The Cluster Mass Function and the $\sigma_8$-tension

We use a large set of halo mass function (HMF) models in order to investigate their ability to represent the observational Cluster Mass Function (CMF), derived from the $\mathtt{GalWCat19}$ cluster catalogue, within the $\Lambda$CDM cosmology. We apply the $\chi^2$ minimization procedure to constrain the free parameters of the models, namely $\Omega_m$ and $\sigma_8$. We find that all HMF models fit well the observational CMF, while the Bocquet et. al. model provides the best fit, with the lowest $\chi^2$ value. Utilizing the {\em Index of Inconsistency} (IOI) measure, we further test the possible inconsistency of the models with respect to a variety of {\em Planck 2018} $\Lambda$CDM cosmologies, resulting from the combination of different probes (CMB - BAO or CMB - DES). We find that the HMF models that fitted well the observed CMF provide consistent cosmological parameters with those of the {\em Planck} CMB analysis, except for the Press $\&$ Schechter, Yahagi et. al., and Despali et. al. models which return large IOI values. The inverse $\chi_{\rm min}^2$-weighted average values of $\Omega_m$ and $\sigma_8$, over all 23 theoretical HMF models are: ${\bar \Omega_{m,0}}=0.313\pm 0.022$ and ${\bar \sigma_8}=0.798\pm0.040$, which are clearly consistent with the results of {\em Planck}-CMB, providing $S_8=\sigma_8\left(\Omega_m/0.3\right)^{1/2}= 0.815\pm 0.05$. Within the $\Lambda$CDM paradigm and independently of the selected HMF model in the analysis, we find that the current CMF shows no $\sigma_8$-tension with the corresponding {\em Planck}-CMB results.


INTRODUCTION
The current cosmic structure formation paradigm assumes that dark matter (DM) halos form via gravitational instabilities of a primordial density field.Halos follow a roughly hierarchical merging process where halos with small masses form first and then more massive halos form through merging (e.g., Springel 2005;Lacey & Cole 1993, 1994;Tormen et al. 2004).Galaxies and galaxy clusters, residing within these DM halos, arise from high peaks of an underlying initially Gaussian density fluctuation field (Kaiser 1984;White & Frenk 1991; Bardeen et al. 1986).Although the details of the formation of galaxies within DM halos and the various feedback mechanisms are not yet fully understood, one can obtain insight into structure formation by studying the properties of DM halos and their evolution.The Halo Mass Function (HMF1 ), defined as the abundance of halos as a function of mass and redshift (e.g.Press & Schechter 1974;Sheth et al. 2001;Vogelsberger et al. 2014), is one of these properties that play an important role in observational cosmology.The HMF is particularly sensitive to the matter density of the universe, Ωm and the root mean square mass fluctuation at a scale of 8h −1 Mpc at z = 0 (e.g., Wang & Steinhardt 1998), as well as the evolution of bound structures from an early epoch to the late universe (Press & Schechter 1974;Bond et al. 1991;Knox et al. 2006)).
One of the biggest challenges in utilizing the CMF as a cosmological probe is the difficulty of measuring cluster masses accurately.A variety of methods have been used such as, the virial mass estimator (e.g., Binney & Tremaine 1987) under the assumption of virial equilibrium, weak gravitational lensing (e.g., Wilson et al. 1996;Holhjem et al. 2009), or the application of Euler's equation on X-ray images of galaxy clusters (e.g., Sarazin 1988), under the assumption of hydrostatic equilibrium and spherical symmetry.However, these methods are observationally rather expensive since they require high-quality data and are also uncertain owing to the necessary assumptions made.Cluster masses can also be indirectly inferred from other observables, the so called mass proxies, which scale rather tightly with cluster mass.Among these mass proxies are X-ray luminosity, temperature, the product of X-ray temperature and gas mass (e.g., Pratt et al. 2009;Vikhlinin et al. 2009a;Mantz et al. 2016), optical luminosity (e.g., Yee & Ellingson 2003), the velocity dispersion of member galaxies (e.g., Biviano et al. 2006;Bocquet et al. 2015), and cluster richness (e.g., Pereira et al. 2018;DES Col-laboration et al. 2020;Abdullah et al. 2022).However, mass proxies are also hampered by uncertainties and scatter which may also introduce systematics in the estimation of cluster masses.Furthermore, understanding the relationship between mass proxies and true mass is fundamental and using weak lensing mass calibration can aid to this effort (e.g., Bocquet et al. 2019).
On the other hand, the determination of the theoretical HMF is also quite challenging.There are many approaches to calculate the HMF either analytically or via N-body simulations.The analytical predictions of the HMF are hampered by the difficulties of the non-linear gravitational collapse which have not been fully addressed, except in the case of simple symmetries, while N-body simulations, in addition to being time consuming, do not provide a detailed understanding of the physical aspects of the mass accretion procedure.
Press & Schechter (1974) (P-S) provided the first model of a linear analytic form of the HMF, under the assumption of spherical collapse and a primordial Gaussian density field in which structures grow via small perturbations.Early numerical simulations showed good agreements with the P-S formalism (cf.Efstathiou et al. 1979).However, the advent of significant computer power and novel numerical techniques revealed deviations from the P-S formalism with overestimating/underestimating the number of halos at the high/low mass range (e.g.Efstathiou et al. 1988;Gross et al. 1998;Governato et al. 1999;Jenkins et al. 2001;White 2002), even if merging processes are included in the calculations (e.g.Bond et al. 1991;Lacey & Cole 1993).Since then many authors attempted to provide an accurate HMF by extending the P-S formalism or by using N-body simulations.We briefly present some of these HMFs below.Sheth & Tormen (1999) and Sheth et al. (2001) (S-T) used an ellipsoidal collapse model to overcome the inaccuracies of P-S HMF.Jenkins et al. (2001) found discrepancies with the S-T model and proposed an analytic fitting formula for the HMF.Their N-body simulations showed that the HMF was in agreement with the S-T analytic formula up to redshift z = 5 in the mass range from galaxies to clusters.Tinker et al. (2008) provided a HMF formula that improved previous approximations by 10 − 20% using a large set of collisionless cosmological simulations.Crocce et al. (2010) also proposed another HMF formula with an accuracy of 2% up to redshift z = 1.Klypin et al. (2011) used the Bolshoi simulation and found that although there was an agreement with the S-T model at low redshifts there were discrepancies at high redshifts.Also Bhattacharya et al. (2011) provided a HMF that was accurate to 2% at redshift z = 0 and 10% at redshift z = 2. Angulo et al. (2012) used the Millenium-XXL simulation to obtain a HMF which was accurate to better than 5% over the mass range used in this study.Comparat et al. (2017aComparat et al. ( , 2019) ) used the Multidark simulation to revisit the HMF which was accurate at < 2% level.Bocquet et al. (2016) used hydrodynamical and dark matter only simulations in order to calibrate the HMF.Additionally, Bocquet et al. (2020) constructed an emulator for estimating the HMF, which was better than < 2% for DM halo masses of 10 13 −10 14 h −1 M⊙ and < 10% for 10 15 h −1 M⊙.Recently, Shirasaki et al. (2021) used high-resolution N-body simulations and provided a new fitting formula for HMF.Their model appears accurate at a 5% level, except for the large masses at z ⩽ 7.They showed that the S-T model overestimates the halo abundance at z = 6 by 20 − 30%.
An important claim regarding the HMF is its apparent universality in the sense that the same HMF parameters fit different redshifts and cosmologies (e.g.Jenkins et al. 2001;Sheth & Tormen 1999;Reed et al. 2003;Warren et al. 2006;Watson et al. 2013).However, various authors have rejected this claim (e.g.White 2002;Reed et al. 2007;Crocce et al. 2010;Bhattacharya et al. 2011;Courtin et al. 2011;Tinker et al. 2008).Recently, Diemer (2020), studying the universality of the HMF, seem to confirm that for the ΛCDM and for a wide range of halo mass definitions there is a varying level of non-universality that increases with peak height and therefore also with cluster mass.However, they also found that splashback radius-based HMF is universal to 10% at z ⩽ 2 in agreement with some previous studies that adopted more traditional definitions of the halo radius (e.g. Warren et al. 2006) 2 Thus, it appears that the universality or non-universality of the HMF parameters depends on how halos are defined.Halos identified using the Friend-of-Friend algorithm (FoF) provide almost a universal HMF, while halos identified using the spherical overdensity (SO) approach returns non-universal HMF, especially at high redshifts.These discrepancies might be caused by the fact that the FOF algorithm links objects before they merge.
Although the ΛCDM model fits a wide variety of cosmological data with high accuracy, tensions have emerged between the values of the Hubble constant, H0, and the normalization of the matter power-spectrum, σ8 (ie., the root mean square amplitude of the matter fluctuations in spheres of 8h −1 Mpc) derived, within the ΛCDM paradigm, using the Planck CMB temperature fluctuation data (corresponding to redshifts z ∼ 1100)) and low-redshift (z ≲ 0.15) data (e.g.Cepheid-based distance indicator, cosmic shear, clusters of galaxies etc).
The tension in S8, together with that of the Hubble constant, H0, could be an indication for the necessity of a new cosmological paradigm, beyond the canonical ΛCDM.However, as far as the S8 tension is concerned, it is evident from the above that it shows up in some datasets and not in others, which hints towards unknown systematic errors entering in some analyses procedures.Amon & Efstathiou (2022) argue for a possible consistency of the weak-lensing analyses results with those of Planck CMB if the matter power spectrum is suppressed more strongly on non-linear scales.
In the present paper we compare a large set of 23 HMF models with the observationally determined CMF from Abdullah et al. (2020b), in order to explore the validity of these HMF models, within the ΛCDM cosmological background, and also to constrain the two free cosmological parameters of the models, namely, the matter density parameter, Ωm and the σ8 normalization factor.Recently, a different procedure was followed by Driver et al. (2022), who reconstructed the HMF through a model-free empirical approach using the GAMA galaxy group data, finding a good agreement with the expectations of the ΛCDM model.
The outline of this paper is as follows.In Section 2 we describe the CMF obtained from Abdullah et al. (2020b).In Section 3 we introduce the main elements of the theoretical HMF models.In Section 4 we present the results of our statistical analysis regarding the cosmological parameters Ωm,0 and σ8, while in Section 5 we draw our conclusions.Throughout this work we use, when necessary, the value of h = H0/100 = 0.678.We have, however, verified that our results are insensitive to the exact value of h within the current range defined by the so-called Hubble constant tension.

CLUSTER OF GALAXIES MASS FUNCTION -CMF
There have been various attempts to derive the observational CMF either using the 2PIGG galaxy groups of the 2 degree Field Galaxy Redshift Survey by Eke et al. (2006), the SDSS-DR10 galaxy groups by Tempel et al. (2014a), the ROSATbased REFLEX-II X-ray cluster catalogue by Böhringer et al. (2017), or recently the GAMA galaxy group data by Driver et al. (2022).
In our current work we use the recent SDSS-DR13 (Albareti et al. 2017) GalWCat19 cluster catalogue (Abdullah et al. 2020a).Using photometric and spectroscopic databases from SDSS-DR13, Abdullah et al. (2020a) extracted data for 704,200 galaxies.These galaxies satisfied the following set of criteria: spectroscopic detection, photometric and spectroscopic classification as a galaxy (by the automatic pipeline), spectroscopic redshift between 0.01 and 0.2 (with a redshift completeness >0.7; Yang et al. 2007;Tempel et al. 2014b), r-band magnitude (reddening-corrected) <18, and the flag SpecObj.zWarninghaving a value of zero, indicating a wellmeasured redshift.Abdullah et al. 2020a identified galaxy clusters utilizing the Finger-of-God effect.Then, they assigned cluster membership using the GalWeight technique (Abdullah et al. 2018).The GalWeight technique has been tested with N-body simulations (MDPL2 and Bolshoi) and it has been shown that it is 98% accurate in assigning cluster membership.Additionally, the masses of each cluster were calculated by applying the virial theorem (e.g.Binney & Tremaine 1987;Abdullah et al. 2011) and correcting for the surface pressure term (The & White 1986).The virial theorem avoids any assumption about the internal physical processes associated with baryons (gas and galaxies).More specifically, the cluster mass was calculated at the virial radius at which the density equals ρ = ∆200ρc, where ρc is the critical density of the universe and ∆200 = 200.The uncertainty of the virial mass is calculated using the limiting fractional uncertainty = (1/π) 2 ln N/N (Bahcall & Tremaine 1981).The systematic uncertainties in the computation of mass are due to the: (i) assumption of virial equilibrium, projection effects, velocity anisotropies in galaxy orbits, etc, (ii) presence of substructure and/or nearby structures, (iii) presence of interlopers in the cluster frame, (iv) uncertainties in the identification of the cluster center.
The GalWCat19 cluster catalogue is incomplete in redshift at z > 0.085 (or comoving distance D > 265h −1 ) and to correct for this incompleteness, each cluster with a comoving distance D > 265h −1 Mpc is weighted by the inverse of the selection function S(D), estimated in Abdullah et al. (2020b) by using the normalized cluster number density and fitting it to an exponential function.The final selection function used is given by: where a = 1.1 ± 0.12, b = 293.4± 20.7h −1 Mpc and γ = 2.97±0.9.Systematic effects due to the uncertainties in the selection function were investigated by Abdullah et al. (2020b) to conclude that they leave mostly unaffected the main results of the analysis.Furthermore, they studied the effect of the inverse selection function weighting at larger redshifts, where the cluster sampling is low and the shot-noise effects large, an analysis which resulted in identifying the optimum redshiftrange within which the possible overcorrection is minimized.The CMF mass function is estimated using: where Di is the comoving distance and V is the comoving volume given by: where Ω sky = 41.253deg 2 , Ωsurvey = 11.000deg 2 and D1, D2 are the minimum and maximum comoving distances of the sample.
It is worth mentioning that the GalWCat19 catalogue is one of the largest available spectroscopic cluster samples for which the determination of the cluster center, cluster redshift and membership assignment are of high accuracy.In Table 1 we present the precise numerical values of the observationally determined SelFMC mass function data, with the corresponding errors, that are used in our current analysis.The width of each logarithmic mass bin is equal to 0.155.

HALO MASS FUNCTION MODELS
In this section we present a large number of the mass function models found in the literature.The number of dark matter halos per comoving volume per unit mass is given by: where σ 2 (M, z) is the mass variance of the smoothed linear density field, given in Fourier space by: where D(z) is the growth factor of matter fluctuations in the linear regime, W (kR) = 3[sin(kR)−kR cos(kR)]/(kR) 3 is the Fourier transform of the top-hat smoothing kernel of radius R. The radius is given by R = [3M h /(4πρm)] 1/3 with M h the mass of the halo and ρm the mean matter density of the universe at the present time.The quantity P (k, z) is the CDM linear power spectrum given by P where n is the spectral index of the primordial power spectrum and T (k) is the CDM transfer function provided by Eisenstein & Hu (1998): with L0 = ln(2e + 1.8q), e = 2.718, C0 = 14.2 + 731 1+62.5q and q = k/Γ, with Γ being the shape parameter given by Sugiyama (1995): The normalization of the power spectrum is given by: where σ8 ≡ σ(R8, 0) is the present value of the mass variance at 8h −1 Mpc.In the original P-S formalism the term f (σ), which appears in eq.( 4), is given by: fP −S (σ) = 2/π(δc/σ) exp (−δ 2 c /2σ 2 ) with δc the linearly extrapolated density threshold above which structures collapse.For the HMF models considered here, the form of f (σ) and its parameters are listed in Table 2.
Since the majority of the HMF models appear to be either universal or the deviations from universality are rather small for the ΛCDM cosmology, for simplicity we use the exact parameter values from each corresponding paper.
Additionally, it is important to note that (a) the GalWCat19 clusters are basically local, with a narrow redshift range and thus there is no significant redshift evolution (Abdullah et al. 2020b), and (b) wherever we had the option we used HMF parameters relevant to the M200 halo mass definition.However, the majority of the HMF models use an FOF halo mass definition (with b = 0.2), corresponding to ∆ = 178, which indeed provides halo masses almost identical to M200; see for example White (2001).Furthermore, using the subset of HMF models for which the values of halo mass were provided for both ∆crit and ∆mean, we found that the differences of the HMF ranges between 0.5 − 1.5% for low masses and 4.5−5.5% for high masses, depending on the cosmological parameters.Repeating our cosmological analysis does not alter significantly our results (e.g. for the Tinker model we found δΩm ∼ 0.3% and δσ8 = 0.7%).

Fitting HMF Models to the observed CMF
To test the validity and to quantify the free parameters of the aforementioned mass function models we perform a standard χ 2 -minimization procedure between the observational CMF derived from the SelFMC cluster subsample and the expected theoretical HMF in the specific mass range covered by the CMF (c.f.Abdullah et al. 2020b;Hung et al. 2021).The χ 2 function is defined as follows: where the vector p contains the main free parameters of the theoretical HMF, namely p = (Ωm, σ8).Note that σi is the CMF uncertainty (see Table 1) and σM = 0.075 corresponds to the half-width of the logarithmic cluster mass-bin.Since we have non-symmetric uncertainties for the CMF we use a weighted uncertainty scheme in the χ 2 function, following Barlow (2004), according to which: where σ1 = 2σpσn/(σp + σn) and σ2 = (σp − σn)/(σp + σn).Note that for σp = σn we get the symmetric error weighting (see Barlow 2004 for details).

Bagla et al. (2009) f
with N the number of data points and k is the number of free parameters.A small value of AIC indicates a better modeldata fit.Furthermore, we also examine the Akaike information criterion difference between a HMF model pair, namely ∆AIC=AICc,x-AICc,y, in order to evaluate the relative ability of any two HMF models to reproduce the data.The larger the difference |∆AIC|, the higher the evidence of inconsistency (with the comparison model) of the model with higher value of AICc.A difference between 4 ⩽ |∆AIC| ⩽ 7 is a positive indication of the previous statement (Burnham & Anderson 2004), |∆AIC| ⩾ 10 indicates a strong such evidence, and |∆AIC| ⩽ 2 is an indication that the two comparison models are rather consistent.
In  3 we present the statistical results of the χ 2minimization procedure, with the first and second columns listing the derived Ωm and σ8 and their corresponding 1σ uncertainties.It is evident that the values of both parameters, for the majority of the HMF models, are consistent with the recent CMB-Planck results (Planck Collaboration et al. 2020b) (see Section 5 for further discussion).
In the following columns we present the goodness of fit statistics, namely the χ 2 min and the AICc.As it is evident by taking into account the resulting χ 2 min values, all the HMF models fit the data at an acceptable level (with a reduced χ 2 min ∼ 1.3 − 1.5), with the best model (lowest χ 2 min value) being that of Bocquet et al (2016).Note that if the CMF uncertainties were larger by 15% we would have obtained a reduced χ 2 min ≃ 1.In the last column we present the difference ∆AIC=AICc,Bocquet-AICc,y between the best HMF model (Bocquet et al) and any other model (y) and find for all HMF models that ∆AIC=AICc,Bocquet-AICc,y < 2, which means that all the models are consistent with the reference model.We conclude that with the current observational CMF data we cannot distinguish, at a statistically acceptable level, between the available HMF models.
To have a visual appreciation of the results, we present in Figure (2) the comparison of the observed CMF (Table 1) with the most and least successful HMF models, i.e., the Bocquet et al. (2016) and the Press & Schechter (1974) models, respectively, utilizing the best fit parameters values provided in Table 3.

Comparison of the CMF and Planck-CMB parameters. Is
there a σ8 tension?
Although the statistical comparison of the observed CMF (Table 1) with the different theoretical HMF models cannot allow us to constrain the range of viable models, it provided a relatively narrow range of the fitted Ωm,0 and σ8 cosmological parameters, where 0.238 ⩽ Ωm ⩽ 0.362 and 0.737 ⩽ σ8 ⩽ 0.918.The weighted (with ωi = 1/χ 2 i,min ) mean values, over all the HMF models, are: where the uncertainties represent 1σ error of the mean values over all HMF models.The corresponding S8-value is S8 = 0.815 ± 0.05 which is fully consistent, within ∼ 0.3σ, with the Planck-CMB results.Note also that our best fit HMF model (Bocquet et al.) provides almost identical results, ie., S8 = 0.811 ± 0.045.Furthermore, we wish to identify those HMF models that are consistent (or inconsistent) with the current Planck-CMB results (Planck Collaboration et al. 2020b), at a statistically significant level.To achieve that, we compare the best fit results of Ωm and σ8 parameters, that we obtain for each HMF model (see Table 3), to the Planck-CMB results.More specifically, we compare the results of: (i) Planck (TT, TE, EE+lowE+lensing) cosmology [Planck], which provides Ωm = 0.315 ± 0.007 and σ8 = 0.811 ± 0.006; (ii) the Planck (TT, TE, EE+lowE+lensing) cosmology, combined with BAO [Planck + BAO], which provides Ωm = 0.311 ± 0.0056 and σ8 = 0.810 ± 0.006; and finally (iii) the Planck (TT, TE, EE+lowE+lensing) cosmology combined with DES [Planck + DES], which provides Ωm = 0.304±0.006and σ8 = 0.8062±0.0057(Planck Collaboration et al. 2020b).We utilize the index of inconsistency (IOI) method to perform this comparison.The IOI is sensitive to the separation distance of the means, the size of the parameter space, and the orientation of the parameters space (Lin & Ishak (2017)).The IOI is given by where δ = µ2 − µ1 is the difference of the means of the pa- rameters provided by the HMF model and by the cosmology, while G = (C1 + C2) −1 is the inverse sum of the covariance matrices of the HMF model parameters and the cosmology.Higher IOI implies higher inconsistency.More specifically, for IOI < 1 there is no significant inconsistency, for 1 < IOI < 2.5 there is a weak inconsistency and for 2.5 < IOI < 5 and IOI > 5 there is a moderate and strong inconsistency, respectively.In Table 4 we present the IOI results and as it can be seen, using the Planck, [Planck-BAO] and [Planck-DES] cosmologies there are twenty out of the twenty-three models that show no significant inconsistency.In the case of Yahagi et al. (2004) and Despali et al. (2015) HMFs there is a weak inconsistency, while the Press & Schechter (1974) model shows a strong inconsistency.
Thus out of 23 HMF models there are 20 models that are consistent with both the observed CMF, obtained from Abdullah et al. (2020b), and the joint Planck CMB-BAO and CMB-DES cosmological probes.The inverse-χ 2 min weighted average values of the two cosmological parameters are Ωm,0 = 0.312 ± 0.011 and σ8 = 0.791 ± 0.027.We also obtain similar results but with a significantly smaller scatter when we use the inverse-IOI weighting scheme.We find Ωm,0 = 0.312 ± 0.003 and σ8 = 0.805 ± 0.007, corresponding to S8 = 0.821 ± 0.008.
Our results present no evidence for a σ8-tension between the Planck-CMB and the Cluster Mass Function probe, based on the large majority of the theoretical HMF models, used in this paper.Rather, the σ8-tension between the lower redshift cosmological probes and the CMB could be due to systematic uncertainties of the lower redshift probes and it should probably not be attributed to a weakness of the ΛCDM model.As discussed in Abdullah et al. (2020b), studies that use galaxy clusters to constrain the cosmological parameters Ωm and σ8, provide conflicting results.Some of these studies are in agreement with the Planck data and some other studies are not.These discrepancies could be caused by many reasons among which the method used to calculate the cluster mass, misidentification of the cluster locations and cluster center, or due to redshift uncertainties.

Robustness of our results: dependence on h and Ω b h 2
Although the cluster abundances as encapsulated in the HMF modelling are mostly sensitive to the Ωm and σ8 cosmological parameters, there are also other parameters, such as h and Ω b h 2 , that enter in such modelling via the Power-Spectrum of matter fluctuations.
In this section we investigate the sensitivity of our results on the variations of h and Ω b h 2 .We use our best fit model (Bocquet et. al.) and repeat our analysis by allowing h to vary within the range of 0.68 ⩽ h ⩽ 0.74, in steps of 0.01.We also allow Ω b h 2 to vary within the range of 0.018 ⩽ Ω b h 2 ⩽ 0.027, in steps of 0.001.The fitted values of the two main cosmological parameters, Ωm and σ8, show insignificant dependence on the specified range of the h and Ω b h 2 parameters (see Figure 3).This indicates that the effects of varying h and Ω b h 2 on the constraints of Ωm (∼ 4%) and σ8 (∼ 2%) is minimal.

Comparison with other Cluster MF studies
Utilizing the probe of the cluster mass function, the constraints on the cosmological parameters Ωm and σ8, has been extensively studied in the literature based on a variety of galaxy surveys in the optical, infrared, X-ray, and mm-wave bands.The basic result is that the inferred (or directly estimated) value of cluster normalization parameter, S8, is systematically lower than that of the CMB temperature fluctuation analyses.Although, due to the relatively large uncertainties of the derived values of Ωm and σ8 and/or S8, the individual significance is rather low, unless one would combine jointly with other cosmological probes.
Note that for the studies where the results of Ωm and σ8 were presented individually, we transformed them to the pivot S8 = σ8 Ωm/0.3 relation.Furthermore, in order to have a consistent comparison of the different studies, we ignored results obtained from the joint analyses of the Cluster Mass Function and other cosmological probes.

CONCLUSIONS
In this work we investigate the ability of 23 theoretical halo mass function models to represent the observational mass function of galaxy clusters, obtained from Abdullah et al. (2020b).To this end we apply a standard χ 2 minimization procedure between the models and the observational cluster  mass function and constrain the two free cosmological parameters of the models, namely Ωm and σ8.Utilizing the best-χ 2 and the value of the Akaike information criterion, we find that all the models fit the data at a statistically acceptable level.We find that the Bocquet et al. 2016 halo mass function model returns the smaller χ 2 value and therefore it presents the best fit model to the CMF data.We compare all other theoretical HMFs to the reference HMF model (Bocquet et al. 2016) and find that all models are consistent with it.This fact implies that in order to constrain further among the HMF models we need more observational data of wider cluster-mass and redshift ranges.Since all our models pass the CMF-HMF comparison test at a statistically acceptable level, we determine the average values of Ωm, 0 and σ8 over all HMF models, weighted by the inverse χ 2 min value.We find that Ωm,0 = 0.313 ± 0.022 and σ8 = 0.798 ± 0.040, with corresponding S8 = 0.815 ± 0.05.Furthermore in order to identify possible HMF models which are in disagreement with the recent Planck-CMB results, we utilized the IOI method to compare the two fitted free parameters of the models (Ωm, σ8) with the corresponding parameters obtained from the Planck CMB analysis in addition to the joint Planck+BAO and Planck+DES analyses and find that only three HMF models are inconsistent.These models are the Yahagi et al. (2004), Despali et al. (2015), and Press & Schechter (1974) HMFs, with the latter being the most inconsistent one.Using the 20 models which are consistent with the Planck-CMB results, we find that the average values of Ωm,0 and σ8, weighted by the inverse-IOI, are Ωm,0 = 0.312 ± 0.003 and σ8 = 0.805 ± 0.007.
Therefore, we conclude that the current cluster mass function, obtained by Abdullah et al. (2020b) at redshift z ∼ 0.1, provides results that show no σ8-tension with the Planck-CMB results within the ΛCDM cosmological model frame, a result which holds for a large majority of the theoretical halo mass functions used to model the CMF data.
To this endAbdullah et al. (2020b) restricted the sample to a maximum comoving distance of D ⩽ 365h −1 Mpc, ie., z ⩽ 0.125.As a further testAbdullah et al. (2020a) compared the CMF derived from the GalWCat19 (after weighting to recover volume completeness) with the HMF derived from the MDPL2 simulation.The comparison indicated that the sample is complete in mass for log(M ) ⩾ 13.9h −1 M⊙.Additionally, as shown in Figure1ofAbdullah et al. (2020b), this mass threshold is roughly the same for a variety of cosmologies.The final cluster subsample (SelFMC, presented byAbdullah et al. 2020b) contains 843 galaxy clusters in the redshift range 0.01 < z < 0.125 with log M ⩾ 13.9h −1 M⊙.

Figure 1 .
Figure 1.Contour plot of the 1σ and 3σ confidence levels for the two fitted parameters, σ 8 and Ωm.The left plot presents the contours of Bocquet et al. (2016) (black line) and the Watson et al. (2013) (FOF) (red line) HMFs which have the lowest χ 2 min .The right panel presents the contour plots of Bocquet et al. (2016) HMF (black line), which has the lowest χ 2 min , and Press & Schechter (1974) HMF (red line), which has the highest χ 2 min .The blue contour presents the 1σ and 2σ confidence levels of the CMB analysis of the Planck Collaboration et al. (2020b).
Figure (1) we present our results of the χ 2 minimization procedure.The figure shows the contours of the 1σ and 3σ confidence levels for the two fitted parameters, σ8 and Ωm.The left panel presents the contours of the Bocquet et al. (2016) (black line) and the Watson et al. (2013) (FOF) (red line) HMF models, which have the lowest χ 2 min values.The right panel presents the contours of the former model (black line) and of the Press & Schechter (1974) model (red line), which has the highest χ 2 min value.In Table

Figure 2 .
Figure 2. Comparison of the Cluster Mass Function data with the theoretical HMF fits of Bocquet et.al. and Press & Schechter models.

Figure 3 .
Figure 3. Dependence of the cosmological parameters, Ωm and σ 8 derived by fitting the current CMF to the Bocquet et.al. HMF, on the parameters h and Ω b h 2 that enter in the definition of the matter perturbations power-spectrum.The left plot the 1σ and 3σ confidence levels of the two fitted parameters, for different values of h.The right plot is corresponding plot, but for different values of Ω b h 2 .

Figure 4 .
Figure 4. Comparison of the derived or inferred S 8 values based on the CMF or counts analysis from a number of studies (indicated by the cluster sample analysed) with the Planck 2018 CMB results (shaded area).

Table 1 .
Abdullah et al. (2020b)ss Function data of the SelFMC subsample fromAbdullah et al. (2020b).Note that the width of each logarithmic mass bin is equal to 0.155.

Table 2 .
Compilation of Fitting Functions

Table 3 .
Results for the fit of the observed CMF to a large range of theoretical HMF: The 1 st column indicates the HMF model, the 2 nd and the 3 rd column provides the fitted Ωm and σ 8 respectively.The remaining columns present the goodness-of-fit statistics χ 2 min , AICc and |∆AIC| = |AIC c,Bocquet − AICc,y| with the index y corresponding to the indicated comparison model.

Table 4 .
IOI results for the mass function data (see Table1): The 1 st column indicates the mass function model, the 2 nd , 3 rd and the 4 th column present the IOI results between the models and the Planck, Planck+DES and Planck+BAO cosmologies, respectively.