The IR-resummed Effective Field Theory of Large Scale Structures

TL;DR

The authors address BAO-induced residuals in equal-time EFT predictions for the matter power spectrum by introducing an IR-resummation that treats large-scale displacements non-perturbatively within a Lagrangian framework. The method reformulates Eulerian EFT predictions through a displacement-mixing kernel K0 and a mode-mixing matrix M_{||}, yielding IR-safe equal-time correlators and improved BAO damping, with extensions to momentum spectra to account for velocity effects. At two loops, the IR-resummed EFT matches numerical simulations up to k ≈ 0.6 h Mpc^{-1} to within 1%, and one-loop results extend accurate predictions further, while momentum spectra gain enhanced UV reach thanks to velocity treatments and optimized resummation. Real-space BAO predictions and their dependence on EFT parameters are shown to be robust, supporting the method’s potential for precise cosmological inferences from BAO and related statistics.

Abstract

We present a new method to resum the effect of large scale motions in the Effective Field Theory of Large Scale Structures. Because the linear power spectrum in $Λ$CDM is not scale free the effects of the large scale flows are enhanced. Although previous EFT calculations of the equal-time density power spectrum at one and two loops showed a remarkable agreement with numerical results, they also showed a 2% residual which appeared related to the BAO oscillations. We show that this was indeed the case, explain the physical origin and show how a Lagrangian based calculation removes this differences. We propose a simple method to upgrade existing Eulerian calculations to effectively make them Lagrangian and compare the new results with existing fits to numerical simulations. Our new two-loop results agrees with numerical results up to $k\sim 0.6 h/$Mpc to within 1% with no oscillatory residuals. We also compute power spectra involving momentum which is significantly more affected by the large scale flows. We show how keeping track of these velocities significantly enhances the UV reach of the momentum power spectrum in addition to removing the BAO related residuals. We compute predictions for the real space correlation function around the BAO scale and investigate its sensitivity to the EFT parameters and the details of the resummation technique.

Figures (9)

• Figure 1: Parameters measuring the amplitude of non-linear correction on a mode of wavenumber $k$. They quantify the motions created by modes longer ($\epsilon_{s<}$) and shorter ($\epsilon_{s>}$) than $k$ and the tides from larger scales ($\epsilon_{\delta <}$).
• Figure 2: For $q=100\, {\rm Mpc}/h$, we plot $P_{{\rm int}||_{0}}(r|q;t_1,t_2)$ in magenta, $P_{{\rm int}||_{1}}(r|q;t_1,t_2)$ in blue, and $P_{{\rm int}||_{2}}(r|q;t_1,t_2)$ in red.
• Figure 3: The value of $c_{s (2)}^2$ from (\ref{['eq:c2equation']}) as a function of the renormalization scale $k_\text{ren}$. We see that as we move $k_\text{ren}\in[0.10,0.35] \,h\, {\rm Mpc}^{-1}\,$, the value of $c_{s (2)}^2$ changes by just order $10\%$.
• Figure 4: Top: The prediction of the IR-resummed EFT at one-loop (in thick red) and two-loops (in thick blue). In thin dashed are the predictions from the Eulerian EFT, that is without IR-resummation, with the same colors respectively. The green band represents the estimated theoretical error from three-loops. The two-loops results have been renormalized at $k_\text{ren}=0.2\,h\, {\rm Mpc}^{-1}\,$, and $c_{s (1)}^2$ has been approximately fit up to $k\simeq 0.5\,h\, {\rm Mpc}^{-1}\,$. Since the equal-time matter power spectrum is IR-safe, we see that the effect of the IR-resummation is just to affect the oscillations, which are indeed now correctly taken into account. We see that the one-loop result matches to percent level the data up to $k\simeq0.34\,h\, {\rm Mpc}^{-1}\,$, while at two-loop matches all the way up to $k\simeq0.6\,h\, {\rm Mpc}^{-1}\,$. The spike at $k\simeq 0.05 \,h\, {\rm Mpc}^{-1}\,$ is due to the numerical interpolator, against which we compare, not to the EFT. It is also important to notice that the match stops exactly the three-loop term is estimated to become relevant. Bottom: We compare the predictions of the IR-resummed EFT with the ones of SPT. In thick magenta, red and blue we plot respectively the IR-resummed linear, one-loop and two-loops predictions of the EFT. With the same colors, but dashed, the same quantities in SPT. As we go to higher orders, SPT does not increase the agreement with the data. Furthermore, we notice that SPT has the same residual oscillatory features as the Eulerian EFT. In contrast, the IR-resummed EFTofLSS correctly predicts the size of the oscillations, and, at each order in perturbation theory, it improves the UV match to the data. Importantly, in the EFTofLSS, order by order in perturbation theory, it is possible to estimate up to where the theory should match the data.
• Figure 5: From top, anticlockwise, the predictions of the EFT for $P_{\delta\pi}$, $P_{\pi\pi}$ and $P_{\delta\delta}$. In Magenta we have the one-loop SPT, in red the one-loop Eulerian EFT, and in blue we have the IR-resummed one-loop EFT with optimized IR-resummation, while in blue dashed we plot the results of the IR-resummed one-loop EFT with non-optimized IR-resummation. The band around each line represents the $1$-$\sigma$ cosmic variance of the simulations. For $P_{\delta\delta}$, the results are analogous to the ones obtained in the former section. In particular the EFT fits the data up to $k\simeq 0.35\,h\, {\rm Mpc}^{-1}\,$. For $P_{\delta\pi}$, the results are very similar to $P_{\delta\delta}$: the UV reach is about the same, which is a confirmation of the validity of the EFT, as the UV reach should be more or less the same at a given order in perturbation theory for every quantity. Furthermore, the effect of the IR-resummation is simply to better reconstruct the oscillations, as $P_{\delta\pi}$ is an IR-safe quantity. Finally, in $P_{\pi\pi}$ we have two effects. Passing from SPT to the Eulerian EFT, the result agree more with the simulations, but the UV reach is much smaller than for $P_{\delta\delta}$. Passing to the IR-resummed EFT, we see that we achieve two effects. First, the reach in the UV is restored to be approximately the one for matter, and no more no less, as it should be. Second, the oscillations are now correctly computed, especially with the optimized procedure. The results are quite satisfactory, even though a more accurate check is limited by the large cosmic variance of the simulations.