A Universe of Ultra-Diffuse Galaxies: Theoretical Predictions from $\Lambda$CDM Simulations

TL;DR

None

Contribution

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Abstract

A particular population of galaxies have drawn much interest recently, which are as faint as typical dwarf galaxies but have the sizes as large as $L^*$ galaxies, the so called "ultra-diffuse galaxie" (UDGs). The lack of tidal features of UDGs in dense environments suggests that their host halos are perhaps as massive as that of the Milky Way. On the other hand, galaxy formation efficiency should be much higher in the halos of such masses. Here we use the model galaxy catalog generated by populating two large simulations: the Millennium-II cosmological simulation and Phoenix simulations of 9 big clusters with the semi-analytic galaxy formation model. This model reproduces remarkably well the observed properties of UDGs in the nearby clusters, including the abundance, profile, color, and morphology, etc. We search for UDG candidates using the public data and find 2 UDG candidates in our Local Group and 23 in our Local Volume, in excellent agreement with the model predictions. We demonstrate that UDGs are genuine dwarf galaxies, formed in the halos of $\sim 10^{10}M_{\odot}$. It is the combination of the late formation time and high-spins of the host halos that results in the spatially extended feature of this particular population. The lack of tidal disruption features of UDGs in clusters can also be explained by their late infall-time.

Tables

Table 1. Table 1: Parameters of the 23 UDG candidates selected from the dwarf catalog of Karachentsev et al. (2013). Col. (1): Dwarf Name. Col. (2): Distance (Mpc) to the Milky Way. Col. (3): The linear Holmberg diameter in unit of kpc (Karachentsev et al. 2004; Karachentsev et al. 2013). Col. (4): B 𝐵 B -band absolute magnitude. Col. (5): Mean surface brightness. Col. (6): Logarithm of K 𝐾 K -band luminosity ( L ⊙ subscript 𝐿 direct-product L_{\odot} ).

Name	D	$A_{26}$	$M_B$	${⟨\mu ⟩}_B$	$\log L_K$
Col. (1)	Col. (2)	Col. (3)	Col. (4)	Col. (5)	Col. (6)
Sag dSph	0.02	3.08	-12.67	26.08	8.02
And XXXII	0.78	3.60	-11.53	27.56	7.56
NGC3109	1.34	7.73	-15.75	25.01	8.58
DDO099	2.65	3.24	-13.53	25.34	7.42
KK35	3.16	3.91	-14.30	25.00	7.97
KKH12	3.48	3.44	-13.35	25.65	7.80
MB3	3.48	5.19	-13.97	25.92	8.22
Cam A	3.56	4.61	-13.85	25.78	7.79
CenA-MM-Dw1	3.63	3.13	-12.56	26.23	7.98
IKN	3.75	3.15	-11.63	27.17	7.60
ESO269-058	3.75	5.52	-15.04	25.00	8.86
KK77	3.80	3.15	-12.22	26.58	7.84
HolmIX	3.85	3.15	-13.75	25.06	7.75
HolmI	4.02	5.54	-14.59	25.44	8.05
LV J1228+4358	4.07	4.56	-13.94	25.67	7.83
UGC A442	4.37	7.52	-14.71	25.98	8.03
DDO169	4.41	3.65	-13.80	25.32	7.73
IC3687	4.57	6.92	-14.60	25.91	8.19
CenA-MM-Dw3	4.61	6.63	-12.32	28.10	7.88
DDO226	4.92	3.12	-13.63	25.15	7.71
DDO126	4.97	4.14	-14.42	25.00	8.09
KK208	5.01	8.77	-14.39	26.64	8.71
ESO115-021	5.08	10.14	-15.58	25.76	8.75

Full text

1 Introduction

A population of low surface brightness galaxies has been observed in spatial regions of, e.g., Coma (van Dokkum et al., 2015a, b; Koda et al., 2015; Yagi et al., 2016), Virgo (Mihos et al., 2015), Fornax (Muñoz et al., 2015), A168 (Román & Trujillo, 2016a), A2744 (Janssens et al., 2017), eight other clusters with redshifts $z\sim 0.044\--0.063$ (van der Burg et al., 2016), and Pisces-Perseus supercluster (Martinez-Delgado et al., 2016), as well as several galaxy groups (Makarov et al., 2015; Toloba et al., 2016; Román & Trujillo, 2016b; Trujillo et al., 2017). While their stellar masses are similar to typical dwarf galaxies, their effective radii are similar to the $L^{*}$ galaxies (van Dokkum et al., 2015a; Beasley et al., 2016). These galaxies are generally referred to as ultra-diffuse galaxies (UDGs). Except for several blue ones (e.g., Román & Trujillo, 2016a), the majority of the observed UDGs populate the red sequence, suggesting that the star formation in UDGs has been quenched before $z\sim 2$ (van der Burg et al., 2016).

UDGs are ubiquitously distributed from the cores of galaxy clusters to the surrounding large-scale filaments (Koda et al., 2015; Yagi et al., 2016; Román & Trujillo, 2016a, b). Since they can withstand the strong tidal forces in the cluster cores without significant features of tidal disruption, one scenario is that UDGs are dark matter dominated galaxies, for instance, the failed $L_{\star}$ galaxies which lost their gas content after the first generation of stars (van Dokkum et al. 2015a; Scannapieco et al. 2008; Stinson et al. 2013; Trujillo-Gomez et al. 2015). Using stellar kinematics of Dragonfly 44, van Dokkum et al. (2016) measured its dynamical mass as $\sim 10^{12}\ M_{\odot}$, similar to the mass of the Milky Way; this is unexpected from prediction of subhalo abundance matching (SHAM) in which galaxy formation efficiency reaches its maximum at this halo mass (Guo et al., 2010; Conroy & Wechsler, 2009; Simha et al., 2012). Another scenario is that UDGs are spatially extended dwarf galaxies (e.g., Amorisco & Loeb, 2016; Di Cintio et al., 2017; Dalcanton et al., 1997a, b; Mo et al., 1998; Huang et al., 2012). Using the abundance and kinematics of globular clusters around two UDGs, VCC 1287 and DF17, Beasley et al. (2016) and Beasley & Trujillo. (2016) estimated the corresponding dynamical masses to be around $m_{\rm{vir}}\sim(8\pm 4)\times 10^{10}\ M_{\odot}$ and $m_{\rm{vir}}\sim(9\pm 2)\times 10^{10}\ M_{\odot}$ respectively, similar to that of the typical dwarf galaxies. Using the relation between the mass of the globular cluster system and the halo mass (Harris et al., 2013, 2015), Peng & Sungsoon (2016) also inferred the total mass of DF17 to be $(9.3\pm 4.7)\times 10^{10}\ M_{\odot}$; Amorisco et al. (2016) estimated the dynamical masses of 54 Coma UDGs to be lower than $1.3\times 10^{11}\ M_{\odot}$. Román & Trujillo (2016a) also found that the distribution of UDGs around A168 is similar to the normal dwarfs, but significantly different from the distribution of massive galaxies with masses similar to the Milky Way. Using the width of the HI line, Trujillo et al. (2016) found a UDG in the very local Universe with a virial mass of 8 $\times 10^{10}M_{\odot}$. Theoretical work also suggest that UDGs might be genuine dwarf galaxies possibly with high spins (Amorisco & Loeb 2016, Yozin & Bekki 2015), or spatially extended stellar components caused by feedback driven gas outflows (Di Cintio et al., 2017).

In this paper, we will use a publicly available semi-analytic galaxy catalog (Guo et al., 2013) to investigate whether UDGs can naturally emerge from the $\Lambda$CDM hierarchical structure formation model. In section 2, we briefly describe the simulation and the semi-analytic models, as well as the selection criteria of the possible UDGs. In section 3, we compare the model predictions with observational results. In section 4, we study the distributions of the model UDGs, and dependence of UDG properties on environments. In section 5, we investigate the origin of this particular population. Conclusions are presented in section 6.

2 UDG selection from simulations

2.1 Simulations

The galaxy catalogs used here are based on two simulations, the Millennium-II simulation (MS-II; Boylan-Kolchin et al. 2009), and Phoenix simulation (Gao et al. 2012). MS-II is a high-resolution cosmological $N$-body simulations, following $2160^{3}$ particles from $z=127$ to $z=0$ in a periodic box of $100\ {\rm{Mpc}}/h$ on a side. Each dark matter particle has a mass of $6.88\times 10^{6}\ M_{\odot}/h$. Particle data were stored at 68 logarithmically spaced output times. The MS-II simulation adopted the cosmological parameters consistent with the first-year Wilkinson Microwave Anisotropy Probe (WMAP) result; it was then rescaled to that consistent with WMAP seven-year parameters (Guo et al., 2013): $\Omega_{\rm{m}}=0.272$, $\Omega_{\rm{b}}=0.0455$, $h=0.704$, $\sigma_{8}=0.81$, $n=0.967$. The Phoenix simulation is a high-resolution re-simulation of nine individual rich clusters and their surroundings. Each Phoenix cluster has been simulated at different resolution levels for numerical convergence studies. Here we adopt the simulation with level 2 resolution. Particle mass is of $\sim 10^{6}\ M_{\odot}/h$, which varies from cluster to cluster slightly. The Phoenix simulation adopted cosmological parameters from a combination of Two-degree-Field Galaxy Redshift Survey (Colless et al. 2001) and first-year Wilkinson Microwave Anisotropy Probe data (Spergel et al. 2003): $\Omega_{\rm{m}}=0.25$, $\Omega_{\rm{b}}=0.045$, $h=0.73$, $\sigma_{8}=0.9$, $n=1$. Although the cosmological parameters adopted by these two simulations are slightly different, this has negligible effect on our main results.

At each snapshot, dark matter halos are identified with the friends-of-friends (FOF) group algorithm by linking particles separated by 0.2 times the mean inter-particle separation (Davis et al. 1985). The SUBFIND algorithm (Springel et al. 2001) was then used to identify self-bound subhalos; merger trees were constructed by linking each subhalo at different output times to its unique descendant using the algorithm described in Springel et al. (2005) and Boylan-Kolchin et al. (2009).

MS-II contains millions of halos from $10^{10}\ M_{\odot}/h$ to $10^{14}\ M_{\odot}/h$, allowing us to study the possible UDGs in different environments in a statistical way. Yet limited by the box size, MS-II has no clusters as massive as the Coma clusters, $\sim 2\times 10^{15}\ M_{\odot}$, and A2744, $\sim 5\times 10^{15}\ M_{\odot}$, where the largest samples of UDGs are discovered. The Phoenix simulation suits compensate this by providing more massive cluster samples, and the largest cluster in the Phoenix suits has a mass $\sim 3.4\times 10^{15}\ M_{\odot}$ (at $z=0$). The minimum resolved halo in this Phoenix cluster is of mass $3.6\times 10^{8}\ M_{\odot}/h$, well below the mass limit ($\sim 10^{9}M_{\odot}$) below which haloes can not form any galaxies. We thus use the combination of these two simulation sets to study UDGs in various environments.

2.2 Galaxy formation model

In order to populate dark matter halos with galaxies, we applied the semi-analytic galaxy formation models (Guo et al. 2011, 2013) to the stored subhalos merger trees extracted from these $N$-body simulations. This model has been proved successful in reproducing many galaxies properties both in the local Universe and at high redshifts, and particularly it provides convincing results for galaxy size vs. stellar mass relations. Here we briefly summarize the main physical processes relevant to the formation of galaxies as faint as UDGs and the models of galaxy stellar component sizes.

As discussed in Guo et al. (2011; here after Guo11), two processes are crucial for the formation of low mass galaxies: UV reionization and supernova (SN) feedback. The capability to capture baryons is reduced in low mass systems due to the UV reionization. Guo et al. (2011, 2013) adopted results given by Okamoto et al. (2008) to quantify the fraction of baryons as a function of halo mass. As demonstrated in Guo11 this effect becomes significant for galaxies less luminous than $M_{V}=$ -11. Vast amount of energy is released during SN explosion which can reheat the surrounding gas and even eject gas out of its dark halo. Guo11 introduced a SN feedback model which depends on the maximum velocity of the host halo, leading to a relatively more efficient feedback in low mass halos than their high mass counterparts. This significantly changes the slope of the stellar mass function at the low mass end (e.g. stellar mass $m_{\rm{st}}<10^{9.5}\ M_{\odot}$).

Guo11 uses the stellar population synthesis models from Bruzual & Charlot (2003), and adopts a Chabrier initial function to calculate the photometric properties of galaxies. For low-redshift galaxies, the slab dust model introduced in De Lucia & Blaizot (2007) is then implemented to account for the extinction of star light. Further comparison with observations suggests that the fiducial model of Guo11 can well predict the luminosity function of galaxies at low redshifts (e.g., Guo et al., 2011; Nierenberg et al., 2012), particularly in the faint end.

Our model galaxies contain two components, disks and bulges. Guo11 assumed the stellar disk to have an exponential surface density profile. Its size is determined by the specific angular momentum and the circular velocity (here using $V_{\rm{max}}$ as a proxy). The angular momentum is obtained from its gas disk during the star formation. The gas disk acquires its angular momentum during the cooling process, i.e. the cooling gas is assumed to have the same specific angular momentum as its host halo. Bulges are formed by mergers and disk instability. In Guo11, bulge sizes were calculated by assuming energy conservation and virial equilibrium. For mergers, the relevant components are the binding energy and interaction energy of the two merging galaxies. For disk instability, they are the binding energy and interaction energy of the existing bulges and the part of mass which is transferred into bulges during disk instability.

In order to compare with the observations directly, we convert the 3-D radius of our model galaxies to effective radius $r_{\rm{e}}$ (also referred to as the projected half-mass radius) by assuming the stellar bulge and stellar disk to have the Jaffe (Jaffe, 1983) and exponential density profiles, respectively (see Xie et al. 2015 for details). For each galaxy, we divide its projected radius (from $10\ \rm{pc}$ to $10\ \rm{kpc}$) into 100 bins ($r_{i}$, where $i$ is from 1 to 100) in logarithm scale and calculate the projected absolute magnitude $M_{i}^{\rm{pro}}$ of each bin. The central surface brightness $\mu_{0}$ is obtained by fitting the surface brightness $\mu_{i}\sim M_{i}^{\rm{pro}}+2.5~{}{\rm{log}}_{10}(s_{i})$, where $s_{i}$ is the area of the bin, using a Sérsic model. We test the dependence of our results on bin sizes and find that this effect is minor.

2.3 Sample selection

There are a few ways to define UDGs in the literature. van Dokkum et al. (2015a, b) defined UDGs as the galaxies with $\mu_{0,B}>24\ \rm{mag/arcsec^{2}}$ and $r_{\rm{e}}>1.5$ kpc; some others used a slightly different quantity, the mean surface brightness within $r_{\rm{e}}$, $\langle\mu\rangle_{{\rm{e}},r}\geq 24\ \rm{mag/arcsec^{2}}$ (e.g. Yagi et al., 2016; van der Burg et al., 2016; Janssens et al., 2017). Given that the Sérsic indices, $n$, of most UDGs are around 1 (Yagi et al., 2016; Koda et al., 2015; Román & Trujillo, 2016a) and colors $g-r\sim 0.6$ (van der Burg et al., 2016), $\langle\mu\rangle_{{\rm{e}},r}\geq 24\ \rm{mag/arcsec^{2}}$ is approximately equivalent to $g$-band $\mu_{0,g}>23.5\ \rm{mag/arcsec^{2}}$ (Graham & Driver, 2005). In this work, we adopt the criteria as follows

• $1.5<r_{\rm{e}}<4.6\ {\rm{kpc}}$,
• -17 $<M_{g}<$-11.5 mag,
• and $\mu_{0,g}>23.5\ \rm{mag/arcsec^{2}}$,

where $M_{g}$ is the $g$-band absolute magnitude.

We show the $r_{\rm{e}}\--M_{g}$ relation for the model galaxies in Fig. 1. Different mean surface brightness $\langle\mu\rangle_{\rm{e}}$ thresholds are highlighted with the dotted lines. We find that almost all of the UDG candidates (red dots) are distributed above $\langle\mu\rangle_{\rm{e}}=25\ \rm{mag/arcsec^{2}}$. The luminosities of the observed UDGs are similar to those of the typical dwarf galaxies; however, their sizes are much larger. Interestingly, Fig. 1 shows that UDGs are not an isolated population, rather they exist as a continuous extension of typical dwarf galaxies. For galaxies of similar magnitudes, the UDGs occupy the large-size tail of the size distribution, suggesting that UDGs are indeed a subsample of dwarf galaxies.

In order to understand the properties of UDGs and compare them with the typical dwarf galaxies more clearly, we select a counterpart sample of the dwarfs within the same luminosity range as the UDGs, but different sizes of $r_{\rm{e}}\in(0.5,1.0)\ \rm{kpc}$ (van der Burg et al., 2016; Misgeld & Hilker, 2011). According to the studies of Graham & Guzmán (2003) and Mo et al. (2010), these dwarf counterparts primarily include the dwarf ellipticals (dE) and dwarf spheroidals (dSph). Note that the ultra-compact dwarfs are not included in this sample. In total, we have $4.4\times 10^{4}$ UDGs and $1.3\times 10^{5}$ dwarf counterparts, corresponding to 11% and 32% of the faint galaxies ($M_{g}\sim$-17 $\--$ -11.5) respectively, at $z=0$ in MS-II.

3 UDGs in simulations and observations

In this section we will firstly compare the properties of the model UDGs with the observations, and then explore how they vary with the different environments.

3.1 UDGs in clusters

Most of the observed UDGs are discovered in clusters, e.g., in Coma (van Dokkum et al. 2015a), A168 (Román & Trujillo, 2016a), A2744 (Janssens et al., 2017), and other 8 low-redshift clusters (vdB16; van der Burg et al., 2016). Here we focus on the comparison of the UDGs in clusters between the model predictions and observations.

Abundance

The abundances of UDGs are observed to correlate with the mass of their host cluster (Janssens et al., 2017; van der Burg et al., 2016): $N\propto M_{200}^{0.93\pm 0.16}$, where $N$ is the number of UDGs in a cluster and $M_{200}$ is the cluster mass within a radius, $r_{200}$, within which the average density is 200 times the cosmic critical density. In Fig. 2, we show the observed results, as well as our model predictions of the abundances of UDGs as a function of their host cluster masses. In order to compare with the observations directly, here we discard the model UDGs fainter than $\mu_{0,g}\sim 26.5\ \rm{mag/arcsec^{2}}$ (approximately corresponding to the $r$-band $\langle\mu\rangle_{{\rm{e}},r}\leq 27\ \rm{mag/arcsec^{2}}$). It shows that the model predictions are in excellent agreement with the observed abundance-mass relation, from groups to rich clusters.

Surface number density (SND) profile

Spatial distribution of the galaxies in clusters provides important clues to their evolution. In Fig. 3, we compare the SND profile of the observed UDGs in 8 observed clusters (van der Burg et al., 2016) with the average SND profile of the model UDGs in the 10 simulated clusters ($M_{200}>10^{14}\ M_{\odot}/h$) selected from the MS-II simulation. The observed SND profile is rescaled by a constant factor to account for the different normalization methods. Clearly, the predicted SND profile fits very well with the observed one over all the observed scales from 0.03$r_{200}$ to $r_{200}$, including the flatting feature in the inner part.

For completeness, we also show the SND profile of the typical dwarf counterparts (the black histogram) in Fig. 3. The SND of UDGs is similar to the profile of the dwarfs at $r/r_{\rm{200}}>0.15$ ($r/r_{\rm{200}}=0.15$ is denoted by the vertical dashed line in Fig. 3), while it is significantly lower at $r/r_{\rm{200}}<0.15$. Moreover, UDGs are absent in the innermost region $r/r_{\rm{200}}<0.03$. The lack of UDGs in the inner regions of clusters could be caused by two possible reasons: (i) UDGs might have been disrupted and dissociated by the strong tidal forces in the inner regions; (ii) UDGs might have fallen into the clusters more recently than the dwarf counterparts so that they have not arrived in the inner regions yet. Observationally, the evidence of tidal disruption for UDGs is very rare (Mihos et al., 2015; Toloba et al., 2016). We will show in section 4 that our model indeed supports the second explanation.

Color

Previous work found that except for several UDGs in groups (e.g., Román & Trujillo, 2016a), most of the observed UDGs are red (e.g., van Dokkum et al., 2015a; Koda et al., 2015; van der Burg et al., 2016). Fig. 4 displays the color-magnitude diagrams for the model UDGs in our simulated clusters. Analogous to the observations, most of the model UDGs are red, except for several relatively faint ones. The left panels show the color distributions of the model UDGs (black histograms) and the 1$\sigma$ range of the observed ones (colored regions) in clusters (van der Burg et al., 2016; van Dokkum et al., 2015a). Most of the model UDGs are located at $g-r\sim 0.6\pm 0.1$ and $g-i\sim 0.8\pm 0.1$, in good agreement with the observations.

Morphology

Observationally, UDGs are found to have low Sérsic indices, $n\sim 0.6\--1$ (e.g., van Dokkum et al., 2015a; van der Burg et al., 2016; Muñoz et al., 2015; Koda et al., 2015). Limited by the capability of semi-analytic models, we cannot measure the profiles directly. Instead here we use the bulge-to-total mass ratio, $B/T$ (Weinzirl et al., 2009), as a proxy, i.e., a lower Sérsic index corresponds to a lower value of $B/T$. The upper panel of Fig. 5 shows that most (95.7%) of the model UDGs have extremely low $B/T$ ($B/T<0.1$), in line with the observed low Sérsic indices.

Most of our model UDGs present $B/T<0.1$, suggesting that the model UDGs perhaps have an oblate, disk-like geometry. This result may conflict with the conclusion of Burkert (2016), who claimed that UDGs are more likely to be prolate rather than oblate because of their observed axial ratio range $q\sim 0.4\--1.0$. However, this conflict may be due to the fact that most of the observed UDGs have relatively small inclination angles $\theta$, i.e., they are more likely to be “face-on” rather than “edge-on”. This is because the UDGs with large inclination angles may be too bright to be identified as ‘UDGs’. For instance, for a UDG with $\langle\mu\rangle_{\rm{e}}\sim 26\ \rm{mag/arcsec^{2}}$ and $\theta\sim 70^{\circ}$, its projected area decreases by a factor of $(\cos\theta)^{2}$, implying that the surface brightness changes about $2.5\log_{10}(\cos\theta)^{2}\simeq 2.3$ mag; therefore its apparent surface brightness becomes $\langle\mu\rangle_{\rm{e}}\sim 23.7\ \rm{mag/arcsec^{2}}$, which would be beyond the criterion to select UDGs.

Total mass

The observed UDGs are very diffuse, and can reside in the dense environments without significant evidence of tidal disruption, suggesting that they are highly dark-matter dominated systems. Observationally, we usually use the abundance of the member globular clusters to infer UDG virial mass, $m_{\rm{vir}}$. In the lower panel of Fig. 5, we show the distribution of the virial masses of the model UDGs. Most of them are in the range of $10^{9}\--10^{11}\ M_{\odot}$, consistent with the recently reported total masses for VCC 1287, DF17, and UGC2162 (Beasley et al., 2016; Beasley & Trujillo., 2016; Trujillo et al., 2017), and Fornax UDGs (Zaritsky, 2016), as well as 54 Coma UDGs (Amorisco et al., 2016). The peak of the model predictions is lower than the observations, which is primarily because that the measured UDGs are observationally brighter. Note that one particular case, DF44, is reported (van Dokkum et al. 2016) to be hosted in a dark halo as massive as $m_{\rm{vir}}\sim 8\times 10^{11}\ M_{\odot}$, an order of magnitude more massive than the typical mass found for the other UDG hosts. Zaritsky (2016) argued that DF44 lies at the upper-end in the size-enclosed mass relation of the observed UDGs and thus may not be a typical UDG.

3.2 UDGs in the Local Group and Local Volume

Most UDGs are discovered in rich clusters and their surrounding. A very interesting issue is whether UDGs could exist in the Local Group ($\sim 5\times 10^{12}\ M_{\odot}$, e.g. Li & White 2008). In the literature, two UDG candidates are found in the census of faint galaxies in the Local Group: Sagittarius dSph (McConnachie, 2012) and Andromeda XXXII (Martin et al., 2013). Sagittarius dSph is 26 kpc away from us, with $V$-band absolute magnitude of $M_{V}=-13.5$ mag, $r_{\rm{e}}=2.6$ kpc, and $V$-band $\mu_{0}\simeq 25.2\ \rm{mag/arcsec^{2}}$. This UDG candidate has been reported by Yagi et al. (2016), who are the first authors to identify UDGs in the Local Group. Andromeda XXXII is 0.78 Mpc away from us, with $M_{V}=-12.3$, $r_{\rm{e}}=1.46$ kpc (slightly smaller than 1.5 kpc), and $\mu_{0}=26.4\ \rm{mag/arcsec^{2}}$.

To compare the model predictions with the data, we first define the Local Group analogues in the simulation according to the observable properties of the Local Group. Here we adopt the selection criteria similar to those described in Xie et al. (2014). We first select the Milky Way analogue using the criteria: $B/T<0.5$ and $5.4\times 10^{10}<M_{\rm{MW}}<7.4\times 10^{10}\ M_{\odot}$, where $M_{\rm{MW}}$ is the stellar mass of the Milky Way analogue, and then request that there is only one bright companion (M31 analogue) within 1 Mpc from each Milky Way analogue, with stellar mass $M_{\rm{MW}}<m_{\rm{st}}<2M_{\rm{MW}}$ (this mass restriction is slightly different from the criterion used in Xie et al. 2014). We further require no galaxy clusters with masses $\geq 10^{14}\ M_{\odot}$ within 10 Mpc of the Local Group catalog. In total, we find 69 “Local Group” analogues in the model galaxy catalog. 207 model UDGs are found within 1 Mpc of the 69 “Local Groups” analogue, i.e. 3 UDGs in each system on average. The $r_{\rm{e}}$ vs. $M_{g}$ relation of these 207 UDGs are overplotted with the red diamonds in Fig. 6.

We further extend the searching radius from 1 Mpc to 5 Mpc as the Local Volume analogue. There are in total 1654 model UDGs (cyan triangles in Fig. 6) in the simulated Local Volumes, corresponding to 24 UDGs in each system. Observationally, we use the dwarf catalog by Karachentsev et al. (2013)111http://www.sao.ru/lv/lvgdb/ to search for the UDG candidates and find 23 possible UDGs residing in the Local Volume, as listed in Table. 1. Note that these galaxies are observed in a different wavelength and their $\mu_{0}$ are not given by Karachentsev et al. (2013), we thus use the selection criteria slightly different from those in Sec 2.3 by requiring: a linear Holmberg diameter $A_{26}>3$ kpc, $\langle\mu\rangle_{B}\geq 25\ \rm{mag/arcsec^{2}}$, and $10^{6}<L_{K}<10^{9}\ L_{\odot}$. Among these possible UDG candidates, CenA-MM-Dw3 has been reported (Crnojević et al., 2016).

In summary, the predicted abundances of UDGs in the simulated Local Group analogue and Local Volume analogue agree very well with those in the real Universe. In addition, we find that the model UDGs comprise less than 10% of the total faint populations ($M_{g}\sim$-17 $\--$ -11.5 mag) in these two systems and thus will not significantly affect the corresponding conditional luminosity functions by including/excluding UDGs.

4 Distribution of the UDGs in the Universe

In the last section, we demonstrated that the model reproduces most of the available observational properties of UDGs. This encourages us to use our model to make prediction of the distribution of UDGs in the Universe.

Fig. 7 shows the accumulative fraction of the model UDGs as a function of the group masses. We find that only 4% model UDGs reside in the clusters more massive than $10^{14}\ M_{\odot}$, while most of the model UDGs (80%) reside in the groups with $M_{200}<10^{12}\ M_{\odot}$ or in the fields. We further find that most of the model UDGs are red in color regardless of their environments, yet with an expected positive correlation between the red fractions and densities of environments.

Comparing the distributions of the model dwarf galaxies and UDGs, in the clusters and groups (more massive than $10^{12}\ M_{\odot}$), 7% of the dwarf galaxies are classified as UDGs, and the fraction is nearly independent of the host halo mass. However in the fields, this fraction is as high as 14%, suggesting that UDGs tend to stay in the lower-density environment. The general low fractions of UDGs also suggest that the luminosity function at the faint end will not be significantly affected by including/excluding UDGs; the effect of UDGs on the conditional luminosity functions is even weaker in the groups and clusters.

In order to study the environmental dependence of the model UDGs in more details, we further divide the model UDGs into four subsamples according to their host halo masses, clusters ($m\geq 10^{14}\ M_{\odot}/h$), groups ($m\sim 10^{12}\--10^{14}\ M_{\odot}/h$), galaxy systems ($m<10^{12}\ M_{\odot}/h$), and fields. UDGs in the former three systems exist as satellite galaxies, while in the fields, they are central galaxies of their own halos (i.e., isolated UDGs). In Fig. 8, we show the probability distributions of six different physical properties for the 4 UDG subsamples. The stellar masses, $m_{\rm{st}}$, of the model UDGs increase with the densities of environments (panel A), e.g., $m_{\rm{st}}$ of the UDGs in the clusters is higher by about 0.2 dex than those in the fields. This is consistent with the increasing fraction of the red populations with the increasing environmental density, as the red galaxies are usually older and more massive (for a given luminosity). UDGs in clusters tend to have lower specific star-formation rates (SSFR=star-formation rate/$m_{\rm{st}}$; panel F) and be relatively older (panel D). Galaxies formed earlier (with higher mass-weighted-ages, $\tau$) are usually more compact, which is reflected by the distributions of $r_{e}$ (panel E). Different from the other properties, dependences of the virial mass $m_{\rm{vir}}$ (panel B) and morphology (panel C) on the environments are very weak. Regardless of the environments, most of the model UDGs are disk-dominated systems and formed in the halos of virial mass $\sim 10^{10}M_{\odot}$, very similar to that of a typical dwarf galaxy.

5 Formation of UDGs

As discussed above, our model predicts that UDGs have the similar dark matter halos to those of the typical dwarf galaxies. In this section, we explore why the stellar components of UDGs are so extended.

In order to investigate the differences between the model UDGs and typical dwarf galaxies more clearly, in Fig. 9 we compare the distributions of the stellar mass (panel A), stellar mass-to-light ratio (panel B), $B/T$ (panel C) and host halo mass (panel D) between the two samples. The typical dwarfs in general have larger stellar masses than UDGs (panel A). This is primarily because that the model UDGs are younger (see the lower panel in Fig.10) and bluer. Consistently, the stellar mass-to-light ratio of a UDG is typically smaller than that of a typical dwarf galaxy by around 50% (panel B). One needs to pay particular attention when converting the luminosities of UDGs to their stellar masses. UDGs are much more extended than the regular dwarf galaxies, and thus we expect that the internal structures of UDGs differ from the dwarf counterparts as well. As shown in the panel C, $B/T$ of the model UDGs are significantly lower than those of the typical dwarf counterparts. 96.6% of the model UDGs present $B/T<0.1$; whereas about 27% of the dwarf counterparts are dEs with $B/T>0.5$. Regardless of the differences shown above, the distributions of the virial masses, $m_{\rm{vir}}$, of the model UDGs and dwarf counterparts are very similar to each other. The host halos of UDGs are only slightly less massive than those of their dwarf counterparts by 0.1 dex.

There are two reasons which may account for the unique feature of UDGs: 1) UDGs may form much later than the typical dwarfs, since the objects formed later are usually more extended because of the diluted Universe at a low redshift; 2) UDGs may have much higher spin parameters as naively expected from the standard galaxy formation scenario in which the galaxy size $r_{\rm{e}}\propto\lambda R_{\rm{vir}}$ (e.g., Mo et al., 1998; Amorisco & Loeb, 2016), where $\lambda$ and $R_{\rm{vir}}$ are the spin parameter and virial radius of the host halo, respectively. We will examine these below with our model. Note that in the modern galaxy formation models, e.g., Guo11 and Bower et al. (2010), the size of a present galaxy is not uniquely determined by the spin parameter of its host halo at $z=0$ or any specific redshift, rather it is a cumulative consequence of the angular momentum evolution of its parent halo and star formation. The size of a galaxy is largely determined by the rotational states of its host halo when the galactic star formation rate was high. We use the specific angular momentum of the main progenitor of a halo at the epoch ($t_{\rm{half}}$) when half of its stellar mass was assembled to take into account this integral effect. Note, the exact choice of the redshift only changes our result quantitatively but not qualitatively.

In Fig. 10, we present the distributions of the specific angular momenta $j$ of the progenitor halos at $t_{\rm{half}}$ and galaxy ages $\tau$ for the model UDGs and dwarf counterparts. Clearly, compared with the typical dwarfs, the specific angular momenta of the model UDGs are larger by a factor of 2.5 at $t_{\rm{half}}$. Also as shown in the lower panel of Fig. 10, the UDGs are indeed much younger with a median age of 7.1 Gyr, compared with the typical dwarfs which have a median age of 9.6 Gyr. Further, these suggest that it is indeed the combination of the late formation of UDGs and high-spins of the host halos that result in the large sizes of UDGs. Therefore, the high-spin tail origin of UDGs proposed by Amorisco & Loeb (2016) is not the complete story to explain the formation of UDGs. Besides, we find that almost all of the model UDGs in the clusters fall in directly from the field.

Di Cintio et al. (2017) developed an alternative strong-outflow model to explain the extended sizes of UDGs. Although this model can also reproduce the broad color range and low Sersic indics of observed UDGs, their simulated UDGs do not live in particularly high-spin halos, which conflicts with both of our model prediction and the recent observational spin parameters of UDGs from the ALFALFA HI survey (Leisman et al., 2017). Besides, their simulation contains about 40 galaxies with halo masses of $10^{10}\--10^{11}\ M_{\odot}$ (Wang et al., 2015), among which 8-21 (depending on the effective radius threshold $r_{\rm{e}}=1$ kpc or 2 kpc) of them are UDGs, suggesting a much higher fraction of UDGs than our model prediction; in the sense that the outflow model may overestimate the adundance of UDGs.

Another interesting phenomenon is that the number density profile of UDGs is flat towards the center in the observed clusters, quite different from that of the typical dwarfs. One possible reason may be that the UDGs fell into the clusters later than the typical dwarfs. We examine it in Fig. 11 by comparing the distributions of the infall-time $t_{\rm{infall}}$ (the time at which a galaxy was accreted into a cluster; $t_{\rm{infall}}\simeq 13.75$ Gyr corresponding to $z=0$) of the two dwarf populations in the 10 simulated clusters. As expected the infall-time of the model UDGs is on average significantly later than that of the dwarf counterparts, with a median value of $\langle t_{\rm{infall}}\rangle\sim 8.9$ Gyr and $\langle t_{\rm{infall}}\rangle\sim 5.2$ Gyr for the model UDGs and dwarf counterparts, respectively. Therefore, the lack of UDGs in the inner regions of clusters as shown in Fig. 3, as well as lack of the tidal disruption features in observations are the natural consequences of this late infall-time.

6 Conclusion and Discussion

As a special subset of low surface brightness population, UDGs draw much attention recently because they are as faint as the typical dwarf galaxies, yet have the sizes similar to those of the $L^{*}$ galaxies. The origin of UDGs is a mystery: are they the genuine dwarf galaxies with extremely large sizes or failed $L^{*}$ galaxies?

We use galaxy formation models (Guo et al. 2011, 2013) to study this special galaxy population. The predicted properties of UDGs in the clusters and groups (where most of the UDGs are discovered) agree very well with the observational results, including the abundance, number density profile, color distribution, and morphology, etc. Our model predicts about 4 and 24 UDG candidates in the Local Group and Local Volume analogues, respectively. When searching for such candidates in the Local group and Local Volume with existing observational data, we find that the numbers of UDGs in these two systems agree remarkably well with the model predictions.

We demonstrate that UDGs are genuine dwarf galaxies and can naturally emerge from the $\Lambda$CDM model. It is the combination of the later formation of UDGs and the relatively larger spins of their host halos that results in the more extended feature of this particular population. The lack of UDG candidates in the inner regions of clusters and the lack of tidal disruption features can be naturally explained by the later infall of the UDGs.

Compared to the typical dwarf galaxies, UDGs tend to reside in the low density regions consistent with their later formation. However, in the fields where there is no environmental effect, UDGs are redder than the typical dwarf galaxies. This is because the UDGs are more extended and the star formation ceases when the densities of the gas disks drop below a certain threshold. The red colors of UDGs suggest that it is even harder to detect UDGs than the typical dwarf galaxies. Fortunately, the model predicts only 7% of dwarf galaxies in clusters and 14% in fields are identified as UDGs, suggesting that it will not significantly affect the global luminosity function at the faint end, neither the conditional luminosity functions. Although most of UDGs are discovered in dense environments, we anticipate to discover a much higher fraction in under-dense regions in the future.

Acknowledgments

We thank Prof. Karachentsev I. D. for his helpful discussion on the UDG candidates in the Local Volume. QG and LG acknowledges support from NSFC grants (nos. 11425312), and two Newton Advanced Fellowships, as well as the hospitality of the Institute for Computational Cosmology at Durham University. THP acknowledges support in form of the FONDECYT Regular Project No. 1161817 and by the BASAL Center for Astrophysics and Associated Technologies (PFB-06).