Statistics of Smoothed Cosmic Fields in Perturbation Theory. I. Formulation and Useful Formulae in Second‐Order Perturbation Theory

Abstract

We formulate a general method for perturbative evaluations of statistics of smoothed cosmic fields and provide useful formulae for application of the perturbation theory to various statistics. This formalism is an extensive generalization of the method used by Matsubara, who derived a weakly nonlinear formula of the genus statistic in a three-dimensional density field. After describing the general method, we apply the formalism to a series of statistics, including genus statistics, level-crossing statistics, Minkowski functionals, and a density extrema statistic, regardless of the dimensions in which each statistic is defined. The relation between the Minkowski functionals and other geometrical statistics is clarified. These statistics can be applied to several cosmic fields, including three-dimensional density field, three-dimensional velocity field, twodimensional projected density field, and so forth. The results are detailed for second-order theory of the formalism. The effect of the bias is discussed. The statistics of smoothed cosmic fields as functions of rescaled threshold by volume fraction are discussed in the framework of second-order perturbation theory. In CDMlike models, their functional deviations from linear predictions plotted against the rescaled threshold are generally much smaller than that plotted against the direct threshold. There is still a slight meatball shift against rescaled threshold, which is characterized by asymmetry in depths of troughs in the genus curve. A theory-motivated asymmetry factor in the genus curve is proposed.

Statistical Decoupling of a Lagrangian Fluid Parcel in Newtonian Cosmology

“…For example, the Gaussian expression of x Y | á ñ is well known in cosmology since Bardeen et al (1986). And for weakly non-Gaussian distributed random variables, the formula adopted here is very similar to the one adopted in, e.g., estimating the Minkowski functional (Matsubara 1994(Matsubara , 2003Pogosyan et al 2009). To help readers who are unfamiliar with the subject, we will provide a detailed derivation in this Appendix.…”

“…Of course, this is under the assumption that the conditional average of the tidal tensor is estimated in the nonlinear regime as well. While it is obviously complicated to evaluate in the deeply nonlinear regime, we will first utilize the cumulant expansion theorem (Ma 1985;Matsubara 2003) to calculate the corrections to the next order, i.e., up to the thirdorder cumulants. The cumulant expansion theorem states that the logarithm of the partition function could be expanded as the nth order of cumulants…”

“…Transforming back to the PDF and substituting the current with partial derivative i l = ¶ ¶G a a , one obtains the expansion of an arbitrary PDF ( )  G in terms of Gaussian distribution G ( )  G (Cramer 1946;Kendall & Stuart 1958;Matsubara 1994Matsubara , 2003Pogosyan et al 2009):…”

“…For a weakly non-Gaussian field, we could apply the cumulant expansion theorem and expand an arbitrary distribution function  in terms of nth-order cumulants (Cramer 1946;Kendall & Stuart 1958;Matsubara 1994Matsubara , 2003Pogosyan et al 2009 …”

Studying topological structure of 21-cm line fluctuations with 3D Minkowski functionals before reionization

“…For a non-gaussian distribution, this symmetry is broken. When the non-gaussinaity is weak, the MFs can be described analytically (Matsubara 1994(Matsubara , 2003. As we will see below, the MFs of the δT b are significantly different from those of the gaussian case because the δT b follows a highly non-gaussian distribution.…”