arXiv subjects

Li-Zhi Fang

Publications and source records attributed to Li-Zhi Fang.

At least 19 recordsLinked to original sources

arXiv2013

Turbulence in the Inter-galactic Medium: Solenoidal and Dilatational Motions, and the Impact of Numerical Viscosity

We use a suite of cosmological hydrodynamic simulations, run by two fixed grid codes, to investigate the properties of solenoidal and dilatational motions of the intergalactic medium (IGM), and the impact of numerical viscosity on turbulence in a LCDM universe. The codes differ only in the spatial difference discretization. We find that (1) The vortical motion grows rapidly since $z=2$, and reaches $\sim 10 km/s -90 km/s$ at $z=0$. Meanwhile, the small-scale compressive ratio $r_{CS}$ drops from 0.84 to 0.47, indicating comparable vortical and compressive motions at present. (2) Power spectra of the solenoidal velocity possess two regimes, $\propto k^{-0.89}$ and $\propto k^{-2.02}$, while the total and dilatational velocity follow the scaling $k^{-1.88}$ and $k^{-2.20}$ respectively in the turbulent range. The IGM turbulence may contain two distinct phases, the supersonic and post-supersonic phases. (3) The non-thermal pressure support, measured by the vortical kinetic energy, is comparable with the thermal pressure for $\rho_b \simeq 10-100$, or $T <10^{5.5} K$ at $z=0.0$. The deviation of the baryon fraction from the cosmic mean shows a preliminary positive correlation with the turbulence pressure support. (4) A relatively higher numerical viscosity would dissipate both the compressive and vortical motions of the IGM into thermal energy more effectively, resulting in less developed vorticity, remarkably shortened inertial range, and leading to non-negligible uncertainty in the thermal history of gas accretion. Shocks in regions outside of clusters are significantly suppressed by numerical viscosity since $z=2$, which may directly cause the different levels of turbulence between two codes.

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arXiv2011

Time-Dependent Behavior of Lyman$α$ Photon Transfer in High Redshift Optically Thick Medium

With Monte Carlo simulation method, we investigate the time dependent behavior of Ly$α$ photon transfer in optically thick medium of the concordance $Λ$CDM universe. At high redshift, the Ly$α$ photon escaping from optically thick medium has a time scale as long as the age of the luminous object, or even comparable to the age of the universe. In this case, time-independent, or stationary solutions of the Ly$α$ photon transfer with resonant scattering will overlook important features of the escaped Ly$α$ photons in physical and frequency spaces. More seriously, the expansion of the universe leads to that the time-independent solutions of the Ly$α$ photon transfer may not exist. We show that time-dependent solutions sometimes are essential for understanding the Ly$α$ emission and absorption at high redshifts. For Lyαphotons from sources at redshift 1+z=10 and being surrounded by neutral hydrogen IGM of the $Λ$CDM universe, the escape coefficient is found to be always less, or much less than one, regardless of the age or life time of the sources. Under such environment, we also find that even when the Ly$α$ photon luminosity of the sources is stable, the mean surface brightness is gradually increasing in the first 10^6 years, and then decreasing with a power law of time, but never approaches a stable, time-independent state. That is, all 1+z=10 sources in a neutral Hubble expanding IGM with Ly$α$ luminosity L have their maximum of mean surface brightness ~ 10^{-21}(L/(10^{43}erg/s)) erg s^{-1} cm^{-2} arcsec^{-2} at the age of about 10^6 years. The time-dependent effects on the red damping wing profile are also addressed.

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arXiv2011

Effect of Dust on Lyman-alpha Photon Transfer in Optically Thick Halo

We investigate the effects of dust on Lyα photons emergent from an optically thick medium by solving the integro-differential equation of the radiative transfer of resonant photons. To solve the differential equations numerically we use the Weighted Essentially Non-oscillatory method (WENO). Although the effects of dust on radia-tive transfer is well known, the resonant scattering of Lyα photons makes the problem non-trivial. For instance, if the medium has the optical depth of dust absorption and scattering to be τa>>1, τ>>1, and τ>>τa, the effective absorption optical depth in a random walk scenario would be equal to \surd τa(τa+τ). We show, however, that for a resonant scattering at frequency ν0, the effective absorption optical depth would be even larger than τ(ν0). If the cross section of dust scattering and absorption is frequency-independent, the double-peaked structure of the frequency profile given by the resonant scattering is basically dust-independent. That is, dust causes neither narrowing nor widening of the width of the double peaked profile. One more result is that the time scales of the Lyα photon transfer in the optically thick halo are also basically independent of the dust scattering, even when the scattering is anisotropic. This is because those time scales are mainly determined by the transfer in the frequency space, while dust scattering, either isotropic or anisotropic, does not affect the behavior of the transfer in the frequency space when the cross section of scattering is wavelength-independent. This result does not support the speculation that dust will lead to the smoothing of the brightness distribution of Lyα photon source with optical thick halo.

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arXiv2011

Intermittence of the Map of Kinetic Sunyaev-Zel'dovich Effect and Turbulence of IGM

We investigate the possibility of detecting the turbulent state of the IGM with the kinetic Sunyaev-Zel'dovich (kSZ) effect. Being sensitive to the divergence-free component of the momentum field of the IGM, the kSZ effect might be used to probe the vorticity of the turbulent IGM. With cosmological hydrodynamical simulation in the concordance $Λ$CDM universe, we find that the structure functions of 2D kSZ maps show strong intermittence, and the intermittent exponents follow a law similar to the She-Leveque scaling formula of fully developed turbulence. We also find that the intermittence is weak in the maps of thermal Sunyaev-Zel'dovich (tSZ) effect. Nevertheless, the superposition of the kSZ and tSZ effects still contain significant intermittence. We conclude that the turbulent behavior of the IGM may be revealed by the observation of SZ effect on angular scales equal to or less than 0.5 arcminute, corresponding to the multipole parameter $l\geq 2 \times10^4$.

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arXiv2010

Statistical and dynamical decoupling of the IGM from Dark Matter

The mean mass densities of cosmic dark matter is larger than that of baryonic matter by a factor of about 5 in the $Λ$CDM universe. Therefore, the gravity on large scales should be dominant by the distribution of dark matter in the universe. However, a series of observations incontrovertibly show that the velocity and density fields of baryonic matter are decoupling from underlying dark matter field. This paper shows our attemps to unveil the physics behind this puzzle. In linear approximation, the dynamics of the baryon fluid is completely governed by the gravity of the dark matter. Consequently, the mass density field of baryon matter $ρ_b({\bf r},t)$ will be proportional to that of dark matter $ρ_{\rm dm}({\bf r},t)$, even though they are different from each other initially. In weak and moderate nonlinear regime, the dynamics of the baryon fluid can be sketched by Burgers equation. A basic feature of the Burgers dynamics is to yield shocks. When the Reynolds number is large, the Burgers fluid will be in the state of Burgers turbulence, which consists of shocks and complex structures. On the other hand, the collisionless dark matter may not show such shock, but a multivalued velocity field. Therefore, the weak and moderate nonlinear evolution leads to the IGM-dark matter deviation. Yet, the velocity field of Burgers fluid is still irrotational, as gravity is curl-free. In fully nonlinear regime, the vorticity of velocity field developed, and the cosmic baryonic fluid will no longer be potential, as the dynamics of vorticity is independent of gravity and can be self maintained by the nonlinearity of hydrodynamics. In this case, the cosmic baryon fluid is in the state of fully developed turbulence, which is statistically and dynamically decoupling from dark matter. This scenario provides a mechanism of cohenent explanation of observations.

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arXiv2010

Log-Poisson Non-Gaussianity of Ly$α$ Transmitted Flux Fluctuations at High Redshift

We investigate the non-Gaussian features of the IGM at redshift $z\sim 5 - 6$ using Ly$α$ transmitted flux of quasar absorption spectra and cosmological hydrodynamic simulation of the concordance $Λ$CDM universe. We show that the neutral hydrogen mass density field and Ly$α$ transmitted flux fluctuations possess all the non-Gaussian features predicted by the log-Poisson hierarchy, which depends only on two dimensionless parameters $β$ and $γ$, describing, respectively, the intermittence and singularity of the random fields. We find that the non-Gaussianity of the Ly$α$ transmitted flux of quasars from $z=4.9$ to $z=6.3$ can be well reconstructed by the hydrodynamical simulation samples. Although the Gunn-Peterson optical depth and its variance underwent a significant evolution in the redshift range of $5 - 6$, the intermittency measured by $β$ is almost redshift-independent in this range. More interesting, the intermittency of quasar's absorption spectra on physical scales $0.1-1$ h$^{-1}$Mpc in redshift $5 - 6$ are found to be about the same as that on physical scales $1-10$ h$^{-1}$Mpc at redshifts $2 - 4$. Considering the Jeans length is less than 0.1 h$^{-1}$Mpc at $z\sim 5$, and $1$ h$^{-1}$Mpc at $z\sim 2$, these results imply that the nonlinear evolution in high and low redshifts will lead the cosmic baryon fluid to a state similar to fully developed turbulence. The log-Poisson high order behavior of current high redshift data of quasar's spectrum can be explained by uniform UV background in the redshift range considered. We also studied the log-Poisson non-Gaussianity by considering inhomogeneous background. With several simplified models of inhomogeneous background, we found the effect of the inhomogeneous background on the log-Poisson non-Gaussianity is not larger than 1-sigma.

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arXiv2010

Resonant Scattering and Ly-alpha Radiation Emergent from Neutral Hydrogen Halos

With a state-of-the-art numerical method for solving the integral-differential equation of radiative transfer, we investigate the flux of the Ly$α$ photon $ν_0$ emergent from an optically thick halo containing a central light source. Our focus is on the time-dependent effects of the resonant scattering. We first show that the frequency distribution of photons in the halo are quickly approaching to a locally thermalized state around the resonant frequency, even when the mean intensity of the radiation is highly time-dependent. Since initial conditions are forgotten during the thermalization, some features of the flux, such as the two peak structure of its profile, actually are independent of the intrinsic width and time behavior of the central source, if the emergent photons are mainly from photons in the thermalized state. In this case, the difference $|ν_{\pm}-ν_0|$, where $ν_{\pm}$ are the frequencies of the two peaks of the flux, cannot be less than $2$ times of Doppler broadening. We then study the radiative transfer in the case where the light emitted from the central source is a flash. We calculate the light curves of the flux from the halo. It shows that the flux is still a flash. The time duration of the flash for the flux, however, is independent of the original time duration of the light source but depends on the optical depth of the halo. Therefore, the spatial transfer of resonant photons is a diffusion process, even though it is not a purely Brownian diffusion. This property enables an optically thick halo to trap and store thermalized photons around $ν_0$ for a long time after the cease of the central source emission. The photons trapped in the halo can yield delayed emission, of which the profile also shows typical two peak structure as that from locally thermalized photons. Possible applications of these results are addressed.

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arXiv2010

Vorticity of IGM Velocity Field on Large Scales

We investigate the vorticity of the IGM velocity field on large scales with cosmological hydrodynamic simulation of the concordance model of LCDM. We show that the vorticity field is significantly increasing with time as it can effectively be generated by shocks and complex structures in the IGM. Therefore, the vorticity field is an effective tool to reveal the nonlinear behavior of the IGM, especially the formation and evolution of turbulence in the IGM. We find that the vorticity field does not follow the filaments and sheets structures of underlying dark matter density field and shows highly non- Gaussian and intermittent features. The power spectrum of the vorticity field is used to measure the development of turbulence in Fourier space. We show that the relation between the power spectra of vorticity and velocity fields is perfectly in agreement with the prediction of a fully developed homogeneous and isotropic turbulence from 0.2 to 3 h^{-1} Mpc at z~0. This indicates that cosmic baryonic field is in the state of fully developed turbulence on scales less than about 3 h^{-1} Mpc. The random field of the turbulent fluid yields turbulent pressure to prevent the gravitational collapsing of the IGM. The vorticity and turbulent pressure are strong inside and even outside of high density regions. In IGM regions with 10 times mean overdensity, the turbulent pressure can be on an average equivalent to the thermal pressure of the baryonic gas with a temperature of 10^5 K. The fully developed turbulence would prevent the baryons in the IGM from falling into the gravitational well of dark matter halos. Moreover, turbulent pressure is dynamical and non-thermal, which makes it different from pre-heating mechanism as it does not affect the thermal state and ionizing process of hydrogen in the IGM.

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arXiv2009

A wavelet-Galerkin algorithm of the E/B decomposition of CMB polarization maps

We develop an algorithm of separating the $E$ and $B$ modes of the CMB polarization from the noisy and discretized maps of Stokes parameter $Q$ and $U$ in a finite area. A key step of the algorithm is to take a wavelet-Galerkin discretization of the differential relation between the $E$, $B$ and $Q$, $U$ fields. This discretization allows derivative operator to be represented by a matrix, which is exactly diagonal in scale space, and narrowly banded in spatial space. We show that the effect of boundary can be eliminated by dropping a few DWT modes located on or nearby the boundary. This method reveals that the derivative operators will cause large errors in the $E$ and $B$ power spectra on small scales if the $Q$ and $U$ maps contain Gaussian noise. It also reveals that if the $Q$ and $U$ maps are random, these fields lead to the mixing of the $E$ and $B$ modes. Consequently, the $B$ mode will be contaminated if the powers of $E$ modes are much larger than that of $B$ modes. Nevertheless, numerical tests show that the power spectra of both $E$ and $B$ on scales larger than the finest scale by a factor of 4 and higher can reasonably be recovered, even when the power ratio of $E$- to $B$-modes is as large as about 10$^2$, and the signal-to-noise ratio is equal to 10 and higher. This is because the Galerkin discretization is free of false correlations, and keeps the contamination under control. As wavelet variables contain information of both spatial and scale spaces, the developed method is also effective to recover the spatial structures of the $E$ and $B$ mode fields.

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arXiv2009

A WENO algorithm for radiative transfer with resonant scattering: the time scale of the Wouthuysen-Field Coupling

We develop a numerical solver for the integral-differential equations, which describes the radiative transfer of photon distribution in the frequency space with resonant scattering of Lyalpha photons by hydrogen gas in the early universe. The time-dependent solutions of this equation is crucial to the estimation of the effect of the Wouthuysen-Field (WF) coupling in relation to the 21 cm emission and absorption at the epoch of reionization. The resonant scattering leads to the photon distribution in the frequency space to be piecewise smooth containing sharp changes. The weighted essentially nonoscillatory (WENO) scheme is suitable to handle this problem, as this algorithm has been found to be highly stable and robust for solving Boltzmann equation. We test this numerical solver against analytic solutions of the evolution of the photon distribution in rest background, analytic solution in expanding background without resonant scattering and formation of local Boltzmann distribution around the resonant frequency with the temperature same as that of atom for recoil. We find that evolution of photon distribution undergoes three phases; profile is similar to the initial one, a flat plateau (without recoil) or local Boltzmann distribution (with recoil) forms around the resonant frequency, and finally the distribution around the resonant frequency is saturated when the photons from the source is balanced by the redshift of the expansion. This result indicates that the onset of the W-F coupling should not be determined by the third phase, but by the time scale of the second phase. We found that the time scale of the W-F coupling is equal to about a few hundreds of the mean free flight time of photons with resonant frequency, and is independent of the Sobolev parameter if this parameter is much less than 1.

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arXiv2008

Log-Poisson Hierarchical Clustering of Cosmic Neutral Hydrogen and Ly-alpha Transmitted Flux of QSO Absorption Spectrum

we study, in this paper, the non-Gaussian features of the mass density field of neutral hydrogen fluid and the Ly-alpha transmitted flux of QSO absorption spectrum from the point-of-view of self-similar log-Poisson hierarchy. It has been shown recently that, in the scale range from the onset of nonlinear evolution to dissipation, the velocity and mass density fields of cosmic baryon fluid are extremely well described by the She-Leveque's scaling formula, which is due to the log-Poisson hierarchical cascade. Since the mass density ratio between ionized hydrogen to total hydrogen is not uniform in space, the mass density field of neutral hydrogen component is not given by a similar mapping of total baryon fluid. Nevertheless, we show, with hydrodynamic simulation samples of the concordance $Λ$CDM universe, that the mass density field of neutral hydrogen, is also well described by the log-Poisson hierarchy. We then investigate the field of Ly$α$ transmitted flux of QSO absorption spectrum. Due to redshift distortion, Ly$α$ transmitted flux fluctuations are no longer to show all features of the log-Poisson hierarchy. However, some non-Gaussian features predicted by the log-Poisson hierarchy are not affected by the redshift distortion. We test these predictions with the high resolution and high S/N data of quasars Ly$α$ absorption spectra. All results given by real data, including $β$-hierarchy, high order moments and scale-scale correlation, are found to be well consistent with the log-Poisson hierarchy. We compare the log-Poisson hierarchy with the popular log-normal model of the Ly$α$ transmitted flux. The later is found to yield too strong non-Gaussianity at high orders, while the log-Poisson hierarchy is in agreement with observed data.

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arXiv2008

Scaling Relation between Sunyaev-Zel'dovich Effect and X-ray Luminosity and Scale-Free Evolution of Cosmic Baryon Field

It has been revealed recently that, in the scale free range, i.e. from the scale of the onset of nonlinear evolution to the scale of dissipation, the velocity and mass density fields of cosmic baryon fluid are extremely well described by the self-similar log-Poisson hierarchy. As a consequence of this evolution, the relations among various physical quantities of cosmic baryon fluid should be scale invariant, if the physical quantities are measured in cells on scales larger than the dissipation scale, regardless the baryon fluid is in virialized dark halo, or in pre-virialized state. We examine this property with the relation between the Compton parameter of the thermal Sunyaev-Zel'dovich effect, $y(r)$, and X-ray luminosity, $L_{\rm x}(r)$, where $r$ being the scale of regions in which $y$ and $L_{\rm x}$ are measured. According to the self-similar hierarchical scenario of nonlinear evolution, one should expect that 1.) in the $y(r)$-$L_x(r)$ relation, $y(r)=10^{A(r)}[L_{\rm x}(r)]^{α(r)}$, the coefficients $A(r)$ and $α(r)$ are scale-invariant; 2.) The relation $y(r)=10^{A(r)}[L_{\rm x}(r)]^{α(r)}$ given by cells containing collapsed objects is also available for cells without collapsed objects, only if $r$ is larger than the dissipation scale. These two predictions are well established with a scale decomposition analysis of observed data, and a comparison of observed $y(r)$-$L_x(r)$ relation with hydrodynamic simulation samples. The implication of this result on the characteristic scales of non-gravitational heating is also addressed.

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arXiv2007

Ly$α$ Leaks in the Absorption Spectra of High Redshift QSOs

Spectra of high redshift QSOs show deep Gunn-Peterson absorptions on the blue sides of the \Lya emissions lines. They can be decomposed into components called \Lya leaks, defined to be emissive regions in complementary to otherwise zero-fluxed absorption gaps. Just like \Lya absorption forests at low redshifts, \Lya leaks are both easy to find in observations and containing rich sets of statistical properties that can be used to study the early evolution of the IGM. Among all properties of a leak profile, we investigate its equivalent width in this paper, since it is weakly affected by instrumental resolution and noise. Using 10 Keck QSO spectra at $z\sim6$, we have measured the number density distribution function $n(W,z)$, defined to be the number of leaks per equivalent width $W$ and per redshift $z$, in the redshift range $5.4 - 6.0$. These new observational statistics, in both the differential and cumulative forms, fit well to hydro numerical simulations of uniform ionizing background in the $Λ$CDM cosmology. In this model, Ly $α$ leaks are mainly due to low density voids. It supports the early studies that the IGM at $z\simeq6$ would still be in a highly ionized state with neutral hydrogen fraction $\simeq 10^{-4}$. Measurements of $n(W,z)$ at $z>6$ would be effective to probe the reionization of the IGM.

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arXiv2007

Ly$α$ Leaks and Reionization

Ly$α$ absorption spectra of QSOs at redshifts $z\simeq6$ show complete Gunn-Peterson absorption troughs (dark gaps) separated by tiny leaks. The dark gaps are from the intergalactic medium (IGM) where the density of neutral hydrogen are high enough to produce almost saturated absorptions, however, where the transmitted leaks come from is still unclear so far. We demonstrate that leaking can originate from the lowest density voids in the IGM as well as the ionized patches around ionizing sources using semi-analytical simulations. If leaks were produced in lowest density voids, the IGM might already be highly ionized, and the ionizing background should be almost uniform; in contrast, if leaks come from ionized patches, the neutral fraction of IGM would be still high, and the ionizing background is significantly inhomogeneous. Therefore, the origin of leaking is crucial to determining the epoch of inhomogeneous-to-uniform transition of the the ionizing photon background. We show that the origin could be studied with the statistical features of leaks. Actually, Ly$α$ leaks can be well defined and described by the equivalent width $W$ and the full width of half area $W_{\rm H}$, both of which are less contaminated by instrumental resolution and noise. It is found that the distribution of $W$ and $W_{\rm H}$ of Ly$α$ leaks are sensitive to the modeling of the ionizing background. We consider four representative reionization models. It is concluded that the leak statistics provides an effective tool to probe the evolutionary history of reionization at $z\simeq5-6.5$. Similar statistics would also be applicable to the reionization of He II at $z \simeq 3$(Abridged)

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arXiv2007

Non-Gaussianity of the Cosmic Baryon Fluid: Log-Poisson Hierarchy Model

In the nonlinear regime of cosmic clustering, the mass density field of the cosmic baryon fluid is highly non-Gaussian. It shows different dynamical behavior from collisionless dark matter. Nevertheless, the evolved field of baryon fluid is scale-covariant in the range from the Jeans length to a few ten h^{-1} Mpc, in which the dynamical equations and initial perturbations are scale free. We show that in the scale-free range, the non-Gaussian features of the cosmic baryon fluid, governed by the Navier-Stokes equation in an expanding universe, can be well described by a log-Poisson hierarchical cascade. The log-Poisson scheme is a random multiplicative process (RMP), which causes non-Gaussianity and intermittency even when the original field is Gaussian. The log-Poisson RMP contains two dimensionless parameters: $β$ for the intermittency and $γ$ for the most singular structure. All the predictions given by the log-Poisson RMP model, including the hierarchical relation, the order dependence of the intermittent exponent, the moments, and the scale-scale correlation, are in good agreement with the results given by hydrodynamic simulations of the standard cold dark matter model. The intermittent parameter $β$ decreases slightly at low redshift and indicates that the density field of baryon fluid contains more singular structures at lower redshifts. The applicability of the model is addressed.

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arXiv2007

A WENO Algorithm for the Growth of Ionized Regions at the Reionization Epoch

We investigate the volume growth of ionized regions around UV photon sources with the WENO algorithm, which is an effective solver of photon kinetics in the phase space described by the radiative transfer equation. We show that the volume growth rate, either of isolated ionized regions or of clustered regions in merging, generally consists of three phases: fast or relativistic growth phase at the early stage, slow growth phase at the later stage, and a transition phase between the fast and slow phases. We also show that the volume growth of ionized regions around clustered sources with intensity $\dot{E}_i$ ($i=1, 2, ...$) would have the same behavior as a single source with intensity $\dot{E}=\sum_i\dot{E}_i$, if all the distances between nearest neighbor sources $i$ and $j$ are smaller than $c(t^i_c+t^j_c)$, $t^i_c$ being the time scale $t_c$ of source $i$. Therefore, a tightly clustered UV photon sources would lead to a slow growth of ionized volume. This effect would be important for studying the redshift-dependence of 21cm signals from the reionization epoch.

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arXiv2007

21 cm Signals from Early Ionizing Sources

We investigate the 21 cm signals from the UV ionizing sources in the reionization epoch. The formation and evolution of 21 cm emission and absorption regions depend essentially on the kinetics of photons in the physical and frequency spaces. To solve the radiative transfer equation, we use the WENO algorithm, which is effective to capture the sharp ionization profile and the cut-off at the front of light $(r=ct)$ and to handle the small fraction of neutral hydrogen and helium in the ionized sphere. We show that a spherical shell of 21 cm emission and absorption will develop around a point source once the speed of the ionization front (I-front) is significantly lower than the speed of light. The 21 cm shell extends from the I-front to the front of light; its inner part is the emission region and its outer part is the absorption region. The 21 cm emission region depends strongly on the intensity, frequency-spectrum and life-time of the UV ionizing source. For a source of short life-time, no 21 cm emission region can be formed if the source dies out before the I-front speed is significantly lower than the speed of light. Yet, a 21 cm absorption region can form and develop even after the emission of the source ceases.

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arXiv2007

Estimating Power Spectrum of Sunyaev-Zeldovich Effect from the Cross-Correlation between WMAP and 2MASS

We estimate the power spectrum of SZ(Sunyaev-Zel'dovich)-effect-induced temperature fluctuations on sub-degree scales by using the cross correlation between the three-year WMAP maps and 2MASS galaxy distribution. We produced the SZ effect maps by hydrodynamic simulation samples of the $Λ$CDM model, and show that the SZ effect temperature fluctuations are highly non-Gaussian. The PDF of the temperature fluctuations has a long tail. More than 70% power of the SZ effect temperature fluctuations attributes to top $\sim 1%$ wavelet modes (long tail events). On the other hand, the CMB temperature fluctuations basically are Gaussian. Although the mean power of CMB temperature fluctuations on sub-degree scales is much higher than that of SZ effect map, the SZ effect temperature fluctuations associated with top 2MASS clusters is comparable to the power of CMB temperature fluctuations on the same scales. Thus, from noisy WMAP maps, one can have a proper estimation of the SZ effect power at the positions of the top 2MASS clusters. The power spectrum given by these top wavelet modes is useful to constrain the parameter of density fluctuations amplitude $σ_8$. We find that the power spectrum of these top wavelet modes of SZ effect on sub-degree scales basically is consistent with the simulation maps produced with $σ_8=0.84$. The simulation samples of $σ_8=0.74$ show, however, significant deviation from detected SZ power spectrum. It can be ruled out with confidence level 99% if all other cosmological parameters are the same as that given by the three-year WMAP results.

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