The 2-loop matter power spectrum and the IR-safe integrand

TL;DR

This paper tackles the challenge of predicting the 2-loop matter power spectrum within the EFTofLSS by addressing infrared (IR) divergences that cancel only after summing all diagrams. The authors introduce an IR-safe global integrand that unifies the four 2-loop diagrams into a single integrand with carefully chosen variable changes and Theta functions, so IR divergences are mapped to the origin and cancel at the integrand level. They apply the method to scaling universes with linear power spectra $P_{11}(k) \propto k^n$, examining $n=-3/2$ (UV-divergent) and $n=-5/2$ (UV-finite) cases, and perform renormalization using EFT counterterms; the results exhibit IR- and UV-independence after renormalization, with well-behaved finite parts that scale as predicted. The approach substantially improves numerical stability and accuracy for higher-loop calculations and is argued to generalize to even higher loops, paving the way for percent-level predictions up to moderate $k$ in the EFTofLSS and informing future analyses of real cosmologies.

Abstract

Large scale structure surveys are likely the next leading probe of cosmological information. It is therefore crucial to reliably predict their observables. The Effective Field Theory of Large Scale Structures (EFTofLSS) provides a manifestly convergent perturbation theory for the weakly non-linear regime, where dark matter correlation functions are computed in an expansion of the wavenumber k over the wavenumber associated to the non-linear scale knl. To push the predictions to higher wavenumbers, it is necessary to compute the 2-loop matter power spectrum. For equal-time correlators, exactly as with standard perturturbation theory, there are IR divergences present in each diagram that cancel completely in the final result. We develop a method by which all 2-loop diagrams are computed as one integral, with an integrand that is manifestly free of any IR divergences. This allows us to compute the 2-loop power spectra in a reliable way that is much less numerically challenging than standard techniques. We apply our method to scaling universes where the linear power spectrum is a single power law of k, and where IR divergences can particularly easily interfere with accurate evaluation of loop corrections if not handled carefully. We show that our results are independent of IR cutoff and, after renormalization, of the UV cutoff, and comment how the method presented here naturally generalizes to higher loops.

Figures (7)

• Figure 1: Left: The 2-loop power spectrum $P_{\text{2-loop}}$ for various values of the UV cutoff $\Lambda$ and IR cutoff $k_{\rm min}$. We show $\Lambda=2k_{\rm NL}$ in blue, $10\,k_{\rm NL}$ in red, $20\,k_{\rm NL}$ in yellow, and $80\,k_{\rm NL}$ in green, The result is $k_{\rm min}$ independent, as we demonstrate in detail in Fig. \ref{['fig:kmin_dependence']}, so here we plot only results with $k_{\rm min}=5\times10^{-3}k_{\rm NL}$, as the results with $k_{\rm min}=5\times10^{-4}k_{\rm NL}$ are visually indistinguishable. Right: same as above but on a logarithmic scale. When the UV cutoff is high enough so that finite terms are negligible, we can see that all the curves have the same slope, corresponding to $k^{1/2}$ dependence, and that they depend linearly on $\Lambda$. This agrees with what is expected from the divergent term in (\ref{['eq:p51_div3']}).
• Figure 2: The points represents the finite contributions to $P_{\text{2-loop}}$ for $\Lambda=2 \, k_{\rm NL}$ (blue) $\Lambda=10\,k_{\rm NL}$ (red), $\Lambda=20\,k_{\rm NL}$ (yellow) and $\Lambda=80\,k_{\rm NL}$ (green) obtained after the divergent part has been subtracted. As we go to low enough $k$'s, the three curves approach the same one, as indicated by the continuous fitting functions of Eq. (\ref{['eq:finite']}), plotted as solid lines. As we move to higher $k$'s the $\Lambda$ dependence, strongest for the $\Lambda=2\,k_{\rm NL}$ case, becomes clearly visible.
• Figure 3: Left: 2-loop power spectrum $P_{\text{2-loop}}$ for ideal scaling of tilt $n=-5/2$ for various values of the IR cutoff $k_{\rm min}$. This universe is UV cutoff ($\Lambda$) independent. Right: We demonstrate the functional form of the IR-dependence by comparing to the fit given in eq. (\ref{['eqn:neg5o2Fit']}).
• Figure 4: Fractional difference between $P_{\text{2-loop},n=-3/2}$ for $k_{\rm min}=5 \times 10^{-3}\, k_{\rm NL}$ and $k_{\rm min}=5 \times 10^{-4}\, k_{\rm NL}$ for $\Lambda=10$. We see that for $k\lesssim 1$, the dependence is less than 1/3 of a percent.
• Figure 5: Plot of the deviation between the "quasi-Monte-Carlo" and "random Monte-Carlo" results from CUBA's Vegas integration algorithm, for the IR safe integrand for scaling universe $n=-3/2$. We see that it is sub-percent for the range considered.