Primordial non-Gaussianity: large-scale structure signature in the perturbative bias model

TL;DR

Primordial non-Gaussianity in the local form imprints a scale-dependent bias on large-scale structure that is not captured by standard linear bias. The author employs renormalized perturbation theory to compute mass and tracer power spectra, identifies infrared divergences, and introduces a minimal linear-in-$\phi$ bias term to absorb them. The mass power spectrum corrections from non-Gaussianity are renormalizable and largely negligible for realistic $f_{NL}$, while the tracer bias must include a $\phi$-term, yielding $P_{gg}(k)=b_\delta^2 P_{\delta\delta}(k) + 2 b_\delta b_\phi f_{NL} P_{\phi\delta}(k) + b_\phi^2 f_{NL}^2 P_{\phi\phi}(k) + \ldots$ plus standard nonlinear-bias contributions. The halo-model provides a way to predict $b_\phi$, but with caveats about galaxy-halo relations, and forecasts indicate that large-volume surveys could constrain $f_{NL}$ near unity, guiding future CMB and LSS observational strategies.

Abstract

I compute the effect on the power spectrum of tracers of the large-scale mass-density field (e.g., galaxies) of primordial non-Gaussianity of the form Phi=phi+fNL (phi-<phi^2>)+gNL phi^3+..., where Phi is proportional to the initial potential fluctuations and phi is a Gaussian field, using beyond-linear-order perturbation theory. I find that the need to eliminate large higher-order corrections necessitates the addition of a new term to the bias model, proportional to phi, i.e., delta_g=b_delta delta+b_phi fNL phi+..., with all the consequences this implies for clustering statistics, e.g., P_gg(k)=b_delta^2 P_deltadelta(k)+2 b_delta b_phi fNL P_phidelta(k)+b_phi^2 fNL^2 P_phiphi(k)+... . This result is consistent with calculations based on a model for dark matter halo clustering, showing that the form is quite general, not requiring assumptions about peaks, or the formation or existence of halos. The halo model plays the same role it does in the usual bias picture, giving a prediction for b_phi for galaxies known to sit in a certain type of halo. Previous projections for future constraints based on this effect have been very conservative -- there is enough volume at z<~2 to measure fNL to ~+-1, with much more volume at higher z. As a prelude to the bias calculation, I point out that the beyond-linear (in phi) corrections to the power spectrum of mass-density perturbations are naively infinite, so it is dangerous to assume they are negligible; however, the infinite part can be removed by a renormalization of the fluctuation amplitude, with the residual k-dependent corrections negligible for models allowed by current constraints.

Figures (2)

• Figure 1: Real-space power spectrum for $f_{\rm NL}=5$, at $z=0.75$, of galaxies with $b_\delta=2$, $b_\phi=3$ (black, solid line). Also shown are the components of the power spectrum, $b_\delta^2 P_{\delta\delta}(k)$ (blue, dotted line), $2 f_{\rm NL} b_\phi b_\delta P_{\phi\delta}(k)$ (green, short-dashed), and $f_{\rm NL}^2 b_\phi^2 P_{\phi\phi}(k)$ (red, long-dashed line). The arrow indicates the approximate minimum $k$ probed by an all-sky survey out to $z=1$.
• Figure 2: Rms error on $f_{\rm NL}$ for a well-sampled all-sky survey out to $z$, ignoring redshift-space, geometric, and evolutionary distortions. Rms detection significance is essentially linear in $f_{\rm NL}$ (out to a $\sim 5 \sigma$ level of detection), and scales roughly as ${\rm volume}^{2/3}$ (as long as the volume remains compact).