A Spherical Harmonic Approach to Redshift Distortion: Implications for $\Omega$ and the Power Spectrum

We examine the nature of galaxy clustering in redshift space using a method based on an expansion of the galaxian density field in Spherical Harmonics and linear theory. We derive a compact and self-consistent expression for the distortion when applied to flux limited redshift surveys. The amplitude of the distortion is controlled by the combination of the density and bias parameters, $\beta\equiv\Omega_\circ^{0.6}/b$ as well as the shape of the real space power spectrum, $P(k)$ (characterized by a shape parameter $\Gamma$), and its normalization, $\sigma_8$; we exploit this fact to derive a maximum likelihood estimator for $\beta$, $\Gamma$, and $\sigma_8$. We check our formalism using $N$-body simulations and demonstrate it provides an unbiased estimate of $\beta$ when the amplitude and shape of the galaxy power spectrum is known. Application of the technique to the 1.2 Jy \iras\ redshift survey yields $\beta =0.94\pm 0.17$ and $\Gamma=0.17\pm0.05$ (1-$\sigma$) when $\sigma_8$ is held fixed at its best value as determined from the real space correlation function. Allowing $\sigma_8$ to be a free parameter, we find $\beta=0.47\pm 0.25$, $\Gamma=0.15\pm0.05$, and $\sigma_8=0.81\pm0.06$.