Cosmological perturbation theory at three-loop order

TL;DR

The paper investigates the cosmological matter power spectrum by pushing standard perturbation theory to three-loop order and finds that the loop expansion behaves like an asymptotic series, failing to converge even in the linear regime at late times. To address this, the authors introduce a Padé resummation scheme based on the small-$k$ asymptotics, encapsulated in a universal kernel K(x) with x = σ_l^2(q,z), and match coefficients C_L from the computed loops. They develop and test Padé approximants K^{pade}_{nm} to replace the divergent small-$k$ series, obtaining a well-behaved integrand and significantly improved convergence, which extends into the BAO regime and yields good agreement with N-body data across several redshifts. The approach is shown to be robust to UV cutoffs and does not introduce free parameters, highlighting a practical path to reconcile perturbation theory with non-linear growth on mildly non-linear scales. Overall, the work provides a convergent, resummed perturbative framework for the power spectrum that enhances accuracy in the BAO range and offers a foundation for higher-order extensions.

Abstract

We analyze the dark matter power spectrum at three-loop order in standard perturbation theory of large scale structure. We observe that at late times the loop expansion does not converge even for large scales (small momenta) well within the linear regime, but exhibits properties compatible with an asymptotic series. We propose a technique to restore the convergence in the limit of small momentum, and use it to obtain a perturbative expansion with improved convergence for momenta in the range where baryonic acoustic oscillations are present. Our results are compared with data from N-body simulations at different redshifts, and we find good agreement within this range.

Figures (8)

• Figure 1: One, two and three-loop contributions to the equal-time power spectrum obtained from a numerical Monte Carlo integration within standard perturbation theory at $z=0$. The linear power spectrum is obtained from the initial power spectrum from CAMB Lewis:1999bs using the $\Lambda$CDM model with WMAP5 parameters. For the three-loop order, the error bars show an estimate for the numerical error obtained by multiplying the error output of the CUBA routine Suave by a factor of two. The relative error is $\leq 0.002$ for $k\leq 0.55\, h/$Mpc. The black diamonds and grey crosses correspond to two different parametrizations of the absolute loop momenta (see App. \ref{['app:numerics']}).
• Figure 2: Comparison at redshifts $z = \{0, 0.375, 0.833, 1.75\}$ of SPT up to one loop (black dashed lines), two loops (black dot-dashed) and three loops (black diamonds) with N-body results of the Horizon Run 2 Kim:2011ab (red dots, see App. \ref{['app:hr3']}). The black line corresponds to the linear result. We also show the results of Padé resummation (same styles as for SPT but in blue, see Sec. \ref{['sec:Pade']}); at $z=0$ the blue and black dashed line lie on top of each other.
• Figure 3: Same as Fig. \ref{['fig:nbody_lowz']} for redshifts $z = \{2.67, 4.5 \}$
• Figure 4: Ratio $P_{L-loop}(k,z=0)/P_{lin}(k,z=0)/k^2$ for the one- two- and three-loop contributions (line styles as in Fig. \ref{['fig:power3L']}).
• Figure 5: Integrand kernel $k\, P_{lin}(k) K_L(\sigma_l^2(k,z))$ for the power spectrum as obtained in SPT at one-loop (black dashed), two loops (black dot-dashed), three loops (black dotted). The solid lines are the integrand kernels obtained after Padé resummation, $K^{pade}_{01}$ (green), $K^{pade}_{02}$ (blue) and $K^{pade}_{11}$ (magenta). The factor $k\, P_{lin}(k)$ is chosen such that the area under the curves represents the integral when using a logarithmic integration measure.