Deriving an Accurate Formula of Scale-dependent Bias with Primordial Non-Gaussianity: An Application of the Integrated Perturbation Theory

TL;DR

This paper derives a comprehensive, nonlocal-bias framework for the scale-dependent bias induced by primordial non-Gaussianity using integrated perturbation theory (iPT). It expresses the biased power spectrum via multipoint propagators and renormalized bias functions, separating Gaussian and non-Gaussian contributions driven by the linear bispectrum, and shows the large-scale slope is dictated by the squeezed limit of PNG, largely independent of the detailed bias model. The authors further develop the shapes of renormalized bias functions within halo bias, derive their behavior beyond peak-background split, and connect their general results to PBS as a limiting case for universal mass functions, while highlighting corrections for non-PS mass functions. They extend the analysis to redshift space, deriving the full non-Gaussian contribution to the power spectrum multipoles and showing that redshift-space distortions have modest impact on the monopole but influence the quadrupole in a PNG-sensitive way. Numerical comparisons illustrate the dominance of the $Q_2$ term on large scales, the agreement and limitations of PBS approximations for different mass functions (PS, ST, MICE), and emphasize the importance of nonlocal bias modeling for accurate PNG constraints from galaxy surveys.

Abstract

We apply the integrated perturbation theory (Matsubara 2011, PRD 83, 083518) to evaluate the scale-dependent bias in the presence of primordial non-Gaussianity. The integrated perturbation theory is a general framework of nonlinear perturbation theory, in which a broad class of bias models can be incorporated into perturbative evaluations of biased power spectrum and higher-order polyspectra. Approximations such as the high-peak limit or the peak-background split are not necessary to derive the scale-dependent bias in this framework. Applying the halo approach, previously known formulas are re-derived as limiting cases of a general formula in this work, and it is implied that modifications should be made in general situations. Effects of redshift-space distortions are straightforwardly incorporated. It is found that the slope of the scale-dependent bias on large scales is determined only by the behavior of primordial bispectrum in the squeezed limit, and is not sensitive to bias models in general. It is the amplitude of scale-dependent bias that is sensitive to the bias models. The effects of redshift-space distortions turn out to be quite small for the monopole component of the power spectrum, while the quadrupole component is proportional to the monopole component on large scales, and thus also sensitive to the primordial non-Gaussianity.

Figures (11)

• Figure 1: The diagrammatic representation of the power spectrum in terms of multipoint propagators. Details of diagrammatic rules are described in Ref. mat11.
• Figure 2: Diagrammatic representations of the first two multipoint propagators with renormalized Lagrangian bias functions in the lowest-order approximation. Details of diagrammatic rules are described in Ref. mat11.
• Figure 3: The scale-independent Lagrangian bias parameters derived from the Sheth-Tormen (solid lines), Press-Schechter (dashed lines) and MICE (dotted lines) mass functions for $z=0,1,2,4$ (from bottom to top on large mass scales). The unity is added to each parameter to show the negative values in this logarithmic plot.
• Figure 4: The functions $A_1(M)$ and $A_2(M)$ derived from the Sheth-Tormen (solid lines), Press-Schechter (dashed lines) and MICE (dotted lines) mass functions for $z=0,1,2,4$ (from bottom to top on large mass scales).
• Figure 5: The functions $Q_n(k)$ divided by $f_{\rm NL} P_{\rm L}(k)$ (dashed: $n=0$, dotted: $n=1$, solid: $n=2$) at redshift $z=1$. Negative values are shown in thin lines. The halo model with mass $M=10^{14}M_\odot$ is assumed in calculating $Q_1(k)$ and $Q_2(k)$. Different panels correspond to different models of primordial non-Gaussianity as indicated.