Perturbation theory for the non-linear halo power spectrum: the renormalized bias and halo bias

We revisit an analytical model to describe the halo-matter cross-power spectrum and the halo auto-power spectrum in the weakly non-linear regime, by combining perturbation theory (PT) for matter clustering, the local bias model and the halo bias. Non-linearities in the power spectra arise from the non-linear clustering of matter as well as the non-linear relation between the matter and halo density fields. By using the 'renormalization' approach, we express the non-linear power spectra as a sum of two contributions: the non-linear matter power spectrum with effective linear bias parameter and the higher-order PT spectra having the halo bias parameters as coefficients. The halo auto-power spectrum includes residual shot-noise contamination, which needs to be treated as an additional free parameter. The higher order PT spectra and the residual shot noise cause a scale-dependent bias function relative to the non-linear matter power spectrum in the weakly non-linear regime. We show that model predictions are in good agreement with the spectra measured from a suite of high-resolution N-body simulations up to k similar or equal to 0.2 h Mpc(-1) at z = 0.35 for different halo mass bins.