Renormalisation of linear halo bias in N-body simulations

The interpretation of redshift surveys requires modeling the relationship between large-scale fluctuations in the observed number density of tracers, $\delta_\mathrm{h}$, and the underlying matter density, $\delta$. Bias models often express $\delta_\mathrm{h}$ as a truncated series of integro-differential operators acting on $\delta$, each weighted by a bias parameter. Due to the presence of `composite operators' (obtained by multiplying fields evaluated at the same spatial location), the linear bias parameter measured from clustering statistics does not coincide with that appearing in the bias expansion. This issue can be cured by re-writing the expansion in terms of `renormalised' operators. After providing a pedagogical and comprehensive review of bias renormalisation in perturbation theory, we generalize the concept to non-perturbative dynamics and successfully apply it to dark-matter haloes extracted from a large suite of N-body simulations. When comparing numerical and perturbative results, we highlight the effect of the window function employed to smooth the random fields. We then measure the bias parameters as a function of halo mass by fitting a non-perturbative bias model (both before and after applying renormalisation) to the cross spectrum $P_{\delta_\mathrm{h}\delta}(k)$. Finally, we employ Bayesian model selection to determine the optimal operator set to describe $P_{\delta_\mathrm{h}\delta}(k)$ for $k<0.2\,h$ Mpc$^{-1}$ at redshift $z=0$. We find that it includes $\delta, \nabla^2\delta, \delta^2$ and the square of the traceless tidal tensor, $s^2$. Considering higher-order terms (in $\delta$) leads to overfitting as they cannot be precisely constrained by our data. We also notice that next-to-leading-order perturbative solutions are inaccurate for $k\gtrsim 0.1\,h$ Mpc$^{-1}$.