Abstract
We calculate the redshift space correlation function and the power spectrum of density peaks of a Gaussian random field. Our derivation, which is valid on linear scales k≲0.1hMpc-1, is based on the peak biasing relation given by Desjacques [Phys. Rev. DPRVDAQ1550-7998, 78, 103503 (2008)10.1103/PhysRevD.78.103503]. In linear theory, the redshift space power spectrum is Ppks(k,μ)=exp(-f2σvel2k2μ2)[bpk(k)+bvel(k)fμ2]2Pδ(k), where μ is the angle with respect to the line of sight, σvel is the one-dimensional velocity dispersion, f is the growth rate, and bpk(k) and bvel(k) are k-dependent linear spatial and velocity bias factors. For peaks, the value of σvel depends upon the functional form of bvel. When the k dependence is absent from the square brackets and bvel is set to unity, the resulting expression is assumed to describe models where the bias is linear and deterministic, but the velocities are unbiased. The peak model is remarkable because it has unbiased velocities in this same sense—peak motions are driven by dark matter flows—but, in order to achieve this, bvel must be k dependent. We speculate that this is true in general: k dependence of the spatial bias will lead to k dependence of bvel even if the biased tracers flow with the dark matter. Because of the k dependence of the linear bias parameters, standard manipulations applied to the peak model will lead to k-dependent estimates of the growth factor that could erroneously be interpreted as a signature of modified dark energy or gravity. We use the Fisher formalism to show that the constraint on the growth rate f is degraded by a factor of 2 if one allows for a k-dependent velocity bias of the peak type. Our analysis also demonstrates that the Gaussian smoothing term is part and parcel of linear theory. We discuss a simple estimate of nonlinear evolution and illustrate the effect of the peak bias on the redshift space multipoles. For k≲0.1hMpc-1, the peak bias is deterministic but k dependent, so the configuration-space bias is stochastic and scale dependent, both in real and redshift space. We provide expressions for this stochasticity and its evolution.
Desjacques, V