Combining perturbation theories with halo models for the matter bispectrum

Abstract

Aims. We investigate how unified models should be built to be able to predict the matter-density bispectrum (and power spectrum) from very large to small scales and that are at the same time consistent with perturbation theory at low k and with halo models at high k. Methods. We use a Lagrangian framework to decompose the bispectrum into "3-halo", "2-halo", and "1-halo" contributions, related to "perturbative" and "non-perturbative" terms. We describe a simple implementation of this approach and present a detailed comparison with numerical simulations. Results. We show that the 1-halo and 2-halo contributions contain counterterms that ensure their decay at low k, as required by physical constraints, and allow a better match to simulations. Contrary to the power spectrum, the standard 1-loop perturbation theory can be used for the perturbative 3-halo contribution because it does not grow too fast at high k. Moreover, it is much simpler and more accurate than two resummation schemes investigated in this paper. We obtain a good agreement with numerical simulations on both large and small scales, but the transition scales are poorly described by the simplest implementation. This cannot be amended by simple modifications to the halo parameters, but we show how it can be corrected for the power spectrum and the bispectrum through a simple interpolation scheme that is restricted to this intermediate regime. Then, we reach an accuracy on the order of 10% on mildly and highly nonlinear scales, while an accuracy on the order of 1% is obtained on larger weakly nonlinear scales. This also holds for the real-space two-point correlation function.

Modeling of weak-lensing statistics

“…(15) and (16) ensure that the 2-halo and 1-halo contributions decay on large scales so that the bispectrum goes to the perturbative prediction B pert . However, as found in Valageas & Nishimichi (2011b) and contrary to the situation encountered for the power spectrum, the standard oneloop perturbation theory prediction for B pert is well-behaved at high k (i.e. it is significantly smaller than the 1-halo contribution) and it is more accurate than two alternative resummations that have been investigated.…”

“…As described in Valageas & Nishimichi (2011b), the 3D matter bispectrum is obtained in a similar fashion as the sum of 3-halo, 2-halo and 1-halo terms,…”

“…(4) significantly underestimates the power spectrum on the transition scales (by ∼20%) even though it gives a good accuracy on larger scales (∼1%) and smaller scales (∼10%). In Valageas & Nishimichi (2011b) we devised a simple geometrical modification to Eq. (4), P tang (k), such that…”

“…As explained in Valageas & Nishimichi (2011b), the wavenumbers k − and k + are determined by the constraint that the reduced equilateral bispectrum,…”

“…The main point here is that the range [k − , k + ] is automatically determined from the shape of P 2H+1H (k) and B (k, k, k), and it evolves with redshift and with the initial power. We refer the reader to Valageas & Nishimichi (2011b) for further details.…”