Monochromatic Globular Clusters as a Critical Test of Formation Models for the Dark Matter–deficient Galaxies NGC 1052-DF2 and NGC 1052-DF4

NGC 1052-DF2 (or DF2) and NGC 1052-DF4 (DF4) share two unusual properties that set them apart from all other known galaxies. First, their globular clusters are, on average, a factor of ≈4 brighter and a factor of ≈2 larger than canonical values (van Dokkum et al. 2018a, 2019; Ma et al. 2020; Shen et al. 2021a). Furthermore, their velocity dispersions are consistent with their stellar mass alone and much smaller than expected from a normal dark matter halo (van Dokkum et al. 2018b; Wasserman et al. 2018; Danieli et al. 2019; Emsellem et al. 2019).

Initially, the main question was whether the dynamical masses and distances were measured correctly (see, e.g., Martin et al. 2018; Laporte et al. 2019; Trujillo et al. 2019), but as the anomalous properties of the galaxies were gradually confirmed (and corroborated with independent evidence; see Dutta Chowdhury et al. 2019; Keim et al. 2022), the focus shifted to the question of how they were formed. Proposals include assembly out of tidally removed gas (Fensch et al. 2019), stripping of dark matter by close encounters with NGC 1052 (Ogiya 2018; Carleton et al. 2019; Nusser 2020; Moreno et al. 2022; Ogiya et al. 2022) or NGC 1035 (Montes et al. 2020), jet- or outflow-induced star formation, like Minkowski's object (van Breugel et al. 1985; Natarajan et al. 1998), and extreme feedback in low-mass halos (Trujillo-Gomez et al. 2022).

Silk (2019) suggested that DF2 could be the result of a "mini bullet cluster" event, where the dark matter and the baryons became separated in a nearly head-on encounter between two gas-rich progenitor galaxies. Such a collision produces two dark matter–dominated remnants with a globular cluster-rich, dark matter–free object in between them that formed from the shocked gas. This scenario was further explored with simulations in Shin et al. (2020) and Lee et al. (2021), who showed that collisions between an unbound object and a satellite could explain many of the observed properties of the galaxies and occur with some regularity in cosmological simulations. The main issue with this model—as with many of the alternative explanations listed above—is the presence of two dark matter–deficient galaxies, seemingly requiring lightning to strike twice in the same group.

Recently, we suggested that a single bullet dwarf collision may have produced both DF2 and DF4 (van Dokkum et al. 2022). This joint formation is consistent with the striking similarities between the two galaxies, their radial velocities and line-of-sight distances, the emergence of multiple clumps in at least some bullet collision simulations (Shin et al. 2020), and with the discovery that DF2 and DF4 are part of a remarkable trail of ≈10 galaxies in the group.

As noted in van Dokkum et al. (2022), this formation model for DF2 and DF4 is falsifiable, as it makes a very specific prediction for their globular clusters. Both galaxies formed out of the gas that was left behind by the progenitor galaxies at the same time. This gas mixed efficiently during the collision and should have a uniform metallicity. While overall star formation likely lasted for ≳500 Myr (Shin et al. 2020; Lee et al. 2021), the globular clusters formed almost instantaneously before further enrichment could substantially change the abundance of the gas (Lee et al. 2021). As a result, the stellar populations of all the globular clusters, in both galaxies, should be extremely similar. ¹⁰ This is not what is typically observed in dwarf galaxies. In a comprehensive study of the colors of globular clusters in Virgo dwarf galaxies, Peng et al. (2006) found that there is substantial cluster-to-cluster and galaxy-to-galaxy scatter.

Previous measurements of the colors of the globular clusters in DF2 and DF4 do in fact suggest differences between them, in potential conflict with the bullet model. Tables 1 and 2 in Shen et al. (2021a) imply a mean V₆₀₆ − I₈₁₄ globular cluster color of 0.37 ± 0.01 AB mag in DF2 and 0.41 ± 0.01 in DF4. This difference seems small, but it corresponds to ≈0.3 dex in age or ≈0.4 dex in metallicity. Furthermore, the cluster-to-cluster scatter is nonzero, with σ = 0.04 ± 0.01 mag for both galaxies. There is also a hint that the galaxies themselves might have different V₆₀₆ − I₈₁₄ colors: Cohen et al. (2018) find 0.40 for the diffuse light in DF2 and 0.32 for DF4, albeit with an uncertainty of 0.1 for both.

These measurements were performed using standard techniques (such as Source Extractor; Bertin & Arnouts 1996) and are largely based on single-orbit HST/ACS images that were reduced with the default STScI flat fields. Here we remeasure the colors of the clusters and the diffuse light in both DF2 and DF4 using custom techniques and well-calibrated, much deeper data, as a stringent test of the bullet dwarf model. Where needed, we use a distance of D = 21.7 Mpc for DF2 and D = 20.0 Mpc for DF4 (Danieli et al. 2020; Shen et al. 2021b; Z. Shen et al., in preparation).

We make use of deep HST/ACS data that were obtained in programs GO-15695 and GO-15851. The aim of these programs was to measure distances to DF4 (GO-15695) and DF2 (GO-15851) from the tip of the red giant branch (TRGB). For DF4, the exposure times were seven and three orbits in I₈₁₄ and V₆₀₆, respectively. In Danieli et al. (2020), these were combined with the 1+1 orbits that had been obtained in GO-14644 (see Cohen et al. 2018), and the location of the TRGB was measured from these 8+4 orbit depth data. The DF2 observations of GO-15851 were deeper, at 19 orbits in I₈₁₄ and 19 orbits in V₆₀₆. Shen et al. (2021b) used these data to measure the TRGB distance of DF2, again after adding the GO-14644 data for a total exposure of 20+20 orbits.

In this study we redrizzle the ACS data, with several changes. First, we discard the 1+1 orbits that were obtained for DF2 and DF4 in GO-14644. The orientation of these observations was very different from the more recent data, which means the point-spread function (PSF) and charge transfer efficiency corrections are different, and there is no need to maximize depth: The fluxes of the globular clusters are typically ∼30 e⁻ s⁻¹, which means that Poisson errors are ≲0.5% even in a single orbit.

More importantly, we apply a flat-fielding correction to the flc files prior to drizzling. The ACS flat fields were last updated in 2006, and they no longer provide optimal corrections. As described in the instrument science report ¹¹ ACS 2017-09, ultradeep stacks of Frontier Fields images show that there are systematic flat-fielding residuals at the level of 1%. This effect is also described in ISR ACS 2020-08. The residuals correlate with the local thickness of the CCD, and as the correlation between quantum efficiency and thickness reverses at ≈700 nm, the residuals in V₆₀₆ and I₈₁₄ are spatially anticorrelated. As a result, flat-fielding errors in the V₆₀₆ − I₈₁₄ color reach 2%, even though they are only half that in each filter individually.

The ACS team at STScI provided us with preliminary correction flats, and we applied these to the flc files. The flc files were aligned with each other using tweakreg. The tweakreg algorithm is sensitive to cosmic rays; we addressed this by running the code on versions of the data where cosmic rays were removed (with L.A.Cosmic van Dokkum 2001). I₈₁₄ images of DF2 and DF4 are shown in Figure 1. Visually comparing the default and flat-field-corrected images, there is a clear improvement in the flatness of the sky background, particularly for DF2. From the remaining variation in the background, we estimate that residual flat-fielding errors are 0.5% ± 0.2%. Assuming that the errors are no longer (anti)correlated, the uncertainty in the color due to spatially varying flat-fielding uncertainties is then 0.7% ± 0.3% in our images.

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3.1. Sample

3.2. Cleaned Globular Cluster Images

3.3. Aperture Photometry

The distribution of the DF2 and DF4 globular clusters in the color–magnitude plane is shown in the top panel of Figure 2. Error bars are a combination of the measurement uncertainty and the 0.007 mag flat-fielding uncertainty. For reference, the grayscale background shows the parameterized distribution of globular clusters in low-luminosity galaxies in the Virgo cluster. This is a combination of the two-component decomposition of the color distribution for the faintest galaxies ¹² in Peng et al. (2006) and the Gaussian fit to the globular cluster luminosity function of the faintest galaxies ¹³ in Jordán et al. (2007).

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The globular clusters in DF2 and DF4 are much brighter than those in Virgo dwarfs, as has been discussed extensively in earlier papers (see van Dokkum et al. 2018a; Trujillo et al. 2019; Shen et al. 2021a), and they are also somewhat bluer. There is no evidence for a systematic trend with magnitude when the full sample of confirmed and candidate clusters is considered. The photometric uncertainties increase sharply at fainter magnitudes, and in the following, we only consider the 18 clusters with errors <0.015. This sample corresponds to the full sample of clusters with M₆₀₆ < −8.6.

The mean color of the DF2 and DF4 clusters is nearly identical: 〈V₆₀₆ − I₈₁₄〉 = 0.374 ± 0.004 for DF2 and 〈V₆₀₆ − I₈₁₄〉 = 0.377 ± 0.003 for DF4, ¹⁴ as determined with the biweight estimator (Beers et al. 1990). The mean color difference is Δ_DF2−DF4 = −0.003 ± 0.005. The mean colors are compared to those of globular clusters in Virgo galaxies ¹⁵ in Figure 2 (lower left). The clusters in DF2 and DF4 are bluer than those in Virgo galaxies with similar N_GC. ¹⁶ Furthermore, in the Virgo sample with N_GC < 20 the median color difference between any two data points is 0.036, an order of magnitude larger than the difference between DF2 and DF4.

The cluster-to-cluster scatter is also very small and within the errors is the same for the two galaxies: σ_obs = 0.015 ± 0.003 for DF2 and σ_obs = 0.010 ± 0.003 for DF4. The observed scatter in the combined sample of 18 bright globular clusters in DF2 and DF4 is σ_obs = 0.015 ± 0.002 (all determined with the biweight estimator; the rms is also 0.015). The intrinsic scatter ${\sigma }_{\mathrm{intr}}$ can be determined by constructing the likelihood function,

$$\begin{eqnarray}&&{ \mathcal L }=\prod _{i=1}^{i\leqslant 18}\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{eff}}}\exp \left[-0.5{\left(\displaystyle \frac{{c}_{i}-\mu }{{\sigma }_{\mathrm{eff}}}\right)}^{2}\right],\end{eqnarray} \tag{ 1 }$$

with c_i the colors of the individual clusters, μ the mean, and ${\sigma }_{\mathrm{eff}}^{2}={\sigma }_{\mathrm{intr}}^{2}+{e}_{i}^{2}$. We find an intrinsic scatter of ${\sigma }_{\mathrm{intr}}={0.012}_{-0.003}^{+0.004}$. As shown in Figure 2 (lower right), the typical scatter in Virgo galaxies is σ ≈ 0.1 mag, and there are no galaxies with σ < 0.05 mag. We conclude that the globular clusters in DF2 and DF4 form a remarkably homogeneous population, as predicted by the bullet dwarf collision model. This level of homogeneity is not observed in normal dwarf galaxies.

The color variation is small but not zero. As noted above, observational uncertainties explain some of the observed scatter. This is shown in the top-left panel of Figure 3, where the error bar for each data point is split into several distinct contributions. Measurement uncertainties are shown in black and the total photometric uncertainties, which include the 0.007 ± 0.003 mag flat-fielding errors, are shown in dark gray.

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The intrinsic scatter is so small that the effects of stochastic sampling of the isochrone need to be taken into account. Red giants contribute significantly to the integrated light, and the Poisson variation in their number causes some clusters to be redder and brighter than others. The same effect causes the well-known pixel-to-pixel surface brightness fluctuations in galaxy images (Tonry & Schneider 1988; Greco et al. 2021). We quantify this effect by generating artificial globular clusters with the ArtPop code (Greco & Danieli 2021) and measuring their integrated colors. The clusters have $\mathrm{log}(\mathrm{age})=9.9$, −1.8 ≤ [Fe/H] ≤ −0.8, and cover a factor of 20 in mass (parameterized by $\mathrm{log}({n}_{\mathrm{stars}})$, which ranges from 5 to 6.3 with steps of 0.2). At each mass, 500 clusters are generated with different random seeds, and the rms scatter in the V₆₀₆ − I₈₁₄ colors is measured.

The results are shown in the bottom-right panel of Figure 3. The cluster-to-cluster V₆₀₆ − I₈₁₄ scatter is ∼1% in the relevant luminosity range, of the same order as the intrinsic scatter in the DF2/DF4 globular clusters, with a modest dependence on metallicity. Two example clusters that illustrate the effect are shown at left (see also Figure 6 in Greco & Danieli 2021). A polynomial fit to the luminosity-dependent variation has the form

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{\mathrm{stoch}} & = & 0.019+0.0087({M}_{606}+7.5)\\ & & +0.00153{\left({M}_{606}+7.5\right)}^{2}\end{array}\end{eqnarray} \tag{ 2 }$$

and is shown by the black line in Figure 3.

Light-gray error bars in the top panel of Figure 3 show the effect of including this uncertainty for each cluster. The broken orange lines show where 68% of the points are expected to fall due to the stochastic sampling effect alone. Solid orange lines include the measurement error; for a normal distribution entirely defined by these lines, 12 out of 18 would fall within them. They encompass 10, only slightly fewer. Using the total errors (that is, the combination of the measurement error, the flat-fielding error, and the stochastic variation) in the likelihood analysis gives the "stellar population scatter", ${\sigma }_{\mathrm{sp}}={0.008}_{-0.006}^{+0.005}$. This scatter is not significantly different from zero. The likelihood function, marginalized over μ, is shown by the solid line in Figure 3.

We use the flexible stellar population synthesis (FSPS) framework (Conroy et al. 2009) with the MIST isochrones (Choi et al. 2016) to determine the (limits on) variation in age and metallicity that is implied by the stellar population scatter. For ages near 8 Gyr and metallicities near [Fe/H] = −1.3, we find Δ(V₆₀₆ − I₈₁₄) = 0.10Δ[Fe/H] and Δ(V₆₀₆ − I₈₁₄) = 0.0064Δ(age) to good approximation, with age in Gyr. The observed stellar population scatter σ_sp implies ${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}={0.08}_{-0.06}^{+0.05}$ if there is no variation in age or ${\sigma }_{\mathrm{age}}={1.3}_{-1.0}^{+0.8}$ Gyr if there is no variation in metallicity. In the simulations of Lee et al. (2021), the globular clusters can have a spread of up to ≈0.1 dex in metallicity and ≈150 Myr in age, and we conclude that the bullet hypothesis cannot be ruled out.

The bullet model also makes predictions for the global colors of DF2 and DF4, although these are more model dependent than the predictions for the globular clusters (Shin et al. 2020; Lee et al. 2021). The galaxies are predicted to have higher metallicities than the globular clusters as they form stars over a longer time period (see Lee et al. 2021). Therefore, while the colors of DF2 and DF4 should be very similar to each other, they are predicted to be redder than those of the globular clusters.

We measure the average colors of the galaxies in the following way. An object mask is created by comparing the summed V₆₀₆ + I₈₁₄ image to a binned and median-filtered version of itself. A first model for the galaxy is made by median-filtering the V₆₀₆ and I₈₁₄ images, not taking masked pixels into account. This model is subtracted from the data, and the object mask is optimized with a lower threshold, taking care not to include giants in the mask. Then, the median-filtering is repeated to create a final model in each filter. There is a background gradient in all images, and at this stage, a surface is fitted to the background and subtracted. The rms of this background model (that is, 68% of the gradient that is removed) is taken as the uncertainty in each filter. Finally, the I₈₁₄ model is divided by the V₆₀₆ model, multiplied by the object mask, and a flux threshold is applied so that the faint outskirts are excluded.

These color images are shown in Figure 4 and compared to the mean color of the globular clusters. The variations within each galaxy are caused by stochastic variation in the number of red giants (surface brightness fluctuations). We measure the mean colors of the galaxies to be ${({V}_{606}-{I}_{814})}_{\mathrm{DF}2}=0.420\pm 0.024$ and ${({V}_{606}-{I}_{814})}_{\mathrm{DF}4}=0.436\pm 0.075$, consistent with the Cohen et al. (2018) measurements with smaller uncertainties. We infer that the colors of the galaxies are identical within the errors, as predicted by the bullet model.

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The galaxies are redder than the luminous globular clusters. If the actual color of both galaxies is V₆₀₆ − I₈₁₄ = 0.375 and the errors are Gaussian, the probability that we measure ${({V}_{606}-{I}_{814})}_{\mathrm{DF}2}\geqslant 0.420$ and ${({V}_{606}-{I}_{814})}_{\mathrm{DF}4}\geqslant 0.436$ is <1%. The color difference between the galaxies and the clusters of 0.05 mag is qualitatively consistent with the metallicity difference predicted by the hydrodynamical model of Lee et al. (2021). Using the relation in Section 5, we find Δ[Fe/H] ≈ 0.5, somewhat larger than the ≈0.2 predicted by Lee et al. (2021) but in good agreement with the spectroscopically determined value of Δ[Fe/H] = 0.56 ± 0.15 of Fensch et al. (2019).

The central result of this paper is that the bright globular clusters in DF2 and DF4 have extremely similar colors. The observed scatter is ≈0.015 mag, and this can be explained by a combination of measurement uncertainties (≈0.01) and stochastic variations in the number of red giants (≈0.01). The remaining scatter among the 18 luminous clusters in DF2 and DF4 is ${\sigma }_{\mathrm{ssp}}={0.008}_{-0.006}^{+0.005}$, that is, not significantly different from zero. The diffuse light is redder in both galaxies, with again no significant difference between DF2 and DF4. These results are expected in the bullet dwarf scenario (see Silk 2019; Shin et al. 2020; Lee et al. 2021; van Dokkum et al. 2022), and we conclude that this model survives an important falsification test. Returning to the formation models listed in Section 1, no other published explanation for the lack of dark matter in DF2 and DF4 also "naturally" produces the extreme uniformity of their globular cluster populations.

The globular clusters in DF2 and DF4 are different from those in Virgo galaxies; as detailed in Section 4, they are brighter, bluer, and have a much smaller scatter. DF2 and DF4 are ultradiffuse galaxies (UDGs; van Dokkum et al. 2015) and at least two other UDGs also have very homogeneous globular cluster populations, NGC 5846-UDG1 (Müller et al. 2021; Danieli et al. 2022) and DGSAT i (Janssens et al. 2022). In other respects they are different; NGC 5846-UDG1 has a canonical globular cluster luminosity function, and DGSAT i's clusters are significantly redder than those in DF2/DF4. A small scatter indicates synchronized formation in a dense, homogeneous medium, and it may arise generically in any formation scenario that results in the extreme (factor of 10⁹) density contrast between the globular clusters and the galaxy light that is seen in UDGs (see, e.g., van Dokkum et al. 2018a; Trujillo-Gomez et al. 2021). Specifically, intense feedback from the formation of the globular clusters may have caused the galaxies to expand and turn into UDGs (Trujillo-Gomez et al. 2021; Danieli et al. 2022). Further deep HST or JWST studies of globular clusters in UDGs in various environments are needed to investigate this further.

Our analysis focuses on the brightest clusters as these have the smallest uncertainties. Shen et al. (2021a) showed that the globular cluster luminosity function in DF2 and DF4 can be modeled as a combination of a bright peak of overluminous clusters plus a "normal" luminosity function with a normalization and mean that are typical for the galaxies' luminosities. In this context, it is interesting that the clusters with M₆₀₆ > −8.6 are somewhat redder than the brighter ones, with 〈V₆₀₆ − I₈₁₄〉 = 0.400 ± 0.011. Perhaps the faint clusters formed together with the diffuse light, with a higher mean metallicity than the brighter ones. ¹⁷ The observed scatter is also higher for the faint clusters at σ_606–814 = 0.034 ± 0.008, although this can be explained by a combination of measurement errors (0.028) and stochastic sampling (0.016). Extremely deep spectroscopy may shed more might on these questions.

Further tests of the bullet model are possible. The most straightforward next step is probably obtaining radial velocities and line-of-sight distances of other galaxies along the trail, as the bullet model predicts that these follow a regular sequence (with some contamination from unrelated objects). Dynamical mass measurements of other trail galaxies will also be highly constraining and interesting in a broader context as bullet dwarf events can, in principle, constrain the self-interaction cross section of dark matter. Modeling of the bullet cluster has provided an upper limit (Randall et al. 2008), but as self-interacting dark matter was introduced to explain the "cored" dark matter density profiles of low-mass galaxies (Spergel & Steinhardt 2000), it is important to measure the cross section on those scales (Tulin & Yu 2018).

We thank Yotam Cohen and the rest of the STScI ACS team for their help with the data reduction. Support from HST grants GO-14644, GO-15695, and GO-15851 is gratefully acknowledged. S.D. is supported by NASA through Hubble Fellowship grant HST-HF2-51454.001-A.