Very Massive Tracers and Higher Derivative Biases

TL;DR

The paper investigates biased tracers in the EFTofLSS and demonstrates that higher derivative bias operators restore comparable predictive accuracy across halo masses. By computing halo-matter and halo-halo two-point functions at one loop and halo-halo-halo, halo-halo-matter, halo-matter-matter three-point functions at tree level within a BoD-based operator basis, the authors show that including these higher derivative terms suppresses tracer-dependent errors. Renormalization and IR-resummation are applied to control UV and IR contributions, respectively, enabling robust comparisons to $N$-body simulations. The main finding is that, up to $k\sim0.17\,h\mathrm{Mpc}^{-1}$ at $z=0$, the theory matches simulations for all halo mass bins when higher derivative biases are included, underscoring the EFTofLSS framework’s predictive power for upcoming large-scale structure surveys.

Abstract

Most of the upcoming cosmological information will come from analyzing the clustering of the Large Scale Structures (LSS) of the universe through LSS or CMB observations. It is therefore essential to be able to understand their behavior with exquisite precision. The Effective Field Theory of Large Scale Structures (EFTofLSS) provides a consistent framework to make predictions for LSS observables in the mildly non-linear regime. In this paper we focus on biased tracers. We argue that in calculations at a given order in the dark matter perturbations, highly biased tracers will underperform because of their larger higher derivative biases. A natural prediction of the EFTofLSS is therefore that by simply adding higher derivative biases, all tracers should perform comparably well. We implement this prediction for the halo-halo and the halo-matter power spectra at one loop, and the halo-halo-halo, halo-halo-matter, and halo-matter-matter bispectra at tree-level, and compare with simulations. We find good agreement with the prediction: for all tracers, we are able to match the bispectra up to $k\simeq0.17\,h/$Mpc at $z=0$ and the power spectra to a higher wavenumber.

Figures (8)

• Figure 1: $p$-values of the fit as a function of $k_{\rm max,B}$ for each different bin, without any higher derivative term taken into account. The dashed lines represent the results that we would have gotten if we had not correct the typo described in Sec. \ref{['subsec:fits']}, which was present in the numerical implementation of Angulo:2015eqa. For the full lines, the $k$-reach is approximately the same for Bin0 and Bin1, and then becomes worse for Bin2 and Bin3, which motivates us to add higher derivative terms for those two bins. Our goal is to show that each bin acquires approximately the same $k$-reach as Bin0 and Bin1 just by adding higher derivative bias terms.
• Figure 2: $p$-values of the fit for Bin2 as a function of $k_{\rm max,B}$, using the numerical values for the bias coefficients obtained at $k_{\rm fit}$. The blue line represents the curve of the $p$-values for Bin0, which defines the $k_{\rm max}$ that we want to achieve. We conclude that the addition of the higher derivative terms allows to fit Bin2 up to the same $k_{\rm max}$ as Bin0.
• Figure 3: $p$-values of the fit for Bin3 as a function of $k_{\rm max,B}$, using the numerical values for the bias coefficients obtained at $k_{\rm fit}$. The blue line represents the curve of the $p$-values for Bin0, which defines the $k_{\rm max}$ that we want to approximately achieve. Since our numerical technique does not seem able to handle the large number of parameters that enter in the fit, we restrict ourselves to a subset of the higher derivative biases. The result is shown by the brown curve, which is quite close to the reach of Bin0. To give a sense if the slight under-reach of Bin3 is due to the residual theoretical error of the prediction or to our non-optimal numerical fitting procedure, in light blue we present the curve obtained by somewhat arbitrarily raising the $k_{\rm fit}$ to $0.15h\,$Mpc$^{-1}$ from $0.12h\,$Mpc$^{-1}$, which leads to a reach in $k$ as large as for Bin0.
• Figure 4: Plots of the values of the bias coefficients for Bin0. The points at $k<0.08 \, h \, {\rm Mpc}^{-1}$ present some unexpected behavior which is hard to justify given the fact thart the EFTofLSS is expected to work well at low wavenumber. We therefore ignore those points for the procedure to fix the values of the coefficients, described in Sec. \ref{['procedure']}. Following this procedure, we find $k_{\rm fit} = 0.15 \, h \, {\rm Mpc}^{-1}$.
• Figure 5: Plots of the values of the bias coefficients for Bin1. The points at $k<0.08 \, h \, {\rm Mpc}^{-1}$ present some unexpected behavior which is hard to justify given the fact that the EFTofLSS is expected to work well at low wavenumber. We therefore ignore those points for the procedure to fix the values of the coefficients, described in Sec. \ref{['procedure']}. Following this procedure, we find $k_{\rm fit} = 0.18 \, h \, {\rm Mpc}^{-1}$.