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Simon Glover

Publications and source records attributed to Simon Glover.

At least 19 recordsLinked to original sources

Toward Unbiased Abundance Measurements in Inhomogeneous $\mathrm{H\,II}$ Regions

Probing the chemical content of the interstellar medium (ISM) in nearby galaxies provides key insight into their chemical evolution and informs our interpretation of galaxies at higher redshift. However, nonlinear structure in the ISM, including density and temperature inhomogeneities, can bias chemical abundance measurements and systematically affect empirical calibrations derived from them. In this work, we investigate biases in $T_e$-derived oxygen abundance determinations and explore the physical properties that correlate with them. We combine $\mathrm{[O\,II]}\lambda\lambda3726, 3729$ measurements from SITELLE with a full suite of optical emission lines obtained with MUSE. From auroral emission lines ($\mathrm{[N\,II]}\lambda5755$, $\mathrm{[S\,III]}\lambda6312$, and $\mathrm{[O\,II]}\lambda\lambda7320, 7330$) and nebular emission lines (including $\mathrm{[N\,II]}\lambda6584$ and $\mathrm{[S\,III]}\lambda9069$), we derive electron densities, temperatures, and chemical abundances for a sample of $\mathrm{H\, II}$ regions in five galaxies. We find that densities derived from the $\mathrm{[O\,II]}$ auroral-to-nebular ratio are $\sim10^3$ cm$^{-3}$, which is higher than the standard $\mathrm{[S\,II]}$ densities derived from nebular doublet ratios. We demonstrate that combining the $\mathrm{[N,II]}$ electron temperature with the density inferred from the $\mathrm{[O\,II]}$ auroral-to-nebular line ratio yields singly ionized oxygen abundances consistent with literature expectations for a prescription insensitive to density inhomogeneities. We also find that the $\mathrm{[S\,III]}$ temperature provides a reliable estimate of $T_{e,\mathrm{[O\,III]}}$, enabling robust measurements of doubly ionized oxygen abundances. Overall, these results indicate that the abundance discrepancy factor could be higher in more chemically evolved $\mathrm{H\, II}$ regions.

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Surveying the Whirlpool at Arcseconds with NOEMA (SWAN). IV. Extent of active galactic nucleus feedback on the interstellar medium

Active Galactic Nuclei (AGN) are intertwined with galaxy evolution, injecting energy into the interstellar medium (ISM) that could regulate star formation as a galaxy evolves. However, the phenomena through which we observe AGN are multiphase and multiscale, which can lead to conflicting results for how significantly AGN influence the ISM. We endeavor to characterize the spatial extent and dominant modes of AGN feedback in M51, which hosts a low-luminosity Seyfert nucleus and multi-phase outflow. We identified regions dominated by AGN ionization using an emission line ratio (ELR) function constructed from VENGA integral field spectroscopy. We then investigated how AGN feedback influences the ISM using cloud-scale mapping of dense molecular gas tracers HCN(1-0), HNC(1-0), HCO+(1-0), and N2H+(1-0) provided by SWAN. This combined dataset has a resolution of 180pc, providing a clear demarcation of where AGN feedback dominates the ISM. If we assume that N2H+ is the best tracer of dense, cold gas in SWAN, then AGN-dominated regions defined by the ELR all have greater emission in (1-0) transitions in HCN, HNC, and HCO+ than expected if they traced dense gas alone, implying excitation of these lines from the AGN. The ELR better selects these regions than molecular tracers of AGN activity like HCN/HCO+. The highest ELR values are also associated with optical and molecular shock tracers (HNCO/CO), indicating a potential dense molecular outflow in the nucleus that agrees with the heightened N2H+ emission in this limited region. All tracers of AGN activity point to a "two-stage" feedback scenario, whereby mechanical feedback from the jet-ISM interaction spurs soft X-ray emission that excites molecules such as HCN. Dense gas entrenched in a molecular outflow may also lead to a greater chemical abundance of multiple tracers measured with SWAN, but to a lesser extent than excitation from AGN feedback.

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Accelerated size evolution in the FirstLight simulations from z=14 to z=5

Galaxies grow very rapidly during the first Gyr of the Universe, mostly driven by high galaxy efficiencies, particularly relevant at $z>5$. This efficiency is related to high gas densities and/or compact gas distributions within these early galaxies. We want to understand the evolution of the size of galaxies at cosmic dawn, from $z=14$ to $z=5$ and its main drivers. We use the FirstLight database of 430 zoom-in cosmological simulations and radiative transfer calculations to generate synthetic images in seven JWST bands. We add observational effects, inspired by recent JWST deep extragalactic surveys. The size-mass relation is already in place at $z\simeq14$ and it shows a large diversity of galaxy sizes at a fixed mass. Extended (compact) galaxies tend to have higher (lower) specific star-formation rate (sSFR). The mass-dependent slope does not evolve significantly. This is driven by a complex interaction between stellar light and dust. Differential dust attenuation dims galaxy centers and it makes larger sizes, modifying the mass-size slope even in the rest-frame optical. At a fixed mass, galaxy size evolves very fast, as the normalization of the size-mass relation increases by 0.5 dex between $z\simeq14$ and $z\simeq6$, in 600 Myr. The SFR surface density increases with redshift, driven by higher sSFRs and smaller sizes at higher redshifts. Size evolution at a fixed stellar mass accelerates at cosmic dawn, driven by an increasing galaxy efficiency at $z\geq5$.

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Sloan Digital Sky Survey-V: Pioneering Panoptic Spectroscopy

The Sloan Digital Sky Survey-V (SDSS-V) is pioneering panoptic spectroscopy: it is the first all-sky, multi-epoch, optical-to-infrared spectroscopic survey. SDSS-V is mapping the sky with multi-object spectroscopy (MOS) at telescopes in both hemispheres (the 2.5-m Sloan Foundation Telescope at Apache Point Observatory and the 100-inch du Pont Telescope at Las Campanas Observatory), where 500 zonal robotic fiber positioners feed light from a wide-field focal plane to an optical (R$\sim 2000$, 500 fibers) and a near-infrared (R$\sim 22,000$, 300 fibers) spectrograph. In addition to these MOS capabilities, the survey is pioneering ultra wide-field ($\sim$ 4000~deg$^2$) integral field spectroscopy enabled by a new dedicated facility (LVM-I) at Las Campanas Observatory, where an integral field spectrograph (IFS) with 1801 lenslet-coupled fibers arranged in a 0.5 degree diameter hexagon feeds multiple R$\sim$4000 optical spectrographs that cover 3600-9800 angstroms. SDSS-V's hardware and multi-year survey strategy are designed to decode the chemo-dynamical history of the Milky Way Galaxy and tackle fundamental open issues in stellar physics in its Milky Way Mapper program, trace the growth physics of supermassive black holes in its Black Hole Mapper program, and understand the self-regulation mechanisms and the chemical enrichment of galactic ecosystems at the energy-injection scale in its Local Volume Mapper program. The survey is well-timed to multiply the scientific output from major all-sky space missions. The SDSS-V MOS programs began robotic operations in 2021; IFS observations began in 2023 with the completion of the LVM-I facility. SDSS-V builds upon decades of heritage of SDSS's pioneering advances in data analysis, collaboration spirit, infrastructure, and product deliverables in astronomy.

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PHANGS-ML: the universal relation between PAH band and optical line ratios across nearby star-forming galaxies

The structure and chemistry of the dusty interstellar medium (ISM) are shaped by complex processes that depend on the local radiation field, gas composition, and dust grain properties. Of particular importance are Polycyclic Aromatic Hydrocarbons (PAHs), which emit strong vibrational bands in the mid-infrared, and play a key role in the ISM energy balance. We recently identified global correlations between PAH band and optical line ratios across three nearby galaxies, suggesting a connection between PAH heating and gas ionization throughout the ISM. In this work, we perform a census of the PAH heating -- gas ionization connection using $\sim$700,000 independent pixels that probe scales of 40--150 pc in nineteen nearby star-forming galaxies from the PHANGS survey. We find a universal relation between $\log$PAH(11.3 \mic/7.7 \mic) and $\log$([SII]/H$\alpha$) with a slope of $\sim$0.2 and a scatter of $\sim$0.025 dex. The only exception is a group of anomalous pixels that show unusually high (11.3 \mic/7.7 \mic) PAH ratios in regions with old stellar populations and high starlight-to-dust emission ratios. Their mid-infrared spectra resemble those of elliptical galaxies. AGN hosts show modestly steeper slopes, with a $\sim$10\% increase in PAH(11.3 \mic/7.7 \mic) in the diffuse gas on kpc scales. This universal relation implies an emerging simplicity in the complex ISM, with a sequence that is driven by a single varying property: the spectral shape of the interstellar radiation field. This suggests that other properties, such as gas-phase abundances, gas ionization parameter, and grain charge distribution, are relatively uniform in all but specific cases.

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Redshift-dependent galaxy formation efficiency at $z=5-13$ in the FirstLight simulations

Current models of the formation of first galaxies predict low masses and faint objects at extremely high redshifts, z=9-15. However, the first observations of this epoch indicate a higher-than-expected number of bright (sometimes massive) galaxies. Numerical simulations can help to elucidate the mild evolution of the bright end of the UV luminosity function and they can provide the link between the evolution of bright galaxies and variations of the galaxy formation efficiency across different redshifts. We use the FirstLight database of 377 zoom-in cosmological simulations of a mass-complete sample of galaxies. Mock luminosities are estimated by a dust model constrained by current observations of an evolution of the beta-MUV relation at high-z. FirstLight contains a high number of bright galaxies, MUV<-20, consistent with current data at z=6-13. The evolution of the UV cosmic density is driven by the evolution of the galaxy efficiency and the relation between MUV and halo mass. The efficiency of galaxy formation increases significantly with redshift at a fixed halo mass because galactic halos at extremely high redshifts convert gas into stars at a higher rate than at lower redshifts. The high gas densities in galaxies at z>9 enable these high efficiencies. Our simulations predict higher number densities of massive galaxies, Ms=10^9 Msun, than other models with constant efficiency. Cosmological simulations of galaxy formation with self-consistent models of star formation and feedback can reproduce the different regimes of galaxy formation across cosmic history.

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Wide-field CO isotopologue emission and the CO-to-H$_2$ factor across the nearby spiral galaxy M101

Carbon monoxide (CO) emission is the most widely used tracer of the bulk molecular gas in the interstellar medium (ISM) in extragalactic studies. The CO-to-H$_2$ conversion factor, $\alpha_{\rm CO}$, links the observed CO emission to the total molecular gas mass. However, no single prescription perfectly describes the variation of $\alpha_{\rm CO}$ across all environments across galaxies as a function of metallicity, molecular gas opacity, line excitation, and other factors. Using resolved spectral line observations of CO and its isotopologues, we can constrain the molecular gas conditions and link them to a variation in the conversion factor. We present new IRAM 30-m 1mm and 3mm line observations of $^{12}$CO, $^{13}$CO, and C$^{18}$O} across the nearby galaxy M101. Based on the CO isotopologue line ratios, we find that selective nucleosynthesis and opacity changes are the main drivers of the variation in the line emission across the galaxy. Furthermore, we estimated $\alpha_{\rm CO(1-0)}$ using different approaches, including (i) the dust mass surface density derived from far-IR emission as an independent tracer of the total gas surface density and (ii) LTE-based measurements using the optically thin $^{13}$CO(1-0) intensity. We find an average value of $\alpha_{\rm CO}=4.4{\pm}0.9\rm\,M_\odot\,pc^{-2}(K\,km\,s^{-1})^{-1}$ across the galaxy, with a decrease by a factor of 10 toward the 2 kpc central region. In contrast, we find LTE-based values are lower by a factor of 2-3 across the disk relative to the dust-based result. Accounting for $\alpha_{\rm CO}$ variations, we found significantly reduced molecular gas depletion time by a factor 10 in the galaxy's center. In conclusion, our result suggests implications for commonly derived scaling relations, such as an underestimation of the slope of the Kennicutt Schmidt law, if $\alpha_{\rm CO}$ variations are not accounted for.

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Improving Star Cluster Age Estimates in PHANGS-HST Galaxies and the Impact on Cluster Demographics in NGC 628

A long-standing problem when deriving the physical properties of stellar populations is the degeneracy between age, reddening, and metallicity. When a single metallicity is used for all star clusters in a galaxy, this degeneracy can result in $`$catastrophic$'$ errors for old globular clusters. Typically, approximately 10 - 20 % of all clusters detected in spiral galaxies can have ages that are incorrect by a factor of ten or more. In this paper we present a pilot study for four galaxies (NGC 628, NGC 1433, NGC 1365, and NGC 3351) from the PHANGS-HST survey. We describe methods to correct the age-dating for old globular clusters, by first identifying candidates using their colors, and then reassigning ages and reddening based on a lower metallicity solution. We find that young $`$interlopers$'$ can be identified from their Halpha flux. CO (2-1) intensity or the presence of dust can also be used, but our tests show that they do not work as well. Improvements in the success fraction are possible at the $\sim$ 15 % level (reducing the fraction of catastrophic age-estimates from between 13 - 21 % to 3 - 8 %). A large fraction of the incorrectly age-dated globular clusters are systematically given ages around 100 Myr, polluting the younger populations as well. Incorrectly age-dated globular clusters significantly impact the observed cluster age distribution in NGC 628, which affects the physical interpretation of cluster disruption in this galaxy. For NGC 1365, we also demonstrate how to fix a second major age-dating problem, where very dusty young clusters with E(B-V) $>$ 1.5 mag are assigned old, globular-cluster like ages. Finally, we note the discovery of a dense population of $\sim$ 300 Myr clusters around the central region of NGC 1365. and discuss how this results naturally from the dynamics in a barred galaxy.

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Primordial magnetic fields in Population III star formation: a magnetised resolution study

Population III stars form in groups due to the fragmentation of primordial gas. While uniform magnetic fields have been shown to support against fragmentation in present day star formation, it is unclear whether realistic k^3/2 primordial fields can have the same effect. We bypass the issues associated with simulating the turbulent dynamo by introducing a saturated magnetic field at equipartition with the velocity field when the central densities reaches 10-13 g cm-3. We test a range of sink particle creation densities from 10-10-10-8 g cm-3. Within the range tested, the fields did not suppress fragmentation of the gas and hence could not prevent the degree of fragmentation from increasing with increased resolution. The number of sink particles formed and total mass in sink particles was unaffected by the magnetic field across all seed fields and resolutions. The magnetic pressure remained sub-dominant to the gas pressure except in the highest density regions of the simulation box, where it became equal to but never exceeded gas pressure. Our results suggest that the inclusion of magnetic fields in numerical simulations of Pop III star formation is largely unimportant.

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FirstLight IV: Diversity in sub-L$_*$ galaxies at cosmic dawn

Using a large sample of sub-L$_*$ galaxies, with similar UV magnitudes, M$_{\rm UV}\simeq -19$ at $z\simeq6$, extracted from the FirstLight simulations, we show the diversity of galaxies at the end of the reionization epoch. We find a factor $\sim$40 variation in the specific star-formation rate (sSFR). This drives a $\sim$1 dex range in equivalent width of the [OIII]$\lambda$5007 line. Variations in nebular metallicity and ionization parameter within HII regions lead to a scatter in the equivalent widths and [OIII]/H$\alpha$ line ratio at a fixed sSFR. [OIII]-bright emitters ([OIII]/H$\alpha$>1) have higher ionization parameters and/or higher metallicities than H$\alpha$-bright ([OIII]/H$\alpha$<1) galaxies. According to the surface brightness maps in both [OIII] and H$\alpha$, [OIII]-bright emitters are more compact than H$\alpha$-bright galaxies. H$\alpha$ luminosity is higher than [OIII] if star formation is distributed over extended regions. OIII dominates if it is concentrated in compact clumps. In both cases, the H$\alpha$-emitting gas is significantly more extended than [OIII].

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The Gas-Star Formation Cycle in Nearby Star-Forming Galaxies I. Assessment of Multi-scale Variations

The processes regulating star formation in galaxies are thought to act across a hierarchy of spatial scales. To connect extragalactic star formation relations from global and kpc-scale measurements to recent cloud-scale resolution studies, we have developed a simple, robust method that quantifies the scale dependence of the relative spatial distributions of molecular gas and recent star formation. In this paper, we apply this method to eight galaxies with roughly 1 arcsec resolution molecular gas imaging from the PHANGS-ALMA and PAWS surveys that have matched resolution, high quality narrowband Halpha imaging. At a common scale of 140pc, our massive (log(Mstar/Msun)=9.3-10.7), normally star-forming (SFR/Msun/yr=0.3-5.9) galaxies exhibit a significant reservoir of quiescent molecular gas not associated with star formation as traced by Halpha emission. Galactic structures act as backbones for both molecular and HII region distributions. As we degrade the spatial resolution, the quiescent molecular gas disappears, with the most rapid changes occurring for resolutions up to about 0.5kpc. As the resolution becomes poorer, the morphological features become indistinct for spatial scales larger than about 1kpc. The method is a promising tool to search for relationships between the quiescent or star-forming molecular reservoir and galaxy properties, but requires a larger sample size to identify robust correlations between the star-forming molecular gas fraction and global galaxy parameters.

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The Molecular Gas Reservoirs of $z\sim 2$ Galaxies: A comparison of CO(1-0) and dust-based molecular gas masses

We test the use of long-wavelength dust continuum emission as a molecular gas tracer at high redshift, via a unique sample of 12, z~2 galaxies with observations of both the dust continuum and CO(1-0) line emission (obtained with the Atacama Large Millimeter Array and Karl G. Jansky Very Large Array, respectively). Our work is motivated by recent, high redshift studies that measure molecular gas masses (\ensuremath{\rm{M}_{\rm{mol}}}) via a calibration of the rest-frame $850μ$m luminosity ($L_\mathrm{850μm,rest}$) against the CO(1-0)-derived \ensuremath{\rm{M}_{\rm{mol}}}\ of star-forming galaxies. We hereby test whether this method is valid for the types of high-redshift, star-forming galaxies to which it has been applied. We recover a clear correlation between the rest-frame $850μ$m luminosity, inferred from the single-band, long-wavelength flux, and the CO(1-0) line luminosity, consistent with the samples used to perform the $850μ$m calibration. The molecular gas masses, derived from $L_\mathrm{850μm,rest}$, agree to within a factor of two with those derived from CO(1-0). We show that this factor of two uncertainty can arise from the values of the dust emissivity index and temperature that need to be assumed in order to extrapolate from the observed frequency to the rest-frame at 850$\mathrm{μm}$. The extrapolation to 850$\mathrm{μm}$ therefore has a smaller effect on the accuracy of \Mmol\ derived via single-band dust-continuum observations than the assumed CO(1-0)-to-\ensuremath{\rm{M}_{\rm{mol}}}\ conversion factor. We therefore conclude that single-band observations of long-wavelength dust emission can be used to reliably constrain the molecular gas masses of massive, star-forming galaxies at $z\gtrsim2$.

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FirstLight III: Rest-frame UV-optical spectral energy distributions of simulated galaxies at cosmic dawn

Using the FirstLight database of 300 zoom-in cosmological simulations we provide rest-frame UV-optical spectral energy distributions of galaxies with complex star-formation histories that are coupled to the non-uniform gas accretion history of galactic halos during cosmic dawn. The population at any redshift is very diverse ranging from starbursts to quiescent galaxies even at a fixed stellar mass. This drives a redshift-dependent relation between UV luminosity and stellar mass with a large scatter, driven by the specific star formation rate. The UV slope and the production efficiency of Lyman continuum photons have high values, consistent with dust-corrected observations. This indicates young stellar populations with low metallicities. The FirstLight simulations make predictions on the rest-frame UV-optical absolute magnitudes, colors and optical emission lines of galaxies at z=6-12 that will be observed for the first time with JWST and the next generation of telescopes in the coming decade.

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FirstLight II: Star formation rates of primeval galaxies from z=5-15

In the FirstLight project, we have used ~300 cosmological, zoom-in simulations to determine the star-formation histories of distinct first galaxies with stellar masses between Ms=10^6 and 3 x 10^9 Msun during cosmic dawn (z=5-15). The evolution of the star formation rate (SFR) in each galaxy is complex and diverse, characterized by bursts of star formation. Overall, first galaxies spend 70% of their time in SF bursts. A sample of 1000 of these bursts indicates that the typical burst at z=6 has a specific SFR (sSFR) maximum of 5-15 Gyr^-1 with an effective width of ~100 Myr, one tenth of the age of the Universe at that redshift. A quarter of the bursts populate a tail with very high sSFR maxima of 20-30 Gyr^-1 and significantly shorter timescales of ~40-80 Myr. This diversity of bursts sets the mean and the mass-dependent scatter of the star-forming main sequence. This scatter is driven by a population of low-mass, Ms < 10^8 Msun, quiescent galaxies. The mean sSFR and the burst maximum at fixed mass increase with redshift, with the later always being a factor ~2 higher than the former. This implies sSFR maxima of ~20-60 Gyr^-1 at z=9-10. The SFR histories are publicly available at the FirstLight website.

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Phantom: A smoothed particle hydrodynamics and magnetohydrodynamics code for astrophysics

We present Phantom, a fast, parallel, modular and low-memory smoothed particle hydrodynamics and magnetohydrodynamics code developed over the last decade for astrophysical applications in three dimensions. The code has been developed with a focus on stellar, galactic, planetary and high energy astrophysics and has already been used widely for studies of accretion discs and turbulence, from the birth of planets to how black holes accrete. Here we describe and test the core algorithms as well as modules for magnetohydrodynamics, self-gravity, sink particles, H_2 chemistry, dust-gas mixtures, physical viscosity, external forces including numerous galactic potentials as well as implementations of Lense-Thirring precession, Poynting-Robertson drag and stochastic turbulent driving. Phantom is hereby made publicly available.

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Introducing the FirstLight project: UV luminosity function and scaling relations of primeval galaxies

We introduce the FirstLight project that aims to generate a large database of high-resolution, zoom-in simulations of galaxy formation around the epoch of reionisation ($z\geq6$). The first results of this program agree well with recent observational constraints at z=6-8, including the UV luminosity function and galaxy stellar mass function, as well as the scaling relationships between halo mass, stellar mass, and UV magnitude. The UV luminosity function starts to flatten below MUV>-14 due to stellar feedback in halos with maximum circular velocities of V=30-40 km/s. The power-law slope of the luminosity function evolves rapidly with redshift, reaching a value of alpha=-2.5 at z=10. On the other hand, the galaxy stellar mass function evolves slowly with time between z=8-10, in particular at the low-mass end.

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The Survey of Lines in M31 (SLIM): The Drivers of the [CII]/TIR Variation

The ratio of the [CII] 158$\,μ$m emission line over the total infrared emission (TIR) is often used as a proxy for the photoelectric (PE) heating efficiency ($ε_{\rm PE}$) of the far-ultraviolet (FUV) photons absorbed by dust in the interstellar medium. In the nearby galaxy M31, we measure a strong radial variation of [CII]/TIR that we rule out as being due to an intrinsic variation in $ε_{\rm PE}$. [CII]/TIR fails as a proxy for $ε_{\rm PE}$, because the TIR measures all dust heating, not just the contribution from FUV photons capable of ejecting electrons from dust grains. Using extensive multiwavelength coverage from the FUV to far-infrared (FIR), we infer the attenuated FUV emission ($\rm UV_{att}$), and the total attenuated flux ($\rm TOT_{att}$). We find [CII]/TIR to be strongly correlated with $\rm UV_{att}$/$\rm TOT_{att}$, indicating that, in M31 at least, one of the dominant drivers for [CII]/TIR variation is the relative hardness of the absorbed stellar radiation field. We define $\rm{ ε_{PE}^{UV}}$, [CII]/$\rm{ UV_{att}}$ which should be more closely related to the actual PE efficiency, which we find to be essentially constant ($1.85 \pm 0.8 \%$) in all explored fields in M31. This suggests that part of the observed variation of [CII]/TIR in other galaxies is likely due to a change in the relative hardness of the absorbed stellar radiation field, caused by a combination of variations in the stellar population, dust opacity and galaxy metallicity, although PE efficiency may also vary across a wider range of environments.

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Effects of binary stellar populations on direct collapse black hole formation

The critical Lyman--Werner flux required for direct collapse blackholes (DCBH) formation, or J$_{crit}$, depends on the shape of the irradiating spectral energy distribution (SED). The SEDs employed thus far have been representative of {realistic} single stellar populations. We study the effect of binary stellar populations on the formation of DCBH, as a result of their contribution to the Lyman--Werner radiation field. Although binary populations with ages $>$ 10 Myr yield a larger LW photon output, we find that the corresponding values of J$_{crit}$ can be up to 100 times higher than single stellar populations. We attribute this to the shape of the binary SEDs as they produce a sub--critical rate of H$^-$ photodetaching 0.76 eV photons as compared to single stellar populations, reaffirming the role that H$^-$ plays in DCBH formation. This further corroborates the idea that DCBH formation is better understood in terms of a critical region in the H$_2$--H$^-$ photo--destruction rate parameter space, rather than a single value of LW flux.

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