Looking for Traces of Nonminimally Coupled Dark Matter in the X-COP Galaxy Clusters Sample

Here we provide a short theoretical background for the NMC DM model, referring the reader to Gandolfi et al. (2021) and Gandolfi et al. (2022) for further information. A very basic NMC model can be built with the addition of a coupling term S_int between DM and gravity in the total Einstein–Hilbert action (in the Jordan frame) with shape:

$$\begin{eqnarray}&&{S}_{\mathrm{int}\ }\left[{\tilde{g}}_{\mu \nu },\varphi \right]=\epsilon {L}_{\mathrm{NMC}}^{2}\int {{\rm{d}}}^{4}x\,\sqrt{-\tilde{g}}\,{\widetilde{G}}^{\mu \nu }\,{{\rm{\nabla }}}_{\mu }\,\varphi {{\rm{\nabla }}}_{\nu }\varphi \,;\end{eqnarray} \tag{ 1 }$$

here φ is the (real) DM scalar field, = ± 1 is the polarity of the coupling, ${\widetilde{G}}^{\mu \nu }$ is the Einstein tensor, and L_NMC is the NMC characteristic length scale. From a purely theoretical perspective, such a form of the NMC is allowed by the Einstein equivalence principle (e.g., Bekenstein 1993; Di Casola et al. 2015). In our approach, however, the length L_NMC does not need to be a new fundamental constant of nature as it is indeed suggested by its virial mass-dependent scaling observed in Gandolfi et al. (2022). Instead, L_NMC could emerge dynamically from some collective behavior of the coarse-grained DM field (e.g., Bose–Einstein condensation). We hence remark that our NMC model does not consist in a modified gravity theory but simply in a formalization of an emergent behavior of cold DM inside halos. Furthermore, the bookkeeping parameter will be set to = −1 (repulsive coupling) based on the findings of Gandolfi et al. (2021) and Gandolfi et al. (2022).

We also stress that the NMC DM model hereby discussed could in principle share features with other prospective DM models, such as self-interacting DM scenarios. Nonetheless, the NMC DM framework contemplates not only a self-interaction term for DM in the action but also a scale-dependent geometric interaction term between the DM field and the baryonic component, which is sourced by the NMC of the DM to gravity.

Adopting the fluid approximation for the field φ (as in Bettoni et al. 2012) and taking the Newtonian limit, the NMC translates into a simple modification of the Poisson equation (Bettoni et al. 2014)

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{\rm{\Phi }}=4\pi G\left[\left(\rho +{\rho }_{\mathrm{bar}}\right)-\epsilon {L}^{2}{{\rm{\nabla }}}^{2}\rho \right],\end{eqnarray} \tag{ 2 }$$

where Φ is the Newtonian potential, and ρ_bar and ρ are the baryonic and DM densities. In spherical symmetry, Equation (2) implies that the total gravitational acceleration writes

$$\begin{eqnarray}&&{g}_{\mathrm{tot}}(r)=-\displaystyle \frac{G\,M(\lt r)}{{r}^{2}}+4\pi \,G\,\epsilon {L}^{2}\,\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}r},\end{eqnarray} \tag{ 3 }$$

where M(<r) is the total mass enclosed in the radius r; the first term is the usual Newtonian acceleration, and the second term is the additional contribution from the NMC.

In Gandolfi et al. (2021) we have highlighted that Equation (2) gives rise to some interesting features for strongly DM-dominated systems in self-gravitating equilibria. First of all, the NMC can help to develop an inner core in the DM density profile. This enforces a shape for the density profile, which closely follows the phenomenological Burkert profile (Burkert 1995) out to several core scale radii. Moreover, DM-dominated halos with NMC are consistent with the core-column density relation (see, e.g., Salucci & Burkert 2000; Donato et al. 2009; Behroozi et al. 2013; Burkert 2015, 2020), i.e., with the observed universality of the product between the core radius r₀ and the core density ρ₀. In Gandolfi et al. (2022) we tested the NMC hypothesis using a diverse sample of spiral galaxies. The NMC DM model proved to yield fits to the stacked rotation curves of such objects with a precision always superior to pure NFW model fits and in several instances comparable to or even better than the Burkert model ones. Furthermore, we observed an interesting power-law scaling relation between the halo virial mass M₂₀₀ and the NMC length scale L_NMC for the fitted galaxies. By assuming such mass-dependent scaling of L_NMC, the NMC DM model was also able to reproduce the radial acceleration relation up to the regime of dwarf spheroidal galaxies. Yet the NMC DM model awaits testing on scales larger than galactic ones, and this is precisely the scope of the present work.

The thermal pressure profiles ⁵ of galaxy clusters are defined as functions of the gravitational potential in play. In the framework of the NMC DM model this reads as

$$\begin{eqnarray}&&{P}^{\mathrm{th}}(R)={P}^{\mathrm{th}}(0)-1.8\mu {m}_{{\rm{p}}}{\int }_{0}^{R}{n}_{{\rm{e}}}(r)\left[\displaystyle \frac{{{GM}}_{\mathrm{DM}}(r)}{{r}^{2}}-4\pi \,G\,\epsilon {L}_{\mathrm{NMC}}^{2}\,\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}r}\right]{\rm{d}}r,\end{eqnarray} \tag{ 4 }$$

where we model the electron density (ED) profile through the Vikhlinin profile (Vikhlinin et al. 2006),

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{e}(r)}{{n}_{0}}=\displaystyle \frac{{\left(r/{r}_{c}\right)}^{-\alpha /2}{[1+{\left(r/{r}_{s}\right)}^{\gamma }]}^{-\varepsilon /(2\gamma )}}{{[1+{\left(r/{r}_{c}\right)}^{2}]}^{(3/2)\beta -\alpha /4}}.\end{eqnarray} \tag{ 5 }$$

To specify the dark mass distribution in Equation (4) we adopt the same perturbative approach of Gandolfi et al. (2022), considering the NMC as a small perturbation over the standard cold DM NFW profile

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\delta }_{{\rm{c}}}{\rho }_{{\rm{c}}}{r}_{s}^{3}}{r{\left(r+{r}_{s}\right)}^{2}}.\end{eqnarray} \tag{ 6 }$$

Here, r_s is the reference scale radius, δ_c is the dimensionless characteristic overdensity of the halo, and ${\rho }_{{\rm{c}}}=3{H}_{0}^{2}/8\pi G$ is the local critical density. The NFW profile can also be written in terms of the halo virial mass M₅₀₀ (i.e., the mass value at which the interior mean density is 500 times the critical density of the universe), and the halo concentration c ≡ r₅₀₀/r_s , with ${r}_{500}\approx 260{\left({M}_{500}/{10}^{12}{M}_{\odot }\right)}^{1/3}$ being the virial radius, and ${\delta }_{c}{\rho }_{c}\,={M}_{500}{c}^{3}g(c)/4\pi {r}_{500}^{3}$ with $g(c)\equiv {[\mathrm{ln}(1+c)-c/(1+c)]}^{-1}$. The DM mass profile in Equation (4) will then coincide with the NFW mass distribution, and the term d ρ/dr will be the gradient of the NFW density profile. We remark that in this analysis the perturbative parameter is L_NMC/r_s , a quantity that is always small for the range of masses probed in our study, as we will show with our results.

We test the aforementioned formalism for the NMC using the X-COP ⁶ catalog (Eckert et al. 2017) with joint X-ray temperature and SZ pressure observations. The methodology we adopt here is equivalent to the one earlier implemented in Haridasu et al. (2021; please refer to it for further details). To constrain the characteristic length scale (L_NMC) alongside the parameters of the mass profile (Θ_M ) and the ED (Θ_e ), we write a joint likelihood ${ \mathcal L }$ as

$$\begin{eqnarray}&&{ \mathcal L }={{ \mathcal L }}_{{{\rm{P}}}_{X}}+{{ \mathcal L }}_{{{\rm{P}}}_{{SZ}}}+{{ \mathcal L }}_{\mathrm{ED}},\end{eqnarray} \tag{ 7 }$$

where the pressure is computed through Equation (4) and the ED is modeled as Equation (5). Here the first term accounts for the likelihood corresponding to the X-ray temperature P_X data, the second term denotes the likelihood for the covarying SZ pressure data, and the last term in Equation (7) accounts for the modeled ED data.

Alongside these primary parameters of the model we also include an additional intrinsic scatter Σ_{P, int}, following the approach in Ghirardini et al. (2018) and Ettori et al. (2019). We refer to Haridasu et al. (2021) for an elaborate discussion on the mild differences between our approach here and the analysis performed in Ettori et al. (2019).

We perform a Bayesian analysis through MCMC sampling using the publicly available emcee ⁷ package (Foreman-Mackey et al. 2013; Hogg & Foreman-Mackey 2018), which implements an affine-invariant ensemble sampler and GetDist ⁸ package (Lewis 2019), to perform analysis of the chains and plot the contours. We utilize flat uniform priors on all the parameters Θ_e = {n₀, α, β, ε, r_c, r_s}, Θ_M = {M₅₀₀, c}, and the NMC characteristic length scale L_NMC in the MCMC analysis. Note here that we utilize the analytical form for the M(<r) of the cluster, which is expressed as a function of Θ_M. Finally, we also perform a model comparison through the Bayesian evidence ${ \mathcal B }$ (Trotta 2008, 2017; Heavens et al. 2017b) using the MCEvidencepackage (Heavens et al. 2017a). ⁹ Comparing the Bayesian evidence, one can assess the preference for a given model ${{ \mathcal M }}_{1}({{\rm{\Theta }}}_{1})$ over the base model, i.e., the NFW model. Also, the Bayesian evidence is contrasted on the Jeffreys scale (Jeffreys 1961), where ${\rm{\Delta }}\mathrm{log}({ \mathcal B })$ < 2.5 and ${\rm{\Delta }}\mathrm{log}({ \mathcal B })$ > 5, implying either a weak or a strong preference for the extended model, respectively.

We report the results of our MCMC parameter estimation in Table 2 and the respective statistical comparison in Table 1. The reduced chi-squared (${\chi }_{\mathrm{red}}^{2}$) values in Table 1 indicate that for the majority of the clusters, the NMC DM model generally provides a description of the data comparable to and often even better than the NFW model. Nevertheless, we point out that the value of the NMC length scale L_NMC is partially guided by the availability of data at the innermost radii, and X-COP cluster pressure profiles are not well characterized in these regions. This lack of data at small radii relaxes the constraints on the higher end of the possible values for L_NMC, and it is ultimately responsible for the production of a hole-like feature (corresponding to low or negative values of pressure) observed in our analysis for a certain fraction of the cluster pressure profiles at inner radii. We, however, anticipate that these features could be erased just by adding one or more data points at inner radii for the pressure profiles. Unfortunately, such data are yet to be available for the X-COP cluster sample. In light of this, the reader should interpret values of the NMC length scale L_NMC obtained in this work for clusters exhibiting a hole in their pressure profiles just as upper bounds on the real values of L_NMC. We also note that our NMC DM model does not modify the estimation of pressure profiles in the outskirts of the cluster, essentially implying that the results presented here are not degenerate with any additional physics that can potentially affect the pressure profile estimation at outer radii, such as nonthermal pressure support, which, for example, could be important for cluster A2319 (Eckert et al. 2019). In the last column of Table 1 we show estimates of the Bayesian evidence ${{\rm{\Delta }}}_{{ \mathcal B }}$ exploited to further compare the two models, assuming standard NFW to be the base model. The NMC DM model is preferred for half of the clusters in the sample, and likewise it is mildly disfavored by the other half (up to the more striking case of RXC1825, for which ${{\rm{\Delta }}}_{{ \mathcal B }}=-3.53$).

Table 1. Reduced χ² and Bayesian Evidence ${{\rm{\Delta }}}_{{ \mathcal B }}$ from the MCMC Analysis for the NFW and NMC DM Models

Cluster	z	${\chi }_{\mathrm{red},\mathrm{NFW}}^{2}$	${\chi }_{\mathrm{red},\mathrm{NMC}}^{2}$	${{\rm{\Delta }}}_{{ \mathcal B }}$
A85	0.0555	2.9	2.7	−0.89
A644	0.0704	2.4	2.2	0.11
A1644	0.0473	3.9	3.4	1.01
A1759	0.0622	1.7	1.6	1.34
A2029	0.0773	1.6	1.6	−0.15
A2142	0.0909	3.3	3.3	−1.32
A2255	0.0809	6.7	1.8	2.64
A2319	0.0557	7.8	7.1	2.05
A3158	0.0597	2.3	2.1	2.81
A3266	0.0589	6.7	6.8	−1.89
RXC1825	0.0650	3.3	6.1	−3.53
ZW1215	0.0766	0.97	0.86	−0.81

Note. The Bayesian evidence ${{\rm{\Delta }}}_{{ \mathcal B }}$ is calculated in favor of the NMC DM model.

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In Table 2 we have reported the concentration c and virial mass M₅₀₀ values from our MCMC analysis for the NFW and the NMC DM models. Estimates for these values from the two models are always compatible within the displayed uncertainties, with the exception of cluster RXC1825's concentration (slightly larger in the NMC framework than the NFW case) and M₅₀₀ (conversely slightly smaller in the NMC case). Despite this overall compatibility, we note that the NMC model predicts concentration values systematically larger than the NFW ones. Table 2 also features the MCMC estimations for the NMC length scale L_NMC. Overall, these values of L_NMC exceed by 2 orders of magnitude, on average the same values obtained for spiral galaxies in Gandolfi et al. (2022). This result is remarkably consistent with the increasing trend observed for spiral galaxies in Gandolfi et al. (2022) between the mass of DM halos and the L_NMC associated with them, as we will show more in detail in Section 3.2.

Table 2. Results of the MCMC Parameter Estimation for the NFW Models and the NMC DM Model

Cluster	c500,GR	M500,GR	c500,NMC	M500,NMC	LNMC
		(1014 M⊙)		(1014 M⊙)	(kpc)
A85	${2.0}_{-0.1}^{+0.1}$	${6.2}_{-0.3}^{+0.2}$	2.0 ± 0.2	${6.2}_{-0.3}^{+0.3}$	${12}_{-10}^{+6}$
A644	${5}_{-2}^{+1}$	4.6 ± 0.4	${6.2}_{-1.7}^{+1.0}$	4.6 ± 0.4	${26}_{-4}^{+11}$
A1644	${1.1}_{-0.4}^{+0.2}$	${3.2}_{-0.4}^{+0.3}$	${1.4}_{-0.5}^{+0.3}$	${3.3}_{-0.3}^{+0.3}$	${27}_{-3}^{+9}$
A1759	3.0 ± 0.2	${4.7}_{-0.3}^{+0.2}$	${3.4}_{-0.3}^{+0.3}$	${4.6}_{-0.2}^{+0.2}$	${20}_{-3}^{+7}$
A2029	${3.3}_{-0.3}^{+0.2}$	7.7 ± 0.4	${3.6}_{-0.5}^{+0.3}$	7.5 ± 0.4	${28}_{-7}^{+15}$
A2142	${2.3}_{-0.2}^{+0.2}$	8.4 ± 0.4	${2.4}_{-0.3}^{+0.2}$	${8.4}_{-0.4}^{+0.4}$	${17.3}_{-15}^{+7}$
A2255	${1.6}_{-0.9}^{+0.4}$	4.7 ± 0.4	${2.3}_{-0.7}^{+0.3}$	4.8 ± 0.3	${109}_{-8}^{+11}$
A2319	${3.8}_{-0.6}^{+0.4}$	7.4 ± 0.2	${4.6}_{-0.8}^{+0.6}$	7.4 ± 0.2	${60}_{-5}^{+17}$
A3158	${2.0}_{-0.4}^{+0.3}$	${4.0}_{-0.3}^{+0.3}$	${2.6}_{-0.4}^{+0.4}$	4.0 ± 0.2	${36}_{-3}^{+6}$
A3266	${1.7}_{-0.2}^{+0.2}$	6.6 ± 0.2	1.7 ± 0.2	${6.5}_{-0.3}^{+0.3}$	${15}_{-13}^{+5}$
RXC1825	${2.6}_{-0.4}^{+0.4}$	${4.1}_{-0.3}^{+0.3}$	${4.0}_{-0.9}^{+0.5}$	3.5 ± 0.3	${9}_{-8}^{+3}$
ZW1215	${1.5}_{-0.3}^{+0.2}$	7.1 ± 0.7	${1.6}_{-0.3}^{+0.2}$	7.0 ± 0.6	${22}_{-19}^{+9}$

Note. The full set of parameter values and the related contours are available in the figure set.

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In Figures 1 and 2 we show two exemplificative profiles (clusters A644 and A2142) obtained with our MCMC analysis, alongside the posterior contour plots for the {M₅₀₀, c, L_NMC}. As in the other clusters, both the NFW and the NMC DM models provide a good description of the general trend of the data. However, the NMC DM model is able to provide a better fit for the clusters whose data at the innermost radii are tracing a flattening in the shape of the pressure profiles. Such flattening seems to arise right within the area in which the NMC effect is active (i.e., within a distance of L_NMC from the center of the dark haloes, represented as a blue shaded area in both Figures 1 and 2). As mentioned before, such an NMC effect should be read with caution, given the limitation of the temperature data available in the innermost regions of the cluster.

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Figure 3 shows the one-dimensional posterior distribution of the L_NMC parameter from our MCMC analysis for the X-COP cluster sample. Consistently with the galactic dark halos analyzed in Gandolfi et al. (2022), L_NMC has different values in different halos, depending on their characteristics (in particular on their virial mass). Some halos (e.g., RXC1825 or A85) show a one-dimensional posterior converging toward L_NMC = 0, suggesting that the DM density profile for these halos may have a cuspy shape, well reproduced by the NFW model. In other halos (e.g., A2319 and A2255) the NMC produces typical scale lengths capable of reaching fractions of megaparsecs. These values are likely to be slightly overestimated since, as previously discussed, some of these clusters exhibit an NMC DM pressure profile featuring a central hole. Despite this, the peak of such a one-dimensional posterior is clearly far from L_NMC = 0, indicating that the shape of the density profile of these dark halos could be less cuspy and different from that of the NFW profile. As can be seen in the right panel of Figure 1, the nonzero values for L_NMC are essentially accompanied by a mild positive correlation with M₅₀₀ and subsequently a non-Gaussian degeneracy with the concentration c. Also, for all the clusters that have a nonzero posterior for the L_NMC, we do not observe any such correlation with the M₅₀₀ parameter, as in the case of A2142, shown in the right panel of Figure 2. In this context, clusters A2255 and A2319 show a slightly larger value of the length scale L_NMC in the posteriors. We also note that for the clusters A2255 and RXC1825, we find a strong bimodal behavior, from which we select the maximum posterior region. As can be seen also from the corresponding Bayesian evidence in favor of the NMC DM model, the clusters A3158, A2319, and A2255 show a moderate preference (${{\rm{\Delta }}}_{\mathrm{log}({ \mathcal B })}\gtrsim 2$), owing to the slightly larger values of L_NMC. As can be seen in Figure 6 in the Appendix, this evidence in favor of the NMC DM in these three clusters is essentially driven by the improvement of the fit accounting for the innermost data point in the X-ray pressure observations. And on the contrary, the cluster RXC1825 shows a preference for the standard NFW scenario at a similar level of Bayesian evidence.

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In this section, we investigate the relation between the NMC length scale L_NMC and the dark halo virial mass M₅₀₀ observed as a result of our analysis. We remark that this relationship is an important feature of the NMC DM model which, as previously stated, is not to be considered as a modified theory of gravity, and therefore L_NMC should not be thought of as a new proposed fundamental constant of nature. The observed relationship between L_NMC and M₅₀₀ shows that L_NMC indeed does not have a universal value, and it depends on at least one property of the dark haloes under consideration. The L_NMC–M₅₀₀ relationship was first observed in Gandolfi et al. (2022) to hold for the galactic dark halos, analyzed therein. A remarkable result of this earlier analysis is that one can describe such a relationship with a simple power law. In this work, we investigate the validity of this relation up to the virial mass ranges typical of galaxy clusters. The results of our analysis are shown in Figure 4. Here, the virial masses of the spiral galaxies and their errors are rescaled from M₂₀₀ to M₅₀₀ to homogenize the results. Remarkably, the X-COP clusters data point derived by our MCMC analysis are seemingly in agreement with the power-law trend of the L_NMC–M₅₀₀ relationship observed in Gandolfi et al. (2022). We performed an MCMC fit using the model ${\mathrm{log}}_{10}{L}_{\mathrm{NMC}}=a{\mathrm{log}}_{10}({{bM}}_{500})$ to fit both galactic and clusters data simultaneously, obtaining as parameter values a = 0.542 ± 0.005 and b = 0.807 ± 0.005. The slope a found in this analysis is compatible with the slope found by fitting a similar power law to galaxies only, as done in Gandolfi et al. (2022; 0.7 ± 0.2). The best-fit line in this work is shown in Figure 4 as a solid black line together with a gray shaded area representing a 1σ confidence limit of the fit. In the same figure, we also show as a gray dotted line the relation ${L}_{\mathrm{NMC}}={M}_{200}^{0.8}$, utilized in Gandolfi et al. (2022) as a reference relation to study the capacity of the NMC DM model in reproducing the radial acceleration relation (RAR). In the galactic virial mass regime, the two power laws are consistent within a 1σ confidence limit, and their slopes are compatible within the errors. The updated scaling law retrieved in this work translates into an average variation of the RAR with respect to the one computed in Gandolfi et al. (2022) by a mere 0.33%, with the average of such variation being taken for every radial acceleration bin in which the RAR of Gandolfi et al. (2022) is computed (spanning from a minimum variation of 0.004%–1.4% among all the bins). We stress that such variation is well within the errors associated to the RAR computed in Gandolfi et al. (2022) for every single bin of radial acceleration. In fact, for the RAR of Gandolfi et al. (2022) the minimum and maximum percentage relative uncertainties are 0.67% and 3.27%, respectively, and the average one is 1.85%. We thus conclude that the updated L_NMC–M₅₀₀ relation retrieved in this work, albeit different from the one considered in Gandolfi et al. (2022), is still able to reproduce the RAR in the galactic dark haloes mass regime. That being said, from Figure 4 it is possible to appreciate the significant difference between the two power laws when approaching the cluster dark halo mass regime. This essentially constitutes an improvement over the previous analysis, which utilized only the galaxies to assess the same relation. As previously mentioned, for some of the clusters, the L_NMC values could be slightly overestimated, and hence it is possible that the real best-fit power law could be even less steep than what is found in our analysis. Moreover, we expect that including a galaxy cluster dataset that probes the innermost regions of the halo could help reduce the scatter in the L_NMC–M₅₀₀ relation.

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In Figure 5 we test the correlation between concentration c and M₅₀₀ values inferred from our MCMC analysis against the relationship between c₂₀₀ and M₂₀₀ of dark halos found in Dutton & Macciò (2014), namely:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{c}_{200}=0.905-0.101{\mathrm{log}}_{10}\left({M}_{200}/{10}^{12}{h}^{-1}\,{M}_{\odot }\right).\end{eqnarray} \tag{ 8 }$$

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To make this comparison, we rescale the value of the virial mass M₅₀₀ of the clusters to M₂₀₀, recalculating the corresponding concentrations accordingly. We then perform an MCMC fit to find the best-fit power law that best describes the data obtained by exploiting both the NFW model and the NMC DM model. In both these cases, there is some visible difference between the two best-fit power laws and the relationship found in Dutton & Macciò (2014). This is true at least up to the cluster mass regime, where both the best-fit power laws of the NFW and NMC DM model intersect the report of Dutton & Macciò (2014). Comparing the best-fit power laws with each other, we do not identify important differences between the two models since the corresponding data have a rather similar scatter around the Dutton & Macciò (2014) relation. This is something we expected following the previous examination of the tabulated results of our MCMC analysis. Figure 5 can provide interesting qualitative hints on the expected concentrations of sub-haloes in galaxy clusters within this framework. As shown in Meneghetti et al. (2020), the ΛCDM is at variance with the observed density and compactness of the DM sub-haloes in galaxy clusters. From our analysis, the NMC DM model predicts galaxy-sized DM substructures in clusters featuring overall higher concentrations associated with lower halo mass values with respect to the standard CDM paradigm. However, we give the caveat that only future analysis relying on high-quality data and exploiting a larger sample of galaxy clusters can confirm this prediction. In this context, the observed tensions at galaxy clusters scales present a promising way to further test the NMC DM scenario and its phenomenology.