ar X iv : 0 71 1 . 07 79 v 2 [ as tr o - ph ] 1 0 Ja n 20 08 Nonlinear Evolution of Anisotropic Cosmological Power

There has been growing interest in the possibility of testing more precisely the assumption of statistical isotropy of primordial density perturbations. If it is to be tested with galaxy surveys at distance scales < ∼ 10 Mpc, then nonlinear evolution of anisotropic power must be understood. To this end, we calculate the angular dependence of the power spectrum to third order in perturbation theory for a primordial power spectrum with a quadrupole dependence on the wavevector direction. Our results suggest that primordial power anisotropies will be suppressed by < ∼ 7% in the quasilinear regime. We also show that the skewness in the statistically anisotropic theory differs by no more than 1% from that in the isotropic theory. It is commonly assumed that primordial density perturbations are statistically isotropic, and statistical isotropy is a prediction of most structure-formation theories. The notion of statistical isotropy can be quantified, though, by considering models in which it is broken, and a growing literature has discussed physical models that produce primordial perturbations that are not statistically isotropic [1]. The manifestations of departures from statistical isotropy can take on many forms, but one simple example [2] predicts a primordial power spectrum P (k) with a quadrupole dependence on the directionˆk of the wavevector k, P (k) = A(k) [1 + g * P 2 (µ k)] , (1) where P 2 (x) = (3x 2 − 1)/2 is the second Legendre polynomial , µ k = ˆ k · ˆ e, andê is a preferred direction. 1 Reference [3] constructed a CMB minimum-variance estimator for the power-anisotropy amplitude g * and showed that the Planck satellite will be sensitive to a value of g * as small as ∼0.02, a number which can be estimated roughly by ∼10 N −1/2 pix , where N pix is the number of statistically independent pixels on the sky detected by Planck. Similar arguments suggest that the sensitivity of a galaxy survey, like the Sloan Digital Sky Survey [4] or Two-Degree Field [5], should have a comparable sensitivity. One issue that will arise, though, in testing statistical isotropy of primordial perturbations is the quasilinear evolution of the power spectrum. The root-mean-square density-perturbation amplitude becomes of order unity at distance scales ∼10 h −1 Mpc, and so quasilinear evolution of the density field must be taken into account if the mass distribution measured on these scales in the Universe today …