An analytic implementation of the IR-resummation for the BAO peak

TL;DR

This paper introduces a fully analytic implementation of IR-resummation for the BAO peak within the EFTofLSS, leveraging FFTLog to decompose the linear power spectrum into complex power laws and performing analytic one-loop integrals. The authors derive explicit tree-level and one-loop resummed expressions for the correlation function, expressible as sums over cosmology-dependent coefficients and fixed analytic kernels, with three practical evaluation strategies: exact evaluation, saddle-point approximation, and a fixed-displacements fast approximation. The analytic method achieves sub-percent accuracy (≈0.2%) relative to standard numerical integration while delivering substantial speedups (≈6–10× when using fixed displacements) and is generalizable to higher loops. The results, validated against Dark Sky simulations, demonstrate accurate BAO reconstruction and robust applicability to data-driven cosmological parameter fitting, with potential extensions to the bispectrum and redshift-space distortions.

Abstract

We develop an analytic method for implementing the IR-resummation of arXiv:1404.5954, which allows one to correctly and consistently describe the imprint of baryon acoustic oscillations (BAO) on statistical observables in large-scale structure. We show that the final IR-resummed correlation function can be computed analytically without relying on numerical integration, thus allowing for an efficient and accurate use of these predictions on real data in cosmological parameter fitting. In this work we focus on the one-loop correlation function and the BAO peak. We show that, compared with the standard numerical integration method of IR-resummation, the new method is accurate to better than 0.2 %, and is quite easily improvable. We also give an approximate resummation scheme which is based on using the linear displacements of a fixed fiducial cosmology, which when combined with the method described above, is about six times faster than the standard numerical integration. Finally, we show that this analytic method is generalizable to higher loop computations.

Figures (11)

• Figure 1: The BAO peak in Eulerian perturbation theory at tree level (dark blue), one loop (red), and two loops (purple), along with the individual one-loop contribution (dashed red). We see that the convergence is quite slow since the difference between the tree-level and one-loop curves is about the same as the difference between the one-loop and two-loop curves. In this plot, we also show the two-loop resummed correlation function (thin black) as a proxy for the correct answer, see Sec. \ref{['resultssec']} for a more thorough discussion.
• Figure 2: We plot the functions $\alpha_0 (r )$ and $\alpha_2 ( r )$ which are related to the long-wavelength displacements, see App. \ref{['irresumexpsec']}, for the cosmology C1 described in Sec. \ref{['resultssec']}. The forms of the functions plotted here are the ones which are most relevant for the calculations that we do in this paper.
• Figure 3: Here we show the difference between the two cosmologies used in this paper, C1 and C2, described in Footnote \ref{['2cosmos']}, by plotting both the linear power spectra and the linear correlation functions. The cosmologies differ by about $4.8\%$ in $\Omega_b$ and $1.5 \%$ in $\sigma_8$, both deviations which are greater than those allowed by the $68\%$ confidence limits of Planck Planck:2015xua. Notice that while the power spectra are different by about $5\%$, as expected from the change in $\Omega_b$, this causes an almost $15\%$ change in the BAO peak. We note that the correlation function goes through zero near $r \approx 120 \, h^{-1} \text{Mpc}$, which is why the purple curve diverges above.
• Figure 4: In this plot, we show the expansion parameters $\epsilon_{s <}$ and $\epsilon_{\delta <}$ for the cosmologies C1 and C2 described in Footnote \ref{['2cosmos']}. Here, the unprimed quantities come from C1 and the primed quantities come from C2. The blue curve is $\epsilon_{s<}$ and $\epsilon_{s<}'$ (they are indistinguishable in this plot), the red curve is $\epsilon_{\delta <}$ and $\epsilon_{\delta <}'$ (they are also indistinguishable), and the green curve is the difference $\epsilon_{s<} - \epsilon_{s<}'$. We see that the difference in the displacements between the two cosmologies, which have a different $\Omega_b$ by about $4.8\%$, is at the percent level for $k \sim 0.12 \, h \, \text{Mpc}^{-1} \equiv \Lambda_{\rm IR}$, which is the scale up to which we resum the IR modes.
• Figure 5: In this plot, we show the convergence of the fixed-displacements approximation as a function of the number of loops, using the two cosmologies C1 and C2 described in Footnote \ref{['2cosmos']}, which have values of $\Omega_b$ which are different by approximately $4.8\%$. Here, $\xi_{\rm resum}$ is the correctly computed IR-resummed correlation function using the base cosmology C1. On the other hand, for $\xi_{\rm resum}'$, the Eulerian loops are computed in the cosmology C1, but the displacements from C2 are used in the IR-resummation. As we see, the approximation gets better with the number of loops, in reasonable agreement with what is expected from eq. (\ref{['errorest']}).