Quadratic shape biases in three-dimensional halo intrinsic alignments

Abstract

Understanding the nonlinear relation between the shapes of halos or galaxies and the surrounding matter distribution is essential in accurate modeling of their intrinsic alignments. In the perturbative treatment, such nonlinear relation of the intrinsic alignments appears as higher-order shape bias parameters. In this paper, we present accurate measurements of the quadratic shape bias parameters by combining the full three-dimensional power spectrum of the intrinsic alignments (i.e., without any projection) with the quadratic field method. In order to benefit from the full three-dimensional power spectrum we employ the spherical tensor decomposition of the three-dimensional shape field and measure their power spectra for the first time. In particular, we detect the vector and tensor power spectra in this basis, which cannot be explained by the widely-used nonlinear alignment model. Further, by cross-correlating the three-dimensional halo shape field with the quadratic shape bias operators from the initial condition of the same simulation to cancel cosmic variance, we effectively extract bispectrum information and detect quadratic shape bias parameters in the intrinsic alignments with high significance for the first time. We also compare these measurements with the prediction where quadratic shape biases are dynamically generated from the linear Lagrangian shape bias through the large-scale bulk flow. We find general agreement for all three biases with small deviations, which in practice could be negligible for the current photometric surveys. This implies that the advection prediction for the higher-order shape biases can be used as a prior in the cosmological analyses of intrinsic alignments.

A Lagrangian theory for galaxy shape statistics

“…The first column shows shape-shape correlators with dependence on the linear tidal field s ab , the second those with dependence on the quadratic term δs ab and finally those with dependence on s ac s cb . The (δs ab , δs ab ) seem to have the highest amplitude for higher helicity spectra, and indeed we will see in section 7 that this is also the most important contribution to the higher helicity spectra in the so-called local Lagrangian bias ansatz [1]. We will return to this point shortly when we compare our results to N -body simulations.…”

“…Additionally, these operators do not have m = 1 helicity modes prior to advection, as was first pointed out by ref. [1]. Finally we have…”

“…As a warm-up to the general structure of the calculations we will carry out in the rest of this paper, we begin by examining only the terms which depend on the Zel'dovich displacement and the linear density and tidal fields. As seen in equation (3.3) we need to compute correlators of these fields (Ψ (1) , δ, s ab ) in Lagrangian coordinates q. These correlators between Lagrangian fields and displacements are the building blocks of predictions in LPT and have been well-studied and computed in previous works [25].…”

“…Assuming the Zel'dovich approximation (Ψ = Ψ (1) ), from the generating function formalism we see that this is equivalent to…”

“…In particular we will use the helicity auto spectra measured in ref. [1], which were taken from 8 boxes each with volume (1.5 h −1 Gpc) 3 . Unlike the projected E and B mode statistics used in previous works validating perturbation theory predictions for galaxy shapes, these spectra fully describe the three-dimensional clustering without mixing independent components, and moreover do not contain projection effects which break the isotropy of galaxy shapes in real space [22].…”