Competitive accretion and the baryonic fraction in galaxies

ABSTRACT

The baryonic fraction of galaxies is observed to vary with the mass of their dark matter (DM) halo. Low-mass galaxies have low baryonic fractions that increase to a maximum for masses near |$10^{12}\ {\rm M}_\odot$|⁠, and decrease thereafter with increasing galaxy mass. This trend is generally attributed to the action of feedback from star formation at the low end and of active galactic nuclei at the high-mass end. An alternative is that the baryonic fraction is at least partially due to the ability of galaxies to competitively accrete gas in a group or clustered environment. Most galaxies in a group including those of lower masses orbit the cluster centre at significant speeds and hence their accretion is limited by a Bondi–Hoyle-type process, |$\dot{M}_{\rm acc} \propto M_{\rm DM}^2$|⁠. In contrast, the few high-mass galaxies reside in the core of the cluster and accrete in a tidal accretion process, |$\dot{M}_{\rm acc} \propto M_{\rm DM}^{2/3}$|⁠. These two mechanisms result in a baryonic mass fraction that increases as |$M_{\rm DM}$| at low masses and decreases as |$M_{\rm DM}^{-1/3}$| at high masses. This model predicts that lower mass haloes in small-N groups should have higher baryonic fractions relative to those in large clusters.

1 INTRODUCTION

Galaxy formation is understood in terms of the hierarchical merger of dark matter (DM) haloes and subsequent infall of baryons to form the luminous components of galaxies (White & Frenk 1991; Lacey & Cole 1993; Navarro, Frenk & White 1995; Cole et al. 2000; Springel et al. 2005; De Lucia & Blaizot 2007; Vogelsberger et al. 2014; Bose et al. 2023). The mass of baryons in galaxies is observed to depend on the mass of the DM halo, with lower mass haloes having a smaller fraction of their total mass in stars and gas (Papastergis et al. 2012; Zaritsky & Behroozi 2023; Dev et al. 2024). The mass fraction in baryons increases with DM halo mass up to masses of |$M_{\rm \rm DM} \approx 10^{12}$| M|$_\odot$| and decreases at higher halo masses.

This variation in the efficiency of DM haloes to accumulate baryons is often attributed to the effects of feedback that remove baryons from the DM haloes. Star formation and feedback are primary internal processes that can regulate galaxy evolution (Dubois & Teyssier 2008; Schaye et al. 2015; Rey-Raposo et al. 2017), suggesting that feedback from supernova could be responsible for removing baryons in lower mass, and hence lower escape velocity haloes (Efstathiou 2000; Kay et al. 2002; Marri & White 2003; Scannapieco et al. 2006; Dalla Vecchia & Schaye 2008, 2012; Keller et al. 2014; Nelson et al. 2019; Tollet et al. 2019; Mina et al. 2021). Feedback from active galactic nuclei (AGNs) is significant in higher mass galaxies suggesting that AGN feedback can remove baryons from these galaxies (Mead et al. 2010; Silk & Nusser 2010; Martizzi et al. 2012; Wright et al. 2020; Cui et al. 2021). The challenge with feedback is that it tends to find weak points in the surrounding environment through which to escape while leaving the bulk of the mass unaffected (Dale et al. 2005; Dale, Ercolano & Bonnell 2012; Rogers & Pittard 2013; Körtgen et al. 2016; Lucas, Bonnell & Dale 2020; Lau & Bonnell 2025).

While feedback is an important process and can play an important role in galactic evolution including removing baryons, it is worthwhile exploring what other physical processes can affect the efficiency of baryon accretion in DM haloes. Many to most galaxies are found to be in groups or clusters (Makarov & Karachentsev 2011; Courtois et al. 2013; Kourkchi & Tully 2017; Lambert et al. 2020) with local estimates having only 11 per cent of galaxies in isolation and a further 10 per cent in double or triple systems (Argudo-Fernández et al. 2015). In small groups to large clusters, gravitational forces from neighbours and the full cluster can affect the dynamics. Accretion of baryons can be affected and limited by these competing forces as the gravitational radius of influence depends on the mass of individual objects. In such a scenario, DM haloes would compete to accrete from the baryon reservoir present in the cluster.

Competitive accretion in star formation has been advanced as a potential mechanism to explain the stellar initial mass function (IMF; Zinnecker 1982; Bonnell et al. 2001a; Bonnell, Larson & Zinnecker 2007). Accretion in a cluster environment is also a prime mechanism to form high-mass stars and result in a mass-segregated system where the highest mass stars form in the deepest part of the cluster potential (Bonnell, Vine & Bate 2004). The requirements for competitive accretion to occur are that the stars form in groups and clusters, and that there exists a shared reservoir of gas to accrete (Bonnell & Bate 2006). The process for galaxy baryon accretion is similar in that galaxies are generally in groups or clusters and acquire their baryonic components by infall into DM haloes. The main difference is that the accretion does not significantly increase the overall mass of the objects as that is determined by the DM halo merger process.

This letter investigates the role that competitive accretion in galaxy clusters can play in determining the efficiency of baryon acquisition by DM haloes. Section 2 reviews the physics of competitive accretion, while Section 3 details how this is adapted to the case of baryon accretion in galaxy clusters. The results are presented in Section 4 and conclusions in Section 5.

2 COMPETITIVE ACCRETION

Competitive accretion in stellar clusters has been advanced as a method to explain the origin of the stellar IMF (Zinnecker 1982; Bonnell et al. 1997, 2001a, b, 2007; Bonnell, Bate & Vine 2003). The basic concept is that stars form in a clustered environment with a communal gravitational potential and a shared reservoir of gas from which they compete to accrete. The competition arises through the gravitational influence of each object and how this depends on its mass. The two possibilities are that (1) the accretion is limited by the kinetic motions of the gas and star, and (2) that the accretion is limited by the tidal forces due to all the other objects in the cluster.

An object can accrete gas at a rate given by

where |$\rho$| is the gas density and |$v_{\rm rel}$| is the relative velocity of the gas. The accretion radius, |$R_{\rm acc}$|⁠, is the radius at which gas is bound to the accreting object considering both the relative kinetic energy of the gas and the tidal forces from the full cluster. Considering only the relative kinetic motions results in the classical Bondi–Hoyle radius

For roughly spherical systems, the tidal radius can be approximated as being due to the enclosed mass (⁠|$M_{\rm enc}$|⁠) inside the object’s position in the cluster |$R_*$|⁠, a tidal radius is given by

Numerical simulations of accretion in stellar clusters (Bonnell et al. 2001a) have shown that the majority of stars residing outside the core of the cluster have relatively high velocities resulting in the Bondi–Hoyle radius being smaller and hence determinant for accretion rates. In contrast, stars in the core of the cluster have relatively low velocities and hence the tidal radius is the limiting radius and hence determines the accretion rates. These objects have the highest accretion rates and end up forming the most massive stars (Bonnell et al. 2004, 2011; Bonnell & Bate 2006). Lower mass objects formed (at the local Jeans mass) in the outer parts of the cluster accrete little from the intracluster gas and hence lose out in this competition and remain with low masses reproducing the peak and the breadth of the stellar IMF (Bonnell & Bate 2006; Bonnell, Clarke & Bate 2006; Bonnell et al. 2011; Klos et al. 2025).

3 COMPETITIVE ACCRETION FOR DARK MATTER HALOES

DM haloes residing in clusters and groups can accrete baryons from their shared communal reservoir. In what follows, I develop a simple scenario where all baryons are assumed to accrete on to already formed DM haloes in such a system. The primary difference with the stellar IMF scenario above is that the overall galaxy mass is given by the mass of the DM halo. The accretion of baryons does not increase the mass of the DM halo, and hence the accretion rate remains nearly constant as baryons accumulate in the DM halo. The two cases for accretion in groups are that in the core of a group or cluster the lower velocities present result in larger Bondi–Hoyle radii and hence accretion is instead limited by the tidal radii. Outwith the core, the Bondi-Hoyle radii are smaller and hence limits the accretion for DM halos located there. The mass of baryons acquired over a time |$t_{\rm acc}$| is then

with the lower mass DM haloes having accretion rates given by

The final baryonic fraction then scales with halo mass as

The higher mass haloes are generally located in the cluster cores and thus have accretion rates

and hence a baryonic mass fraction that scales as

These scalings are promising as they roughly agree with the observed distributions, yet we need a model that includes the halo mass function and dependences within the galaxy clusters.

4 BARYON ACCRETION IN GALAXY CLUSTERS

In order to assess the potential role for accretion in determining the baryonic fraction of galaxies, we use a simple model for how baryon accretion occurs in galaxy clusters. It neglects any baryons included in the initial halo formation or any subsequent baryon removal through stripping or ejection. The model contains 1000 clusters based on observed properties (Makarov & Karachentsev 2011; Kourkchi & Tully 2017; Singh et al. 2025), with each cluster composed of between 5 and 2000 DM haloes, following a log-uniform |${\rm d}N_{\rm gals} \propto N_{\rm gals}^{-1}$| distribution. The individual haloes are randomly chosen from a DM halo mass function (Driver et al. 2022), with minimum and maximum masses of |$10^{9.5}$| and |$10^{14.5}$| M|$_\odot$|⁠, respectively:

where |$M_*=10^{14.43}\ {\rm M}_\odot$|⁠, |$\alpha =-1.85$|⁠, and |$\beta =0.77$|⁠.

The clusters have a fiducial radius of |$R_{\rm clust} = 1$| Mpc, with near-uniform density cores inside 0.05|$R_{\rm clust}$|⁠. The cores contain 5 per cent of the cluster members, with a minimum of 2 and a maximum of 25. The DM haloes in the core are chosen to be the most massive members in the cluster as is commonly found in clusters. The rest of the haloes are located at distances outside the core in either a mass-segregated or a non-mass-segregated distribution. The total baryonic mass is taken to be 10 per cent of the DM mass in the cluster. The cluster cores are relatively small as they represent where the mass density is near-uniform.

For DM haloes in the core, their accretion radii are calculated as the minimum of their tidal and Bondi–Hoyle radii (as above) and then they accrete accordingly with uniform velocities given by the core velocity dispersion and uniform core densities. Similarly, haloes outwith the core have their tidal and Bondi–Hoyle radii evaluated, with a ubiquitous smaller Bondi–Hoyle radii, which is then used to evaluate the accretion along with a uniform cluster velocity dispersion (⁠|$v_{\rm clust} = \sqrt{2G M_{\rm clust} / R_{\rm clust}}$|⁠) and a power-law density |$\rho = \rho _{\rm core} \left(R/R_{\rm core}\right)^{-\gamma }$| with |$1 \le \gamma \le 2$| (Vikhlinin et al. 2006; Meneghetti et al. 2014). Haloes accrete over two cluster crossing times (⁠|$t_{\rm acc} = 2 t_{\rm cross}$|⁠) to attain a final baryonic mass and hence a baryonic mass fraction as plotted in Fig. 1 for cluster gas density profiles |$\rho \propto r^{-1}$| and cluster number functions that are |${\rm d}N_{\rm gals} \propto N_{\rm gals}^{-1}$|⁠.

The galaxy halo mass function is such that haloes near or above |$10^{12}$| M|$_\odot$| are always among the most massive objects in the cluster and hence reside in the core. Lower mass haloes are generally found outwith the cluster core unless the cluster or group consists of a small number of objects. Fig. 1 shows the mean baryonic fraction of galaxies as a function of DM halo mass. As haloes with |$M_{\rm DM} \ge 10^{12}$| M|$_\odot$| are located in the core of the cluster, they accrete via tidal-lobe accretion and hence result in baryonic mass fractions that decrease as |$f_{\rm baryon} \propto {(M_{\rm DM}})^{-1/3}$|⁠.

The majority of lower mass haloes (⁠|$M_{\rm DM} \le 10^{12}$| M|$_\odot$|⁠) are located outside the core and hence accrete via Bondi–Hoyle accretion. This results in an average baryonic mass fraction that increases with mass at a somewhat steeper rate than |$f_{\rm baryon} \propto M_{\rm DM}$|⁠, the pure Bondi–Hoyle expectation discussed above. The complications are due to (1) a cluster gas density profile that decreases with radius and a degree of mass segregation along with (2) a number of cases where lower mass haloes are still the most massive member of a small-N group. The former can result in baryonic mass fractions as steep as |$f_{\rm baryon} \propto {(M_{\rm DM})}^{\nu }$| with |$\nu = 1 + \gamma$| and |$\gamma$| is the gas density profile power-law index (see above). The latter results in haloes with high |$f_{\rm DM}$| as they are in the core of low-N groups. Together, this results in a wide spread of |$f_{\rm DM}$| at low halo masses and an average |$f_{\rm DM} \propto (M_{\rm DM})^{1+\epsilon }$| with |$\epsilon \approx 0.5$|⁠.

4.1 Model parameters

Fig. 1 plots the mean baryonic fraction for DM haloes for clusters that are mass-segregated, have gas densities |$\rho \propto r^{-1}$|⁠, and follow galaxy number distributions of |${\rm d}N_{\rm gals} \propto N_{\rm gals}^{-1}$|⁠. The baryonic fraction varies significantly at a given halo mass depending on the cluster properties and especially at lower masses. The middle panel of Fig. 2 plots a 2D histogram of the full baryonic mass fractions across the full 1000-cluster simulation. At low halo masses, we see a large variation in resultant baryonic fractions depending on the cluster sizes with low-|$N_{\rm gals}$| clusters likely to have a lower mass halo as their most massive member. In these cases, the central DM halo accretes more than its neighbours and has a relatively high baryonic mass fraction. In high-|$N_{\rm gals}$| clusters, lower mass haloes have accretion rates that are lower by |$f_{\rm baryon} \propto \rho M_{\rm DM}$| and as |$\rho \propto r^{-1}$| in the mass-segregated system, this results in a lower limit of |$f_{\rm baryon} \propto (M_{\rm DM})^2$|⁠. Taken together, this produces a mean baryonic mass function at low |$M_{\rm DM}$| that has |$f_{\rm baryon} \propto (M_{\rm DM})^{1.2}$|⁠. The high-mass |$M_{\rm DM}$| follows an |$f_{\rm baryon} \propto (M_{\rm DM})^{-1/3}$| distribution in all three panels of Fig. 2.

Fig. 2 explores how the resultant baryonic fraction depends on our chosen parameters, the cluster density profile, and the distribution of the number of clusters with given galaxy populations (Vikhlinin et al. 2006; Meneghetti et al. 2014). The left-hand panel shows the baryonic mass fractions for DM haloes in clusters where the gas density is taken to be uniform and the distribution of clusters with given |$N_{\rm gals}$| galaxies is flat |${\rm d}N_{\rm gals} \propto N_{\rm gals}^0$|⁠. In this case, we have a mean |$f_{\rm baryon} \propto M_{\rm DM}$| and a variation around this where the mean and mode are similar. Most haloes are in high-|$N_{\rm gals}$| clusters with lower mass haloes accreting as |$\dot{M}_*\propto (M_{\rm DM})^2$|⁠.

The right-hand panel of Fig. 2 explores the other extreme where the cluster number density distribution is |${\rm d}N_{\rm gals} \propto N_{\rm gals}^{-2}$| and the cluster gas follows a steep profile of |$\rho \propto r^{-2}$|⁠. In this case, the lower mass haloes in high-|$N_{\rm gals}$| clusters follow a steep baryonic mass fraction with |$f_{\rm baryon} \propto (M_{\rm DM})^3$|⁠. This is offset by the larger number of low-|$N_{\rm gals}$| clusters in which lower mass haloes are more likely to be among the more massive members and hence accrete following tidal-lobe accretion. The relatively high number density at low |$M_{\rm DM}$| and high |$f_{\rm baryon}$| is apparent in this panel. Other model parameters have been explored including the proportion of haloes in the core, the core radius, and the degree of mass segregation outside the halo. These all have small effects that are within the results for the models presented in Fig. 2.

5 DISCUSSION

The results of the simple toy model presented in this letter show that accretion on to DM haloes can conceivably produce baryonic mass fractions that are similar to what are observed. The underlying assumption is that most galaxies are in groups or clusters during this accretion phase. The fraction of galaxies in clusters and groups is uncertain due to completeness and the definition of group or cluster size. Published estimates show that of the order of 50–80 per cent are in larger groups or clusters or in the process of infalling into the cluster potential (Makarov & Karachentsev 2011; Courtois et al. 2013; Lambert et al. 2020). Argudo-Fernández et al. (2015) find that only 10 per cent of the galaxies in the nearby Universe are truly isolated with a further 10 per cent in double or triple systems. Competitive accretion can operate in small-N systems such that the process described here may be applicable to a large majority of galaxies. Galaxies that are not found in groups and clusters would neither gain baryons from the higher accretion rates in the cluster cores nor lose the competition for baryons in the periphery of the group or cluster. Such galaxies, presumably of lower halo masses, would likely have a higher baryonic fraction than their counterparts that are in groups and clusters.

In addition to producing steeply increasing baryonic fraction at low halo masses and a shallow decreasing baryonic fraction at high halo masses, the model results in a peak near halo masses of |$10^{12}$| M|$_\odot$| due to the halo mass function (Driver et al. 2022) and having higher mass haloes predominantly located in cluster cores. The model’s greatest attraction is its simplicity in relying entirely on gravitational processes. This simplicity harbours unknown challenges in terms of the missing physics such as feedback from AGN and supernova. A complete picture also requires exploration of the dynamics of cluster formation and the interior dynamics where accretion occurs. This is well beyond the scope of this letter and its intention to highlight the role that competitive accretion can play in determining the baryonic mass fraction of galaxies.

The resulting baryonic fractions from this model are simply due to baryon accretion on to already formed DM haloes. In reality, baryons can accumulate at all stages of the hierarchical merger process from the formation of the first low-mass haloes through the subsequent mergers and dynamics in groups and clusters. This adds a significant degree of complexity to this simple picture, with processes such as tidal and ram-pressure stripping (Merritt 1983; Abadi, Moore & Bower 1999; Mayer et al. 2006; Read et al. 2006; McCarthy et al. 2008; Boselli, Fossati & Sun 2022) and feedback (Silk & Nusser 2010; Dalla Vecchia & Schaye 2012) all likely to play a role in removing baryons, although tidal stripping may play a larger role in removing mass from the DM haloes rather than the more concentrated baryons (Smith et al. 2016). Feedback during the accretion phase is most likely to have an important role in limiting accretion rather than removing baryons from deep inside the haloes.

The hierarchical merger of DM haloes implies that the most massive haloes are just accumulations of many lower mass haloes. In the absence of baryon accretion during this process, the final baryonic mass fraction would simply reflect the primordial baryonic mass fraction in each small-mass halo. Numerical simulations of galaxy and large-scale structure formation (Vogelsberger et al. 2014; Matthee et al. 2016; Artale et al. 2018; Bose et al. 2019; Cui et al. 2021; Camargo & Casas-Miranda 2025) highlight the importance of baryon accretion on to the DM haloes and can be used to explore how this accretion depends on local properties such as locations in the groups, relative velocities, and tidal effects (Pillepich et al. 2018; Springel et al. 2018). The complexity present in full dynamical simulations will undoubtedly be more involved than the simple picture presented here but the baryon accretion rates onto DM halos can be analysed with respect to the predicted physical processes of tidal and Bondi–Hoyle accretion.

What we have highlighted in this letter is that accretion in clusters is likely to follow two regimes that produce marked differences in the amount of baryonic gas that can be accreted on to DM haloes. The model finds a robust |$f_{\rm baryon} \propto (M_{\rm DM})^{-1/3}$| in all models explored independent of the cluster properties. This is due to the tidal-lobe accretion, which dominates in the core of clusters. Lower mass haloes have mean baryonic mass fractions that decrease below |$M_{\rm DM}\approx 10^{12}$| M|$_\odot$| but with significant variations. The exact slope of |$f_{\rm baryon} \propto (M_{\rm DM})^{\mu }$| varies but with a mean in the ballpark of |$1\le \mu \le 3$| with most likely values near |$1 \le \mu \le 1.5$|⁠. A prediction for the model is that lower mass haloes in small-N groups, or in isolation, should have higher baryonic mass fractions. Conversely, lower mass haloes in larger groups, or where they are significantly less massive than most group halo, should have very low baryonic mass fractions.

6 CONCLUSIONS

This letter presents a simple model for baryon accretion on to DM haloes in galaxy groups and clusters in order to explain the observed baryonic fraction as a function of halo mass. The model uses simple prescriptions for accretion in groups whereby more massive haloes reside in the cores of clusters and accrete via a tidal process. In contrast, lower mass haloes reside outside the cores and accrete via a Bondi–Hoyle process determined by the virial velocity in the cluster. These two processes result in an increasing baryonic mass fraction at low halo masses with |$f_{\rm baryon} \propto (M_{\rm DM})^{\mu }$| with |$\mu \approx 1 \!\!-\!\! 1.5$|⁠, a peak baryonic fraction for |$M_{\rm DM} \approx 10^{12}$| M|$_\odot$|⁠, and a decreasing |$f_{\rm baryon} \propto (M_{\rm DM})^{1/3}$| for high-mass haloes. Low-mass haloes have significant dispersion in their |$f_{\rm baryon}$| as they can be members of large-N clusters or equally small-N groups of galaxies in which they are more likely to be the most massive members. A prediction of this model is that lower mass haloes in small-N groups or in isolation should have larger baryonic fractions than those in large groups and clusters.

The model presented here is a simple toy model and a complete model would require the incorporation of the halo formation and merger process in the context of a cosmological model. Feedback from supernova and AGN would also be important additional physics to include in order to definitively ascertain the role of competitive accretion in determining the baryonic fraction in galaxies.

ACKNOWLEDGEMENTS

IAB would like to thank Keith Horne, Jim Pringle, Paul Clark, and Kat Klos and the referee for comments and discussions, which helped improve the letter.

DATA AVAILABILITY

The data generated in this work can be made available upon request to the author.

REFERENCES