Biased Tracers in Redshift Space in the EFT of Large-Scale Structure

TL;DR

This work extends the EFTofLSS to biased tracers in redshift space by computing the one-loop power spectrum, including IR resummation for BAO, and comparing to Dark Sky simulations at $z\simeq 0.67$. The formalism introduces a tractable set of bias, counter-term, and stochastic parameters (ten in total) governing the real-space halo expansion and its redshift-space mapping, with velocity divergences treated as a special tracer. The authors demonstrate percent-level agreement with simulations up to $k\approx 0.43\,h\mathrm{Mpc}^{-1}$ for multiple tracer populations, validating the EFTofLSS as a practical framework for interpreting large-scale structure data and enabling robust cosmological inferences from long-wavelength statistics. This establishes the groundwork for applying EFTofLSS to observational data, including higher multipoles and more complex tracers, in the coming years.

Abstract

The Effective Field Theory of Large-Scale Structure (EFTofLSS) provides a novel formalism that is able to accurately predict the clustering of large-scale structure (LSS) in the mildly non-linear regime. Here we provide the first computation of the power spectrum of biased tracers in redshift space at one loop order, and we make the associated code publicly available. We compare the multipoles $\ell=0,2$ of the redshift-space halo power spectrum, together with the real-space matter and halo power spectra, with data from numerical simulations at $z=0.67$. For the samples we compare to, which have a number density of $\bar n=3.8 \cdot 10^{-2}(h \ {\rm Mpc}^{-1})^3$ and $\bar n=3.9 \cdot 10^{-4}(h \ {\rm Mpc}^{-1})^3$, we find that the calculation at one-loop order matches numerical measurements to within a few percent up to $k\simeq 0.43 \ h \ {\rm Mpc}^{-1}$, a significant improvement with respect to former techniques. By performing the so-called IR-resummation, we find that the Baryon Acoustic Oscillation peak is accurately reproduced. Based on the results presented here, long-wavelength statistics that are routinely observed in LSS surveys can be finally computed in the EFTofLSS. This formalism thus is ready to start to be compared directly to observational data.

Figures (8)

• Figure 1: Plot of $p$-values calculated up to a given $k$ for the IR-resummed fit depicted in Fig. \ref{['fits']} with $k_{\rm fit}=0.39 \ h \ { \rm Mpc^{-1}}$. The solid blue curve shows the $p$-value, neglecting the data points with $k<0.06 \ {\rm h \ Mpc^{-1}}$, and the dotted blue curve includes all of the low-$k$ points. The horizontal red line shows $p=0.05$.
• Figure 2: Results of the fits of the IR-resummed EFT power spectra at $z=0.67$ to the power spectra of halos and dark matter extracted from simulations. The halos have masses of $M_{200} > 1\times 10^{11}$$h^{-1}M_{\odot}$, with a number density $\bar{n}=3.8 \cdot 10^{-2}(\,h\, {\rm Mpc}^{-1}\,)^3$. The fits were performed in the $k$-range $k_{\rm min}=0.01 \ h \ { \rm Mpc^{-1}}$ to $k_{\rm fit}=0.39 \ h \ { \rm Mpc^{-1}}$ and resulted in the best-fit parameters $\{ b_1=0.98 \pm 0.01, \ b_2=0.01\pm 2.73, \ b_3=-0.62 \pm 1.43, \ b_4=0.58 \pm 2.33, \ c_{\rm ct}^{(\delta_h)}=(5.3\pm 4.7) \left( k_{\rm NL} \ h^{-1} {\rm Mpc} \right)^{2} , \ \tilde{c}_{r,1}=(-14 \pm 5) \left( k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2}, \ \tilde{c}_{r,2}=(-0.69 \pm 1.67 ) \left( k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2}, \ c_{\epsilon,1}=0.76 \pm 14.74, \ c_{\epsilon,2}=(8.9 \pm 3.4) \left( k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2}, \ c_{\epsilon,3}=(8.0 \pm 7.8 ) \left( k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2} \}$ for the halos and $c_s^2=(-0.61 \pm 0.02) \left( k_{\rm NL} \ h^{-1} \ {\rm Mpc} \right)^{2}$ for the dark matter. The shaded regions show the 1$\sigma$ error on the simulation data, which includes the error on the halo spectra from simulations described in Feldman:1993ky and a $1\%$ error that we add in quadrature to account for unknown systematic effects. The expected theoretical error is given by the dotted lines.
• Figure 3: Left: Results of the fits of the EFT power spectra at $z=0.67$ after IR-resummation to the power spectra of LRGs in the $v_{m_{peak}}$ sample Jennings:2015lea, which has a number density $\bar{n}=3.9 \cdot 10^{-4}(\,h\, {\rm Mpc}^{-1}\,)^3$, and dark matter extracted from simulations. The fits were performed in the $k$-range $k_{\rm min}=0.01 \ h \ { \rm Mpc^{-1}}$ to $k_{\rm fit}=0.42 \ h \ { \rm Mpc^{-1}}$ and resulted in the best-fit parameters $\{ b_1=1.86 \pm 0.04,\ b_2=0.99\pm 7.59, \ b_3=-4.5 \pm 2.8, \ b_4=0.68 \pm 6.01,\ c_{\rm ct}^{(\delta_h)}=(0.69 \pm 8.35) \left( k_{\rm NL} \ h^{-1} \ {\rm Mpc} \right)^{2}, \ \tilde{c}_{r,1}=(-30 \pm 6)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2}, \ \tilde{c}_{r,2}=(4.6 \pm 1.3)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2}, \ c_{\epsilon,1}=13 \pm 33, \ c_{\epsilon,2}=(30 \pm 12)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2},\ c_{\epsilon,3}=(14 \pm 25)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2} \}$ for the LRGs and $c_s^2=(-0.61 \pm 0.02) \left( k_{\rm NL} \ h^{-1} \ {\rm Mpc} \right)^{2}$ for the dark matter. $P_{\rm real}$ is plotted in red, $P_{l=0}$ in blue, $P_{l=2}$ in green, and $P_{\rm DM}$ in orange. The shaded region shows the 1$\sigma$ error on the simulation data, which includes the error on the spectra from simulations described in Feldman:1993ky and a $1\%$ error added in quadrature to account for unknown systematics. The expected theoretical error is given by the dotted lines. Right: Plot of $p$-values calculated up to a given $k$ for the IR-resummed fit to the $v_{m_{peak}}$ power spectra with $k_{\rm fit}=0.42 \ h \ { \rm Mpc^{-1}}$. The solid blue curve shows the $p$-value, neglecting the data points with $k<0.06 \ {\rm h \ Mpc^{-1}}$, and the dotted blue curve includes all of the low-$k$ points. The horizontal red line shows $p=0.05$.
• Figure 4: Results of the fits of the EFT power spectra at $z=0.67$ before IR-resummation to the power spectra of halos and dark matter extracted from simulations, which were performed in the $k$-range $k_{\rm min}=0.01 \ h \ { \rm Mpc^{-1}}$ to $k_{\rm fit}=0.39 \ h \ { \rm Mpc^{-1}}$ and resulted in the best-fit parameters $\{ b_1=0.98 \pm 0.01 ,\ b_2=1.4 \pm 1.9, \ b_3=-0.84 \pm 0.88, \ b_4=-0.83\pm 1.63,\ c_{\rm ct}^{(\delta_h)}=(9.6 \pm 3.0)\left( k_{\rm NL} \ h^{-1} {\rm Mpc} \right)^{2}, \ \tilde{c}_{r,1}=(-12 \pm 4) \left(k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2}, \ \tilde{c}_{r,2}=(-0.45 \pm 1.26)\left(k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2}, \ c_{\epsilon,1}=-1.4 \pm 10.7, \ c_{\epsilon,2}=(11 \pm 2)\left(k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2},\ c_{\epsilon,3}=(-7.1\pm 8.2)\left(k_{\rm M} \ h^{-1} {\rm Mpc} \right)^{2}\}$ for the halos and $c_s^2=(-0.61 \pm 0.02) \left( k_{\rm NL} \ h^{-1} {\rm Mpc} \right)^{2}$ for the dark matter. $P_{\rm real}$ is plotted in red, $P_{l=0}$ in blue, $P_{l=2}$ in green, and $P_{\rm DM}$ in orange. The shaded region shows the 1$\sigma$ error on the simulation data, which includes the error on the halo spectra from simulations described in Feldman:1993ky and a $1\%$ error added in quadrature to account for unknown systematics. The expected theoretical error is given by the dotted lines.
• Figure 5: Left: Results of the fits, including theoretical error in quadrature, of the EFT power spectra at $z=0.67$ after IR-resummation to the power spectra of halos Jennings:2015lea, which has a number density $\bar{n}=3.8 \cdot 10^{-2}(\,h\, {\rm Mpc}^{-1}\,)^3$, and dark matter extracted from simulations. The fits were performed in the $k$-range $0.01 \ h \ { \rm Mpc^{-1}}$ to $0.54 \ h \ { \rm Mpc^{-1}}$ and resulted in the best-fit parameters $\{ b_1=0.98 \pm 0.01, b_2= 0.04 \pm 1.28, b_3 = 0.06 \pm 0.90, b_4 = 0.56 \pm 1.05, c_{\rm ct}^{(\delta_h)}= (6.2 \pm 3.2)\left( k_{\rm NL} \ h^{-1} \ {\rm Mpc} \right)^{2}, \tilde{c}_{r,1} = (-15 \pm 2)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2}, \tilde{c}_{r,2} =( 0.64 \pm 0.85)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2}, c_{\epsilon,1} = 1.1 \pm 8.4, c_{\epsilon,2} = (4.6 \pm 1.0)\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2}, c_{\epsilon,3} = (12 \pm 1 )\left( k_{\rm M} \ h^{-1} \ {\rm Mpc} \right)^{2}\}$ for the halos and $c_s^2=(-0.49 \pm 0.01)\left( k_{\rm NL} \ h^{-1} \ {\rm Mpc} \right)^{2}$ for the dark matter. $P_{\rm real}$ is plotted in red, $P_{l=0}$ in blue, $P_{l=2}$ in green, and $P_{\rm DM}$ in orange. The shaded region shows the 1$\sigma$ error on the simulation data, which includes the error on the spectra from simulations described in Feldman:1993ky and a $1\%$ error added in quadrature to account for unknown systematics. The expected theoretical error is given by the dotted lines. Right: Plot of $p$-values calculated up to a given $k$ for the IR-resummed fit to the $v_{m_{peak}}$ power spectra. The solid blue curve shows the $p$-value, neglecting the data points with $k<0.06 \ {\rm h \ Mpc^{-1}}$, and the dotted blue curve includes all of the low-$k$ points. The horizontal red line shows $p=0.05$.