Michal J. Chodorowski

arXiv1997

Large-scale density from velocity expansion and shear

I derive up to second order in Eulerian perturbation theory a new relation between the weakly nonlinear density and velocity fields. In the case of unsmoothed fields, density at a given point turns out to be a purely local function of the expansion (divergence) and shear of the velocity field. The relation depends on the cosmological parameter Omega, strongly by the factor f(Omega) = Omega^{0.6} and weakly by the factors K(Omega) and C(Omega) proportional to Omega^{-2/63} and Omega^{-1/21} respectively. The Gramann solution is found to be equivalent to the derived relation with the weak Omega-dependence neglected. To make the relation applicable to the real world, I extend it for the case of smoothed fields. The resulting formula, when averaged over shear given divergence, reproduces up to second order the density-velocity divergence relation of Chodorowski & Lokas; however, it has smaller spread. It makes the formula a new attractive local estimator of large-scale density from velocity.

astro-ph↗

arXiv1997

Weakly Nonlinear Density-Velocity Relation

We rigorously derive weakly nonlinear relation between cosmic density and velocity fields up to third order in perturbation theory. The density field is described by the mass density contrast, $\de$. The velocity field is described by the variable $\te$ proportional to the velocity divergence, $\te = - f(Ω)^{-1} H_0^{-1} \nabla\cdot\bfv$, where $f(Ω) \simeq Ω^{0.6}$, $Ω$ is the cosmological density parameter and $H_0$ is the Hubble constant. Our calculations show that mean $\de$ given $\te$ is a third order polynomial in $\te$, $\lan \de \ran|_{\te} = a_1 \te + a_2 (\te^2 - \s_\te^2) + a_3 \te^3$. This result constitutes an extension of the formula $\lan \de \ran|_{\te} = \te + a_2 (\te^2 - \s_\te^2)$, found by Bernardeau~(1992) which involved second order perturbative solutions. Third order perturbative corrections introduce the cubic term. They also, however, cause the coefficient $a_1$ to depart from unity, in contrast with the linear theory prediction. We compute the values of the coefficients $a_p$ for scale-free power spectra, as well as for standard CDM, for Gaussian smoothing. The coefficients obey a hierarchy $a_3 \ll a_2 \ll a_1$, meaning that the perturbative series converges very fast. Their dependence on $Ω$ is expected to be very weak. The values of the coefficients for CDM spectrum are in qualitative agreement with the results of N-body simulations by Ganon et al. (1996). The results provide a method for breaking the $Ω$-bias degeneracy in comparisons of cosmic density and velocity fields such as IRAS-POTENT.

astro-ph↗