Precision Comparison of the Power Spectrum in the EFTofLSS with Simulations

TL;DR

This paper demonstrates that a two-loop Effective Field Theory of Large Scale Structures (EFTofLSS) power spectrum, when augmented with three counterterms—linear $\delta$, quadratic $\delta^2$, and a higher-derivative term—achieves UV-insensitive, high-precision agreement with Dark Sky simulations up to $k\sim0.34\,h\mathrm{Mpc}^{-1}$ at $z=0$ and extends the usable range substantially at higher redshift. It introduces a UV-insensitive renormalization approach, uses IR-resummation, and provides a careful fitting strategy that controls cosmic variance and theoretical errors. The authors show that two additional counterterms beyond the single one commonly used are necessary to account for short-distance nonlinearities, and they quantify the time evolution of the counterterms with a quasi-two-parameter model guided by UV expectations. Across redshifts, the EFTofLSS offers a dramatic increase in the number of accessible modes and, therefore, cosmological information, while remaining mindful of potential overfitting and simulation systematics. This work strengthens the case for high-precision analytical control of nonlinear structure formation and sets the stage for exploiting larger, more detailed datasets in the coming decade.

Abstract

We study the prediction of the dark matter power spectrum at two-loop order in the Effective Field Theory of Large Scale Structures (EFTofLSS) using high precision numerical simulations. In our universe, short distance non-linear perturbations, not under perturbative control, affect long distance fluctuations through an effective stress tensor that needs to be parametrized in terms of counterterms that are functions of the long distance fluctuating fields. We find that at two-loop order it is necessary to include three counterterms: a linear term in the over density, $δ$, a quadratic term, $δ^2$, and a higher derivative term, $\partial^2δ$. After the inclusion of these three terms, the EFTofLSS at two-loop order matches simulation data up to $k\simeq 0.34 \,h\, {\rm Mpc}^{-1}$ at redshift $z=0$, up to $k\simeq 0.55\,h\, {\rm Mpc}^{-1}$ at $z=1$, and up to $k\simeq 1.1\,h\, {\rm Mpc}^{-1}$ at $z=2$. At these wavenumbers, the cosmic variance of the simulation is at least as small as $10^{-3}$, providing a high precision comparison between theory and data. The actual reach of the theory is affected by theoretical uncertainties associated to not having included higher order terms in perturbation theory, for which we provide an estimate, and by potentially overfitting the data, which we also try to address. Since in the EFTofLSS the coupling constants associated with the counterterms are unknown functions of time, we show how a simple parametrization gives a sensible description of their time-dependence. Overall, the $k$-reach of the EFTofLSS is much larger than previous analytical techniques, showing that the amount of cosmological information amenable to high-precision analytical control might be much larger than previously believed.

Figures (18)

• Figure 1: Ratio of the predictions of the EFT at two and one loops. By choosing a value of $k_\text{ren}$ small enough, we are able to make this ratio very flat, demonstrating that the higher order terms contained in $P_\text{EFT-2-loop}$ are negligible below at the wavenumber at which we impose the renormalization condition.
• Figure 2: The two-loop EFTofLSS prediction for the $z=0$ power spectrum, when one includes only one counterterm (associated with the speed of sound $c_{s (1)}^2$), along with various other theory predictions. The EFT curves use a value of $c_{s (1)}^2 \simeq 0.53\left({k_{\rm NL}} / (2 \,h\, {\rm Mpc}^{-1}\,) \right)^2$. We can see that the theory performs better and better as higher order contributions are included. The blue shading represents the variation of the result if we perform the fit to determine $c_{s (1)}^2$ up to $0.75k_{\rm fit}$ and choose the two values $1\sigma$ away from the central value, where $k_{\rm fit}$ is the wavenumber beyond which $c_{s (1)}^2$ begins to deviate from the value determined at lower $k$. For $k<0.1\,h\, {\rm Mpc}^{-1}\,$, the linear power spectrum in the theory prediction is replaced with the power spectrum measured from the initial conditions of the simulations, allowing for a dramatic reduction in the variance of the $P_\text{theory}/P_\text{NL}$ curves at these wavenumbers, but also implying that the cosmic variance errorbars (represented by the grey shading) do not reflect the uncertainty on the curves for $k<0.1\,h\, {\rm Mpc}^{-1}\,$ (as discussed in Sec. \ref{['sec:darksky']}). The ${k_{\rm reach}}$ of the EFT at two loops is about ${k_{\rm reach}}\simeq 0.15\,h\, {\rm Mpc}^{-1}\,$, where the cosmic variance is about 0.4%, even though there is large theoretical uncertainty. The $k$-reach is smaller than what was previously presented in Carrasco:2013mua, where the errorbars were taken to be $\sim$2%, because the much higher precision of the available numerical data allows the choice of a lower $k_\text{ren}$, which eliminates the strong cancellation between various two-loop terms that was seen in Carrasco:2013mua.
• Figure 3: Same as Fig. \ref{['fig:two-loop-prediction']}, but using the procedure from previous papers (e.g. Carrasco:2013muaSenatore:2014viaForeman:2015uva) to fix $c_{s (1)}^2$ and $c_{s (2)}^2$ (see the main text for details). If we allow for a uniform 2% error budget on $P_{\rm NL}$ (as was required in previous work, due to the use of the Coyote emulator), and use a $k_\text{ren}$ comparable to previous work ($k_\text{ren}\sim 0.23\,h\, {\rm Mpc}^{-1}\,$), we find results that are consistent with Carrasco:2013muaSenatore:2014viaForeman:2015uva (the $\sim$1.5% offset between the one- and two-loop curves was actually about $0.7\%$ in the cosmologies and data considered in Carrasco:2013muaSenatore:2014viaForeman:2015uva). One sees that at low $k$, there is an offset between theory and data. Such an offset cannot be ruled out by using power spectra from the Coyote emulator, since its output has a systematic errorbar of at least $1\%$. Instead, for Dark Sky, we cannot allow for that level of systematic error, and forcing the absence of the offset at low $k$ affects the $k$-reach of the theory, leading to the results in Fig. \ref{['fig:two-loop-prediction']}.
• Figure 4: The red dashed line shows the ratio of the calculations of $P_\text{2-loop}$ with cutoff $\Lambda=\infty$ and $\Lambda=2 \, k_{\rm reach}(z=0) =0.68\,h\, {\rm Mpc}^{-1}\,$, while the blue dotted and black solid lines show the same ratio but adding the two or three counterterms to the $\Lambda=\infty$$P_\text{2-loop}$ calculation, respectively. We see that the difference between the $\Lambda=0.68\,h\, {\rm Mpc}^{-1}\,$ and $\Lambda=\infty$ calculation of $P_\text{2-loop}$ can be absorbed by the counterterms. This is an important consistency check of the EFTofLSS, indicating that the unknown short-distance physics affecting the loop corrections can be accounted for by the EFT counterterms. It also tells us that the two-loop power spectrum in the EFTofLSS should minimally include the counterterms corresponding to $c_1$ and $c_4$ in order to be insensitive to our assumptions about the UV behavior of the theory.
• Figure 5: The correspondence between the terms we add to the effective stress tensor and a schematic representation of the lowest-order diagrams (in either the power spectrum or bispectrum) they are associated with via renormalization.