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The E-MOSAICS Project: simulating the formation and co-evolution of galaxies and their star cluster populations

Joel Pfeffer, J. M. Diederik Kruijssen, Robert A. Crain, Nate Bastian

arXiv:1712.00019v1astro-ph.GA

TL;DR

The paper addresses how star clusters form and evolve alongside their host galaxies in a cosmological setting. E-MOSAICS couples cluster formation, evolution, and disruption models to EAGLE simulations, finding that the model broadly reproduces young-cluster and globular-cluster properties while linking GCs to massive clusters formed early. The simulations also expose insufficient disruption of low-mass clusters under their resolved gas conditions.

  • Problem

    The paper addresses the need to model star-cluster formation, evolution, and disruption together with galaxy formation in a fully cosmological context.

  • Method

    E-MOSAICS couples the MOSAICS semi-analytic cluster model to EAGLE simulations, setting cluster initial properties from natal gas and evolving mass loss through stellar and dynamical processes.

  • Results

    The simulations broadly reproduce observed young-cluster and globular-cluster characteristics, with GCs predominantly forming at z ≳1 in L⋆ galaxies.

  • Takeaways & Limitations

    The results support GCs as survivors of massive clusters formed at early cosmic times under higher gas pressures and surface densities.

  • Takeaways & Limitations

    Low-mass clusters are not disrupted efficiently enough to transform the initial power-law mass function into the Milky Way's log-normal GCMF, reflecting insufficient disruption in the simulated gas conditions.

Abstract

from arXiv · show

We introduce the MOdelling Star cluster population Assembly In Cosmological Simulations within EAGLE (E-MOSAICS) project. E-MOSAICS incorporates models describing the formation, evolution and disruption of star clusters into the EAGLE galaxy formation simulations, enabling the examination of the co-evolution of star clusters and their host galaxies in a fully cosmological context. A fraction of the star formation rate of dense gas is assumed to yield a cluster population; this fraction, and the population's initial properties, are governed by the physical properties of the natal gas. The subsequent evolution and disruption of the entire cluster population is followed accounting for two-body relaxation, stellar evolution, and gravitational shocks induced by the local tidal field. This introductory paper presents a detailed description of the model and initial results from a suite of 10 simulations of $\sim L^\star$ galaxies with disc-like morphologies at $z=0$. The simulations broadly reproduce key observed characteristics of young star clusters and globular clusters (GCs), without invoking separate formation mechanisms for each population. The simulated GCs are the surviving population of massive clusters formed at early epochs ($z\gtrsim1-2$), when the characteristic pressures and surface densities of star-forming gas were significantly higher than observed in local galaxies. We examine the influence of the star formation and assembly histories of galaxies on their cluster populations, finding that (at similar present-day mass) earlier-forming galaxies foster a more massive and disruption-resilient cluster population, while galaxies with late mergers are capable of forming massive clusters even at late cosmic epochs. (Abridged)

1 INTRODUCTION

Globular clusters are widespread tracers of galaxy formation, but their connection to host-galaxy histories and their evolution from young massive clusters remains uncertain. E-MOSAICS addresses this by coupling cluster formation and disruption models to cosmological galaxy simulations.

  • Globular clusters occur across galaxy masses, from a few in dwarf galaxies to tens of thousands in brightest cluster galaxies.
  • GC properties correlate broadly with host galaxies, but unimodal and multimodal GC metallicity distributions complicate simple formation-history interpretations.
  • Young massive clusters are proposed analogues of proto-GCs, but direct observations of GC formation are precluded by their old ages and small sizes.
  • YMCs have power-law mass functions with high-mass truncations, whereas GCs have peaked mass functions relatively insensitive to environment.
  • Previous models often treated cluster formation, disruption, or environmental effects separately, with gas-driven disruption frequently omitted.
  • E-MOSAICS couples semi-analytic cluster formation and disruption models to cosmological hydrodynamical simulations, using ambient gas properties to set cluster initial conditions.

2 NUMERICAL METHODS

E-MOSAICS couples the MOSAICS star-cluster model to EAGLE, using local gas properties to form sub-grid cluster populations and following their evolution and disruption within cosmological galaxy simulations.

  • Model framework: E-MOSAICS attaches MOSAICS star-cluster populations to stellar particles formed in EAGLE, combining cluster and galaxy formation models without recalibrating EAGLE's sub-grid parameters.The model runs on-the-fly because the temporal resolution needed to identify tidal shocks makes post-processing impractical.
  • The EAGLE simulations: EAGLE is a hydrodynamical ΛCDM simulation suite with radiative cooling, photoionization heating, pressure-based star formation, stellar evolution, and calibrated feedback prescriptions.Gas eligible for star formation is converted stochastically into stellar particles according to a pressure-dependent star formation rate.
  • Cluster formation: The cluster model forms unresolved populations from stellar-particle formation events, assigning clusters the particle's phase-space coordinates and metallicity while deriving natal properties from local gas conditions.This YMC-based, sub-grid treatment is required because current cosmological simulations cannot resolve individual star clusters.
  • Cluster formation: The cluster formation efficiency Γ increases with turbulent gas pressure, while the initial cluster mass scale depends on the molecular-cloud mass and Γ.The molecular-cloud mass is linked to the Toomre mass through a feedback-limited collapse fraction evaluated in the sub-grid model.
  • Cluster formation: Clusters are sampled stochastically from the initial cluster mass function, with populations below 5×10^3 M⊙ discarded because they disrupt on short timescales.The expected total cluster mass equals ΓM_* on average, although individual stellar particles can contain no clusters or more cluster mass than the particle mass.
  • Cluster evolution: Cluster evolution includes two-body relaxation and tidal-field-driven mass loss, but the simulations do not model dynamical friction on-the-fly.Dynamical friction is omitted because one stellar particle can host clusters with different masses while most particle mass belongs to field stars.

3 THE SIMULATIONS

The simulations use zoomed, high-resolution EAGLE resimulations to follow 10 Milky Way-like L⋆ galaxies and their star-cluster populations from high redshift to z = 0. They combine cosmological galaxy formation, cluster evolution, merger-tree tracking, and visual diagnostics of galaxy and cluster assembly.

  • Simulation sample: 10 L⋆ galaxies are followed with zoomed resimulations based on the EAGLE Recal-L025N0752 parent volume.The simulations target typical spiral galaxies similar to the Milky Way while retaining their cosmological environments.
  • Initial conditions: High-resolution regions extend to at least 600 pkpc around each target, while surrounding large-scale structure is represented with collisionless particles.This setup follows the target galaxies and progenitors hydrodynamically while limiting high-resolution computation to their immediate environments.
  • Resolution: The runs use gas particles of approximately 2.25×10^5 M⊙ and dark-matter particles of approximately 1.2×10^6 M⊙, with 0.35 pkpc proper softening after z = 2.8.The standard-resolution simulations marginally resolve Jeans scales at the star-formation threshold in warm interstellar gas.
  • Galaxy tracking: Galaxies and their descendants are identified across 29 snapshots using FoF and SUBFIND-based subhalo merger trees.The descendant-selection procedure weights the binding-energy ranks of linked particles to resolve ambiguous descendant assignments.
  • Galaxy and cluster histories: The 10 galaxies typically peak in star formation at z ≈ 2–3, with maximum SFRs of 2–10 M⊙ yr^-1, while their cluster populations are visualised by origin, metallicity, and mass.Figure 1 distinguishes in-situ and accreted massive clusters, and Figure 2 compares SFR and sSFR histories across the sample.

4 CLUSTER FORMATION PROPERTIES

E-MOSAICS reproduces observed present-day cluster-formation properties and predicts that pressure-driven formation evolves strongly with redshift and galaxy assembly history. Massive clusters preferentially form early, although mergers and centrally concentrated star formation can sustain massive-cluster formation later.

  • 4.1 Cluster formation efficiency: Γ ranges from approximately 1% to unity across birth pressures, with the lowest efficiencies restricted to metal-rich, late-forming stars.The model links cluster formation efficiency to the pressure and metallicity-dependent star-formation conditions of natal gas.
  • 4.2 Radial distributions at z = 0: At z = 0, Γ typically exceeds 10% within approximately 2 kpc and falls to a few percent farther out, matching nearby-disc observations.Global efficiencies span 1.5–30% across the ten galaxies, within the observed approximately 1–50% range.
  • 4.3 Redshift evolution of the cluster formation physics: For Gal004, birth pressure peaks near z ≈ 2.5 and z ≈ 0.7, with the latter associated with gas-rich mergers.The overall decline toward late times is driven partly by decreasing characteristic star-formation pressures and metallicity-dependent star-formation thresholds.
  • 4.3 Redshift evolution of the cluster formation physics: MT changes only 0.5 dex while pressure changes 3 dex, because centralized star formation makes the Toomre mass strongly limited by the epicyclic frequency κ.The median MT rises from 10^8 M⊙ at z = 6 to 2 × 10^8 M⊙ at z = 0.
  • 4.3 Redshift evolution of the cluster formation physics: Γ and MGMC decline below z ≲ 2, while Mc,∗ reaches a broad maximum between z = 1.5 and 5; high-redshift masses are more often Coriolis- or centrifugal-limited.At low redshift, stellar feedback more often prevents the Toomre-limited volume from collapsing into one unit, making massive clusters less prevalent.
  • 4.4 Galaxy to galaxy diversity of the evolving cluster populations: Across galaxies, cluster-formation properties generally peak at z = 1–4, but quenching, central star formation, and assembly history produce substantial diversity.Gal005 declines from 70% at z = 2 to 2% at z = 0 after AGN-driven quenching, whereas Gal008 peaks at z = 0 from central high-pressure star formation.
  • 4.4 Galaxy to galaxy diversity of the evolving cluster populations: Most galaxies form their most massive clusters before z ∼ 1, yielding predominantly old surviving massive clusters with mean formation redshift z ∼ 2.Along assembly histories, CFE and MGMC peak near major star-formation episodes and can be elevated by mergers, including at late times.

5 TIDAL HISTORIES

Tidal histories vary substantially among galaxies and evolve with environment and cosmic time. Tidal shocks are strongest in dense, actively star-forming regions, while cluster migration into diffuse environments supports long-term survival.

  • Environmental dependence: Tidal shocks are a strong function of the ambient gas density experienced after cluster formation.The evolving gas density and pressure therefore affect both cluster formation and subsequent mass-loss.
  • Evolution with cosmic time: At lookback times > 13 Gyr, strongly negative median tidal strengths indicate very central star formation in both Gal004 and Gal005.The median and maximum tidal field strengths peak near z = 2 as star formation becomes more spatially extended.
  • Disc–halo contrast: Maximum tidal heating for halo clusters is typically 1–2 orders of magnitude lower than for disc clusters, reaching nearly 4 dex lower in Gal005 at a lookback time of ≈11 Gyr.This indicates that tidal shocks are much more effective in high-density gas than in diffuse environments.
  • Tidal heating: Median tidal heating peaks first at lookback times > 13 Gyr and again near peak star formation, when gas densities are highest.The median peak occurs later for halo clusters than disc clusters by 2 Gyr in Gal004 and 1.5 Gyr in Gal005, consistent with migration from dense regions.
  • Galaxy-to-galaxy variation: Gal005 reaches significantly greater tidal field strengths and tidal heating parameters than Gal004 despite their similar present-day masses and morphologies.The difference largely reflects their differing gas density distributions and star formation rates.
  • Galaxy-to-galaxy variation: Across all 10 galaxies, median and mean tidal histories span a large range, reflecting differences in cluster radial distributions and underlying galaxy mass profiles.Mean tidal strengths often reach negative minima during z = 1–4 central star formation, while mean tidal heating generally peaks during high-redshift cluster formation.

6 PROPERTIES OF THE PRESENT DAY CLUSTER POPULATIONS

The simulated present-day cluster populations are shaped by mass-dependent disruption, formation-environment physics, and galaxy assembly histories. They broadly reproduce observed high-mass GC properties, but retain excess low-mass clusters and show radial metallicity discrepancies.

  • Cluster masses: Most low-mass clusters are disrupted by z = 0, especially in Gal008 at formation redshifts z > 3 where nearly all clusters below 10^4 M⊙ disappear.Final cluster masses are therefore strongly reduced relative to initial masses at low masses and high formation redshifts.
  • The GC mass function: High-mass simulated GCMFs above 10^5.5 M⊙ agree well with observed Milky Way and M31 GCMFs.The most massive simulated clusters reach 10^6–10^7 M⊙, consistent with observed massive clusters.
  • The GC mass function: At birth pressures above 10^6 K cm−3, tidal shocks account for 75 per cent of dynamical mass-loss and transform the initial power-law mass function into a peaked distribution.Clusters formed at lower pressures retain a power-law mass function because they experience little mass-loss beyond stellar evolution.
  • Radial cluster properties: Simulated cluster metallicities match Milky Way GCs near galaxy centres and beyond 20 kpc but are too metal-rich at intermediate radii.The authors attribute this radial discrepancy to inefficient disruption of clusters formed at low-to-intermediate gas pressures.
  • Radial cluster properties: Maximum cluster masses typically peak at 3–10 kpc because dynamical friction removes massive central clusters and low birth pressures limit massive-cluster formation at large radii.Without cluster formation physics, modelled maximum masses are about an order of magnitude too high.
  • Formation physics: Omitting physically motivated cluster formation physics produces populations in qualitative disagreement with the Milky Way GC population.Simple particle tagging cannot simultaneously reproduce present-day young clusters and high-redshift GCs in Milky Way-like galaxies.

7 SUMMARY

E-MOSAICS couples cosmological galaxy formation with star-cluster formation and evolution, modelling their co-formation across cosmic history. Its simulations reproduce broad young-cluster and GC properties while identifying environmental and assembly-history drivers of cluster populations.

  • Project and approach: E-MOSAICS couples the semi-analytic MOSAICS model of star-cluster formation and disruption to EAGLE cosmological hydrodynamical simulations.The project follows cluster populations and their host galaxies across cosmic history.
  • Project and approach: The reference study uses cosmological zoom-in simulations of 10 MW-like, L⋆ galaxies to examine cluster and galaxy co-formation and co-evolution.
  • Cluster formation: Low-redshift clusters are broadly compatible with observations of young clusters, with mean CFE declining from > 10 per cent within 2 kpc to a few per cent beyond 4 kpc.The maximum cluster mass is set by environmentally dependent physical limits rather than stochastic SFR sampling alone.
  • Cluster formation: GCs in L⋆ galaxies are predicted to be predominantly old, with formation redshifts z ≳1, because early gas pressures and surface densities raise CFE and Mc,∗.These quantities peak at z = 1–4, limiting when massive clusters mostly form.
  • Cluster formation: Massive clusters rarely form locally, but mergers can boost their formation by elevating gas pressures from P/k = 103–105 K cm−3 to 105–108 K cm−3.
  • Galaxy assembly: At similar present-day stellar masses, earlier-forming, lower-metallicity galaxies produce more efficient and more massive cluster populations than later-forming galaxies.
  • Cluster evolution: Tidal disruption is strongest during z ≈ 1–4, when natal gas densities peak, and low-pressure populations are disrupted too weakly in the simulations.High-pressure populations can evolve from power-law to peaked log-normal mass functions resembling the Milky Way GC population.
  • Cluster evolution: The simulations generally reproduce the high-mass end of observed GCMFs, with most massive surviving clusters typically in the interval 106–107 M⊙.

APPENDIX A: TENSOR METHOD FOR CIRCULAR AND EPICYCLIC FREQUENCIES

The appendix derives circular and epicyclic frequencies directly from the tidal tensor for use in calculating Toomre masses and tidal-field strengths. Comparisons show good agreement with standard post-processing estimates across the tested radial range.

  • Method: The tidal-tensor method derives circular frequency Ω and epicyclic frequency κ at simulation positions for subsequent Toomre-mass and tidal-field calculations.
  • Circular frequency: The circular frequency is defined as Ω ≡ v_c/r, with v_c the circular speed at radius r.
  • Enclosed density: The method estimates enclosed density from the tidal-tensor eigenvalues, whose sum fits the mean enclosed density well for Gal009 and other tested galaxies.The verification spans galaxy stellar masses from 10^8 to 10^10.5 M⊙.
  • Validation: The tidal-tensor estimates of Ω and κ agree well with values calculated by projecting the galaxy in post-processing.

APPENDIX B: LOCAL CALCULATION OF GAS SURFACE DENSITY AND TOOMRE MASS

The appendix evaluates local estimates of gas surface density and Toomre mass from simulation variables. The method follows radial trends reasonably well but underestimates surface density, especially near galactic centres.

  • Method: The local method estimates gas surface density from pressure under hydrostatic equilibrium and uses it to calculate the Toomre mass.It also determines φP and fgas as required inputs.
  • Gas properties: The locally determined φP decreases from ∼6 at the galactic centre to ∼1 beyond 10 kpc, consistent with expected ranges and M31 estimates.
  • Gas properties: The local gas-surface-density calculation underestimates projected values by a factor ∼2 around star-forming particles and by a factor ∼1.5 at radii < 2 kpc.The authors attribute this partly to treating SPH-particle pressure as mid-plane pressure.
  • Toomre mass: The locally calculated Toomre mass in Gal009 rises from ∼10^6 M⊙ inside 1 kpc to 5 × 10^7 M⊙ at 10 kpc, then decreases beyond 15 kpc.It is typically 0.5 dex below projected values inside 2 kpc but agrees better at larger radii.
  • Toomre mass: Inside 2 kpc, molecular-cloud masses are limited by the epicyclic frequency, whereas beyond 2 kpc they are feedback-limited and decline with radius.

APPENDIX C: TIDAL FIELD STRENGTH

The appendix refines the tidal-field calculation used for cluster evolution by including the circular-frequency term. This correction matters mainly in galactic centres, where the previous maximum-eigenvalue prescription can become negative.

  • Tidal-field definition: The tidal field strength governing a cluster’s tidal radius is ∂^2Φ/∂r^2 + Ω^2, rather than only the maximum tidal-tensor eigenvalue.
  • Validation: For a Plummer model, the calculated tidal-tensor eigenvalues follow the expected radial and tangential tidal-field components.
  • Correction: Including Ω^2 prevents the tidal field from becoming negative in the inner region where the maximum eigenvalue is negative.
  • Correction: The updated MOSAICS prescription sets T = max(λ) + Ω^2; the correction is negligible beyond ≳5 kpc but changes the result inside 2 kpc.It affects the mass-loss rate from two-body relaxation.

APPENDIX D: CLUSTER DISRUPTION TESTS

The tests show that timestep resolution and cluster radius mainly affect tidal-shock mass loss in intermediate-mass clusters, while stellar-evolution expansion has little effect on final population properties.

  • Timestepping: 0.35 dex: increasing stellar-particle timesteps from ×1 to ×10 depletes the mass function most strongly near 3 × 10^4 M⊙.The effect spans final masses of 10^4–10^5.5 M⊙ and does not increase beyond ×10 resolution.
  • Timestepping: For masses ≳5 × 10^5 M⊙, increasing timestep resolution does not significantly affect cluster mass loss.The ×20 run is nearly identical to the ×10 run.
  • Cluster radius: Larger cluster radii make tidal shocks more effective and extend disruption to more massive clusters, whereas smaller radii reduce shock-driven mass loss.The tested radii are 1.5, 4, and 6 pc; the mass-loss rate scales as dM/dt ∝ r_h^3.
  • Stellar-evolution expansion: Clusters formed with r_h,init = 2.2 pc expand to approximately 4 pc after most stellar-evolution mass loss.Expansion reaches 3.1 pc at 0.3 Gyr and 3.6 pc at 2 Gyr, with larger radii at higher metallicity for a given age.
  • Stellar-evolution expansion: Initial cluster expansion delays disruption by typically a few hundred Myr and has little effect on final cluster population properties.The resulting mass function is nearly indistinguishable from that of a constant r_h = 4 pc model.

APPENDIX E: CONSTANT STAR FORMATION DENSITY THRESHOLD

This appendix compares EAGLE’s metallicity-dependent star-formation density threshold with a constant threshold to assess its effect on cluster formation properties.

  • Threshold comparison: The comparison uses the standard metallicity-dependent threshold and a constant density threshold of nH = 0.1 cm−3.The metallicity-dependent threshold is motivated by thermogravitational collapse occurring at lower densities and pressures in metal-rich gas.

APPENDIX F: ISM EQUATION OF STATE

Changing the equation-of-state exponent leaves some cluster-formation trends similar but alters the evolution and peak values of the maximum cluster mass, especially at earlier epochs and low metallicity.

  • Cluster formation efficiency: CFE evolution with redshift is relatively similar across EOS exponents, but its peaks are higher toward lower γEOS.For γEOS = 5/3, the CFE peak at z ≈2.5 is absent; the weak dependence reflects CFE’s scaling with gas pressure.
  • Maximum cluster mass: At z < 0.5, median maximum cluster-mass evolution is similar across EOS exponents because star formation mainly occurs below nH = 10−1 cm−3.At z > 0.5, the evolution differs between EOS choices.
  • Maximum cluster mass: For an isothermal EOS, maximum-cluster-mass peaks are reduced relative to the fiducial model, except at z = 2–3.5 where merger-driven pressures produce a peak.For an adiabatic EOS, median and peak values at z > 2 are nearly unchanged from the fiducial case.
  • Maximum cluster mass: Maximum cluster masses depend on the chosen EOS, particularly for [Z/H] < −1 at z > 3.The dependence arises because higher γEOS makes density variations produce larger excursions in the model’s mass scale.
  • Pressure relation: For feedback-limited masses on the EOS, the truncation mass scales with pressure as approximately log10(Mc,∗) ∝ γ3/2.The comparison relates Mc,∗ to particle birth pressure under specified temperature-floor and φP assumptions.

APPENDIX G: ENVIRONMENT DENSITY

The environment-density analysis estimates the gas surrounding young stars over a kinematic travel scale and compares it with their birth density across redshift and galactocentric radius.

  • Method: The environment radius is estimated as renv ≈ σg/κ from the gas velocity dispersion and epicyclic frequency.The density is then measured within this radius, with a maximum renv of 5 kpc imposed in practice.
  • Method: At z = 0, renv is approximately 0.5 kpc inside 2 kpc galactocentric radius and approximately 2 kpc beyond 5 kpc, with large scatter.The exact 5 kpc maximum does not affect the results.
  • Density comparison: The surrounding environment density scales with stellar-particle birth density but is typically lower by a factor of ≈3.Figure G1 shows this comparison for young stars younger than 100 Myr in Gal009 at redshifts 2, 1, and 0.
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