Multi-tracer beyond linear theory

TL;DR

This work extends multi-tracer analyses into the non-linear regime of galaxy clustering using the EFT-based large-scale bias expansion. It shows that splitting tracers by non-linear bias parameters can outperform splits in the linear bias, significantly tightening constraints on $A_s$, $h$, and $\omega_{\rm cdm}$, and it explores the roles of cross-spectra, cross-stochastic terms, and the number of tracers. Assembly bias emerges as a promising mechanism to realize large non-linear bias differences, while the information gains persist even with realistic tracer densities and when considering FoG differences across subsamples. The findings yield practical guidance for upcoming surveys on tracer design, FoG treatment, and potential sub-sample splits to maximize cosmological information from galaxy clustering.

Abstract

The multi-tracer (MT) technique has been shown to outperform single-tracer analyses in the context of galaxy clustering. In this paper, we conduct a series of Fisher analyses to further explore MT information gains within the framework of non-linear bias expansion. We examine how MT performance depends on the bias parameters of the subtracers, showing that directly splitting the non-linear bias generally leads to smaller error bars in $A_s$, $h$, and $ω_{\rm cdm}$ compared to a simple split in $b_1$. This finding opens the door to identifying subsample splits that do not necessarily rely on very distinct linear biases. We discuss different total and subtracer number density scenarios, as well as the possibility of splitting into more than two tracers. Additionally, we consider how different Fingers-of-God suppression scales for the subsamples can be translated into different $k_{\rm max}$ values. Finally, we present forecasts for ongoing and future galaxy surveys.

Figures (15)

• Figure 1: Relative errors in $\omega_{\rm cdm}$, $h$ and $A_s$ as a function of the difference in each bias parameter, $\Delta b$. The solid (dashed) black line represents the single-tracer EFT (linear) results, while the dashed blue line corresponds to a multi-tracer split in $b_1$ in linear theory. The $b_{\rm all}^{++-+}$ curve indicates a simultaneous split in all four bias parameters, with the $\pm$ signs denoting the sign of $b_1$, $b_{\delta^2}$, $b_{\mathcal{G}_2}$, $b_{\Gamma_3}$ for the $t_2$ sample, as described in Eq. (\ref{['eq:deltab2']}). We consider $k_{\rm max} = 0.15 \,h/$Mpc (see Fig. \ref{['fig:deltab_all']} for $k_{\rm max} = 0.05 \,h/$Mpc) and assume a tracer number density $\bar{n}^T = 10^{-3} \,h^3 {\rm Mpc}^{-3}$.
• Figure 2: Monopole, quadrupole and hexadecapole normalized by the ST spectra for a bias split in $b_1$ (left), $b_{\mathcal{G}_2}$ (middle) and $b_{\rm all}$ (right). Different colors correspond to various $\Delta b$ value. Dotted, solid and dashed lines correspond to the auto AA, cross AB and auto BB spectra, respectively.
• Figure 3: In the left panel, the difference in the mean values of $b_1$ (blue) and $b_{\delta^2}$ (red) between the lower and higher mass samples defined by a cut $M_{\rm cut}$. Different redshifts are represented by different line styles (solid, dashed and dotted lines). We show $f_{t_1}$, the density fraction of the lower mass tracer $t_1$ [see Eq. (\ref{['eq:frac']})], as black lines, with its values indicated on the right y-axis. In the right panel, the same differences are plotted as a function of $f_{t_1}$.
• Figure 4: Relative error in $\omega_{\rm cdm}$, $h$ and $A_s$ as a function of the maximum mode $k_{\rm max}$ included in the analysis. The solid lines represent the full MT analysis, which includes both the cross spectra and the cross stochastic terms. When the cross-stochastic parameters are removed (dotted lines), the error bars decrease, while completely removing the cross spectra (dash-dotted lines) leads to an increase in the error bars. We fix either $\Delta b_1 = 0.6$ or $\Delta b_{\rm all} = 0.6$.
• Figure 5: Correlation matrices between the parameters for the ST case (top left), and MT both without (top right, with $\Delta b_{\rm all} = 0.6$) and with (bottom left with $\Delta b_{\rm all} = 0.0$ and bottom right with $\Delta b_{\rm all} = 0.6$) the cross-stochastic operator. To avoid cluttering the MT matrices with labels, we denote the bias, counter-terms, and stochastic parameters by numbers, arranged in the same order as in the ST case (i.e., $b_{1}^{t_1}, b_{1}^{t_2}, b_{\delta^2}^{t_1}, b_{\delta^2}^{t_2}, \dots, c_{\rm st, 22}^{t_1t_1}, c_{\rm st, 22}^{t_1t_2}, c_{\rm st, 22}^{t_2t_2}$, with the stochastic terms in the end).