# The ALMaQUEST Survey XI: A strong but non-linear relationship between star
formation and dynamical equilibrium pressure.

Sara L. Ellison1, Hsi-An Pan2, Asa F. L. Bluck3, Mark R. Krumholz4,5, Lihwai
Lin6, Leslie Hunt7, Edvige Corbelli7, Mallory D. Thorp1,8, Jorge Barrera-
Ballesteros9, Sebastian F. Sánchez9, Jillian M. Scudder10, Salvatore Quai11,12
1 Department of Physics & Astronomy, University of Victoria, Finnerty Road,
Victoria, British Columbia, V8P 1A1, Canada
2 Department of Physics, Tamkang University, No.151, Yingzhuan Road, Tamsui
District, New Taipei City 251301, Taiwan
3 Department of Physics, Florida International University, 11200 SW 8th
Street, Miami, FL, USA
4 Research School of Astronomy and Astrophysics, Australian National
University, Canberra ACT 2600, Australia
5 ARC Center of Excellence for Astronomy in Three Dimensions (ASTRO-3D),
Canberra ACT 2600, Australia
6 Institute of Astronomy & Astrophysics, Academia Sinica, Taipei 10617, Taiwan
7 INAF— Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Florence, Italy
8 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
53121, Bonn, Germany
9 Instituto de Astronomía, Universidad Nacional Autonoma de Mexico, A. P.
70-264, C.P. 04510, Mexico, D.F., Mexico
10 Department of Physics and Astronomy, Oberlin College, Oberlin, Ohio, OH
44074, USA
11 INAF-Osservatorio di Astrofisica e Scienze dello Spazio di Bologna, Via
Gobetti 93/3, I-40129 Bologna, Italy
12 Dipartimento di Fisica e Astronomia ‘Augusto Righi’, Universita degli Studi
di Bologna, Via Gobetti 93/2, I-40129 Bologna, Italy

###### Abstract

We present the extended ALMA MaNGA QUEnching and STar formation survey, a
combination of the original 46 ALMaQUEST galaxies plus new ALMA observations
for a further 20 interacting galaxies. Three well-studied scaling relations
are fit to the 19,999 star-forming spaxels in the extended sample, namely the
resolved Schmidt-Kennicutt (rSK) relation, the resolved star forming main
sequence (rSFMS) and the resolved molecular gas main sequence (rMGMS). We
additionally investigate the relationship between the dynamical equilibrium
pressure ($P_{DE}$) and star formation rate surface density ($\Sigma_{\rm
SFR}$), which we refer to as the resolved $P_{DE}$ (rPDE) relation. Contrary
to previous studies that have focussed on normal star-forming galaxies and
found an approximately linear rPDE relation, the presence of more vigourously
star-forming galaxies in the extended ALMaQUEST sample reveals a marked
turnover in the relation at high pressures. Although the scatter around the
linear fit to the rPDE relation is similar to the other three relations, a
random forest analysis, which can extract non-linear dependences, finds that
$P_{DE}$ is unambiguously more important than either $\Sigma_{\rm H_{2}}$ or
$\Sigma_{\star}$ for predicting $\Sigma_{\rm SFR}$. We compare the observed
rPDE relation to the prediction of the pressure-regulated feedback-modulated
(PRFM) model of star formation, finding that galaxies residing on the global
SFMS do indeed closely follow the rPDE relation predicted by the PRFM theory.
However, galaxies above and below the global SFMS show significant deviations
from the model. Galaxies with high SFR are instead consistent with models that
include other contributions to turbulence in addition to the local star
formation feedback.

###### keywords:

Galaxies: evolution, galaxies: interactions, galaxies: starburst, galaxies:
star formation, galaxies: ISM

##  1 Introduction

Scaling relations provide a valuable tool for understanding the underlying
drivers in physical processes. In astronomy, correlations such as the Faber-
Jackson relation (Faber & Jackson 1976), the $M-\sigma$ relation (Ferrarese &
Merritt 2000) and the mass metallicity relation (Lequeux et al. 1979), have
been used as both theoretical supports and interpretative tools for various
aspects of galaxy evolution. Examining the dependence on additional variables
(e.g. Ellison et al. 2008b; Mannucci et al. 2010; Peng et al. 2010; Hunt et
al. 2020; Morselli et al. 2020) provides further insight, with machine
learning techniques now being employed to distinguish ‘fundamental’ relations
from mere correlations (e.g. Teimoorinia et al. 2016; Dey et al. 2019; Bluck
et al. 2020; Baker & Maiolino 2023).

In the field of star formation, an extensive literature has established a
tight correlation between the surface densities of star formation rate and
molecular gas, $\Sigma_{\rm SFR}$ and $\Sigma_{\rm H_{2}}$ respectively (e.g.
Wong & Blitz 2002; Bigiel et al. 2008; Schruba et al. 2011), with possibly an
even tighter relationship between star formation and the denser gas phase
traced by HCN (Wu et al. 2005; Lada et al. 2010; Jimenez-Donaire et al. 2019).
Ostensibly, these scaling relations indicate that star formation is set by the
abundance of available fuel (i.e. gas content). However, the non-universality
of star formation efficiencies on both global (e.g. Daddi et al. 2010;
Saintonge et al. 2011) and local scales (e.g. Leroy et al. 2008; Usero et al.
2015; Ellison et al. 2021a; Thorp et al. 2022; Jimenez-Donaire et al. 2023)
hints at additional physics that regulates star formation.

Early studies suggested that the fraction of the interstellar medium (ISM) in
the molecular phase was driven by the internal pressure of the ISM, which then
formed stars at an approximately constant efficiency (e.g. Wong & Blitz 2002;
Blitz & Rosolowsky 2004, 2006; Kim & Ostriker 2007; Leroy et al. 2008). These
theories have been expanded in the last decade to describe star formation as a
‘self-regulating’ process, that is set by the balance between energy/momentum
input from massive stars and the weight of the ISM (e.g. Ostriker et al. 2010;
Ostriker & Shetty 2011; Shetty & Ostriker 2012; Kim et al. 2013; Ostriker &
Kim 2022). As a result of this balance, the SFR is predicted to scale
approximately linearly with the gravity felt by the ISM. Such models are
referred to as pressure-regulated feedback-modulated (PRFM) theories.

Several contemporary observational studies have supported the PRFM formalism,
that star formation is not only strongly correlated with the ISM pressure
(e.g. Herrera-Camus et al. 2017; Fisher et al. 2019; Sun et al. 2020a, 2023),
but that relations between the SFR and combinations of both gas and stellar
mass are tighter and stronger than the resolved star-forming main sequence
(rSFMS), resolved Schmidt-Kennicutt (rSK) relation or resolved molecular gas
main sequence (rMGMS) (e.g. Shi et al. 2011; Barrera-Ballesteros et al.
2021a,b). Moreover, Barrera-Ballesteros et al. (2021a) found that the
hydrostatic ISM pressure and $\Sigma_{\rm SFR}$ correlate consistently over a
wide range of galaxy morphologies, thus proposing it as the main parameter
that regulates star formation on kpc scales. Most recently, the data
compilation and comparison with the latest generation of high resolution
hydrodynamical simulations presented by Ostriker & Kim (2022) show strong
support for the PRFM model. However, to date, the data that have been included
in these comparisons have been drawn almost exclusively from local spiral
galaxies on the star-forming main sequence. Unlike earlier star formation
models (e.g., Tan 2000; Krumholz et al. 2018, Semenov et al. 2019), the PRFM
formalism has not yet been tested against a wider range of galaxies that
display differing star formation behaviour, such as starbursts (e.g., Daddi et
al. 2010, Sharda et al. 2019), metal-poor dwarfs (e.g., Jameson et al. 2016),
and early types that retain gas (e.g., Davis et al. 2014). Here, we aim to
further the comparison of the PRFM model with an additional dataset that
includes a wider range of galaxies than those previously assessed.

The ALMA MaNGA QUEnching and STar formation (ALMaQUEST) survey aims to
understand the kpc-scale processes that regulate star formation in the nearby
universe. The original ALMaQUEST sample consists of 46 galaxies (Lin et al.
2020), spanning a wide range of SFRs from the green valley (e.g. Lin et al.
2022) to the starburst regime (Ellison et al. 2020a). Although other molecular
gas surveys have either higher spatial resolution such as the Physics at High
Resoution in Nearby Galaxies survey (PHANGS; Leroy et al. 2021), or a larger
sample such as the Extragalactic Database for Galaxy Evolution (EDGE-CALIFA;
Bolatto et al. 2017), ALMaQUEST’s diversity of star-forming properties
provides vital leverage for studying the mechanisms that both boost and quench
star formation. Moreover, we have recently enlarged the ALMaQUEST sample by
almost 50% by observing a further 20 galaxies in ALMA’s Cycle 7 (PI Pan). In
keeping with the survey’s goal of probing a wide range of environments, the
new sample focuses on interacting galaxies, allowing us to study the relative
role of fuel supply and star formation efficiency in boosting star formation
(Thorp et al. 2022). We will refer to the combined sample of 46 original
ALMaQUEST galaxies plus the 20 mergers as the extended ALMaQUEST sample.

In this paper we have five main goals. First, we re-visit the three scaling
relations (rSK, rSFMS and rMGMS) that we have studied in our previous works,
in order to present a complete assessment for the extended dataset. Second, we
provide a public release of the star formation rate, stellar mass and
molecular gas surface densities for all star-forming spaxels in the extended
sample of 66 galaxies to permit a reproduction of these relations. Third, we
separate the sample into subsets of control galaxies that represent normal,
face-on orientations that we might expect to manifest fiducial relations, then
mergers and central starbursts which represent more extreme populations. The
star formation scaling relations are examined separately in each of these
subsets. Fourth, we investigate the relation between dynamical equilibrium
pressure ($P_{DE}$) and $\Sigma_{\rm SFR}$ (hereafter the resolved dynamical
equilibrium pressure relation, rPDE) in ALMaQUEST galaxies for the first time,
in order to assess whether (as suggested by Barrera-Ballesteros et al. 2021a)
this relation might be the fundamental regulator of star formation. Finally,
we compare the observed rPDE relation with contemporary theories of star
formation in order to assess whether (and under what conditions) the models
can reproduce the data.

The paper is organized as follows. In Section 2 we describe the extended
ALMaQUEST sample, as well as three sub-samples that are defined for the
purposes of this paper. In Section 3 we describe the MaNGA and ALMA data
products that are released with this paper. The star formation scaling
relations for the extended ALMaQUEST sample, with a particular focus on the
rPDE relation, are presented in Section 4 with a more detailed discussion of
results in Section 5. Our conclusions are presented in Section 6.

##  2 The extended ALMaQUEST sample

The extended ALMaQUEST sample consists of a total of 66 galaxies, all of which
were drawn from the MaNGA DR15. ALMA observations (described below) were
obtained through five separate regular proposals. In ALMA Cycles 3, 5 and 6
proposals 2015.1.01225.S, 2017.1.01093.S, 2018.1.00558.S (PI: Lin) obtained
CO(1-0) data for a sample of galaxies that spanned both the star-forming main
sequence and the green valley, in order to primarily investigate quenching
(Ellison et al. 2021b; Lin et al. 2022). The Cycle 6 proposal 2018.1.00541.S
(PI: Ellison) complemented this sample by focussing on galaxies with central
starbursts, again with main sequence galaxies included as comparison targets,
in order to investigate the physical processes that lead to enhanced star
formation rates (Ellison et al. 2020a, b). The combination of these four
proposals includes 46 unique galaxies and represents the main ALMaQUEST sample
that is described by Lin et al. (2020) and that has been used to investigate
the scaling relations of star forming regions (Lin et al. 2019; Ellison et al.
2021a, Lin et al. 2022). More recently, in ALMA Cycle 7 (2019.1.00260.S, PI
Pan), we obtained CO(1-0) data for a further 20 MaNGA selected galaxies. The
Cycle 7 sample focused on interacting galaxies and contains targets that have
close companions and/or show evidence for tidal features (Thorp et al. 2022).
Taken together, the final extended ALMaQUEST sample therefore spans a broad
range of galactic properties that can be used to investigate a variety of
environments and star formation regimes.

In addition to the complete sample of 66 galaxies, we here define three galaxy
sub-samples that will be used in this paper’s analysis. First, we select a
sample of galaxies that we expect to be minimally affected by issues such as
inclination, starbursts or mergers. This ‘control’ sample consists of 16
galaxies which have axial ratios $b/a\geq 0.35$ and are not in either of the
central starburst, nor merger samples (see below). Second, we define a sample
of central starburst galaxies. Following Ellison et al. (2020a), central
starburst galaxies are identified by plotting the radial profiles of
$\Sigma_{\rm SFR}$ (as determined from H$\alpha$ emission; see Section 3.1 for
more details) and selecting galaxies with enhancements of at least 0.2 dex
within the inner 0.5 $R/R_{e}$. Although this selection process is identical
to that of Ellison et al. (2020a), here we enforce the additional criterion
that the galaxy not be in a merger, which excludes three of the central
starbursts from the Ellison et al. (2020a) sample (8081-9101, 8156-3701,
8615-3703). However, the Cycle 7 observations that have been obtained since
the publication of Ellison et al. (2020a) include two galaxies that qualify as
central starbursts and do not show merger features (8085-12701, 8085-3704). As
a result of these choices, there are 11 central starburst galaxies in our
sample.

Finally, we define a sample of galaxy mergers. Although our Cycle 7 proposal
focussed exclusively on mergers, there are also some interacting galaxies in
the main (original) ALMaQUEST sample. Moreover, the Cycle 7 sample includes
galaxies that span a wide range of interaction stages, including pairs that do
not show any tidal features and therefore may either yet to have experienced a
pericentric passage, or those whose features are either fundamentally weaker
(due to the orbital configuration) or fainter (due to fading). We therefore
elect to define our merger sample from scratch, by visually inspecting both
the SDSS imaging, as well as the deeper Dark Energy Camera Legacy Survey
(DECaLS) imaging that exists for the full sample. Galaxies that exhibit clear
morphological disturbances, such as bridges, tidal arms or shells, are
included in the merger sample. 19 galaxies fulfill this requirement, the
majority of which are post-mergers, i.e. single galaxies assumed to be
observed after coalescence (e.g. Ellison et al. 2013; Thorp et al. 2019;
Bickley et al. 2021, 2022), but some are still readily identifiable as
interacting pairs (e.g. Thorp et al. 2022). Although we did not allow mergers
to be in the central starburst sample, we do allow mergers to have central
starbursts (there are six such galaxies in the merger sample). That is, all
galaxies in Table 1 identified as mergers are in the merger sample, but the
galaxies in Table 1 with central starbursts and also identified as mergers are
not in our central starburst sample. Images for each galaxy in the extended
ALMaQUEST sample are available in either the original ALMaQUEST survey
description paper (Lin et al. 2020) or in the presentation of the Cycle 7 data
for the mergers (Thorp et al. 2022).

Table 1 summarizes the targets in the extended ALMaQUEST sample and presents
their integrated stellar masses and SFRs taken from the PIPE3D (Sánchez et
al., 2016a, 2016b) value-added catalog (VAC, Sánchez et al. 2018), derived
through summing individual spaxel values across the MaNGA data cubes and the
integrated molecular gas masses described in Section 3.2 from our ALMA data.
The molecular gas masses are calculated assuming a conversion factor
$\alpha_{CO}$ = 4.3 M⊙ pc-2 (K km s-1)-1 (including the contribution from
helium). In contrast with many of our previous studies that have used a
Salpeter initial mass function (IMF), which is the default option adopted by
PIPE3D, in the work presented here we have converted $\Sigma_{\rm SFR}$ and
$\Sigma_{\star}$ values to a Chabrier IMF in order to be more readily
comparable to other work in the literature (e.g. Sun et al. 2020a; Barrera-
Ballesteros et al. 2021a). This conversion from Salpeter to Chabrier IMFs is
achieved by multiplying the former by a factor of 0.61, see Madau & Dickinson
(2014).

Table 1: Summary of global (within the IFU) properties for the extended ALMaQUEST sample, taken from either the PIPE3D Value-added catalog, or derived from our ALMA observations. The Proposal ID indicates the initial of the PI’s surname (E=Ellison, L=Lin, P=Pan) and the ALMA Cycle in which the data were obtained. Plate-IFU | $z$ | log M⋆ | log SFR | Log M(H2) | Axial | Proposal | Merger? | Central
---|---|---|---|---|---|---|---|---
|  | (VAC) | (VAC) | (within IFU) | ratio | ID |  | starburst?
|  | (M⊙) | (M⊙ yr-1) | (M⊙) |  |  |  |
7977$-$3704 | 0.02724 | 10.14 |  $-$0.60 | 8.73 | 0.66 | L3 | 0 | 0
7977$-$12705 | 0.02724 | 10.64 | 0.25 | 9.41 | 0.67 | L3 | 1 | 0
7977$-$9102 | 0.06331 | 10.71 | 0.61 | 9.85 | 0.63 | P7 | 1 | 0
7977$-$3703 | 0.02781 | 10.14 | 0.18 | 9.16 | 0.80 | E6 | 0 | 1
7977$-$9101 | 0.02656 | 11.02 |  $-$0.08 | 8.79 | 0.55 | L3 | 1 | 0
7975$-$6104 | 0.07920 | 10.80 | 0.40 | 9.96 | 0.88 | P7 | 1 | 0
7968$-$12705 | 0.08638 | 11.08 |  $-$1.63 |  $<$7.13 | 0.47 | P7 | 1 | 0
8153$-$12702 | 0.03823 | 9.68 |  $-$0.95 |  $<$7.06 | 0.47 | P7 | 1 | 0
8077$-$9101 | 0.04323 | 10.20 |  $-$0.31 | 8.90 | 0.18 | L6 | 0 | 0
8082$-$12701 | 0.02703 | 10.27 |  $-$0.09 | 8.90 | 0.62 | E6 | 0 | 0
8081$-$3704 | 0.05400 | 10.34 | 0.75 | 9.26 | 0.96 | E6 | 0 | 1
8082$-$9102 | 0.03652 | 10.53 | 0.51 | 9.82 | 0.31 | P7 | 1 | 1
8085$-$12701 | 0.02980 | 10.22 | 0.37 | 8.71 | 0.52 | P7 | 0 | 1
8081$-$9101 | 0.02846 | 10.39 | 0.11 | 9.45 | 0.55 | L5 | 1 | 1
8083$-$12702 | 0.02104 | 11.01 | 0.45 | 9.55 | 0.74 | L5 | 0 | 0
8085$-$3704 | 0.03707 | 10.51 | 0.44 | 9.74 | 0.52 | P7 | 0 | 1
8156$-$3701 | 0.05272 | 10.31 | 0.66 | 9.02 | 0.73 | E6 | 1 | 1
8086$-$9101 | 0.04003 | 10.73 |  $-$0.10 | 9.20 | 0.50 | L5 | 0 | 0
8078$-$6103 | 0.02859 | 10.53 | 0.40 | 9.62 | 0.59 | L5 | 0 | 0
8081$-$6102 | 0.03719 | 10.58 |  $-$0.61 | 8.72 | 0.54 | L5 | 0 | 0
8081$-$12703 | 0.02558 | 10.12 |  $-$1.00 | 8.93 | 0.20 | L5 | 0 | 0
8084$-$3702 | 0.02206 | 10.01 | 0.21 | 9.30 | 0.66 | E6 | 0 | 0
8085$-$6101 | 0.05180 | 10.85 |  $-$0.95 |  $<$7.56 | 0.64 | P7 | 1 | 0
8078$-$12701 | 0.02698 | 10.73 | 0.19 | 9.56 | 0.38 | L5 | 0 | 0
8078$-$6104 | 0.04449 | 10.20 | 0.02 | 9.18 | 0.44 | P7 | 1 | 0
8081$-$9102 | 0.03407 | 10.45 | 0.03 | 9.28 | 0.17 | L5 | 0 | 0
8155$-$6102 | 0.03081 | 10.14 | 0.14 | 9.27 | 0.89 | E6 | 0 | 0
8241$-$12705 | 0.02719 | 10.18 |  $-$0.21 | 9.20 | 0.49 | P7 | 1 | 0
8241$-$3703 | 0.02911 | 9.89 | 0.04 | 8.83 | 0.94 | E6 | 0 | 0
8241$-$3704 | 0.06617 | 10.79 | 1.00 | 10.03 | 0.95 | E6 | 0 | 1
8083$-$6101 | 0.02677 | 10.09 |  $-$1.09 | 9.32 | 0.23 | L5 | 0 | 0
8084$-$12705 | 0.02545 | 10.24 |  $-$0.24 | 8.85 | 0.22 | L5 | 0 | 0
8155$-$6101 | 0.03740 | 10.71 |  $-$0.64 | 9.11 | 0.74 | L5 | 0 | 0
8077$-$6104 | 0.04601 | 10.52 | 0.43 | 9.53 | 0.93 | E6 | 0 | 0
8082$-$12704 | 0.13214 | 11.20 | 0.55 | 10.21 | 0.44 | L5 | 0 | 0
8083$-$9101 | 0.03847 | 10.92 | 0.38 | 9.65 | 0.33 | L6 | 1 | 0
8078$-$12703 | 0.02832 | 10.60 |  $-$0.08 | 9.12 | 0.77 | P7 | 0 | 0
8084$-$6103 | 0.03593 | 10.30 |  $-$0.35 | 9.40 | 0.31 | L5 | 0 | 0
8083$-$12703 | 0.02466 | 10.24 | 0.10 | 9.04 | 0.30 | P7 | 1 | 1
8082$-$6103 | 0.02416 | 10.09 | 0.17 | 8.94 | 0.74 | E6 | 0 | 1
8615$-$12702 | 0.02095 | 9.99 |  $-$0.19 | 8.55 | 0.27 | E6 | 0 | 0
8623$-$6104 | 0.09704 | 11.07 | 0.96 | 10.12 | 0.47 | E6 | 0 | 0
8616$-$9102 | 0.03039 | 10.23 | 0.44 | 9.42 | 0.82 | E6 | 0 | 1
8615$-$3703 | 0.01845 | 9.98 | 0.19 | 9.28 | 0.55 | E6 | 1 | 1
8952$-$6104 | 0.02843 | 10.12 | 0.24 | 9.11 | 0.93 | L5 | 0 | 0
8950$-$12705 | 0.02528 | 10.32 |  $-$0.62 | 9.44 | 0.15 | L5 | 0 | 0
8655$-$3701 | 0.07149 | 10.94 | 0.90 | 10.43 | 0.55 | L6 | 0 | 0
9194$-$3702 | 0.07454 | 10.84 | 0.75 | 10.32 | 0.82 | P7 | 1 | 0
9195$-$3703 | 0.02724 | 10.18 | 0.26 | 9.32 | 0.56 | P7 | 1 | 0
8728$-$3701 | 0.02833 | 10.42 |  $-$0.59 | 8.70 | 0.73 | L5 | 1 | 0
7815$-$12705 | 0.02955 | 10.62 | 0.20 | 9.72 | 0.34 | L5 | 0 | 1
8616$-$12702 | 0.03083 | 10.54 |  $-$0.45 | 8.41 | 0.78 | L5 | 1 | 0
8615$-$9101 | 0.03346 | 10.37 |  $-$0.18 | 9.12 | 0.26 | L5 | 0 | 0
8616$-$9101 | 0.09197 | 10.88 | 0.53 | 9.81 | 0.50 | P7 | 1 | 0
8952$-$12701 | 0.02856 | 10.52 |  $-$0.60 | 8.91 | 0.60 | L5 | 1 | 0
8450$-$6102 | 0.04200 | 10.21 | 0.43 | 9.23 | 0.76 | E6 | 0 | 1
Table 1: Continued Plate-IFU | $z$ | log M⋆ | log SFR | Log M(H2) | Axial | Proposal | Merger? | Central
---|---|---|---|---|---|---|---|---
|  | (VAC) | (VAC) | (within IFU) | ratio | ID |  | starburst?
|  | (M⊙) | (M⊙ yr-1) | (M⊙) |  |  |  |
9195$-$3702 | 0.06434 | 10.93 | 0.89 | 9.98 | 0.61 | P7 | 1 | 0
9195$-$9101 | 0.05687 | 10.53 | 0.17 | 9.45 | 0.81 | P7 | 1 | 0
8618$-$9102 | 0.04334 | 10.24 | 0.21 | 9.27 | 0.33 | L6 | 0 | 1
8623$-$1902 | 0.02495 | 9.95 |  $-$1.22 | 8.30 | 0.92 | P7 | 1 | 0
8615$-$1901 | 0.02019 | 9.47 |  $-$0.29 |  $<$7.30 | 0.85 | P7 | 1 | 1
8623$-$12702 | 0.02691 | 10.49 |  $-$0.57 | 9.49 | 0.17 | L5 | 0 | 0
8616$-$6104 | 0.05426 | 10.56 | 0.04 | 9.37 | 0.45 | L5 | 0 | 0
8655$-$12705 | 0.04557 | 10.12 |  $-$0.93 | 8.78 | 0.49 | L5 | 1 | 0
8655$-$9102 | 0.04509 | 10.20 | 0.06 | 8.82 | 0.71 | E6 | 0 | 1
9512$-$3704 | 0.05463 | 10.48 |  $-$0.07 | 8.96 | 0.68 | P7 | 1 | 0

##  3 Data

Since the acquisition and processing of ALMaQUEST data have been described
extensively in our previous works (e.g. Lin et al. 2019, 2020; Ellison et al.
2020a, b, 2021a, b), we give only a brief overview of salient details here and
refer readers requiring more detail to the aforementioned works.

###  3.1 MaNGA data

We use MaNGA data from the Data Release 15 as the source of all resolved
quantities such as emission line fluxes and stellar mass surface densities.
Spaxels are 0.5 arcseconds on a side and are thus over-sampled compared with
both the fibre size (2 arcseconds) and typical seeing. Although the analysis
presented here (and in all of our previous papers) uses the 0.5 arcsecond
spaxels, we have repeated all of our analysis smoothing to a range of angular
scales (up to 3 arcseconds) and find no qualitative difference in our results.

Optical emission line fluxes are taken from the public PIPE3D data cubes
(Sánchez et al. 2016a,b, 2018) and corrected for internal extinction by
assuming an intrinsic H$\alpha$/H$\beta$=2.85 and a Milky Way extinction curve
(Cardelli, Clayton & Mathis 1989). Star formation rate surface densities
($\Sigma_{\rm SFR}$) are computed from H$\alpha$ luminosities using Equation 2
from Kennicutt (1998), a technique that has been shown to reproduce the UV and
IR SFRs well in integral field unit (IFU) data (Catalan-Torrecilla et al.
2015). We discuss the potential limitations of these $\Sigma_{\rm SFR}$
measurements for the work presented here in more detail in Section 5.3.5.
Surface densities of stellar mass ($\Sigma_{\star}$) are also taken from the
PIPE3D catalog. All surface density quantities are inclination corrected using
the axial ratio ($b/a$) listed in Table 1.

In the next section, we will investigate the scaling relations of star-forming
spaxels. To qualify as star-forming, any given spaxel must meet three
conditions. First, we require that the MaNGA spaxel has a S/N$>$2 in each of
the four optical emission lines used in the Baldwin, Phillips & Terlevich
(1981; hereafter BPT) diagram, i.e., H$\alpha$, H$\beta$, [OIII]$\lambda$5007,
[NII]$\lambda$6584\. In practice, this means that the strongest line
(H$\alpha$) is observed with a much higher S/N, but this does not affect the
results of this work. Second, the emission line ratios of the spaxel must lie
below the criterion defined by Kauffmann et al. (2003), thus identifying it as
being dominated by stellar photoionization. Third, we impose an H$\alpha$
equivalent width (EW) cut H$\alpha>6$ Å (e.g. Cid-Fernandes et al. 2011) to
remove potential ‘retired’ spaxels (which have been studied explicitly for the
ALMaQUEST sample by Ellison et al. 2021b and Lin et al. 2022).

###  3.2 ALMA data

The acquisition and processing of ALMA data for the main sample of 46 galaxies
is described in detail in Lin et al. (2020). The extended ALMaQUEST sample,
including the 20 new galaxies observed in Cycle 7, follows identical
procedures (described in more detail in Thorp et al. 2022). Below we review
the details relevant for the current work.

CO(1-0) (hereafter, simply CO) spectral line observations were obtained
between 2016-2020 in the array’s second most compact configuration (C43-2).
The single pointing primary beam size for this configuration is $\sim$ 50
arcsec with an angular resolution $\sim$ 2.5 arcsec. Integration times ranged
from 0.2 to 2.5 hours on source, using one high resolution spectral window
focused on the CO line and one to three additional low resolution continuum
windows for calibration. The data cubes were all processed using the Common
Astronomy Software Applications (CASA; McMullin et al. 2007) package. The
final cubes have channel widths of 11 km s-1 and root mean square (RMS) noise
of $\sigma_{rms}$ = 0.2 – 2 mJy beam-1. To permit a mapping of the ALMA data
cubes onto the MaNGA data products, the ALMA data were first trimmed to the
size of that galaxy’s MaNGA cube (MaNGA IFU bundles range in size from 12 – 32
arcsec chosen to match the galaxy size). A fixed restoring beam size of 2.5
arcsec with pixel size of 0.5 arcsec was then applied to the ALMA cube. These
two steps resulted in ALMA data cubes with the same size and sampling as the
MaNGA data products.

CO luminosities in each pixel (L’(CO) in Jy km s-1 pc2 pix-1) were converted
to molecular gas surface densities ($\Sigma_{\rm H_{2}}$) using a conversion
factor ($\alpha_{CO}$) such that $\Sigma_{\rm H_{2}}$ (M⊙ pix-1) =
$\alpha_{CO}\times$ L’(CO). In Table 2 we provide $\Sigma_{\rm H_{2}}$ both
per pixel and per kpc2. In keeping with our previous ALMaQUEST papers, we
assume a fixed conversion factor $\alpha_{CO}$ = 4.3 M⊙ pc-2 (K km s-1)-1
(including the contribution from helium) which is a typical value adopted for
the Milky Way (e.g. Bolatto et al. 2013). As with other surface densities used
in this work, $\Sigma_{\rm H_{2}}$ is corrected for inclination using the
$b/a$ axial ratio given in Table 2. Typical uncertainties in $\Sigma_{\rm
H_{2}}$ are $<$ 0.1 dex.

In Table 1 we report the integrated molecular gas mass within the MaNGA IFU
footprint (assuming our fiducial $\alpha_{CO}$=4.35 M⊙ pc-2 (K km s-1)-1 )
that can be used to compute a ‘global’ gas fraction or star formation
efficiency by comparing with the VAC stellar mass or star formation rate. Four
galaxies in the Cycle 7 sample are not detected in the integrated maps. In
these cases the 3$\sigma$ upper limit of the CO flux is calculated as
$3\sigma_{RMS}\times{\sqrt{\delta v\Delta V}}$, where $\sigma_{RMS}$ is the
RMS noise from the spectral line data cube, $\delta v$ $=$ 11 km s-1 is the
velocity resolution, and $\Delta V$ is the assumed 500 km s-1 width.

There is a total of 19,999 spaxels that fulfill both the star-forming criteria
described in Section 3.1 and have detections in CO with a S/N$>$3, that we
henceforth refer to as the ‘full’ star-forming spaxel sample. These spaxels
have all of the necessary measurements required to investigate the various
star formation scaling relations. The control, central starburst and merger
samples contain 8321, 5357 and 4176 star-forming spaxels respectively (these
numbers do not sum to 19,999 because some galaxies appear in more than one
sub-sample, and some galaxies appear in none). We note that one of the control
galaxies and two of the merger galaxies have no star-forming spaxels. The
$\Sigma_{\rm SFR}$, $\Sigma_{\star}$ and $\Sigma_{\rm H_{2}}$ values for the
full sample of (CO-detected) 19,999 star-forming spaxels are listed in Table
2.

There exist numerous alternative prescriptions for variable conversion
factors, taking into account parameters such as metallicity and SFR (e.g.
Narayanan et al. 2012; Bolatto 2013; Sandstrom et al. 2013; Accurso et al.
2017; Hunt et al. 2020; Gong et al. 2020). As an alternative to using a fixed
conversion factor, we therefore also compute $\Sigma_{\rm H_{2}}$ with the
metallicity dependent formalism of Sun et al. (2020b):

$\alpha_{CO,Z}=4.35Z^{-1.6}M_{\odot}pc^{-2}(K~{}km~{}s^{-1})^{-1}.$ (1)

In order to determine $\alpha_{CO,Z}$ from Equation 1, an accurate gas phase
metallicity must be available for the spaxel. We therefore only compute
$\alpha_{CO,Z}$ (and the subsequent values of $\Sigma_{\rm H_{2}}$) when the
emission lines of H$\alpha$, H$\beta$, [OIII]$\lambda$5007, [NII]$\lambda$6584
all have S/N$>5$ and the spaxel lies below the Kauffmann et al. (2003) line
that separates star-forming and AGN dominated zones. Metallicities are
computed using the O3N2 calibration of Pettini & Pagel (2004) normalized to a
solar value of 12+log(O/H)=8.69. Since the S/N requirement for accurate
metallicity determination (S/N$>$5) is stricter than for our nominal star-
forming sample (S/N$>$2) of the full sample of 19,999 star-forming spaxels,
only 16,254 have metallicity measurements (with values mostly in the range
8.55 $<$ 12 + log O/H $<$ 8.75) and hence have metallicity dependent
determinations of $\Sigma_{\rm H_{2}}$. The impact of the choice of conversion
factor on our work (as well as tests with other possible values) is discussed
extensively in Section 5.

Table 2: Spaxel properties for all CO detected (S/N$\geq$3) star-forming spaxels in the extended ALMaQUEST sample. Spaxel X and Y coordinates are given as unitless quantities (starting at 0,0) to enable a reconstruction of the 2 dimensional maps. All surface density quantities are inclination corrected. The first 10 rows are given here as an example; the full dataset is available from the online journal. Plate-IFU | X, Y | Log L’(CO) | CO | Log $\Sigma_{\rm H_{2}}$ | Log $\Sigma_{\rm H_{2}}$ | Log $\Sigma_{\star}$ | Log $\Sigma_{\rm SFR}$
---|---|---|---|---|---|---|---
|  | (Jy km s-1 pc2 pix-1) | S/N | (M⊙ pix-1) | (M⊙ kpc-2) | (M⊙ kpc-2) | (M⊙ yr-1 kpc-2)
7977$-$3704 | 28,10 | 5.27 | 3.21 | 5.90 | 6.85 | 8.08 | -2.35
7977$-$3704 | 29,10 | 5.27 | 3.17 | 5.90 | 6.84 | 8.08 | -2.37
7977$-$3704 | 29,11 | 5.40 | 4.32 | 6.03 | 6.98 | 8.08 | -2.29
7977$-$3704 | 30,11 | 5.36 | 3.92 | 5.99 | 6.94 | 8.08 | -2.44
7977$-$3704 | 31,11 | 5.27 | 3.20 | 5.90 | 6.85 | 7.88 | -2.67
7977$-$3704 | 26,12 | 5.31 | 3.51 | 5.94 | 6.89 | 8.25 | -2.15
7977$-$3704 | 27,12 | 5.39 | 4.24 | 6.03 | 6.97 | 8.25 | -2.15
7977$-$3704 | 29,12 | 5.46 | 4.96 | 6.09 | 7.04 | 8.12 | -2.21
7977$-$3704 | 30,12 | 5.43 | 4.64 | 6.06 | 7.01 | 8.12 | -2.34
7977$-$3704 | 31,12 | 5.37 | 3.99 | 6.00 | 6.95 | 7.88 | -2.56

##  4 Star formation scaling relations

Figure 1: The resolved Schmidt-Kennicutt relation for the four galaxy samples
defined in this work. In each panel the ODR fit is shown by the dashed line.
The dotted line reproduces the fit for the full star-forming sample as a
reference.

Figure 2: The resolved molecular gas main sequence for the four galaxy samples
defined in this work. In each panel the ODR fit is shown by the dashed line.
The dotted line reproduces the fit for the full star-forming sample as a
reference.

One of the main focuses of this paper is to present the rPDE relation for
star-forming spaxels in the ALMaQUEST survey, and to assess whether a) it is
universal (i.e. invariant between galaxies) and b) whether it is more
fundamental than the previously studied rSK, rMGMS and rSFMS relations.
Although these latter three scaling relations have already been presented in
several previous ALMaQUEST papers (e.g. Lin et al. 2019; Ellison et al. 2021a;
Lin et al. 2022), it is useful to re-visit them here for several reasons.
First, we are presenting the extended ALMaQUEST sample, including 20 new
galaxies to complement the original 46, for the first time. Second, in the
previous section we introduced three sub-samples of galaxies (controls,
central starbursts and mergers) to capture the diversity in the dataset.
Before investigating the rPDE relation for these samples, as well as for the
full extended ALMaQUEST sample, we therefore briefly present the rSK, rMGMS
and rSFMS relations for context and comparison.

###  4.1 The Resolved Schmidt-Kennicutt Relation

Figure 3: The resolved star forming main sequence for the four galaxy samples
defined in this work. In each panel the ODR fit is shown by the dashed line.
The dotted line reproduces the fit for the full star-forming sample as a
reference.

In the top left panel of Figure 1 we present the rSK relation for all 19,999
star-forming spaxels in the extended ALMaQUEST sample of 66 galaxies. In the
remaining three panels we show the rSK relation for the other galaxy sub-
samples considered in this work: the ‘control’ sample of non-merger, non-
central starburst galaxies with moderate inclinations in the top right panel,
as well as the mergers and central starbursts in the bottom two panels. The
dashed line in each panel shows the orthogonal distance regression (ODR) fit,
with fit coefficients given explicitly in each panel. The dotted line in the
top right and bottom two panels reproduces the ODR fit to the full sample
(i.e. the dashed line in the top left panel) and is shown for reference. A
strong rSK exists for all of these galaxy sub-samples with scatter of 0.2–0.3
dex, with the smallest scatter measured in the control galaxies and in the
central starbursts, and the largest in the mergers. Thorp et al. (2022)
presented a dedicated study of mergers in ALMaQUEST (although their sample is
defined slightly differently from ours and includes some pre-coalescence
galaxy pairs). One of their conclusions was that some galaxy mergers have high
SFRs driven by an elevated star formation efficiency (SFE = $\Sigma_{\rm
SFR}$/ $\Sigma_{\rm H_{2}}$), which would naturally increase the scatter in
the rSK relation (see also Garay-Solis et al. 2023 for a study of the central
molecular gas content in CALIFA-selected mergers). However, elevated SFEs were
also found by Ellison et al. (2020a) in the sample of ALMaQUEST central
starbursts, and yet the rSK relation remains relatively tight in the lower
left panel of Figure 1, indicating that deviations are typically smaller in
this sample. The central starbursts therefore have broadly self-similar SFEs
(even if they are enhanced), whereas Thorp et al. (2022) found that, in
mergers, enhanced gas fractions could also lead to SFR enhancements.

###  4.2 The Resolved Molecular Gas Main Sequence

In Figure 2 we show the rMGMS for the four galaxy sub-samples; once again, a
strong relationship is seen for each one with the central starbursts
exhibiting the tightest relationship. Indeed, the rMGMS for the central
strburst sample has the smallest scatter of all of the relations and samples
studied in this work. Therefore, despite central starbursts being selected to
be deviant (in their central star formation), their gas fractions are
apparently quite self-similar. In contrast to the rSK relation, the rMGMS for
the control sample exhibits the largest scatter, and the ODR fit yields a
flatter slope than for the other samples. The precise details of the relation
in a given sample is likely to be driven by variations in the individual
galaxies (e.g. Ellison et al. 2021a; Pessa et al. 2022). Overall though, we
again find a scatter that is 0.2 - 0.3 dex across the various sub-samples,
consistent with values in previous works, not only for ALMaQUEST (Lin et al.
2019; Ellison et al. 2021a), but also for other galaxy samples studied at kpc-
scales (e.g. Pessa et al. 2021; Sánchez et al. 2021; Casasola et al. 2022).

###  4.3 The Resolved Star Forming Main Sequence

Figure 3 completes the trio of the standard star formation scaling relations
by showing the rSFMS. As found in previous studies (e.g. Lin et al. 2019;
Morselli et al. 2020; Pessa et al. 2021) the scatter is larger than the
previous two relations, up to 0.34 dex. Indeed, it has been suggested that the
rSFMS (and, by extension, its global counterpart) is simply a by-product of
other physical correlations (e.g. Lin et al. 2019; Ellison et al. 2021a; Baker
et al. 2022, 2023). Once again though, we note the relative tightness of the
rSFMS exhibited by the central starbursts, only 0.21 dex in the ODR fit. We
note also the wide range of best fit slopes obtained for the rSFMS amongst the
different samples, ranging between 0.65 and 1.55. These results serve to
underline the caution required when comparing the scaling relations between
different works; even when a consistent fitting method is used, the details of
the galaxy sample can significantly impact the best fit relation.

###  4.4 The Resolved Dynamical Equilibrium Pressure Relation

Figure 4: The resolved dynamical equilibrium pressure relation for the four
galaxy samples defined in this work. In each panel the ODR fit is shown by the
dashed line. The red dot-dashed line shows the fit to the TIGRESS simulation
of Ostriker & Kim (2022). The dotted line in the upper left panel shows the
broken power law fit described in Equations 5 and 6. In the other three panels
the dotted line shows the single power law fit to the full star forming spaxel
sample.

Having reviewed the three principal star formation scaling relations
previously discussed in the literature and their characteristics in the
ALMaQUEST sample, we now turn to the rPDE relation which has been assessed in
both the PHANGS (Sun et al. 2020a, 2023) and EDGE-CALIFA surveys (Barrera-
Ballesteros et al. 2021a), as well as in the nearby KINGFISH sample (Herrera-
Camus et al. 2017). Each of these surveys has their own benefits and
limitations. PHANGS has an order of magnitude better physical resolution than
either EDGE-CALIFA or ALMaQUEST, but companion IFU data exists for only 19
galaxies (e.g. Groves et al. 2023). The EDGE-CALIFA sample is the largest of
the three (126 galaxies in the original sample; Bolatto et al. 2017), but is
limited almost exclusively to galaxies close to the SFMS. The KINGFISH sample
had additional [CII] measurements that allowed a complementary analysis of the
thermal pressure. The niche of ALMaQUEST is its dynamic range in galactic
properties, both in terms of a dedicated merger component, as well as
populations of both starbursts and green valley galaxies.

ISM pressure can be evaluated in numerous different ways, but recent papers
have favoured $P_{DE}$ which evaluates the mid-plane dynamical equilibrium
pressure in the galactic disk accounting for contributions from both gas and
stars. The general framework for computing the kpc-scale dynamical equilibrium
pressure is well established in the literature (e.g. Ostriker et al. 2010; Kim
et al. 2011; Gallagher et al. 2018; Fisher et al. 2019; Schruba et al. 2019),
where the gas and stars in the galactic disk are modelled as isothermal fluids
in a plane-parallel geometry. Specifically, we follow the implementation of
Sun et al. (2020a) who express $P_{DE}$ as

$P_{DE}=\frac{\pi
G}{2}\Sigma_{gas}^{2}+\Sigma_{gas}\sqrt{2G\rho_{\star}}\sigma_{gas,z}.$ (2)

The first term represents the weight of the ISM due to self gravity and the
second term is the weight of the ISM due to stellar gravity. $\Sigma_{gas}$ is
the total gas surface density, i.e. the sum of the atomic and molecular
components, where $\Sigma_{gas}$ = $\Sigma_{\rm H_{2}}$ \+ $\Sigma_{HI}$.
$\rho_{\star}$ and $\sigma_{gas,z}$ are the mid-plane stellar mass volume
density and the gas velocity dispersion perpendicular to the disk,
respectively. All three of these terms require some assumptions for our
dataset.

In the absence of a measurement of $\Sigma_{HI}$, we assume a fixed value of
$\Sigma_{HI}$ = 7 M⊙ pc-2, which is a typical value in galactic disks (e.g.
Bigiel et al. 2008). For most of the spaxels in our sample the molecular gas
surface density is considerably higher than this (e.g. Figure 1), so that the
precise value of $\Sigma_{HI}$ for our sample is not expected to have a
significant impact on our result. Indeed, even using a value as large as
$\Sigma_{HI}$ = 9 M⊙ pc-2 (which is the value at which the atomic gas surface
density is found to saturate in nearby star-forming galaxies; Bigiel et al.
2008) does not strongly impact our results. Nonetheless, we return to the
assumption of a fixed $\Sigma_{HI}$ in the Discussion.

In the absence of a robust measurement of the vertical gas velocity disperson,
we follow Barrera-Ballesteros et al. (2021a) and assume a fixed value of
$\sigma_{gas,z}$ = 11 km s-1 (see also Blitz & Rosolowsky 2004, 2006; Leroy et
al. 2008; Ostriker et al. 2010) which is a fairly universal value found for
normal star-forming galaxies in the local universe (Kennicutt & Evans 2012;
Caldu-Primo et al. 2013). The assumption of a fixed $\sigma_{gas,z}$ was
assessed in Sun et al. (2020a) to lead to a small over-estimate in $P_{DE}$,
but with a scatter that was generally within 0.2 dex. Again, we will return to
the assumption of a fixed $\sigma_{gas,z}$ in the Discussion.

In order to estimate $\rho_{\star}$, we again follow the previous works of
Blitz & Rosolowsky (2006), Leroy et al. (2008), Ostriker et al. (2010) and Sun
et al. (2020a), and calculate the mid-plane stellar mass density as:

$\rho_{\star}=\frac{\Sigma_{\star}}{4H_{\star}}=\frac{\Sigma_{\star}}{0.54R_{\star}}$
(3)

where $R_{\star}$ and $H_{\star}$ are the disk scale length and height
respectively. The first step in Equation 3 assumes an isothermal density
profile in the vertical direction. The second step assumes a fixed disk
flattening of $R_{\star}/H_{\star}$=7.3. We determine $R_{\star}$ by taking
the half-light (effective) radius ($R_{50}$) from the NASA Sloan Atlas (NSA)
and then convert to the scale length using

$R_{\star}=R_{50}/1.68.$ (4)

The above formalism, although widely adopted in previous literature, ignores
locally enhanced gravity due to small scale structures. Clumpy sub-structure
contributes additional weight to the mid-plane pressure compared to the
assumption (used here) of a smooth disk. This effect is well demonstrated in
the analysis of PHANGS data at various resolutions, where Sun et al. (2020a)
find that the over-pressurization of the disk (as inferred by comparing
$P_{DE}$ to the turbulent pressure) depends on the physical resolution at
which the terms are evaluated. Since molecular gas is expected to be clumpy on
scales below the resolution of our data, the impact of sub-structure may lead
to an under-estimate of $P_{DE}$ in the ALMaQUEST data.

With these caveats and assumptions in mind (which we will return to with a
more extensive discussion in Section 5.3), in Figure 4 we present the rPDE
relation for the extended ALMaQUEST sample. The figure format follows that of
Figures 1 – 3, with all star-forming spaxels presented in the top left panel
and spaxels in the control, central starburst and merger samples in the
remaining three panels. In addition to the ODR fit (black dashed line), we
also show (red dot-dashed line) the theoretical prediction of the PRFM model
of Ostriker & Kim (2022), derived by fitting to the output of the high
resolution TIGRESS simulation (their Equation 26c).

Figure 4 demonstrates that there exists a tight relationship between $P_{DE}$
and $\Sigma_{\rm SFR}$ in the ALMaQUEST data with a scatter whose magnitude of
$\sim$ 0.2 – 0.3 dex is broadly consistent with that of the other three star
formation scaling relations. The rPDE relation also shares the same general
characteristic of exhibiting the greatest scatter in the merger sample and the
least in the central starburst sample. This is perhaps not surprising, since
the calculation of $P_{DE}$ derives from $\Sigma_{\star}$ and $\Sigma_{\rm
H_{2}}$ that appear individually in the rKS relation, the rMGMS and the rSFMS.
We will return to a quantitative comparison of the four relations in the next
section.

Figure 5: The rPDE relation for all star-forming spaxels in the extended
ALMaQUEST dataset (the same data as shown in the top left panel of Figure 4;
green 2D histogram) shown with data from the EDGE-CALIFA survey (Barrera-
Ballesteros et al. 2021a; blue crosses) and PHANGS (Sun et al. 2020; purple
points). The Ostriker & Kim (2022) PRFM theory prediction is shown with the
red dot-dashed line. All three observational datasets are in broad agreement
with one another and all of them fall systematically below the PRFM model at
high pressures.

In terms of comparison to the PRFM formalism of Ostriker & Kim (2022), Figure
4 shows that there is generally broad agreement with data; the red dot-dashed
line (representing the model) typically passes through the data and (with the
exception of the central starburst sample, which we return to below) the best
fit relation is in reasonable agreement with the theoretical one.

However, a closer scrutiny of Figure 4 reveals some disparities between the
PRFM model and the ALMaQUEST data. For example, for the complete sample of
19,999 star-forming spaxels (top left panel of Figure 4) it can be seen that
the data do not follow a simple linear relation. Instead, there is a break at
log $P_{DE}/k_{B}>4.5$ K cm-3 beyond which the relationship flattens. Such a
turnover is not predicted by the Ostriker & Kim (2022) model despite the fact
that the TIGRESS simulations include the full range of pressures and star
formation rates sampled by our data. The flattening at high $P_{DE}$ can also
be seen in the control sample (top right panel of Figure 4). Finally, the
central starbursts also show a poorer match to the PRFM model; at low values
of $P_{DE}$ the star formation rate surface densities are systematically
higher than predicted by the model. Conversely, at high $P_{DE}$, $\Sigma_{\rm
SFR}$ seems to fall short of the theory. These two effects lead to a rPDE
relation for the central starbursts that is significantly flatter than
predicted by Ostriker & Kim (2022). In Section 5.3.4 we discuss some of the
possible caveats in our measurements, although we will ultimately fail to
identify any obvious culprit in the treatment of the data, or calculation of
$P_{DE}$ that lead to this systematic disagreement.

Given the apparent turnover in the rPDE relation at log $P_{DE}/k_{B}\sim 4.5$
K cm-3 we experimented with fitting a broken power law to the full star-
forming spaxel sample, leaving both the exponents and the break point as free
parameters. The resulting fits are

$\log\Sigma_{SFR}=-7.09+1.12\log(P_{DE}/k_{B})$ (5)

and

$\log\Sigma_{SFR}=-4.87+0.63\log(P_{DE}/k_{B})$ (6)

for values of log $P_{DE}/k_{B}$ below and above 4.528 K cm-3 respectively.
The broken power law is shown as a dotted line in the top left panel of Figure
4. Despite the visual improvement when using a double power law, the residuals
of the best fit broken power law are only 0.01 dex smaller than the single
power law fit, so statistically a two component description does not offer an
improvement over our original fit. However, in the next section, we look more
closely at the data that contribute to the rPDE relation in different regimes,
which reveals distinct behaviour that depends on SFR.

##  5 Discussion

Building on earlier work (Ostriker et al. 2010; Ostriker & Shetty 2011; Shetty
& Ostriker 2012; Kim et al. 2013), Ostriker & Kim (2022) have presented the
pressure-regulated feedback-modulated theory of star formation as a framework
to relate both the availability of gas, as well as its physical state, to the
production of stars. By running a series of high resolution, multi-phase,
magnetohydrodynamic simulations, Ostriker & Kim (2022) predict that dynamical
equilibrium pressure will scale with an approximately unity slope with the
surface density of SFR. The predicted scaling relation (referred to herein as
the rPDE relation) showed good agreement with several previous observational
studies (e.g. Leroy et al. 2008; Barrera-Ballesteros et al. 2021a; Sun et al.
2020a). Moreover, Barrera-Ballesteros et al. (2021a) found that the
relationship between hydrostatic pressure (a slight modification to the
dynamical equilibrium pressure, but one that can be considered broadly
equivalent) and $\Sigma_{\rm SFR}$ was largely robust to variations in galaxy
properties (e.g. morphology) in the EDGE-CALIFA sample. The data presented in
these papers support the PRFM model’s fundamental tenet of a balance between
star formation-driven feedback and ISM pressure that is widely applicable in
local star-forming galaxies.

In contrast to these previous works, Figure 4 shows that, for the more diverse
galaxy sample in the extended ALMaQUEST sample, the PRFM model is an
incomplete representation of the dataset as a whole, and particularly so for
the central starbursts. Hints of a similar departure between theory and
observations, which manifests as values of low $\Sigma_{\rm SFR}$ at high
$P_{DE}$, can in fact be glimpsed in other datasets. For example, the rPDE
relation in the EDGE-CALIFA dataset, although dominated by spaxels with lower
$P_{DE}$ values than in ALMaQUEST, does in fact show a mild curvature (e.g
Fig. 9 in Barrera-Ballesteros et al. 2021a and Fig. 15 in Ostriker & Kim
2022). This is demonstrated in Figure 5 where we overplot the EDGE-CALIFA
dataset (blue crosses; Barrera-Ballesteros et al. 2021a) as well as the PHANGS
data (purple points, taken from Sun et al. 2020a) on top of the full set of
star-forming spaxels in the extended ALMaQUEST sample (green scale). Although
these datasets have different selection functions and sample the gas at
different physical scales, it can be seen that, over the majority of the range
of pressures sampled by ALMaQUEST, all three surveys are in broad agreement
with one another. It can also be seen that, although neither the PHANGS nor
EDGE-CALIFA datasets have extensive sampling of the high $P_{DE}$ regime,
these samples fall under the Ostriker & Kim (2022) prediction in the same way
as seen for ALMaQUEST when log $P_{DE}/k_{B}>5.0$ K cm-3. Significant
deviations from both the theoretical prediction and low $z$ field galaxy
samples have also been reported by Fisher et al. (2019) who studied highly
turbulent star-forming galaxies in the DYNAMO sample. Taken together, these
results raise the question of how universal the rPDE relation (and by
extension, its underpinning PRFM theory) is. In the following sub-sections we
discuss this point, as well as possible caveats to our data.

Figure 6: The rPDE relation for galaxies in three bins of $\Delta$ SFR. The
red dot-dashed line shows the prediction from the Ostriker & Kim (2022) PRFM
model. Panels from bottom to top show star-forming spaxels located in galaxies
with a global SFR that is at least a factor of two below the SFMS, within a
factor of two of the SFMS, or at least a factor of two above the SFMS.

###  5.1 Is the rPDE relation universal?

In order to further investigate whether (as indicated by the overall flattened
rPDE relation seen in central starbursts) it is the presence of highly star-
forming galaxies in our sample that leads to a departure from the theoretical
expectation of the PRFM formalism of Ostriker & Kim (2022), we separate the
extended ALMaQUEST sample into three bins of SFR. Specifically, we calculate
an offset from the global SFMS (i.e. using total stellar masses and star
formation rates) on a galaxy-by-galaxy basis. This is achieved by comparing a
given galaxy’s SFR to a control sample matched within 0.1 dex in total stellar
mass, within 0.1 dex of 5th nearest neighbour density and within 0.005 in
redshift, such that $\Delta$SFR captures (in log units) the excess, or
deficit, of star formation compared to the ‘norm’ for a given stellar mass,
environment and redshift (see Ellison et al. 2018 for more details).

In Figure 6 we show the rPDE relation for galaxies in three bins of
$\Delta$SFR, representing galaxies that are either at least a factor of two
below the SFMS (bottom panel), within a factor of two of the SFMS (middle
panel) or at least a factor of two above the SFMS (top panel). Looking first
at the ‘normal’ star-forming galaxies that lie within a factor of two of the
SFMS (Figure 6, middle panel), the spaxels in these galaxies are well
described by a linear slope (as determined from the ODR fit), in excellent
agreement with the Ostriker & Kim (2022) prediction. We conclude that for
typical star-forming galaxies in the extended ALMaQUEST sample the PRFM model
agrees with the data, consistent with previous studies of normal star-forming
galaxies at low redshift (e.g. Herrera-Camus et al. 2017; Sun et al. 2020a,
2023; Barrera-Ballesteros et al. 2021a).

However, significant deviations from the model are seen for ALMaQUEST galaxies
that are either below (Figure 6, bottom panel) or above (Figure 6, top panel)
the SFMS. For galaxies located below the global SFMS (Figure 6, bottom panel),
we see that the $\Sigma_{\rm SFR}$ values lie below the PRFM model for all
$P_{DE}$ (although the range is limited to values of log $P_{DE}/k_{B}<5.0$ K
cm-3). In the context of the PRFM model, this suggests that there will be
insufficient feedback from stellar processes to balance the pressure in the
disk. However, there are relatively few spaxels in the $\Delta$SFR$<-$0.3 dex
bin and they hence contribute relatively little to the full sample show in the
top left panel of Figure 4.

On the other hand, over half of the 19,999 spaxels in the full star-forming
sample are located in galaxies with $\Delta$SFR$>$+0.3 dex. Half of the
galaxies in this sample are central starbursts and one is a merger (i.e. as
defined in Section 2), with several others exhibiting possible disturbances
that were sufficiently ambiguous to not be classified as mergers. However, the
majority of the $\Delta$SFR$>$+0.3 dex sample do not show signs of
interactions. Containing over 11,000 spaxels, the $\Delta$SFR$>$+0.3 dex
sample therefore contributes significantly to the deviation between the data
and the PRFM model in the top left panel of Figure 4.

From the top panel of Figure 6 we can see that the model and the data diverge
in two distinct ways in different regimes. First, at log $P_{DE}/k_{B}<5.0$ K
cm-3 the $\Sigma_{\rm SFR}$ values are significantly higher than the model
would predict. Conversely, at higher values of $P_{DE}$, the observed
$\Sigma_{\rm SFR}$ falls below the model prediction. Combined, these effects
lead to a gradient in the rPDE relation that is significantly more shallow in
the high SFR sample than predicted by the model. A hint of these low
$\Sigma_{\rm SFR}$ values at high $P_{DE}$ is even present in the ‘normal’
star forming sample (Figure 6, middle panel), as well as in the EDGE-CALIFA
sample (Barrera-Ballesteros et al. 2021a and our Figure 5), but it is only
with the significant sample of highly star-forming galaxies in ALMaQUEST that
the signal becomes very clear.

To further dissect the dependence of the rPDE relation on $\Delta$SFR in finer
detail than the three bins shown in Figure 6, for each galaxy we compute the
median spaxel offset from the Ostriker & Kim (2022) relation. Figure 7 shows
that there is indeed an anti-correlation between the offset between the data
and the PRFM prediction and $\Delta$ SFR (Pearson correlation test results are
given in the lower left of the panel), indicating that whilst the pressure
regulated feedback modulated formalism is a good approximation for main
sequence galaxies, it breaks down for galaxies with more extreme star
formation rates. Figure 7 also therefore indirectly shows that a single rPDE
relation is not representative of the entire galaxy sample.

Figure 7: For all star-forming spaxels in a given galaxy, we compute the
median offset of the measured $P_{DE}$ compared to the prediction of Ostriker
& Kim (2022) and plot the result as a function of $\Delta$ SFR. The anti-
correlation (Pearson correlation test $p$ and $\rho$ values reported in the
lower left of the figure) supports the results in Figure 6 that whilst the
pressure regulated feedback modulated formalism is a good description of
normal star-forming galaxies, it breaks down for galaxies above and below the
SFMS.  Figure 8: The rPDE relation for a random selection of 30 galaxies in
the extended ALMaQUEST sample. The MaNGA plate-IFU of each galaxy is noted in
the top left corner. The grey scale background shows the full sample of
$\sim$20,000 star forming spaxels as a visual reference. The red dot-dashed
line shows the prediction from the TIGRESS simulation of Ostriker & Kim
(2022).

The non-universality of the rPDE relation can be seen most effectively by
examining individual galaxies. In Figure 8 we show a selection of galaxies
from the extended ALMaQUEST sample; these are chosen randomly, but with the
(arbitrary, for visual presentation purposes) requirement that the galaxy have
at least 100 star-forming spaxels. Although this minimum spaxel count means
that green valley galaxies are under-represented in this montage, it does not
affect the qualitative point that different galaxies exhibit different rPDE
relations. In each panel we also show the theoretical relation from Ostriker &
Kim (2022) as a red dot-dashed line and the ensemble of all star-forming
spaxels, reproduced from the top left panel of Figure 4 as background
greyscale for reference. As has been previously shown for the other three star
formation scaling relations (rSK, rMGMS and rSFMS; Ellison et al. 2021a; Pessa
et al. 2021, 2022; Casasola et al. 2022) the rPDE for the galaxies in
ALMaQUEST can lie significantly above or below both the ensemble and the
linear theoretical relation. Considerable differences in slope are also
observed, although the flattening at high values of $P_{DE}$ is a common
feature. We conclude that the rPDE relation is not universal between galaxies.
Nonetheless, it may still be the case that there is less variation in the rPDE
relation than for other star formation scaling relations (rSK, rSFMS and
rMGMS). We turn to this point in the next sub-section.

Figure 9: The RMS scatter around the ODR fit derived for each of four
different sub-samples (distinguished by symbols) and four different scaling
relations (distinguished by colours). All of the relations have scatters
within $\sim$ 0.2 – 0.3 dex (depending on the galaxy sub-sample), with the
rPDE exhibiting no tighter scatter than the others.  Figure 10: The relative
importance of six variables in predicting $\Sigma_{\rm SFR}$ as determined
from a random forest regression analysis. $P_{DE}$ is a far superior predictor
of $\Sigma_{\rm SFR}$ than either $\Sigma_{\star}$ or $\Sigma_{\rm H_{2}}$,
and similarly better than other combinations of these variables.

###  5.2 Is the rPDE relation the most fundamental star formation scaling
relation?

Scaling relations are often used as a way to motivate the understanding of
physical processes. However, we are well-drilled in the caveat that
correlation does not imply causation, stimulating considerable effort in
disentangling fundamental correlations from those that arise as by-products.
This has been achieved with a variety of statistical methods, including
classical analyses such as the assessment of scatter and correlation,
dependence on additional variables and partial correlation coefficients, to
more sophisticated approaches that attempt to rank the relative importance of
variables to a target (e.g. Ellison et al. 2008b; Teimoorinia et al. 2016; Dey
et al. 2019; Bluck et al. 2020, 2022; Ellison et al. 2020a,b; Baker & Maiolino
2023).

Specifically, there has been active discussion in the literature concerning
the star formation scaling relations studied in this paper. Lin et al. (2019)
were the first to suggest that the rSFMS is not fundamental, but rather a
result of combining the rMGMS and rSK. Lin et al. (2019) propose that the
rMGMS and the rSK relation set-up a 3-dimensional correlation that projects
onto the plane of $\Sigma_{\star}$– $\Sigma_{\rm SFR}$ to produce the rSFMS,
despite there being no direct causal connection between these two variables.
Several other authors have since supported these conclusions, by replicating
the 3-dimensional structure in other datasets (e.g. Sánchez et al. 2021)
finding larger scatter in the rSFMS compared with the other relations (Ellison
et al. 2021a; Morselli et al. 2020; Pessa et al. 2021) and no statistical
evidence for a relation between $\Sigma_{\star}$ and $\Sigma_{\rm SFR}$ once
the rMGMS and rSK are accounted for (Baker et al. 2022).

The PRFM theory moves beyond these simple two-variable scaling relations by
predicting that the ISM pressure captures the fundamental regulator of star
formation rate. This conclusion was supported by Barrera-Ballesteros et al.
(2021a) in their study of EDGE-CALIFA data and they conclude ‘Our results also
suggest that hydrostatic pressure is the main parameter that modulates star
formation at kpc scales, rather than individual components of the baryonic
mass.’ In order to test this conclusion with the extended ALMaQUEST data, in
Figure 9 we plot the RMS scatter of the data around the ODR fit for the four
different scaling relations (shown in distinct colours in vertical groupings)
and in four different galaxy sub-samples (shown as different symbols). We find
that the rPDE relation shows no less scatter (around the linear fit) than the
other three relations. Moreover, as expected based on the considerable galaxy
to galaxy variation seen in Figure 8, the scatter is variable depending on the
sample, because each one (even the control sample) has a range of star
formation properties.

Although a comparison of scatters has been previously used to assess the
‘fundamental’ nature of scaling relations (e.g. Lin et al. 2019; Morselli et
al. 2020; Ellison et al. 2021a; Pessa et al. 2021), the relationship between
$P_{DE}$ and $\Sigma_{\rm SFR}$ is clearly not linear. This non-linearity,
which manifests as a flattening of the relation at high pressures (which tend
to be associated with highly star-forming galaxies) was not obvious in
previous studies which were limited to main sequence galaxies (e.g. Herrera-
Camus et al. 2017; Sun et al. 2020a; Barrera-Ballesteros et al. 2021a). Using
the scatter around a linear fit is therefore unlikely to be the optimal way to
assess the fundamental nature of the rPDE relation, motivating a more
sophisticated treatment of the data.

We therefore turn to a random forest analysis, which is powerful at extracting
non-linear dependences in data. A random forest consists of a series of
decision trees which can be used to assess the relative importance of input
variables in determining a given target variable. Our approach follows closely
the analyses presented in our previous works (e.g. Baker et al. 2022, 2023);
in particular, we refer the curious reader to Appendix B of Bluck et al.
(2022) for a detailed explanation of the random forest methodology and
mathematical formulation.

The target variable of our random forest is $\Sigma_{\rm SFR}$. We first
assess the parameters that represent the scaling relations examined in this
paper, namely $\Sigma_{\rm H_{2}}$ (i.e. testing the rKS relation),
$\Sigma_{\star}$ (i.e. testing the rSFMS) and $P_{DE}$ (i.e. testing the rPDE
relation). This random forest is therefore an extension of the work presented
by Baker et al. (2022) who assessed only the relative importances of
$\Sigma_{\rm H_{2}}$ and $\Sigma_{\star}$, finding that the former was more
important for predicting $\Sigma_{\rm SFR}$. The relative performance of
$P_{DE}$, $\Sigma_{\rm H_{2}}$ and $\Sigma_{\star}$ and for predicting
$\Sigma_{\rm SFR}$ is shown in the first three bars of Figure 10, where the
error bars represent the variance in 100 independent realizations of the
training, validation and testing data. The mean squared error (MSE) is
reported in the top right of the figure for both the training and testing
data. From the first three bars of Figure 10 it is clear that the $P_{DE}$ is
much more important than either $\Sigma_{\rm H_{2}}$ or $\Sigma_{\star}$ for
determining $\Sigma_{\rm SFR}$, and that this result is highly significant and
stable. We emphasize that the results presented in Figure 10 only rank the
relative importances of the features included in our test set, and by design,
add up to a total of one. Therefore, these relative importances are not
absolute values, and their quantitative values are only relevent for the fixed
feature set included in our test. For example, the ratio of the relative
importances of any two variables shown in Figure 10 would change if more (or
less, or different) features were included.

In order to assess whether the successful performance of $P_{DE}$ as a
predictor of $\Sigma_{\rm SFR}$ is a ‘trivial’ result, in the sense that
$P_{DE}$ is itself a combination of $\Sigma_{\rm H_{2}}$ and $\Sigma_{\star}$,
we include in the random forest two additional variables that are also
combinations of $\Sigma_{\star}$ and $\Sigma_{\rm H_{2}}$. The first is a
modified version of $P_{DE}$, in that it combines $\Sigma_{\rm H_{2}}$ and
$\Sigma_{\star}$ in the same relative proportions as $P_{DE}$ (Equation 2),
but without the addition of $\Sigma_{\rm HI}$ (which we recall we have set to
be a constant), velocity dispersion nor information on $R_{50}$ and without
any coefficients (such as physical constants). That is, we compute
$\Sigma_{\rm H_{2}}$2 \+ $\Sigma_{\rm H_{2}}$ $\Sigma_{\star}$0.5. The other
extra variable we include is also a combination of $\Sigma_{\star}$ and
$\Sigma_{\rm H_{2}}$ that represents the ‘extended’ Kennicutt-Schmidt relation
of Shi et al. (2011). These authors showed that $\Sigma_{\rm H_{2}}$
$\Sigma_{\star}$0.5 correlated more tightly with $\Sigma_{\rm SFR}$ than
$\Sigma_{\rm H_{2}}$ alone. These two additional variables are shown in the
4th and 5th bars in Figure 10; for reference, a random variable is shown in
the final bar. All three of these additional variables have a very small
relative importance compared with $P_{DE}$.

Taken together, the results in Figure 10 show that $P_{DE}$ represents a
physically meaningful combination of $\Sigma_{\star}$ and $\Sigma_{\rm H_{2}}$
for the prediction of $\Sigma_{\rm SFR}$. Neither the extended KS law, nor a
modification of the $P_{DE}$ equation (which also combines $\Sigma_{\star}$
and $\Sigma_{\rm H_{2}}$) perform anywhere near as well as $P_{DE}$. Indeed,
after PDE it is the single variable $\Sigma_{\rm H_{2}}$ that has the next
highest relative importance (but it is in a very distant second place). These
results reflect the ability of the random forest analysis to extract non-
linear relations. Our random forest therefore demonstrates that $P_{DE}$ is a
far superior predictor of $\Sigma_{\rm SFR}$ than $\Sigma_{\rm H_{2}}$ or
$\Sigma_{\star}$ alone, a result that was not seen when assessing the scatter
around the linear relations alone (Figure 9).

It is also important to recognize that, whilst there are a number of
uncertainties associated with our calculation of $P_{DE}$ (which we explore in
detail in the next section) any improvement in the accuracy of the measurement
of the pressure will only increase the dominance shown in Figure 10. Put
another way, $P_{DE}$ is the most predictive parameter (of those tested) for
$\Sigma_{\rm SFR}$, despite some uncertainties in its derivation. As a final
comment, we note that our analysis says nothing about whether $P_{DE}$ is an
optimal combination of $\Sigma_{\star}$ and $\Sigma_{\rm H_{2}}$ (or any other
additional variables), only that it is better than the individual variables
(or the other combinations) we have tested.

###  5.3 Uncertainties in the measurement of $P_{DE}$

We have so far shown that dynamical equilibrium pressure is the more
predictive of the star formation rate surface density than either $\Sigma_{\rm
H_{2}}$ or $\Sigma_{\star}$ (Figure 10), but that the relationship between
$P_{DE}$ and $\Sigma_{\rm SFR}$ shows significant deviations from the
prediction of the PRFM model (Figures 4 – 8) . In this sub-section, we
consider whether the assumptions used in our calculation of $P_{DE}$ might
explain the observed discrepancy between the PRFM model and the extended
ALMaQUEST data, with a particular focus on the highly star-forming galaxies in
the sample shown in the top panel of Figure 6. We remind the reader that the
nature of the disagreement is that the slope of the rPDE relation is much
flatter in the data than predicted by the model, such that $\Sigma_{\rm SFR}$
is too high for its $P_{DE}$ at values of log $P_{DE}/k_{B}<5$ K cm-3 and
$\Sigma_{\rm SFR}$ is lower than the model at higher pressures.

####  5.3.1 Assumption of a fixed $\Sigma_{\rm HI}$

First, we consider our assumption of a constant value of $\Sigma_{\rm HI}$ = 7
M⊙ pc-2 for all spaxels. At high values of $P_{DE}$ we do not expect our
choice of $\Sigma_{\rm HI}$ to be a significant issue. In order to bring the
data into agreement with the model in the high $P_{DE}$ regime, the pressure
needs to be reduced from its current value. This requires the atomic surface
density to be lower than our currently assumed value of $\Sigma_{\rm HI}$ = 7
M⊙ pc-2. However, as noted earlier, the typical H2 surface densities for
ALMaQUEST spaxels are at least $\Sigma_{\rm H_{2}}$ = 10 M⊙ pc-2, meaning that
our assumed value of $\Sigma_{\rm HI}$ does not significantly contribute to
$\Sigma_{gas}$. Indeed, in the extended ALMaQUEST sample, $\Sigma_{gas}$ will
be dominated by $\Sigma_{\rm H_{2}}$ for the majority of our spaxels for any
reasonable choice of $\Sigma_{\rm HI}$ below the saturation threshold
($\Sigma_{\rm HI}$ $\sim$ 9 M⊙ pc-2) observed in nearby disks (Bigiel et al.
2008).

However, at low $P_{DE}$ values, where the pressure needs to be increased in
order to bring it into agreement with the data, it is reasonable to
contemplate whether our assumed value of $\Sigma_{\rm HI}$ might be too low.
We experimented with changes in the adopted value of $\Sigma_{\rm HI}$, as
well as including a randomized element to emulate a range of HI surface
densities (e.g. following Barrera-Ballesteros et al. 2021a). For values of
$\Sigma_{\rm HI}$ up to the locally observed saturation threshold, there is
little impact and the data remain inconsistent with the model. We have to
increase the atomic contribution to $\Sigma_{\rm HI}$ $\sim$ 14 M⊙ pc-2 before
there is some reasonable agreement between the data and the model at low
$P_{DE}$. However, such high values of $\Sigma_{\rm HI}$ simultaneously
exacerbate the discrepancy between the model and the data at high pressures,
and so are not a good solution for the data as a whole. Moreover, there is
little evidence to empirically (or theoretically) motivate such a high value
of $\Sigma_{\rm HI}$ for our sample. Despite the more intense UV radiation
fields that come with high SFRs (that might be expected to dissociate H2),
this is offset by higher gas volume and surface densities, which shield
against far ultra-violent radiation and promote H2 formation. Consequently,
models and observations alike find that $\Sigma_{\rm HI}$ is predominantly
sensitive to metallicity, rather than SFR, with $\Sigma_{\rm HI}$ $>$10 M⊙
pc-2 only expected in regions of much lower metallicity than in our sample
(Krumholz, McKee & Tumlinson 2009; Fumagalli, Krumholz & Hunt 2010; Schruba,
Bialy & Sternberg 2018). Indeed, Bigiel et al. (2008) find that, even in the
centres of galaxies with locally enhanced $\Sigma_{\rm SFR}$, the $\Sigma_{\rm
HI}$ is rarely in excess of $\sim$ 8 M⊙ pc-2.

As a final comment on the choice of a fixed $\Sigma_{\rm HI}$, we note that
Barrera-Ballesteros et al. (2021a) made a similar assumption for the EDGE-
CALIFA sample, although those authors additionally added a random component to
the value of $\Sigma_{\rm HI}$ = 7 M⊙ pc-2 to emulate the range of observed
values. Despite this similar approach, Barrera-Ballesteros et al. (2021a) find
that their data do agree with the model at low pressures, in much the same way
that the ‘normal’ star-forming galaxies in ALMaQUEST do (middle panel of
Figure 6). Overall, we conclude that the choice of a fixed $\Sigma_{\rm HI}$
is unlikely to be the reason that our data disagree with the PRFM model.

####  5.3.2 Assumption of a fixed CO-to-H2 conversion factor

Next, we consider the impact of using a fixed CO conversion factor. This is
almost certainly an incorrect assumption, as $\alpha_{CO}$ is known to vary as
a function ISM conditions, including density, galactocentric radius and
metallicity (Narayanan et al. 2012; Bolatto 2013; Sandstrom et al. 2013;
Accurso et al. 2017; Hunt et al. 2020; Gong et al. 2020). However, before
embarking on an exploration of alternative values of $\alpha_{CO}$, we begin
by considering the general impact that this alteration would have on our
results (beyond the change to $P_{DE}$). Adopting a different (or variable)
$\alpha_{CO}$ has the obvious direct result of changing $\Sigma_{\rm H_{2}}$,
which we (and others) have shown has a strong and tight correlation with
$\Sigma_{\star}$ (e.g. Fig 2). Changing the conversion factor will therefore
have a direct impact on our observed rMGMS. As shown in Fig 2, unlike the rPDE
relation, the rMGMS of the extended ALMaQUEST is very similar for central
starburst and control galaxies. Moreover, the slope of the rMGMS is identical
for galaxies with $\Delta$ SFR$>0.3$ dex and those with $-0.3<\Delta$ SFR
$<0.3$ dex, and yet the rPDE relation is very different in these two regimes
(Fig. 6). Although there is a small offset to higher $\Sigma_{\rm H_{2}}$ at
fixed $\Sigma_{\star}$ (i.e. higher gas fractions) in the $\Delta$ SFR$>0.3$
dex sample by 0.1 dex, this offset is modest and higher gas fractions are in
any case frequently observed for galaxies above the main sequence (e.g.
Saintonge et al. 2012, 2016). Any significant reduction in $\alpha_{CO}$ (as
might be more appropriate for a starburst sample, and necessary to bring the
high $P_{DE}$ values into line with the Ostriker & Kim 2022 relation) would
lead to starburst galaxies with suppressed gas fractions, which is not
consistent with either other observations or the fuel requirements for their
high SFRs. In short, the rMGMS gives us a prior hint that significant
reductions to $\alpha_{CO}$ are likely inappropriate for our sample.

Nonetheless, we next embark on a quantitative assessment of alternative values
of $\alpha_{CO}$. First, we re-compute $P_{DE}$ using the metallicity
dependent conversion factor given in Equation 1. Since the majority of the
spaxels in our sample have 8.55 $<$ 12 + log O/H $<$ 8.75, adopting Equation 1
leads to conversion factors slightly higher than our fiducial value of 4.3. As
a result, the recomputed values of $\Sigma_{\rm H_{2}}$ (and hence $P_{DE}$)
are slightly larger than those derived from our original values with fixed
$\alpha_{CO}$, bringing the spaxels with low $P_{DE}$ into marginally better
agreement with the model, but the change is small and the disagreement with
the model persists. In the regime of high $P_{DE}$, where the pressure is too
large compared with the model, the metallicity dependent conversion factor
exacerbates the discrepancy between the model and the data. We also test the
Accurso et al. (2017) conversion prescription which, in addition to a primary
dependence on metallicity, has a mild secondary dependence on galaxy offset
from the main sequence. The addition of this second term has a minimal effect
on our data.

As an alternative to the metallicity dependent conversion factor, we next
consider whether choosing a lower $\alpha_{CO}$ (than our fiducial value of
4.3) could resolve the discrepancy. Values as low as $\alpha_{CO}\sim 1$ (or
even smaller) are seen in some parts of disks as well as in extreme starbursts
(e.g. Bolatto et al. 2013; Sandstrom et al. 2013; Teng et al. 2023).
Artificially lowering the conversion factor by only 50% is sufficient to bring
the high $P_{DE}$ spaxels into line with the model; but the disagreement at
low $P_{DE}$ is then exacerbated. However, it is likely that different values
of $\alpha_{CO}$ are applicable in different galactic regions, such that a
blanket adjustment of $\alpha_{CO}$ is also probably not correct (even for
starbursts). There are several formulations for a variable $\alpha_{CO}$ in
the literature that we can potentially adopt (e.g. Narayanan et al. 2012;
Bolatto et al. 2013; Gong et al. 2020; Teng et al. 2023). We test the
implementation of two of these, namely a dependence of $\alpha_{CO}$ on the
total surface density $\Sigma_{tot}$ = $\Sigma_{\star}$ \+ $\Sigma_{\rm
H_{2}}$ \+ $\Sigma_{\rm HI}$ with index of $-0.5$ (green line in Fig. 12 of
Bolatto et al. 2013) and a dependence on the CO line width (Equation 6 from
Teng et al. 2023). Both of these prescriptions result in values of lower
$P_{DE}$, but to the extent that the starburst data now lie a factor of 2–3 to
the left of the PRFM model (essentially ‘over-correcting’ the original
discrepancy seen between the model and the data).

Finally, we assess the impact of simply removing spaxels with physical
properties that might be indicative of more extreme conversion factors. We try
removing spaxels with line widths larger than 20 km s-1 (as Teng et al. 2023
show that $\alpha_{CO}$ decreases when the line is broad) as measured from the
moment-2 maps, removing spaxels where $\Sigma_{tot}>$ 300 M⊙ pc-2 (Bolatto et
al. 2013’s $\alpha_{CO}$ prescription scales as $\Sigma_{tot}^{-0.5}$) and
removing spaxels within 2 kpc of the galaxy centre (Sandstrom et al. 2013 and
Teng et al. 2023 identify central regions as most deviant from Galactic
conversion factors). The resulting rPDE relation for starburst galaxies after
each of these three purges still shows the same general trend (albeit with
fewer spaxels) as shown in the top panel of Figure 6. We conclude that whilst
alternative values of $\alpha_{CO}$ could be hand-picked to bring any given
spaxel in our sample into alignment with the PRFM model, we have not
identified a physically motivated formalism that can achieve this.

All of the factors considered thus far in our discussion have focussed on
elements that contribute to the calculation of $\Sigma_{\rm gas}$, i.e.
uncertainties in either $\Sigma_{\rm HI}$ or $\Sigma_{\rm H_{2}}$. Although we
have recognized some limitations in our ability to compute $\Sigma_{\rm gas}$
to high accuracy, we have also not found evidence that the uncertainties in
the gas contribution is driving the disagreement between the PRFM model and
the data. We also note that in the calculation of $P_{DE}$ (i.e. Equation 2)
the first term (the contribution to the pressure from gas self-gravity) is
almost always smaller (by a factor of two, on average) than the second
(stellar) term. Therefore, modest changes to either $\Sigma_{\rm HI}$ or
$\Sigma_{\rm H_{2}}$ have a minor impact on the calculation of $P_{DE}$.

####  5.3.3 Assumption of a fixed gas velocity dispersion

The next uncertainty we consider is our assumption of a fixed $\sigma_{gas,z}$
= 11 km s-1. Although we can not measure this parameter accurately in our own
data, the assumed value is consistent with the fairly universal value for
normal star-forming disks at low redshift (e.g. Kennicutt & Evans 2012; Caldu-
Primo et al. 2013). Thanks to the availability of higher resolution data, Sun
et al. (2020a) were able to explicitly assess the impact of this choice on
kpc-scale data, finding that it could lead to a small over-estimate in
$P_{DE}$, but with a scatter that was generally within 0.2 dex. Although a
downward correction by 0.2 dex in $P_{DE}$ somewhat relieves the tension
between the data and the model at high pressures, the $P_{DE}$ would need to
be reduced by values closer to 0.5 – 1 dex in order to be fully in line with
the theory. Moreover, a downward correction in $P_{DE}$ acts in the contrary
direction needed to resolve the discrepancy between the data and the model in
the low $P_{DE}$ regime. In order to match the data to the theory in the low
pressure regime, the calculated pressure needs to increase, a result which
might be achieved if our assumed value of $\sigma_{gas,z}$ = 11 km s-1 is an
under-estimate. Since elevated velocity dispersions are indeed expected for
galaxies with high SFRs (Krumholz & Burkhart 2016; Krumholz et al. 2018), we
re-compute $P_{DE}$ using the actual (line of sight) values of the velocity
dispersions as measured from the CO(1-0) line. Although the median value in
the data is $\sim$ 11 km s-1 (which is also the spectral resolution of the
ALMaQUEST data), there is a tail to values as large as 80 km s-1. Of course,
since we have not modelled and removed disk rotation, these velocity
dispersions represent an upper limit to the possible value of
$\sigma_{gas,z}$, but this nonetheless serves our purpose of assessing whether
our assumption of $\sigma_{gas,z}$ = 11 km s-1 is causing the discrepany
between the data and the model at low pressures. We find that using the
measured values of the CO velocity dispersion does not reduce the offset
between the model and theory in high SFR galaxies.

####  5.3.4 Uncertainties in $\Sigma_{\rm SFR}$

Finally, we can also consider whether the discrepancy between the PRFM model
and the ALMaQUEST data might be due to incorrect values of $\Sigma_{\rm SFR}$.
At high values of $P_{DE}$ the discrepancy between the model and the data
could be explained if $\Sigma_{\rm SFR}$ has been under-estimated, for example
due to high extinction. Indeed, it has been suggested that the Balmer
decrement method (although very widely used) might significantly under-
estimate extinction (Inoue et al. 2001; Dopita et al. 2003). However, in
general, the method we have adopted to determine $\Sigma_{\rm SFR}$, which is
based on dust-corrected H$\alpha$ emission, has been found to be a good match
to those determined from the UV and IR in the CALIFA DR2 (Garcia-Benito et al.
(2015) which includes $\sim$ 10 percent mergers in various stages (Catalan-
Torrecilla et al. 2015). A visual inspection of the high SFR galaxies reveals
that they are not obviously dusty or highly inclined. Moreover, high values of
$\Sigma_{\rm SFR}$ do exist in our sample for some galaxies (e.g. Ellison et
al. 2021a), demonstrating that we can identify such regions and note that the
distinctive flattening seen in the rPDE relation (top left panel of Figure 4)
is not seen in the rSK relation (top left panel of Figure 1), indicating that
the raw variables of $\Sigma_{\rm SFR}$ and $\Sigma_{\rm H_{2}}$ are well-
behaved. In terms of a possible over-estimate of $\Sigma_{\rm SFR}$ that might
cause disagreement with the PRFM model at low values of $P_{DE}$ we re-iterate
that we have removed spaxels with contributions from AGN. Therefore, the
measured H$\alpha$ flux should not have significant contributions from non-
stellar sources, indicating that an over-estimate of $\Sigma_{\rm SFR}$ is
unlikely to be the cause of the disagreement between the model and data in the
low $P_{DE}$ regime.

In summary, we have not identified any methodological cause for the
disagreement between the observed rPDE relation in the extended ALMaQUEST
sample and the PRFM model. It appears that, whereas the PRFM model is a good
representation of the data for galaxies on the SFMS (middle panel of Figure
6), there is a persistent tension between the extended ALMaQUEST sample and
the theory for high SFR galaxies (top panel of Figure 6). And yet we have also
found that $P_{DE}$ is the most (of the variables tested) relevant predictor
of star formation in our sample. Taken together, these results indicate that
whilst dynamical equilibrium pressure is a key variable in regulating star
formation, the PRFM model is an incomplete formulation of the relation. In the
final sub-section below, we consider alternative models that might better
represent the full diversity present in the extended ALMaQUEST sample.

Table 3: Symbol definitions used in the Krumholz et al. (2018) star formation model (Equations 7 – 13) and their adopted values, where relevant. Symbol | value | Meaning
---|---|---
$\Omega$ | … | Galaxy angular velocity
$v_{\phi}$ | … | Galaxy rotation curve velocity
$t_{orb}$ | … | Galaxy orbital period
$f_{sf}$ | … | Fraction of ISM in molecular phase
$\rho_{min}$ | … | Minimum midplane density required to produce rotation curve
$\beta$ | 0 | Rotation curve index
$f_{g,Q}$ | 0.5 | Fractional contribution of gas to Toomre $Q$
$f_{g,P}$ | 0.33 | Fractional contribution of gas to mid-plane pressure
$\phi_{mp}$ | 1.4 | Ratio of total pressure to turbulent pressure at midplane
$\epsilon_{ff}$ | 0.015 | Star formation efficiency per free-fall time
$t_{sf,max}$ | 2 Gyr | Maximum star formation timescale

###  5.4 What causes the discrepancy between the PRFM model and the ALMaQUEST
data?

We have found that the extended ALMaQUEST data deviate from the PRFM model
most dramatically for spaxels in highly star-forming galaxies. Hassan,
Ostriker & Kim (in prep) have recently presented a new analytic expression for
$P_{DE}$ specifically designed to account for ISM conditions in starburst
galaxies and/or those with particularly turbulent ISM conditions (e.g. Fisher
et al. 2019). In particular, the new formulation guards against an over-
prediction of the stellar (second) term in Equation 2 if the velocity
dispersion is very large and/or the gas scale height exceeds that of the
stars. We have re-computed $P_{DE}$ using the new formulation of Hassan et al.
(in prep) and find that $P_{DE}$ is typically reduced by $\sim$ 5% compared
with the fiducial calculation using Equation 2. The disagreement between our
highly star-forming galaxies and the PRFM model is therefore not resolved by
using this new formalism.

Having explored a number of possible avenues for reconciliation between the
ALMaQUEST data and the PRFM theory, we have found no obvious culprit amongst
observational assumptions, nor in the most recent version of the PRFM
expression that should be most suitable for our high SFR galaxies. We are
therefore left with the conclusion that, although the PRFM model is a good
representation of the data for galaxies that lie on the SFMS (see also
Barrera-Ballesteros et al. 2021a; Sun et al. 2020a, 2023), the model exhibits
a fundamental disagreement with galaxies in the high (and low) SFR regime. In
the remainder of this section we consider why this might occur.

The PRFM model is predicated on the tenet that feedback from star formation is
the sole source of turbulence in the ISM, and that this in turn balances the
pressure across the galactic disk. Starting with Ostriker & Shetty (2011), the
generations of models that have built on this assumption all ignore (by
design) the larger scale physics of the galaxy. If there are additional
sources of turbulence, the underlying assumptions of the PRFM model are
undermined. Large scale motions of gas within galaxies could readily provide
such additional turbulence, both through on-going gas accretion as well as
radial flows, and there is an abundance of evidence that both of these effects
must be at play in real galaxies. For example, the well-established
observation that galaxies have depletion times $\sim$ 1-2 Gyr (Leroy et al.
2008, 2013; Bigiel et al. 2008, 2011; Saintonge et al. 2011), i.e. much
shorter than the Hubble time, and yet continue to form stars over an extended
period demonstrates that on-going gas accretion/disk fuelling must be
occuring. A similar conclusion is reached from the observation that the HI gas
mass density in damped Lyman $\alpha$ systems (gas-rich galaxies seen in
absorption in quasar spectra) is flat over a large range in redshift, at least
$1<z<4$ (e.g. Zafar et al. 2013; Sánchez-Ramirez et al. 2016), despite high
rates of cosmic star formation in this period. Inflows of gas are responsible
for central starbursts in both models (Barnes & Hernquist 1991; Blumenthal &
Barnes 2018; Moreno et al. 2015, 2021) and observations (Ellison et al. 2008a,
2013; Scudder et al. 2012; Thorp et al. 2019; Bickley et al. 2021; Garay-Solis
et al. 2023) of galaxy mergers, and starbursts in general (not just those that
are merger driven) have preferentially enhanced SFRs in their centres (Ellison
et al. 2018; Medling et al. 2018; Wang et al. 2019). Multiphase outflows are
also an apparently ubiquitous feature of low $z$ galaxies, driven by both AGN
and star formation (Fluetsch et al. 2019; Roberts-Borsani & Saintonge 2019;
Avery et al. 2021). Krumholz et al. (2018) have argued that these large scale
effects can dominate over star formation as the primary source of turbulence.
Indeed, it is in the high SFR regime, where we see the strongest disagreement
between the data and the PRFM model, that we might expect that these large
scale effects to dominate. Likewise, in galaxy mergers where there are not
only radial gas flows but galaxy-wide disruptions, we see that the data are in
relatively poor agreement with the PRFM model (lower right panel in Figure 4).

Although Krumholz et al. (2018) do not make explicit predictions for the rPDE
relation, it is nonetheless possible to extract the expected relationship from
their formalism. Our starting point is Equation 30 of Krumholz et al. (2018)
who express the star formation rate as

$\Sigma_{SFR}=f_{sf}\Sigma_{gas}\frac{\epsilon_{ff}}{t_{ff}}$ (7)

where $f_{sf}$ is the fraction of the total gas ($\Sigma_{\rm gas}$) that is
in the star-forming (molecular) phase and $t_{ff}$ and $\epsilon_{ff}$ are the
free-fall time and star formation rate per free-fall time in this gas.
$\epsilon_{ff}$ is set to be 0.015, consistent with a wide range of
environments and the best fit to the data derived by Krumholz, Dekel & McKee
(2012). The ratio of free-fall time and SFR per free-fall time is parametrized
in Krumholz et al. (2018) as the Toomre timescale (their Equation 31):

$t_{sf,T}=\frac{t_{ff}}{\epsilon_{ff}}=\frac{\pi
Q}{4f_{g,Q}\epsilon_{ff}}\sqrt{\frac{3f_{g,P}\phi_{mp}}{2(1+\beta)}}\frac{1}{\Omega}$
(8)

where the description of the variables and their adopted values are given in
Table 3. For the majority of these variables we have adopted the default
values from Krumholz et al. (2018) with the exception that we set the
fractional contribution of gas to the pressure, $f_{g,P}$, to be 0.33 (instead
of the fiducial value of 0.5 used in Krumholz et al. 2018) since this is the
median value in the extended ALMaQUEST dataset. We also use a fixed value of
$\beta=0$, i.e. a flat rotation curve.

Since the angular velocity at radius $r$ is given by $\Omega=v_{\phi}/r$ and
the orbital period is $t_{orb}=2\pi r/v_{\phi}$, substituting in Equation 8
therefore yields the requisite expression for $t_{sf,T}$ for different
$t_{orb}$:

$t_{sf,T}=\frac{\pi
Q}{4f_{g,Q}\epsilon_{ff}}\sqrt{\frac{3f_{g,P}\phi_{mp}}{2(1+\beta)}}\frac{t_{orb}}{2\pi}.$
(9)

In the Krumholz et al. (2018) model, if $t_{sf,T}$ is shorter than
$t_{sf,max}=$ 2 Gyr then stars form in a continuous medium, otherwise the gas
appears to break up into individual molecular clouds (Bigiel et al. 2008;
Leroy et al. 2008, 2013). In order to capture these two different ‘modes’, the
SFR in Equation 7 can thus be re-written as

$\Sigma_{SFR}=f_{sf}\Sigma_{gas}max(t_{sf,T}^{-1},t_{sf,max}^{-1}).$ (10)

Next we must evaluate $f_{sf}$. This is achieved using the theoretical model
originally laid out by Krumholz, McKee & Tumlinson (2009), with improvements
as presented in McKee & Krumholz (2010) and Krumholz (2013), referred to as
the KMT+ model. These models require input values of a clumping factor
(following Krumholz et al. 2009, we adopt a value of 5) and $\rho_{min}$, the
minimum mid-plane pressure required to produce the rotation curve.
$\rho_{min}$ is computed using equation 51 of Krumholz et al. (2018):

$\rho_{min}=\frac{v_{\phi}^{2}}{4\pi Gr^{2}}(2\beta+1).$ (11)

Since $v_{\phi}/r=2\pi/t_{orb}$, Equation 11 can be re-written as

$\rho_{min}=\left(\frac{2\pi}{t_{orb}}\right)^{2}\frac{(2\beta+1)}{4\pi G}.$
(12)

Finally, we need to express $P_{DE}$ in terms of $\Sigma_{\rm gas}$, which is
Equation 20 in Krumholz et al. (2018):

$P_{DE}=\frac{\pi G}{2f_{g,P}}\Sigma_{gas}^{2}.$ (13)

$f_{g,P}$ packages the contributions of the pressure from gas, stars and dark
matter into a single term (see Equation 20 of Krumholz et al. 2018), such that
Equation 13 is equivalent to the formalism used by Ostriker & Kim (2022).

Figure 11: The rPDE relation for all star-forming spaxels in the extended
ALMaQUEST dataset (the same data as shown in the top left panel of Figure 4;
green 2D histogram), data from the EDGE-CALIFA survey (Barrera-Ballesteros et
al. 2021a; blue crosses) and PHANGS (Sun et al. 2020; purple points). The
Ostriker & Kim (2022) PRFM theory prediction is shown with the red dot-dashed
line; the data from all three datasets diverge from this model at high values
of $P_{DE}$. The orange curves show three examples of an alternative star
formation model presented by Krumholz et al. (2018) model, for different
choices of orbital times that are expected to encapsulate the range of
conditions in the ALMaQUEST sample. The flatter slope of the Krumholz et al.
(2018) models is a better representation of the data at high $P_{DE}$ than the
Ostriker & Kim (2022) model. Our results therefore support the idea that
turbulence is injected in the ISM not only by feedback from star formation,
but additionally through radial gas inflows that act to decrease the orbital
time (Krumholz et al. 2018). This effect is seen most strongly in mergers and
starbursts, where the orbital times are shorter than in normal star-forming
disks.

With these equations we can now evaluate $\Sigma_{\rm SFR}$ in Equation 10 for
different choices of the orbital time which essentially capture the relative
contribution of feedback from non-stellar sources. In Figure 11 we show the
Krumholz et al. (2018) model predictions for the rPDE relation for three
representative values of $t_{orb}$, where the longer values are representative
of normal disks and higher values more appropriate for mergers and central
starbursts. It can be seen that the flatter slope predicted by the Krumholz et
al. (2018) model is in much better agreement with the data at high $P_{DE}$
than the PRFM model of Ostriker & Kim (2022). Indeed, reasonable choices of
the orbital time result in predictions that encompass all of the data.
Adopting different choices for other variables modifies the curves shown in
Figure 11 slightly, but we find them to generally bracket the spread of the
data. We conclude that the extended ALMaQUEST dataset, in sampling more
extreme galaxy environments, particularly in the high pressure regime, can
provide important leverage for distinguishing star formation scenarios and
favour a picture in which turbulence does not (always) come from star
formation alone.

##  6 Conclusions

We have presented the extended ALMaQUEST sample of 66 galaxies containing
19,999 kpc-scale star-forming spaxels. The main distinguishing feature of the
extended ALMaQUEST sample is its diversity, since it contains galaxies on,
below and above the main sequence, as well as central starbursts and mergers.
This diverse sample allows us to investigate how the star formation scaling
relations respond to a broad range of physical conditions. For the first time,
we include the resolved dynamical equilibrium pressure (rPDE) relation amongst
those investigated for the ALMaQUEST dataset. We compare the rPDE relation in
the extended ALMaQUEST sample to the prediction from the pressure-regulated
feedback-modulated (PRFM) theory of star formation (e.g. Ostriker & Kim 2022).

Our main conclusions are as follows.

  * •

Star formation scaling relations in the extended ALMaQUEST sample:  In
addition to the strong scaling relations seen in the full sample of star-
forming spaxels (as found in previous papers in the ALMaQUEST series), the rSK
relation, the rSFMS and the rMGMS all persist even in galaxies experiencing
more extreme conditions, such as in mergers and those with central starbursts
(Figures 1 – 3).

  * •

The resolved $P_{DE}$ relation in the extended ALMaQUEST sample and comparison
with the pressure regulated feedback modulated model:  There is a strong
correlation between $P_{DE}$ and $\Sigma_{\rm SFR}$, as expected from the PRFM
model. However, rather than a linear relation, the data exhibit a flattening
at high values of $P_{DE}$ that is not predicted by the model (Figure 4).
Although previous datasets only sparsely sample the high pressure regime, both
the PHANGS and EDGE-CALIFA surveys show similar deviations from the PRFM model
(Figure 5).

  * •

Diversity in the rPDE relation:  There is significant galaxy-to-galaxy
variation in the rPDE relation (Figure 8). By dissecting the extended
ALMaQUEST sample into different subsets we find that the rPDE relation behaves
differently depending on the global SFR of the galaxy. Galaxies that lie
within $\pm$0.3 dex of the global SFMS show good agreement with the PRFM model
(as has been previously found for other samples of normal star-forming
galaxies in the nearby universe), middle panel of Figure 6. Conversely,
galaxies above and below the main sequence show significant offsets from the
PRFM (top and bottom panels of Figure 6). Moreover, we find that the magnitude
and direction of the offset between the rPDE of a given galaxy and the PRFM
prediction is directly anti-correlated with $\Delta$SFR (Figure 7). The
flattening of the rPDE relation in our full sample (top left panel of Figure
4) can therefore be understood by the superposition of normal star-forming
galaxies and high SFR galaxies, that each contribute $\sim$ 50% of the sample
(top and middle panels of Figure 6).

  * •

Uncertainties in the data: We discuss a range of possible caveats in our
calculation of $P_{DE}$. Although the precise value of $\alpha_{CO}$ is likely
to be the largest source of uncertainty, adopting various alternative
prescriptions (e.g. metallicity or density dependent conversion factors) does
not reconcile our data with the predictions from the PRFM model.

  * •

Dynamical equilibrium pressure as a star formation rate regulator:  We compare
the scatters in the four star formation scaling relations and find that they
vary depending on the sample of galaxies chosen, but are typically 0.2 – 0.3
dex; the rPDE relation is no tighter than any of the other relations (Figure
9). However, given the clear non-linearity of the observed rPDE relation, a
comparison to a linear fit is inadequate to properly quantify the relationship
between $P_{DE}$ and $\Sigma_{\rm SFR}$. A random forest analysis, which is
capable of capturing non-linear dependences in the data, shows unambiguously
that $P_{DE}$ is a better predictor of $\Sigma_{\rm SFR}$ than either
$\Sigma_{\rm H_{2}}$ or $\Sigma_{\star}$ alone (Figure 10). In this sense,
$P_{DE}$ is more ‘fundamental’ for regulating star formation than
$\Sigma_{\star}$ or $\Sigma_{\rm H_{2}}$ alone (even though the form of this
relation deviates from the PRFM theory).

The work presented here thus extends previous comparisons of the PRFM model in
normal star-forming disks (where it has been seen to work well, e.g. Herrera-
Camus et al. 2017; Sun et al. 2020a, 2023; Barrera-Ballesteros et al. 2021a)
into more extreme environments, where it apparently breaks down. Conversely,
the data are qualitatively consistent with a model in which inflows of gas
contribute to (and potentially dominate) the ISM turbulence (Krumholz et al.
2018) when SFRs are elevated above fiducial SFMS values (Figure 11).

## Acknowledgements

The authors acknowledge an NSERC Discovery Grant (SLE), MOST
108-2628-M-001-001-MY3 and NSTC 111-2112-M-001-044- (LL), National Science and
Technology Council of Taiwan grant 110-2112-M-032-020-MY3 (HAP), Australian
Research Council Laureate Fellowship award FL220100020 (MRK), DGAPA-
PAPIIT/UNAM grant IA-101522 and CONACYT grant CF19-39578 (JBB) and National
Science Foundation Grant No 2205551 (JMS).

SLE is grateful to the European Southern Observatory, the University of
Bologna and INAF Arcetri Observatory for funding extended visits during which
work on this project took place. The stimulating environments of these
institutes was critical for the conception and development of this research.
In particular, SLE thanks Munan Gong for seeding the idea of comparing the
ALMaQUEST to the PRFM model - it was a productive coffee time chat! We are
also grateful to Eve Ostriker and Chang-Goo Kim for discussions of the PRFM
model and for sharing the formalism of their forthcoming work (Hassan et al.
in prep) in advance of publication. Conversations with Yu-Hsuan (Eltha) Teng
and Adam Leroy also contributed significantly to the discussion presented in
this work.

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.01225.S,
ADS/JAO.ALMA#2017.1.01093.S, ADS/JAO.ALMA#2018.1.00558.S,
ADS/JAO.ALMA#2018.1.00541.S, ADS/JAO.ALMA#2019.1.00260.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. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. Funding
for the SDSS IV has been provided by the Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of Science, and the Participating Institutions.
SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah.

## Data Availability

The data used in this paper are all released in a supplementary table in the
electronic version of this paper.

## References

  * [Accurso et al. (2017)] Accurso, G., et al., 2017, MNRAS, 470, 4750
  * [Avery et al. (2021)] Avery, C., et al. 2021, MNRAS, 503, 5134
  * [Baker et al. (2022)] Baker, W. M., Maiolino, R., Bluck, A. F. L., Lin, L., Ellison, S. L., Belfiore, F., Pan, H.-A., Thorp, M., 2022, MNRAS, 510, 3622
  * [Baker et al. (2023)] Baker, W. M., et al., 2023, MNRAS, 519, 1149
  * [Baker & Maiolino (2023)] Baker, W., & Maiolino, 2023, MNRAS, 521, 4173
  * [Baldwin, Phillips & Terlevich (1981)] Baldwin, J. A,; Phillips, M. M., Terlevich, R., 1981, PASP, 93, 5
  * [Barnes & Hernquist (1991)] Barnes, J. E., & Hernquist, L., 1991, ApJ, 370, L65
  * [Barrera-Ballesteros et al. (2021)] Barrera-Ballesteros, J. K., et al., 2021a, MNRAS, 503, 3643
  * [Barrera-Ballesteros et al. (2020)] Barrera-Ballesteros, J. K., et al., 2021b, ApJ, 909, 131
  * [Bickley et al. (2021)] Bickley, R. W., et al., 2021, MNRAS, 504, 372
  * [Bickley et al. (2022)] Bickley, R. W., Ellison, S. L., Patton, D. R., Bottrell, C., Gwyn, S., Hudson, M., 2022, MNRAS, 514, 3294
  * [Bigiel et al. (2008)] Bigiel, F., Leroy, A., Walter, F., Brinks, E., de Blok, W. J. G., Madore, B., Thornley, M. D., 2008, AJ, 136, 2846
  * [Bigiel et al. (2011)] Bigiel, F., et al., 2011, ApJ, 730, L13
  * [Blitz & Rosolowsky (2004)] Blitz, L., & Rosolowsky, E., 2004, ApJ, 612, L29
  * [Blitz & Rosolowsky (2006)] Blitz, L., & Rosolowsky, E., 2006, ApJ, 650, 933
  * [Bluck et al. (2020)] Bluck, A. F. L., et al., 2020, MNRAS, 499, 230
  * [Bluck et al. (2022)] Bluck, A. F. L., Maiolino, R., Brownson, S., Conselice, C. J., Ellison, S. L., Piotrowska, J. M., Thorp, M. D., 2022, A&A, 659, 160
  * [Blumenthal & Barnes (2018)] Blumenthal, K. A., & Barnes, J. E, 2018, MNRAS, 479, 3952
  * [Bolatto et al. (2017)] Bolatto, A. D., et al. 2017, ApJ, 846, 159
  * [Bolatto et al. (2013)] Bolatto, A. D., Wolfire, M., Leroy, A. K., 2013, ARA&A, 51, 207
  * [Caldu-Primo et al. (2013)] Caldu-Primo, A., Schruba, A., Walter, F., Leroy, A., Sandstrom, K., de Blok, W. J. G., Ianjamasimanana, R., Mogotsi, K. M., 2013, AJ, 146, 150
  * [Cardelli, Clayton & Mathis (1989)] Cardelli, J. A., Clayton, G. C., & Mathis, J. S., 1989, ApJ, 345, 245
  * [Casasola et al. (2022)] Casasola, V., et al., 2022, A&A, 668, 130
  * [Catalan-Torrecilla et al (2015)] Catalan-Torrecilla, C., et al., 2015, A&A, 584, 87
  * [Cid-Fernandes et al. (2011)] Cid Fernandes, R., Stasinska, G., Mateus, A., Vale Asari, N., 2011, MNRAS, 413, 1687
  * [Daddi et al. (2010)] Daddi, E., et al. 2010, ApJ, 714, L118
  * [Davis et al. (2014)] Davis, T. A., et al., 2014, MNRAS, 444, 3427
  * [Dey et al. (2019)] Dey, B., et al., 2019, MNRAS, 488, 1926
  * [Dopita et al. (2003)] Dopita, M. A., Groves, B. A., Sutherland, R. S., Kewley, L. J., 2003, ApJ, 583, 727
  * [Ellison et al., (2021a)] Ellison, S. L., Lin, L., Thorp, M. D., Pan, H.-A. Scudder, J. M., Sanchez, S. F., Bluck, A. F. L., Maiolino, R., 2021a, MNRAS, 501, 4777
  * [Ellison et al., (2021b)] Ellison, S. L., Lin, L., Thorp, M. D., Pan, H.-A., Sanchez, S. F., Bluck, A. F. L., Belfiore, F., 2021b, MNRAS, 502, L6
  * [Ellison et al. (2013)] Ellison, S. L., Mendel, J. T., Patton, D. R., Scudder, J. M., 2013, MNRAS, 453, 3627
  * [Ellison et al. (2008)] Ellison, S. L., Patton, D. R., Simard, L., McConnachie, A. W., 2008a AJ, 135, 1877
  * [Ellison et al. (2008)] Ellison, S. L., Patton, D. R., Simard, L., McConnachie, A. W., 2008b, ApJ, 672, L107
  * [Ellison et al., (2018)] Ellison, S. L., Sanchez, S. F., Ibarra-Medel, H., Antonio, B., Mendel, J. T., Barrera-Ballesteros, J., 2018, MNRAS, 474, 2039
  * [Ellison et al., (2020a)] Ellison, S. L., Thorp, M. D., Pan, H.-A. Lin, L., Scudder, J. M., Bluck, A. F. L.; Sanchez, S. F., Sargent, M., 2020a, MNRAS, 492, 6027
  * [Ellison et al., (2020b)] Ellison, S. L., Thorp, M. D., Lin, L., Pan, H.-A. Bluck, A. F. L.; Scudder, J. M., Sanchez, S. F., Sargent, M., 2020b, MNRAS, 493, L39
  * [Faber & Jackson (1976)] Faber, S. M., & Jackson, R. E., 1976, ApJ, 204, 668
  * [Ferrarese & Merritt (2000)] Ferrarese, L., & Merritt, D., 2000, ApJ, 539, L9
  * [Fisher et al. (2019)] Fisher, D. B., Bolatto, A. D., White, H., Glazebrook, K., Abraham, R. G., Obreschkow, D., 2019, ApJ, 870, 46
  * [Fluetsch et al. (2019)] Fluetsch, A., et al., 2019, MNRAS, 483, 4586
  * [Fumagalli et al. (2010)] Fumagalli, M., Krumholz, M. R., Hunt, L. K., 2010, ApJ, 722, 919
  * [Gallagher et al. (2018)] Gallagher, M. J., et al., 2018, ApJ, 858, 90
  * [Garay-Solis et al. (2023)] Garay-Solis, Y., Barrera-Ballesteros, J. K., Colombo, D., Sanchez, S. F., Lugo-Aranda, A. Z., Villanueva, V., Wong, T., Bolatto, A. D., 2023, ApJ, 952, 122
  * [Garcia-Benito et al. (2015)] Garcia-Benito, R., et al. ., 2015, A&A, 576, 135
  * [Gong et al. (2020)] Gong, M., Ostriker, E. C., Kim, C.-G., Kim, J.-G., 2020, ApJ, 903, 142
  * [Groves et al. (2023)] Groves, B., et al., 2023, MNRAS, 520, 4902
  * [Herrera-Camus et al. (2017)] Herrera-Camus, R., et al., 2017, ApJ, 835, 201
  * [Hunt et al. (2020)] Hunt, L. K., Tortora, C., Ginolfi, M., Schneider, R., 2020, A&A, 643, 180
  * [Inoue et al. (2001)] Inoue, A. K., Hirashita, H., Kamaya, H., 2001, ApJ, 555, 613
  * [Jameson et al., (2016)] Jameson, K. E., et al., 2016, ApJ, 825, 12
  * [Jimenez-Donaire et al (2019)] Jimenez-Donaire, M. J., et al., 2019, ApJ, 880, 127
  * [Jimenez-Donaire et al (2023)] Jimenez-Donaire, M. J., et al., 2023, A&A, 671, 3
  * [Kauffmann et al. (2003)] Kauffmann, G., et al., 2003, MNRAS, 346, 1055
  * [Kennicutt (1998)] Kennicutt, R.C. 1998, ARA&A, 36, 189
  * [Kennicutt & Evans (2012)] Kennicutt, R.C., Evans, N. J., 2012, ARA&A, 50, 531
  * [Kim & Ostriker (2007)] Kim, W.-T., & Ostriker, E., 2007, ApJ, 660, 1232
  * [Kim et al. (2011)] Kim, C.-G., Kim, W.-T., & Ostriker, E., 2011, ApJ, 743, 25
  * [Kim et al. (2013)] Kim, C.-G., Ostriker, E., Kim, W.-T., 2013, ApJ, 776, 1
  * [Krumholz (2013)] Krumholz, M. R., 2013, MNRAS, 436, 2747
  * [Krumholz et al. (2018)] Krumholz, M. R., Burkhart, B., Forbes, J. C., Crocker, R. M., 2018, MNRAS, 477, 2716
  * [Krumholz & Burkhart (2016)] Krumholz, M. R., & Burkhart, B., 2016, MNRAS, 458, 1671
  * [Krumholz et al. (2012)] Krumholz, M. R., Dekel, A., McKee, C. F., 2012, ApJ, 745, 69
  * [Krumholz et al. (2009)] Krumholz, M. R., McKee, C. F., Tumlinson, J., 2009, ApJ, 693, 216
  * [Lada, Lombardi & Alves (2010)] Lada, C. J., Lombardi, M., & Alves, J. F., 2010, ApJ, 724, 687
  * [Lequeux et al. (1979)] Lequeux, J., Peimbert, M., Rayo J. F., Serrano, A., Torres-Peimbert, S., 1979, A&A, 80, 155
  * [Leroy et al. (2013)] Leroy, A. K., et al., 2013, AJ, 146, 19
  * [Leroy et al. (2021)] Leroy, A. K., et al., 2021, ApJS, 257, 43
  * [Leroy et al. (2008)] Leroy, A. K., Walter, F., Brinks, E., Bigiel, F., de Blok, W. J. G., Madore, B., Thornley, M. D., 2008, AJ, 136, 2782
  * [Lin et al. (2019)] Lin, L., et al., 2019, ApJ, 884, L33
  * [Lin et al. (2020)] Lin, L., et al., 2020, ApJ, 903, 145
  * [Lin et al. (2022)] Lin, L., et al., 2022, ApJ, 926, 175
  * [Madau & Dickinson (2014)] Madau, P., & Dickinson, M., 2014, ARA&A, 52, 415
  * [Mannucci et al. (2010)] Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., Gnerucci, A., 2010, MNRAS, 408, 2115
  * [McKee Krumholz (2010)] McKee, C. F., & Krumholz, M. R., 2010, ApJ, 709, 308
  * [McMullion et al (2007)] McMullin, J. P., Waters, B., Schiebel, D., Young, W., Golap, K., 2007, ASPC, 376, 127
  * [Medling et al. (2018)] Medling, A. M., et al., 2018, MNRAS, 475, 5194
  * [Moreno et al. (2021)] Moreno, J., et al 2021, MNRAS, 503, 3113
  * [Moreno et al. (2015)] Moreno, J., Torrey, P., Ellison, S. L., Patton, D. R., Bluck, A. F. L., Bansal, G., Hernquist, L., 2015, MNRAS, 448, 1107
  * [Morselli et al. (2020)] Morselli, L., et al., 2020, 496, 4606
  * [Narayanan et al. (2012)] Narayanan, D., Krumholz, M., Ostriker, E. C., Hernquist, L., 2012, MNRAS, 421, 3127
  * [Ostriker & Kim (2022)] Ostriker, E., & Kim, C.-G., 2022, ApJ, 936, 137
  * [Ostriker, McKee, Leroy (2010)] Ostriker, E. C., McKee, C. F., Leroy, A. K., 2010, ApJ, 721, 975
  * [Ostriker & Shetty (2011)] Ostriker, E., & Shetty, R., 2011, ApJ, 731, 41
  * [Peng et al. (2010)] Peng, Y., et al., 2010, ApJ, 721, 193
  * [Pessa et al. (2021)] Pessa, I., et al., 2021, A&A, 650, 134
  * [Pessa et al. (2022)] Pessa, I., et al., 2022, A&A, 663, 61
  * [Pettini & Pagel (2004)] Pettini, M., & Pagel, B. E. J., 2004, MNRAS, 348, L59
  * [Roberts-Borsani & Saintonge (2019)] Roberts-Borsani, G., & Saintonge, A., 2019, MNRAS, 482, 4111
  * [Saintonge et al. (2011)] Saintonge, A., et al., 2011, MNRAS, 415, 61
  * [Saintonge et al. (2012)] Saintonge, A., et al., 2012, ApJ, 758, 73
  * [Saintonge (2016)] Saintonge, A., et al., 2016, MNRAS, 462, 1749
  * [Sanchez et al. (2016a)] Sánchez, S. F., et al., 2016a, RMxAA, 52, 21
  * [Sanchez et al. (2016b)] Sánchez, S. F., et al., 2016b, RMxAA, 52, 171
  * [Sánchez et al. (2018)] Sánchez, S. F., et al., 2018, RMxAA, 54, 217
  * [Sánchez et al. (2021)] Sánchez, S. F., et al., 2021, MNRAS, 503, 1615
  * [Sanchez-Ramirez et al. (2016)] Sánchez-Ramirez, R., et al., 2016, MNRAS, 456, 4488
  * [Sandstrom et al. (2013)] Sandstrom, K. M., et al., 2013, ApJ, 777, 5
  * [Schruba et al. (2011)] Schruba, A., et al., 2011, AJ, 142, 37
  * [Schruba et al. (2018)] Schruba, A., Bialy, S., Sternberg, A., 2018, ApJ, 862, 110
  * [Schruba et al. (2019)] Schruba, A., Kruijssen, J. M. D., Leroy, A. K., 2019, ApJ, 883, 2
  * [Scudder et al. (2012b)] Scudder, J. M., Ellison, S. L., Torrey, P., Patton, D. R., Mendel, J. T., 2012, MNRAS, 426, 549
  * [Semenov et al. (2019)] Semenov, V. A., Kravtsov, A. V. Gnedin, N. Y., 2019, ApJ, 870, 79
  * [Sharda et al. (2019)] Sharda, P., da Cunha, E., Federrath, C., Wisnioski, E., Di Teodoro, E. M., Tadaki, K., Yun, M. S., Aretxaga, I., Kawabe, R., 2019, MNRAS, 487, 4305
  * [Shetty & Ostriker (2012)] Shetty, R., & Ostriker, E., 2012, ApJ, 754, 2
  * [Shi et al., (2011)] Shi, Y., Helou, G., Yan, L., Armus, L., Wu, Y., Papovich, C., Stierwalt, S., 2011, ApJ, 733, 87
  * [Sun et al. (2020)] Sun, J., et al., 2020a, ApJ, 892, 148
  * [Sun et al. (2020)] Sun, J., et al., 2020b, ApJ, 901, L8
  * [Sun et al. (2023)] Sun, J., et al., 2023, ApJ, 945, L19
  * [Tan (2000)] Tan, J. C., 2000, ApJ, 536, 173
  * [Teimoorinia, Bluck & Ellison (2016)] Teimoorinia, H., Bluck, A. F. L., & Ellison, S. L., 2016, MNRAS, 457, 2086
  * [Teng et al (2023)] Teng, Y-H., et al., 2023, ApJ, 950, 119
  * [Thorp et al. (2022)] Thorp, M. D., Ellison, S. L., Pan, H.-A., Lin, L., Patton, D. R., Bluck, A. F. L., Walters, D., Scudder, J. M., 2022, MNRAS, 516, 1462
  * [Thorp et al. (2019)] Thorp, M. D., Ellison, S. L., Simard, L., Sanchez, S. F., Antonio, B., 2019, MNRAS, 482, L55
  * [Usero et al., (2015)] Usero, A., et al., 2015, AJ, 150, 115
  * [Wang et al. (2019)] Wang, E., Lilly, S. J., Pezzulli, G., Matthee, J., 2019, ApJ, 877, 132
  * [Wong & Blitz (2002)] Wong, T., & Blitz, L., 2002, ApJ, 569, 157
  * [Wu et al. (2005)] Wu, J., Evans, N. J., Gao, Y., Solomon, P. M., Shirley, Y. L., Vanden Bout, P. A., 2005, ApJ, 635, 173
  * [Zafar et al. (2013)] Zafar, T., Peroux, C., Popping, A., Milliard, B., Deharveng, J.-M., Frank, S.