Optimal dataset combining in f_nl constraints from large scale structure in an idealised case

TL;DR

This work addresses constraining local-type non-Gaussianity through large-scale structure by formulating a Fisher-matrix analysis around $f_{NL}=0$ and introducing an optimal two-weighting scheme that uses a single tracer spanning a range of biases. It derives an analytic expression for the Fisher information from infinitesimal bias slices and proves that a two-weight scheme with weight functions $\alpha(M)$ and $\beta(M)$ can reproduce the maximal information, with practical choices such as $\alpha(M) = (b(M)-b_{min})/(b_{max}-b_{min})$ and $\beta(M) = (b_{max}-b(M))/(b_{max}-b_{min})$. The paper also connects these results to optimal weighting in power-spectrum estimation and discusses implementation using observable bias proxies, highlighting substantial gains in $f_{NL}$ constraints in well-sampled regimes and noting the assumptions under which the results hold. Overall, the proposed weighting scheme offers a simple, data-driven path to tighten $f_{NL}$ limits from biased tracers while remaining robust to certain systematic and modeling uncertainties.

Abstract

We consider the problem of optimal weighting of tracers of structure for the purpose of constraining the non-Gaussianity parameter f_NL. We work within the Fisher matrix formalism expanded around fiducial model with f_NL=0 and make several simplifying assumptions. By slicing a general sample into infinitely many samples with different biases, we derive the analytic expression for the relevant Fisher matrix element. We next consider weighting schemes that construct two effective samples from a single sample of tracers with a continuously varying bias. We show that a particularly simple ansatz for weighting functions can recover all information about f_NL in the initial sample that is recoverable using a given bias observable and that simple division into two equal samples is considerably suboptimal when sampling of modes is good, but only marginally suboptimal in the limit where Poisson errors dominate.

Figures (2)

• Figure 1: This figure shows scaling of $F_{{f_{\rm NL}} {f_{\rm NL}}}$ with $\left<b\right>$, $\left<\Delta b^2\right>$ and $\bar{n}P$. Panels from the top to bottom correspond to values of $\bar{n}P$ of $10^{-2}$, $1$ and $10^2$, where $\bar{n}$ is the tracer's number density and $P$ the underlying power spectrum. In each panel, thin solid lines correspond to values of $\left<\Delta b^2\right>=0,1,2,4$ (bottom up) and $p=1$. Solid dashed lines are for $\left<\Delta b^2\right>=0,4$ and $p=1.6$.
• Figure 2: This figure shows the relative performance of the weighting methods that divide samples into two compared to optimal weighting for a model survey discussed in Section \ref{['sec:fundamental-limit']}. Top set of lines are for $q=b(M)$, while bottom are for the numerically determined optimal choice of $M_b$. Different line-styles represent density of objects to that of the halos: 1 (solid), 0.1 dashed and 0.01 (dotted).