# Death of the Immortal Molecular Cloud: Resolution Dependence of the Gas-Star
Formation Relation Rules out Decoupling by Stellar Drift

J. M. Diederik Kruijssen⋆ 1,2 0000-0002-8804-0212 Mélanie Chevance 3,2
0000-0002-5635-5180 Steven N. Longmore 4,2 0000-0001-6353-0170 Adam Ginsburg 5
Lise Ramambason 3 Andrea Romanelli 3 1Technical University of Munich, School
of Engineering and Design, Department of Aerospace and Geodesy, Chair of
Remote Sensing Technology, Arcisstr. 21, 80333 Munich, Germany  2Cosmic
Origins Of Life (COOL) Research DAO, https://coolresearch.io 3Institut für
Theoretische Astrophysik, Zentrum für Astronomie der Universität Heidelberg,
Albert-Überle-Str. 2, 69120 Heidelberg, Germany  4Astrophysics Research
Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146
Brownlow Hill, Liverpool L3 5RF, UK  5Department of Astronomy, University of
Florida, P.O. Box 112055, Gainesville, FL, USA

(2024 April 24; Xxxxx XX; Xxxxx XX)

###### Abstract

Recent observations have demonstrated that giant molecular clouds (GMCs) are
short-lived entities, surviving for the order of a dynamical time before
turning a few percent of their mass into stars and dispersing, leaving behind
an isolated young stellar population. The key question has been whether this
GMC dispersal actually marks a point of GMC destruction by stellar feedback
from the new-born stars, or if GMCs might be ‘immortal’ and only dynamically
decouple from their nascent stars due to stellar drift. We address this
question in six nearby galaxies, by quantifying how the gas-star formation
relation depends on the spatial scale for scales between the GMC diameter and
the GMC separation length, i.e. the scales where an excess of GMCs would be
expected to be found in the stellar drift scenario. Our analysis reveals a
consistent dearth of GMCs near young stellar populations regardless of the
spatial scale, discounting the notion of ‘immortal’ GMCs that decouple from
their nascent stars through stellar drift. Instead, our findings demonstrate
that stellar feedback destroys most GMCs at the end of their lifecycle.
Employing a variety of statistical techniques to test both hypotheses, we find
that the probability that stellar feedback concludes the GMC lifecycle is
about 2,000 times higher than the probability that stellar drift separates
GMCs and young stellar regions. This observation strengthens the emerging
picture that galaxies consist of dynamic building blocks undergoing vigorous,
feedback-driven lifecycles that collectively regulate star formation and drive
the baryon cycle within galaxies.

###### keywords:

stars: formation – ISM: clouds – ISM: structure – galaxies: evolution –
galaxies: ISM – galaxies: star formation

††thanks: ⋆E-mail: kruijssen [at] coolresearch.io

##  1 Introduction

The vast majority of star formation in galaxies takes place in giant molecular
clouds (GMCs; Chevance et al. 2023). The cycle of GMC collapse, star
formation, GMC dispersal, and stellar feedback represents the engine driving
galaxy evolution. As a result, understanding the GMC lifecycle has been one of
the key open questions in modern studies of star formation and galaxy
evolution.

Until recently, the key unknown was on what timescales the GMC lifecycle takes
place, with qualitative implications for the physical processes driving the
matter cycle within galaxies. The traditional picture of long-lived GMCs (e.g.
Scoville & Hersh, 1979; Koda et al., 2009), with lifetimes only a factor of
several shorter than the galaxy wide gas depletion time of $\sim
1~{}\mbox{${\rm Gyr}$}$ (e.g. Leroy et al., 2013), would imply that GMCs turn
a large fraction of their mass into stars over their lifetimes, requiring
modest mass loading factors and mass outflow rates. By contrast, the dynamical
picture of star formation taking place within short-lived GMCs over the course
of a crossing time (e.g. Elmegreen, 2000; Hartmann, 2001) would imply
considerably higher mass loading factors and mass outflow rates – not
necessarily driving gas out of the galaxy at large (Keller et al., 2022), but
certainly requiring a violent matter cycle on sub-galactic scales.
Distinguishing between these two perspectives requires knowledge of the GMC
lifetime, measured from the initial GMC condensation out of the atomic phase
to the termination of star formation within the GMC and its eventual
dispersal.

Methods for inferring GMC lifetimes have existed for a long time (e.g.
Scoville & Hersh, 1979; Leisawitz et al., 1989), but they often relied on
strong assumptions, such as requiring GMCs to form at special locations within
the host galaxy and follow evolutionary streamlines (Koda et al., 2009; Meidt
et al., 2015; Kruijssen et al., 2015), or requiring GMCs and Hii regions to be
easily identifiable entities that can be separated from their environment
(Kawamura et al., 2009; Corbelli et al., 2017), despite the hierarchical
structure of the interstellar medium (ISM) and young stellar populations.
Either of these approaches carry a high degree of subjectivity. Similarly
worryingly, the estimates of the resulting GMC lifetimes reported by these
studies varied by (more than) two orders of magnitude, from
$1{-}100~{}\mbox{${\rm Myr}$}$. The heterogeneity of the techniques for
measuring the GMC lifetime made it challenging to assess whether this spread
was due to physical variations or differences in methodology.

Obtaining robust measurements of the GMC lifetime has recently been possible
thanks to two key developments. First, the spatial resolution and sensitivity
required to resolve the molecular gas distribution of galaxies (usually traced
by CO) into individual GMCs is now routinely achieved (e.g. Leroy et al.,
2021a). Second, statistically robust and objective measurement techniques have
now been developed to measure GMC lifetimes. Specifically, the ‘uncertainty
principle for star formation’ initially developed by Kruijssen & Longmore
(2014) and Kruijssen et al. (2018) (with important extensions by Hygate et al.
2019 and Haydon et al. 2020a, b) has enabled routine measurements of the GMC
lifetime, the gas clearance timescale after the emergence of the first massive
stars (also referred to as the ‘feedback timescale’), the integrated star
formation efficiency per GMC, as well as many other derived quantities, across
$50{-}100$ galaxies (e.g. Kruijssen et al., 2019a; Chevance et al., 2020a,
2022; Ward et al., 2020, 2022; Zabel et al., 2020; Kim et al., 2021, 2022,
2023; Lu et al., 2022). These measurements have resulted in GMC lifetimes
ranging from $5{-}35~{}\mbox{${\rm Myr}$}$ across the nearby galaxy
population. Typical uncertainties are a few 10s of percent, implying that this
range indicates physical variation of the GMC lifetime. Regardless of these
quantitative galaxy-to-galaxy variations, these studies have painted a picture
of low-efficiency star formation in short-lived GMCs, with high mass loading
factors and outflow rates driven by strong stellar feedback.

The uncertainty principle methodology is inspired by the discovery of Schruba
et al. (2010) that the tight relation between the gas mass and the star
formation rate (SFR) on large ($\sim\mbox{${\rm kpc}$}$) scales in galaxies
(e.g. Kennicutt, 1998; Bigiel et al., 2008; Leroy et al., 2008) breaks down on
the small ($\sim 100~{}\mbox{${\rm pc}$}$) scales of GMCs and Hii regions
(also see e.g. Onodera et al., 2010). This ‘decorrelation’ was quantified by
Schruba et al. (2010) in the form of a ‘tuning fork diagram’, in which the
gas-to-SFR tracer flux ratio deviates from the galactic average towards small
spatial scales. The deviation is upward or downward, depending on whether the
measurement apertures are centred on gas emission peaks (e.g. GMCs) or SFR
tracer peaks (e.g. Hii regions), respectively. The magnitude and (a)symmetry
of these deviations in the gas-to-SFR tracer flux ratio directly probe the
relative rarities of emission peaks in both tracers, and thus can be used to
infer the ratios of their lifetimes, as well as the time for which both
tracers coexist within these emission peaks. In practice, these quantities are
obtained by fitting the size scale dependence of the gas-to-SFR tracer flux
ratio with a mathematical expression that depends on the key evolutionary
timescales (GMC lifetime and the GMC-Hii region coexistence timescale) and the
region separation length. Kruijssen & Longmore (2014) and Kruijssen et al.
(2018) formalised this timescale dependence and provided a robust statistical
framework for measuring the timescales that define the GMC lifecycle using the
‘tuning fork diagrams’ of Schruba et al. (2010).

No other technique for measuring GMC lifetimes has been subjected to the level
of scrutiny that the uncertainty principle methodology has received, ranging
from simplified numerical experiments (e.g. Kruijssen et al., 2018; Hygate et
al., 2019; Haydon et al., 2020a, b) to state-of-the-art numerical simulations
(e.g. Fujimoto et al., 2019; Jeffreson et al., 2021; Semenov et al., 2021;
Keller et al., 2022) and extensive observational tests (e.g. Kruijssen et al.,
2019a; Chevance et al., 2020a; Ward et al., 2022). The analysis code (named
Heisenberg) is publicly available at https://github.com/mustang-
project/heisenberg. As a result, the measurements themselves are robust and
have well-defined uncertainties, but their physical interpretation has spurred
lively discussion in the literature.

The GMC lifetimes obtained with the uncertainty principle methodology are
similar to a dynamical time, indicating rapid star formation, and the short-
lived ($1{-}6~{}\mbox{${\rm Myr}$}$) spatial coincidence between GMCs and
unembedded massive stars suggests that GMC lifetimes are curtailed by early,
pre-supernova feedback mechanisms (see discussions in e.g. Kruijssen et al.,
2019a; Chevance et al., 2020a, 2022; Kim et al., 2022). The important role of
feedback in driving GMC dispersal is further supported by the close match
between the outflow velocities implied by these short timescales and direct
measurements of the expansion velocity of Hii regions (e.g. McLeod et al.,
2019, 2020, 2021; Ward et al., 2022) and the implied chemical mixing scale in
the ionised medium (Kreckel et al., 2020).

An alternative hypothesis for explaining the decorrelation between GMCs and
their nascent stellar populations was recently proposed by Koda & Tan (2023,
hereafter KT23). These authors performed a simple numerical experiment to show
that the observed decorrelation may also be produced if the GMCs are in fact
immortal, i.e. they have ‘infinite’ lifetimes (i.e. much longer than a star
formation cycle), but the young stars experience a kinetic drift relative to
their natal GMC. As the physical agent for this drift, the authors propose
cloud-cloud collisions. The feasibility of cloud-cloud collisions as an
important mechanism for driving star formation in galaxies has been questioned
in numerous studies due to the low predicted incidence of cloud-cloud
collisions, as well as other fine-tuning issues (e.g. McKee & Ostriker, 2007;
Dobbs et al., 2015; Chevance et al., 2020b). However, regardless of the
physical mechanism causing stellar drift, the drift itself is a valid
hypothesis for the observed decorrelation between GMCs and Hii regions.

In earlier papers, stellar drift was disregarded as the cause for the small-
scale gas-SFR decorrelation, because either the typical drift velocity was
considered too slow (Kruijssen & Longmore, 2014), or the obtained GMC
lifetimes were found to be independent of the presence of spiral arms
(Chevance et al., 2020a), contrary to the expectation of stellar drift caused
by cloud-cloud collisions. However, the renewed interest in the stellar drift
hypothesis by KT23 warrants a more rigorous test of this idea. In this paper,
we carry out a simple observational experiment using existing data, which
enables a statistically robust distinction between GMC destruction by stellar
feedback and GMC isolation due to stellar drift. A description of the
experiment and the observational data used are provided in §2. In §3, we
present the results of the experiment, showing that the decorrelation cannot
be driven by stellar drift, and instead GMCs must be destroyed by stellar
feedback. We conclude in §4 with a short discussion of the implications of
these results.

##  2 Method

###  2.1 Experiment design

Figure 1:  Schematic representation of the experiment performed in this paper.
By placing apertures of increasing size on peaks of molecular gas (GMCs) and
SFR tracer emission (Hii regions), we measure the coexistence timescale (i.e.
the ‘overlap timescale’) of these phases using the uncertainty principle
methodology (Kruijssen et al., 2018). We then worsen the resolution at which
this measurement is made, from the GMC scale ($\ell\approx 2\mbox{$R_{\rm
GMC}$}$) to the region separation length ($\ell\approx\lambda$). If GMCs and
young stellar regions decouple by feedback-driven GMC destruction, then the
overlap timescale should be independent of the size scale (i.e. aperture size
or spatial resolution). If GMCs and young stellar regions decouple by stellar
drift, then the overlap timescale should linearly increase with the size
scale. On size scales $\ell>\lambda$, the measured overlap timescale may
increase regardless of the physics due to blending with neighbouring regions.
See §2.1 for further details.

We consider a population of GMCs that all form a young stellar population
after some time delay. After a further time delay, during which the GMC and
stars coincide and which we refer to as the ‘overlap timescale’ or $t_{\rm
over}$, the GMC is removed and the stars reside in isolation. We examine the
following two physical scenarios for the removal of the GMC.

  1. 1.

Stellar feedback: the GMC is destroyed by stellar feedback, either through a
phase transition or by fragmentation into many small parts that
observationally give the impression of a diffuse molecular phase. In this
case, the GMC lifecycle ends.

  2. 2.

Stellar drift: the GMC is not destroyed, but is displaced from the stellar
population due to kinetic drift, caused by an unspecified physical mechanism
(cloud-cloud collisions in KT23). In this case, the GMC is ‘immortal’ and
continues its lifecycle.

Fundamentally, the ‘GMC lifetime’ obtained through the uncertainty principle
methodology as described by Kruijssen et al. (2018) measures the time spent by
GMCs prior to and during the appearance of a young stellar population.
Likewise, the ‘overlap timescale’ measures the time during which a GMC and its
nascent stars are co-spatial. Therefore, the methodology cannot directly
distinguish between the two scenarios outlined above.111The GMC lifecycle
measurements obtained so far do directly rule out the existence of immortal
GMCs that do not dynamically decouple from their nascent stellar populations,
because isolated Hii regions exist. In the tuning fork representation, this
decoupling manifests itself as a non-zero excess of the gas-to-SFR tracer flux
ratio when focusing small apertures on peaks of gas emission. If the young
stellar populations were to remain co-spatial with their parent GMC and
disappear by stellar evolutionary fading, then the apertures focused on GMCs
would always capture all of the SFR tracer flux, and there would be no excess
in the gas-to-SFR tracer flux ratio (e.g. Fujimoto et al., 2019). However, the
dependence of the measured overlap timescale on the size scale fundamentally
differs in both cases.

  1. 1.

If stellar feedback destroys GMCs after the stars have formed, then placing
increasingly large apertures on isolated stellar populations (i.e. post-GMC
destruction) would not capture any gas emission from their natal GMCs, because
these have either disappeared altogether or have been dispersed into a diffuse
phase (which is filtered out in the uncertainty principle methodology and can
no longer be identified as a GMC, see Hygate et al. 2019). As a result, the
measured overlap timescale would be independent of the size scale for aperture
sizes between the GMC diameter and the typical separation length between
independent GMCs or Hii regions.

  2. 2.

If stellar drift causes a spatial displacement between the stellar population
and their natal GMCs, then placing increasingly large apertures on isolated
stellar populations (i.e. post-displacement) would capture the gas emission
from their natal GMCs for an increasing fraction of the total number of
stellar regions as the aperture size grows. As a result, the measured overlap
timescale would increase with the size scale for aperture sizes in between the
GMC diameter and the typical separation length between independent GMCs or Hii
regions.

Both situations are illustrated schematically in Figure 1. Mathematically, the
dependence of the overlap timescale on the size scale can be expressed as
follows. For stellar feedback-driven GMC destruction, the overlap timescale
measured on a size scale $\ell$ is identical to the overlap timescale measured
at the GMC diameter $\ell_{\rm 0}\equiv 2\mbox{$R_{\rm GMC}$}$, as long as the
size scale remains smaller than the region separation length $\lambda$.222For
reference, the region separation length is defined as the mean geometric
distance between regions (of any type) in the vicinity of identified emission
peaks. As such, it is larger than the nearest-neighbour distance between
regions, but smaller than the separation length obtained by uniformly
distributing regions across the entire field of view. See Kruijssen & Longmore
(2014) and Kruijssen et al. (2018) for a formal definition, and the Methods
section of Kruijssen et al. (2019a) for a comparison to other characteristic
size scales. Beyond this size scale, contamination by neighbouring regions
artificially increases the overlap timescale (see §4.3.6 of Kruijssen et al.
2018 and fig. 3 of Kruijssen et al. 2019a). Therefore, we expect

$\left[\frac{\mbox{$t_{\rm over}$}(\ell)}{\mbox{$t_{\rm
over}$}(\ell_{0})}\right]_{\rm fb}=1~{}~{}~{}~{}~{}{\rm
for}~{}~{}2\mbox{$R_{\rm GMC}$}\leq\ell\leq\lambda,$ (1)

where the subscript ‘fb’ indicates stellar feedback and $R_{\rm GMC}$
represents the GMC radius. The GMC radius is obtained as a byproduct of the
uncertainty principle methodology and is retrieved directly from the emission
structure in the molecular gas maps. This means that if the GMC radius is not
resolved, it will be approximately equal to half the resolution scale by
definition, i.e. $\mbox{$R_{\rm GMC}$}\approx\ell/2$ (see §3.2.11 of Kruijssen
et al. 2018). Throughout this paper, we define $\ell_{0}=2\mbox{$R_{\rm
GMC}$}$, where $R_{\rm GMC}$ is obtained by performing the uncertainty
principle analysis at the best available resolution.

By contrast, the measured overlap timescale in the case of stellar drift is
equal to the time it takes a moving stellar population to clear a distance
equal to the aperture radius, or half the aperture size $\ell$, at the drift
velocity $v_{\rm drift}$.333For simplicity, we assume a constant drift
velocity here, but we will relax this assumption in §3.3. As before, the
requirement is that the size scale is larger than the GMC diameter and remains
smaller than the region separation length $\lambda$. In this regime, we
physically expect $\mbox{$t_{\rm over}$}(\ell)=\ell/2v_{\rm drift}$ and can
take the ratio relative to the overlap timescale at the resolution equal to
the GMC diameter ($\ell_{0}$) to obtain the simple expectation

$\left[\frac{\mbox{$t_{\rm over}$}(\ell)}{\mbox{$t_{\rm
over}$}(\ell_{0})}\right]_{\rm drift}=\frac{\ell}{\ell_{0}}~{}~{}~{}~{}~{}{\rm
for}~{}~{}2\mbox{$R_{\rm GMC}$}\leq\ell\leq\lambda,$ (2)

where the subscript ‘drift’ indicates stellar drift.

The qualitatively different behaviour of $\mbox{$t_{\rm over}$}(\ell)$ for
stellar feedback-driven GMC destruction and stellar drift-driven decoupling
means that we can distinguish between both scenarios by measuring the overlap
timescale using the uncertainty principle methodology with increasing minimum
aperture sizes, in the range $2\mbox{$R_{\rm GMC}$}\leq\ell\leq\lambda$, and
testing whether the resulting data are more consistent with equation (1) or
with equation (2). As shown in §3, the uncertainties in the measurements are
small enough to unambiguously distinguish between both cases.

Table 1: Galaxy sample Galaxy | $\ell_{\rm min}$ | $\ell_{0}$ | $\lambda$ | $\lambda/\ell_{\rm min}$ | H2 tracer
---|---|---|---|---|---
| [${\rm pc}$] | [${\rm pc}$] | [${\rm pc}$] |  |
LMC | $25$ | $24.1^{+0.5}_{-0.3}$ | $71^{+13}_{-8}$ | $2.8^{+0.6}_{-0.3}$ | CO($1{-}0$)
NGC300 | $20$ | $30.9^{+2.3}_{-1.6}$ | $113^{+25}_{-17}$ | $5.7^{+1.2}_{-0.9}$ | CO($1{-}0$)
M33 | $49$ | $68.9^{+2.8}_{-3.1}$ | $155^{+30}_{-24}$ | $3.2^{+0.6}_{-0.5}$ | CO($1{-}0$)
NGC628 | $54$ | $52.0^{+2.3}_{-2.9}$ | $96^{+13}_{-10}$ | $1.8^{+0.2}_{-0.2}$ | CO($2{-}1$)
NGC5068 | $35$ | $35.0^{+0.9}_{-0.5}$ | $118^{+19}_{-14}$ | $3.7^{+0.5}_{-0.4}$ | CO($2{-}1$)
M83 | $45$ | $46.6^{+2.1}_{-1.7}$ | $86^{+9}_{-7}$ | $1.9^{+0.2}_{-0.1}$ | CO($2{-}1$)

The overlap timescales are obtained at different size scales by simply
omitting the observed gas-to-SFR tracer flux ratios at aperture sizes smaller
than $\ell$. This effectively implies truncating the tuning fork diagram
before re-fitting the mathematical expression through which the GMC lifetime,
overlap timescale, and region separation length are obtained. When carrying
out the fit at lower resolutions, we fix the region separation length to the
value that was obtained when including all aperture sizes (down to
$\ell_{0}$), to ascertain that the lower-resolution fits are not affected by
any degeneracies between the timescales and the region separation length. The
resulting overlap timescales are then normalised to the overlap timescale
obtained at the best resolution as in equations (1) and (2).

###  2.2 Observational data

The experiment described in §2.1 requires observations at a resolution
sufficient to probe the gas-to-SFR tracer ratio at scales smaller than the
region separation length $\lambda$. All previous applications of the
uncertainty principle methodology have yielded measurements of $\lambda$,
making it possible to use the ratio $\lambda/\ell_{\rm min}\ga 2$, where
$\ell_{\rm min}$ is the highest available resolution, as a selection criterion
to guarantee a sufficient dynamic range. Additionally, we prioritise absolute
spatial resolutions of $\ell_{\rm min}\la 50~{}\mbox{${\rm pc}$}$ (implying an
effective distance cut given the current state-of-the-art in submillimeter
observations of the molecular ISM). As the molecular gas tracer, we use CO
emission (either the $1{-}0$ or $2{-}1$ transition), and we use H$\alpha$ to
trace the SFR. The resulting sample of six galaxies is given in Table 1,
listing the best spatial resolution $\ell_{\rm min}$, the estimated GMC
diameter $\ell_{0}$, the region separation length $\lambda$, and the dynamic
range ratio $\lambda/\ell_{\rm min}$. The sample contains five galaxies that
have been previously analysed – the LMC by Ward et al. (2022), NGC300 by
Kruijssen et al. (2019a), M33 by Kim et al. (2021), and NGC628 and NGC5068 by
Chevance et al. (2020a). We refer to these papers for descriptions of the
observational data used. In addition, we also include M83, because of its
favourable properties (high resolution and sensitivity), and the fact that
Koda et al. (2023) recently estimated a GMC lifetime of $\sim
100~{}\mbox{${\rm Myr}$}$ in this galaxy by assuming that GMCs form in spiral
arms and follow evolutionary streamlines. Because the uncertainty principle
methodology does not make such stringent assumptions, it is a worthwhile
exercise to also obtain a GMC lifetime measurement for this galaxy. To achieve
this, we use the CO($2{-}1$) map from ALMA programmes 2013.1.01161.S,
2015.1.00121.S, and 2016.1.00386.S (PI K. Sakamoto, processed as described by
Leroy et al., 2021b), and the H$\alpha$ map from SINGG (Meurer et al., 2006),
and perform the uncertainty principle analysis in exactly the same way as in
our previous work (e.g. Kruijssen et al., 2019a; Chevance et al., 2020a; Kim
et al., 2022).

##  3 Results

###  3.1 Change of the measured overlap timescale as a function of the spatial
resolution

For each of the six sample galaxies, Table 2 lists the GMC lifetime, overlap
timescale (in other papers often referred to as the ‘feedback timescale’), and
the region separation length. Galaxies with ‘updated’ in their reference
column have had the analysis repeated with the latest version of the analysis
pipeline from Kim et al. (2022). The changes relative to the originally
published numbers fall within the uncertainties. For M83, the table lists new
results that are presented for the first time in this paper. We see that the
GMC lifetime ($\mbox{$t_{\rm GMC}$}=33.2^{+5.6}_{-3.7}~{}\mbox{${\rm Myr}$}$)
and overlap timescale ($\mbox{$t_{\rm over}$}=3.9^{+1.1}_{-0.6}~{}\mbox{${\rm
Myr}$}$) fall within the range of previous results ($\mbox{$t_{\rm
GMC}$}=5{-}35~{}\mbox{${\rm Myr}$}$ and $\mbox{$t_{\rm
over}$}=1{-}6~{}\mbox{${\rm Myr}$}$, respectively, see Kim et al. 2022),
indicating that GMC in M83 experience rapid lifecycles, like GMCs in all other
nearby galaxies. These results also imply that the small-scale decorrelation
of GMCs and Hii regions in M83 is inconsistent with the GMC lifetimes of $\sim
100~{}\mbox{${\rm Myr}$}$ estimated by Koda et al. (2023).444There are two
galaxies for which $\sim 100~{}\mbox{${\rm Myr}$}$ GMC lifetimes have been
claimed in the literature. Next to M83, this also includes M51 (Koda et al.,
2009). Interestingly, our GMC lifetime measurements for both of these galaxies
reside towards the upper end of the observed range of $\mbox{$t_{\rm
GMC}$}=5{-}35~{}\mbox{${\rm Myr}$}$. While for M83 we here obtain
$\mbox{$t_{\rm GMC}$}=33.2^{+5.6}_{-3.7}~{}\mbox{${\rm Myr}$}$, the GMC
lifetime in M51 is found to be $\mbox{$t_{\rm
GMC}$}=30.5^{+9.5}_{-4.8}~{}\mbox{${\rm Myr}$}$ (Chevance et al., 2020a).
Consulting the environmental trends of $t_{\rm GMC}$ identified by Kim et al.
(2022), these galaxies may fall towards the top of the range because they both
have relatively high molecular gas surface densities. In such environments,
the observed GMC lifetimes are typically longer.

Table 2: Uncertainty principle measurement results Galaxy |  $t_{\rm GMC}$ [${\rm Myr}$] |  $t_{\rm over}$ [${\rm Myr}$] |  $\lambda$ [${\rm pc}$] | Reference
---|---|---|---|---
LMC | $11.1^{+1.6}_{-1.7}$ | $1.2^{+0.2}_{-0.2}$ | $71^{+13}_{-8}$ | 1
NGC300 | $10.6^{+1.4}_{-1.5}$ | $1.2^{+0.2}_{-0.1}$ | $113^{+25}_{-17}$ | 2, updated
M33 | $14.5^{+1.6}_{-1.5}$ | $3.3^{+0.6}_{-0.5}$ | $155^{+30}_{-24}$ | 1
NGC628 | $24.0^{+2.4}_{-2.4}$ | $3.2^{+0.5}_{-0.6}$ | $96^{+13}_{-10}$ | 3
NGC5068 | $9.0^{+1.5}_{-1.3}$ | $1.3^{+0.3}_{-0.3}$ | $118^{+19}_{-14}$ | 3, updated
M83 | $33.2^{+5.6}_{-3.7}$ | $3.9^{+1.1}_{-0.6}$ | $86^{+9}_{-7}$ | this work
References: | (1) Kim et al. (2021), (2) Kruijssen et al. (2019a),
| (3) Kim et al. (2022)

The key question then is whether the short GMC lifetimes listed in Table 2
reflect truly short lifetimes due to GMC destruction by feedback, or whether
the GMCs might be ‘immortal’ and decouple from the nascent stellar populations
by stellar drift. In the latter case, the short lifetimes would actually
represent short periods of time in between bursts of star formation. To answer
this question, we perform the experiment outlined in §2.1 and Figure 1. For
each of the six galaxies, we investigate how the overlap timescale changes if
we increase the smallest aperture size down to which we perform the
uncertainty principle fit, for aperture sizes in between the GMC diameter
($2\mbox{$R_{\rm GMC}$}$) and the region separation length ($\lambda$). If the
decoupling between GMCs and Hii regions is caused by stellar drift, then
larger apertures focused on Hii regions are more likely to contain GMCs, which
would result in a proportional increase of the overlap timescale. By contrast,
if the decoupling is caused by feedback-driven GMC destruction, then larger
apertures focused on Hii regions would not have an increased probability of
containing GMCs, and the overlap timescale would be independent of the
aperture size.

Figure 2:  Relative change of the measured overlap timescale as a function of
the spatial resolution, shown for the six galaxies in our sample. The data
points with error bars show the measurements. The relevant range of spatial
resolutions is bracketed by the two grey-shaded areas – the transparent data
points in the grey-shaded ranges either probe sub-GMC scales (towards the
left) or suffer from blending with neighbouring clouds by exceeding the region
separation length (towards the right). The small relevant range of resolutions
illustrates the technical challenge of carrying out this experiment. Within
this range bracketed by the grey-shaded regions, the horizontal solid line
indicates the expectation if stellar feedback causes the GMC-Hii region
decorrelation, whereas the inclined solid line indicates the expectation if
stellar drift drives apart GMCs and their nascent stellar populations. The
figure shows that the data are more consistent with stellar feedback-driven
GMC dispersal than with stellar drift. This is statistically quantified in
§3.2 and in Table 3.

The results of our experiment are shown in Figure 2. Per galaxy, only a few
($2{-}3$) data points fall within the range of spatial resolutions where they
help distinguish both the feedback and drift scenarios, but across all
galaxies a total of 16 data points are suitable for the comparison. Despite
the data scarcity, we clearly see that most galaxies have constant overlap
timescales of the relevant resolution range, indicating that feedback-driven
GMC dispersal is favoured over stellar drift-driven decoupling between GMCs
and Hii regions. The overlap timescales only start to increase at resolutions
coarser than the region separation length ($\ell>\lambda$), reflecting an
artificial increase due to the blending of neighbouring regions (see e.g.
Appendix B2 of Kruijssen et al. 2018). This behaviour is expected when GMCs
are destroyed by stellar feedback.

###  3.2 Statistical tests

We quantify the agreement between the data and the expectations for the
feedback and drift hypotheses using a variety of statistical tests. Although
we note that these tests are not strictly independent, we present the results
of all relevant tests commonly used in hypothesis testing to avoid making an
arbitrary (and potentially biasing) selection of one or two of these. These
tests are performed in logarithmic space, i.e. by using the quantity
$\log{[\mbox{$t_{\rm over}$}(\ell)/\mbox{$t_{\rm over}$}(\ell_{0})]}$. Visual
inspection of the probability distribution function of $t_{\rm over}$ reveals
close-to-Gaussian distributions. For several of the tests, we therefore make
use of $z$-values, which express by how many standard errors the data are
separated from the model expectation:

$z=\frac{x-\mu}{\sigma},$ (3)

where $x$ is the observation, $\mu$ is the value expected for a given model
(i.e. feedback or drift), and $\sigma$ is the $1\sigma$ uncertainty on the
observation. When calculating the $z$-values, we account for the asymmetry of
the error bars. The results of all statistical tests are presented in Table 3.
In the following, we first describe the full set of tests before discussing
the results.

####  3.2.1 Kolmogorov-Smirnov test

We first evaluate the null hypotheses that the $z$-values for the feedback and
drift scenarios are drawn from a standard normal distribution ${\cal N}(0,1)$,
by employing the two-sample Kolmogorov-Smirnov (KS) test. While it is
generally recommended for larger sample sizes and subject to caveats555See
e.g. https://asaip.psu.edu/Articles/beware-the-kolmogorov-smirnov-test/, the
KS test is very commonly used for hypothesis testing due to its simplicity and
flexibility. To perform the test, a reference sample of $10^{5}$ data points
is generated from ${\cal N}(0,1)$, against which the empirical cumulative
distribution functions (CDFs) of the $z$-values are compared. The KS statistic
quantifies the maximum distance between the empirical and the reference CDFs,
and the associated $p$-value expresses the probability of observing such a
statistic under the null hypothesis.

####  3.2.2 Paired $t$-test

We continue our analysis with a paired sample $t$-test for the feedback and
drift scenarios, which provides a statistical evaluation of the mean
differences between the observed data points and the respective expectations
for either scenario. The $t$-statistic is calculated as
$t=\bar{d}/(s_{d}/\sqrt{n})$, where $\bar{d}$ is the mean of the paired
differences, $s_{d}$ is their standard deviation, and $n$ is the number of
observations. The $t$-statistic effectively represents the number of standard
deviations that the sample mean deviates from the null hypothesis that there
is no difference between the observations and expectation. The associated
$p$-value indicates the probability of observing a $t$-statistic at least as
extreme as the one calculated, assuming that the null hypothesis is true.

The $t$-test relies on the standard deviation of the paired differences rather
than on the individual error bars of the data points. Therefore, a scenario
for which the data exhibit greater variations in $d$ may appear more
consistent with the data according to the $t$-test, simply because the data
have a larger standard deviation. This counterintuitive property of the
$t$-test should be kept in mind when evaluating the results.

####  3.2.3 Combined probability test

We conduct further statistical analysis to assess whether the mean and
standard deviation of the $z$-values for both the feedback and drift scenarios
significantly deviate from the expected values of 0 and 1, respectively, under
the assumption of a standard normal distribution ${\cal N}(0,1)$. We use a
one-sample $t$-test to determine if the mean of the $z$-values differs
significantly from 0, resulting in the usual $t$-statistic and $p$-value for
each of the scenarios.

Additionally, we apply a $\chi^{2}$ test to determine if the variance of the
$z$-values differs significantly from unity. This is done by first computing
the sample variance of the $z$-values and dividing this quantity by the number
of degrees of freedom ($n-1$, where $n$ is the sample size). The resulting
$\chi^{2}$ statistic is then compared to the CDF of the $\chi^{2}$
distribution with $n-1$ degrees of freedom. We calculate the $p$-value by
finding the probability that a $\chi^{2}$ distribution with $n-1$ degrees of
freedom would yield a value at least as extreme as the observed $\chi^{2}$
statistic. This probability is given by the integral of the tail of the
$\chi^{2}$ distribution beyond the observed $\chi^{2}$ statistic.

To synthesise the results from both the mean and variance tests, we combine
both $p$-values using Fisher’s method. We first calculate the quantity
$-2\sum\ln(p_{i})$, where $p_{i}$ are the individual $p$-values, and then
compare the result to a $\chi^{2}$ distribution for four degrees of freedom
(corresponding to twice the number of tests, which here are constituted by the
$t$-test and the $\chi^{2}$ test). The integral of the tail of the
distribution beyond this value then yields the combined $p$-value, which
provides a single metric to evaluate the overall deviation of the mean and
standard deviation of the $z$-values from a normal distribution.

####  3.2.4 Sample z-score

To evaluate the deviation of the sample means from the expectations for both
the feedback and drift scenarios, we calculate the sample $z$-scores. The
sample $z$-score is calculated as $z=\bar{z}\sqrt{n}$, where $\bar{z}$ is the
mean of the $z$-values. This transformation yields the $z$-score of the sample
mean, which indicates how many standard deviations the sample mean is from the
null hypothesis mean.

####  3.2.5 $\chi^{2}$ test

To assess the goodness of fit between the observed data points and the
expectations for the feedback and drift scenarios, we employ a $\chi^{2}$
test. The $\chi^{2}$ statistic is calculated for each scenario by summing the
squared differences between the observed and expected values, normalised by
the squared error terms. This calculation incorporates the asymmetric error
bars associated with each data point. The resulting $\chi^{2}$ statistic is
then divided by the number of observations $n$. The corresponding $p$-value is
calculated as described above, using a $\chi^{2}$ distribution with $n$
degrees of freedom, which is appropriate here because the expectations for
both scenarios are fixed and involve no free parameters. This $p$-value
reflects the probability of observing a $\chi^{2}$ statistic at least as
extreme as the one obtained under the null hypothesis that the observed data
are consistent with the expectations.

####  3.2.6 Bayesian comparison

Table 3: Statistical test results Galaxy | Number | KS-test $\log p$-value |  $t$-test $\log p$-value |  $z$-values $\log p$-value | sample $z$-score |  $\chi^{2}$-test $\log p$-value | Bayes $\log$ post. prob.
---|---|---|---|---|---|---|---
| of data | feedback | drift | feedback | drift | feedback | drift | feedback | drift | feedback | drift | feedback | drift
LMC | 3 | -0.40 | -2.34 | -0.05 | -1.02 | -0.01 | -1.00 | -0.03 | -3.45 | -0.00 | -2.23 | -0.01 | -1.66
NGC300 | 3 | -1.19 | -0.47 | -1.97 | -0.56 | -1.04 | -0.32 | 1.14 | -1.00 | -0.14 | -0.18 | -0.58 | -0.13
M33 | 3 | -0.60 | -1.82 | -0.98 | -0.95 | -0.42 | -1.22 | 0.65 | -1.82 | -0.04 | -0.47 | -0.09 | -0.73
NGC628 | 2 | -0.32 | -1.49 | -0.69 | -0.72 | -0.21 | -0.73 | 0.16 | -1.90 | -0.01 | -0.79 | -0.09 | -0.73
NGC5068 | 3 | -0.59 | -2.09 | -0.74 | -1.43 | -0.45 | -1.07 | 0.43 | -2.64 | -0.01 | -1.18 | -0.13 | -0.58
M83 | 2 | -0.30 | -1.24 | -0.71 | -0.69 | -0.22 | -0.46 | -0.09 | -1.70 | -0.00 | -0.66 | -0.09 | -0.74
All | 16 | -2.49 | -6.83 | -2.20 | -5.20 | -1.54 | -5.38 | 0.97 | -5.12 | -0.00 | -1.91 | -0.00 | -3.60

We conclude the suite of statistical tests by comparing the feedback and drift
scenarios in a Bayesian context. This is done by calculating their respective
posterior probabilities based on the observed data. We assign prior
probabilities $p_{\rm fb}=0.5$ and $p_{\rm drift}=0.5$, to reflect a lack of
preference for either scenario. The likelihood of each scenario given the
data, ${\cal L}_{\rm fb}$ and ${\cal L}_{\rm drift}$, is computed as the
product of the probability density functions of the normal distribution, with
means corresponding to the expectations and standard deviations by the
observed error bars. These likelihoods represent the probability of the
observed data under the assumption that a particular scenario is true. The
marginal likelihoods for each scenario, ${\cal M}_{\rm fb}$ and ${\cal M}_{\rm
drift}$, are then calculated by weighting the likelihoods by the respective
prior probabilities (which are both equal to $0.5$ in this case). The total
evidence across both scenarios is then defined as $p_{\rm tot}={\cal M}_{\rm
fb}+{\cal M}_{\rm drift}$. Finally, the posterior probabilities are obtained
as $p_{\rm fb}={\cal M}_{\rm fb}/p_{\rm tot}$ and $p_{\rm drift}={\cal M}_{\rm
drift}/p_{\rm tot}$. These posterior probabilities reflect the degree of
belief in each scenario given the data and the prior information.

####  3.2.7 Test results

Table 3 lists the results of these different tests, showing that we obtain
qualitatively consistent results across the full suite of statistics. The
small number of data points per galaxy causes the comparison to be
inconclusive in some cases, but when performing the tests described above on
the full sample of 16 data points, the probabilities differ significantly
between the stellar feedback and drift scenarios. Across all tests and
galaxies, the range of probability that the data are consistent with stellar
feedback-driven GMC dispersal is $p_{\rm fb}=3.2\times 10^{-3}{-}1$, with a
logarithmic average of $p_{\rm fb}=5.7\times 10^{-2}$. By contrast, the range
of probability that the data are consistent with the stellar drift-driven
decoupling of GMCs and Hii regions is $p_{\rm fb}=1.5\times
10^{-7}{-}1.2\times 10^{-2}$, with a logarithmic average of $p_{\rm
fb}=2.6\times 10^{-5}$. Comparing the probability ratios between all tests
shows that the small-scale decorrelation between GMCs and Hii regions is
approximately 100 to 20,000 times more likely to be driven by stellar feedback
than by stellar drift, with the logarithmic average across all tests
indicating that it is about 2,000 times more likely to be feedback-driven.

###  3.3 Variable drift velocities

One of the key assumptions made in this work is that the drift velocity is
constant. This need not be the case, as the velocity difference between a GMC
and its nascent Hii region might decelerate (e.g. by the mutual gravitational
attraction) or accelerate (e.g. by differential rotation in the galactic
potential, or by stellar feedback from the Hii region acting on the GMC). We
now test whether the data presented in Figure 2 could be reconciled with the
drift scenario by adopting a suitable acceleration law.

For this purpose, we assume a simple power law acceleration relation for
describing the drift out of an aperture with a diameter $\ell\geq
2\mbox{$R_{\rm GMC}$}$:

$v_{\rm drift}(\ell)=v_{0}\left(\frac{\ell}{\ell_{0}}\right)^{\alpha},$ (4)

where the initial conditions are defined as before, i.e.
$\ell_{0}=2\mbox{$R_{\rm GMC}$}$ and $t_{0}=\mbox{$t_{\rm over}$}(\ell_{0})$.
For simplicity, we assume that the velocity at $t<t_{0}$ is constant, such
that $v_{0}=\ell_{0}/t_{0}$. Assuming the acceleration law of equation (4),
the expected relative increase of the overlap time with the spatial resolution
follows by integration of the equation of motion as

$\displaystyle\left[\frac{\mbox{$t_{\rm over}$}(\ell)}{\mbox{$t_{\rm
over}$}(\ell_{0})}\right]_{\rm drift}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{t_{0}}\int_{\ell_{0}}^{\ell}\frac{{\rm d}\ell^{\prime}}{v_{\rm
drift}(\ell^{\prime})}$ (5) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{1}{1-\alpha}\left[\left(\frac{\ell}{\ell_{0}}\right)^{1-\alpha}-1\right]+1&\mbox{if
}\alpha\neq 1\\\ \ln{\left(\frac{\ell}{\ell_{0}}\right)}+1&\mbox{if
}\alpha=1.\end{array}\right.$ (8)

This is a generalised form of equation (2), to which it reduces for a constant
drift velocity ($\alpha=0$). These expressions allow us to determine which
values of $\alpha$ are needed to make the data from Figure 2 consistent with
the stellar drift scenario.

Figure 3 compares the observational data from Figure 2 to the solutions of
equation (5). The lack of change in $t_{\rm over}$ towards coarser resolutions
($\ell>\ell_{0}$) means that making the data consistent with the stellar drift
scenario would require high values of $\alpha>1$ (and all galaxies are
actually consistent with $\alpha\rightarrow\infty$). This implies that the
drift velocity would need to increase superlinearly with distance, and that
the acceleration itself would also increase with distance. Differential
rotation in the galactic potential (‘shear’) is known to drive a velocity
differential that increases linearly with the radial distance ($\alpha=1$;
e.g. Kruijssen et al. 2019b), whereas a possible acceleration due to stellar
feedback acting on the GMC would decrease with distance ($\alpha<1$), rather
than exhibiting the increase with distance ($\alpha>1$) seen here.

Figure 3:  Relative change of the overlap timescale as a function of the
spatial resolution, comparing the observational data (coloured points) to the
expectations in the stellar drift scenario (lines). The comparison is made for
a variety of different acceleration laws, parameterised through $\alpha$ from
equation (4) as indicated by the legend. The colours of the data points are
the same as in Figure 2, enabling the straightforward identification of each
galaxy. Only data points for the relevant range of spatial resolutions
($2\mbox{$R_{\rm GMC}$}\leq\ell\leq\lambda$) are shown. The lack of change in
$t_{\rm over}$ towards coarser resolutions ($\ell>\ell_{0}$) means that making
the data consistent with the stellar drift scenario would require a
superlinear increase of the drift velocity with distance ($\alpha>1$). There
is no known physical agent for such an acceleration.

The result for a variable drift velocity is further visualised in Figure 4,
which repeats the Bayesian comparison of §3.2.6 as a function of
$\alpha$.666We symmetrise the error bars in logarithmic space before carrying
out the comparison. This is done because the probability would otherwise
exhibit discontinuities at values of $\alpha$ for which the lines in Figure 3
precisely match a data point. At such a value, the error bar used in the
comparison changes between the upward and downward error, implying that
asymmetric error bars would cause a jump in probability. The stellar drift
scenario is disfavoured ($p_{\rm drift}/p_{\rm fb}<1$) for all physical values
of $\alpha\leq 1$. In summary, using a variable drift velocity to reconcile
the observations with the stellar drift scenario would require conditions that
are unparalleled in galactic dynamics or stellar feedback physics. This means
that stellar drift is ruled out by the observations even if the drift velocity
is allowed to vary.

##  4 Discussion

The reasonable hypothesis put forward by KT23 is that the decorrelation
between molecular gas and star formation at $\sim 100$-pc scales may originate
from stellar drift. The measurements presented in the current work disprove
this stellar drift hypothesis at the $3{-}4\sigma$ level and show that the
decorrelation is feedback-driven. This conclusion holds even when the drift
velocity is allowed to vary because the data would require the stellar drift
to follow an unphysical acceleration law. Due to the limited spatial dynamic
range over which our measurements can be made, we can draw these conclusions
only for one galaxy individually (the LMC). However, when all measurements are
combined, the data enable an unambiguous interpretation. Quantitatively, we
find that the drift hypothesis is inconsistent with the data, with a
likelihood of roughly 1% (as indicated by a $\chi^{2}$ test), whereas a
Bayesian likelihood comparison between both scenarios shows that the feedback
scenario is 4,000 times more likely to explain the data than drift. Across all
statistical tests, we find that the small-scale decorrelation between GMCs and
Hii regions is on average 2,000 times more likely to be driven by stellar
feedback than by stellar drift. The experiment presented here adds the
necessary empirical and statistical evidence to earlier discussions on this
topic by Kruijssen & Longmore (2014) and Chevance et al. (2020a), who gave
physical reasons why stellar drift is likely a negligible contributor to the
gas-star formation decorrelation compared to stellar feedback-driven GMC
dispersal.

Figure 4:  Posterior probability ratio of the drift and feedback scenarios,
visualising when the drift scenario with a variable drift velocity might be
favoured over the feedback scenario, shown as a function of the acceleration
law slope $\alpha$ (solid line). Equal probability of the feedback and drift
scenarios ($p_{\rm drift}/p_{\rm fb}=1$) is indicated by the horizontal dashed
line. Unphysical values of $\alpha>1$ are indicated by the shaded region,
highlighting that stellar feedback-driven GMC dispersal is favoured ($p_{\rm
drift}/p_{\rm fb}<1$) for all physical drift acceleration laws ($\alpha\leq
1$). As expected, the posterior probability ratio converges to $1$ when
$\alpha\rightarrow\infty$.

In previous studies, we did not distinguish between GMC destruction by
feedback or kinetic dispersal (Kruijssen et al., 2019a). Our new results
suggest that GMCs undergo feedback-driven dispersal at the end of their
lifetimes to the point that they either dissolve altogether or are dispersed
into a diffuse molecular phase. This paper also includes the first use of the
uncertainty principle methodology to characterise the GMC lifecycle in the
nearby spiral galaxy M83, for which Koda et al. (2023) estimated GMC lifetimes
$\sim 100~{}\mbox{${\rm Myr}$}$. Our analysis of the decorrelation between
GMCs and Hii regions shows that the GMC lifetime is considerably shorter, at
$\mbox{$t_{\rm GMC}$}=33.2^{+5.6}_{-3.7}~{}\mbox{${\rm Myr}$}$, and the
‘feedback timescale’ needed for stellar feedback to clear the parent GMC (in
this paper referred to as the ‘overlap timescale’, by exception to other
papers using this method) is $\mbox{$t_{\rm
over}$}=3.9^{+1.1}_{-0.6}~{}\mbox{${\rm Myr}$}$. These numbers are consistent
with the range of values found in $50{-}100$ other nearby galaxies (e.g.
Kruijssen et al., 2019a; Chevance et al., 2020a; Kim et al., 2022), for which
the GMC lifetimes range from $5{-}35~{}\mbox{${\rm Myr}$}$ and the feedback
timescales range from $1{-}6~{}\mbox{${\rm Myr}$}$.

While the main body of their paper presents the stellar drift hypothesis, KT23
conclude by attempting to identify further reasons why the GMC lifetimes
measured with the uncertainty principle methodology might possibly be
underestimated. Unfortunately, these further reasons are rooted in factual
misrepresentations of the uncertainty principle methodology. Because an
underestimation of the measured timescales would also call into question the
physical conclusions of the experiment presented in this paper, we take the
opportunity to briefly correct these misrepresentations here, with references
to earlier studies where these potential issues have been dispelled more
extensively.

  1. 1.

The reference timescale $t_{\rm star,ref}$ from Haydon et al. (2020b) would
“[be] estimated under an assumption that all stars in a [region] form
simultaneously”, and therefore would cause the duration of the entire GMC
lifecycle to be underestimated (KT23). — Fundamentally, the uncertainty
principle methodology measures the relative timescale ratio between the
lifetimes of GMCs and the SFR tracer (usually referring to ionised emission
from Hii regions). The emission timescale of the SFR tracer is known from
stellar evolution and provides the calibration to which all other timescales
are anchored (Haydon et al., 2020a, b). As discussed by Haydon et al. (2020b,
§3.1; and also in previous observational applications, e.g. the Methods
section of  and §3.2 of ), the reference timescale $t_{\rm star,ref}$ refers
exclusively to the lifetime of the Hii region after it is devoid of molecular
gas and the possibility of further star formation has ceased. Any time during
which the GMC may be forming stars over an extended period of time thus
precedes the duration of the reference timescale. Because the physical
definition of $t_{\rm star,ref}$ specifically refers to the GMC-less lifetime
of Hii regions, it enables preceding age spreads of any duration. The total
duration of the stellar timescale is defined as $t_{\rm star}=\mbox{$t_{\rm
over}$}+t_{\rm star,ref}$, where $t_{\rm star,ref}$ is fixed, but there is no
limit placed on any age spread encapsulated by $t_{\rm over}$. Therefore, the
concern raised by KT23 is not justified.

  2. 2.

The uncertainty principle methodology “uses the [gas-to-SFR tracer] flux
ratio, and thus is biased toward the population of the brighter clouds in
spiral arms”, resulting in short cloud lifetimes compared to “long-lived”
inter-arm clouds (KT23). — While using the gas-to-SFR tracer flux ratio in
itself does not lead to a flux bias, it is true that its specific use in the
uncertainty principle methodology results in flux-weighted GMC lifetimes.
However, it has been used to measure GMC lifetimes across $50{-}100$ galaxies
and individual subregions of these, many of which do not contain spiral arms.
This motivated Chevance et al. (2020a) to write “[we] do not find any
dependence of the measured evolutionary timelines on the strength or the
number of spiral arms in the galaxies of our sample. […] As a result, we
suggest that the offsets between molecular clouds and Hii regions
perpendicular to spiral arms that have been used to infer evolutionary
timescales are driven primarily by cloud evolution and feedback rather than by
dynamical drift”. Galaxies with more pronounced spiral arms (e.g. M51 and M83)
have longer GMC lifetimes (see §3 and Chevance et al. 2020a), contrary to the
suggestion made by KT23. Finally, Romanelli et al. (in prep.) have applied the
uncertainty principle methodology separately to GMCs in spiral arms and those
in inter-arm regions, and find no statistically significant difference in GMC
lifetimes between both environments. In conclusion, any bias towards GMCs in
spiral arms does not cause the GMC lifetime measurements to be underestimated.

  3. 3.

The region separation length $\lambda$ would “[depend] on the global
distribution of the star formation activity”, therefore it “does not trace the
local physical processes of star formation” (KT23). — While $\lambda$ is
referred to as the ‘region separation length’, it is important to realise that
this is measured locally, as it measures the divergence scale of the tuning
fork diagram, i.e. the size scale at which GMCs and Hii regions decorrelate
(see e.g. §3.2.11 and §7.2.1 of Kruijssen et al. 2018). Therefore, it differs
fundamentally from the geometric mean separation length that may be obtained
as $l=2\sqrt{A\pi N}$, with $A$ the total area and $N$ the number of GMCs or
Hii regions. In galaxies with strong morphological features such as spiral
arms or bars, $\lambda$ stays close to the actual separation length in the
local environment of the GMCs, whereas $l$ would be inflated by the presence
of voids. As a result, the uncertainty principle methodology only assumes
approximate statistical isotropy on scales $\leq 2\lambda$ (which is
$150{-}400~{}\mbox{${\rm pc}$}$ for the galaxies in this work), which is much
smaller than the size scales of the galactic-morphological features observed
in nearby galaxies. Indeed, Kruijssen et al. (2019a) explicitly demonstrate
that the nearest-neighbour distances of the emission peaks identified in
NGC300 are consistent with the measured region separation lengths (see their
Methods section, eq. (9), and Extended Data Figure 7). As such, the
uncertainty principle methodology is designed specifically to extract the
local physical processes of star formation, independently of the global
distribution of mass or (star formation) activity.

This work provides a further example of how the observed decorrelation of GMCs
and Hii regions on sub-kpc scales holds key information to help constrain the
physical processes governing the GMC lifecycle. The result that stellar
feedback rather than stellar drift marks the end of the GMC lifecycle does not
only settle a crucial aspect of this cycle, but also provides key insights
into the processes that drive galactic ecosystems. The inference that GMCs
have finite lifespans and are destroyed by stellar feedback lends further
credence to the emerging paradigm in which galaxies are composed of building
blocks undergoing vigorous, feedback-driven lifecycles that collectively drive
the baryon cycle and regulate star formation within galaxies. The quantifiable
difference in probabilities favouring feedback over drift provides a robust
statistical basis for this picture, and the ongoing advances in data quality
and statistical methodologies promise to refine these insights further. Such
advancements, building on the uncertainty principle methodology (e.g. by
integrating modern machine learning techniques; Chevance & Kruijssen in
prep.), may provide an increasingly nuanced understanding of the interplay
between the GMC lifecycle and galaxy evolution.

###### Acknowledgements.

Acknowledgments: We thank Jaeyeon Kim for providing the source files from Kim
et al. (2021, 2022), and Jin Koda, Jonathan Tan, and Mark Krumholz for helpful
discussions. J.M.D.K. gratefully acknowledges funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme via the ERC Starting Grant MUSTANG (grant agreement
number 714907). M.C., L.R. and A.R. gratefully acknowledge funding from the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through an
Emmy Noether Research Group (grant number CH2137/1-1). A.G. acknowledges
support from the NSF through AST 2008101, AST 220651, and CAREER 2142300. COOL
Research DAO is a Decentralised Autonomous Organisation supporting research in
astrophysics aimed at uncovering our cosmic origins.
Software: matplotlib (Hunter, 2007), numpy (van der Walt et al., 2011), pandas
(Reback et al., 2020) scipy (Jones et al., 2001) seaborn (Waskom et al.,
2020).
Author contributions: J.M.D.K. carried out the experiment design and wrote the
text, to which M.C. and S.N.L. contributed. M.C. carried out the physical
analysis of the data and produced the figures and tables, to which J.M.D.K.
and S.N.L. contributed. All authors contributed to aspects of the analysis,
the interpretation of the results, and the preparation of the manuscript.

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