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The Nonlinear Redshift Space Power Spectrum: Omega from Redshift Surveys

We examine the anisotropies in the power spectrum by the mapping of real to redshift space. Using the Zel'dovich approximation, we obtain an analytic expression for the nonlinear redshift space power spectrum in the distant observer limit. For a given unbiased galaxy distribution in redshift space, the anisotropies in the power spectrum depend on the parameter $f(Ω)\approx Ω^{0.6}$, where $Ω$ is the density parameter. We quantify these anisotropies by the ratio, $R$, of the quadrupole to monopole angular moments of the power spectrum. In contrast to linear theory, the Zel'dovich approximation predicts a decline in $R$ with decreasing scale. This departure from linear theory is due to nonlinear dynamics and not a result of incoherent random velocities. The rate of decline depends strongly on $Ω$ and the initial power spectrum. However, we find a {\it universal} relation between the quantity $R/R_{lin}$ (where $R_{lin}$ the linear theory value of $R$) and the dimensionless variable $k/k_{nl}$, where $k_{nl}$ is a wavenumber determined by the scale of nonlinear structures. The universal relation is in good agreement with a large N-body simulation. This universal relation greatly extends the scales over which redshift distortions can be used as a probe of $Ω$. A preliminary application to the 1.2 Jy IRAS yields $Ω\sim 0.4$ if IRAS galaxies are unbiased.