tokens: in the magnitude range where our analysis is conducted ( g 24). This procedure was repeated 10 4 times and the corresponding distributions of N syn and μ V, syn were computed. The number of stars contained within the same magnitude limits defined above was counted in the observed CMD and the density of stars per square degree Σ obs was calculated by dividing by the sampled area. The fractions f N of random extractions with N syn Σ obs and the fractions with μ V, mod > μ V, max were then computed. If the MS star density estimated around the sample of observed GCs were compatible with that predicted by models, the distribution of percentiles would be uniform. In Fig. 7 , the cumulative distributions of percentiles of the eight GCs at d ⊙ 45 kpc with respect to the N syn distribution of the five Aquarius simulations are shown. On average, the distribution of percentiles is consistent with a uniform distribution (although a vast range of lumpiness is covered by the five considered simulations), indicating a general agreement with model predictions for three out of five simulations. A Kolmogorov-Smirnov (KS) test indicates probabilities 0.1 P 0.2 for these simulations . However, the simulations Aquarius D and Aquarius E show opposite trends with distributions of percentiles that are incompatible with observations. Note that Aquarius D is the most massive halo among the five simulations of the Aquarius sample, with a virial mass M = 2.52 × 10 12 M ⊙ , exceeding the current estimate for the Milky Way (Patel, Besla Mandel 2017 ). It is therefore likely that the number density of the halo is also overestimated in this simulation. On the other hand, simulation Aquarius E is the one with the lowest fraction of stars in the diffuse halo at distances d GC > 20 kpc, with the majority of stars in this distance range contained in bound satellites. For this reason, the number density outside satellites turns out to be lower than those in other haloes. Figure 7. Cumulative distribution of percentiles in the number density domain for the eight GCs at d ⊙ 45 kpc. Different lines refer to the various cosmological simulations. The probabilities associated with the KS test are also indicated. The same conclusion can be drawn when the entire sample of 14 GCs is considered in the μ V space. The distributions of extracted surface brightness from each of the considered simulations are shown as a function of heliocentric distance in Fig. 8 . We note that the background surface brightness distribution is peaked at values in the range 30 μ V, syn 35 mag arcsec −2 following a declining trend with distance. At large distances, the surface brightness distribution becomes skewed with a tail towards bright values at μ V, syn 34 mag arcsec −2 . This is a consequence of the increasing degree of lumpiness of the halo at large distances: GCs forming within satellites in an early stage of dissolution are still surrounded by the compact remnant of their host galaxy. The upper limits
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