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Monthly Notices of the Royal Astronomical Society: Letters
, Volume 475 (1) – Mar 1, 2018

6 pages

/lp/ou_press/the-virial-mass-distribution-of-ultradiffuse-galaxies-in-clusters-and-4KhR61DItc

- Publisher
- Oxford University Press
- Copyright
- © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
- ISSN
- 1745-3925
- eISSN
- 1745-3933
- D.O.I.
- 10.1093/mnrasl/sly012
- Publisher site
- See Article on Publisher Site

Abstract We use the observed abundances of ultradiffuse galaxies (UDGs) in clusters and groups and Λ cold dark matter subhalo mass functions to put constraints on the distribution of present-day halo masses of satellite UDGs. If all of the most massive subhaloes in the cluster host a UDG, UDGs occupy all subhaloes with log Msub/M⊙ ≳ 11. For a model in which the efficiency of UDG formation is higher around some characteristic halo mass, higher fractions of massive UDGs require larger spreads in the UDG mass distribution. In a cluster with a virial mass of 1015 M⊙, the 90 per cent upper limit for the fraction of UDGs with log Msub/M⊙ > 12 is 7 per cent, occupying 70 per cent of all cluster subhaloes above the same mass. To reproduce the observed abundances, however, the mass distribution of satellite UDGs has to be broad, with > 30 per cent having log Msub/M⊙ < 10.9. This strongly supports that UDGs are part of a continuous distribution in which a majority are hosted by low-mass haloes. The abundance of satellite UDGs may fall short of the linear relation with the cluster/group mass Mhost in low-mass hosts, log Mhost/M⊙ ∼ 12. Characterizing these deviations – or the lack thereof – will allow for stringent constraints on the UDG virial mass distribution. galaxies: dwarf, galaxies: formation, galaxies: haloes, galaxies: structure 1 INTRODUCTION The suggestion that surface brightness limited surveys may significantly underestimate the total number of galaxies is at least 20 yr old. Using a very simple model based on a standard Λ cold dark matter framework, Dalcanton et al. (1997) predicted the existence of an extremely abundant population of low surface brightness (LSB) galaxies, potentially extending to surface brightness levels of μ ≳ 30 mag arcsec−2. Reaching these depths remains exceptionally challenging and great effort is currently being devoted to push to ever fainter limits, to probe a yet largely unexplored regime of the galaxy formation process. LSB galaxies, including faint cluster galaxies, have been the object of numerous studies (e.g. Binggeli et al. 1985; Ferguson 1989; Davies et al. 1994; Impey & Bothun 1997; Conselice et al. 2002, 2003; Adami et al. 2006; Penny, Conselice & de Rijcke 2009; Ferrarese et al. 2012; Yamanoi et al. 2012), and the significant abundance of ultradiffuse galaxies (UDGs) in clusters (e.g. Koda, Yagi & Yamanoi 2015; Muñoz et al. 2015; van Dokkum et al. 2015; van der Burg et al. 2016; Janssens et al. 2017; Lee, Kang & Lee 2017; Román & Trujillo 2017a; Venhola et al. 2017) has confirmed the above prediction. With mean surface brightnesses within their effective radius Re of 〈μ〉 ≳ 24.5 mag arcsec−2, UDGs have stellar masses of dwarf galaxies (log M*/M⊙ ∼ 7.5/8.5), but Re > 1.5 kpc, quite larger than what is common among bright galaxies with similar stellar mass. Within the framework considered by Dalcanton et al. (1997), UDGs owe their remarkable sizes to the high angular momentum of their dark matter haloes (see also e.g. Mo et al. 1998; Dutton, van den Bosch & Dekel 2007). Although simplistic, this scenario reproduces the abundance of UDGs in clusters and their size distribution (Amorisco & Loeb 2016), and predicts that most UDGs are hosted by low-mass haloes (log Msub/M⊙ ∼ 10.3/11.3 for a normal stellar-to-halo mass relation; Amorisco & Loeb 2016). Stellar feedback has been proposed as alternative cause of the UDGs extended sizes (Di Cintio et al. 2017), which would also require that most UDGs are hosted by low-mass haloes (log Msub/M⊙ ∼ 11). Galaxies with the properties of UDGs have been obtained in recent hydrodynamical simulations (Chan et al. 2017; Di Cintio et al. 2017), but clear predictions for their population properties are not yet available. It remains possible, however, that a fraction of UDGs is hosted by haloes that are considerably more massive than suggested by the scenarios above, and potentially as massive as the Milky Way (MW) halo. It has been proposed that some UDGs may be ‘failed’ L* galaxies, with a different formation pathway. Failure could be caused by gas stripping and/or extreme feedback processes, which may have halted their star formation. UDGs would then fall significantly short of the stellar-to-halo mass relation and be hosted by ‘overmassive haloes’ (e.g. van Dokkum et al. 2015, 2016; Beasley et al. 2016). The only direct measurement of the virial mass of UDGs, based on the stacked weak lensing signal of >700 systems, cannot rule out this possibility (Sifón et al. 2018). Indirect arguments on the mass of UDG hosting haloes appear to confirm that a majority have low-mass haloes (≲2 × 1011 M⊙; Amorisco, Monachesi & Agnello 2016; Beasley et al. 2016; Beasley & Trujillo 2016; Peng & Lim 2016; Román & Trujillo 2017a). A few notable exceptions could however be interpreted as due to a fraction of systems with higher virial masses. These exceptions include the high Globular Cluster (GC) abundances of some Coma UDGs (van Dokkum et al. 2016, 2017, together with some less extended LSB galaxies; Amorisco et al. 2016) and the central stellar velocity dispersion of a couple of Coma UDGs (van Dokkum et al. 2016, 2017). In fact, if UDGs comply with the same scaling relations of normal galaxies, the heterogeneity of surface brightness values and sizes would suggest a mix of halo masses (Zaritsky 2017). It is therefore important to try and constrain the distribution of UDG virial masses. In this letter, we concentrate on satellite UDGs, and constrain their present-day subhalo masses using literature measurements of their cluster and group abundances (Koda et al. 2015; Muñoz et al. 2015; van der Burg et al. 2016, 2017; Janssens et al. 2017; Román & Trujillo 2017a,b). Section 2 describes the data and our simple model. Section 3 details our statistical analysis and collects results. Section 4 discusses them and lays out the conclusions. 2 THE ABUNDANCE OF UDGs Significant effort has been put into measuring the abundance of UDGs in galaxy clusters (e.g. van der Burg et al. 2016; Janssens et al. 2017; Lee et al. 2017; Román & Trujillo 2017a; Venhola et al. 2017). These results have confirmed the initial finding of van der Burg et al. (2016) that the relation between the number of UDGs, NUDG, and the virial mass of the host cluster, Mhost, is compatible with being linear. As discussed in Amorisco et al. (2016), an approximately linear relation between NUDG and Mhost is straightforwardly reproduced if the following two conditions are satisfied: (i) the physical mechanism that is responsible for the properties of UDGs is independent of environment, (ii) the majority of satellite UDGs have low-mass haloes. For small values of the subhalo-to-host mass ratio, Msub/Mhost ≪ 1, the mean subhalo abundance per unit parent mass is independent of the mass of the host (e.g. Gao et al. 2004; van den Bosch et al. 2005; Giocoli et al. 2008). Therefore, the two conditions above are sufficient to ensure an approximately linear relation between NUDG and Mhost. Interestingly though, abundant UDG populations have been detected in galaxy groups (Román & Trujillo 2017b; van der Burg et al. 2017, hereafter RT17 and vdB17), extending the same approximately linear relation valid in massive clusters (see also Trujillo et al. 2017). Fig. 1 reproduces the collection of data presented by vdB17: over ∼4 orders of magnitude in Mhost, NUDG is approximately proportional to the mass of the host (the dashed line shows the slope of a linear relation in this plane). As we will show, these high abundances put strong constraints on the distribution halo masses of satellite UDGs, and so does the apparent linearity of this relation. The subhalo abundance per unit parent mass is not independent of parent mass when Msub/Mhost ∼ 1: abundances are strongly suppressed for subhaloes above Msub/Mhost ∼ 0.1 (e.g. Gao et al. 2004; Giocoli et al. 2008). This causes potentially observable deviations from linearity in the relation between Mhost and NUDG when the mix of UDG halo masses includes a significant fraction of massive haloes. We formalise these concepts in the following. Figure 1. View largeDownload slide UDG abundances in groups and clusters as measured by Koda et al. (2015), Muñoz et al. (2015), van der Burg et al. (2016, 2017), Román & Trujillo (2017a,b), and Janssens et al. (2017). The dashed line shows the slope of a linear relation. The orange shading displays the 10-to-90 per cent confidence region for the mean NUDG–Mhost relation obtained from our analysis; the yellow shaded region shows the 1σ scatter around it. Figure 1. View largeDownload slide UDG abundances in groups and clusters as measured by Koda et al. (2015), Muñoz et al. (2015), van der Burg et al. (2016, 2017), Román & Trujillo (2017a,b), and Janssens et al. (2017). The dashed line shows the slope of a linear relation. The orange shading displays the 10-to-90 per cent confidence region for the mean NUDG–Mhost relation obtained from our analysis; the yellow shaded region shows the 1σ scatter around it. 2.1 Model UDG abundances We model the mean differential mass function of subhaloes with mass Msub, 〈N(Msub)〉, in a parent halo or cluster of mass Mhost with a fitting function based on: (i) the results of Boylan-Kolchin, Springel & White (2010, hereafter BK10) and (ii) the mentioned independence of the subhalo abundance per unit parent mass on the parent mass itself. The subhalo mass function in MW-mass haloes (12 ≤ log Mhost/M⊙ ≤ 12.5) has been measured with high precision by BK10, using the Millennium-II Simulation (Boylan-Kolchin et al. 2009). We adopt the functional form and parameters suggested by these authors (their equation 8) and refer to this function as \begin{equation} \left. {{{\rm d}\langle N\rangle }\over {{\rm d}\log M_{\rm sub}}}\right|_{M_{\rm host}=M_{\rm MW}}(M_{\rm sub})=\mathcal {N}(M_{\rm sub}). \end{equation} (1)This is a power law with an index1a = −1.935 for Msub/MMW ≪ 1 with an exponential truncation for subhalo masses Msub ≳ 1011 M⊙. To calculate the differential mass function of subhaloes with mass Msub in any parent of mass Mhost, we scale equation (1) using that the subhalo mass function is independent of the parent mass: \begin{equation} {{{\rm d}\langle N\rangle }\over {{\rm d}\log M_{\rm sub}}}={M_{\rm host}\over {M_{\rm MW}}} \mathcal {N}\left(M_{\rm MW} {M_{\rm sub}\over M_{\rm host}}\right){\mathcal {N}(m_0)\over {\mathcal {N}(M_{\rm MW} {m_0/M_{\rm host}})}}. \end{equation} (2)This uses that the shape of the truncation at a high subhalo to parent mass ratio is also independent of the parent mass (see e.g. Giocoli et al. 2008). In equation (2), m0 is any subhalo mass satisfying m0 ≪ MMW. The mass function $$\mathcal {N}$$ has been measured using all central haloes with virial mass between 1012 and 1012.5 M⊙. We therefore estimate the mean value MMW (needed in equation 2) and the relative uncertainty (necessary for our statistical analysis, see the following section) using a standard halo mass function with a slope of −1.9, which returns log MMW/M⊙ = 12.23 ± 0.14. For any cluster or group, equation (2) allows us to calculate the mean number of subhaloes in any given mass interval. A fraction of these subhaloes will host UDGs. Before introducing a model for the fraction of UDG as a function of halo mass, in Fig. 2 we compare the measured UDG abundances with the total mean number of massive subhaloes above some threshold mass2, 〈Nsub(>Msub)〉. Data points are the same as in Fig. 1 and the coloured lines display 〈Nsub(>Msub)〉 for the threshold masses log Msub = {10.5, 11, 11.5, 12, 12.5}. In order to roughly reproduce the observed abundances, all available subhaloes more massive than log Msub ∼ 11 need to be occupied by UDGs. This would leave no halo with log Msub > 11 for the non-UDG galaxies that also populate the same clusters/groups, showing that many UDG hosting subhaloes ought to have lower masses. In this extreme case, the fraction of UDGs hosted by haloes with Msub > 1012 M⊙ is of 11.5 per cent. The same upper limit is of 39 per cent for the largest UDGs, with Re > 2.5 kpc, the abundance of which we estimate using the observed abundances in Fig. 2 and the size distribution measured by van der Burg et al. (2016). Figure 2. View largeDownload slide Model UDG abundances. Coloured lines show the total number of subhaloes above mass Msub, with values as indicated by the legend in the lower right. Black lines with different styles are obtained from our model for the UDG mass distribution (see the text for details). Figure 2. View largeDownload slide Model UDG abundances. Coloured lines show the total number of subhaloes above mass Msub, with values as indicated by the legend in the lower right. Black lines with different styles are obtained from our model for the UDG mass distribution (see the text for details). Next, we introduce a simple model for the fraction of subhaloes hosting UDGs as a function of the present-day subhalo mass. We take that the physical mechanism responsible for forming UDGs is more efficient around some particular mass: the fraction of subhaloes with mass Msub hosting UDGs has a Gaussian shape, $$\mathcal {G}(M_{\rm sub})$$. Therefore, the differential UDG mass function is \begin{equation} \begin{array}{ll}{{\rm d}\langle N_{\rm UDG}\rangle }\over {{\rm d} M_{\rm sub}}&={{{\rm d}\langle N\rangle }\over {{\rm d} M_{\rm sub}}}\ \mathcal {G}(M_{\rm sub})\\ &={{{\rm d}\langle N\rangle }\over {{\rm d} M_{\rm sub}}}\ f_{\rm max} \exp \left[-{1\over 2}\left({{\log M_{\rm sub}/\bar{m}}\over {\varsigma }}\right)^2\right], \end{array} \end{equation} (3)where $$\bar{m}$$, ς, and fmax are free parameters of the model. The parameter $$\bar{m}$$ is the mass at which the fraction of UDGs is largest, fmax, with fmax ≤ 1. Notice that the value $$\bar{m}$$ is strictly larger than the mean UDG halo mass, and their difference quickly increases with the spread ς. Due to the steepness of the subhalo mass function, at fixed $$\bar{m}$$, an increase in ς implies higher counts of satellite UDGs through a larger fraction of low-mass subhaloes. By taking the model parameters to be constant across parent haloes, by construction, equation (3) results in a linear relation between NUDG and Mhost when $$\bar{m}\ll M_{\rm host}$$. NUDG may however drop below the linear relation when considering parent haloes with low enough mass. Fig. 2 shows the mean UDG abundances, 〈NUDG〉, corresponding to our model mass distribution (3) for a selection of pairs $$(\log \bar{m},\varsigma )$$. All displayed models adopt fmax = 1. By comparing with the observed abundances: we confirm there are more than enough low-mass haloes to host the observed cluster and group UDGs. If $$(\log \bar{m},\varsigma )=(10, 0.3)$$, then a fraction fmax < 1 is needed. This remains true if $$(\log \bar{m},\varsigma )=(10.75, 0.3)$$, for which some deviation from linearity in the NUDG–Mhost relation can be noticed at log Mhost/M⊙ ≲ 12. A model with $$(\log \bar{m},\varsigma )=(12, 0.3)$$, corresponding to a median UDG halo mass log MUDG,50/M⊙ = 11.8, cannot reproduce the observed abundances, despite fmax = 1. While keeping $$\log \bar{m}=12$$, this can be ameliorated by increasing the value of the spread ς, as shown by the model $$(\log \bar{m},\varsigma )=(12, 0.75)$$. This however corresponds to a dramatic decrease in the median UDG halo mass, with log MUDG,50/M⊙ = 10.8. 3 STATISTICAL ANALYSIS We now quantify the qualitative constraints above within a proper statistical framework. We take that, as shown by BK10, the scatter in the subhalo mass function is wider than Poissonian, and that it approaches a fractional intrinsic scatter of sI = 18 per cent for large values of 〈N〉. Here, sI = σI/〈N〉, where σI is the intrinsic scatter in 〈N〉. As suggested by BK10, we adopt that the probability distribution of observing NUDG UDGs in a cluster of mass Mhost, P(NUDG|〈NUDG〉(Mhost)), is a Negative Binomial3 (see equations 13–15 in BK10). The observed abundances we use in this analysis are either abundances for a single cluster or group, or mean abundances in samples of clusters or groups of similar mass. For the latter, we numerically construct the relevant probability distribution starting from the P above (the parent samples are often not large enough to invoke the central limit theorem). For all of the used measurements, we take account of the uncertainty in the group/cluster mass (as well as of the uncertainty in MWM, see Section 2.1). We take an uncertainty of 0.1 dex for the groups in vdB17 and, for the one group from RT17 lacking a mass uncertainty we adopt the same fractional uncertainty of the lowest mass group in the same study. If, for simplicity, we still refer to the resulting probability distributions with the symbol P, the likelihood of the measured abundances NUDG,i is \begin{equation} \mathcal {L}=\prod _i P(N_{{\rm UDG},i}| \langle N_{\rm UDG}(M_{{\rm c},i})\rangle ), \end{equation} (4)where the function 〈NUDG(Mc)〉 depends on the model parameters $$(f_{\rm max}, \bar{m}, \varsigma )$$. 3.1 Results As discussed in the previous section, the model parameters $$\bar{m}$$ and ς are not readily interpreted. We therefore start by casting our results in terms of MUDG,30 and MUDG,85, respectively, the 30 and 85 per cent quantile of the UDG virial mass distribution. As these are a function of the cluster mass Mhost, unless otherwise specified, we take log Mhost = 15. In other words, we refer to the case in which the tail at high masses of the UDG virial mass distribution is fully populated. The left-hand panel of Fig. 3 shows the distribution of models accepted by our Markov chain Monte Carlo chains in the (MUDG,85, MUDG,30) plane. We only accept models that have log MUDG, 30 > 9.5. Models that lie close to the line MUDG,85 = MUDG,30, shown as a black dashed line, have negligible scatter in the distribution of UDG halo masses (i.e. small values of ς). As a consequence, high values of fmax are required, as indicated by the colour coding. As MUDG,85 increases from log MUDG,85 = 9.5, even when fmax = 1, some minimum spread is necessary to reproduce the high observed abundances, and models depart from the MUDG,85 = MUDG,30 locus. While log MUDG,85 ≲ 11, all values 0 < fmax < 1 are allowed, corresponding to different values of MUDG,30, in a one-to-one relation. All of these models result in identical – and very close to linear – NUDG–Mhost relations. In this analysis, these models are degenerate because the observed abundances do not suggest significant deviations from linearity. When log MUDG,85 > 11.3, only high UDG fractions fmax ≳ 0.7 are allowed, and the mass distribution is required to be broad, with log MUDG,30 < 10.9 (or log MUDG,30 < 11.15 if a = −1.86). This is mirrored in the right-hand panel of the same figure, which shows the accepted models in the plane of the model parameters, $$(\log \bar{m}, \varsigma )$$, colour coded by MUDG,85. The UDG abundances alone cannot constrain these parameters, but the paucity of massive subhaloes imposes a marked correlation between them, corresponding to a tight upper limit on the fraction of massive UDGs. Figure 3. View largeDownload slide Left-hand panel: the distribution of accepted models in the plane of the 30 and 85 per cent quantiles of the UDG virial mass distribution, colour coded according to the maximum fraction fmax; the dashed line is the one-to-one relation. Right-hand panel: the correlation between the characteristic mass $$\bar{m}$$ and the spread ς. The colour coding is according to the 85 per cent quantile of the UDG virial mass distribution. Figure 3. View largeDownload slide Left-hand panel: the distribution of accepted models in the plane of the 30 and 85 per cent quantiles of the UDG virial mass distribution, colour coded according to the maximum fraction fmax; the dashed line is the one-to-one relation. Right-hand panel: the correlation between the characteristic mass $$\bar{m}$$ and the spread ς. The colour coding is according to the 85 per cent quantile of the UDG virial mass distribution. In Fig. 1, we show the resulting marginalized NUDG–Mhost relation. The orange shading identifies the 10-to-90 per cent confidence region for the mean UDG abundance. The data are fully consistent with an exactly linear relation, although some deviation in low-mass groups is allowed. The yellow shaded region shows the scatter around the mean, comprising both Poisson and intrinsic scatter. Fig. 4 shows the marginalized posteriors for the cumulative mass distribution of subhaloes hosting UDGs (10, 50, and 90 per cent quantiles). The three panels refer to hosts of different virial masses, respectively, log Mhost = 13, 14, and 15 in the left-hand, central, and right-hand panels. In a massive cluster with log Mhost = 15, at 90 per cent probability, 50 per cent of all UDGs are hosted by subhaloes with Msub < 10.8 (Msub < 11.05 if a = −1.86), 90 per cent by subhaloes with Msub < 11.8 (Msub < 12.1 if a = −1.86). In a group with log Mhost = 13, at 90 per cent probability, 50 per cent of all UDGs are hosted by subhaloes with Msub < 10.7 (Msub < 10.8 if a = −1.86), 90 per cent by subhaloes with Msub < 11.6 (Msub < 11.75 if a = −1.86). Figure 4. View largeDownload slide The cumulative distribution of the subhalo masses of UDGs in clusters with virial masses log Mhost/M⊙ ∈ {13, 14, 15} (respectively, left-hand, central, and right-hand panel). The sets of lines show the 10, 50, and 90 per cent quantiles of the posterior distribution. Figure 4. View largeDownload slide The cumulative distribution of the subhalo masses of UDGs in clusters with virial masses log Mhost/M⊙ ∈ {13, 14, 15} (respectively, left-hand, central, and right-hand panel). The sets of lines show the 10, 50, and 90 per cent quantiles of the posterior distribution. Fig. 5 shows the 10, 50, and 90 per cent quantiles for the inferred fraction of subhaloes with log Msub > 12 that are occupied by UDGs, NUDG(log Msub > 12)/N(>12), as a function of the fraction of UDGs with similarly massive subhaloes, NUDG(log Msub > 12)/NUDG. As in Fig. 4, panels refer to hosts with virial mass log Mhost/M⊙ ∈ {13, 14, 15}. In a massive cluster with log Mhost = 15, if more than 5 per cent (more than 8.5 per cent if a = −1.86) of all UDGs have massive subhaloes, log Msub > 12, more than 50 per cent of subhaloes with the same mass are occupied by UDGs. In a group with mass log Mhost = 13, no accepted model has NUDG( > 12)/NUDG > 4 per cent [NUDG( > 12)/NUDG > 6 per cent if a = −1.86], and if more than 2 per cent (more than 2.5 per cent if a = −1.86) of all UDGs are similarly massive, more than 50 per cent of massive haloes in groups are occupied by UDGs. Figure 5. View largeDownload slide The mean fraction of subhaloes with present-day mass log Msub > 12 that host UDGs, NUDG(Msub > 12)/Nsub(Msub > 12), against the mean fraction of satellite UDGs with similarly massive subhaloes, NUDG(Msub > 12)/NUDG, as obtained from our analysis for hosts with virial masses log Mhost/M⊙ ∈ {13, 14, 15} (respectively, left-hand, central, and right-hand panel). The sets of lines show the 10, 50, and 90 per cent quantiles of the posterior distribution. Figure 5. View largeDownload slide The mean fraction of subhaloes with present-day mass log Msub > 12 that host UDGs, NUDG(Msub > 12)/Nsub(Msub > 12), against the mean fraction of satellite UDGs with similarly massive subhaloes, NUDG(Msub > 12)/NUDG, as obtained from our analysis for hosts with virial masses log Mhost/M⊙ ∈ {13, 14, 15} (respectively, left-hand, central, and right-hand panel). The sets of lines show the 10, 50, and 90 per cent quantiles of the posterior distribution. 4 SUMMARY AND CONCLUSIONS In this letter, we have used the observed abundances of satellite UDGs in clusters and groups to constrain the present-day mass distribution of their dark matter subhaloes. If all of the most massive subhaloes available in the cluster host a UDG, all subhaloes with log Msub/M⊙ ≳ 11 would be occupied by UDGs, leaving no room for the non-UDG galaxies in the cluster. This implies a sharp upper limit to the fraction of UDGs hosted by massive haloes with log Msub > 12, which is of 11.5 per cent. We introduce a model in which the efficiency of UDG formation is a function of halo mass, and the probability for a subhalo to host a UDG is maximum around some characteristic subhalo mass, taken to be constant across clusters and groups. This simple assumption may more easily describe the scenario in which UDGs are formed in the field (e.g. Amorisco & Loeb 2016; Chan et al. 2017; Di Cintio et al. 2017) rather than the case in which UDGs are normal galaxies at first and expand after infall due to satellite-specific processes such a harassment and tidal stirring. However, we find that the currently available UDG abundances cannot constrain the parameters of this model, so that our specific choice has negligible impact on our results. Instead, as a consequence of the limited number of massive subhaloes, we find that the fraction of UDGs with high virial mass and the spread in the UDG mass distribution are strongly correlated. For instance, if 15 per cent of all UDGs in a massive cluster have log Msub > 11.5, the spread of the distribution is such that >30 per cent have log Msub < 10.9. No model in which 15 per cent of all UDGs in a massive cluster have log Msub > 11.8 can reproduce the observed abundances. This translates in a fraction of UDGs with log Msub > 12 that is <7 per cent at 90 per cent probability, and corresponding to a cluster in which ∼70 per cent of all subhaloes with log Msub > 12 are occupied by UDGs. An analysis that folds in constraints for the fraction of satellite galaxies in clusters and groups that are UDGs versus non-UDGs is beyond the scope of this letter and is left for future studies. If we take that 50 per cent of all massive subhaloes in Coma host UDGs, <16 out of the 332 UDGs counted by Koda et al. (2015) may be massive. If so, the mass distribution has to be broad, with >110 UDGs having log Msub < 10.8. This strongly supports a number of observational arguments suggesting that UDGs are part of a continuous distribution in which a majority have low-mass haloes. These include: the seamless continuity of the properties of UDGs with respect to those of the numerous – though relatively more compact – LSB dwarfs (e.g. Koda et al. 2015; Venhola et al. 2017; Wittmann et al. 2017); the fact that a majority of UDGs have normal GC systems for their stellar mass (Amorisco et al. 2016; Beasley et al. 2016; Beasley & Trujillo 2016; Peng & Lim 2016) and that a minority of systems with enhanced GC abundances exist among UDGs as well as among LSB dwarfs (Amorisco et al. 2016); the fact they do not appear to significantly deviate from the mass–metallicity relation of bright dwarf galaxies (Gu et al. 2017; Pandya et al. 2017). Finally, this analysis shows that it is extremely useful to better assess the properties of UDGs in low-mass groups, as UDG abundances in this regime constrain the actual shape of the UDG virial mass distribution. We have shown that, in proceeding towards lower mass groups, the linearity of the relation between the UDG abundance NUDG and the group mass Mhost is expected to break, with mean abundances falling short of the linear relation. This discrepancy quantifies the weight and shape of the high-mass tail of the UDG virial mass distribution. Interestingly, the results of vdB17 appear to hint to similar deviations from linearity, with low-mass groups (log Mhost ∼ 12) featuring a UDG in only 1 out of ∼10 cases. However, as confirmed by our analysis, this is currently not statistically significant. Larger samples will elucidate the behaviour of the NUDG–Mhost relation at low group masses, allowing for better constraints on the UDG virial mass distribution and therefore more stringent tests for formation models. Acknowledgements It is a pleasure to thank Remco van der Burg and Adriano Agnello for useful comments and the anonymous referee for a constructive report. 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Monthly Notices of the Royal Astronomical Society: Letters – Oxford University Press

**Published: ** Mar 1, 2018

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