Amplitude Of Mass Fluctuations Research Articles

Large-scale motions and growth rate from forward-modelling Tully–Fisher peculiar velocities

ABSTRACT Peculiar velocities are an important probe of the mass distribution in the Universe and the growth rate of structure, directly measuring the effects of gravity on the largest scales and providing a test for theories of gravity. Comparing peculiar velocities predicted from the density field mapped by a galaxy redshift survey with peculiar velocities measured using a distance estimator such as the Tully–Fisher relation yields the growth factor for large-scale structure. We present a method for forward modelling a sample of galaxy magnitudes and velocity widths that simultaneously determines the parameters of the Tully–Fisher relation and the peculiar velocity field. We apply this to the Cosmicflows-4 Tully–Fisher data set, using the peculiar velocities predicted from the 2M++ redshift survey. After validating the method on mock surveys, we measure the product of the growth rate and mass fluctuation amplitude to be fσ8 = 0.35 ± 0.03 at an effective redshift of z = 0.017. This is consistent at 3σ with the Planck CMB prediction, even though the uncertainty does not fully account for all sources of sample variance. We find the residual bulk flow from gravitational influences outside the 2M++ survey volume to be |V| = 227 ± 11 km s−1, (l, b) = (303°, −1°) in Galactic polar coordinates and the CMB frame. Using simulations, we show that applying our methodology to the large new sample of Tully–Fisher peculiar velocities expected from the WALLABY H I survey of the southern sky can improve the constraints on the growth rate by a factor of 2–3.

Cosmological constraints from the power spectrum of eBOSS quasars

We present the effective-field theory (EFT)-based cosmological full-shape analysis of the anisotropic power spectrum of eBOSS quasars at the effective redshift $z_{\rm eff}=1.48$. We perform extensive tests of our pipeline on simulations, paying a particular attention to the modeling of observational systematics, such as redshift smearing, fiber collisions, and the radial integral constraint. Assuming the minimal $\Lambda$CDM model, and fixing the primordial power spectrum tilt and the physical baryon density, we find the Hubble constant $H_0=(66.7\pm 3.2)~$km~s$^{-1}$Mpc$^{-1}$, the matter density fraction $\Omega_m=0.32\pm 0.03$, and the late-time mass fluctuation amplitude $\sigma_8=0.95\pm 0.08$. These measurements are fully consistent with the Planck cosmic microwave background results. Our eBOSS quasar $S_8$ posterior, $0.98\pm0.11$, does not exhibit the so-called $S_8$ tension. Our work paves the way for systematic full-shape analyses of quasar samples from future surveys like DESI.

The information content of projected galaxy fields

ABSTRACT The power spectrum of the non-linearly evolved large-scale mass distribution recovers only a minority of the information available on the mass fluctuation amplitude. We investigate the recovery of this information in 2D ‘slabs’ of the mass distribution averaged over ≈100 h−1 Mpc along the line of sight, as might be obtained from photometric redshift surveys. We demonstrate a Hamiltonian Monte Carlo method to reconstruct the non-Gaussian mass distribution in slabs, under the assumption that the projected field is a point-transformed Gaussian random field, Poisson-sampled by galaxies. When applied to the Quijote N-body suite at z = 0.5 and at a transverse resolution of 2 h−1 Mpc, the method recovers ∼30 times more information than the 2D power spectrum in the well-sampled limit, recovering the Gaussian limit on information. At a more realistic galaxy sampling density of 0.01 h3 Mpc−3, shot noise reduces the information gain to a factor of 5 improvement over the power spectrum at resolutions of 4 h−1 Mpc or smaller.

Cosmology with the redshift-space galaxy bispectrum monopole at one-loop order

We study the cosmological information content of the redshift-space galaxy bispectrum monopole at one-loop order in perturbation theory. We incorporate all effects necessary for comparison to data: fourth-order galaxy bias, infrared resummation (accounting for the nonlinear evolution of baryon acoustic oscillations), ultraviolet counterterms, nonlinear redshift-space distortions, stochastic contributions, projection, and binning effects. The model is implemented using FFTLog, and validated with the PT Challenge suite of $N$-body simulations, whose large volume allows for high-precision tests. Focusing on the mass fluctuation amplitude, ${\ensuremath{\sigma}}_{8}$, and galaxy bias parameters, we find that including one-loop corrections allow us to significantly extend the range of scales over which the bispectrum can be modeled, and greatly tightens constraints on bias parameters. However, this does not lead to noticeable improvements in the ${\ensuremath{\sigma}}_{8}$ error bar due to the necessary marginalization over a large number of nuisance parameters with conservative priors. Analyzing a BOSS-volume likelihood, we find that the addition of the one-loop bispectrum may lead to improvements on primordial non-Gaussianity constraints by $\ensuremath{\lesssim}30%$ and on ${\ensuremath{\sigma}}_{8}$ by $\ensuremath{\approx}10%$, though we caution that this requires pushing the analysis to short scales where the galaxy bias parameters may not be correctly recovered; this may lead to biases in the recovered parameter values. We conclude that restrictive priors from simulations or higher-order statistics such as the bispectrum multipoles will be needed in order to realize the full information content of the galaxy bispectrum.

Precision analysis of the redshift-space galaxy bispectrum

We study the information content of the angle-averaged (monopole) redshift space galaxy bispectrum. The main novelty of our approach is the use of a systematic tree-level perturbation theory model that includes galaxy bias, IR resummation, and also accounts for nonlinear redshift space distortions, binning, and projection effects. We analyze data from the PT challenge simulations, whose cumulative volume of 566 $h^{-3}$Gpc$^3$ allows for a precise comparison to theoretical predictions. Fitting the power spectrum and bispectrum of our simulated data, and varying all necessary cosmological and nuisance parameters in a consistent Markov chain Monte Carlo analysis, we find that our tree-level bispectrum model is valid up to $k_{\max}=0.08~h{\rm Mpc}^{-1}$ (at $z=0.61$). We also find that inclusion of the bispectrum monopole improves constraints on cosmological parameters by $(5-15)\%$ relative to the power spectrum. The improvement is more significant for the quadratic bias parameters of our simulated galaxies, which we also show to deviate from biases of the host dark matter halos at the $\sim 3\sigma$ level. Finally, we adjust the covariance and scale cuts to match the volume of the BOSS survey, and estimate that within the minimal $\Lambda$CDM model the bispectrum data can tighten the constraint on the mass fluctuation amplitude $\sigma_8$ by roughly $10\%$.

Misinterpreting modified gravity as dark energy: a quantitative study

Standard cosmological data analyses typically constrain simple phenomenological dark-energy parameters, for example the present-day value of the equation of state parameter, w 0, and its variation with scale factor, w a. However, results from such an analysis cannot easily indicate the presence of modified gravity. Even if general relativity does not hold, experimental data could still be fit sufficiently well by a phenomenological w 0 w aCDM, unmodified-gravity model. Hence, it would be very useful to know if there are generic signatures of modified gravity in standard analyses. Here we present, for the first time to our knowledge, a quantitative mapping showing how modified gravity models look when (mis)interpreted within the standard unmodified-gravity analysis. Scanning through a broad space of modified-gravity (Horndeski) models, and assuming a near-future survey consisting of CMB, BAO, and SNIa observations, we report values of the best-fit set of cosmological parameters including (w 0, w a) that would be inferred if modified gravity were at work. We find that modified gravity models that can masquerade as standard gravity lead to very specific biases in standard-parameter spaces. We also comment on implications for measurements of the amplitude of mass fluctuations described by the parameter S 8.

Cosmological constraints from the power spectrum of eBOSS emission line galaxies

We present cosmological parameter measurements from the effective field theory-based full-shape analysis of the power spectrum of emission line galaxies (ELGs). First, we perform extensive tests on simulations and determine appropriate scale cuts for the perturbative description of the ELG power spectrum. We study in detail non-linear redshift-space distortions (``fingers-of-God'') for this sample and show that they are somewhat weaker than those of luminous red galaxies. This difference is not significant for current data, but may become important for future surveys like Euclid/DESI. Then we analyze recent measurements of the ELG power spectrum from the extended Baryon acoustic Oscillation Spectroscopic Survey (eBOSS) within the $\nu\Lambda$CDM model. Combined with the BBN baryon density prior, the ELG pre- and post-reconstructed power spectra alone constrain the matter density $\Omega_m=0.257_{-0.045}^{+0.031}$, the current mass fluctuation amplitude $\sigma_8=0.571_{-0.076}^{+0.052}$, and the Hubble constant $H_0=84.5_{-7}^{+5.8}$ km/s/Mpc (all at 68\% CL). Combining with other full-shape and BAO data we measure $\Omega_m=0.327_{-0.016}^{+0.014}$, $\sigma_8=0.69_{-0.045}^{+0.038}$, and $H_0=68.6_{-1.1}^{+1}$ km/s/Mpc. The total neutrino mass is constrained to be $M_{\rm tot}<0.63$ eV (95\% CL) from the BBN, full-shape and BAO data only. Finally, we discuss the apparent $\sim 3\sigma$ discrepancy in the inferred clustering amplitude between our full shape analysis and the cosmic microwave background data.

A Chi-Squared Analysis of the Measurements of Two Cosmological Parameters over Time

The aim of this analysis was to determine whether or not the given error bars truly represented the dispersion of values in a historical compilation of two cosmological parameters: the amplitude of mass fluctuations (σ8) and Hubble’s constant (H0) parameters in the standard cosmological model. For this analysis, a chi-squared test was executed on a compiled list of past measurements. It was found through analysis of the chi-squared (χ2) values of the data that for σ8 (60 data points measured between 1993 and 2019 and χ2 between 182.4 and 189.0) the associated probability Q is extremely low, with Q=1.6×10−15 for the weighted average and Q=8.8×10−15 for the best linear fit of the data. This was also the case for the χ2 values of H0 (163 data points measured between 1976 and 2019 and χ2 between 480.1 and 575.7), where Q=1.8×10−33 for the linear fit of the data and Q=1.0×10−47 for the weighted average of the data. The general conclusion was that the statistical error bars associated with the observed parameter measurements have been underestimated or the systematic errors were not properly taken into account in at least 20% of the measurements. The fact that the underestimation of error bars for H0 is so common might explain the apparent 4.4σ discrepancy formally known today as the Hubble tension.

Joint analysis of 6dFGS and SDSS peculiar velocities for the growth rate of cosmic structure and tests of gravity

ABSTRACT Measurement of peculiar velocities by combining redshifts and distance indicators is a powerful way to measure the growth rate of a cosmic structure and test theories of gravity at low redshift. Here we constrain the growth rate of the structure by comparing observed Fundamental Plane peculiar velocities for 15 894 galaxies from the 6dF Galaxy Survey (6dFGS) and Sloan Digital Sky Survey (SDSS) with predicted velocities and densities from the 2M++ redshift survey. We measure the velocity scale parameter $\beta \equiv {\Omega _{\rm m}^\gamma }/b = 0.372^{+0.034}_{-0.050}$ and $0.314^{+0.031}_{-0.047}$ for 6dFGS and SDSS, respectively, where Ωm is the mass density parameter, γ is the growth index, and b is the bias parameter normalized to the characteristic luminosity of galaxies, L*. Combining 6dFGS and SDSS, we obtain β = 0.341 ± 0.024, implying that the amplitude of the product of the growth rate and the mass fluctuation amplitude is fσ8 = 0.338 ± 0.027 at an effective redshift z = 0.035. Adopting Ωm = 0.315 ± 0.007, as favoured by Planck and using γ = 6/11 for General Relativity and γ = 11/16 for DGP gravity, we get $S_8(z=0) = \sigma _8 \sqrt{\Omega _{\rm m}/0.3} =0.637 \pm 0.054$ and 0.741 ± 0.062 for GR and DGP, respectively. This measurement agrees with other low-redshift probes of large-scale structure but deviates by more than 3σ from the latest Planck CMB measurement. Our results favour values of the growth index γ &amp;gt; 6/11 or a Hubble constant H0 &amp;gt; 70 km s−1 Mpc−1 or a fluctuation amplitude σ8 &amp;lt; 0.8 or some combination of these. Imminent redshift surveys such as Taipan, DESI, WALLABY, and SKA1-MID will help to resolve this tension by measuring the growth rate of cosmic structure to 1 per cent in the redshift range 0 &amp;lt; z &amp;lt; 1.

Cosmological parameters from the BOSS galaxy power spectrum

We present cosmological parameter measurements from the publicly available Baryon Oscillation Spectroscopic Survey (BOSS) data on anisotropic galaxy clustering in Fourier space. Compared to previous studies, our analysis has two main novel features. First, we use a complete perturbation theory model that properly takes into account the non-linear effects of dark matter clustering, short-scale physics, galaxy bias, redshift-space distortions, and large-scale bulk flows. Second, we employ a Markov-Chain Monte-Carlo technique and consistently reevaluate the full power spectrum likelihood as we scan over different cosmologies. Our baseline analysis assumes minimal ΛCDM, varies the neutrino masses within a reasonably tight range, fixes the primordial power spectrum tilt, and uses the big bang nucleosynthesis prior on the physical baryon density ωb. In this setup, we find the following late-Universe parameters: Hubble constant H0=(67.9± 1.1) km s−1Mpc−1, matter density fraction Ωm=0.295± 0.010, and the mass fluctuation amplitude σ8=0.721± 0.043. These parameters were measured directly from the BOSS data and independently of the Planck cosmic microwave background observations. Scanning over the power spectrum tilt or relaxing the other priors do not significantly alter our main conclusions. Finally, we discuss the information content of the BOSS power spectrum and show that it is dominated by the location of the baryon acoustic oscillations and the power spectrum shape. We argue that the contribution of the Alcock-Paczynski effect is marginal in ΛCDM, but becomes important for non-minimal cosmological models.

Using the Tilted flat-ΛCDM and the Untilted Non-flat ΛCDM Inflation Models to Measure Cosmological Parameters from a Compilation of Observational Data

Abstract We use the physically consistent tilted spatially flat and untilted non-flat ΛCDM inflation models to constrain cosmological parameter values with the Planck 2015 cosmic microwave background (CMB) anisotropy data and recent SNe Ia measurements, baryonic acoustic oscillations (BAO) data, growth rate observations, and Hubble parameter measurements. The most dramatic consequence of including the four non-CMB data sets is the significant strengthening of the evidence for non-flatness in the non-flat ΛCDM model, from 1.8σ for the CMB data alone to 5.1σ for the full data combination. The BAO data is the most powerful of the non-CMB data sets in more tightly constraining model-parameter values and in favoring a spatially closed universe in which spatial curvature contributes about a percent to the current cosmological energy budget. The untilted non-flat ΛCDM model better fits the large-angle CMB temperature anisotropy angular spectrum and is more consistent with the Dark Energy Survey constraints on the current value of the rms amplitude of mass fluctuations (σ 8) as a function of the current value of the nonrelativistic matter-density parameter (Ω m ) but does not provide as good a fit to the smaller-angle CMB temperature anisotropy data, as does the tilted flat-ΛCDM model. Some measured cosmological parameter values differ significantly between the two models, including the reionization optical depth and the baryonic matter density parameter, both of whose 2σ ranges (in the two models) are disjointed or almost so.

No Slip CMB

No Slip Gravity is a simple modified gravity theory with only one free function and the interesting characteristic that— unlike many modified gravity theories—it suppresses growth. This allows it to fit current redshift space distortion and σ8 mass fluctuation amplitude data better than ΛCDM in general relativity, while retaining a ΛCDM background expansion and hence distances. Since it has no gravitational slip it alters equally CMB lensing and matter density growth, and in addition affects the CMB gravitational wave B-mode polarization power spectrum. We investigate and compute the effects of No Slip Gravity for CMB lensing and B-modes, and present a simple analytic approximation. Using a Monte Carlo analysis, we place constraints on the theory from current CMB data.

Observational Constraints on the Tilted Spatially Flat and the Untilted Nonflat ϕCDM Dynamical Dark Energy Inflation Models

Abstract We constrain spatially flat tilted and nonflat untilted scalar field (ϕ) dynamical dark energy inflation (ϕCDM) models by using Planck 2015 cosmic microwave background (CMB) anisotropy measurements and recent baryonic acoustic oscillation distance observations, Type Ia supernovae apparent magnitude data, Hubble parameter measurements, and growth rate data. We assume an inverse power-law scalar field potential energy density . We find that the combination of the CMB data with the four non-CMB data sets significantly improves parameter constraints and strengthens the evidence for nonflatness in the nonflat untilted ϕCDM case from 1.8σ for the CMB measurements only to more than 3.1σ for the combined data. In the nonflat untilted ϕCDM model, current observations favor a spatially closed universe with spatial curvature contributing about two-thirds of a percent of the present cosmological energy budget. The flat tilted ϕCDM model is a 0.4σ better fit to the data than is the standard flat tilted ΛCDM model: current data allow for the possibility that dark energy is dynamical. The nonflat tilted ϕCDM model is in better accord with the Dark Energy Survey bounds on the rms amplitude of mass fluctuations now (σ 8) as a function of the nonrelativistic matter density parameter now (Ω m ) but it does not provide as good a fit to the larger-multipole Planck 2015 CMB anisotropy data as does the standard flat tilted ΛCDM model. A few cosmological parameter value measurements differ significantly when determined using the tilted flat and the untilted nonflat ϕCDM models, including the cold dark matter density parameter and the reionization optical depth.

Noise estimates for measurements of weak lensing from the Ly α forest

We have proposed a method for measuring weak lensing using the Lyman-alpha forest. Here we estimate the noise expected in weak lensing maps and power spectra for different sets of observational parameters. We find that surveys of the size and quality of the ones being done today and ones planned for the future will be able to measure the lensing power spectrum at a source redshift of z~2.5 with high precision and even be able to image the distribution of foreground matter with high fidelity on degree scales. For example, we predict that Lyman-alpha forest lensing measurement from the Dark Energy Spectroscopic Instrument survey should yield the mass fluctuation amplitude with statistical errors of 1.5%. By dividing the redshift range into multiple bins some tomographic lensing information should be accessible as well. This would allow for cosmological lensing measurements at higher redshift than are accessible with galaxy shear surveys and correspondingly better constraints on the evolution of dark energy at relatively early times.

Testing ΛCDM at the lowest redshifts with SN Ia and galaxy velocities

Peculiar velocities of objects in the nearby universe are correlated due to the gravitational pull of large-scale structure. By measuring these velocities, we have a unique opportunity to test the cosmological model at the lowest redshifts. We perform this test, using current data to constrain the amplitude of the ``signal'' covariance matrix describing the velocities and their correlations. We consider a new, well-calibrated ``Supercal'' set of low-redshift SNe Ia as well as a set of distances derived from the fundamental plane relation of 6dFGS galaxies. Analyzing the SN and galaxy data separately, both results are consistent with the peculiar velocity signal of our fiducial ΛCDM model, ruling out the noise-only model with zero peculiar velocities at greater than 7σ (SNe) and 8σ (galaxies). When the two data sets are combined appropriately, the precision of the test increases slightly, resulting in a constraint on the signal amplitude of A = 1.05−0.21+0.25, where A = 1 corresponds to our fiducial model. Equivalently, we report an 11% measurement of the product of the growth rate and amplitude of mass fluctuations evaluated at zeff = 0.02, f σ8 = 0.428−0.045+0.048, valid for our fiducial ΛCDM model. We explore the robustness of the results to a number of conceivable variations in the analysis and find that individual variations shift the preferred signal amplitude by less than ∼0.5σ. We briefly discuss our Supercal SN Ia results in comparison with our previous results using the JLA compilation.

Large-scale clustering of Lyman α emission intensity from SDSS/BOSS

(Abridged) We detect the large-scale structure of Lya emission in the Universe at redshifts z=2-3.5 by measuring the cross-correlation of Lya surface brightness with quasars in SDSS/BOSS. We use a million spectra targeting Luminous Red Galaxies at z<0.8, after subtracting a best fit model galaxy spectrum from each one, as an estimate of the high-redshift Lya surface brightness. The quasar-Lya emission cross-correlation we detect has a shape consistent with a LambdaCDM model with Omega_M =0.30^+0.10-0.07. The predicted amplitude of this cross-correlation is proportional to the product of the mean Lya surface brightness, <mu_alpha>, the amplitude of mass fluctuations, and the quasar and Lya emission bias factors. Using known values, we infer <mu_alpha>(b_alpha/3) = (3.9 +/- 0.9) x 10^-21 erg/s cm^-2 A^-1 arcsec^-2, where b_alpha is the Lya emission bias factor. If the dominant sources of Lya emission are star forming galaxies, we infer rho_SFR = (0.28 +/- 0.07) (3/b_alpha) /yr/Mpc^3 at z=2-3.5. For b_alpha=3, this value is a factor of 21-35 above previous estimates from individually detected Lya emitters, although consistent with the total rho_SFR derived from dust-corrected, continuum UV surveys. 97% of the Lya emission in the Universe at these redshifts is therefore undetected in previous surveys of Lya emitters. Our measurement is much greater than seen from stacking analyses of faint halos surrounding previously detected Lya emitters, but we speculate that it arises from similar Lya halos surrounding all luminous star-forming galaxies. We also detect redshift space anisotropy of the quasar-Lya emission cross-correlation, finding evidence at the 3.0 sigma level that it is radially elongated, consistent with distortions caused by radiative-transfer effects (Zheng et al. (2011)). Our measurements represent the first application of the intensity mapping technique to optical observations.

Galaxy cluster number count data constraints on cosmological parameters

[Abridged] We use data on massive galaxy clusters ($M_{\rm cluster} > 8 \times 10^{14} h^{-1} M_\odot$ within a comoving radius of $R_{\rm cluster} = 1.5 h^{-1}\Mpc$) in the redshift range $0.05 \lesssim z \lesssim 0.83$ to place constraints, simultaneously, on the nonrelativistic matter density parameter $\Omega_m$, on the amplitude of mass fluctuations $\sigma_8$, on the index $n$ of the power-law spectrum of the density perturbations, and on the Hubble constant $H_0$, as well as on the equation-of-state parameters $(w_0,w_a)$ of a smooth dark energy component. For the first time, we properly take into account the dependence on redshift and cosmology of the quantities related to cluster physics: the critical density contrast, the growth factor, the mass conversion factor, the virial overdensity, the virial radius and, most importantly, the cluster number count derived from the observational temperature data. We show that, contrary to previous analyses, cluster data alone prefer low values of the amplitude of mass fluctuations, $\sigma_8 \leq 0.69 (1\sigma C.L.)$, and large amounts of nonrelativistic matter, $\Omega_m \geq 0.38 (1\sigma C.L.)$, in slight tension with the $\Lambda$CDM concordance cosmological model, though the results are compatible with $\Lambda$CDM at $2\sigma$. In addition, we derive a $\sigma_8$ normalization relation, $\sigma_8 \Omega_m^{1/3} = 0.49 \pm 0.06 (2\sigma C.L.)$.

Radiation-induced large-scale structure during the reionization epoch: the autocorrelation function

The structures produced during the epoch of reionization by the action of radiation on neutral hydrogen are in principle different from those that arise through gravitational growth of initially small perturbations. We explore the difference between the two mechanisms using high-resolution cosmological radiative transfer. Our computations use a Monte Carlo code which ray-traces directly through smoothed particle hydrodynamics (SPH) kernels without a grid, preserving the high spatial resolution of the underlying hydrodynamic simulation. Because the properties of the first sources of radiation are uncertain, we simulate a range of models with different source properties and recombination physics. We examine the morphology of the neutral hydrogen distribution and the reionization history in these models. We find that at fixed mean neutral fraction, structures are visually most affected by the existence of a lower limit in source luminosity, then by galaxy mass-to-light ratio, and are minimally affected by changes in the recombination rate and amplitude of mass fluctuations. We concentrate on the autocorrelation function of the neutral hydrogen, ξ Hl (r), as a basic quantitative measure of radiation-induced structure. All the models we test exhibit a characteristic behaviour, with ξ Hl becoming initially linearly antibiased with respect to the matter correlation function, reaching a minimum bias factor b ∼ 0.5 when the universe is ∼ 10-20 per cent ionized. After this ξ Hl increases rapidly in amplitude, overtaking the matter correlation function. It keeps a power-law shape, but flattens considerably, reaching an asymptotic logarithmic slope of γ Hl ≃ -0.5. The growth rate of H i fluctuations is exponentially more rapid than gravitational growth over a brief interval of redshift Δ z ∼ 2-3.

Systematic Bias in an Estimate of the Cluster Mass and the Fluctuation Amplitude from Cluster Abundance Statistics

Abstract We revisit an estimate of the mass fluctuation amplitude, $\sigma_{8}$, from the observational X-ray cluster abundance. In particular, we examine the effect of the systematic difference between the cluster virial mass estimated from the X-ray spectroscopy, $M_\mathrm{vir,\, spec}$, and the true virial mass of the corresponding halo, $M_\mathrm{vir}$. Mazzotta et al. (2004, MNRAS, 354, 10) recently pointed out the possibility that $\alpha_{M} = M_\mathrm{vir,\, spec} / M_\mathrm{vir}$ is systematically lower than unity. We performed a statistical analysis by combining the latest X-ray cluster sample and the improved theoretical models, and found that $\sigma_8 \sim 0.76 \pm 0.01 + 0.50 \, (1-\alpha_{M})$ for $0.5 \le \alpha_{M} \le 1$, where the quoted errors are only statistical. Thus, if $\alpha_{M} \sim 0.7$, the value of $\sigma_8$ from cluster abundance alone is now in better agreement with other cosmological data, including the cosmic microwave background, the galaxy power spectrum and the weak lensing data. The current study also illustrates the importance of possible systematic effects in mapping real clusters to underlying dark halos, which changes the interpretation of cluster abundance statistics.

Cluster Ellipticities as a Cosmological Probe

We investigate the dependence of the ellipticities of clusters of galaxies on cosmological parameters using large-scale cosmological simulations. We determine cluster ellipticities out to redshift unity for λCDM models with different mean densities Ωm and amplitudes of mass fluctuation σ8,0. The mean ellipticity increases monotonically with redshift for all models. Larger values of σ8,0, i.e., earlier cluster formation times, produce lower ellipticities. The dependence of ellipticity on Ωm is relatively weak in the range 0.2 ≤ Ωm ≤ 0.5 for high-mass clusters. The mean ellipticity (z) decreases linearly with the amplitude of mass fluctuations at the cluster redshift z, nearly independent of Ωm; on average, older clusters are more relaxed and are thus less elliptical. The distribution of ellipticities about the mean is approximated by a Gaussian, allowing a simple characterization of the evolution of ellipticity with redshift as a function of cosmological parameters. At z = 0, the mean ellipticity of high-mass clusters is approximated by (z = 0) = 0.245 - 0.070σ8,0 + 0.020Ωm,0. This relation opens up the possibility that, when compared with future observations of large cluster samples, the mean cluster ellipticity and its evolution could be used as a new, independent tool to constrain cosmological parameters, especially the amplitude of mass fluctuations, σ8,0.