Combining perturbation theories with halo models

We investigate the building of unified models that can predict the matter-density power spectrum and the two-point correlation function from very large to small scales, being consistent with perturbation theory at low $k$ and with halo models at high $k$. We use a Lagrangian framework to re-interpret the halo model and to decompose the power spectrum into "2-halo" and "1-halo" contributions, related to "perturbative" and "non-perturbative" terms. We describe a simple implementation of this model and present a detailed comparison with numerical simulations, from $k \sim 0.02$ up to $100 h$Mpc$^{-1}$, and from $x \sim 0.02$ up to $150 h^{-1}$Mpc. We show that the 1-halo contribution contains a counterterm that ensures a $k^2$ tail at low $k$ and is important not to spoil the predictions on the scales probed by baryon acoustic oscillations, $k \sim 0.02$ to $0.3 h$Mpc$^{-1}$. On the other hand, we show that standard perturbation theory is inadequate for the 2-halo contribution, because higher order terms grow too fast at high $k$, so that resummation schemes must be used. We describe a simple implementation, based on a 1-loop "direct steepest-descent" resummation for the 2-halo contribution that allows fast numerical computations, and we check that we obtain a good match to simulations at low and high $k$. Our simple implementation already fares better than standard 1-loop perturbation theory on large scales and simple fits to the power spectrum at high $k$, with a typical accuracy of 1% on large scales and 10% on small scales. We obtain similar results for the two-point correlation function. However, there remains room for improvement on the transition scale between the 2-halo and 1-halo contributions, which may be the most difficult regime to describe.