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 $\sigma$ 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 $\sigma$ as mere fortune, but not as a universal form of density PDF in general.Comment: 30 pages in Plain Tex, 1 table and 11 figures available as postscript files by anonymous ftp from ftp.cita.utoronto.ca in directory /cita/francis/lev, IFA-94-1

Cross-correlating CMB polarization with local large scale structures

We study heterogeneous quantities that efficiently cross-correlate the lensing information encoded in CMB polarization and large-scale structures recovered from weak lensing galaxy surveys. These quantities allow us to take advantage of the special features of weak lensing effect on CMB B-polarization and of the high (40%) cross-correlation between the two data sets. We show that these objects are robust to filtering effects, have a low intrinsic cosmic variance (around 8% for small 100 square degrees surveys) and can be used as an original constraint on the vacuum energy density.Comment: 4 pages, use moriond.sty, to appear in the proceedings of the XXXVth Rencontres de Moriond "Energy densities in the Universe

Cosmology with Weak Lensing Surveys

Weak gravitational lensing surveys measure the distortion of the image of distant sources due to the deflections of light rays by the fluctuations of the gravitational potential along the line of sight. Since they probe the non-linear matter power spectrum itself at medium redshift such surveys are complimentary to both galaxy surveys (which follow stellar light) and cosmic microwave background observations (which probe the linear regime at high redshift). Ongoing CMB experiments such as WMAP and the future Planck satellite mission will measure the standard cosmological parameters with unprecedented accuracy. The focus of attention will then shift to understanding the nature of dark matter and vacuum energy: several recent studies suggest that lensing is the best method for constraining the dark energy equation of state. During the next 5 year period ongoing and future weak lensing surveys such as the Joint Dark Energy Mission (JDEM, e.g. SNAP) or the Large-aperture Synoptic Survey Telescope (LSST) will play a major role in advancing our understanding of the universe in this direction. In this review article we describe various aspects of weak lensing surveys and how they can help us in understanding our universe.Comment: 15 pages, review article to appear in 2005 Triennial Issue of Phil. Trans.

Lensing effect on the relative orientation between the Cosmic Microwave Background ellipticities and the distant galaxies

The low redshift structures of the Universe act as lenses in a similar way on the Cosmic Microwave Background light and on the distant galaxies (say at redshift about unity). As a consequence, the CMB temperature distortions are expected to be statistically correlated with the galaxy shear, exhibiting a non-uniform distribution of the relative angle between the CMB and the galactic ellipticities. Investigating this effect we find that its amplitude is as high as a 10% excess of alignement between CMB and the galactic ellipticities relative to the uniform distribution. The relatively high signal-to-noise ratio we found should makes possible a detection with the planned CMB data sets, provided that a galaxy survey follow up can be done on a sufficiently large area. It would provide a complementary bias-independent constraint on the cosmological parameters.Comment: 7 pages, 3 figures; uses emulateapj.sty; submitted to Ap

Extended Perturbation Theory for the Local Density Distribution Function

Perturbation theory makes it possible to calculate the probability distribution function (PDF) of the large scale density field in the small variance limit. For top hat smoothing and scale-free Gaussian initial fluctuations, the result depends only on the linear variance, sigma_linear, and its logarithmic derivative with respect to the filtering scale -(n_linear+3)=dlog sigma_linear^2/dlog L (Bernardeau 1994). In this paper, we measure the PDF and its low-order moments in scale-free simulations evolved well into the nonlinear regime and compare the results with the above predictions, assuming that the spectral index and the variance are adjustable parameters, n_eff and sigma_eff=sigma, where sigma is the true, nonlinear variance. With these additional degrees of freedom, results from perturbation theory provide a good fit of the PDFs, even in the highly nonlinear regime. The value of n_eff is of course equal to n_linear when sigma << 1, and it decreases with increasing sigma. A nearly flat plateau is reached when sigma >> 1. In this regime, the difference between n_eff and n_linear increases when n_linear decreases. For initial power-spectra with n_linear=-2,-1,0,+1, we find n_eff ~ -9,-3,-1,-0.5 when sigma^2 ~ 100.Comment: 13 pages, 6 figures, Latex (MN format), submitted to MNRA

Scaling in Gravitational Clustering, 2D and 3D Dynamics

Perturbation Theory (PT) applied to a cosmological density field with Gaussian initial fluctuations suggests a specific hierarchy for the correlation functions when the variance is small. In particular quantitative predictions have been made for the moments and the shape of the one-point probability distribution function (PDF) of the top-hat smoothed density. In this paper we perform a series of systematic checks of these predictions against N-body computations both in 2D and 3D with a wide range of featureless power spectra. In agreement with previous studies, we found that the reconstructed PDF-s work remarkably well down to very low probabilities, even when the variance approaches unity. Our results for 2D reproduce the features for the 3D dynamics. In particular we found that the PT predictions are more accurate for spectra with less power on small scales. The nonlinear regime has been explored with various tools, PDF-s, moments and Void Probability Function (VPF). These studies have been done with unprecedented dynamical range, especially for the 2D case, allowing in particular more robust determinations of the asymptotic behaviour of the VPF. We have also introduced a new method to determine the moments based on the factorial moments. Results using this method and taking into account the finite volume effects are presented.Comment: 13 pages, Latex file, 9 Postscript Figure