Efficient exploration of cosmology dependence in the EFT of LSS

TL;DR

This paper addresses efficient, high-precision predictions of LSS clustering in the mildly nonlinear regime using the EFTofLSS. It introduces CosmoEFT and ResumEFT for fast two-loop power-spectrum calculations across nearby cosmologies, and TaylorEFT for rapid cosmology-parameter expansions, all validated against simulations and emulators. The results show that sub-percent accuracy is achievable within a few minutes on a laptop for cosmologies within $3\sigma$ of Planck, with Taylor expansions offering instant predictions and potential extensions to broader cosmologies. Collectively, the work provides practical, publicly available tools to explore cosmology dependence in EFTofLSS and to calibrate EFT parameters against simulations or observations, enabling efficient parameter studies and survey analyses.

Abstract

The most effective use of data from current and upcoming large scale structure~(LSS) and CMB observations requires the ability to predict the clustering of LSS with very high precision. The Effective Field Theory of Large Scale Structure (EFTofLSS) provides an instrument for performing analytical computations of LSS observables with the required precision in the mildly nonlinear regime. In this paper, we develop efficient implementations of these computations that allow for an exploration of their dependence on cosmological parameters. They are based on two ideas. First, once an observable has been computed with high precision for a reference cosmology, for a new cosmology the same can be easily obtained with comparable precision just by adding the difference in that observable, evaluated with much less precision. Second, most cosmologies of interest are sufficiently close to the Planck best-fit cosmology that observables can be obtained from a Taylor expansion around the reference cosmology. These ideas are implemented for the matter power spectrum at two loops and are released as public codes. When applied to cosmologies that are within 3$σ$ of the Planck best-fit model, the first method evaluates the power spectrum in a few minutes on a laptop, with results that have 1\% or better precision, while with the Taylor expansion the same quantity is instantly generated with similar precision. The ideas and codes we present may easily be extended for other applications or higher-precision results.

Figures (22)

• Figure 1: Left: The ratio $\Delta \tilde{P}_\alpha(k)/P_\alpha^\text{target}$ for the one-loop terms that enter the two-loop EFTofLSS matter power spectrum (see Sec. \ref{['sec:Results']} for more details): $P_\text{1-loop}$ (blue), $P_\text{1-loop}^{(c_{\rm s})}$ (red) and $P_\text{1-loop}^{(\text{quad},1)}$ (green). $\Delta\tilde{P}_\alpha$ is given by Eq. \ref{['eq:adj_DeltaPloop']} with $L=1$, and we show results for the cosmo_5 test cosmology (given in Table \ref{['tab:Cosmo_list']}). The black dashed line is the actual $k$-independent conservative value used in CosmoEFT to set the relative precision required for one-loop integrations (see Eq. \ref{['eq:target_prec_modified']}). Right: The same ratio for $P_\text{2-loop}$, again for the cosmo_5 cosmology. The dashed red curve uses Eq. \ref{['eq:diff_integration']}, while the solid blue curve uses the adjusted form, Eq. \ref{['eq:adj_DeltaPloop']}, with $P_\text{2-loop}^{\text{target/ref}}$ evaluated through direct integration. For general cosmologies, Eq. \ref{['eq:adj_DeltaPloop']} shows a similar improvement over Eq. \ref{['eq:diff_integration']} in removing most of the difference associated with the cosmological parameter $A_s$.
• Figure 2: Comparison of estimate (Eq. \ref{['eq:ratioestimate']}, with the modifications described in the main text; black points) and exact calculation of $|\Delta \tilde{P}_\text{2-loop} / P_\text{2-loop}^\text{target}|$ (blue lines) for two test cosmologies, cosmo_1 (left) and cosmo_5 (right), given in Table \ref{['tab:Cosmo_list']}. On average, for $k\gtrsim 0.5\,h\, {\rm Mpc}^{-1}\,$, the estimate slightly over-predicts the exact calculation, but this only means that the precision requested for the integration of $\Delta \tilde{P}_\text{2-loop}$ is slightly more conservative than necessary. For lower wavenumbers, the estimate is less precise, but, as we describe in the text, the required precision is also lower. Also, the estimate has the desirable feature of automatically limiting the precision requested close to the zero-crossings of $P_\text{2-loop}^\text{target}(k)$, by setting a ceiling on the value of $|\Delta \tilde{P}_\text{2-loop} / P_\text{2-loop}^\text{target}|$ in Eq. \ref{['eq:ratioestimate']}.
• Figure 3: $P_{11}(k)_{\| 2}$ (green), $P_\text{2-loop}(k)_{\| 0}$ from CosmoEFT (dashed blue), and $P_\text{2-loop}(k)_{\| 0}$ evaluated through full integration (red), all for the cosmo_5 test cosmology (see Table \ref{['tab:Cosmo_list']}). We show the IR-resummed version of each term, as indicated by the subscripts. The $P_\text{2-loop}$ curves in the left panel are $P_\text{2-loop}^\text{(UV-improved)}$, while the curves in the right panel are $P_\text{2-loop}^\text{(full)}$. The fact that $P_\text{2-loop}^\text{(UV-improved)}\ll P_{11}$ at low $k$ makes the small deviations we observe in the left panel acceptable. On the other hand, $P_\text{2-loop}^\text{(full)}$ amounts to $\sim 1\%$ of $P_{11}$ already at $k \approx 0.05 \,h\, {\rm Mpc}^{-1}\,$, leading to more demanding requirements on the precision of $P_\text{2-loop}$ if $P_\text{2-loop}^\text{(full)}$ is used.
• Figure 4: IR-resummed CosmoEFT outputs for our $3\sigma$-cosmologies cosmo_1-6 (in order: blue, red, green, purple, orange, cyan) relative to the direct calculation of the full integrand with precision $\epsilon=0.1\%$. Here $\epsilon_{\text{target}}=0.5\%$ for all panels, and spikes are due to zero-crossing. Subscripts $\|0,\|1$ denotes the IR-resummation order as in Senatore:2014via. Dashed lines mark 1% departures from direct calculations. For $P_\text{2-loop}$ we opt for showing two quantities relevant on two disjoint scale intervals, $k < 0.5 \,h\, {\rm Mpc}^{-1}\,$ and $k > 0.5 \,h\, {\rm Mpc}^{-1}\,$. $\Delta P_{\text{2-loop} \| 0}/P_{11 \| 2}$ indicates the error on the matter power spectrum predictions, and $\Delta P_{\text{2-loop} \| 0}/P_{\text{2-loop} \| 0}$ confirms the performance of our estimates for smaller scales.
• Figure 5: Two-dimensional compact stencil for numerical evaluation of first and second derivatives at $\{\theta_i^{\text{ref}},\theta_j^{\text{ref}}\}$. The increments in the two parameters correspond to their Planck standard deviations. Loop integrals $P_{\alpha}^{i,j}$ for the reference cosmology (blue box) are calculated through direct integration with 0.1% precision. The $k$-dependence has been omitted for clarity.