Combining perturbation theories with halo models

¹ Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France
e-mail: patrick.valageas@cea.fr
² Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, 277-8568 Chiba, Japan

Received: 3 September 2010
Accepted: 11 December 2010

Abstract

Aims. 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.

Methods. 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 ~ 0.02 up to 100   h   Mpc^-1, and from x ~ 0.02 up to 150   h^-1   Mpc.

Results. We show that the 1-halo contribution contains a counterterm that ensures a k² tail at low k and this contribution is important not to spoil the predictions on the scales probed by baryon acoustic oscillations, k ~ 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. Moreover, we explain why such a model, which is based on the combination of perturbation theories and halo models, remains consistent with standard perturbation theory up to the order of the resummation scheme. 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. We also study the dependence of such predictions on the details of the underlying model, such as the choice of the perturbative resummation scheme or the properties of halo profiles. Our simple implementation already fares better than both 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.