arXiv subjects

F. Bernardeau

Publications and source records attributed to F. Bernardeau.

80 records · Page 5Linked to original sources

arXiv1995

The angular correlation hierarchy in the quasilinear regime.

For Gaussian initial conditions the perturbation theory predicts a very specific hierarchy for the projected matter $p$-point correlation functions. In the small angle approximation and assuming a power-law spectrum I derive the exact expressions of the coefficients $s_p$ relating the averaged $p$-order angular correlation function, $\omb_p$ to the second one, $\omb_p=s_p\ \omb_2^{p-1}$. These results are valid for any selection function, but for a top-hat angular filter only. These coefficients are found to be significantly higher than their 3D counterparts, $S_p=\xib_p/\xib_2^{p-1}$. For the coefficient $s_3$ I discussed the accuracy of the small angle approximation by computing, for particular examples, its angular dependence with Monte-Carlo numerical integrations. It is found that the accuracy of the small angle approximation for $θ\approx 1^o$ slightly depends on the selection function. Using the selection function expected for galaxy catalogues the approximation is found to be reasonably good. The measurements of the $s_p$ parameters made in the APM angular survey are found to give systematic lower values than the theoretical predictions. How significant this discrepancy is and what the implications would be for galaxy formation models is discussed in the last section.

astro-ph↗

arXiv1994

Omega from the skewness of the cosmic velocity divergence

We propose a method for measuring the cosmological density parameter $Ω$ from the statistics of the divergence field, $θ\equiv H^{-1} ÷v$, the divergence of peculiar velocity, expressed in units of the Hubble constant, $H \equiv 100 h km/s/Mpc$. The velocity field is spatially smoothed over $\sim 10 h^{-1} Mpc$ to remove strongly nonlinear effects. Assuming weakly-nonlinear gravitational evolution from Gaussian initial fluctuations, and using second-order perturbative analysis, we show that $ <θ^3> \propto -Ω^{-0.6} <θ^2>^2$. The constant of proportionality depends on the smoothing window. For a top-hat of radius R and volume-weighted smoothing, this constant is $26/7-γ$, where $γ=-d\log <θ^2> / d\log R$. If the power spectrum is a power law, $P(k)\propto k^n$, then $γ=3+n$. A Gaussian window yields similar results. The resulting method for measuring $Ω$ is independent of any assumed biasing relation between galaxies and mass. The method has been successfully tested with numerical simulations. A preliminary application to real data, provided by the POTENT recovery procedure from observed velocities favors $Ω\sim 1$. However, because of an uncertain sampling error, this result should be treated as an assessment of the feasibility of our method rather than a definitive measurement of $Ω$.

astro-ph↗

arXiv1994

Properties of the Cosmological Density Distribution Function

The properties of the probability distribution function of the cosmological continuous density field are studied. We present further developments and compare dynamically motivated methods to derive the PDF. One of them is based on the Zel'dovich approximation (ZA). We extend this method for arbitrary initial conditions, regardless of whether they are Gaussian or not. The other approach is based on perturbation theory with Gaussian initial fluctuations. We include the smoothing effects in the PDFs. We examine the relationships between the shapes of the PDFs and the moments. It is found that formally there are no moments in the ZA, but a way to resolve this issue is proposed, based on the regularization of integrals. A closed form for the generating function of the moments in the ZA is also presented, including the smoothing effects. We suggest the methods to build PDFs out of the whole series of the moments, or out of a limited number of moments -- the Edgeworth expansion. The last approach gives us an alternative method to evaluate the skewness and kurtosis by measuring the PDF around its peak. We note a general connection between the generating function of moments for small r.m.s $σ$ and the non-linear evolution of the overdense spherical fluctuation in the dynamical models. All these approaches have been applied in 1D case where the ZA is exact, and simple analytical results are obtained. The 3D case is analyzed in the same manner and we found a mutual agreement in the PDFs derived by different methods in the the quasi-linear regime. Numerical CDM simulation was used to validate the accuracy of considered approximations. We explain the successful log-normal fit of the PDF from that simulation at moderate $σ$ as mere fortune, but not as a universal form of density PDF in general.

astro-ph↗

arXiv1994

The Effects of Smoothing on the Statistical Properties of the Large-Scale Cosmic Fileds

It has been shown that the large--scale correlation functions of the density field (and velocity divergence field) follow a specific hierarchy in the quasilinear regime and for Gaussian initial conditions (Bernardeau 1992). The exact relationships between the cumulants of the probability distribution functions (the so-called $S_p$ parameters) are however sensitive to the smoothing window function applied to the fields. In this paper, I present a method to derive the whole series of the $S_p$ parameters when the density field is smoothed with a top--hat window function. The results are valid for any power spectrum and any cosmological parameters. Similar calculations are presented for the velocity divergence field. The resulting shapes of the one--point probability distribution functions of the cosmic density and the velocity divergence fields are given as a function of the power spectrum and $Ω$. Simple analytical fits are proposed when the index of the power spectrum is -1. Comparisons with numerical simulations prove these analytical results to be extremely accurate.

astro-ph↗

arXiv1993

Skewness and Kurtosis in Large-Scale Cosmic Fields

In this paper, I present the calculation of the third and fourth moments of both the distribution function of the large--scale density and the large--scale divergence of the velocity field, $θ$. These calculations are made by the mean of perturbative calculations assuming Gaussian initial conditions and are expected to be valid in the linear or quasi linear regime. The moments are derived for a top--hat window function and for any cosmological parameters $Ω$ and $Λ$. It turns out that the dependence with $Λ$ is always very weak whereas the moments of the distribution function of the divergence are strongly dependent on $Ω$. A method to measure $Ω$ using the skewness of this field has already been presented by Bernardeau et al. (1993). I show here that the simultaneous measurement of the skewness and the kurtosis allows to test the validity of the gravitational instability scenario hypothesis. Indeed there is a combination of the first three moments of $θ$ that is almost independent of the cosmological parameters $Ω$ and $Λ$, $${(<θ^4>-3<θ^2>^2) <θ^2> \over <θ^3>^2}\approx 1.5,$$ (the value quoted is valid when the index of the power spectrum at the filtering scale is close to -1) so that any cosmic velocity field created by gravitational instabilities should verify such a property.

astro-ph↗

arXiv1993

The Nonlinear Evolution of Rare Events

In this paper I consider the nonlinear evolution of a rare density fluctuation in a random density field with Gaussian fluctuations, and I rigorously show that it follows the spherical collapse dynamics applied to its mean initial profile. This result is valid for any cosmological model and is independent of the shape of the power spectrum. In the early stages of the dynamics the density contrast of the fluctuation is seen to follow with a good accuracy the form $$δ=(1-δ_L/1.5)^{-1.5}-1,$$ where $δ_L$ is the linearly extrapolated overdensity. I then investigate the validity domain of the rare event approximation in terms of the parameter $ν=δ_L/σ$ giving the initial overdensity scaled by the rms fluctuation at the same mass scale, and find that it depends critically on the shape of the power spectrum. When the power law index $n$ is lower than $-1$ the departure from the spherical collapse is expected to be small, at least in the early stages of the dynamics, and even for moderate values of $ν$ ($\vertν\vert\ge 2$). However, for $n\ge -1$ and whatever the value of $ν$, the dynamics seems to be dominated by the small--scale fluctuations and the subsequent evolution of the peak may not be necessarily correlated to its initial overdensity. I discuss the implications of these results for the nonlinear dynamics and the formation of astrophysical objects.

astro-ph↗

arXiv1993

The Quasi-Linear Evolution of the Density Field in Models of Gravitational Instability

Two quasi-linear approximations, the frozen flow approximation (FFA) and the frozen potential approximation (FPA), have been proposed recently for studying the evolution of a collisionless self-gravitating fluid. In the FFA it is assumed that the velocity field remains unchanged from its value obtained from the linear theory whereas in FPA the same approximation is made for the gravitational potential. In this paper we compare these and the older Zel'dovich approximation by calculating the evolution of the density in perturbation theory. In particular we compute the skewness, including the smoothing effects, and the kurtosis for the FFA, FPA and Zel'dovich approximation and compare their relative accuracy.

astro-ph↗

arXiv1993

The Fundamental Plane of Galaxy Clusters

Velocity dispersion $σ$, radius $R$ and luminosity $L$ of elliptical galaxies are known to be related, leaving only two degrees of freedom and defining the so-called ``fundamental plane". In this {\em Letter} we present observational evidence that rich galaxy clusters exhibit a similar behaviour. Assuming a relation $L \propto R^ασ^{2 β}$, the best-fit values of $α$ and $β$ are very close to those defined by galaxies. The dispersion of this relation is lower than 10 percent, i.e. significantly smaller than the dispersion observed in the $L-σ$ and $L-R$ relations. We briefly suggest some possible implications on the spread of formation times of objects and on peculiar velocities of galaxy clusters.

astro-ph↗