stars and HII regions
(e.g. Luck et al., 1998; Hunter et al., 2007; Trundle et al., 2007; Toribio
San Cipriano et al., 2017) to estimate $\delta_{X}$ from equation 3. In
principle, there could be variations of the metallicity of the neutral
interstellar medium in the Magellanic Clouds. We stress that the approach
described in this section is therefore different from the ones reported in
other sections for the Milky Way and for high-redshift galaxies.

####  2.2.2 Observational Results in the Large Magellanic Cloud

The Large Magellanic Cloud lies just 50 kpc from us (Subramanian &
Subramaniam, 2009). Its dust content is probed through its extinction map
(Furuta et al., 2022), while dust reddening (Chen et al., 2022b) and
extinction (Gordon et al., 2003) have been measured in both the Large and
Small Magellanic Clouds as well as dust emission (Chastenet et al., 2017).
Recently, the Large Magellanic Cloud has been the focus of a HST Large Program
dubbed ”The Metal Evolution, Transport, and Abundance in the Large Magellanic
Cloud” (METAL) and introduced in Roman-Duval et al. (2019). Roman-Duval et al.
(2021) demonstrates that the depletion of different elements in these data are
tightly correlated with the gas (hydrogen) surface density. Roman-Duval et al.
(2022a) further make a new appraisal of the dust estimates in the Milky Way,
Large and Small Magellanic Clouds. The Galaxy is more strongly affected by
dust depletion than the Large Magellanic Cloud, and even more than the Small
Magellanic Cloud. Nevertheless, the way different elements deplete into dust
is very similar between these various environments (De Cia, 2018a;
Konstantopoulou et al., 2022). Here, we use the data shown in Figure 7 of
Roman-Duval et al. (2022a), which also include results from Tchernyshyov et
al. (2015). We use a constant metallicity throughout the cloud, taken to be
[X/H]=$-$0.30 (Roman-Duval et al., 2022a).

####  2.2.3 Observational Results in the Small Magellanic Cloud

Similarly, depletion of gas-phase metal abundances are clearly seen in the
Small Magellanic Cloud, situated at about 60 kpc (Subramanian & Subramaniam,
2009), though with a smaller degree of depletion reflecting the lower dust-to-
metal mass ratio in this system. Initial works from Tchernyshyov et al.
(2015); Jenkins & Wallerstein (2017) provide the first estimates of dust
depletion in multiple lines-of-sight. Here, we use the observations displayed
in Figure 6 of Roman-Duval et al. (2022a).These works assume a constant
metallicity throughout the Small Magellanic Cloud, taken to be [X/H]=$-$0.70
(Roman-Duval et al., 2022a).

Figure 2: Dust surface density as a function of cold gas surface density. The
grey lines represent constant dust-to-gas ratios in steps of 1 dex. The cold
gas refers to the sum of neutral atomic, H i, and molecular, $N({\rm H}_{2})$,
except for quasar absorbers where the molecular gas is found to be negligible
(Ledoux et al., 2003). Overall, the dust column densities follow the total
hydrogen column densities with a 4-order of magnitude scatter.

###  2.3 Dust in High-Redshift Galaxies

####  2.3.1 Methodology

Given the presence of dust in a broad range of galaxies, depletion is
naturally expected to be also detected in the material traced by high-redshift
absorption lines seen against the background light of bright sources. In
extragalactic systems, there have been numerous studies of diffuse neutral
medium depletions, facilitated by the redshifting of the rest-frame UV
absorption lines in the visible range (Ledoux et al., 2002; Vladilo, 2002; De
Cia et al., 2016; Bolmer et al., 2019a). Similar to the Milky Way, we derive
the dust-to-gas ratios for individual absorption systems from the elemental
depletions. Because the behavior of each metal species varies (Jenkins,
2009a), the estimates of the depletion, $\delta_{X}$, for each individual
element, $X$, are derived from the dust sequences following the approach
highlighted by De Cia et al. (2016, 2018); Péroux & Howk (2020). The
methodology used here is therefore identical to the one described in Section
2.1.1.

####  2.3.2 Observational Results in Gamma-Ray Burst Hosts

Gamma-Ray Burst events in particular probe the insterstellar medium of their
galaxy hosts. These systems have been used to probe the dust and metals in
Gamma-Ray Burst host galaxies (Savaglio & Fall, 2004; Schady et al., 2007; De
Cia et al., 2012; Zafar & Møller, 2019). Here, we report observations from
Bolmer et al. (2019a) which offer a set of dust-depletion estimates based on
the correction from De Cia et al. (2016), hydrogen column density and
metallicity for a sample of Gamma-Ray Bursts.

####  2.3.3 Observational Results in Quasar Absorbers

Quasar absorbers probe the gas inside and around hundreds of foreground
galaxies unrelated to the background quasars (Pettini et al., 1994; Dessauges-
Zavadsky et al., 2004; Rafelski et al., 2012). There is strong evidence for
the differential depletion of metals in the quasar absorbers observed both for
the neutral gas (De Cia et al., 2016, 2018; Péroux & Howk, 2020) as well as
for partially-ionized gas (Quiret et al., 2016; Fumagalli et al., 2016). The
depletion in quasar absorbers is typically smaller than in the Milky Way (De
Cia et al., 2016; Roman-Duval et al., 2022b; Konstantopoulou et al., 2022;
Konstantopoulou et al., 2023). In addition, the abundance ratios change in a
similar way from local to high-redshift galaxies (De Cia, 2018b;
Konstantopoulou et al., 2022), indicating that the depletion of dust evolves
homogeneously all the way to high-redshift systems. These results also imply
that grain growth in the interstellar medium is an important process of dust
production

Here, we use the values of dust depletion summarised by Péroux & Howk (2020),
which are based on results from De Cia et al. (2018) with some additional
updates (see footnote 5 of Péroux & Howk, 2020). We note that only a small
fraction (of the order 10%) of these systems have detections of molecular
hydrogen, $N({\rm H}_{2})$. In addition, when detected, the fraction of
molecular gas in the cold phase is also found to be small ($\sim$0.01%, see
Petitjean et al., 2000; Noterdaeme et al., 2008; Balashev et al., 2019). For
these reasons, we neglect the molecular hydrogen gas in quasar absorbers and
assume N(H)=$N(\mbox{H$\;${{i}}})$.

Figure 3:  Top Panel: Dust surface density as a function of gas metallicity.
There is a tight correlation between dust surface densities and the dust-
corrected metallicity estimates at all cosmic times. For a given metallicity,
quasar absorbers have lower surface density of dust than GRB host galaxies and
the Milky Way, which are closer to the denser and colder parts of their
galaxies. The Milky Way values from Jenkins (2009) and the Large & Small
Magellanic Clouds (Roman-Duval et al., 2022a) are assumed to have one global
metallicity each. Bottom Panels: Distribution of dust surface density in
quasar absorbers, the Magellanic Clouds, and GRB hosts at fixed metallicity.
The bottom left panel displays observations for the SMC, the bottom middle
panel shows data for the LMC, while the bottom right whos measurements from
GRB hosts. At fixed metallicity, the distribution in dust surface density in
the LMC is wider than for quasar absorbers. Similarly, at a given metallicity,
$\Sigma_{\rm dust}$ is higher in GRB sight lines that are likely closer to the
denser and colder parts of their galaxies than quasar absorbers.

##  3 Observed Dust Surface Density

###  3.1 Methodology

The characterisation of the bulk statistical properties of dust involve
assessing the dust-to-gas (DTG) and dust-to-metal (DTM) mass ratios. The
former is the fraction of the interstellar mass locked into dust grains; the
latter is the fraction of the metal mass incorporated into the solid phase.
The dust-to-gas ratio for an individual element, $X$, is related to its
depletion, $\delta_{X}$, and its dust-to-metal ratio as follows:

${\rm DTG}_{X}=(1-10^{\delta_{X}})\,Z_{\rm total}^{\rm X}={\rm
DTM}_{X}\,Z_{\rm total}^{\rm X}$ (4)

where Z${}_{\rm total}^{\rm X}$ is the intrinsic abundance of X expressed by
mass (e.g., Vladilo, 2004; De Cia et al., 2016). We derive the $\delta_{X}$
from the differential depletion of various elements as described in Section 2,
and use them to calculate the individual ${\rm DTG}_{X}$ (equation 4).

${\rm DTG}=\sum_{x}{\rm DTG}_{x}$ (5)

To obtain a global ${\rm DTM}$ ratio expressed for all the elements and in
mass fraction, we then average the ${\rm DTM}_{X}$ and weight them by
elemental abundances and atomic weight for the Milky Way and the Magellanic
clouds (as also done in Roman-Duval et al., 2021). For the high-redshift
galaxies, we derive the global ${\rm DTM}$ from the ${\rm DTG}$ as follows:

$Z_{\rm total}=\sum_{x}Z_{\rm total}^{x}$ (6) ${\rm DTM}={\rm DTG}/Z_{\rm
total}=\frac{\sum_{x}{\rm DTG}_{x}}{\sum_{x}Z_{\rm total}^{x}}$ (7)

For this calculation, we include the 18 elements which depletion have been
characterized by Konstantopoulou et al. (2022), though C, O, Si, Mg, and Fe
contribute a major fraction of the total dust mass (see also Konstantopoulou
et al., 2023). In the calculation we also include all the volatile metals that
have an element abundance higher than $12+\log(X/H)>$3 (Table 1 of Asplund et
al., 2009, see also Asplund et al. (2021)), most notably the N and Ne, which
do not contribute to the dust budget, but contribute to the metal budget.

We then directly calculate the dust surface density which provide a
observationally-based measurements of the dust quantity. Specifically, we use
several UV-based depletion measurements to calculate the dust mass surface
density, $\Sigma_{\rm Dust}$, in various environments. To this end, we couple
observations of the total dust-to-gas ratio, DTG, described above with
estimates of the column density of hydrogen gas, both in its atomic and
molecular phases. We derive the dust surface densities as:

$\Sigma_{\rm Dust}={\rm DTG}\times\Sigma_{\rm Gas}={\rm DTG}\times N(H)\times
m_{\rm H}\times\mu[g/cm^{2}]$ (8)

where the total hydrogen column density is N(H)=$N(\mbox{H$\;${{i}}})$+2N(H2),
the sum of $N(\mbox{H$\;${{i}}})$ and two atoms of hydrogen, $N({\rm H}_{2})$,
expressed in atoms/cm2. The quantity $\Sigma_{\rm Gas}$ is the gas surface
density, $m_{\rm H}$ is the hydrogen mass $m_{\rm H}$=$1.67\times 10^{-24}$ g,
and $\mu$ is the mean molecular weight of the gas which is taken to be 1.3
(76% hydrogen and 24% helium by mass). The dust surface density, $\Sigma_{\rm
Dust}$, is therefore expressed in g/cm2 or alternatively in M⊙/kpc2. The
resulting values for each of the systems are listed in the table available on-
line, an excerpt of which is presented in the Appendix A.

###  3.2 Dust Surface Density as a function of Gas Properties

For completeness, we start by briefly summarising the relations between the
dust surface densities and galaxy physical properties. This work intentionally
refrains to plot dust-to-gas ratio relation with galaxy’s properties since
those have presented elsewhere (e.g. Popping & Péroux, 2022). Figure 1
displays the dust surface density as a function of cosmic time. In this figure
and the followings, the left y-axis displays the dust surface density in units
of g per cm2 and the right y-axis in units of M⊙/kpc2. We stress that all
these dust surface densities are derived from depletion measurements performed
in absorption at UV wavelengths. The shade of colours from light to dark
refers to the Milky Way, the Large and Small Magellanic Clouds, Gamma-Ray
Bursts and quasar absorbers at z$>$0\. There is a clear trend of increasing
dust surface density with cosmic time with a large scatter at any given
redshift, as expected from the build up of metals and associated dust with
time. Gamma-Ray Burst host galaxies overall have higher dust surface densities
than quasar absorbers at the same redshift. This is consistent with Gamma-Ray
Bursts exploding in inner regions of their host galaxies, while quasar
absorbers probe peripheral regions of the intervening galaxies where the sky
cross-section is the largest (Prochaska et al., 2007; Fynbo et al., 2008).

Figure 2 shows the dust surface density as a function of cold gas surface
density. The cold gas refers to the sum of neutral atomic, H i, and molecular,
$N({\rm H}_{2})$, except for quasar absorbers where the molecular gas is found
to be negligible (Ledoux et al., 2003). The grey lines show constant dust-to-
gas ratios in steps of 1 dex. The dust column densities roughly follow the
total hydrogen column densities, though with a 4-order of magnitude spread at
a given $\Sigma_{\rm Gas}$. Roman-Duval et al. (2014) used resolved Herschel
infra-red maps of Magellanic clouds in combination with 21cm, CO and H$\alpha$
observations to infer the relation between the atomic, molecular and ionised
gas with dust surface density. The authors report a clear trend of increasing
dust surface density with increasing gas surface density in the diffuse
interstellar medium, albeit for a medium with consistent metallicity. Our
depletion-derived results show a similar scaling in Figure 2. Roman-Duval et
al. (2017) further explore the relation based on IRAS and Planck observations
and find an increase by a factor three of the dust-to-gas ratio going from the
diffuse to the dense insterstellar medium, in line with elemental depletions
results. In the Milky Way (Jenkins, 2009a) and Small Magellanic Cloud (Jenkins
& Wallerstein, 2017), the fraction of metals in the gas phase decreases with
increasing hydrogen volume density and column density, albeit at different
rates for different elements. Gamma-Ray Burst host galaxies display low dust
surface densities in comparison with quasar absorbers while still bearing
large amounts of gas (Jakobsson et al., 2006; Fynbo et al., 2009), even at
relatively high gas surface densities. One possible reason for this difference
is that Gamma-Ray Burst host galaxies have overall low metallicities and low
dust content (Perley et al., 2016; Krühler et al., 2015; Savaglio et al.,
2009), although dustier systems do exist (Perley et al., 2011, 2013). We
cannot exclude that systems with high dust surface density (even in the low
metallicity surface density regime) are missing from current sample based on
UV dust-depletion observations. Interestingly, there is a dearth of systems
with high gas surface density and low dust surface density which can not be
attributed to an observational bias, because systems lying in this parameter
space - if they exist - would have small extinction.

The top panel of Figure 3 displays the dust surface density as a function of
gas metallicity. There is a relatively tight correlation between the dust
column densities and the dust-corrected metallicity estimates at all cosmic
times. Works from Jenkins (2009a); Roman-Duval et al. (2022b) report dust-
depletion measurements assuming a global metallicity of [X/H]=0 for the Milky
Way, [X/H]=$-$0.30 for the Large Magellanic Cloud and [X/H]=$-$0.70 for the
Small Magellanic Cloud. Additionally, the top panel of Figure 3 shows a clear
correlation between $\Sigma_{\rm dust}$ and metallicity. These results are in
line with the dust-to-metal ratio increasing with metallicity (Wiseman et al.,
2017; De Cia et al., 2013; Péroux & Howk, 2020). The bottom panels of Figure 3
illustrate the distribution in dust surface density at fixed metallicity
materialised by the grey shaded areas in the top panel. In the Large
Magellanic Clouds in particular, the dispersion in $\Sigma_{\rm dust}$ is
larger than in quasar absorbers for a given metallicity, providing further
evidence that the metallicity within the Clouds might vary. Likewise, we note
that the Magellanic Clouds dust distributions are skewed towards higher values
than quasar absorbers with the similar metallicity. This effect might be
related to (i) quasar absorbers probing random position in the galaxy, and
therefore being more likely to probe the outermost parts, and (ii) Magellanic
Clouds are infalling satellites, so the gas compression coming from the ram
pressure might boost dust formation by increasing the densities. We stress
that further differences in $\Sigma_{\rm dust}$ might be not revealed here
because of the assumption of fixed metallicity in the Magellanic Clouds.

The right most panel of Figure 3 shows a comparison of quasar absorbers with
Gamma-Ray Bursts in the range -1.5$<$[M/H]$<$-0.5 indicating that the latter
have larger dust surface density values. Indeed, at a given metallicity,
$\Sigma_{\rm dust}$ is higher in systems such as Gamma-Ray Bursts that are
closer to the denser and colder parts of their galaxies, where the physical
conditions (high density, low temperature and high pressure) favour the
formation of molecules (Blitz & Rosolowsky, 2006) and likely dust.

Figure 4: Dust surface density distribution function. This function is
computed in analogy with the gas column density distribution function. We note
a turn-over at low dust surface densities to the left of the dotted grey line.
We interpret this feature as due to incompleteness in the sample. Indeed, the
data plotted here are focused on the larger $N(\mbox{H$\;${{i}}})$ column
density quasar absorbers. We fit the high dust surface density values,
$\log\Sigma_{\rm Dust}\geq-6$, with a simple power law of the form: $\log
f(\Sigma_{\rm Dust})=-1.92\times\log\Sigma_{\rm Dust}-3.65$. This observed
relation, which can be computed by spatially resolved simulations predicting
dust mass functions through 2D projection, provides new constraints on modern
dust models.

###  3.3 Dust Surface Density Distribution Function

Next, we use these observations to calculate the dust surface density
distribution function. To this end, we propose to use an analogue of the gas
($N(\mbox{H$\;${{i}}})$ or $N({\rm H}_{2})$) column density distribution
functions (Péroux et al., 2003; Zwaan et al., 2005; Zwaan & Prochaska, 2006;
Klitsch et al., 2019; Péroux & Howk, 2020; Szakacs et al., 2022). We express
the function as follows:

$f(\Sigma_{\rm Dust})=\frac{\mathcal{N}}{\Delta\Sigma_{\rm
Dust}{\rm[g/cm^{2}]}}$ (9)

where $\mathcal{N}$ denotes the number of absorbers in the dust surface
density bin $\Delta\Sigma_{\rm Dust}$ (see also Churchill et al., 2003;
Richter et al., 2011). Since the function is not normalised by the redshift
path, it depends on the number of sightlines in the survey. Our quasar
absorber sample comprises a total of 247 systems. The resulting function for
quasar absorbers is displayed in Figure 4. The data appear to follow a power-
law distribution, with the turn-over at small column densities. This turn-over
kicks in at dust surface densities below $\log\Sigma_{\rm Dust}\leq-6$. We
interpret this feature as due to incompleteness in the observed sample. We
stress that the observations reported here are focused on the larger
$N(\mbox{H$\;${{i}}})$ column density quasar absorbers. It is likely that at
low gas surface density, there are a number of lower dust surface density
systems which are currently not included in the sample. Indeed, the sample is
limited to strong quasar absorbers with $\log N(\mbox{H$\;${{i}}})$$>$20.3.
Figure 2 shows that quasar absorbers with $\log$N(H) $\leq$ 20.3 (top x-axis)
will mostly have $\log\Sigma_{\rm Dust}$ $\leq$ -6 (left y-axis). We note that
these systems are not included by construction in the dust surface density
distribution function presented here. For this reason, we choose to fit the
function without taking the low $\Sigma_{\rm Dust}$ values into account and
with a simple power law of the form:

$f(\Sigma_{\rm Dust})=C\Sigma_{\rm Dust}^{-\delta}$ (10)

which we rewrite as:

$\log f(\Sigma_{\rm Dust})=-\delta\times\log\Sigma_{\rm Dust}+\log C\\\
=-1.92\times\log\Sigma_{\rm Dust}-3.65$ (11)

for surface densities with $\log\Sigma_{\rm Dust}\geq-6$ in units of g/cm2.
Here, $\log f(\Sigma_{\rm Dust})$ is the number of systems with dust surface
density $\Sigma_{\rm Dust}$ per unit of dust surface density
$\Delta\Sigma_{\rm Dust}$. It is therefore expressed in number of systems
cm2/g. C is the normalisation factor. The fit is also shown in Figure 4. Rees
(1988) demonstrated that assuming randomly distributed lines-of-sight through
spherical isothermal halos the column density distribution function, $f(N)$,
was shown to have a power law of slope $\delta$=5/3. Interestingly, Kim et al.
(2001) report a slope $\delta\sim$1.5 for HI absorbers and Churchill et al.
(2003); Richter et al. (2011) measure slopes ranging $\delta$=1.5-2.0 for
MgII, FeII, MgI and CaII, respectively. Therefore, the slope of the dust
surface density distribution function, $\delta$=1.92$\pm$0.13, is possibly
steeper than for neutral gas and on the high-end with respected to metal
absorbers. Finally, we note that this observed relation, which simulations
predicting dust mass function (Pozzi et al., 2020; Millard et al., 2020) will
be able to compute through 2D projection, provides new constraints on modern
dust models (McKinnon et al., 2017; Graziani et al., 2020; Li et al., 2019;
Hou et al., 2019; Baes et al., 2020).

We caution that these UV-depletion results could potentially be incomplete due
to observational biases affecting e.g., the dust surface density distribution.
For example, in the Milky Way and the Magellanic Clouds, UV sight-lines are
biased toward the less reddened stars/lower surface densities by the
sensitivity of the Ultra-Violet telescopes. Similarly, such effects apply to
Gamma-Ray Bursts and quasar samples so that the dustier objects might be
missed from current sample. Several results (Ellison et al., 2009) indicate
that these effects are minimum, but one cannot fully exclude that dustier
objects exists and have been missed from dust-depletion studies presented
here.

##  4 Conclusions

In this work we have looked at a observable, namely $\Sigma_{\rm Dust}$, to
put novel constraints on simulations of dust. Indeed, reproducing dust masses
over cosmic times requires that dust grow in the interstellar medium, and
therefore that the dust properties change significantly with environment,
particularly density. To this end, we gathered observations from the Milky
Way, Large and Small Magellanic Clouds and high-redshift galaxies traced by
Gamma-Ray Burst host galaxies and quasar absorbers. By putting all these
results together we can make a new appraisal of the dust surface density (dust
column density) expressed in g per cm2 or alternatively in M⊙/kpc2 across
cosmic times measured through dust depletion. We also contrast the
observational measurements with recent hydrodynamical simulations. Our main
results are:

  * •

The dust surface densities increase with cosmic time with a large scatter at
any given redshift.

  * •

The dust surface densities are also a function of the total gas surface
densities in the same systems, with increasing dust surface density increasing
with total hydrogen surface density, although the scatter in the relation is 4
orders of magnitude.

  * •

There is a tight correlation between the dust column densities and the dust-
corrected metallicity estimates at all cosmic times.

  * •

We introduce the dust surface density distribution function – in analogy with
the cold gas column density distribution function. We note a turn-over at low
dust surface densities. We interpret this feature as due to incompleteness in
the sample. We provide a fit to the observed distribution of the form: $\log
f(\Sigma_{\rm Dust})$ [number of systems per unit dex]
$=-1.92\times\log\Sigma_{\rm Dust}-3.65$ which proves steeper than for neutral
gas and metal absorbers.

## Data Availability

Data directly related to this publication and its figures is available on
request from the corresponding author.

## Acknowledgements

We are grateful to Omima Osman, Enrico Garaldi, Qi Li, Julia Roman-Duval and
Sandra Savaglio for helpful comments. We thank the anonymous referee for their
suggestions which improved the results presented here. This research was
supported by the International Space Science Institute (ISSI) in Bern, through
ISSI International Team project #564 (The Cosmic Baryon Cycle from Space). ADC
acknowledges support from the Swiss National Science Foundation under grant
185692. JCH recognizes support from the US National Science Foundation through
grant AST-1910255.

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##  Appendix A Observed dust properties at various cosmic times.

Table 1 lists the observed physical properties of different environment
plotted in Figures 1, 2 and 3. We derive these quantities from data of the the
Milky Way (De Cia et al., 2021), the Large and Small Magellanic Clouds
observations (Roman-Duval et al., 2022a), the Gamma-Ray Bursts information
(Bolmer et al., 2019b) and the quasar absorbers data (Péroux & Howk, 2020).
This extract shows the first few entries while the full table is available as
machine-readable on-line material.

Table 1: Summary of the observational data used in this study. The first column refers to the systems under study, the second column lists our computation of the logarithm of the dust surface density in units of g per cm2, the third column provides the redshift, while the fourth column tabulates the total gas surface density and the last column provides the dust-depletion corrected total metallicity with respect to solar. System | log $\Sigma_{\rm Dust}$ | Redshift | log $\Sigma_{\rm Gas}$ | Metallicity
---|---|---|---|---
| [g/cm-2] |  | [g/cm-2] | [M/H]total
Milky Way |  $-$5.05$\pm$0.61 | 0.000 |  $-$2.49$\pm$0.20 |  $-$0.39$\pm$ 0.18
Milky Way |  $-$5.04$\pm$0.52 | 0.000 |  $-$2.31$\pm$0.20 |  $-$0.56$\pm$ 0.09
Milky Way |  $-$5.48$\pm$0.55 | 0.000 |  $-$3.12$\pm$0.20 |  $-$0.18$\pm$ 0.13
Milky Way |  $-$4.87$\pm$0.56 | 0.000 |  $-$2.33$\pm$0.20 |  $-$0.41$\pm$ 0.13
Milky Way |  $-$4.81$\pm$0.62 | 0.000 |  $-$2.12$\pm$0.20 |  $-$0.50$\pm$ 0.20
Milky Way |  $-$5.35$\pm$0.56 | 0.000 |  $-$3.46$\pm$0.20 |  $+$0.28 $\pm$0.13
… |  |  |  |