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Paul Torrey

Publications and source records attributed to Paul Torrey.

At least 163 records · Page 9Linked to original sources

First results from the IllustrisTNG simulations: matter and galaxy clustering

Hydrodynamical simulations of galaxy formation have now reached sufficient volume to make precision predictions for clustering on cosmologically relevant scales. Here we use our new IllustrisTNG simulations to study the non-linear correlation functions and power spectra of baryons, dark matter, galaxies and haloes over an exceptionally large range of scales. We find that baryonic effects increase the clustering of dark matter on small scales and damp the total matter power spectrum on scales up to k ~ 10 h/Mpc by 20%. The non-linear two-point correlation function of the stellar mass is close to a power-law over a wide range of scales and approximately invariant in time from very high redshift to the present. The two-point correlation function of the simulated galaxies agrees well with SDSS at its mean redshift z ~ 0.1, both as a function of stellar mass and when split according to galaxy colour, apart from a mild excess in the clustering of red galaxies in the stellar mass range 10^9-10^10 Msun/h^2. Given this agreement, the TNG simulations can make valuable theoretical predictions for the clustering bias of different galaxy samples. We find that the clustering length of the galaxy auto-correlation function depends strongly on stellar mass and redshift. Its power-law slope gamma is nearly invariant with stellar mass, but declines from gamma ~ 1.8 at redshift z=0 to gamma ~ 1.6 at redshift z ~ 1, beyond which the slope steepens again. We detect significant scale-dependencies in the bias of different observational tracers of large-scale structure, extending well into the range of the baryonic acoustic oscillations and causing nominal (yet fortunately correctable) shifts of the acoustic peaks of around ~5%.

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Mg II Absorption at 2<z<7 with Magellan/FIRE, III. Full Statistics of Absorption Towards 100 High-Redshift QSOs

We present final statistics from a survey for intervening MgII absorption towards 100 quasars with emission redshifts between $z=3.55$ and $z=7.08$. Using infrared spectra from Magellan/FIRE, we detect 279 cosmological MgII absorbers, and confirm that the incidence rate of $W_r>0.3 Å$ MgII absorption per comoving path length does not evolve measurably between $z=0.25$ and $z=7$. This is consistent with our detection of seven new MgII systems at $z>6$, a redshift range that was not covered in prior searches. Restricting to relatively strong MgII systems ($W_r>1$Å), there is significant evidence for redshift evolution. These systems roughly double in number density between $z=0$ and $z=2$-$3$, but decline by an order of magnitude from this peak by $z\sim 6$. This evolution mirrors that of the global star formation rate density, which could reflect a connection between star formation feedback and strong MgII absorbers. We compared our results to the Illustris cosmological simulation at $z=2$-$4$ by assigning absorption to catalogued dark-matter halos and by direct extraction of spectra from the simulation volume. To reproduce our results using the halo catalogs, we require circumgalactic (CGM) MgII envelopes within halos of progressively smaller mass at earlier times. This occurs naturally if we define the lower integration cutoff using SFR rather than mass. MgII profiles calculated directly from the Illustris volume yield far too few strong absorbers. We argue that this arises from unresolved phase space structure of CGM gas, particularly from turbulent velocities on sub-mesh scales. The presence of CGM MgII at $z>6$-- just $\sim 250$ Myr after the reionization redshift implied by Planck--suggests that enrichment of intra-halo gas may have begun before the presumed host galaxies' stellar populations were mature and dynamically relaxed. [abridged]

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Similar star formation rate and metallicity evolution timescales drive the fundamental metallicity relation

The fundamental metallicity relation (FMR) is a postulated correlation between galaxy stellar mass, star formation rate (SFR), and gas-phase metallicity. At its core, this relation posits that offsets from the mass-metallicity relation (MZR) at a fixed stellar mass are correlated with galactic SFR. In this Letter, we quantify the timescale with which galactic SFRs and metallicities evolve using hydrodynamical simulations. We find that Illustris and IllustrisTNG predict that galaxy offsets from the star formation main sequence and MZR evolve over similar timescales, are often anti-correlated in their evolution, evolve with the halo dynamical time, and produce a pronounced FMR. In fact, for a FMR to exist, the metallicity and SFR must evolve in an anti-correlated sense which requires that they evolve with similar time variability. In contrast to Illustris and IllustrisTNG, we speculate that the SFR and metallicity evolution tracks may become decoupled in galaxy formation models dominated by globally-bursty SFR histories, which could weaken the FMR residual correlation strength. This opens the possibility of discriminating between bursty and non-bursty feedback models based on the strength and persistence of the FMR -- especially at high redshift.

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The Size Evolution of Star-forming and Quenched Galaxies in the IllustrisTNG simulation

We analyze scaling relations and evolution histories of galaxy sizes in TNG100, part of the IllustrisTNG simulation suite. Observational qualitative trends of size with stellar mass, star-formation rate and redshift are reproduced, and a quantitative comparison of projected r-band sizes at 0~ ~10^{9.5}Msun, the evolution of the median main progenitor differs, with quenched galaxies hardly growing in median size before quenching, whereas main-sequence galaxies grow their median size continuously, thus opening a gap from the progenitors of quenched galaxies. This is partly because the main-sequence high-redshift progenitors of quenched z=0 galaxies are drawn from the lower end of the size distribution of the overall population of main-sequence high-redshift galaxies. (ii) Quenched galaxies with M_{*,z=0}>~10^{9.5}Msun experience a steep size growth on the size-mass plane after their quenching time, but with the exception of galaxies with M_{*,z=0}>~10^{11}Msun, the size growth after quenching is small in absolute terms, such that most of the size (and mass) growth of quenched galaxies (and its variation among them) occurs while they are still on the main-sequence. After they become quenched, the size growth rate of quenched galaxies as a function of time, as opposed to versus mass, is similar to that of main-sequence galaxies. Hence, the size gap is retained down to z=0.

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The evolution of the mass-metallicity relation in IllustrisTNG

The coevolution of galaxies and their metal content serves as an important test for galaxy feedback models. We analyze the distribution and evolution of metals within the IllustrisTNG simulation suite with a focus on the gas-phase mass-metallicity relation (MZR). We find that the IllustrisTNG model broadly reproduces the slope and normalization evolution of the MZR across the redshift range $0 2$) evolution fit by $\mathrm{d\;log(Z)}/\mathrm{dz} \approx - 0.064$. Our simulations indicate that the metal retention efficiency of the interstellar medium (ISM) is low: a majority of gas-phase metals ($\sim$ 85 per cent at $z=0$) live outside of the ISM, either in an extended gas disk, the circumgalactic medium, or outside the halo. Nevertheless, the redshift evolution in the simulated MZR normalization is driven by the higher gas fractions of high redshift galaxies, not by changes to the metal retention efficiency. The scatter in the simulated MZR contains a clear correlation with the gas-mass or star formation rate of the system, in agreement with the observed fundamental metallicity relation. The scatter in the MZR is driven by a competition between periods of enrichment- and accretion-dominated metallicity evolution. We expect that while the normalization of the MZR declines with redshift, the strength of the correlation between metallicity and gas-mass at fixed stellar mass is not a strong function of redshift. Our results indicate that the "regulator" style models are best suited for simultaneously explaining the shape, redshift evolution, and existence of correlated scatter with gas fraction about the MZR.

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The uniformity and time-invariance of the intra-cluster metal distribution in galaxy clusters from the IllustrisTNG simulations

The distribution of metals in the intra-cluster medium encodes important information about the enrichment history and formation of galaxy clusters. Here we explore the metal content of clusters in IllustrisTNG - a new suite of galaxy formation simulations building on the Illustris project. Our cluster sample contains 20 objects in TNG100 - a ~(100 Mpc)^3 volume simulation with 2x1820^3 resolution elements, and 370 objects in TNG300 - a ~(300 Mpc)^3 volume simulation with 2x2500^3 resolution elements. The z=0 metallicity profiles agree with observations, and the enrichment history is consistent with observational data going beyond z~1, showing nearly no metallicity evolution. The abundance profiles vary only minimally within the cluster samples, especially in the outskirts with a relative scatter of ~15%. The average metallicity profile flattens towards the center, where we find a logarithmic slope of -0.1 compared to -0.5 in the outskirts. Cool core clusters have more centrally peaked metallicity profiles (~0.8 solar) compared to non-cool core systems (~0.5 solar), similar to observational trends. Si/Fe and O/Fe radial profiles follow positive gradients. The outer abundance profiles do not evolve below z~2, whereas the inner profiles flatten towards z=0. More than ~80% of the metals in the intra-cluster medium have been accreted from the proto-cluster environment, which has been enriched to ~0.1 solar already at z~2. We conclude that the intra-cluster metal distribution is uniform among our cluster sample, nearly time-invariant in the outskirts for more than 10 Gyr, and forms through a universal enrichment history.

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Simulating Galaxy Formation with the IllustrisTNG Model

We introduce an updated physical model to simulate the formation and evolution of galaxies in cosmological, large-scale gravity+magnetohydrodynamical simulations with the moving mesh code AREPO. The overall framework builds upon the successes of the Illustris galaxy formation model, and includes prescriptions for star formation, stellar evolution, chemical enrichment, primordial and metal-line cooling of the gas, stellar feedback with galactic outflows, and black hole formation, growth and multi-mode feedback. In this paper we give a comprehensive description of the physical and numerical advances which form the core of the IllustrisTNG (The Next Generation) framework. We focus on the revised implementation of the galactic winds, of which we modify the directionality, velocity, thermal content, and energy scalings, and explore its effects on the galaxy population. As described in earlier works, the model also includes a new black hole driven kinetic feedback at low accretion rates, magnetohydrodynamics, and improvements to the numerical scheme. Using a suite of (25 Mpc $h^{-1}$)$^3$ cosmological boxes we assess the outcome of the new model at our fiducial resolution. The presence of a self-consistently amplified magnetic field is shown to have an important impact on the stellar content of $10^{12} M_{\rm sun}$ haloes and above. Finally, we demonstrate that the new galactic winds promise to solve key problems identified in Illustris in matching observational constraints and affecting the stellar content and sizes of the low mass end of the galaxy population.

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Supermassive black holes and their feedback effects in the IllustrisTNG simulation

We study the population of supermassive black holes (SMBHs) and their effects on massive central galaxies in the IllustrisTNG cosmological hydrodynamical simulations of galaxy formation. The employed model for SMBH growth and feedback assumes a two-mode scenario in which the feedback from active galactic nuclei occurs through a kinetic, comparatively efficient mode at low accretion rates relative to the Eddington limit, and in the form of a thermal, less efficient mode at high accretion rates. We show that the quenching of massive central galaxies happens coincidently with kinetic-mode feedback, consistent with the notion that active supermassive black cause the low specific star formation rates observed in massive galaxies. However, major galaxy mergers are not responsible for initiating most of the quenching events in our model. Up to black hole masses of about $10^{8.5}\,{\rm M}_\odot$, the dominant growth channel for SMBHs is in the thermal mode. Higher mass black holes stay mainly in the kinetic mode and gas accretion is self-regulated via their feedback, which causes their Eddington ratios to drop, with SMBH mergers becoming the main channel for residual mass growth. As a consequence, the quasar luminosity function is dominated by rapidly accreting, moderately massive black holes in the thermal mode. We show that the associated growth history of SMBHs produces a low-redshift quasar luminosity function and a redshift zero black hole mass-stellar bulge mass relation in good agreement with observations, whereas the simulation tends to over-predict the high-redshift quasar luminosity function.

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Black Holes on FIRE: Stellar Feedback Limits Early Feeding of Galactic Nuclei

We introduce massive black holes (BHs) in the Feedback In Realistic Environments project and perform high-resolution cosmological hydrodynamic simulations of quasar-mass halos ($M_{\rm halo}(z=2) \approx 10^{12.5}\,\rm{M}_{\odot}$) down to $z=1$. These simulations model stellar feedback by supernovae, stellar winds, and radiation, and BH growth using a gravitational torque-based prescription tied to resolved properties of galactic nuclei. We do not include BH feedback. We show that early BH growth occurs through short ($\lesssim 1\,$Myr) accretion episodes that can reach or even exceed the Eddington rate. In this regime, BH growth is limited by bursty stellar feedback continuously evacuating gas from galactic nuclei, and BHs remain under-massive relative to the local $M_{\rm BH}$-$M_{\rm bulge}$ relation. BH growth is more efficient at later times, when the nuclear stellar potential retains a significant gas reservoir, star formation becomes less bursty, and galaxies settle into a more ordered state, with BHs rapidly converging onto the scaling relation when the host reaches $M_{\rm bulge} \sim 10^{10}\,\rm{M}_{\odot}$. Our results are not sensitive to the details of the accretion model so long as BH growth is tied to the gas content within $\sim 100\,$pc of the BH. Our simulations imply that bursty stellar feedback has strong implications for BH and AGN demographics, especially in the early Universe and for low-mass galaxies.

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Simulating the dust content of galaxies: successes and failures

We present full volume cosmological simulations using the moving-mesh code AREPO to study the coevolution of dust and galaxies. We extend the dust model in AREPO to include thermal sputtering of grains and investigate the evolution of the dust mass function, the cosmic distribution of dust beyond the interstellar medium, and the dependence of dust-to-stellar mass ratio on galactic properties. The simulated dust mass function is well-described by a Schechter fit and lies closest to observations at $z = 0$. The radial scaling of projected dust surface density out to distances of $10 \, \text{Mpc}$ around galaxies with magnitudes $17 < i < 21$ is similar to that seen in Sloan Digital Sky Survey data, albeit with a lower normalisation. At $z = 0$, the predicted dust density of $Ω_\text{dust} \approx 1.3 \times 10^{-6}$ lies in the range of $Ω_\text{dust}$ values seen in low-redshift observations. We find that dust-to-stellar mass ratio anti-correlates with stellar mass for galaxies living along the star formation main sequence. Moreover, we estimate the $850 \, μ\text{m}$ number density functions for simulated galaxies and analyse the relation between dust-to-stellar flux and mass ratios at $z = 0$. At high redshift, our model fails to produce enough dust-rich galaxies, and this tension is not alleviated by adopting a top-heavy initial mass function. We do not capture a decline in $Ω_\text{dust}$ from $z = 2$ to $z = 0$, which suggests that dust production mechanisms more strongly dependent on star formation may help to produce the observed number of dusty galaxies near the peak of cosmic star formation.

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Log-normal star formation histories in simulated and observed galaxies

Gladders et al. have recently suggested that the star formation histories (SFHs) of individual galaxies are characterized by a log-normal function in time, implying a slow decline rather than rapid quenching. We test their conjecture on theoretical SFHs from the cosmological simulation Illustris and on observationally inferred SFHs. While the log-normal form necessarily ignores short-lived features such as starbursts, it fits the overall shape of the majority of SFHs very well. In particular, 85% of the cumulative SFHs are fitted to within a maximum error of 5% of the total stellar mass formed, and 99% to within 10%. The log-normal performs systematically better than the commonly used delayed-tau model, and is superseded only by functions with more than three free parameters. Poor fits are mostly found in galaxies that were rapidly quenched after becoming satellites. We explore the log-normal parameter space of normalization, peak time, and full width at half maximum, and find that the simulated and observed samples occupy similar regions, though Illustris predicts wider, later-forming SFHs on average. The ensemble of log-normal fits correctly reproduces complex metrics such as the evolution of Illustris galaxies across the star formation main sequence, but overpredicts their quenching timescales. SFHs in Illustris are a diverse population not determined by any one physical property of galaxies, but follow a tight relation, where $\mathrm{width}\propto\mathrm{(peak\ time)}^{3/2}$. We show that such a relation can be explained qualitatively (though not quantitatively) by a close connection between the growth of dark matter halos and their galaxies.

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Metal flows of the circumgalactic medium, and the metal budget in galactic halos

We present an analysis of the flow of metals through the circumgalactic medium (CGM) in the Feedback in Realistic Environments (FIRE) simulations of galaxy formation, ranging from isolated dwarfs to $L*$ galaxies. We find that nearly all metals produced in high-redshift galaxies are carried out in winds that reach $0.25 R_{\rm vir}$. When measured at $0.25 R_{\rm vir}$ the metallicity of outflows is slightly higher than the interstellar medium (ISM) metallicity. Many metals thus reside in the CGM. Cooling and recycling from this reservoir determine the metal budget in the ISM. The outflowing metal flux decreases by a factor of $\sim2-5$ between $0.25 R_{\rm vir}$ and $R_{\rm vir}$. Furthermore, outflow metallicity is typically lower at $R_{\rm vir}$ owing to dilution of the remaining outflow by metal-poor material swept up from the CGM. The inflow metallicity at $R_{\rm vir}$ is generally low, but outflow and inflow metallicities are similar in the inner halo. At low redshift, massive galaxies no longer generate outflows that reach the CGM, causing a divergence in CGM and ISM metallicity. Dwarf galaxies continue to generate outflows, although they preferentially retain metal ejecta. In all but the least massive galaxy considered, a majority of the metals are within the halo at $z=0$. We measure the fraction of metals in CGM, ISM and stars, and quantify the thermal state of CGM metals in each halo. The total amount of metals in the low-redshift CGM of two simulated $L*$ galaxies is consistent with estimates from the COS halos survey, while for the other two it appears to be lower.

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Massive Close Pairs Measure Rapid Galaxy Assembly in Mergers at High Redshift

We compare mass-selected close pairs at z > 1 with the intrinsic galaxy merger rate in the Illustris Simulations. To do so, we construct three 140 arcmin^2 lightcone catalogs and measure pair fractions, finding that they change little or decrease with increasing redshift at z > 1. Consistent with current surveys, this trend requires a decrease in the merger-pair observability time, roughly as (1 + z)^-2, in order to measure the merger rates of the same galaxies. This implies that major mergers are more common at high redshift than implied by the simplest arguments assuming a constant observability time. Several effects contribute to this trend: (1) The fraction of massive, major (4:1) pairs which merge by today increases weakly from ~0.5 at z=1 to ~0.8 at z=3. (2) The median time elapsed between an observed pair and final remnant decreases by a factor of two from z~1 to z~3. (3) An increasing specific star formation rate (sSFR) decreases the time during which common stellar-mass based pair selection criteria could identify the mergers. The average orbit of the pairs at observation time varies only weakly, suggesting that the dynamical time is not varying enough to account by itself for the pair fraction trends. Merging pairs reside in dense regions, having overdensity ~10 to ~100 times greater than the average massive galaxy. We forward model the pairs to reconstruct the merger remnant production rate, showing that it is consistent with a rapid increase in galaxy merger rates at z > 1.

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Simulating galaxy formation with black hole driven thermal and kinetic feedback

The inefficiency of star formation in massive elliptical galaxies is widely believed to be caused by the interactions of an active galactic nucleus (AGN) with the surrounding gas. Achieving a sufficiently rapid reddening of moderately massive galaxies without expelling too many baryons has however proven difficult for hydrodynamical simulations of galaxy formation, prompting us to explore a new model for the accretion and feedback effects of supermassive black holes. For high accretion rates relative to the Eddington limit, we assume that a fraction of the accreted rest mass energy heats the surrounding gas thermally, similar to the `quasar mode' in previous work. For low accretion rates, we invoke a new, pure kinetic feedback model which imparts momentum into the surrounding gas in a stochastic manner. These two modes of feedback are motivated both by theoretical conjectures for the existence of different types of accretion flows as well as recent observational evidence for the importance of kinetic AGN winds in quenching galaxies. We find that a large fraction of the injected kinetic energy in this mode thermalises via shocks in the surrounding gas, thereby providing a distributed heating channel. In cosmological simulations, the resulting model produces red, non star-forming massive elliptical galaxies, and achieves realistic gas fractions, black hole growth histories and thermodynamic profiles in large haloes.

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ALMA Resolves the Nuclear Disks of Arp 220

We present 90 mas (37 pc) resolution ALMA imaging of Arp 220 in the CO (1-0) line and continuum at $λ= 2.6$ mm. The internal gas distribution and kinematics of both galactic nuclei are well-resolved for the first time. In the West nucleus, the major gas and dust emission extends out to 0.2\arcsec radius (74 pc); the central resolution element shows a strong peak in the dust emission but a factor 3 dip in the CO line emission. In this nucleus, the dust is apparently optically thick ($τ_{\rm 2.6mm} \sim1$) at $λ= 2.6$ mm with a dust brightness temperature $\sim147$ K. The column of ISM at this nucleus is $\rm N_{H2} \geq 2\times10^{26}$ cm$^{-2}$, corresponding to $\sim$900 gr cm$^{-2}$. The East nucleus is more elongated with radial extent 0.3\arcsec or $\sim111$ pc. The derived kinematics of the nuclear disks provide a good fit to the line profiles, yielding the emissivity distributions, the rotation curves and velocity dispersions. In the West nucleus, there is evidence of a central Keplerian component requiring a central mass of $8\times10^8$ \msun. The intrinsic widths of the emission lines are $Δ\rm v (FWHM)$ = 250 (West) and 120 (East) \kms. Given the very short dissipation timescales for turbulence ($\lesssim10^5$ yrs), we suggest that the line widths may be due to semi-coherent motions within the nuclear disks. The symmetry of the nuclear disk structures is impressive -- implying the merger timescale is significantly longer than the rotation period of the disks.

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The role of mergers and halo spin in shaping galaxy morphology

Mergers and the spin of the dark matter halo are factors traditionally believed to determine the morphology of galaxies within a $Λ$CDM cosmology. We study this hypothesis by considering approximately 18,000 central galaxies at $z=0$ with stellar masses $M_{\ast} = 10^{9}-10^{12} \, {\rm M}_{\odot}$ selected from the Illustris cosmological hydrodynamic simulation. The fraction of accreted stars -- which measures the importance of massive, recent and dry mergers -- increases steeply with galaxy stellar mass, from less than 5 per cent in dwarfs to 80 per cent in the most massive objects, and the impact of mergers on galaxy morphology increases accordingly. For galaxies with $M_{\ast} \gtrsim 10^{11} \, {\rm M}_{\odot}$, mergers have the expected effect: if gas-poor they promote the formation of spheroidal galaxies, whereas gas-rich mergers favour the formation and survivability of massive discs. This trend, however, breaks at lower masses. For objects with $M_{\ast} \lesssim 10^{11} \, {\rm M}_{\odot}$, mergers do not seem to play any significant role in determining the morphology, with accreted stellar fractions and mean merger gas fractions that are indistinguishable between spheroidal and disc-dominated galaxies. On the other hand, halo spin correlates with morphology primarily in the least massive objects in the sample ($M_{\ast} \lesssim 10^{10} \, {\rm M}_{\odot}$), but only weakly for galaxies above that mass. Our results support a scenario where (1) mergers play a dominant role in shaping the morphology of massive galaxies, (2) halo spin is important for the morphology of dwarfs, and (3) the morphology of medium-sized galaxies -- including the Milky Way -- shows little dependence on galaxy assembly history or halo spin, at least when these two factors are considered individually.

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Galaxies in the Illustris simulation as seen by the Sloan Digital Sky Survey - II: Size-luminosity relations and the deficit of bulge-dominated galaxies in Illustris at low mass

The interpretive power of the newest generation of large-volume hydrodynamical simulations of galaxy formation rests upon their ability to reproduce the observed properties of galaxies. In this second paper in a series, we employ bulge+disc decompositions of realistic dust-free galaxy images from the Illustris simulation in a consistent comparison with galaxies from the Sloan Digital Sky Survey (SDSS). Examining the size-luminosity relations of each sample, we find that galaxies in Illustris are roughly twice as large and $0.7$ magnitudes brighter on average than galaxies in the SDSS. The trend of increasing slope and decreasing normalization of size-luminosity as a function of bulge-fraction is qualitatively similar to observations. However, the size-luminosity relations of Illustris galaxies are quantitatively distinguished by higher normalizations and smaller slopes than for real galaxies. We show that this result is linked to a significant deficit of bulge-dominated galaxies in Illustris relative to the SDSS at stellar masses $\log\mathrm{M}_{\star}/\mathrm{M}_{\odot}\lesssim11$. We investigate this deficit by comparing bulge fraction estimates derived from photometry \emph{and} internal kinematics. We show that photometric bulge fractions are systematically lower than the kinematic fractions at low masses, but with increasingly good agreement as the stellar mass increases.

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The stellar mass assembly of galaxies in the Illustris simulation: growth by mergers and the spatial distribution of accreted stars

We use the Illustris simulation to study the relative contributions of in situ star formation and stellar accretion to the build-up of galaxies over an unprecedentedly wide range of masses ($M_{\ast} = 10^9-10^{12} \, {\rm M_{\odot}}$), galaxy types, environments, and assembly histories. We find that the `two-phase' picture of galaxy formation predicted by some models is a good approximation only for the most massive galaxies in our simulation -- namely, the stellar mass growth of galaxies below a few times $10^{11} \, {\rm M_{\odot}}$ is dominated by in situ star formation at all redshifts. The fraction of the total stellar mass of galaxies at $z=0$ contributed by accreted stars shows a strong dependence on galaxy stellar mass, ranging from about 10 per cent for Milky Way-sized galaxies to over 80 per cent for $M_{\ast} \approx 10^{12} \, {\rm M_{\odot}}$ objects, yet with a large galaxy-to-galaxy variation. At a fixed stellar mass, elliptical galaxies and those formed at the centres of younger haloes exhibit larger fractions of ex situ stars than disc-like galaxies and those formed in older haloes. On average, $\sim$50 per cent of the ex situ stellar mass comes from major mergers (stellar mass ratio $μ> 1/4$), $\sim$20 per cent from minor mergers ($1/10 < μ< 1/4$), $\sim$20 per cent from very minor mergers ($μ< 1/10$), and $\sim$10 per cent from stars that were stripped from surviving galaxies (e.g. flybys or ongoing mergers). These components are spatially segregated, with in situ stars dominating the innermost regions of galaxies, and ex situ stars being deposited at larger galactocentric distances in order of decreasing merger mass ratio.

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