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A&A
Volume 693, January 2025
Article Number A109
Number of page(s) 10
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/202451898
Published online 07 January 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

In their ambition to elucidate the enigmatic nature of dark energy, responsible for the observed cosmic acceleration, cosmology researchers have increasingly moved beyond the standard ΛCDM paradigm anchored in Einstein’s theory of general relativity (GR; Lue et al. 2004; Clifton et al. 2012; Joyce et al. 2016; Langlois 2019). The emerging extended gravity models offer convincing alternatives, introducing pronounced deviations from GR that generate accelerated cosmic expansion in the later epochs of the Universe without a cosmological constant or an exotic energy component. These deviations not only influence homogeneous cosmic dynamics, but they also considerably modify the growth of density perturbations, leaving unmistakable imprints on the formation of cosmic structures (Song et al. 2007; Bertschinger & Zukin 2008; Baghram & Rahvar 2010; Bernardeau & Brax 2011; Koyama 2016; Heymans & Zhao 2018; Sharma & Verma 2022). Remarkably, in some alternative gravitational frameworks, the fundamental principles of GR can resurface at higher densities through screening mechanisms, ensuring compatibility with high-precision gravitational data derived from Solar System studies (Khoury & Weltman 2004b; Faulkner et al. 2007; Chiba et al. 2007; Brax et al. 2008; Jain & Khoury 2010; Liu et al. 2018; Sakstein 2018). Typical examples of these modified gravity (MG) theories are models of the f(R) type (Starobinsky 1980; Capozziello et al. 2003; Carroll et al. 2004), in which the standard scalar curvature, R, in the Einstein-Hilbert action, is replaced by a more general function, f(R). Within this new theoretical framework, the approach of Hu & Sawicki (2007), hereafter referred to as HS, aims not only to harmonize with the homogeneous cosmic dynamics of the widely accepted ΛCDM model, but also to provide new elements of cosmological structuring through the introduction of an additional fifth force.

It has also been obtained that gravity modifications affect differently the lensing potential and the dynamical potential, so that estimates of cluster mass using either photon deflection or the velocity of their massive constituents no longer match (Schmidt 2010; Gronke et al. 2016). The density profiles of clusters due to matter accumulation in their accretion region in HS models are also modified (Lombriser et al. 2012). However, the search for a distinguishing feature to differentiate models built in the framework of GR from those built in the framework of modified gravity (MG) theories remains a crucial open question.

Llinares & Mota (2013) and Burrage et al. (2015) suggested that the study of the shape of dark matter (DM) halos in MG models could be a more critical way of discriminating between the MG models and models developed in GR. Indeed, due to the screening process inherent to these models, the collapse during the formation of DM halos naturally leads to an anisotropic component that can accentuate the triaxiality of these cosmic structures. The work presented in this article addresses, from a purely statistical perspective, the crucial question of whether such a feature in the shape of DM halos could imprint a specific MG signature in structure formation processes. We investigated the influence of MG-f(R) models on the statistical properties of the shape of DM halos. We also tested whether the cosmological invariance of the shape properties of DM halos, when expressed in terms of the nonlinear fluctuations of the cosmic matter, already verified for wCDM models of GR (as we had demonstrated in Alimi & Koskas 2024 from the Dark Energy Universe numerical simulations Alimi et al. 2010; Rasera et al. 2010; Alimi et al. 2012; Reverdy et al. 2015), can be extended to HS models performed by Giocoli et al. (2018).

It is important to note that the cosmological invariance of the shape properties of DM halos resulted both from the isotropic nature of the gravitational interaction in wCDM models of GR, implying that the more a halo collapses, the more it evolves toward a spherical cosmic structure, and from the fact that the estimate of the nonlinear fluctuations also results from an isotropic calculation of the two-point correlation in the cosmic matter field. Ultimately, this cosmological invariance expressed that the shape of DM halos in wCDM models of GR encapsulates the entire power spectrum of the cosmic matter field (Alimi & Koskas 2024). It remains to be seen whether the anisotropy enhancement due to scale-dependent, and therefore mass-dependent, screening processes during the formation of DM halos in MG models would break, or at least somehow modify, this cosmological universality property on the shape of DM halos Alimi & Koskas (2024).

Our paper is structured as follows. We first present (Sect. 2) a comprehensive review and interpretation of the findings reported by Alimi & Koskas (2024) concerning wCDM models from RG and we discuss the potential physical limitations of such cosmological universality beyond those models. We then present the theoretical framework of the HS theories and DUSTGRAIN-pathfinder simulations carried out by Giocoli et al. (2018), detailing the models simulated and the DM halo catalogs derivable from such simulations. In Sect. 3, we explore the shape properties of the DM halo populations in MG models in terms of mass and terms of the nonlinear fluctuations of the cosmic matter field. We discuss how MG models can break cosmological invariance observed on wCDM models in GR. We then estimate the scale at which universality could be broken analytically, precisely because of the screening effect. We then discuss that, although universality may be broken, the universal correspondence (according to cosmology) between the nonlinear fluctuations of matter within halos and the nonlinear fluctuations of all cosmic matter on the halo scale is preserved. In the final section, we summarize our main findings and their implications, and discuss future directions for this line of research.

2. Methodology

2.1. Motivations: Fundamental cosmological invariance of the shape properties of DM halos

Alimi & Koskas (2024) highlighted the existence of a fundamental cosmological invariance of the shape properties of DM halos in wCDM models from GR. Indeed, they first showed that shape parameters such as the prolaticity p, the triaxiality T, or the ellipticity E of a DM halo strongly depended on the mass of the DM halo and that such dependence was moreover highly sensitive to the underlying cosmology. However, when these shape parameters were re-expressed in terms of the nonlinear fluctuations of cosmic matter, this cosmological dependence completely disappeared.

The shape parameters p, T, E are defined from the eigenvalues a2/5, b2/5, c2/5 of the total inertia tensor ℳFoF:

M 1 ≤ i , j ≤ 3 FoF = ⟨ x i x j ⟩ − ⟨ x i ⟩ ⟨ x j ⟩ , $$ \begin{aligned} \mathcal{M} ^{\mathrm{FoF} }_{1\le i, j\le 3}=\langle x_ix_j\rangle - \langle x_i\rangle \langle x_j\rangle , \end{aligned} $$

where the averages are taken over all the particles included in the DM halo. The prolaticity of halos is

p = a − 2 b + c 2 ( a + b + c ) , $$ \begin{aligned} p=\frac{a-2b+c}{2(a+b+c)}, \end{aligned} $$

which is worth 0 for spheres, +∞ for cigar-shaped halos, and −∞ for pancake-like halos, while the triaxiality of halos,

T = a 2 − b 2 a 2 − c 2 , $$ \begin{aligned} T=\frac{a^2-b^2}{a^2-c^2}, \end{aligned} $$

tends to be 1 for cigar-shaped halos and 0 for pancake-like halos. Finally, the ellipticity,

E = a − c 2 ( a + b + c ) , $$ \begin{aligned} E=\frac{a-c}{2(a+b+c)}, \end{aligned} $$

quantifies the deviation from sphericity. The cosmological invariance has been highlighted from the DM halos catalog, built from the wCDM models’ Dark Energy Universe numerical simulation data (Alimi et al. 2010; Rasera et al. 2010; Alimi et al. 2012; Reverdy et al. 2015). In principle, as shown in Alimi & Koskas (2024), the calculation of the shape parameters p, T, E of DM halos should be carried out with caution to avoid potential numerical and methodological biases. Indeed, such biases can arise from a variety of sources, such as limited numerical resolution, particle sampling effects, halo search algorithms, or assumptions made in shape measurement techniques Zemp et al. (2011). To avoid such potential numerical effects, the set of particles taken into account when calculating the total inertia tensor must be carefully defined (Alimi & Koskas 2024). This is because, as the numerical resolution in mass increases, the substructures inside the halos are better resolved, and because they are generally off-center, they dominate the inertia tensor of the halo in which they are located. The shape of the halo is then artificially dominated by the shape of the substructures on its edge and, therefore, depends on the resolution. This effect can be very significant and can, if not corrected, distort the whole analysis. The correction process is explained in detail in Alimi & Koskas (2024). However, in this article, all the simulations used (see below) have the same resolution (and the same number of particles); consequently, the only remaining precaution to take when measuring the shapes of the DM halos is to consider their ellipsoidal fit, deduced from the diagonalization of the total inertia tensor of all the particles in the halo detected by the FOF algorithm (Alimi & Koskas 2024), which we did.

Giocoli et al. (2018) performed simulations for models of the same nature (where only the equation-of-state parameter w is modified) as those used in Alimi & Koskas (2024). Using the output data, we computed median triaxiality T (or prolaticity p, or ellipticity E) of DM halos of these models1, detected by the FOF algorithm. In Fig. 1, we plot T as function of the nonlinear fluctuations of the total cosmic matter field, smoothed over the mass scale of these halos. The wCDM models of DUSTGRAIN-pathfinder simulations are represented, including models with and without neutrinos. All curves superpose and show the invariance we highlighted in Alimi & Koskas (2024); this means that cosmological dependence of mass-shape relations is again completely encapsulated in the nonlinear power spectrum. We note that the presence of neutrinos does nothing to change the existence of this cosmological invariance. The results are analogous when we plot the ellipticity or prolaticity of the DM halos. It was also shown in Alimi & Koskas (2024) that cosmological invariance is verified either if the shape of the DM halos is expressed in terms of nonlinear fluctuations of the "full" cosmic matter field (smoothed on the halo scale) or in terms of nonlinear fluctuations of the cosmic matter field only within the DM halos. Again, this result is also very well verified here. When we re-express the shape parameters in terms of the nonlinear fluctuations of the cosmic matter only inside the halos, they appear invariant in the same way according to the cosmology. This latter property then induces the existence of a correspondence between the nonlinear fluctuations of the matter inside the DM halos and those of the total cosmic field as in Alimi & Koskas (2024); once again, this is insensitive to the underlying cosmology (see Sect. 3.2).

thumbnail Fig. 1.

Median triaxiality of the DM halos of the wCDM models from the DUSTGRAIN-pathfinder simulations (with or without neutrinos), as function of the non-linear fluctuations of the cosmic matter field, σNL. All the curves are superimposed, which means that the cosmological dependence of the mass-triaxiality relations is once again completely encapsulated in the non-linear power spectrum, as was highlighted in Alimi & Koskas (2024).

These first results obtained with the DUSTGRAIN-pathfinder simulations (Giocoli et al. 2018) also show that the invariance highlighted in Alimi & Koskas (2024) is robust when the numerical method is changed; that is, a different N-body solver is used: the Dark Energy Universe simulations were carried out using an optimized version of the RAMSES code (Teyssier 2002; Alimi et al. 2012), which is an AMR multigrid code. Conversely, the DUSTGRAIN-pathfinder simulations were carried out using an advanced version of the GADGET Tree code (Springel 2005; Puchwein et al. 2013).

In GR, under the effect of gravitation, the forming DM halos tend to become more and more spherical over time. However, along their formation, their shape properties stay independent of cosmology when expressed in terms of the nonlinear fluctuations of the cosmic field. Our goal is to elucidate the fate of this invariance when the gravitational collapse is modified as expected in a theory of MG. Indeed, the sphericization process during the gravitational collapse would be altered, because screening effects can influence the collapse differently along each axis during DM halo triaxial formation. We rely on the DM halos of HS cosmological models, as simulated by Giocoli et al. (2018).

2.2. Hu & Sawicki models in MG theories

In f(R) cosmologies, the behavior of matter is governed by revised Einstein field equations. A direct adaptation of GR that bypasses the cosmological constant, Λ, is achieved by incorporating a function of the curvature scalar into the Einstein-Hilbert action. As a result, a curvature-dependent term appears in the equations of motion, mimicking the effects of dark energy. In these models, the modified Einstein-Hilbert action can be written as

L HS = c 3 16 π G ∫ d 4 x − | g | ( R + f ( R ) ) , $$ \begin{aligned} \mathcal{L} _{HS}=\frac{c^3}{16\pi G}\int \mathrm{d} ^4x\sqrt{-|g|}\left(R+f(R)\right),\; \end{aligned} $$(1)

where g denotes the metric tensor, G is Newton’s gravitational constant, R is the curvature scalar, and f(R) is a scalar function responsible for an additional degree of freedom. For a comprehensive review of GR modifications, we advise consulting Clifton et al. (2012).

To ensure consistency between the different scales, HS considers a specific f function compatible with the strict Solar System constraints while accurately representing the cosmic dynamical expansion of the ΛCDM model. In the HS model, this function takes the following form:

f ( R ) = − m 2 c 1 ( R m 2 ) n c 2 ( R m 2 ) n + 1 . $$ \begin{aligned} f(R) = -m^2\frac{c_1\left(\frac{R}{m^2}\right)^n}{c_2\left(\frac{R}{m^2}\right)^n+1}. \end{aligned} $$(2)

Here, n, c1, and c2 are the model parameters. m2 ≡ ΩmH02, where Ωm is the current matter density parameter and H0 is the present Hubble parameter. In this paper, we consider n = 1. Hu & Sawicki (2007) showed that the constraints imposed by the cosmic background radiation and the late expansion of the Universe force the function f(R) to be:

f ( R ) = − 6 ( 1 − Ω m ) · R · ( R H 0 2 + | f R 0 | ( 12 − 9 Ω m ) 2 6 ( 1 − Ω m ) ) − 1 . $$ \begin{aligned} f(R) = {-6(1-\Omega _{m})\cdot R}\cdot \left(\frac{R}{H_0^2}+\frac{|f_{R0}|\left(12-9\Omega _{m}\right)^2}{6(1-\Omega _{m})}\right)^{-1}. \end{aligned} $$(3)

Therefore, compared to GR, the sole supplementary degree of freedom is fR0. It can be seen in the low curvature limit as the present-day characteristic amplitude of the dimensionless scalar field fR ≡ df/dR. The standard model is recovered when |fR0| tends toward 0.

From the revised Einstein field equations, one obtains the revised Friedmann equations (Song et al. 2007). When this HS f(R) theory is rewritten in the form of a massive scalar-tensor model with potential, an important aspect of such a model emerges: the mass of the dynamical scalar field varies with the local density (Hu & Sawicki 2007). This so-called chameleon mechanism (Khoury & Weltman 2004a,b; Brax et al. 2004) leads to the disappearance of MG effects in high-density regions so that gravity then behaves as GR. We cover this fundamental property and its consequences in more detail later.

Giocoli et al. (2018) performed the DUSTGRAIN-pathfinder simulation suite. These N-body simulations simulate a volume of (750 Mpc/h)3 with Np = 7683 particles (equivalent to a mass resolution of mp ≈ 8 ⋅ 1010M/h). The DUSTGRAIN-pathfinder simulation suite contains flat wCDM models (including ΛCDM concordance model) and HS-f(R) models, optionally including massive neutrinos. We now focus on the simulations where fR0 takes extreme values: successively 10−4 (fR4; high value) and 10−6 (fR6; low value). The other standard cosmological parameters are consistent with the Planck-2015 cosmological data: Ωm = 0.31345, Ωb = 0.0481, H0 = 67.31, km/s/Mpc, ns = 0.9658, and σ8 = 0.847. These simulations were carried out using the MG-gadget simulation code (Puchwein et al. 2013). For an exhaustive overview of these simulations, we refer the reader to Giocoli et al. (2018), Hagstotz et al. (2019a,b), Lee et al. (2023).

First, we discuss the impact of deviations from GR on the formation of cosmic structures through the properties of the DUSTGRAIN-pathfinder power spectra.

Using MGCAMB (Lewis & Challinor 2011; Zhao et al. 2009; Hojjati et al. 2011; Zucca et al. 2019; Wang et al. 2023), we computed the linear spectra of the cosmic matter field for these models at z = 0, denoted PL, and the variance of these linear power spectra:

σ L 2 ( M ) = ∫ 0 + ∞ k 2 P L ( k ) W 2 [ k · ( 3 M 4 π Ω m ρ c ) 1 / 3 ] d k 2 π 2 , $$ \begin{aligned} \sigma ^2_L(M) = \int _{0}^{+\infty }k^2 P_L(k) W^2\left[k\cdot \left(\frac{3M}{4\pi \Omega _m\rho _c}\right)^{1/3}\right]\;\frac{\mathrm{d} k}{2\pi ^2}, \end{aligned} $$(4)

where W is a Gaussian window function: W ( x ) = exp − x 2 10 $ W(x) = \exp -\frac{x^2}{10} $. In fact, σL2(M) measures the level of the collapse of the cosmic field on mass scale M compared to the average mass one expects for a homogeneous field (4πΩmρcR3/3). PL and σL are represented by dotted lines in Fig. 2. We observe that increasing fR0 increases the variance and hence the clustering in the Universe. This is particularly true when we focus on the low-mass range.

thumbnail Fig. 2.

Root-mean-square of nonlinear fluctuations of cosmic matter field as function of the smoothing mass M. The linear fluctuations σL (computed with MGCAMB) are represented by dashed lines, while the solid curves correspond to the σNL, the nonlinear fluctuations of cosmic matter field (computed from the z = 0 snapshots of the DUSTGRAIN-pathfinder simulations). The internal panel contains the corresponding linear and nonlinear power spectra.

We proceeded in the same way for the nonlinear power spectrum, PNL(k); PNL is calculated from the particle positions of the last snapshot (z = 0) of the simulations, by applying a CIC interpolation scheme to a 35003 grid of voxels. The PNLs are represented by solid lines in the inner panel of Fig. 2. We note that the nonlinear dynamics induces much stronger clustering than the linear dynamics; indeed, P NL Λ CDM ( k = 1 ) $ P_{NL}^{\mathrm{\Lambda CDM}}(k=1) $ is four times higher than P L Λ CDM ( k = 1 ) $ P_{L}^{\mathrm{\Lambda CDM}}(k=1) $, and, similarly, σ NL Λ CDM ( M = 10 14 ) ≈ 1.2 σ L Λ CDM ( M = 10 14 ) $ \sigma_{NL}^{\mathrm{\Lambda CDM}}(M=10^{14})\approx 1.2 \sigma_{L}^{\mathrm{\Lambda CDM}}(M=10^{14}) $. For large masses (M > 1015 M⊙/h) or large scales (k < 0.2 h/Mpc), the linear and nonlinear regimes merge as expected. In the DUSTGRAIN-pathfinder simulations suite, the initial power spectra are the same in all models. The ratio PL/PLΛCDM then gives the growth rate of each model relatively to the one of the ΛCDM model. These growth rates are scale-dependent in MG models, and their main feature is the increase in the gravitational constant (Geff = 4G/3) at the smallest spatial scales. The collapse of low-mass halos is therefore enhanced when the amplitude of the scalar field greatly exceeds that of the Newtonian potential. It should also be noted that, when neutrinos are included, they tend to homogenize the distribution of cosmic matter and therefore attenuate this accentuation of the gravitational interaction. For instance, the linear spectrum of fR4 is 60% larger than in the concordance model, whereas fR4-0.3eV only shows an increase of 30% (at k = 1 h/Mpc). However, such linear enhancements combine with the effects of nonlinear dynamics. To distinguish between the contributions of the linear dynamics and those of the nonlinear dynamics due specifically to the change in gravity, we compute, for every model, the pure, nonlinear power spectrum 𝒫 = PNL/PL and, similarly, the pure, nonlinear variance 𝒮 = σNL/σL. The ratio of these last quantities to their value for ΛCDM model, the pure nonlinear boost, is plotted in Fig. 3 for each cosmological model. This gives, for each fR0, the gain in power brought by nonlinear dynamics compared to that obtained in the concordance model. We observe that, for the MG models, the purely nonlinear spectrum quickly decreases and is 20% weaker than that of ΛCDM as early as k = 1 h/Mpc. At the smallest scales or masses, the collapse of cosmic matter generates high-density environments, as the mass of the dynamic scalar field is then very high - this leads it to reach the minimum of the potential in which it evolves. Effects of chameleon-field existence therefore vanish and gravity mimics the Newtonian gravitation (Geff = G); the nonlinear power spectrum no longer carries the accentuated effect of MG, whereas, at the same scale, the linear spectrum keeps doing so. As a result, the screening effect mitigates the pure, nonlinear boost at high k (small scale or small mass). This would explain why the nonlinear boost decreases sharply at large k and why such a decrease is initiated on a scale that depends on the chosen fR0. As the DUSTGRAIN-pathfinder simulations are also designed to study the degeneracy between gravity modification and neutrino mass, we refer the reader to Giocoli et al. (2018) for further details on their joint effects on the power spectrum.

thumbnail Fig. 3.

Purely nonlinear variance relatively to ΛCDM one as function of the smoothing mass M. For a given model of gravity, the purely nonlinear variance is defined as 𝒮 = σNL/σL. In the internal panel, we similarly plot, for each model, the purely nonlinear power spectrum 𝒫 = PNL/PL relatively to the ΛCDM one.

2.3. DM halo datasets of the DUSTGRAIN-pathfinder simulations

Beyond the analysis of the power spectrum of the cosmic density field, we now study the shape of the halos and in particular how the cosmological invariance highlighted in the wCDM models from GR (Alimi & Koskas 2024) can be altered by a non-vanishing fR0; this enables us to go further in our understanding of the impact of the chameleon screening.

To do so, we built halo catalogs from DUSTGRAIN-pathfinder simulations with or without massive neutrinos. The halos were identified using the friends-of-friends (FOF) algorithm, taking the percolation parameter b = 0.2. We chose to stay consistent with the convention adopted in Giocoli et al. (2018), in which massive neutrinos were ignored during the FoF detection. This is justified in Castorina et al. (2014) by the fact that the very large free streaming of massive neutrinos can spuriously link them to CDM particle groups.

Table 1 features these catalogs’ properties for every simulation used in this article. From these new numerical datasets, we tested the possible extension of the cosmological invariance of the distribution of the DM halos shape in terms of the nonlinear fluctuations of the cosmic matter field, particularly for Hu & Sawicki MG models.

Table 1.

Parameters of DUSTGRAIN-pathfinder simulations and the main features of the associated halo catalogs.

3. Triaxiality of DM halos in MG models

3.1. Halo shape measurement

As all the numerical simulations examined in this article share the same resolution (the same number of particles and box size), it is sufficient to incorporate all the particles associated with a given FOF halo in the mass tensor (whose eigenvalues give an ellipsoidal fit of the considered halo). The precautions taken to evaluate the shape of the halos concerning substructure removal would have been necessary if our simulations had been of different resolutions (Alimi & Koskas 2024), but this is not required here.

For the mass interval [1013,1014.5] M⊙/h, in Figs. 4, 5, and 6, we plot the median values of the shape parameters (respectively, T, P, and E) as function of their mass. In the same way as for models of the wCDM type, from Dark Energy Universe simulations (Alimi & Koskas 2024) or from simulations of the DUSTGRAIN-pathfinder suite (see Sect. 2.1), these shape parameters increase with the mass of the DM halo. The lighter the DM halo, the more spherical and flattened it tends to be. The uncertainties in the measurements of the shape parameters of DM halos are tiny, and, consequently, the error bars that could be added to the figures in this article are almost indistinguishable from the line thickness. Indeed, two types of error bars can be distinguished; the first is statistical. It is fixed by the number of halos from which the median of the shape parameters is calculated. This statistical error is negligible as the number of halos is very large in each catalog (> 250,000; see Table 1) in each mass bin. A second kind of uncertainty is systematic. It is fixed by the number of particles on which the inertia tensor (and therefore the shape parameters) of each DM halo are calculated. The selected DM halos contain several hundred particles, so this error is also tiny (cf. Appendix B in Alimi & Koskas (2024)). Therefore, possible uncertainties in shape measurements do not affect the results presented in this article.

thumbnail Fig. 4.

Median triaxiality of DM halos as function of their FoF mass. Each curve corresponds to a given gravitational and cosmological model, with or without massive neutrinos. The most massive halos are the most prolate, and the higher the σ8, the more oblate the halos.

thumbnail Fig. 5.

Median prolaticity of DM halos as function of their FoF mass.

thumbnail Fig. 6.

Median ellipticity of DM halos as function of their FoF mass. Again, DM halo ellipticity increases with mass and decreases with σ8. The DM halos become more and more spherical as they collapse.

The dependence of the shape properties on the mass of the DM halos is a consequence of the radial and isotropic nature of Newtonian gravitational collapse. In the mass domain considered here, the triaxiality increases by about 13% (from 0.73 to 0.82), the prolaticity doubles (i.e., from 0.030 to 0.060), and the ellipticity increases by 25%. Figures 4, 5, and 6 also show that, at a fixed mass, E, p, and T decrease as the σ8 parameter of the current cosmological model increases. The higher σ8, the more the structures are collapsed (at a given redshift) and, consequently, the more spherical they are. The corner piece of gravity modification effects on the halo shapes lies in the fact that the chameleon screening is effective at a given fR0-dependent mass, around which it can alter the gravitational collapse of DM halos. It indeed introduces an anisotropic component in the collapse. To quantify the impact of the latter, we study the shape dependence on cosmic matter field nonlinear fluctuations (rather than on the mass) and compare it to the dependence given by a Newtonian collapse.

3.2. Breaking of the cosmological invariance of DM halos shape in MG models.

We now consider cosmological models that vary the gravity parameter (fR0) as well as the potentially non-vanishing neutrino mass (mν). We want to know if the cosmological invariance of the shape parameters as function of the nonlinear fluctuations of the cosmic matter field highlighted in Alimi & Koskas (2024), and again found for the wCDM models of the DUSTGRAIN-pathfinder suite (Sect. 2.1), disappears for such models. In Fig. 7, we observe that replacing M by σNL(M) again compensates for the cosmological dependence of the triaxiality of the halos: all the curves merge, regardless of the specific cosmological and gravitational parameters, but there is one notable exception. While the fR6 curves, with or without massive neutrinos, indeed superpose well on the other curves at large masses (i.e., at large values of log σ NL − 1 $ \log\sigma_{NL}^{-1} $), they differ from the others if log σ NL − 1 < − 0.15 $ \log\sigma_{NL}^{-1} < -0.15 $. This corresponds for these two models to a typical mass of 1014 M⊙/h. Prolaticity and ellipticity measurements show similar behaviors.

thumbnail Fig. 7.

Median triaxiality of DM halos as function of nonlinear fluctuations of cosmic matter field, σNL. All the curves superpose, which means that the cosmological dependence of the mass-triaxiality relations is completely encapsulated in the nonlinear power spectrum. However, when |fR0| = 10−6, whatever the neutrino mass, the relation between σNL and triaxiality does not identify with the relation obtained for the other models. This occurs specifically from log ( σ NL − 1 ) ≈ − 0.1 $ \log(\sigma_{NL}^{-1})\approx-0.1 $, corresponding to a mass of 1014 h/M⊙ for the fR6 model, which is, for this model, the characteristic mass where the screening starts. The way the halos collapse and sphericize therefore depends on the gravity model. The screening phenomenon modifies the evolution of the shape of the DM halos, which can no longer be encapsulated in the power spectrum. The divergence in the σNL − T relations is then a distinctive marker of the change in the gravity model, which nevertheless remains independent of the neutrino mass.

It seems that the origin of this invariance breaking can be backtracked to the halo collapse process. Indeed, since the gravitational isopotentials of a given ellipsoidal distribution have the remarkable property of being systematically more spherical than the isodensities that give rise to them (Llinares & Mota 2013; Stapelberg et al. 2022), the halos become more and more spherical as their collapse progresses. Meanwhile, as the halos collapse, the cosmic matter field becomes more structured, and its nonlinear fluctuations increase, so that there necessarily exists a relationship (independent of cosmology) between the triaxiality (or prolaticity or ellipticity) of halos of mass M on the one hand, and the nonlinear fluctuations of the cosmic matter field measured here with σNL(M) on the other hand. Adding a screening mechanism will change the picture since it is more or less effective, depending on the potential well depth and (therefore) the halo mass; it will consequently lead to a change in the sphericization process of the halos.

We distinguish three mass regimes for the screening mechanism. High-mass halos, whose absolute Newtonian potential GM/(Rc2) is much larger than |fR0|, are completely screened. Gravity is then indistinguishable from Newtonian gravitation. Such halos become more and more spherical with time, and no significant difference in their shape is expected compared to the halos accreted in a purely Newtonian context. The relation between T and σNL for these halos is then unchanged compared to the one we obtained for the wCDM halos (which assume GR). On the contrary, very low-mass halos, whose potential is low compared to |fR0|, were not (or very slightly) screened. Gravity behaves again like Newtonian gravitation, with a modified effective gravitational constant of Geff ≃ 4G/3. As such a modification of gravity remains perfectly isotropic, the invariance of the shape properties as function of nonlinear fluctuations of the cosmic field is again preserved; In reality, halos sphericize faster, but the cosmic matter field clusters faster too.

In these two cases, the observed cosmological dependence shape according to the mass is then fully absorbed into the power spectrum of the cosmic matter field in which the DM halos formed. In other words, for these halos, the relationship between σNL and T remains identical to that observed in the wCDM models from GR.

The situation is more subtle when the (Newtonian) potential of the halos is comparable, in terms of absolute value, to fR0. As Llinares & Mota (2013) pointed out, these halos can be screened along a single principal axis of their ellipsoid and not be screened along the other axes. In MG, such halos can then become more elliptical along the axis where the screening mechanism was effective. In other words, as the chameleon field is generally as elliptical as the isodensities, its presence tends to alter the efficiency of the "sphericization" expected in Newtonian gravity; the ellipsoidal matter distribution of such halos, therefore, tends to sphericize over time less than is expected in purely Newtonian gravity. At constant density, the critical potential (of the order of fR0) can be recast as a critical mass (Mc). The triaxiality, the prolaticity, or the ellipticity of the DM halos whose σNL is close to σNL(Mc) will be higher in MG than for DM halos of the same σNL in the wCDM models.

To find an explicit expression of Mc, one can use the fact that a DM halo is screened as soon as (Baker et al. 2021)

GM R c 2 ≥ 3 2 | f R 0 | . $$ \begin{aligned} \frac{GM}{Rc^2}\ge \frac{3}{2}| f_{R0}|. \end{aligned} $$(5)

As the average spherical density of FoF halos (with b = 0.2) is generally around 200ρm, which gives

M R ≥ M / ( 3 4 π M 200 ρ m ) 1 / 3 , $$ \begin{aligned} \frac{M}{R}\ge M/\left(\frac{\frac{3}{4\pi } M}{200\rho _m}\right)^{1/3}, \end{aligned} $$

Equation (5) becomes

G M 2 / 3 ≥ 3 c 2 2 | f R 0 | ( 3 800 π ρ m ) 1 / 3 . $$ \begin{aligned} GM^{2/3}\ge \frac{3c^2}{2}| f_{R0}| \left(\frac{3}{800\pi \rho _m}\right)^{1/3}. \end{aligned} $$

Hence, the typical mass of the screened halos is

M c = ( 3 c 2 2 G | f R 0 | ) 3 / 2 ( 3 800 π ρ m ) 1 / 2 . $$ \begin{aligned} M_c=\left(\frac{3c^2}{2G}| f_{R0}|\right)^{3/2}\left(\frac{3}{800\pi \rho _m}\right)^{1/2}. \end{aligned} $$(6)

Assuming Ωm = 0.31, we obtain

M c = | f R 0 | 3 / 2 ( 2.1 · 10 22 M ⊙ / h ) . $$ \begin{aligned} M_c=| f_{R0}|^{3/2}\left(2.1\cdot 10^{22} \,M_\odot /h\right). \end{aligned} $$(7)

For the fR4 model, Mc = 2.1 ⋅ 1016 M⊙/h. For such a model, all the DM halos have a mass well below the critical mass at which the screening effect becomes effective. The halos are then said to have low masses, and the (σNL − T) curve of fR4 merges with that of the models derived from GR (Fig. 8). For the fR6 model, on the other hand, since Mc = 2.1 ⋅ 1013 M⊙/h, the DM halos’ masses are comparable to this critical mass, and they are only likely to be screened during their formation along some of their principal axes (Llinares & Mota 2013). The sphericization process is then altered, and the σNL − T invariance is broken around such a critical mass as that observed in Fig. 7 (i.e., for halos with a mass lower than about 1014 M⊙/h). Above this mass, for more massive halos that are completely screened, the invariance is again well verified (Fig. 7). The modifications of the sphericization process due to the screening effect, according to the value of fR0 and the mass of the DM halos, are summed up in the diagram of Fig. 8.

thumbnail Fig. 8.

Sphericization process of DM halos during their formation, according to their mass and the screening effect of MG models. The mass range of the DM halos detected in the numerical simulation is delimited by the red rectangle. The a-sphericization process affects DM halos around the critical mass depending on the present-day characteristic amplitude of the dimensionless scalar field fR0 of each model (see Eq. (7)). These DM halos are indeed present for the fR6 numerical simulation, but not the fR4 numerical simulation.

Figure 7 shows the relation between triaxiality and nonlinear fluctuations of the cosmic matter field for multiple values of fR0 and mν. The results for prolaticity and ellipticity flow in the same direction. Although our interpretation is confirmed by these results, it cannot be considered sufficient. Indeed, the combined effect of gravitational collapse and the shielding effect due to the chameleon scalar field in modified-gravity models, and how it modifies the sphericization process of the forming halo, is obviously very complex. This combined effect may evolve and even possibly reverse during the formation of the halo, favoring either an a-sphericization or a sphericization. To improve the simplistic and binary nature of our interpretation, we would need a larger number of high-resolution numerical simulations with a wide range of well-sampled fR0 values and a wider range of DM halo masses, screened or not. Such simulations are not yet available, so we defer such a detailed analysis to future work. However, we believe that the interpretation we are proposing, although binary, is a good first proposal compatible with the given data. As observed in Fig. 7, it is worth noticing that the presence of massive neutrinos does not change the previous conclusions: the σNL − T curves of the same gravity models, with or without massive neutrinos, are systematically merged.

The σNL-shaped curves thus become a probe of MG at the limit of the mass domain where the screening effect is initiated. The joint measurement of the nonlinear power spectrum of a given cosmic matter field, on the one hand, and of the shape of the halos it contains, on the other, will then make it possible to interpret any break in the curve, known to be universal for the classical gravity cosmological models, as the manifestation of a change in gravity. The scale of the nonlinear fluctuations where the invariance break occurs also provides an estimate of fR0.

If the shape of the halos is, by definition, calculated on the halos themselves (practically meaning from the particles that make up the halos) and independently of their environment, it would then be logical to only compute the nonlinear fluctuations on the matter inside the halos, rather than the total cosmic matter field. Let σNL, IH denote the nonlinear fluctuations of the cosmic matter inside the halos:

σ N L , I H 2 ( M ) = ∫ R + k 2 P N L , I H ( k ) W 2 [ k · ( 3 M 4 π Ω m ρ c ) 1 3 ] d k 2 π 2 , $$ \begin{aligned} \sigma ^2_{NL, IH}(M) = \int _{\mathbb{R} _+}k^2 P_{NL, IH}(k) W^2\left[k\cdot \left(\frac{3M}{4\pi \Omega _m\rho _c}\right)^{\frac{1}{3}}\right]\;\frac{\mathrm{d} k}{2\pi ^2}, \end{aligned} $$

where the power spectrum PNL, IH(k) is computed by taking into account only the particles belonging to the halos. In Fig. 9, we plot the median triaxiality of the halos in bins of log σ N L , I H − 1 $ \log\sigma_{NL, IH}^{-1} $. The results are quite similar to those obtained with log σ NL − 1 $ \log\sigma_{NL}^{-1} $ - the curves merge for all cosmological models (Fig. 9), whereas the invariance is again broken at low masses for fR6 halos (Fig. 9).

thumbnail Fig. 9.

Median triaxiality of halos as function of variance of non-linear fluctuations of cosmic matter internal to DM halos. All the curves superpose, meaning again that the cosmological dependence of mass-triaxiality relations is completely encapsulated in the nonlinear power spectrum. However, when |fR0| = 10−6, independently of the neutrino mass, the median relation between variance and triaxiality does not identify with the relation obtained for the other models.

The fact that, at high mass, there is a universal relationship between T and σNL on the one hand and between T and σNL, IH on the other, must imply a correspondence (independent of cosmology and the gravity model) between σNL and σNL, IH in the domain corresponding to high mass, similarly to the correspondence highlighted in Alimi & Koskas (2024). To verify this and to find out what happens at lower masses, in Fig. 10 we plot σNL(M) in terms of σNL, IH(M) for each simulation and the entire mass domain covered by the simulation halos (i.e., [1013, 1014.5] M⊙/h). We find that over the whole mass domain, even when including Mc, the correspondence between σNL and σNL, IH is well verified, independently of the cosmology and gravity model.

thumbnail Fig. 10.

Scatter plot of (σNL, IH,σNL) for every value of fR0 and Mν. There is manifestly a correspondence between those two kinds of nonlinear variance, respectively computed from the matter inside the halos and from the whole cosmic matter field. this correspondence seems to be universal, independently of the cosmological parameters and the gravity model.

4. Discussion and conclusion

In this work, we investigated the effect of MG on the shape of DM halos in the special case of the Hu-Sawicki modified-gravity models. We show that the variance of the nonlinear power spectrum, smoothed over the scale of the halos, completely determines the triaxiality of the halos whatever the cosmological model, except the light halos (mass less than 1014 M⊙/h) for the fR0 = −10−6 models from DUSTGRAIN-pathfinder simulation data. We propose the following interpretation of this result: the higher the nonlinear variance smoothed on the scale of the halos, the more structured the corresponding Universe is and the more its halos are collapsed. Under the effect of gravitation, the halos become spherical as they collapse. This confirms the existence of a relationship between the degree of structuring of the Universe (a measure of which is given by the variance of the power spectrum) and the triaxiality of the halos it contains. However, this interpretation of a universal relationship independent of the cosmology, between the power spectrum variance and the triaxiality of DM halos, is broken if we compare different models of MG (and therefore different sphericization processes). The universality relationship linking nonlinear fluctuations of cosmic matter field and shapes of DM halos must finally be understood "for a given gravity model". In the specific context of Hu-Sawicki models, it results from the phenomenon of screening inherent in these models, with a critical mass dependent on ρm and especially the parameter controlling the deviation from GR |fR0|. DM halos whose mass is then close to this critical mass are likely to be screened (by a chameleon mechanism) along one of their principal axes only so that the total gravity field is much more elliptical than the Newtonian isopotentials. The sphericization process is then altered, while the power spectrum (and therefore its variance), which is averaged over the angles, ignores the information linked to the induced anisotropies of this type. Thus, for a given parameter |fR0|, there is still a relationship between the nonlinear fluctuations of cosmic matter field and triaxiality (or any other shape parameter) of the DM halos, which does not depend on the other cosmological parameters (neutrino mass, ΩM, σ8, w0...). However, this relationship differs from that found in pure wCDM models in the vicinity of this critical mass, and the simultaneous measurement of the power spectrum and shape of the halos thus becomes an innovative probe of MG.

It then becomes exciting to study the effects of other MG models on the shape of the halos, with different screening mechanisms. Examples include symmetron screening and nDGP screening. In addition, any change including the nature of DM could also, in principle, be likely to leave a visible imprint on the shape of the halos. The formation of hot DM halos or fuzzy halos differs greatly from those observed in the case of cold DM. This could lead to the relationship between the shape of the halos and the power spectrum being given a fundamental nature, with regard to which any deviation could be seen as a powerful probe of a change in gravity or the nature of DM.

1

All results presented in this paper for median quantities are also recovered for mean quantities.

Acknowledgments

We would like to thank Dr Philippe Brax for fruitful discussions.

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Appendix A: Broken degeneracy on δc

The models we study in this paper admit very different critical spherical overdensities δc at z = 0 (see Table A.1). It is therefore necessary to test whether an invariance in terms of the variables νL(M, z) = δc(z = 0)/σL(M, z) or νNL(M, z) = δc(z = 0)/σL(M, z) that combine both the cosmic matter field fluctuations and the δc effect exists.

Table A.1.

Critical spherical overdensities δc at z = 0 for cosmological models with or without MG.

We plot respectively in Figs. A.1 and A.2 the median triaxiality of DM halos as function of νL and νNL. We see clearly that the cosmological invariance is not recovered. There is no invariance in terms of νL and we observe no rapprochement of the curves when we consider νNL. It is uniquely the re-expression of the shape properties of the DM halos in terms of the non-linear fluctuations of the cosmic matter field that allows the highlighting of a cosmological invariance for a given theory of gravity. The inclusion of massive neutrinos has no bearing on this question.

thumbnail Fig. A.1.

Triaxiality of DM halos at z = 0 as function of the variable νL(M, 0) = δc(z = 0)/σL(M, 0) which takes into account the explicit dependence on the critical spherical overdensities δc, of each model. We do not observe any full cosmological invariance.

thumbnail Fig. A.2.

Triaxiality of DM halos at z = 0 as function of the variable νNL(M, 0) = δc(z = 0)/σNL(M, 0) which takes into account the explicit dependence on the critical spherical overdensities δc, of each model. We note the universality for the cosmological models with the same δc(z = 0) parameter, but We observe no full cosmological invariance.

All Tables

Table 1.

Parameters of DUSTGRAIN-pathfinder simulations and the main features of the associated halo catalogs.

In the text
Table A.1.

Critical spherical overdensities δc at z = 0 for cosmological models with or without MG.

In the text

All Figures

thumbnail Fig. 1.

Median triaxiality of the DM halos of the wCDM models from the DUSTGRAIN-pathfinder simulations (with or without neutrinos), as function of the non-linear fluctuations of the cosmic matter field, σNL. All the curves are superimposed, which means that the cosmological dependence of the mass-triaxiality relations is once again completely encapsulated in the non-linear power spectrum, as was highlighted in Alimi & Koskas (2024).
In the text
thumbnail Fig. 2.

Root-mean-square of nonlinear fluctuations of cosmic matter field as function of the smoothing mass M. The linear fluctuations σL (computed with MGCAMB) are represented by dashed lines, while the solid curves correspond to the σNL, the nonlinear fluctuations of cosmic matter field (computed from the z = 0 snapshots of the DUSTGRAIN-pathfinder simulations). The internal panel contains the corresponding linear and nonlinear power spectra.
In the text
thumbnail Fig. 3.

Purely nonlinear variance relatively to ΛCDM one as function of the smoothing mass M. For a given model of gravity, the purely nonlinear variance is defined as 𝒮 = σNL/σL. In the internal panel, we similarly plot, for each model, the purely nonlinear power spectrum 𝒫 = PNL/PL relatively to the ΛCDM one.
In the text
thumbnail Fig. 4.

Median triaxiality of DM halos as function of their FoF mass. Each curve corresponds to a given gravitational and cosmological model, with or without massive neutrinos. The most massive halos are the most prolate, and the higher the σ8, the more oblate the halos.
In the text
thumbnail Fig. 5.

Median prolaticity of DM halos as function of their FoF mass.
In the text
thumbnail Fig. 6.

Median ellipticity of DM halos as function of their FoF mass. Again, DM halo ellipticity increases with mass and decreases with σ8. The DM halos become more and more spherical as they collapse.
In the text
thumbnail Fig. 7.

Median triaxiality of DM halos as function of nonlinear fluctuations of cosmic matter field, σNL. All the curves superpose, which means that the cosmological dependence of the mass-triaxiality relations is completely encapsulated in the nonlinear power spectrum. However, when |fR0| = 10−6, whatever the neutrino mass, the relation between σNL and triaxiality does not identify with the relation obtained for the other models. This occurs specifically from log ( σ NL − 1 ) ≈ − 0.1 $ \log(\sigma_{NL}^{-1})\approx-0.1 $, corresponding to a mass of 1014 h/M⊙ for the fR6 model, which is, for this model, the characteristic mass where the screening starts. The way the halos collapse and sphericize therefore depends on the gravity model. The screening phenomenon modifies the evolution of the shape of the DM halos, which can no longer be encapsulated in the power spectrum. The divergence in the σNL − T relations is then a distinctive marker of the change in the gravity model, which nevertheless remains independent of the neutrino mass.
In the text
thumbnail Fig. 8.

Sphericization process of DM halos during their formation, according to their mass and the screening effect of MG models. The mass range of the DM halos detected in the numerical simulation is delimited by the red rectangle. The a-sphericization process affects DM halos around the critical mass depending on the present-day characteristic amplitude of the dimensionless scalar field fR0 of each model (see Eq. (7)). These DM halos are indeed present for the fR6 numerical simulation, but not the fR4 numerical simulation.
In the text
thumbnail Fig. 9.

Median triaxiality of halos as function of variance of non-linear fluctuations of cosmic matter internal to DM halos. All the curves superpose, meaning again that the cosmological dependence of mass-triaxiality relations is completely encapsulated in the nonlinear power spectrum. However, when |fR0| = 10−6, independently of the neutrino mass, the median relation between variance and triaxiality does not identify with the relation obtained for the other models.
In the text
thumbnail Fig. 10.

Scatter plot of (σNL, IH,σNL) for every value of fR0 and Mν. There is manifestly a correspondence between those two kinds of nonlinear variance, respectively computed from the matter inside the halos and from the whole cosmic matter field. this correspondence seems to be universal, independently of the cosmological parameters and the gravity model.
In the text
thumbnail Fig. A.1.

Triaxiality of DM halos at z = 0 as function of the variable νL(M, 0) = δc(z = 0)/σL(M, 0) which takes into account the explicit dependence on the critical spherical overdensities δc, of each model. We do not observe any full cosmological invariance.
In the text
thumbnail Fig. A.2.

Triaxiality of DM halos at z = 0 as function of the variable νNL(M, 0) = δc(z = 0)/σNL(M, 0) which takes into account the explicit dependence on the critical spherical overdensities δc, of each model. We note the universality for the cosmological models with the same δc(z = 0) parameter, but We observe no full cosmological invariance.
In the text

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