Mass Function Of Galaxy Clusters Research Articles

An optical observational cluster mass function at z ∼ 1 with the ORELSE survey

ABSTRACT We present a new mass function of galaxy clusters and groups using optical/near-infrared (NIR) wavelength spectroscopic and photometric data from the Observations of Redshift Evolution in Large-Scale Environments (ORELSE) survey. At z ∼ 1, cluster mass function studies are rare regardless of wavelength and have never been attempted from an optical/NIR perspective. This work serves as a proof of concept that z ∼ 1 cluster mass functions are achievable without supplemental X-ray or Sunyaev-Zel’dovich data. Measurements of the cluster mass function provide important contraints on cosmological parameters and are complementary to other probes. With ORELSE, a new cluster finding technique based on Voronoi tessellation Monte Carlo (VMC) mapping, and rigorous purity and completeness testing, we have obtained ∼240 galaxy overdensity candidates in the redshift range 0.55 < z < 1.37 at a mass range of 13.6 < log (M/M⊙) < 14.8. This mass range is comparable to existing optical cluster mass function studies for the local universe. Our candidate numbers vary based on the choice of multiple input parameters related to detection and characterization in our cluster finding algorithm, which we incorporated into the mass function analysis through a Monte Carlo scheme. We find cosmological constraints on the matter density, Ωm, and the amplitude of fluctuations, σ8, of $\Omega _{m} = 0.250^{+0.104}_{-0.099}$ and $\sigma _{8} = 1.150^{+0.260}_{-0.163}$. While our Ωm value is close to concordance, our σ8 value is ∼2σ higher because of the inflated observed number densities compared to theoretical mass function models owing to how our survey targeted overdense regions. With Euclid and several other large, unbiased optical surveys on the horizon, VMC mapping will enable optical/NIR cluster cosmology at redshifts much higher than what has been possible before.

Galaxies and clusters of galaxies as peak patches of the density field

ABSTRACT The mass function of galaxies and clusters of galaxies can be derived observationally based on different types of observations. In this study we test if these observations can be combined to a consistent picture which is also in accord with structure formation theory. The galaxy data comprise the optical galaxy luminosity function and the gravitational lensing signature of the galaxies, while the galaxy cluster mass function is derived from the X-ray luminosity distribution of the clusters. We show the results of the comparison in the form of the mass density fraction that is contained in collapsed objects relative to the mean matter density in the Universe. The mass density fraction in groups and clusters of galaxies extrapolated to low masses agrees very well with that of the galaxies: both converge at the low mass limit to a mass fraction of about 28 per cent if the outer radii of the objects are taken to be r200. Most of the matter contained in collapsed objects is found in the mass range $M_{200} \sim 10^{12}\!-\!10^{14}\, h^{-1}_{70} \, \mathrm{M}_\odot$, while a larger amount of the cosmic matter resides outside of r200 of collapsed objects.

Relation between X-ray and Sunyaev—Zeldovich Galaxy Cluster Mass Measurements

We investigate the possibility of transferring the galaxy cluster mass scale calibrations from weak-lensing measurements to X-ray cluster mass measurements using Planck cluster mass measurements. For this purpose, we compare the mass measurements for 47 galaxy clusters that simultaneously enter the X-ray sample of clusters from Vikhlinin et al. (2009a, 2009b) and the sample of galaxy clusters from the second Planck catalogue of Sunyaev-Zeldovich sources. We show that except for a few outliers in the measurements, most of which arise from cluster projection effects in the Planck data, the scatter in the mass ratio turns out to be small, ~15%, while the accuracy of determining the mean mass ratio for this sample is about 2%. This allows the existing cluster mass measurement calibrations in the Planck survey to be transferred to Chandra X-ray mass measurements almost without any loss of accuracy. Using the derived relations and based on new cluster mass scale calibrations from weak-lensing measurements in clusters, we have obtained corrections to the constraints on cosmological parameters from Vikhlinin et al. (2009b). Given these corrections, σ8 increases approximately by 4%, within one standard deviation of the uncertainty in the constraint on this quantity at constant Ωm. We show that, given the systematic uncertainties, all of the currently available galaxy cluster mass function measurements yield approximately identical constraints on the parameters σ8 and Ωm.

Measurements of the Matter Density Perturbation Amplitude from Cosmological Data

We compare various physically different measurements of linear matter density perturbation amplitude, $sigma_8$, which are obtained from the observations of CMB anisotropy, galaxy cluster mass function, weak gravitation lensing, matter power spectrum and redshift space distortions. We show that $\sigma_8$ measurement from CMB gravitational lensing signal based on Planck CMB temperature anisotropy data at high multipoles, $\ell>1000$, contradict to all other measurements obtained both from remaining Planck CMB anisotropy data and from other cosmological data, at about $3.7\sigma$ significance level. Therefore, these data currently should not be combined with other data to constrain cosmological parameters. With the exception of Planck CMB temperature anisotropy data at high multipoles, all other measurements are in good agreement between each other and give the following measurements of linear density perturbation amplitude: $\sigma_8=0.792\pm0.006$, mean density of the Universe: $\Omega_m=0.287\pm0.007$, and Hubble constant: $H_0 = 69.4\pm 0.6$~km~s$^{-1}$~Mpc$^{-1}$. Taking in account the data on baryon acoustic oscillations and (or) direct measurements of the Hubble constant, one can obtain different constraints on sum of neutrino mass and number of relativistic species.

The extended ROSAT-ESO Flux-Limited X-ray Galaxy Cluster Survey (REFLEX II)

The mass function of galaxy clusters is a sensitive tracer of the gravitational evolution of the cosmic large-scale structure and serves as an important census of the fraction of matter bound in large structures. We obtain the mass function by fitting the observed cluster X-ray luminosity distribution from the REFLEX galaxy cluster survey to models of cosmological structure formation. We marginalise over uncertainties in the cosmological parameters as well as those of the relevant galaxy cluster scaling relations. The mass function is determined with an uncertainty less than 10% in the mass range 3 x 10^12 to 5 x 10^14 M$_\odot$. For the cumulative mass function we find a slope at the low mass end consistent with a value of -1, while the mass rich end cut-off is milder than a Schechter function with an exponential term exp($- M^\delta$) with $\delta$ smaller than 1. Changing the Hubble parameter in the range $H_0 = 67 - 73 km s^-1 Mpc^{-1}$ or allowing the total neutrino mass to have a value between 0 - 0.4 eV causes variations less than the uncertainties. We estimate the fraction of mass locked up in galaxy clusters: about 4.4% of the matter in the Universe is bound in clusters (inside $r_200$) with a mass larger than 10^14 M$_\odot$ and 14% to clusters and groups with a mass larger than 10^13 M$_\odot$ at the present Universe. We also discuss the evolution of the galaxy cluster population with redshift. Our results imply that there is hardly any clusters with a mass > 10^15 M$_\odot$ above a redshift of z = 1.

Mass Function of Galaxy Clusters in Relativistic Inhomogeneous Cosmology

The current model ($\Lambda$CDM) with the underlying FLRW metric relies on the assumption of local isotropy, hence homogeneity of the Universe. Difficulties arise when one attempts to justify this model as an average description of the Universe from first principles of general relativity, since in general, the Einstein tensor built from the averaged metric is not equal to the averaged stress--energy tensor. In this context, the discrepancy between these quantities is called cosmological backreaction and has been the subject of scientific debate among cosmologists and relativists for more than $20$ years. Here we present one of the methods to tackle this problem, i.e. averaging the scalar parts of the Einstein equations, together with its application, the mass function of galaxy clusters.

Cosmology based on f(R) gravity with \U0001d4aa(1) eV sterile neutrino

We address the cosmological role of an additional \U0001d4aa(1) eV sterile neutrino in modified gravity models. We confront the present cosmological data with predictions of the FLRW cosmological model based on a variant of f(R) modified gravity proposed by one of the authors previously. This viable cosmological model which deviation from general relativity with a cosmological constant Λ decreases as R−2n for large, but not too large values of the Ricci scalar R (while no Λ is introduced by hand at small R) provides an alternative explanation of present dark energy and the accelerated expansion of the Universe (the case n=2 is considered in the paper). Various up-to-date cosmological data sets exploited include measurements of the cosmic microwave background (CMB) anisotropy, the CMB lensing potential, the baryon acoustic oscillations (BAO), the cluster mass function and the Hubble constant. We find that the CMB+BAO constraints strongly restrict the sum of neutrino masses from above. This excludes values of the model parameter λ∼ 1 for which distinctive cosmological features of the model are mostly pronounced as compared to the ΛCDM model, since then free streaming damping of perturbations due to neutrino rest masses is not sufficient to compensate their extra growth occurring in f(R) modified gravity. Thus, in the gravity sector we obtain λ>8.2 (2σ) with the account of systematic uncertainties in galaxy cluster mass function measurements and λ>9.4 (2σ) without them. At the same time in the latter case we find for the sterile neutrino mass 0.47 eV < mν, sterile < 1 eV (2σ) assuming that the sterile neutrinos are thermalized and the active neutrinos are massless, not significantly larger than in the standard ΛCDM with the same data set: 0.45 eV < mν, sterile < 0.92 eV (2σ). However, a possible discovery of a sterile neutrino with the mass mν, sterile ≈ 1.5 eV motivated by various anomalies in neutrino oscillation experiments would favor cosmology based on f(R) gravity rather than the ΛCDM model.

Comparing gravitational redshifts of SDSS galaxy clusters with the magnified redshift enhancement of background BOSS galaxies

A clean measurement of the evolution of the galaxy cluster mass function can significantly improve our understanding of cosmology from the rapid growth of cluster masses below z < 0.5. Here we examine the consistency of cluster catalogues selected from the SDSS by applying two independent gravity-based methods using all available spectroscopic redshifts from the DR10 release. First, we detect a gravitational redshift related signal for 20,119 and 13,128 clusters with spectroscopic redshifts contained in the GMBCG and redMaPPer catalogues, respectively, at a level of $\sim - 10$ km s$^{-1}$. This we show is consistent with the magnitude expected using the richness-mass relations provided by the literature and after applying recently clarified relativistic and flux bias corrections. This signal is also consistent with the richest clusters in the larger catalogue of Wen et al. (2012), corresponding to $M_{200m} \gtrsim 2 \times 10^{14}\,\mathrm{M}_\odot\,h^{-1}$, however we find no significant detection of gravitational redshift signal for less riched clusters, which may be related to bulk motions from substructure and spurious cluster detections. Second, we find all three catalogues generate mass-dependent levels of lensing magnification bias, which enhances the mean redshift of flux-selected background galaxies from the BOSS survey. The magnitude of this lensing effect is generally consistent with the corresponding richness-mass relations advocated for the surveys. We conclude that all catalogues comprise a high proportion of reliable clusters, and that the GMBCG and redMaPPer cluster finder algorithms favor more relaxed clusters with a meaningful gravitational redshift signal, as anticipated by the red-sequence colour selection of the GMBCG and redMaPPer samples.

Weighing the giants – IV. Cosmology and neutrino mass

We employ robust weak gravitational lensing measurements to improve cosmological constraints from measurements of the galaxy cluster mass function and its evolution, using X-ray selected clusters detected in the ROSAT All-Sky Survey. Our lensing analysis constrains the absolute mass scale of such clusters at the 8 per cent level, including both statistical and systematic uncertainties. Combining it with the survey data and X-ray follow-up observations, we find a tight constraint on a combination of the mean matter density and late-time normalization of the matter power spectrum, σ8(Ωm/0.3)0.17 = 0.81 ± 0.03, with marginalized, one-dimensional constraints of Ωm = 0.26 ± 0.03 and σ8 = 0.83 ± 0.04. For these two parameters, this represents a factor of 2 improvement in precision with respect to previous work, primarily due to the reduced systematic uncertainty in the absolute mass calibration provided by the lensing analysis. Our new results are in good agreement with constraints from cosmic microwave background (CMB) data, both Wilkinson Microwave Anisotropy Probe (WMAP) and Planck (plus WMAP polarization), under the assumption of a flat ΛCDM cosmology with minimal neutrino mass. Consequently, we find no evidence for non-minimal neutrino mass from the combination of cluster data with CMB, supernova and baryon acoustic oscillation measurements, regardless of which all-sky CMB data set is used (and independent of the recent claimed detection of B modes on degree scales). We also present improved constraints on models of dark energy (both constant and evolving), modifications of gravity, and primordial non-Gaussianity. Assuming flatness, the constraints for a constant dark energy equation of state from the cluster data alone are at the 15 per cent level, improving to ∼6 per cent when the cluster data are combined with other leading probes.

Neutrino constraints: what large-scale structure and CMB data are telling us?

We discuss the reliability of neutrino mass constraints, either active or sterile, from the combination of different low redshift Universe probes with measurements of CMB anisotropies. In our analyses we consider WMAP 9-year or Planck Cosmic Microwave Background (CMB) data in combination with Baryonic Acoustic Oscillations (BAO) measurements from BOSS DR11, galaxy shear measurements from CFHTLenS, SDSS Ly α forest constraints and galaxy cluster mass function from Chandra observations. At odds with recent similar studies, to avoid model dependence of the constraints we perform a full likelihood analysis for all the datasets employed. As for the cluster data analysis we rely on to the most recent calibration of massive neutrino effects in the halo mass function and we explore the impact of the uncertainty in the mass bias and re-calibration of the halo mass function due to baryonic feedback processes on cosmological parameters. We find that none of the low redshift probes alone provide evidence for massive neutrino in combination with CMB measurements, while a larger than 2σ detection of non zero neutrino mass, either active or sterile, is achieved combining cluster or shear data with CMB and BAO measurements. Yet, the significance of the detection exceeds 3σ if we combine all four datasets. For a three active neutrino scenario, from the joint analysis of CMB, BAO, shear and cluster data including the uncertainty in the mass bias we obtain ∑ mν =0.29+0.18-0.21 eV and ∑ mν =0.22+0.17-0.18 eV 95%CL) using WMAP9 or Planck as CMB dataset, respectively. The preference for massive neutrino is even larger in the sterile neutrino scenario, for which we get mseff=0.44+0.28-0.26 eV and Δ Neff=0.78+0.60-0.59 95%CL) from the joint analysis of Planck, BAO, shear and cluster datasets. For this data combination the vanilla ΛCDM model is rejected at more than 3σ and a sterile neutrino mass as motivated by accelerator anomaly is within the 2σ errors. Conversely, the Ly α data favour vanishing neutrino masses and from the data combination Planck+BAO+Ly α we get the tight upper limits ∑ mν <0.14 eV and mseff<0.22 eV—Δ Neff<1.11 95%CL) for the active and sterile neutrino model, respectively. Finally, results from the full data combination reflect the tension between the σ8 constraints obtained from cluster and shear data and that inferred from Ly α forest measurements; in the active neutrino scenario for both CMB datasets employed, the full data combination yields only an upper limits on ∑ mν, while assuming an extra sterile neutrino we still get preference for non-vanishing mass, mseff=0.26+0.22-0.24 eV, and dark contribution to the radiation content, Δ Neff=0.82±0.55.

Modified Baryonic Dynamics: two-component cosmological simulations with light sterile neutrinos

In this article we continue to test cosmological models centred on Modified Newtonian Dynamics (MOND) with light sterile neutrinos, which could in principle be a way to solve the fine-tuning problems of the standard model on galaxy scales while preserving successful predictions on larger scales. Due to previous failures of the simple MOND cosmological model, here we test a speculative model where the modified gravitational field is produced only by the baryons and the sterile neutrinos produce a purely Newtonian field (hence Modified Baryonic Dynamics). We use two-component cosmological simulations to separate the baryonic N-body particles from the sterile neutrino ones. The premise is to attenuate the over-production of massive galaxy cluster halos which were prevalent in the original MOND plus light sterile neutrinos scenario. Theoretical issues with such a formulation notwithstanding, the Modified Baryonic Dynamics model fails to produce the correct amplitude for the galaxy cluster mass function for any reasonable value of the primordial power spectrum normalisation.

Possible indication for non-zero neutrino mass and additional neutrino species from cosmological observations

The constraints on total neutrino mass and effective number of neutrino species based on CMB anisotropy power spectrum, Hubble constant, baryon acoustic oscillations and galaxy cluster mass function data are presented. It is shown that the discrepancies between various cosmological data in Hubble constant and density fluctuation amplitude, measured in standard LCDM cosmological model, can be eliminated if more than standard effective number of neutrino species and non-zero total neutrino mass are considered. This extension of LCDM model appears to be \approx 3 \sigma\ significant when all cosmological data are used. The model with approximately one additional neutrino type, N_eff \approx 4, and with non-zero total neutrino mass, \approx 0.5 eV, provide the best fit to the data. In the model with only one massive neutrino the upper limits on neutrino mass are slightly relaxed. It is shown that these deviations from LCDM model appear mainly due to the usage of recent data on the observations of baryon acoustic oscillations. Larger than standard number of neutrino species is measured mainly due to the comparison of the BAO data with direct measurements of Hubble constant, which was already noticed earlier. As it is shown below, the data on galaxy cluster mass function in this case give the measurement of non-zero neutrino mass.

The impact of global environment on galaxy mass functions at low redshift

We study the galaxy stellar mass function in different environments in the local Universe, considering both the total mass function and that of individual galaxy morphological types. We compare the mass functions of galaxies with $\rm log_{10} M_{\star}/M_{\odot} \geq 10.25$ in the general field and in galaxy groups, binary and single galaxy systems from the Padova-Millennium Galaxy and Group Catalogue at $z=0.04-0.1$ with the mass function of galaxy clusters of the WIde-field Nearby Galaxy-Cluster Survey at $z=0.04-0.07$. Strikingly, the variations of the mass function with global environment, overall, are small and subtle. The shapes of the mass functions of the general field and clusters are indistinguishable, and only small, statistically insignificant variations are allowed in groups. Only the mass function of our single galaxies, representing the least massive haloes and comprising less than a third of the general field population, is proportionally richer in low-mass galaxies than other environments. The most notable environmental effect is a progressive change in the upper galaxy mass, with very massive galaxies found only in the most massive environments. This environment-dependent mass cut-off is unable to affect the Schechter parameters and the K-S test, and can only be revealed by an ad-hoc analysis. Finally, we show how, in each given environment, the mass function changes with morphological type, and that galaxies of the same morphological type can have different mass functions in different environments.

The 400d Galaxy Cluster Survey weak lensing programme

Evolution in the mass function of galaxy clusters sensitively traces both the expansion history of the Universe and cosmological structure formation. Robust cluster mass determinations are a key ingredient for a reliable measurement of this evolution, especially at high redshift. Weak gravitational lensing is a promising tool for, on average, unbiased mass estimates. This weak lensing project aims at measuring reliable weak lensing masses for a complete X-ray selected sample of 36 high redshift (0.35<z<0.9) clusters. The goal of this paper is to demonstrate the robustness of the methodology against commonly encountered problems, including pure instrumental effects, the presence of bright (8--9 mag) stars close to the cluster centre, ground based measurements of high-z (z~0.8) clusters, and the presence of massive unrelated structures along the line-sight. We select a subsample of seven clusters observed with MMT/Megacam. Instrumental effects are checked in detail by cross-comparison with an archival CFHT/MegaCam observation. We derive mass estimates for seven clusters by modelling the tangential shear with an NFW profile, in two cases with multiple components to account for projected structures in the line-of-sight. We firmly detect lensing signals from all seven clusters at more than $3.5\sigma$ and determine their masses, ranging from $10^{14} M_{\odot}$ to $10^{15} M_{\odot}$, despite the presence of nearby bright stars. We retrieve the lensing signal of more than one cluster in the CL 1701+6414 field, while apparently observing CL 1701+6414 through a massive foreground filament. We also find a multi-peaked shear signal in CL 1641+4001. Shear structures measured in the MMT and CFHT images of CL 1701+6414 are highly correlated.

Cosmological parameters constraints from galaxy cluster mass function measurements in combination with other cosmological data

We present the cosmological parameters constraints obtained from the combination of galaxy cluster mass function measurements (Vikhlinin et al. 2009a, 2009b) with new cosmological data obtained during last three years: updated measurements of cosmic microwave background anisotropy with Wilkinson Microwave Anisotropy Probe (WMAP) observatory, and at smaller angular scales with South Pole Telescope (SPT), new Hubble constant measurements, baryon acoustic oscillations and supernovae Type Ia observations. New constraints on total neutrino mass Σm ν and effective number of neutrino species are obtained. In models with free number of massive neutrinos the constraints on these parameters are notably less strong, and all considered cosmological data are consistent with non-zero total neutrino mass Σm ν ≈ 0.4 eV and larger than standard effective number of neutrino species, N eff ≈ 4. These constraints are compared to the results of neutrino oscillations searches at short baselines. The updated dark energy equation of state parameter constraints are presented. We show that taking in account systematic uncertanties, current cluster mass funstion data provide similarly powerful constraints on dark energy equation of state, as compared to the constraints from supernovae Type Ia observations.

LENSING MAGNIFICATION: A NOVEL METHOD TO WEIGH HIGH-REDSHIFT CLUSTERS AND ITS APPLICATION TO SpARCS

We introduce a novel method to measure the masses of galaxy clusters at high redshift selected from optical and IR Spitzer data via the red-sequence technique. Lyman-break galaxies are used as a well understood, high-redshift background sample allowing mass measurements of lenses at unprecedented high redshifts using weak lensing magnification. By stacking a significant number of clusters at different redshifts with average masses of ~1-3x10^14M_sun, as estimated from their richness, we can calibrate the normalisation of the mass-richness relation. With the current data set (area: 6 deg^2) we detect a magnification signal at the >3-sigma level. There is good agreement between the masses estimated from the richness of the clusters and the average masses estimated from magnification, albeit with large uncertainties. We perform tests that suggest the absence of strong systematic effects and support the robustness of the measurement. This method - when applied to larger data sets in the future - will yield an accurate calibration of the mass-observable relations at z>~1 which will represent an invaluable input for cosmological studies using the galaxy cluster mass function and astrophysical studies of cluster formation. Furthermore this method will probably be the least expensive way to measure masses of large numbers of z>1 clusters detected in future IR-imaging surveys.

The400dGalaxy Cluster Survey weak lensing programme

The mass function of galaxy clusters at high redshifts is a particularly useful probe to learn about the history of structure formation and constrain cosmological parameters. We aim at deriving reliable masses for a high-redshift, high-luminosity sample of clusters of galaxies selected from the 400d survey of X-ray selected clusters. Here, we will focus on a particular object, CL0030+2618 at z=0.50 Using deep imaging in three passbands with the MEGACAM instrument at MMT, we show that MEGACAM is well-suited for measuring gravitational shear. We detect the weak lensing signal of CL0030+2618 at 5.8 sigma significance, using the aperture mass technique. Furthermore, we find significant tangential alignment of galaxies out to ~10 arcmin or >2r_200 distance from the cluster centre. The weak lensing centre of CL0030+2618 agrees with several X-ray measurements and the position of the brightest cluster galaxy. Finally, we infer a weak lensing virial mass of M_200=7.5 10^{14} M_sun for CL0030+2618. Despite complications by a tentative foreground galaxy group in the line of sight, the X-ray and weak lensing estimates for CL0030+2618 are in remarkable agreement. This study paves the way for the largest weak lensing survey of high-redshift galaxy clusters to date.

Slicing cluster mass functions with a Bayesian razor

AbstractWe apply a Bayesian “razor” to forecast Bayes factors between different parameterizations of the galaxy cluster mass function. To demonstrate this approach, we calculate the minimum size N‐body simulation needed for strong evidence favoring a two‐parameter mass function over one‐parameter mass functions and visa versa, as a function of the minimum cluster mass (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Mass function of rich galaxy clusters and its constraint on σ8

The mass function of galaxy clusters is a powerful tool to constrain cosmological parameters, e.g., the mass fluctuation on the scale of 8 h^{-1} Mpc, sigma_8, and the abundance of total matter, Omega_m. We first determine the scaling relations between cluster mass and cluster richness, summed r-band luminosity and the global galaxy number within a cluster radius. These relations are then used to two complete volume-limited rich cluster samples which we obtained from the Sloan Digital Sky Survey (SDSS). We estimate the masses of these clusters and determine the cluster mass function. Fitting the data with a theoretical expression, we get the cosmological parameter constraints in the form of sigma_8(Omega_m/0.3)^{alpha}=beta and find out the parameters of alpha=0.40-0.50 and beta=0.8-0.9, so that sigma_8=0.8-0.9 if Omega_m=0.3. Our sigma_8 value is slightly higher than recent estimates from the mass function of X-ray clusters and the Wilkinson Microwave Anisotropy Probe (WMAP) data, but consistent with the weak lensing statistics.

Mass variance and cluster abundance in the context of a Gaussian + lognormal distribution model

Context. We investigate the behavior of the mass variance and the mass function of galaxy clusters in a mixed distribution model. Aims. Our aim is to find a relation between the mass variance at a 8 h -1 Mpc scale, $\sigma_8$, and the parameter controlling the Gaussian deviation in the model, $\alpha_0$, and to constrain the non-Gaussianity using observational data at cluster scales. Methods. By assuming that the statistics of the density field is built as a weighted mixture of two components, a Gaussian + Lognormal distribution, we rewrite the mass variance expression and the mass function for galaxy clusters. Results. We find a relation between the mass variance at a 8 h -1 Mpc scale, $\sigma_8$, and the scale parameter controlling the Gaussian deviation in the model, $\alpha_0$. This result, in conjunction with observational constraints on the mass variance and high- z galaxy clustering, suggests a scenario where structures develop earlier in comparison to strictly Gaussian models, even for $\alpha_0\lesssim 0.003$ Mpc. Our model also indicates that only well selected galaxy cluster samples at $z\gtrsim 1$ can discriminate between Gaussian and non-Gaussian (mixed) distribution models.