Structure formation from non-Gaussian initial conditions: multivariate biasing, statistics, and comparison with N-body simulations

TL;DR

This work develops a multivariate halo bias framework to describe structure formation with local-type primordial non-Gaussianity, showing that halo clustering depends on both the density field and the Gaussian primordial potential. By applying the peak-background split and Eulerian perturbation theory to third order, the authors derive bias coefficients as functions of halo mass and nonlinear parameters $f_{ m NL}$ and $g_{ m NL}$, and they compare predictions for the halo and halo–matter power spectra, including scale-dependent bias, against N-body simulations with excellent agreement up to $k oughly 0.1$–$0.3~h$ Mpc$^{-1}$. The analysis demonstrates that large-scale halo fluctuations trace the primordial potential, yielding a strong scale and shape dependence in the halo bispectrum, which provides a robust avenue to detect primordial non-Gaussianity in future surveys. The framework generalizes to higher-order non-Gaussianity and offers a predictive, physically motivated path to constrain inflationary models via two- and three-point statistics in large-scale structure.

Abstract

We study structure formation in the presence of primordial non-Gaussianity of the local type with parameters f_NL and g_NL. We show that the distribution of dark-matter halos is naturally described by a multivariate bias scheme where the halo overdensity depends not only on the underlying matter density fluctuation delta, but also on the Gaussian part of the primordial gravitational potential phi. This corresponds to a non-local bias scheme in terms of delta only. We derive the coefficients of the bias expansion as a function of the halo mass by applying the peak-background split to common parametrizations for the halo mass function in the non-Gaussian scenario. We then compute the halo power spectrum and halo-matter cross spectrum in the framework of Eulerian perturbation theory up to third order. Comparing our results against N-body simulations, we find that our model accurately describes the numerical data for wavenumbers k < 0.1-0.3 h/Mpc depending on redshift and halo mass. In our multivariate approach, perturbations in the halo counts trace phi on large scales and this explains why the halo and matter power spectra show different asymptotic trends for k -> 0. This strongly scale-dependent bias originates from terms at leading order in our expansion. This is different from what happens using the standard univariate local bias where the scale-dependent terms come from badly behaved higher-order corrections. On the other hand, our biasing scheme reduces to the usual local bias on smaller scales where |phi| is typically much smaller than the density perturbations. We finally discuss the halo bispectrum in the context of multivariate biasing and show that, due to its strong scale and shape dependence, it is a powerful tool for the detection of primordial non-Gaussianity from future galaxy surveys.

Figures (8)

• Figure 1: Comparison of different models for the halo mass function originating from Gaussian (dashed) and non-Gaussian (solid) initial conditions with the N-body data by PPH08. From left to right, the different panels refers to $f_{\mathrm{NL}} = 0, 80, 500$. Note that all non-Gaussian models (with the exception of MR) have been rescaled by the ratio $f_{\mathrm {ST}} / f_{\mathrm {PS}}$, and thus coincide with the ST function in the leftmost panel.
• Figure 2: Lagrangian bias factors at $z=0$ as a function of $f_{\mathrm{NL}}$ for a halo mass of $2 \cdot 10^{14} M_{\odot}/h$ (left), and as a function of the halo mass, for $f_{\mathrm{NL}} = 500$ (right). The results obtained from the PPH and LV mass functions are shown in both cases. For reference, in the right panel we also show the results obtained from the PS mass function, which does not depend on $f_{\mathrm{NL}}$.
• Figure 3: Left: Deviation of the matter power spectrum in models with different $f_{\mathrm{NL}}$ (and $g_{\mathrm{NL}}=0$) from the Gaussian case at $z=0$ (top) and $z=1$ (bottom). The lines indicate our one-loop calculation for different values of $f_{\mathrm{NL}}$ while points with error bars correspond to the N-body simulations by PPH08. Right: Halo-matter cross spectrum at $z=0$ for a narrow bin of halo masses centered around $M = 2 \cdot 10^{14} M_{\odot}/h$ (top) and at $z=1$ for $M = 5 \cdot 10^{13} M_{\odot}/h$ (bottom). The solid and dashed lines have been obtained using our model with the bias parameters from the LV and PPH mass functions, respectively. The dotted lines indicate the model by Dal07, while points with error bars correspond to the simulations by PPH08.
• Figure 4: The halo-matter cross spectrum as a function of halo mass at $z=0$ (left) and $z=1$ (right) is plotted for three different values of the comoving wavenumber $k$ (in $h$ Mpc$^{-1}$). Colored solid and dashed lines indicate the results of our perturbative calculation using the LV and PPH mass functions, respectively. Data points with error bars correspond to the N-body simulations by PPH08. The thin lines show the linear theory by Dal07 (black), also corrected with the factor $\beta$ (magenta) introduced by PPH08 (see Section \ref{['sec:asymp']} for further details). At $z=0$ structure has evolved further into the non-linear regime, so that the range of validity of both the linear and one loop theories is reduced.
• Figure 5: Left: Change in the effective bias $\Delta b$ for fixed $f_{\mathrm{NL}}=500$, $M = 2 \cdot 10^{14} M_{\odot}/h$, at $z=0$. Different models are compared with N-body simulations: the simple $\Delta b_{\mathrm{Dal07}}$ (dotted), $\Delta b_{\mathrm{linear}}$ (dashed) using the LV mass function, and the full one-loop theory (solid), which yields the best match. Right: Fractional difference between the one-loop prediction for $\Delta b$ (with LV and PPH mass functions) and the Dal07 linear theory, compared with the simulations, for different values of $f_{\mathrm{NL}}$, at the same mass and redshift.