What drives the corpulence of galaxies? I. The formation of central compact dwarf galaxies in TNG50
Abhner P. De Almeida Gary A. Mamon Avishai Dekel Gastão B. Lima Neto

    
	
 Formation of compact central dwarf galaxies


   Institut d'Astrophysique de Paris (UMR 7095: CNRS & Sorbonne Université), 98 bis Bd Arago, F-75014 Paris, France 
		 Instituto de Astronomia, Geofísica e Ciências Atmosféricas (Universidade de São Paulo), R. do Matão, 1226, São Paulo - SP, 05508-090, Brasil
    Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
    		 

   



 
  
  
  
  
      Nearby dwarf galaxies display a variety of effective radii (sizes)  at given stellar mass, suggesting different evolution scenarios according to their final `stellar' size. The  TNG hydrodynamical  simulations present a  bimodality in the z=0 size-mass relation (SMRz0) of dwarf galaxies,  at r_1/2,⋆∼450 pc.   
  
  
  	 Using the TNG50 simulation, we explore the evolution of the most massive progenitors of dwarf galaxies (z=0  between 8.4 and 9.2) that ended up as central galaxies of their groups. We split these dwarfs into three classes of the SMRz0:  from the central spine of the main branch and  from the secondary branch as well as from the lower envelope of the main branch.
  
 
  

    Both classes of  see their stellar sizes decrease since z∼1 in contrast to , while  the sizes of the gas and dark matter (DM) components keep increasing (as for ). 
  

  A detailed analysis reveals that  live in poorer environments, thus suffer fewer major mergers since z=0.8 that otherwise pump angular momentum into the gas, allowing strong gas inflows, producing inner star formation, hence the buildup of a stellar core.  are predicted to be rounder and with bluer cores. Compact dwarfs of similar sizes are observed in the GAMA survey, but the bimodality in sizes is less evident and the most compact dwarfs tend to be passive rather than star forming as in TNG50. Therefore, our conclusions should be confirmed with future cosmological hydrodynamical simulations.
      
   







What drives the corpulence of galaxies? I. The formation of central compact dwarf galaxies in TNG50
    Abhner P. de Almeida
 		 1,2abhner.almeida@usp.br
 		 Gary A. Mamon1
        Avishai Dekel 3
 		 Gastão B. Lima Neto2
 		 
    Received 11 March 2024 / Accepted 22 April 2024
==============================================================================================================================






§ INTRODUCTION




Dwarf galaxies (hereafter dwarfs) constitute an important laboratory to study the influence of different physical mechanisms in the formation and evolution of galaxies. They mainly acquire their stellar mass from star formation driven by gas accretion. Indeed, the other path for star formation, starbursts from gas-rich galaxy mergers, should not be important since low-mass galaxies rarely acquire the bulk of their stellar mass from mergers <cit.>.

Dwarfs have difficulty in maintaining the supply of gas by accretion, via several mechanisms, both internal and external. Internal processes include energetic feedback from supernovae <cit.> and from the central supermassive black hole <cit.>, if present. External processes include a variety of environmental effects, such as collisional tides <cit.>; tides from  clusters <cit.> and groups <cit.>; ram pressure <cit.>;  harassment from numerous flybys <cit.> and the effects of a nearby active galactic nucleus <cit.>. These external processes mainly  affect galaxies that end up as satellites. The interactions between galaxies can lead to different results depending on how it occurs and the galaxies involved <cit.>.

One fundamental issue is the size - mass relation (SMR) of galaxies, that is the relation between their effective (projected half-light) radius with their stellar mass. Analyzing galaxies from the Main Galaxy Sample of the Sloan Digital Sky Survey (SDSS), <cit.> found that the SMR of galaxies is not universal: it is steeper for  early-type galaxies than for late-type galaxies, leading to smaller sizes for early-type galaxies once one extrapolates to the low-mass end. 


Low-mass (M_⋆  between 10^7 and 10^9) galaxies are bimodal in size, with a split at a few hundred parsec (e.g., ), with a diffuse class (dwarf ellipticals and dwarf spheroidals) on one hand and a compact class (compact ellipticals, ultra-compact dwarfs and globular clusters) on the other. The compact class systems have sizes less than 10 (respectively, 100) times the median sizes of galaxies with =8 (respectively, 7).[All our logarithms are in base 10.]


The strong bimodality in the sizes of dwarf galaxies indicate that compact dwarfs have scenarios of formation and evolution that are different from those of high-mass galaxies. For example, the observation of compact dwarf galaxies (both ultra compact dwarf galaxies, UCDs, and compact ellipticals, cEs) in galaxy clusters or near massive galaxies suggests formation scenarios associated with tidal stripping of larger galaxies, groups or clusters <cit.>.


But tidal stripping does not affect central galaxies. Several other physical mechanisms have been suggested to form compact dwarfs. For example, UCDs could be extremely massive globular clusters (; ) or the result of mergers of young globular clusters formed in merging galaxies <cit.>. Blue compact dwarfs (BCDs) may be the result of mergers, specifically dwarf-dwarf mergers that can trigger a central starburst <cit.>. 

  
High-mass galaxies can also be compact: <cit.> discovered compact 10^10 galaxies at z=2, with effective radii as low as 500 pc, corresponding to  spherical half-mass radii of roughly 750 pc.
Dekel and collaborators (; )  argued that these high-redshift fairly massive compact galaxies have a three-phase cycle: they form from violent disk instabilities in normal size  galaxies, causing compaction of the gas, which becomes so dense that it creates a compact starburst (blue nugget) and later passively evolves into a red nugget.  For massive compact galaxies observed in the local Universe (probed by the Mapping Nearby Galaxies at Apache Point Observatory, MaNGA), <cit.>  suggested an analogous scenario, occurring between z=2 and 0.4, but less extreme.   In a  followup by the same team on the Illustris TNG100 cosmological hydrodynamical simulation, <cit.> recently  pointed out that compact galaxies  accrete lower angular momentum gas than other galaxies, allowing the galaxy  to increase its mass without significantly growing in size. In a similar analysis of galaxies with resolved spectroscopy (Sydney-AAO Multi-object Integral field spectrograph, SAMI) and with the Illustris TNG50 simulation, <cit.> found that one-third of compact massive dwarfs were stripped by a massive host galaxy.  They also noted that the remaining, isolated ones preferentially formed stars in their inner regions, although no physical mechanism differentiating these galaxies from normal ones was given.

The SMR of galaxies in hydrodynamical simulations are in good agreement with observations for galaxies with M_ stars > 10^9 <cit.>.  A striking feature of the Illustris simulations is the existence of a second, low-size branch in the z=0 SMR, whose slope is, surprisingly, negative (Fig. 10 of  for Illustris, upper-left panel of Fig. 4 of  for TNG100 and lower-right panel of Fig. 4 and Fig. 8 of  for TNG50).  A closer look using the Plot Group/Halo    Catalogs facility on the TNG web site[https://www.tng-project.org/data/groupcat/], reveals that this negative-slope low-size mode extends to stellar masses of 10^9.5 to 10^10  M_⊙ for TNG50, TNG100, and TNG300. This secondary negative-slope branch of the SMR has not yet been explained.

In the present article, we explore the physical mechanisms producing low-corpulence dwarf galaxies that end up as centrals of their hosts.  Since the SMR extends to different stellar mass ranges, we use corpulence to refer to the stellar size given a specific stellar mass.  We consider both the lower-corpulence galaxies in the main branch of the SMR and the lower-corpulence (secondary) branch of the  SMR. 
For this, we will use  the well-resolved TNG50-1 simulation, by following the evolution of the main-progenitors of z=0  galaxies. In two forthcoming studies (de Almeida et al., in prep.), we will use our tools to study the physical mechanisms driving low corpulence dwarfs that end up as satellites as well as those processes leading to high corpulence (highly diffuse) dwarfs.

In Sect. <ref>, we present the TNG50 simulation as well as  our selection of compact galaxies and a control sample of normal galaxies in the same range of stellar masses. In Sect. <ref>, we analyze the median evolution of the main progenitors of present-day compact and normal galaxies. In Sect. <ref>, we explore the respective roles of the physical   mechanisms driving size evolution. We discuss our results in Sect. <ref> and summarize them in Sect. <ref>.




§ SIMULATION AND SAMPLE





 §.§ IllustrisTNG



We study the evolution of dwarf compact galaxies using the IllustrisTNG (hereafter, TNG) suite of cosmological magneto-hydrodynamical simulations <cit.>. These simulations (as the previous  Illustris simulations) were run with the arepo magneto-hydrodynamics moving mesh code <cit.>. The TNG simulations were run with initial conditions drawn from the cosmological parameters values from <cit.>: a flat ΛCDM cosmology with Hubble constant H_0 = 67.74 km  s^-1  Mpc^-1, matter density Ω_ m = 0.3089, baryon density Ω_ b = 0.0486, power spectrum normalization σ_8 = 0.8159, and primordial spectral index n_s = 0.9667.

The TNG suite  has a variety of box sizes and resolutions, and some were run without baryons. For the simulations with baryons, TNG implements prescriptions for star formation, stellar feedback, metal enrichment and black hole (BH) physics (seeding and feedback). All simulations follow the evolution of a comoving box of the Universe (with periodic boundary conditions) from z = 127 to z = 0, with data saved at 100 different snapshots, typically spaced by 150 Myr in time.

The TNG database provides tables of groups, subhalos, particles (gas,   dark matter, stars, black holes, as well as tracer particles not considered  here), subhalo merger trees, as well as 33 (at the time of the writing of this manuscript) additional tables provided by the users. The groups are identified  using the Friends-of-Friends  algorithm <cit.> on the particles, while the subhalos are the structures within groups identified using the subfind algorithm <cit.>. The subhalos are essentially galaxies and the TNG database provides numerous astrophysical attributes for each one. One can have groups with a single subhalo, representing an isolated galaxy. We hereafter use the term galaxies for the subhalos and often use hosts for groups.

In TNG, a subhalo has a bad flag if it forms within a group, within one virial radius of its group, and with less than 80% of its mass in dark matter.  This flag allows to discard  regions incorrectly extracted by subfind. 



 §.§ The TNG50 simulation



We adopt the TNG50-1 (hereafter, TNG50) simulation  <cit.>, as it is the best resolved and most suited to study  the evolution of   dwarf galaxies. The simulation's dark matter and gas resolutions are m_DM = 4.5 × 10^5 an m_gas = 8.5 × 10^4, respectively, in a volume of 51.7^3 Mpc^3. The size resolution is described by  softening lengths of the collisionless components (dark matter and stars) whose physical values increase with cosmic time at 576 comoving pc until z=1, and are then fixed at the z=1 physical size of 288 pc <cit.>. The gas softening
length is 74 comoving pc (the minimum gas cell size is 8 pc). Thus, galactic
disks are reasonably well resolved in TNG50.

The TNG50 simulations have the same subgrid physics as the TNG100 and TNG300 simulations, except for a slightly different criterion for the star formation time, made to avoid instantaneous star formation in dense regions <cit.>.




 §.§ Sample selection




   
Figure <ref> displays the present-day size-mass relation of galaxies in our chosen range of stellar masses.  The bimodality of sizes is clear: there are only few small grey points at  log(r_1/2,⋆/ kpc)≈ -0.25 (i.e. 560 pc), see the red curve found with a   Gaussian mixing model (we hereafter use log(r_1/2/ kpc) for the stellar half-mass radius).[We used  sklearn.mixtureGaussianMixture), considering both good- and bad-flag subhalos.] We denote the high and low-size branches  Main and Secondary, respectively.  The Secondary Branch has a negative slope, as <cit.> and <cit.> previously found in Illustris and  TNG100, respectively.

Since the Secondary Branch of the SMR is polluted by suspicious subhalos, one-third with bad flags, mostly low-DM galaxies, we consider two samples of  galaxies: the lower envelope of the Main Branch of the SMR, and the Secondary Branch, both restricted to good-flag galaxies. 

We select all good-flag subhalos with log ( M_⋆ /  ) > 8.4 to have   sufficient number of particles to resolve the galaxy and be able to separate well the Main and Secondary branches. We also restrict the subhalos to
log ( M_⋆ /  )< 9.2 to limit our selection to  low-mass galaxies and avoid the poorly-populated end of the Secondary Branch.[These stellar masses correspond to those enclosed in a sphere of twice the stellar half-mass radius, to mimic the fact that observers have trouble measuring the outer luminosities (hence stellar masses) of galaxies.] Finally, we only consider subhalos that have an identified progenitor.


We then define a first set of  galaxies taken from a  conservative subset of the Secondary Branch:  are the galaxies whose (3D) stellar half-mass radius satisfies log(r_1/2/ kpc) < -0.35 (i.e. r_1/2  < 447  pc). We also define a second set of  galaxies, hereafter , taken from the lower 5th  percentile of the Main Branch (taking the galaxy sizes for different stellar mass bins and computing the median and the  5th percentile of the residuals). The minimum and maximum sizes of  as a function of stellar mass within 2 r_1/2 can be approximated by

    log(r_1/2,   maxkpc) = 0.29  log(M_⋆M_⊙)  - 2.61  ,


    log(r_1/2,   minkpc) = -0.06  log(M_⋆M_⊙)  + 0.33  ,

for galaxies in 8.4 < log(M_⋆/M_⊙) < 9.2. Finally, we define a conservative control sample of galaxies (hereafter,  ) as the galaxies within 25th - 75th percentiles of the Main Branch. Since the  have a higher median stellar mass than the other two samples, we select subsamples of  and   to have the same stellar mass distribution as the  , with median stellar mass 10^8.75 within two effective radii, i.e. 10^8.9 over the full subhalo. Our selection is shown in Fig. <ref>.

This selection led to 1289   131  and 157   galaxies. We also split these samples (hereafter, size classes) according to their location  within their group at z=0, between z=0 centrals and satellites. We abusively use this term for the main progenitors of z=0 centrals and satellites. We identify centrals as the subhalos whose index is contained in the list of central subhalos of all z=0 groups (obtained from GroupFirstSub). We verify centrals that are not  backsplash  galaxies (those that previously traveled once or several times  through a group and are currently outside that group and identified as the center of another  group) .  For this, we checked if the central's group mass (Group_M_Crit200) was not more than 1.5 dex greater at earlier times.  The fractions of centrals are  54%, 37%, 52% among , , and , respectively. In other words, while roughly half the z=0  and  are centrals, almost two-thirds of the z=0  are satellites. 
   
   


 We also define galaxies that were born later than z = 1, as the young population; galaxies that were born between z=5 and z=1 as the intermediate population; and galaxies that were born before  z=5 as the old population. Most galaxies are old: 90%, 100% and 99.7% for , , and  respectively.  No  are intermediate or younger age. Among the , only three are intermediate-age, while only one is younger. Finally, only one  is intermediate-age, while 9% are younger, all of which are satellites. Since we have few galaxies from the intermediate-age population, we exclude these from our analysis. 

 
Table <ref> lists the numbers of galaxies in each size class and subsample
of our final sample (with percentages in parentheses). None of the central galaxies have young ages or a bad flag, and only three among 810 have intermediate ages. In the present work we only consider  the central old galaxies. We thus end up considering 674 , 48 , and 85 .



 §.§ Components of z=0 galaxies







A more global view of the relative importance of the three components (stars, gas and DM) among central galaxies is given in 
Table <ref>, which shows the median z=0 DM, gas and stellar fractions in each population. In Table <ref> and elsewhere in the present paper,  we verify the significance of different distributions using  5000 random shuffles of the members of the two samples.
The central ,  and  are typically dominated by DM.  and  have the same distribution for each component, while  have slightly more stars (median of 0.02) and less DM (median of 0.91), with distributions statistically different. Fig. <ref> shows  the present-day fractions for the gas and stellar components, with no striking differences between our three size classes. 








§ DIFFERENCES IN THE HISTORIES OF  AND  GALAXIES



We explore how galaxies become compact by studying the backwards  evolution (hereafter, histories) of specific parameters of the most massive progenitors (hereafter, main progenitors) of the z=0 galaxies, separately for the  and . We  follow the main progenitors using the Main Progenitor Branch of the SubLink merger trees from the TNG database.

We first studied the median evolution of specific galaxy parameters to understand the different histories by tracking the histories of each of the 1289 galaxies that end up as centrals (always being centrals throughout their history): 674 , 48 , and 85 , and taking the median at each snapshot containing at least five galaxies (their main progenitors). We used bootstraps to estimate the uncertainties on these medians.




 §.§ Size and mass evolution




Fig. <ref> shows  the median evolution of the sizes and  masses  of the stellar, gas, and DM components, measured over the entire subhalo. We do not use the popular TNG stellar mass within twice the stellar half-mass radius, which is obviously size-dependent and would tend to show decreasing stellar mass evolution for the . 



The solid lines and associated shaded regions of the top panels of Fig. <ref> show the median evolution of the stellar half-mass radius.  One should note that  galaxies that are centrals at z=0 were not satellites at earlier epochs, since we removed the z=0 backsplash centrals.

The top panel of Fig. <ref> indicates that the stellar size of  grows at all times. In contrast, the  and  both start shrinking at z=0.8. This behaviour appears robust, as it is also seen when we split our samples into five bins of final stellar mass.
We then verify that at z = 0, the sizes of the stellar components of  and  are very different, as expected from our selection (with  in between the  and the ). But at z∼0.8, the median stellar half-mass radius of the three size classes are similar.

This decrease in size (hereafter, compaction) is roughly exponential in time (linear in the figure) between z=0.5 (5.2 Gyr ago) and z=0.3 (3.5 Gyr ago). The size reduction is a factor 1.9 for  in this 1.7 Gyr time interval, as well as a factor 1.1 for  centrals. To summarize, on average,  galaxies did not start compact, but evolved to become compact.


Interestingly, as seen in the top panel of Fig. <ref>, while  have a huge decrease in  stellar half-mass radius, the corresponding DM and gas half-mass radii of  (upper panel) evolve fairly similarly to those of the : the z=0 DM sizes are 5% and 20% lower for  and , respectively, while the z=0 gas sizes are 8% and 30% lower, respectively.   

The lower panels of Fig. <ref> show the median total mass evolution of the different components of the different size classes. The total mass corresponds to the mass within the entire subhalo, and not only within 2 stellar half-mass radii. One sees that the sample has very little effect on the typical mass evolution of all three components of the galaxies.  Since we selected our sample using the stellar mass within 2 r_1/2, we can summarize that the outer masses of  are slightly depleted.

Combining the size and mass evolution, in  that end up as centrals, the compaction of their stellar components occurs independently of the gas and DM components, of the  size class, and of the mass evolution of the DM, gas and even stellar components.
 

 





 §.§ Evolution of specific star formation rates







Some of the differences in the evolution of the stellar mass component may be
related to differences in the evolution of the star formation efficiency. Here, we consider the specific star formation rate (star formation rate over stellar mass, hereafter sSFR) as a proxy for star formation efficiency. Figure <ref> compares the histories of the sSFR of  and , showing the sSFR in the inner region (r < r_1/2) and the ratio between the inner to outer (r > r_1/2) ratios of sSFR.  We estimate the sSFRs by using the stellar masses and star formation rates provided in the TNG database in the full subhalo and within r_1/2.

Figure <ref> shows that the main progenitors of the z=0  have a continuous decrease in their inner sSFR (i.e. quenching of star formation) with cosmic time. There are  small gradients in sSFR at z ∼ 1 for this population: the ratio between the inner and outer sSFR is close to the unity. The inner star formation starts to become lower than in the outer region after z ∼ 0.9. The  also continuous decrease in their inner sSFR, although slowly than .
 
In contrast, the inner sSFR in  is almost constant between z ∼ 1.7 and ∼ 0.45. This is related to concentrated star formation, which is verified by the rapid increase in the ratio between the inner and outer sSFR.  also have a slow increase in this ratio, which suggests that these galaxies also have concentrated star formation, but not as strong as in  and also without the almost constant inner sSFR. We will return to these radial-dependent sSFRs in  in Sect. <ref>.





 §.§ Merger history


The star formation in galaxies can be within the main progenitor (in-situ) or within external galaxies that later merged with the main progenitor (ex-situ). The top panel of Fig. <ref> displays the median evolution of the ex-situ stellar mass, normalized by the z=0 ex-situ stellar mass (solid) and by the z=0 total stellar mass (dashed); while the bottom panel shows the stellar mass from major mergers and other mergers. The data was obtained using the Stellar Assembly TNG supplementary data catalog, provided by <cit.>.



The dashed lines in the top panel of Fig. <ref> indicate that the ex-situ fractions of stellar mass for all three size-classes are typically small (<4%) at all times. Furthermore, while the ex-situ stellar mass fraction of  is continuously increasing,  those of the two  size classes decouple from that of the  at z = 0.8, quickly reaching their maxima.  By z = 0, the ex-situ fraction is, respectively for  and  2% and 2.5% compared to 3.5% for . The narrow uncertainties (shaded regions) indicate that this slower ex-situ growth of  is highly significant.
 
The solid lines of the top panel of Fig, <ref> indicate that  and  tend to collect half their ex-situ mass roughly 1.4 and 1.0 Gyr, respectively, earlier than : by, respectively, z = 2.3, (11.0 Gyr ago) and z = 2.0 (10.6 Gyr ago) instead of z = 1.5 (9.6 Gyr ago) for . Given the lower contribution from ex-situ material,  rely more on in-situ star formation to grow in stellar mass,  compared to . In other words, mergers play a smaller role in the evolution of galaxies that end up .

The second and third panels of Fig. <ref> show the split of the ex-situ mass  by the ratio of the secondary to primary stellar masses: major (1/4 to 1), intermediate (1/10 to 1/4) and minor mergers (less than 1/10).[Our merger nomenclature differs  from that of the TNG database, which denotes `minor' mergers as stellar mass ratios from 1/4 to 1/10 (what we call intermediate).] At z = 0,  typically have ∼20% (0.1 dex) lower stellar mass from major mergers (green solid) than  and  (orange and blue solid lines). Also,  have a continuous growth of mass by major mergers, whereas  typically reach a maximum mass from major mergers at z∼ 1.4 () or 1.2 (). In fact, before z=2, the growth of stellar mass from major mergers is   faster in  than in . 

The bottom panel of Fig. <ref> shows that intermediate mergers are even less important for the  relative to the : they produce roughly 3 times less ex-situ stellar mass to the  than to the .

The Stellar Assembly TNG also provides the flyby contribution in ex-situ stellar mass. The flybys contribute less than 1% of the ex-situ stellar mass, for the three size classes. In fact,  the median mass provided by flybys in  is zero.

To summarise, the stellar build up of , especially  is less driven by major and intermediate mergers, while flybys are negligible for all three classes. 

 




Figure <ref> compares the merger histories of  and , showing the epochs of the last major merger, intermediate merger, and any merger. The last mergers of the three mass ratio classes occurs earlier in  compared to : z_ median = 2.74, 2.58, and 0.38 in  vs. 1.6, 1.49, and 0.21 in , for major, intermediate, and minor mergers respectively.  lie in between  and  for intermediate and minor merger, while these galaxies have earlier major and intermediate mergers, with median redshifts of 3.28, 2.21, and 0.27 for the last major, intermediate  and minor merger, respectively. 



Table <ref> shows the significance of the differences of medians for the redshift of the last merger.  and  have statistically identical (P = 0.29) epochs of their last major merger, while the last major merger occurs significantly later  in  (P = 0.0004 and 0.01, respectively). Similarly, the last intermediate merger occurs significantly earlier in   and  compared to  (P = 0.01 and 0.01 respectively). In contrast, while the last minor merger occurs significantly earlier in  compared to , this is not the case for . Disregarding the minor mergers,  thus have more time to evolve without being appreciably perturbed by mergers, which explains why their ex-situ stellar mass is  smaller than in . 






 §.§ Evolution of the environment






The lower ex-situ mass fraction and fewer (major) mergers of  relative to  of the same z=0 stellar mass suggest that  live in lower density regions.  The top panel of Fig. <ref> shows that  central  have lower numbers of satellites[We don't impose a threshold in mass on the satellites around our centrals.] than  since z=0.5: by 25% () or 80% (). The lower panel indicates that central   live in  slightly lower mass halos than . At z=0, the halos of  are 25% () to 60% () lower than those of .   The evolution of the number of satellites of   departs from that of  at z∼2 () and z∼5 (), with a plateau or even a decreasing number until z∼0.3. Similarly, the evolution of halo mass of   departs from that of  at z∼1 () and z∼2 (), with continuously increasing halo mass for all three size classes. In summary,  centrals,  and especially , live in slightly lower mass groups and  have fewer satellites.

 


 §.§ Morphological evolution





We follow the evolution of the flattening of the stellar distribution by using the the Stellar Circularities,  Angular Momenta, Axis Ratios TNG supplementary data catalog <cit.>. This catalog provides the eigenvalues, μ_i, of the mass tensor of the stellar mass within 2 r_1/2, with μ_1 < μ_2 < μ_3. We define a sphericity parameter, μ_1 / √(μ_2  μ_3), where low values indicate a flattened galaxy <cit.>.


Figure <ref> displays the median evolution  of the sphericity parameter  for the  and . The main progenitors of the  centrals typically become progressively flatter in time. Relative to the  centrals, the  centrals follow the same trend of shape until z∼0.6, then become rapidly more spherical. In contrast, the  centrals flatten more quickly than the other two size classes, until z∼ 0.8, then very rapidly become more spherical until 2 Gyr lookback time and remain so until the present.



 §.§ Individual histories







While the median history of a given physical galaxy   parameter allows to view the typical evolution of that parameter  for  and  for , each galaxy has its own evolution. We followed the histories of a few randomly selected individual galaxies to better understand the rapidity of the   decrease in size of the stellar component and its relation to the histories of other galaxy parameters.
 


Figure <ref> displays the individual histories of the half-mass radius and specific star formation  for five ,   and   galaxies. The top panel of Fig. <ref> shows that most  begin shrinking at redshift close to z=0.8 (except for the red one). The compaction of   occurs at different epochs: both with earlier (before z ∼ 1) and late (after z ∼ 1)  compaction. In contrast, the compaction of the different individual  appear  more synchronized in time.  On the other hand, all   have rather constant size  histories after z∼1, with two cases of  centrals that do somewhat shrink  after z ∼ 0.2. All the compaction episodes starting later than 10 Gyr ago last at least 2 Gyr. Nearly all  decrease their stellar size all throughout the last 8 Gyr. Thus, compaction, is not a sudden process,  which is expected from Fig. <ref>.


The compaction of  and  are not affected by the softening of the gravitational interactions between stars and with DM particles, since the softening scale (black curve) is always well below the stellar half-mass radius of the galaxies. On the other hand, some of the  see their sizes reach the softening scale at z ∼ 0.3. This may affect subsequent evolution, since below this scale the calculated parameters are not necessarily reliable. We will discuss this in Sec. <ref>. The  keep shrinking until z=0, except one central (purple)  that suffers a major merger and stops shrinking. Many of the  also keep shrinking until z=0. Two  central (brown and green) stop shrinking when a BH is present (thicker curve).


In general, Fig. <ref> confirms the median trend of Fig. <ref> that major and intermediate mergers are rare in   after z = 1 (only two  out of 10 suffer a major merger). In contrast, mergers are more frequent in  after z=1 (four out of five suffer a major merger). These results are in agreement with Fig. <ref> and Table <ref>, which show that  stop undergoing mergers earlier than .

While AGN activity (thicker lines in Fig. <ref>) is very rare in our ten individual  (only one  case at z < 0.5 and one  case at z < 0.2), it is more frequent in the , in particular at recent times (z < 0.7). In the top panel of Fig. <ref>,  one notices that the size of  increases roughly 1 Gyr after the onset of AGN activity. But since the AGN activity is usually preceded by  major or intermediate mergers, it is not clear if this size increase is  driven by the merger or by the subsequent AGN activity. We will return to this issue in Sect. <ref>.

 

As in Fig. <ref>, the middle panels of Fig. <ref> show that  (especially ) are able to maintain star formation in their inner region after z = 1 almost constant and above the median sSFR evolution for all galaxies in the stellar mass range. One also sees that epochs of late gradual compaction of individual  occurs during times when the inner specific star formation rate is usually enhanced. For example, the green  shrinks between lookback times of 6 to 2.5 Gyr, which corresponds to a higher inner sSFR compared to the median sSFR evolution of all galaxies in our adopted range of stellar masses. This panel is also interesting as it shows both rapid fluctuations in inner sSFR, especially for , and  also  that individual galaxies can move above and below the median trend by typically 0.4 dex, and sometimes downwards by 0.8 dex. 

The bottom panels of Fig. <ref> shows the  evolution of the ratio between the specific star formation rate in the inner and outer region of the galaxy (in terms of r_1/2).  (especially ) have a concentrated star formation after z=1 (ratio above 1). In particular, the periods of gradual compaction of  are even better matched with the ratio of inner to outer sSFR than to the inner sSFR itself (middle panel). For example, the brown  rapidly shrinks between lookback times 7 to 5 Gyr, when the inner sSFR is 1.8 times the outer one; and continues shrinking more gradually in the last 5 Gyr, when the inner sSFR is still 1.4 times the outer sSFR.





 §.§ Radial profiles





 Figure <ref> shows the median radial profiles at different redshifts  for the radial velocity of the gas, gas specific angular momentum normalized by the radius, gas density,  sSFR and stellar density.  
 
  have an intense gas infall compared to the other  galaxies starting at z ∼ 0.5. This infall seems to be related to the lower angular momentum in the inner regions of the  (as shown in the second row of Fig. <ref>): at z ∼ 0.5,  have a specific angular momentum at r = 1 kpc that is only half of that of  and .  also show lower angular momentum, but only at z ∼ 0.2 for r  0.7 kpc. The higher angular momentum of   prevent the gas to efficiently infall.


 As a result of these trends of gas infall, the gas density profiles (third row of Fig. <ref>) of  and  display a relative under-density at r ∼ 1 kpc for z=0.2 (∼ three times lower than the , while these galaxies have instead denser gas than  in the inner regions (below r ∼ 0.7 kpc). 

The gas kinematics should have a strong influence on the star formation rate. The fourth row of Fig. <ref> shows the evolution of the radial profile of sSFR. While  display only small gradients in sSFR at all times,  have a double-peaked sSFR profile at redshifts unity and lower, with a deficiency in the intermediate region (r∼1.5 kpc) at z=1 and 0.5 (by respective factors of three and over ten), relative to peak SFR efficiencies at r∼0.5 and ∼4 kpc.   also display  this relation, but only after z ∼ 0.2. The sSFR gradients agree with those seen in Fig. <ref>. This shows that the compaction of  is related to intense star formation in the core. Central  have the same evolution, but not as strong as for . This suggests that  are extreme cases of central .

The bottom row of Fig. <ref> shows the evolution of the stellar density profile. Compared to the main progenitors of central  (solid orange), the main progenitors of central  (solid green) have lower stellar densities at 1 kpc at all times, but have steeper inner stellar density profiles, catching up with the density of the  at the resolution limit of ≈300 pc.   (solid blue) have similar  stellar densities at all radii as  (solid orange) before z = 0.5, when start to have higher stellar density in the inner region.


Figure <ref> reveals that  the concentrated star formation of  is the consequence of the stronger gas infall, which in turn is due to its lower angular momentum. We saw in Sects. <ref> and <ref> that  tend to undergo fewer mergers and interactions and also live in lower mass halos, with fewer satellites. Therefore, the environment  can have an influence on the angular momentum of dwarf galaxies: a denser environment and more galaxy mergers lead to different gas distributions, where lower angular momentum does not accumulate in the inner regions. 




§ WHAT  PHYSICAL MECHANISMS MAKE TNG50 GALAXIES COMPACT?



We now explore the different  physical mechanisms leading to  galaxies in TNG50.




 §.§ Mergers of globular clusters?



If  were formed by the mergers of globular clusters, one should see an enhanced merger rate at the epochs of globular cluster mergers. However, the median ex-situ fraction of stellar mass is lower in  than in  at all epochs (dashed curves of top panel of Fig. <ref>). Thus, at best only a minority of  could be formed by mergers of globular clusters.



 §.§ Initial compactness?


If central  were initially small at formation of their most massive progenitors, one would see this in the median trends of main progenitor size. However, the early (z>4) sizes of the main progenitors of central  match well those of  (Fig. <ref>). Therefore the origin of  in TNG50 cannot be ascribed to initial compactness.



 §.§ Active galactic nuclei?




In TNG50, a BH is seeded in a subhalo  only if its halo mass exceeds 7.3 × 10^10  <cit.>. Therefore, central galaxies that have a lower mass accretion rate will have more difficulty to host a BH. During the galaxy evolution this accretion can be produced by mergers and interactions, which help a galaxy to reach the TNG halo mass threshold to grow a BH. 




At z=1, the BH occupation fractions of central  are much smaller than for : 0 and 8% of  and  have BHs in contrast with 31% of . By z=0, the BH occupation fractions evolve to 22%, 49% and 60% for , , and  , respectively. These lower occupation fractions are due the lower halo masses of dwarf galaxies, as seen in Fig. <ref> (even  have median z = 0 halo mass slightly close to the threshold for seeding a BH).
 
There are three classes of BH occupation: galaxies that never hosted a BH, galaxies that have a BH at z=0, and galaxies that had a BH in the past and lost it. We  discard this last class of BH occupation, and call the first and second ones “without BH” and “with  BH”, respectively. 

Figure <ref> shows the evolution of size split by z=0 BH occupation. The size evolution is only slightly altered for galaxies with BHs. The sizes of  are significantly larger if they end up with BHs than not, as expected. But for , the z=0 sizes are only weakly affected by the presence of BHs. Therefore the absence of the BH, and henceforth  the absence of AGN feedback cannot be the the main driver in producing . In fact, comparing several non-public higher-resolution TNG simulations with different subgrid physics,  (, lower-right panel of their fig. 8) found that BH feedback has no influence on the sizes of low-mass galaxies at z = 0.





 §.§ Initial halo spin?


Analyzing the Illustris-5 simulation, <cit.> suggested that the angular momentum of the halo is the key parameter to define the morphology of a dwarf galaxy. The second rows of Fig. <ref> shows that, at z=1.5 and z = 1, before the stellar compaction, the gas angular momentum profiles of  are not so different than those of  . The difference in gas spin of  becomes apparent at z = 0.5, especially for the inner regions. At this time,  already have lower sizes than , which shows that the initial halo spin is not the key for  to become compact, although the decrease in size is related to the gas distribution in these galaxies. 



 §.§ Lack of mergers and rapid interactions?


In Sect. <ref>, we found that mergers contribute only a few percent to the z=0 stellar mass of all three size classes of galaxies (3% for  and ∼2% for the , from the top panel of Fig. <ref>). Also,  acquire earlier their ex-situ mass and, after z∼ 2, their ex-situ mass growth is slower than that of  as they evolve “passively” through accretion and in-situ star formation. These results  suggest that the lack of mergers plays a role in the formation of the  galaxies.



For the same galaxies as Fig. <ref>, we computed the  ex-situ contributions for the gas and dark matter components, which are not provided in the TNG supplementary catalogs. Fig. <ref> shows  that starting at z=5,  have only half the ex-situ gas and DM masses than  and , and this ratio remains until z=0. On the other hand,  the ex-situ masses of all three components of  are similar to those of the . This shows, again, a difference between the two size classes of . But while  have similar ex-situ mass as , mergers stopped contributing earlier to their mass growth, as we saw in Fig. <ref>. 




 §.§ Low angular momentum mergers?


Mergers may prevent galaxies from growing in size if the specific orbital angular momentum of the merging galaxies is greater than the internal specific angular momentum of the two galaxies. We checked if the progenitors of the present-day  suffered early from lower orbital angular momentum encounters than , analyzing the same random galaxies used in Fig. <ref>. 

Then, through their merger trees, we computed the angular momentum of the system that encloses the main progenitor and the secondary galaxy. For this, we computed the importance of the orbital angular momentum of the merging system using 

    j_merger/j_max = |r⃗_⃗1⃗2⃗×v⃗_⃗1⃗2⃗|/|r⃗_⃗1⃗2⃗||v⃗_⃗1⃗2⃗|,


where  j_merger is the specific angular momentum of the system, j_max is the maximum value, while r⃗_⃗1⃗2⃗ and v⃗_⃗1⃗2⃗ are respectively the position and velocity in the main progenitor frame. Values close to the unity indicate very high angular momentum mergers. We do not see any statistically difference in the median values for j_merger/j_max:  0.86 and 0.87 respectively for  and , while  have a median of 0.88. Thus,  mergers tend to be high angular momentum, regardless of the galaxy size class.  This indicates that rather than the merger properties, it is the merger frequency and contribution that is the key for , as discussed in Sect. <ref>.




 §.§ Low angular momentum gas infall?



The top panels of Fig. <ref> clearly indicate that  tend to have inner gas infall (which is intense for ), leading to concentrated star formation, and then to a young stellar core, making the galaxy compact. 




More precisely, gas infall leads to  lower angular momentum gas accumulation in the center (second row of panels of Fig. <ref>). This  suggests that  gas infall is caused by lower angular momentum gas accretion onto the galaxy. To verify that, Fig. <ref> shows the angular momentum evolution for the outer (recent infalling) gas particles: v_r < 0 km s^-1 and r > 2 r_1/2, gas.

We computed the median angular momentum for the same 30 centrals galaxies considered in Fig. <ref> at 14 snapshots, roughly regularly spaced in time. Fig. <ref> indicates that, from z=1 (5.5  Gyr lookback time) to z=0.25 (3 Gyr lookback time),  have lower angular momentum of the infalling gas than do , except for central  at one epoch (z=0.5). This time interval corresponds to the epoch of decreasing size for . Therefore, the infall of low angular momentum gas appears linked to the compaction of dwarf galaxies.






§ DISCUSSION






 §.§ Comparison to previous work


First, we agree with <cit.> that  ending up as centrals preferentially form stars in their inner regions. But while these authors explain that these  grew by continuous growth, they do not explain why they became compact.

We now compare with other previous works our main conclusion that  dwarf galaxies that are centrals (or isolated) shrunk as a result of a lack of mergers between redshifts two and unity, allowing gas infall, inner star formation, and the buildup of a stellar core.



Several studies point to different scenarios for massive galaxies. In a series of articles, A. Dekel and collaborators <cit.> argued that massive galaxies at z∼2 went through the following steps: their gas disks, fed by powerful accreting streams and minor mergers, became unstable, creating important clumps that then spiraled into the center by dynamical friction, merging, and creating a compact star forming galaxy that they coined blue nugget, which subsequently passively evolves into a red nugget. Mergers of gas-rich dwarf galaxies were proposed as the mechanism forming blue compact dwarfs <cit.>. The merger will lead to a central gas concentration that results in starburst.

But mergers are rare for dwarf galaxies  <cit.>. The lack of mergers also suppresses the minor mergers as one avenue to generate the violent disk instability mechanism outlined above. The TNG50 simulations point to the opposite scenario:  those  that end up as centrals  are caused by the lack of mergers allowing for gas infall to produce efficient star formation in the inner regions.

Although central  have gas infall (Fig. <ref>) they do not show gas contraction, contrary to  the “red nuggets” scenario. This  suggests that  gas infall maintains the reservoir for the inner star formation without concentrating gas in the galaxy cores.

Finally, <cit.> studied the formation of massive compact galaxies (MCGs) in TNG100 and showed that these galaxies formed by the accretion of low-angular momentum gas. Although this scenario is similar to central  dwarf population evolution (Sect. <ref>), the dwarf galaxies shrink since z=1 in contrast to the MCGs that do not shrink, but have a much slower growth in their sizes than other high-mass galaxies. 



 §.§ Are our results caused by the limitations of the simulation?




  §.§.§ Resolution effects


One may wonder if the 288 pc softening scale of the star and DM particles at z<1 (and constant comoving resolution at earlier epochs) severely affects our results. Since the gas resolution is much finer, then if stars form in a thin disk, they will later diffuse  into a thicker disk. This diffusion occurs  in a numerical two-body relaxation time, which may be faster or slower than the observed or expected from different diffusion mechanisms (molecular clouds, ; spiral arms, ; black holes, ; minor mergers, ; ingested satellites, ; dark matter subhalos, ).



<cit.> argued that the sizes of galaxies with M_⋆ > 10^8 are correctly resolved in TNG50. However, its limited resolution may explain why our  are much bigger than compact ellipticals and ultra-compact dwarfs in our range of stellar masses (8.4 <  < 9.2).

We compared the TNG50 dwarfs to similar-mass dwarfs from a complete observational sample: the fourth data release of the Galaxy and Mass Assembly (GAMA) survey <cit.>. We computed the sizes by taking the galaxy redshifts and masses from the GAMA  table, and the i-band effective (projected half-light) radii from the single-Sérsic fits provided in the  table.  We used the  table for the specific star formation rates.  We selected the galaxies between z = 0.002 to z = 0.025 in our stellar mass range (8.2 <  < 9.4) restricting to galaxies whose log size error were smaller than 0.1. To compare with the TNG50 galaxies, we used the TNG50 supplementary data catalog Stellar Projected Sizes  <cit.> for the z=0 i-band effective radii at a random orientation.


Figure <ref> shows the relation between effective radius and stellar mass for both the TNG50 galaxies and those observed in the GAMA survey.  TNG50 reproduces fairly well the observed  SMR for  and , galaxies.  But one sees that for galaxies with stellar mass with 8.4 <  < 8.8, the GAMA survey shows more very compact galaxies than does TNG50, where the smallest effective radius is ∼ 100 pc. Indeed, the  in TNG50 are at the size resolution of TNG50 at z=0 (black lines in upper panels of Fig. <ref>, and would have probably been smaller in a similar simulation with better spatial resolution.





  §.§.§ The secondary branch


The existence of the clearly distinguished secondary branch may be caused by a physical process causing runaway stellar compaction. Identifying this process is not straightforward.

We noted several differences between  and :
  1) the former tend to end up as satellites, while the latter tend to end up as centrals (Table <ref>);
  2) the  have  lower ex-situ fractions than the  (Fig. <ref>);
  3) the preference for inner star formation is much more pronounced in the  (bottom panel of Fig. <ref>).

Therefore, we found that  are the extreme cases of  , in terms of  concentrated star formation. This concentrated star formation leads to an anticorrelation between size and stellar mass: galaxies with higher stellar mass (that form more stars in their inner regions) have smaller sizes. This is what we see clearly in the secondary branch (Fig. <ref>).

The intensity of the inner star formation depends on the inner gas infall (top panels of Fig. <ref>), which is controlled by the discrete number of major and intermediate mergers: too few mergers lead to uncontrolled inner gas infall.
  
  The small sizes of the stellar components of  are at the limit of the spatial resolution (softening scale) of TNG50. With better resolution, the secondary branch of the SMR would  extend to lower sizes and thus not be so sharp.  However, we have not been able to explain the gap between the two branches, other than the fact that the   are the result of intense concentrated star formation.




Figure <ref> shows a possible gap in the GAMA SMR at effective radii that increase from 140 pc to 450 pc (log(r_eff/kpc) between ≃ -0.85 and -0.35) between log (M_⋆/ M_⊙)=8.4 to 9.2, but it is not as pronounced  as in TNG50. 
This gap in GAMA is also seen as the abrupt rise  in Fig. <ref> of the GAMA counts at log(r_ eff/ kpc) ≃ -0.7 (200 pc). The histogram of log sizes weakens the observed gap, whose location in the SMR increases slightly with stellar mass, as noted above. However, a Hartigan dip test <cit.> of the residuals of log r_1/2 indicates that this gap is not statistically significant. Furthermore, it is less pronounced if we increase the maximum redshift for our sample of GAMA dwarfs.



  §.§.§ High star formation rates in the cores of 


It is difficult to understand why the simulated  tend to be more star-forming than , while the opposite trend appears in the SDSS <cit.>. Fig. <ref> shows that the star forming galaxies  show a  bimodality in TNG50 that is not present in GAMA. Also, passive galaxies are likely to have smaller sizes in GAMA than TNG50.

One possible explanation is that TNG50 is missing low-mass BHs in dwarf galaxies in general and in the  in particular. More precisely, the minimum halo mass to seed a black hole is 7.3 ×10^10 in the TNG simulations, which may be too large (at least in TNG50) and lead to too low BH occupation fractions, especially for  that have a lower contribution from mergers and therefore have more difficulty reaching the threshold to seed a BH.  

As seen in Fig. <ref>, not only  are bluer (they have higher sSFR) than , but they have blue cores (their sSFR is higher in their cores). Although the presence or absence of the BH does not affect the decrease in the size of  (Sect. <ref>), central feedback from the BH could perhaps regulate this higher sSFR. Both TNG50 star formation and BH prescriptions may favor concentrated star formation and thus the formation of compact dwarf galaxies.



§ CONCLUSIONS



In this first study analyzing the TNG50 cosmological hydrodynamical  simulation to explore what drives low and high corpulence galaxies, we explored the physical mechanisms that cause  central dwarf galaxies  to become . We selected dwarf galaxies according to their z=0 properties: stellar log masses were between 8.4 and 9.2 (solar units). We called compact the lower envelope of the main branch of the size-mass relation (), easily selected using Eqs. (<ref>), (<ref>); as well as the dwarfs in the secondary branch,  with stellar half-mass radius below 450 pc (), while the galaxies that ended with larger sizes were called . We furthermore restricted our samples to the  and  that ended up as centrals of their hosts. 


Galaxies that end up as central  typically have a similar size evolution as  until z=0.8, at which point they start shrinking, while the  keep growing (Figs. <ref>).  Central  live in lower density environments (Figs. <ref>), have lower  ex-situ stellar mass fractions (Figs. <ref>, <ref>, <ref>). They are thus the lucky ones that suffered fewer major and intermediate mergers before z=1 (Figs. <ref> and <ref>)  and accreted more lower angular momentum gas (Fig. <ref>).

These mergers would otherwise pump their orbital angular momentum into the remnant and prevent radial gas inflow. Here, the gas infall is ubiquitous and produces  a concentrated star formation (Figs. <ref> and <ref>), which leads to the formation of a stellar core and makes the galaxy more spherical (Fig. <ref>).  While this lack of mergers prevents the formation and growth of black holes, whose AGN could prevent the gas infall, the rare presence or not of a BH has little effect on the evolution of the stellar sizes of , in contrast with   (Fig. <ref>). The compaction of individual central  is gradual and occurs when the ratio of inner to outer specific star formation rate is enhanced (Fig. <ref>). Their gas infall (much stronger in  than in ) occurs while their gas half-mass-radii keep growing (Fig. <ref>).

It is not clear how our conclusions depend on the limitations of the TNG50 simulation. Observed , from the GAMA survey, also show weak but not significant signs of bimodality of sizes (Fig. <ref>). But while the  are typically star-forming, with `blue' cores (Fig. <ref>) as observed in some compact dwarfs, and appear round (Fig. <ref>), GAMA  tend to be passive (Fig. <ref>). This difference may be caused by the absence of black holes in very low-mass halos in the sub-grid physics of TNG50. Therefore, a study like ours ought to be repeated with current (e.g. NewHorizon,  and FIRE-2, ) and future high-resolution cosmological hydrodynamical simulations.



§ ACKNOWLEDGEMENTS

We thank the anonymous referee for useful comments. APA thanks the São Paulo Research Foundation, FAPESP, for
financial support through contracts 2022/05059-2 and  2020/16152-8. GBLN
acknowledges partial financial support from CNPq grant 303130/2019-9.



