Combining perturbation theories with halo models

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 k 2 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.

Modeling of weak-lensing statistics

“…Lagrangian) separation q, that belong to two distinct halos, and P pert (k) is the matter power spectrum obtained by perturbation theory. As discussed in Valageas & Nishimichi (2011a), it is not possible to use standard perturbation theory (beyond linear order) for P pert (k), unless one adds a nonperturbative ad-hoc cutoff, because this would yield a term that keeps growing at high k and prevents a good agreement with numerical simulations. A natural cure to this problem is to use a resummation scheme that remains well-behaved at high k. As in Valageas & Nishimichi (2011a), we use the "direct steepest-descent" resummation developed in Valageas (2007Valageas ( , 2008, going to "one-loop" order.…”

“…(11). We use the usual "NFW" halo profile (Navarro et al 1997) with the massconcentration relation from Valageas & Nishimichi (2011a). The countertermW 2 in Eq.…”

“…Thus, this framework ensures that the power spectrum (4) is consistent with perturbation theory (up to the order of truncation of the resummation scheme) while going smoothly to the 1-halo power at high k. A more detailed derivation of Eqs. (5) and (7) is given in Valageas & Nishimichi (2011a), through a Lagrangian point of view.…”

Combining perturbation theories with halo models for the matter bispectrum

“…This is particularly useful on the weakly nonlinear scales associated with the baryon acoustic oscillations, where the matter power spectrum and bispectrum show small "wiggles" that would be difficult to reproduce with a high accuracy by a simple halo model or fits to simulations (unless one runs a simulation with the required cosmological parameters). As a first step in this direction, we have recently built such a unified model in Valageas & Nishimichi (2011), focusing on the matter power spectrum. In this paper, we extend this study to the matter bispectrum, using the same halo model and perturbative schemes.…”

“…Then, we present in Sect. 3 a simple implementation, based on the halo model and the perturbative approaches used in Valageas & Nishimichi (2011) for the power spectrum. We compare our results for the bispectrum with numerical simulations in Sect.…”