On the Statistics of Biased Tracers in the Effective Field Theory of Large Scale Structures

TL;DR

This work extends the Effective Field Theory of Large Scale Structures to biased tracers by incorporating baryonic effects and primordial non-Gaussianities. It develops a minimal, irreducible basis of seven bias parameters and computes one-loop two-point and tree-level three-point functions for halos, all renormalized and IR-resummed. When confronted with Millennium-XXL simulations, the EFT predictions achieve percent-level accuracy for halo–matter and halo–halo power spectra up to $k\sim0.3\,h\,\mathrm{Mpc}^{-1}$ and up to $k\sim0.10-0.16\,h\,\mathrm{Mpc}^{-1}$ for the bispectra, with the reach decreasing for higher-mass halos. The results demonstrate that analytical control of LSS statistics extends far into quasi-nonlinear scales, providing substantial cosmological information beyond previous expectations and guiding future analyses of large-scale surveys.

Abstract

With the completion of the Planck mission, in order to continue to gather cosmological information it has become crucial to understand the Large Scale Structures (LSS) of the universe to percent accuracy. The Effective Field Theory of LSS (EFTofLSS) is a novel theoretical framework that aims to develop an analytic understanding of LSS at long distances, where inhomogeneities are small. We further develop the description of biased tracers in the EFTofLSS to account for the effect of baryonic physics and primordial non-Gaussianities, finding that new bias coefficients are required. Then, restricting to dark matter with Gaussian initial conditions, we describe the prediction of the EFTofLSS for the one-loop halo-halo and halo-matter two-point functions, and for the tree-level halo-halo-halo, matter-halo-halo and matter-matter-halo three-point functions. Several new bias coefficients are needed in the EFTofLSS, even though their contribution at a given order can be degenerate and the same parameters contribute to multiple observables. We develop a method to reduce the number of biases to an irreducible basis, and find that, at the order at which we work, seven bias parameters are enough to describe this extremely rich set of statistics. We then compare with the output of $N$-body simulations. For the lowest mass bin, we find percent level agreement up to $k\simeq 0.3\,h\,{\rm Mpc}^{-1}$ for the one-loop two-point functions, and up to $k\simeq 0.15\,h\,{\rm Mpc}^{-1}$ for the tree-level three-point functions, with the $k$-reach decreasing with higher mass bins. This is consistent with the theoretical estimates, and suggests that the cosmological information in LSS amenable to analytical control is much more than previously believed.

Figures (16)

• Figure 1: Dark matter power spectrum at redshift $z=0$ divided by the nonlinear simulation data. The one-loop EFTofLSS result is shown after (solid blue line) and before (dashed blue line) performing the IR resummation procedure. For comparison, we also show the one-loop SPT result (solid orange line). The parameter $c^2_{s(1)}$ is determined by fitting the IR-resummed EFT predictions over the range $0.15h\,$Mpc$^{-1} < k < 0.3h\,$Mpc$^{-1}$. The thin gray dashed lines signal the $2\%$ systematics error assumed for the simulation data.
• Figure 2: $\chi^2$ and $p$-values are given for best fit bias parameters procedure for the three mass bins. In each table we indicate the bispectrum fitting $k$-range (for the power spectrum the $k$-range is always $0.04-0.3h\,$Mpc$^{-1}$). Green plus (red minus) signs are indicating which statistics is included (excluded) from the fit. We see that adding the tree level bispectrum significantly improves the constraining power of the fitting procedure.
• Figure 3: Halo-matter (upper panels) and halo-halo (lower panels) power spectrum ratios are shown for the mass bin0 ($b_{\delta}=1.0$) on the left, and for bin1 ($b_{\delta}=1.33$) on the right. The solid blue line is our theoretical prediction containing seven bias parameters (see table \ref{['tb:bias']}), divided by the nonlinear simulation data measurements. The thin gray lines correspond to the theoretical error estimates stemming from our neglecting two-loop corrections. The horizontal thin dashed gray lines signal the $2\%$ error that we input to account for the systematics in simulations. Consistently, we see that the theory stops matching the data when the theoretical error becomes sizable.
• Figure 4: $p$-values corresponding to three halo mass bins, shown as a function of the maximum triangle side length $k_{{\rm max},B}$ for the bispectra (and fixed $k_{{\rm max},P}=0.3h\,$Mpc$^{-1}$). We see the characteristic sharp drop in the $p$-value after the maximal scale $k_{{\rm max},B}$. As expected for higher mass bins results fail earlier, around $k_{{\rm max},B}=0.08$ for bin0 and then at slightly higher scales for middle mass halos (around scales of $k_{{\rm max},B}=0.09$) for bin1 and bin2). For lithest mass bin0 expansion preforms up to scales $\sim k_{{\rm max},B}=0.15$)
• Figure 5: Dark matter correlation function. The subscript ${}_{\rm IR}$ means that we have performed the IR resummation. Error bars from simulations represent the estimated $2\sigma$ (see Sanchez:2008iw). Error bars are correlated though we do not quantify their correlation.