In the context of future large surveys like the Euclid mission extracting the cosmic web from galaxies at higher redshifts with more statistical power will become feasible particularly within the groupcluster mass regime. Therefore it is imperative to enlarge the number of metrics that can used to constrain our cosmological models at these large scales. The number of cosmic filaments surrounding galaxies groups and clusters namely the connectivity has recently emerged as a compelling probe of the largescale structures and has been investigated in various observational and numerical analyses. In this first paper we examine dark matteronly cosmological simulations using the widely used DisPerSE filament finder code under two theories of gravity the Poisson LambdaCDM and the MongeAmpre models in order to quantify how alternative models of gravity alter the properties of the cosmic skeleton. We specifically focused on this alternative gravity theory due to its propensity to enhance the formation of anisotropic structures such as filaments but it also makes them more resistant to collapse which consequently reduces the formation of halos. Indeed our findings reveal that replacing the Poisson equation has a significant impact on the hierarchical formation scenario. This is evidenced by examining the redshift evolution of both the slope and the offset of the connectivity. Additionally we demonstrated that current observations are generally in better agreement with our wellestablished gravity model. Finally our study suggests that filament connectivity in the groupcluster regime could serve as a probe of our gravity model at cosmological scales. We also emphasize that our approach could be extended to alternative theories of dark matter such as warm or fuzzy dark matter given the extraordinary datasets provided by nextgeneration surveys.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']
Distinguish dark matter theories with the cosmic web and nextgeneration surveys I an alternative theory of gravity
{'Distinguish dark matter theories with the cosmic web and next-generation surveys I: an alternative theory of gravity': "Pierre Boldrini 1 and Clotilde Laigle 1  \n1 Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France  \nsubmitted to A&A", 'ABSTRACT': 'In the context of future large surveys like the Euclid mission, extracting the cosmic web from galaxies at higher redshifts with more statistical power will become feasible, particularly within the group-cluster mass regime. Therefore, it is imperative to enlarge the number of metrics that can be used to constrain our cosmological models at these large scales. The number of cosmic filaments surrounding galaxies, groups and clusters, namely the connectivity, has recently emerged as a compelling probe of the large-scale structures, and has been investigated in various observational and numerical analyses. In this first paper, we examine dark matter-only cosmological simulations using the widely used D is P er SE filament finder code under two theories of gravity: the Poisson ( Λ CDM) and the Monge-Ampère models, in order to quantify how alternative models of gravity alter the properties of the cosmic skeleton. We specifically focused on this alternative gravity theory due to its propensity to enhance the formation of anisotropic structures such as filaments, but it also makes them more resistant to collapse, which consequently reduces the formation of halos. Indeed, our findings reveal that replacing the Poisson equation has a significant impact on the hierarchical formation scenario. This is evidenced by examining the redshift evolution of both the slope and the o ff set of the connectivity. Additionally, we demonstrated that current observations are generally in better agreement with our well-established gravity model. Finally, our study suggests that filament connectivity in the group-cluster regime could serve as a probe of our gravity model at cosmological scales. We also emphasize that our approach could be extended to alternative theories of dark matter, such as warm or fuzzy dark matter, given the extraordinary datasets provided by next-generation surveys.  \nKey words. (cosmology:) dark matter, large-scale structures, filaments - galaxies: kinematics and dynamics - methods: numerical', '1. Introduction': "While its existence is inferred from its gravitational e ff ects on baryons, the true nature of dark matter (DM) remains one of the most significant unsolved mysteries in astrophysics and cosmology. As it continues to encounter challenges on galactic and cosmological scales such as the cusp-core problem (Bullock & Boylan-Kolchin 2017; Boldrini 2021)) or the σ 8 tension (Douspis et al. 2019), the standard cosmological model, Λ CDM, remains in competition with a variety of alternative DM models. A first possibility is that the DM is more complex and hotter than simple CDM. A wide range of alternative DM models has been proposed over the last decades to resolve Λ CDM issues. Three DMtheories have gained attention as alternatives to the standard CDM model: warm dark matter (WDM, Bode et al. 2001), selfinteracting dark matter (SIDM, Carlson et al. 1992; Spergel & Steinhardt 2000), and fuzzy dark matter (FDM, Hu et al. 2000). They o ff er potential solutions to some of CDM puzzles, particularly on small scales. It is important to note that the traditional approach has always been to add new properties to DM to address the issues of the CDM paradigm.  \nRather than modifying the nature of dark matter, one could question whether Poisson's gravity provides a complete description of the gravity that dark matter undergoes on a cosmological scale. Even if the Poisson equation describes the large-scale structures of the Universe and agrees remarkably well with observations, alternatives to Poisson's gravity can be considered to explain the Λ CDM problems. Here, we focus on a specific al-  \nSend o ff print requests to : Pierre Boldrini, e-mail: boldrini@iap.fr  \nternative theory of gravity, Monge-Ampère gravity. Replacing Poisson's gravity with Monge-Ampère's simply involves changing the trace operation in the Poisson equation to a determinant operation as:  \nTr( 1 + 1 4 π G ¯ ρ D 2 ϕ ) = ρ ¯ ρ -→ Det( 1 + 1 4 π G ¯ ρ D 2 ϕ ) = ρ ¯ ρ . (1)  \nFor weak gravitational potential ϕ , by expanding the determinant about the identity matrix, one recovers the Poisson model approximately and exactly in one dimension. Brenier (2011); Ambrosio et al. (2020) proposed that the Monge-Ampère equation might provide an alternative model of self-gravitating systems. Brenier (2015) has also shown that Monge-Ampère gravity can arise from a microscopic system in which a finite number of indistinguishable particles move on independent Brownian trajectories without any interaction. Then, gravity emerges through application of a principle of statistical physics, namely the large deviation principle. Monge-Ampère gravity has also been previously studied in the framework of Galileon gravity (Dvali et al. 2000; Luty et al. 2003; Nicolis et al. 2009; De ff ayet et al. 2009) and could be a natural low-energy limit of the Galileon theory, similar to the Poisson equation in general relativity. Moreover, it is quite common in mathematics to replace Poisson with MongeAmpère because it allows solving this nonlinear partial di ff erential equation by approaching it as an optimal transport problem that can be solved with highly e ffi cient algorithms (Mérigot 2011; Jun Kitagawa 2019). In other fields, in particular meteorology, invoking Monge-Ampère equation instead of Poisson equation was a fruitful approach: the semi-geostrophic model  \nA & A proofs: manuscript no. main  \nTable 1. Connectivity in the literature: From left to right, the columns give: data type, the size of the box, the resolution and the cosmic web tracers for simulations, the redshift of observed data, the persistence threshold to detect filaments, and the calibration method to set this free parameter of the DisPerSE code. If the persistence is not specified, it means that the skeleton has not been extracted with the DisPerSE code. One of the calibration methods consists of comparing the positions of peaks (CPmax) to those of the most massive halos such as galaxy groups and clusters ( > 10 13 M ⊙ ) (see section 2.4). Concerning cosmological simulations, all the cosmic web tracers are 3D quantities except for HzAGN simulation. For COSMOS and CFHTLS data, the calibration method involves optimally recovering the filaments in the 2D light-cone catalogues created from 3D cosmological simulations. 1 It corresponds to an absolute persistence cut because DisPerSE is not applied to discrete particles but on a 3d grid for the gas. ∗ These data are not taken into account in our study (for the reasons, see 2.5).  \nworks far better on large scales than the older quasi-geostrophic model which instead uses the Poisson equation (Cullen 2006).  \nOne of the challenges confronting alternative theories is determining e ff ective theoretical experiments within a cosmological context. One major problem is that the contribution of baryons in a gravitational and hydrodynamical fashion on the distribution of DM within galaxies is non-negligible. Besides, the presence of baryons can potentially bias our constraints of the DM properties at small scales (e.g. Gouin et al. 2019; Schneider et al. 2019; Chisari et al. 2019, amongst others). Instead of trying to control for these biases, an alternative is to look at larger scales where gravitational dynamics is dominated by DM to test and confront alternative theories with observations. At these scales, DM is distributed along the anisotropic cosmic web (Bond et al. 1996; Pogosyan et al. 1998), a well-understood consequence of the gravitational growth of primordial overdensities (Zel'dovich 1970). It forms a complex network made of filaments and walls bordering huge regions of low density, called voids, and intersecting at clusters of galaxies (e.g. Davis et al. 1982; de Lapparent et al. 1986; Geller & Huchra 1989). This large-scale cosmic web contains the di ff use material necessary to grow the virialized structures, up to 40% of the total matter in the Universe. The topology of the large-scale density field provides an interesting probe to constrain gravity or cosmology, in complement to standard statistics of galaxies and clusters, like the cluster mass function, the two-point correlation functions, or even higher order moments. This avenue has already been explored e.g. through Minkowski functionals (e.g. Gott et al. 1986; Park & Kim 2010; Hikage et al. 2003; Appleby et al. 2018), statistics of voids (e.g. Sahlén et al. 2016; Hamaus et al. 2016; Contarini et al. 2024), the clustering of critical points of the density fields (e.g. Shim et al. 2023) or the properties of the skeleton (e.g. Ho et al. 2018) amongst other statistics.  \nIn this paper, we propose an approach to quantify how an alternative model of gravity alters the properties of the cosmic  \nnetwork as measured through the connectivity, i.e. the number of cosmic filaments branching out a cluster 1 . Connectivity is a tracer of the growth of structures and of the percolation properties of the cosmic web (Pichon et al. 2010) and as such its evolution throughout cosmic times is expected to informs us both about the equation of state of dark energy and the model of gravity which drives the coalescence of structures (Codis et al. 2018). Connectivity has recently emerged as an interesting probe of the large-scale structure around galaxies, groups and clusters, and was explored in several observational and numerical analyses. Pinpointing clusters, the most massive structures of the Universe, will ultimately mitigate the degeneracy between baryonic physics and the model of gravity. However this requires to observe su ffi ciently large area to beat the statistical uncertainties, and to overcome the fact that connectivity might be dependent on the degree of relaxation of the clusters (e.g. Gouin et al. 2021). We expect that having a relatively large sample size will help minimizing the risk of being contaminated by systematic errors of such nature, or at least controlling their impact.  \nMeasuring connectivity is non-trivial, since it depends on how filaments have been defined. Filament identification is dependent on the choice of tracers (e.g. Zakharova et al. 2023), the sampling of these tracers and the chosen tool and its parameterisation (see e.g. Libeskind et al. 2018). Therefore in this paper, we try to adopt a definition of connectivity as objective as possible, and we focus on specifically measuring the connectivity of groups and clusters (instead of measuring the connectivity of all peaks of the density field, as done in Kraljic et al. 2020), around which cosmic filaments are generally better constrained from observations, due to their high density contrast. Beyond factually checking the connectivity of the cosmic web under the MongeAmpere gravity, our analysis aims at providing a framework for future comparison of the connectivity in alternative models and  \nFig. 1. Halo mass function of Poisson and Monge-Ampère cosmological simulations with L = 205 Mpc / h and 512 3 particles at z = 0, as determined using the adaptaHOP halo finder (Aubert et al. 2004). Only (sub)halos that have more than 100 or more particles are kept. We have also included a Poissonian error normalized by the cosmological box size.  \n<!-- image -->  \nin large observational surveys. Furthermore, we aim to investigate whether our method can e ff ectively exclude an alternative theory of gravity, particularly given its current developmental stage which suggests limited potential to replace Poisson gravity. Our method to measure connectivity is therefore driven by what is expected to be possible in future galaxy surveys. The paper is organized as follows. Section 2 provides a description of our data from cosmological simulations of Poisson and MongeAmpère gravity and of how we measure the connectivity of massive halos. In Section 3, we describe our new metrics, based on the connectivity, designed to di ff erentiate gravity theories. Moreover, these metrics hold the potential for extension to alternative DM theories. Section 4 summarizes our main results and suggests a benchmark for future studies.", '2.1. DM-only cosmological simulations': 'The analysis presented in this paper uses snapshots of DM only simulations with Poisson gravity and Monge-Ampère gravity, starting from the same Λ CDM initial conditions using the NGENIC code (Angulo et al. 2012; Springel et al. 2005). Our Poisson gravity simulation refers to classic Λ CDM cosmological simulations. These cosmological simulations from z = 49 to z = 0 were developed in a first work about Monge-Ampère gravity in a cosmological context (see Lévy et al. (2024)). With a mass resolution of m DM = 8 × 10 9 M ⊙ and a uniform periodicboundary cube of L = 205 Mpc / h ∼ 300 Mpc on a side, these simulations enable us to study the clustering of DM at di ff erent scales, from galaxies to filaments. The cosmological parameters that we used are taken from Planck Collaboration et al. (2016) results, with the matter density Ω m = 0 . 3089, the cosmological constant ΩΛ = 0 . 6911, the Hubble constant H 0 = 67 . 74 km s -1 Mpc -1 and σ 8 = 0.8159. The numerical implementation of this alternative gravity theory for a cosmological box using the latest algorithms based on optimal transport-based theory is detailed in (Lévy et al. 2024). Besides, the derivation of Monge-  \nAmpère gravity from statistical physic principles is given in Brenier (2015) and Lévy et al. (2024).', '2.2. Extracting dark matter halos': 'We used the adaptaHOP structure finder (Aubert et al. 2004; Tweed et al. 2009) in the most massive submaxima mode (MSM) to identify the halos in our simulations. We used a local threshold of 178 times the average matter density, with the local density of individual particles calculated using the 20 nearest neighbours. The MSM method is designed to construct the halo structure tree (host halo along with its subhalos and nested subhalos), ensuring that the halo itself encompasses the most massive local maximum. We require a minimum of 100 particles per (sub)halo. Fig-  \n8 { 8  \nFig. 2. Diversity of dark matter structures for Poisson and MongeAmpère simulations. We show projected density maps of a 20 Mpc / h slice of the full cosmological box (top panels) and of zoomed regions of 25 Mpc / h on a side (middle and bottom panels). On top panels, we over-plotted the filaments identified by DisPerSE using the DM density field. For this visualization, we used the same persistence level (7 σ ) to extract the cosmic web for both gravity theories. It can be seen that this alternative gravity theory enhances more abundant filament structures. On bottom panels, the orange circles are drawn at one virial radius of their halo with M vir > 10 13 M ⊙ , identified by adaptaHOP . For this zoomed region, we also represent the cosmic filaments in blue reconstructed by DisPerSE for the halo distribution using the CPmax-halos calibration with parameters specified in Table 1.  \n<!-- image -->  \nFig. 3. Criterion to choose the persistence threshold for the cosmic skeleton based on DM density fields at z = 0: ratio of the fraction of peaks (CPmax) found within the virial radius (rvir) of any of the massive haloes to the fraction of groups or clusters hosting a CPmax inside their rvir sphere, as function of the persistence parameter σ . We are seeking to determine for which persistence we are closest to unity.  \n<!-- image -->  \nure 1 shows the halo mass function in our simulations at z = 0. The alternative theory of gravity strongly a ff ects the halo formation over cosmic time. Monge-Ampère simulation exhibits a lack of DM bound structures, especially at high masses ( > 10 13 M ⊙ ). We emphasise that Monge-Ampère equation is invariant under a larger group of symmetries, i.e. SL(3) or unimodular a ffi ne symmetry, than the Poisson equation which is invariant only under SO(3) and does not support deformation or shearing transformations. The extra symmetry facilitates the formation of anisotropic structures such as filaments, but it also makes them more resistant to collapse, which reduces the formation of halos (see Figure 1). This result can also be visually inspected on Figure 2. Given the halo mass function induced by the Monge-Ampère gravity, it appears challenging to reconcile it with the observed cluster mass function. This further confirms that this theory is not viable, even if we have not resolved baryonic physics in our simulations with Monge-Ampère gravity. Indeed, the calculated halo mass function in Figure 1 is su ffi cient to exclude this gravity theory. Indeed, there is a di ff erence of one to two orders of magnitude between the predictions of Poisson and MongeAmpère. However, the observations with their error bars regarding the halo mass function are tightly constrained in our considered mass range and are in very good agreement with Λ CDM (Driver et al. 2022). On the contrary, our halo mass function in the Monge-Ampère gravity is clearly in disagreement with Λ CDM, considering the observation error bars. As mentioned before, our approach aims to find new metrics to constrain alternative gravity theories, as well as alternative dark matter theories that are consistent with certain observations, such as the stellar or cluster mass function.', '2.3. Extracting the cosmic web with DisPerSE': 'We detect and extract the skeleton of the cosmic web using the publicly available algorithm Discrete Persistent Structure Extractor ( DisPerSE , Sousbie 2011; Sousbie et al. 2011) applied to snapshots at redshift z = 0, 0.5, 1, 1.5, 2 of both Poisson and Monge-Ampère simulations (see Figure 2). This algorithm iden-  \nFig. 4. Similar to Figure 3 expect that the cosmic skeleton is based on DMhalo distributions at z = 0.  \n<!-- image -->  \nifies the cosmic skeleton using the Discrete Morse Theory and the theory of persistence. In the case of DM-only cosmological simulations, it is possible to extract the cosmic web either from the density field traced by DM particles, or from catalogs of discrete objects such as DM halos. We opt for a dual approach, using these two cosmic web tracers, as leveraging the density field allows us to validate our simulations and methodology against simulation results from the literature. Then, extracting the cosmic skeleton based on halos enhances our alignment with observed data utilizing catalogs of galaxies.', '2.4. Calibration of DisPerSE': 'Each filament is defined as a set of connected small segments linking peaks to saddle points. These endpoints of filaments are considered as critical points. In order to correctly extract filaments from our simulations, we need to assess the robustness of the derived skeleton by DisPerSE . One way to do that is to compare the positions of peaks (CPmax) to those of the most massive halos such as galaxy groups and clusters ( > 10 13 M ⊙ ) identified by adaptaHOP for di ff erent persistence thresholds. The persistence is defined as the ratio of the density of the two critical points in the pair. This parameter, expressed in terms of numbers of σ , controls the significance level of the identified filaments with respect to the noise. Increasing the persistence threshold results in a decrease in the number of filaments as we are filtering low-persistence structures. This rigorous method of calibration was already been employed by Galárraga-Espinosa et al. (2020, 2021, 2023b,a). During the skeleton construction, we also ensure that the filaments are not described by overlapping segments by using -breakdown option of DisPerSE in order to avoid doublecounting filaments in the connectivity measurement. The results of the matching between CPmax and halos can be found in Figure 3 and 4. Instead of plotting the fraction of CPmax found within the virial radius (rvir) of any of the massive haloes, and conversely, the fraction of groups or clusters hosting a CPmax inside their rvir spheres, we show the ratio of the two fractions, which means we are seeking to determine for which persistence we are closest to unity in order to minimize at z = 0 both the number of halos without peak association and the number of peaks without halo association. The persistence thresholds chosen to compute the connectivity from skeleton extracted from  \nFig. 5. Connectivity plotted against halo mass based on DM density distributions at z = 0 with r v ir -2r v ir band. Table 1 provides details on simulated data in the literature. Data from Gouin et al. (2021) are displayed for di ff erent dynamical states of galaxies.  \n<!-- image -->  \nthe density field and halos are summarized in Table 1. Cosmic skeletons at z = 1, 2 were computed using the same DisPerSE parameterisation as for the z = 0 case.', '2.5. Connectivity κ': 'By design, we are mostly concerned here by the connectivity of galaxy groups and clusters given the resolution of our simulations and the next-generation surveys. One of the objectives of this paper is to investigate whether the connectivity can be a good metric to validate or invalidate alternative gravity theories such as Monge-Ampère gravity in a cosmological context and ultimately extend it to alternative DM theories such as WDM, SIDM, and FDM. The local number of filaments that are connected to galaxy groups and clusters in a sphere of 1.5 rvir is the connectivity κ , similarly to Darragh Ford et al. (2019); Gouin et al. (2021); Sarron et al. (2019) 2 . Given that the choice of a radius of 1.5 times the virial radius is arbitrary for calculating the mean connectivity, measurements within 1 and 2 times the virial radius are considered as error bars in all our figures.  \nBefore assessing the impact of our alternative theory on the connectivity, we want to verify that our analysis of our Λ CDM DM-only simulations allows us to reproduce the results found in the literature. Table 1 compiles all key details about simulated and observed data from the literature, along with their analysis parameters regarding connectivity. Figure 5 attests to a good agreement between our Poisson simulation and state-of-the-art hydrodynamical cosmological simulations such as Horizon-AGN (Dubois et al. 2014), Illustris TNG-300 (Nelson et al. 2019) and The Three Hundred (Cui et al. 2022). Even if we obtain similar results to the literature, a rigorous calibration method based on the matching between CPmax and massive halos allows for setting a persistence threshold adapted to the simulation resolution. It would be interesting to see if the observed deviation at high mass ( > 10 14 . 5 M ⊙ ) between our simulations is due to the inclusion of baryonic physics. In our study and on Figure 5, we did not incorporate Galárraga-Espinosa et al. (2023a) data from the MTNG simulation. In their analysis, they  \nFig. 6. Connectivity plotted against halo mass for the cosmic skeleton based on DM halo distributions at z = 0 with r v ir -2r v ir band. Table 1 provides details on observed data in from the literature.  \n<!-- image -->  \nchose to consider only halos with a connectivity of at least 2 3 . This results in a significant deviation between our simulations and the MTNG simulation (Pakmor et al. 2023) for masses less than 10 14 M ⊙ , and this deviation becomes more pronounced as the mass decreases. Applying a similar criterion in our data, we observed this plateau behavior between 10 12 and 10 15 M ⊙ . We also did not include Kraljic et al. (2020) data because they only looked at the connectivity of those galaxies which are located at the nodes of the cosmic web whereas in our case we measure the connectivity of all groups and clusters in the sample regardless the fact that they are sited on a node.  \nFigure 5 also compares the mean connectivity of Poisson and Monge-Ampère theories extracted from the DM particles. As the extra symmetry enhances the formation of filaments in MongeAmpère gravity, it results that the mean connectivity is higher at high halo mass compared to Poisson gravity. We also stress that in both gravity theories, the connectivity increases with mass in agreement with previous studies. Besides, the trend of the mean connectivity approaching zero at low halo mass must be attributed to the lack of resolution in our simulations (see Figure 6). We emphasize that we take into account halos that have zero connectivity according to DisPerSE code. Figure 6 contains the entire set of observed data that we have compiled from the literature (see details in Table 1). In the case of the cosmic web extracted from dark matter halos, there is still a deviation between Poisson connectivity and that of Monge-Ampère at high halo mass (see Figure 2). This behaviour persists because a cosmic web estimated from halos slightly underestimates the actual connectivity revealed by the DM density field (see Figure 7). The connectivity based on halos as a function of connectivity based on DM particles was computed by fitting the mean connectivities presented in the Figures 5 and 6 by excluding the points from the zero-slope region. We also examined whether the connectivity of the same halos identified in both simulations was correlated. This analysis, applied to 742 halos at z = 0, revealed a positive correlation, as expected from Figure 6, but it remains very weak between halos formed in Poisson gravity and MongeAmpère gravity. The simulation data concerning the connectiv-  \nFig. 7. Connectivity based on halos as function of connectivity based on DM particles. Both connectivities were computed as functions of the halo virial mass in Figure 6 and 5. A cosmic web estimated from halos slightly underestimates the actual connectivity revealed by the DM density field. This e ff ect is stronger for our alternative theory of gravity. Despite the small di ff erence between the connectivities, this also strengthens the idea that it is preferable to extract the skeleton from the halos if one wants to follow an observational approach.  \n<!-- image -->  \nused in our study of Poisson and Monge-Ampère models are available here 4 .', '2.6. Evolution with redshift': 'Future large surveys, like the one provided by the Euclid mission will allow to explore cosmic filaments around clusters at higher redshifts with more statistics, very useful especially at high mass where the di ff erence between our two models is the most important, hence providing an extraordinary dataset to robustly test gravity models. This serves as motivation to explore the evolution of the mean connectivity as a function of redshift for both Poisson and Monge-Ampère simulations. In Figure 8, we found in our Poisson simulations that connectivity increases with redshift, as predicted by Galárraga-Espinosa et al. (2023a). We remind that we extracted the high-z skeletons with the same persistence threshold as identified at z = 0. Starting from Figure 8, interpreting the behavior of connectivity with respect to redshift is challenging. However, by calculating the mean connectivity, we observed that it remains almost constant between z = 1 and z = 2, after which it increases until z = 0, similar to the Poisson model. This strongly encourages us to evaluate the connectivity as a function of redshift, as the di ff erences between our gravity models will manifest not only at z = 0 but also over time. Indeed, we remember that the Monge-Ampère gravity influences the formation of structures such as filaments and halos.', '3. Results': 'In light of Figure 6, the question that now interests us is how we can utilize these observed connectivity data to distinguish between our gravity theories and, subsequently, alternative DM theories. As depicted by this Figure, it is not possible to precisely distinguish between our two models with the observed data when considering the connectivity as a function of the virial mass at  \nFig. 8. Connectivity plotted against halo mass for the cosmic skeleton based on halo distributions at z = 0 , 1 and 2.  \n<!-- image -->  \nfixed redshift. However, the redshift evolution of the signal is supposed to carry information on the overall growth of structures governed by gravity.', '3.1. Redshift evolution of the slope and offset of the connectivity': 'The two metrics considered to evaluate our theories are the connectivity slope and o ff set with respect to redshift (see Figures 9 and 10). They are calculated by fitting the mean connectivity as a function of the virial mass for each redshift. In order to mitigate the contamination of numerical noise due to the lack of resolution in our simulations, we exclude points from the zeroslope region for our fits. We only keep those with a slope greater than 0.1. The o ff set corresponds to the y-intercept of the fitted function for a mass of M vir = 10 14 M ⊙ . This arbitrary mass marks approximately the center of the region where there is the highest density of observable data (see Figure 6). Relying on the slope and o ff set of connectivity as robust metrics is motivated by their ease of tracking, particularly when evaluating connectivity across di ff erent redshifts. In order to compare to observations, we have calculated our two metrics for the COSMOS2015 (Laigle et al. 2016) and CFHTLS data (Sarron et al. 2018). We have also chosen to include a data point which takes into account all the observable data reported in Table 1. To do this, we have assigned a weight to the data corresponding to the number of objects identified in the mass bin as reported by the associated publications. In this calculation, we have not included connectivity measurements of the four richest clusters of the Corona Borealis supercluster (Einasto et al. 2021) as they do not have error bars. Regarding the Poisson theory, we observe that the slope and the o ff set are decreasing and increasing functions with respect to redshift, respectively (see Figures 9 and 10). However, it is more challenging to identify a linear behavior as in Λ CDM, even though the slope appears to be almost constant with respect to redshift for Monge-Ampère gravity.  \nAccording to Figure 9, the slope of connectivity clearly distinguishes our gravity theories up to z = 1. Although the considered data do not allow us to favor one gravity theory over the other, it appears that the overall slope, taking into account all the literature data, aligns with the Poisson theory. This result is strengthened by the fact that all our data points for the connec-  \nFig. 9. Connectivity slope as a function of redshift. For simulations, measurements within 1 and 2 times the virial radius are still considered as error bars. The gray point corresponds to the slope that takes into account all the observable data from Table 1.  \n<!-- image -->  \ntivity o ff set are in very good agreement with the Λ CDM gravity theory. It is worth noting that this metric significantly di ff erentiates our two gravity theories between z = 0 and z = 2. Our results highlight that the data from the next generation surveys, probing a larger range of redshift than nowadays observations with increased statistics, will be crucial for robustly validating or invalidating alternative theories through these metrics. The data concerning the connectivity slope and o ff set used in our study are also available here 5 .', '3.2. Alternative DM theories and cosmic web': "Aquestion that now arises concerns the viability of our approach not only for gravity theories a ff ecting the dynamics of matter, but for alternative theories of dark matter whose intrinsic properties are modified. Intuitively, one would expect our new metrics to work within the framework of DM theories that a ff ect the cosmic web and, consequently, the connectivity of filaments. Indeed, we have highlighted that modifying the gravity in Monge-Ampère theory notably impacts large-scale dynamics, and this modification leaves an imprint on the connectivity. Are there alternative DM theories with exotic properties that a ff ect the cosmic skeleton? A wide range of alternative DM models has been proposed over the last decades. Two main classes of alternative DM models have been simulated on large scales: warm dark matter (WDM, Bode et al. 2001) and fuzzy dark matter (FDM, Hu et al. 2000). These theories have gained attention as alternatives to the standard CDM model because they o ff er potential solutions to some of CDM puzzles, particularly on small scales (Bullock & Boylan-Kolchin 2017; Boldrini 2021). That's why there are few studies, especially on large scales, concerning the cosmic web and its properties.  \nSince particles in WDM and FDM models have di ff erent intrinsic properties from the CDM particle candidates, the effect of these particles on structure formation and evolution is expected to be qualitatively di ff erent on both small and large scales and particularly on the overdense structures that connect halos (Paduroiu et al. 2015; Mocz et al. 2020; May & Springel 2023). In WDM cosmology, it has been demonstrated that the  \nFig. 10. Connectivity o ff set as a function of redshift for M vir = 10 14 M ⊙ . For simulations, measurements within 1 and 2 times the virial radius are still considered as error bars. The gray point corresponds to the o ff set that takes into account all the observable data from Table 1.  \n<!-- image -->  \npresence of less substructures is associated with warmer dark matter. Specifically, as the particle velocity dispersion increases, the filaments become larger, and their preservation extends over a longer period. For WDM models, structure formation follows a complex scenario in which both top-down and bottom-up processes play a role (Paduroiu et al. 2015). Due to this hybrid structure formation, we expect that this di ff erence will be evident in the connectivity across redshifts. Like WDM, FDM filaments do not fragment into halos like those of CDM (Mocz et al. 2020; May & Springel 2023). Over time, CDM filaments can fragment and collapse to form gravitationally bound structures known as DM halos, which are seeds for the formation of galaxies and galaxy clusters. This is why WDM and FDM simulations appear similarly filamentary (Mocz et al. 2020). Furthermore, these alternative models can also induce a delay in the formation of structures. In CDM scenario, the first structures begin to form well before reaching z = 9. In contrast, in a WDM universe, the first collapse may be delayed until z = 4 (Paduroiu et al. 2015). All these physical arguments from the literature invite us to conduct an investigation into the properties of filaments in these alternative theories, following a benchmark that we propose.", '3.3. Forecasts for future surveys': "While Fig. 9 and Fig. 10 show that observations are in overall better agreement with the Poisson gravity model, error bars remain large and some data points at specific redshift (e.g. the AMASCFI dataset at z ∼ 0 . 5) are clearly discrepant with respect to the bulk of the observations. This indicates both the need for larger group and cluster samples around which to measure connectivity to beat down the statistical uncertainties, and a consistent measurement of connectivity across redshift ranges. The next-generation of photometry surveys are particularly well suited to address these issues. For example, the European Space Agency's Euclid mission (Laureijs et al. 2011) will provide, over the entire course of the mission, several thousands of clusters (Euclid Collaboration et al. 2019) over ∼ 14000deg 2 to z ∼ 2 and down to 10 13 . 5 M ⊙ . We note that the accuracy of photometric redshifts might be too limited in the Euclid Wide Survey (Euclid Collaboration et al. 2022b) especially above z = 1 to reconstruct  \nwell enough the distribution of cosmic filaments around clusters. However deeper photometric data will be available in the several tens of degrees of the Euclid Deep survey, along with exquisite complimentary ancillary optical and infrared photometry from HSC and Spitzer as part of the Hawaii Two-0 Survey (Euclid Collaboration et al. 2022a), which should enable a photometric redshift accuracy of the same order of the one obtained on the COSMOS field (Weaver et al. 2022). In this case, we expect to multiply by ∼ 20 our number of detections with respect to the data reported from Darragh Ford et al. (2019) while making a consistent extraction across the redshift range to 0 < z < 2, which will significantly improve the observational constraints on connectivity. Besides Euclid , other upcoming surveys which have identified cosmic web mapping as one of their focus science cases could be used, including spectroscopic surveys like e.g. 4MOST (de Jong et al. 2012) and PFS (Takada et al. 2014) or other photometric surveys like LSST (Ivezi'c et al. 2019) or WFIRST (Spergel et al. 2013).  \nWe note that we have not explicitly explored the e ff ect associated to photometric redshift uncertainties and projection, which might impact the extraction of cosmic filaments, hence the connectivity measurements. In addition, di ff erent methods of cluster selections su ff er from di ff erent biases, depending on if they are based either on galaxies probing primarily the stellar mass, or on X-Ray sensitive to the warm gas content, or on lensing measuring the total mass. The selection and cluster mass measurements methods will potentially create an additional source of systematics. We have not quantified these biases in this paper since the measurements presented here were not done in the context of forecasting a particular survey, and the observational data that we reported come both from spectroscopic and photometric samples, which make di ffi cult a thorough analysis of biases. We note however that the good agreement between most of the surveys presented here suggest that observational uncertainties and systematics are sub-dominant with respect to the impact of the considered gravity model. However, a more careful end-to-end analysis of all systematic biases a ff ecting specific surveys is necessary and will be presented in a future analysis.", '4. Conclusions': 'In this work, we have explored the properties of the cosmic web in an alternative theory of gravity, the Monge-Ampère gravity. We used DM-only cosmological simulations of Monge-ampère and Poisson gravity ( Λ CDM) models at redshifts z = 0, 0.5, 1, 1.5, 2. Following an observational approach, we have extracted the cosmic skeleton using the DisPerSE filament finder by employing the calibration method suggested by Galárraga-Espinosa et al. (2020). We then focused on the connectivity of groups and clusters in both theories in order to provide a framework for future comparison of its properties in alternative models and in large observational surveys. We have demonstrated that distinctions between our gravity theories begin to manifest in the measurement of the connectivity of high-mass halos ( M vir > 10 14 M ⊙ ). To precisely di ff erentiate our theories, the redshift evolution of the slope and o ff set of the connectivity can be used to retrieve information on the overall growth of structures, which di ff ers in our two theories. Following our study of connectivity for di ff erent gravity theories, we propose a benchmark for a more e ff ective comparison between simulations and observations:  \n- -Creating mock data, including noise and instrumental properties, to replicate the survey data based on simulations\n- -Extracting cosmic filaments from the mock data and observations with the DisPerSE code  \n- -Calibrating independently the persistence threshold of DisPerSE for observed and simulated data by using CPmaxhalos method\n- -Computing specific metrics to compare synthetic and observed data.  \nThis approach enables unbiased comparisons between the data and theoretical predictions, free from the influence of the theories themselves. This benchmark will also enable us to make the most of the new metrics identified in our study: the slope and o ff set of connectivity across redshift. We have demonstrated that these metrics could serve as reliable probes to validate or invalidate gravity theories such as the Monge-Ampère gravity. Data from next-generation surveys such as Euclid will confirm the success of our approach for alternative models in the mass regime of groups and clusters. As suggested by our results, there is a clear need for larger samples of groups and clusters to measure observed connectivity. This is essential to reduce statistical uncertainties and ensure a consistent measurement across redshift ranges. Finally, we discussed how our approach can be extended to alternative DM theories such as warm or fuzzy DM.', '5. Data Availability': 'The code, which computes the filament connectivity at a given halo mass and then the connectivty slope and o ff -set, is publicly available at https: // github.com / Blackholan. The data results shown in this paper can be found at https: // www.iap.fr / useriap / boldrini / data.html.', '6. Acknowledgements': "The first part of P. Boldrini's work was carried out as part of a post-doc (funded by Inria Nancy and Sorbonne University), cosupervised by R. Mohayaee and B. Lévy, and in collaboration with Y. Brenier. The study of connectivity as a possible cosmological probe, which is the subject of this paper, was conducted outside of and after this post-doc. This work uses data from the Monge-Ampère gravitational numerical simulation (Lévy et al. 2024). We thank Céline Gouin and Daniela Galárraga-Espinosa for valuable discussions. P.B. also thanks Eduardo Vitral and Denis Werth for their help concerning the data visualisations. This work has made use of the Infinity Cluster hosted by the Institut d'Astrophysique de Paris.", 'References': "- Ambrosio, L., Baradat, A., & Brenier, Y. 2020, Monge-Ampère gravitation as a Γ -limit of good rate functions\n- Angulo, R. E., Springel, V., White, S. D. M., et al. 2012, Mon. Not. R. Astron. Soc., 426, 2046\n- Appleby, S., Park, C., Hong, S. E., & Kim, J. 2018, ApJ, 853, 17\n- Aragón-Calvo, M. A., van de Weygaert, R., & Jones, B. J. T. 2010, MNRAS, 408, 2163\n- Aubert, D., Pichon, C., & Colombi, S. 2004, Mon. Not. R. Astron. Soc., 352, 376\n- Bode, P., Ostriker, J. P., & Turok, N. 2001, ApJ, 556, 93\n- Boldrini, P. 2021, Galaxies, 10, 5\n- Bond, J. R., Kofman, L., & Pogosyan, D. 1996, Nature, 380, 603\n- Brenier, Y. 2011, Confluentes Math., 361\n- Brenier, Y. 2015, working paper or preprint\n- Bullock, J. S. & Boylan-Kolchin, M. 2017, ARA&A, 55, 343\n- Carlson, E. D., Machacek, M. E., & Hall, L. J. 1992, ApJ, 398, 43\n- Chisari, N. E., Mead, A. J., Joudaki, S., et al. 2019, The Open Journal of Astrophysics, 2, 4  \nCodis, S., Pogosyan, D., & Pichon, C. 2018, MNRAS, 479, 973  \n- Contarini, S., Pisani, A., Hamaus, N., et al. 2024, A&A, 682, A20\n- Cui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977  \nCullen, M. 2006, A Mathematical Theory of Large-Scale Atmosphere / Ocean Flow (Imperial College Press)  \nDarragh Ford, E., Laigle, C., Gozaliasl, G., et al. 2019, MNRAS, 489, 5695 Davis, M., Huchra, J., Latham, D. W., & Tonry, J. 1982, ApJ, 253, 423 de Jong, R. S., Bellido-Tirado, O., Chiappini, C., et al. 2012, in Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, ed. I. S.  \nMcLean, S. K. Ramsay, & H. Takami, 84460T  \nde Lapparent, V., Geller, M. J., & Huchra, J. P. 1986, ApJ, 302, L1 De ff ayet, C., Deser, S., & Esposito-Farèse, G. 2009, Phys. Rev. D, 80, 064015 Douspis, M., Salvati, L., & Aghanim, N. 2019, arXiv e-prints, arXiv:1901.05289 Driver, S. P., Robotham, A. S. G., Obreschkow, D., et al. 2022, MNRAS, 515, 2138  \nDubois, Y., Pichon, C., Welker, C., et al. 2014, MNRAS, 444, 1453 Dvali, G., Gabadadze, G., & Porrati, M. 2000, Physics Letters B, 485, 208 Einasto, M., Deshev, B., Tenjes, P., et al. 2020, A&A, 641, A172 Einasto, M., Kipper, R., Tenjes, P., et al. 2021, A&A, 649, A51 Euclid Collaboration, Adam, R., Vannier, M., et al. 2019, A&A, 627, A23 Euclid Collaboration, Moneti, A., McCracken, H. J., et al. 2022a, A&A, 658, A126  \nEuclid Collaboration, Scaramella, R., Amiaux, J., et al. 2022b, A&A, 662, A112 Galárraga-Espinosa, D., Aghanim, N., Langer, M., Gouin, C., & Malavasi, N. 2020, A&A, 641, A173  \nGalárraga-Espinosa, D., Aghanim, N., Langer, M., & Tanimura, H. 2021, A&A, 649, A117  \nGalárraga-Espinosa, D., Cadiou, C., Gouin, C., et al. 2023a, arXiv e-prints, arXiv:2309.08659  \nGalárraga-Espinosa, D., Garaldi, E., & Kau ff mann, G. 2023b, A&A, 671, A160 Geller, M. J. & Huchra, J. P. 1989, Science, 246, 897 Gott, J. Richard, I., Melott, A. L., & Dickinson, M. 1986, ApJ, 306, 341 Gouin, C., Bonnaire, T., & Aghanim, N. 2021, A&A, 651, A56 Gouin, C., Gavazzi, R., Pichon, C., et al. 2019, A&A, 626, A72 Hamaus, N., Pisani, A., Sutter, P. M., et al. 2016, Phys. Rev. Lett., 117, 091302 Hikage, C., Schmalzing, J., Buchert, T., et al. 2003, PASJ, 55, 911 Ho, A., Gronke, M., Falck, B., & Mota, D. F. 2018, A&A, 619, A122 Hu, W., Barkana, R., & Gruzinov, A. 2000, Phys. Rev. Lett., 85, 1158 Ivezi'c, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111 Jun Kitagawa, Quentin Mérigot, B. T. 2019, J. Eur. Math. Soc., 21 Kraljic, K., Pichon, C., Codis, S., et al. 2020, MNRAS, 491, 4294 Laigle, C., McCracken, H. J., Ilbert, O., et al. 2016, ApJS, 224, 24 Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, arXiv:1110.3193 Lévy, B., Brenier, Y., & Mohayaee, R. 2024, arXiv e-prints, arXiv:2404.07697 Libeskind, N. I., van de Weygaert, R., Cautun, M., et al. 2018, MNRAS, 473, 1195  \nLuty, M. A., Porrati, M., & Rattazzi, R. 2003, Journal of High Energy Physics, 2003, 029  \nMalavasi, N., Aghanim, N., Douspis, M., Tanimura, H., & Bonjean, V. 2020,  \nA&A, 642, A19 May, S. & Springel, V. 2023, MNRAS, 524, 4256 Mérigot, Q. 2011, Comput. Graph. Forum, 30, 1583 Mocz, P., Fialkov, A., Vogelsberger, M., et al. 2020, MNRAS, 494, 2027 Nelson, D., Springel, V., Pillepich, A., et al. 2019, Computational Astrophysics and Cosmology, 6, 2  \nNicolis, A., Rattazzi, R., & Trincherini, E. 2009, Phys. Rev. D, 79, 064036 Paduroiu, S., Revaz, Y., & Pfenniger, D. 2015, arXiv e-prints, arXiv:1506.03789 Pakmor, R., Springel, V., Coles, J. P., et al. 2023, MNRAS, 524, 2539 Park, C. & Kim, Y.-R. 2010, ApJ, 715, L185  \nPichon, C., Gay, C., Pogosyan, D., et al. 2010, in American Institute of Physics Conference Series, Vol. 1241, Invisible Universe, ed. J.-M. Alimi & A. Fuözfa, 1108-1117  \nPlanck Collaboration, Ade, P., Aghanim, N., et al. 2016, A&A, 594, A13  \nPogosyan, D., Bond, J. R., Kofman, L., & Wadsley, J. 1998, in Wide Field Sur-  \nveys in Cosmology, ed. S. Colombi, Y. Mellier, & B. Raban, 61  \nSahlén, M., Zubeldía, Í., & Silk, J. 2016, ApJ, 820, L7  \nSantoni, S., De Petris, M., Ferragamo, A., Yepes, G., & Cui, W. 2023, arXiv e-prints, arXiv:2310.07494  \nSarron, F., Adami, C., Durret, F., & Laigle, C. 2019, A&A, 632, A49 Sarron, F., Martinet, N., Durret, F., & Adami, C. 2018, A&A, 613, A67 Schneider, A., Teyssier, R., Stadel, J., et al. 2019, J. Cosmology Astropart. Phys., 2019, 020  \nShim, J., Pichon, C., Pogosyan, D., et al. 2023, arXiv e-prints, arXiv:2311.09886 Sousbie, T. 2011, MNRAS, 414, 350  \nSousbie, T., Pichon, C., & Kawahara, H. 2011, MNRAS, 414, 384  \nSpergel,  \nD.,  \nGehrels,  \nN.,  \nBreckinridge,  \nJ.,  \net  \nal.  \n2013,  \narXiv  \ne-prints,  \narXiv:1305.5422  \nSpergel, D. N. & Steinhardt, P. J. 2000, Phys. Rev. Lett., 84, 3760 Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629 Takada, M., Ellis, R. S., Chiba, M., et al. 2014, PASJ, 66, R1 Tweed, D., Devriendt, J., Blaizot, J., Colombi, S., & Slyz, A. 2009, A&A, 506, 647  \nWeaver, J. R., Kau ff mann, O. B., Ilbert, O., et al. 2022, ApJS, 258, 11 Zakharova, D., Vulcani, B., De Lucia, G., et al. 2023, MNRAS, 525, 4079 Zel'dovich, Y. B. 1970, A&A, 5, 84"}