The Effective Field Theory of Large Scale Structures at Two Loops

TL;DR

The paper demonstrates that the EFTofLSS, extended to two-loop order, predicts the nonlinear matter power spectrum with percent accuracy up to $k\approx 0.6\,h\,\mathrm{Mpc}^{-1}$ at $z=0$ using a single primary counterterm (plus a theoretically determined two-loop counterterm), vastly extending the usable quasi-linear regime beyond standard perturbation theory. By modeling short-distance physics through a nonlocal-in-time framework and employing IR-safe loop integrals, the authors obtain a self-consistent expansion whose finite two-loop piece scales as $(k/k_{\rm NL})^{3}$ relative to the nonlinear baseline, enabling predictions for momentum-related observables as well. The results, validated against $N$-body simulations, indicate significant gains in extractable cosmological information and potential to constrain primordial physics, while highlighting observational and modeling challenges such as velocity-related measurements and baryonic effects. Overall, the work provides a robust pathway to harnessing large-scale structure data with a controlled, renormalized perturbative approach that substantially broadens the accessible scale range.

Abstract

Large scale structure surveys promise to be 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 of dark matter, where correlation functions are computed in an expansion of the wavenumber k of a mode over the wavenumber associated with the non-linear scale k_nl. Since most of the information is contained at high wavenumbers, it is necessary to compute higher order corrections to correlation functions. After the one-loop correction to the matter power spectrum, we estimate that the next leading one is the two-loop contribution, which we compute here. At this order in k/k_nl, there is only one counterterm in the EFTofLSS that must be included, though this term contributes both at tree-level and in several one-loop diagrams. We also discuss correlation functions involving the velocity and momentum fields. We find that the EFTofLSS prediction at two loops matches to percent accuracy the non-linear matter power spectrum at redshift zero up to k~0.6 h/Mpc, requiring just one unknown coefficient that needs to be fit to observations. Given that Standard Perturbation Theory stops converging at redshift zero at k~0.1 h/Mpc, our results demonstrate the possibility of accessing a factor of order 200 more dark matter quasi-linear modes than naively expected. If the remaining observational challenges to accessing these modes can be addressed with similar success, our results show that there is tremendous potential for large scale structure surveys to explore the primordial universe.

Figures (12)

• Figure 1: Describing the linear power spectrum. We compare the linear power spectrum $P_{11}$ (black) and two fit scaling descriptions, $P_{\rm fit}\sim\left(\frac{k}{k_{\rm NL}} \right)^n$, with $n\approx-1.70$ using $k_{\rm NL}\approx1.80h\,$Mpc$^{-1}$ (dashed, red), and $n\approx-2.12$ using $k_{\rm NL}\approx4.64 h\,$Mpc$^{-1}$ (solid, blue). The left plot is shown as a ratio to $P_{11}$ while the right shows the absolute values on a log scale. We see that a piecewise scaling description using both scales could be a useful approximation for power counting estimates.
• Figure 2: Examples of reducible two-loop diagrams. The upper diagram is included in $P_{51}$ and is 1PI. The lower diagram is included in $P_{33}$ and is not 1PI.
• Figure 3: Components of the one-loop EFT correction.There are three functions $\tilde{P}_{1,2,3}$ (given by Eqs. (\ref{['eq:ptilde1']}) to (\ref{['eq:ptilde3']}) in App. \ref{['app:solutions']}) relevant to the one-loop perturbative correction to the matter power spectrum. We plot these functions $\tilde{P}_{1,2,3}$ as solid (black), dashed (red) and dotted (blue) lines respectively. Note that $\tilde{P}_3$ is subdominant for much of the range.
• Figure 4: Fit range for one-loop EFT. We plot $P_{\text{EFT-1-loop}}$ normalized to non-linear data over the range to which the sole EFT parameter is fit. The red curve is the best fit value of $c_{s (1)}^2$ and the red band shows the 2-$\sigma$ error on $c_{s (1)}^2$. We also show the data points that are fit, along with their 2-$\sigma$ errors (assuming 1 percent error on all points).
• Figure 5: Contributions to the power specturm.Left: We show every term appearing in Eq. (\ref{['equ:peft2loop']}) separately. We plot $P_{11}$, $P_{\text{1-loop}}$ and $P_{\text{2-loop}}$ in solid, dashed and dotted blue respectively. The contributions of $-2(2\pi) (c_{s (1)}^2+c_{s (2)}^2) \frac{k^2}{k_{\rm NL}^2} P_{11}$, $(2\pi) c_{s (1)}^2 P_{\text{1-loop}}^{(c_{\rm s}, p)}$ and $(2\pi)^2c_{s(1)}^4 \frac{k^4}{k_{\rm NL}^4} P_{11}$ have been plotted in red, green and orange respectively. Notice that many of the terms are of the same order of magnitude. Right: We plot the sums of terms appearing in Eq. (\ref{['equ:peft2loop']}) that appear together at each loop order: tree-level ($P_{11}$), one-loop ($P_{\text{1-loop}}-{2\,(2 \pi)} c_{s (1)}^2 \frac{k^2}{k_{\rm NL}^2} P_{11}$) and two-loop (everything else) in solid blue, dashed red and dotted green respectively. Notice now that each group of terms is smaller than the previous group, as required for a consistent perturbative expansion.