On the EFT of Large Scale Structures in Redshift Space

TL;DR

This paper extends the EFTofLSS to redshift-space distortions by incorporating baryonic effects and primordial non-Gaussianities into the redshift-space counterterms and by performing IR-resummation for the redshift-space power spectrum. It develops a computationally efficient, controlled one-loop framework that yields multipole predictions up to $\ell=6$, with BAO features accurately captured. Comparisons to large-volume N-body simulations show percent-level agreement for the lowest multipoles up to $k$ values around $0.13$–$0.18\,h\mathrm{Mpc}^{-1}$ at $z=0.56$, highlighting a longer velocity nonlinear scale and the necessity of higher-derivative counterterms; data limitations and noise temper precise $k$-reach estimates. Overall, the results strongly support the EFTofLSS in redshift space and point to future work with higher-order calculations and larger simulations to sharpen the $k$-reach and parameter constraints.

Abstract

We further develop the description of redshift space distortions within the Effective Field Theory of Large Scale Structures. First, we generalize the counterterms to include the effect of baryonic physics and primordial non-Gaussianity. Second, we evaluate the IR-resummation of the dark matter power spectrum in redshift space. This requires us to identify a controlled approximation that makes the numerical evaluation straightforward and efficient. Third, we compare the predictions of the theory at one loop with the power spectrum from numerical simulations up to $\ell=6$. We find that the IR-resummation allows us to correctly reproduce the BAO peak. The $k$-reach, or equivalently the precision for a given $k$, depends on additional counterterms that need to be matched to simulations. Since the non-linear scale for the velocity is expected to be longer than the one for the overdensity, we consider a minimal and a non-minimal set of counterterms. The quality of our numerical data makes it hard to firmly establish the performance of the theory at high wavenumbers. Within this limitation, we find that the theory at redshift $z=0.56$ and up to $\ell=2$ matches the data to percent level approximately up to $k \sim 0.13 \, h { \rm Mpc^{-1}}$ or $k \sim 0.18 \, h { \rm Mpc^{-1}}$, depending on the number of counterterms used, with potentially large improvement over former analytical techniques.

Figures (10)

• Figure 1: Here, we compare the EFT prediction with with $c_s^2 = 0.31 \, (k_{\rm NL}/ \, h { \rm Mpc^{-1}})^2$ to the BigMultiDark dark-matter power spectrum Klypin:2014kpa at $z=0.56$ . The dark blue curve is the one-loop IR-resummed EFT prediction, the red dot-dashed curve is the non-resummed EFT prediction, the gray dot-dashed curve is one-loop SPT, and the green region is the error on the data, where we have included a $1\%$ systematic error. We have also plotted the cosmic variance separately as a dashed grey curve to show that this largely explains the fluctuations in the data.
• Figure 2: Here, we present the results in redshift space at $z=0.56$ for $\ell = 0,2,4,6$, with $\bar{c}_1^2=-3.47 \, ( k^r_{ \rm NL} / \, h { \rm Mpc^{-1}})^2$ and $\bar{c}_2^2 = 0.81 \, ( k^r_{ \rm NL} / \, h { \rm Mpc^{-1}}) ^2$. The dark blue curve is the one-loop IR-resummed EFT prediction, the dot-dashed red curve is the non-resummed EFT prediction, the gray dot-dashed curve is one-loop SPT, and the green region is the error in the data, where we have included a $2\%$ systematic error. The plots fit until about $k \approx 0.13 \, h { \rm Mpc^{-1}}$. In the last plot, the non-linear data is very noisy, so we do not present the ratio of power spectra. However, it is still clear that the prediction is within the errors. We have also plotted the cosmic variance separately as a dashed grey curve to show that this largely explains the fluctuations in the data. Note that for $\ell = 4$ and $\ell = 6$ the error is completely dominated by the cosmic variance.
• Figure 3: Here, we present the results in redshift space at $z=0.56$ for $\ell = 0,2,4,6$, with $\bar{c}_1^2=-3.47 \, ( k^r_{ \rm NL} / \, h { \rm Mpc^{-1}})^2$ and $\bar{c}_2^2 = 0.81 \, ( k^r_{ \rm NL} / \, h { \rm Mpc^{-1}}) ^2$, the same as in Figure \ref{['pls']}, but we plot $k^2 P^r_\ell (k)$. The dark blue curve is the one-loop IR-resummed EFT prediction, the dot-dashed red curve is the non-resummed EFT prediction, the gray dot-dashed curve is one-loop SPT, the thin black line is the data, and the green region is the error in the data, where we have included a $2\%$ systematic error, which, for the higher multipoles, is probably an underestimate (see text for additional comments). The plots fit until about $k \approx 0.13 \, h { \rm Mpc^{-1}}$.
• Figure 4: Here, we present the resummation results in redshift space at $z=0.56$ for the real-space power spectrum and $\ell = 0,2,4$. The dark blue curve is the one-loop IR-resummed EFT prediction and the dot-dashed red curve is the non-resummed EFT prediction. The values of the speed of sound parameters have been chosen so that the curves only change by about $10\%$ up to $k = 0.5 \, h { \rm Mpc^{-1}}$ in order to emphasize the result of the resummation. As expected, the resummation does not change the UV part of the theory; it just captures the BAO oscillations.
• Figure 5: Here, we show two possible improvements in the small $k^r_{ \rm NL}$ limit at $z=0.56$. The dark blue curve is the one-loop IR-resummed EFT prediction with $\bar{c}_1^2=-0.56\,(k_{\rm NL} / \, h { \rm Mpc^{-1}})^2$ and $\bar{c}_2^2=0.13\,( k_{\rm NL} / \, h { \rm Mpc^{-1}}) ^2$, the same as in Figure \ref{['pls']} but we have set $k^r_{ \rm NL} =k_{\rm NL} / 2.5$. The dashed green curve includes the $( k / k^r_{ \rm NL} )^4 P_{11}$ counter terms with no stochastic piece and the remaining parameters given by Eq. (\ref{['eq:param1']}). The dashed red curve includes the stochastic piece and has parameters $\bar{c}_1^2=-0.54\,(k_{\rm NL} / \, h { \rm Mpc^{-1}})^2$, $\bar{c}_2^2=0.072\,( k_{\rm NL} / \, h { \rm Mpc^{-1}})^2$ and the remaining parameters given by Eq. (\ref{['eq:param2']}). Thus, including these higher order terms can extend the fits until $k \approx 0.18 \, h { \rm Mpc^{-1}}$ without the stochastic term, and to $k \approx 0.2 \, h { \rm Mpc^{-1}}$ with the stochastic term. The dot-dashed red curve, the gray dot-dashed curve, and the green region are the same as in Figure \ref{['pls']}.