Perturbation theory, effective field theory, and oscillations in the power spectrum

TL;DR

The paper investigates how nonlinear matter power spectra can be modeled by perturbation theories (LPT and SPT) augmented with effective-field-theory (EFT) corrections. Using a 1-D exactly tractable case, it shows that 1LPT plus a transfer-function EFT expansion achieves broad accuracy and that EFT has a larger convergence radius than SPT; in 3-D, EFT parameters acquire strong k-dependence and SPT with IR resummation generally yields more stable broadband predictions. The study demonstrates BAO damping can be described by IR-resummed LPT, while residual wiggles in primordial spectra persist only to higher wavenumbers, with a crossover region around $k\sim0.2\,h/{\rm Mpc}$ where EFT-based and halo-model descriptions overlap. The results support using a transfer-function-like EFT framework to quantify perturbative ignorance, relate higher-order correlators to the power spectrum, and connect perturbative approaches with halo-model terms for a more complete picture of nonlinear structure formation.

Abstract

We explore the relationship between the nonlinear matter power spectrum and the various Lagrangian and Standard Perturbation Theories (LPT and SPT). We first look at it in the context of one dimensional (1-d) dynamics, where 1LPT is exact at the perturbative level and one can exactly resum the SPT series into the 1LPT power spectrum. Shell crossings lead to non-perturbative effects, and the PT ignorance can be quantified in terms of their ratio, which is also the transfer function squared in the absence of stochasticity. At the order of PT we work, this parametrization is equivalent to the results of effective field theory (EFT), and can thus be expanded in terms of the same parameters. We find that its radius of convergence is larger than the SPT loop expansion. The same EFT parametrization applies to all SPT loop terms and, if stochasticity can be ignored, to all N-point correlators. In 3-d, the LPT structure is considerably more complicated, and we find that LPT models with parametrization motivated by the EFT exhibit running with $k$ and that SPT is generally a better choice. Since these transfer function expansions contain free parameters that change with cosmological model their usefulness for broadband power is unclear. For this reason we test the predictions of these models on baryonic acoustic oscillations (BAO) and other primordial oscillations, including string monodromy models, for which we ran a series of simulations with and without oscillations. Most models are successful in predicting oscillations beyond their corresponding PT versions, confirming the basic validity of the model.

Figures (11)

• Figure 1: Error of various models of 1-d power spectrum shown relative to the nonlinear simulations results. Show are PT results in black: 1LPT/Zeldovich (dashed line), 1-loop SPT (dot-dashed line), 2-loop SPT (double dot-dashed line), 5-loop SPT (long-dashed line) and linear theory result (dotted line). In addition we apply the transfer functions to these results giving us EFT+SPT and EFT+1LPT models in 1-d. Results for three different transfer functions are shown: going up to $\alpha_1$ (in blue), $\alpha_2$ (in red) and $\alpha_3$ (in orange) in expansion given by Eq. \ref{['alphak1d2']}. Thin grey horizontal dotted and dashed lines represent respectively 1% and 2% errors. Thin grey vertical solid lines represent maximal $k$ values up to where EFT+1LPT models acheve 1% errors. Results are shown at redshift $z=0$.
• Figure 2: Running of $\alpha(k)$ for severals different models. On the left panel we show the running of the LPT models (in green) related to Eq. \ref{['eq:ilpttf']}: 1LPT (solid line), 2LPT (dashed line), and 3LPT (dot-dashed line). We also show the CLPTs model in Eq. \ref{['eq:clptstf']} (purple solid line). On the same panel we show the running of the $\alpha$'s related to the hybrid models from Eq. \ref{['eq:Hymodels']}: Hy1 (blue solid line), Hy2 (orange solid line), Hy3 (blue dashed line) and Hy4 (orange dashed line). On the right panel we show one loop (red dashed line) and two loop (orange dashed line) results for SPT EFT models, and also the IR resummed verisins of the same lines (solid red and orange lines). One loop (blue solid line) and two loop (blue dashed line) Hy1 results also shown, as well as one loop results of LEFT 2015JCAP...09..014V.
• Figure 3: Scale dependence of the linear two point functions of displacement field, which contribute to the cumulant expansion, Eq. \ref{['eq:XYex']}. We have split the contributions into the wiggle (left panel) and no-wiggle part (right panel). All results are shown at redshift $z=0.0$. We see that the wiggle part has a most of the support at scale $\sim$100 Mpc/$h$.
• Figure 4: BAO wiggles, i.e. ratio of the wiggle and non-wiggle power spectrum, is shown for four different models: $\Lambda$cdm (top left panel), Monodromy model (see 2010JCAP...06..009F2014arXiv1406.0548M) (top right panel) and two other models labeled $V_3$ and $V_4$ with additional wiggles relative to $\Lambda$cdm (bottom panels). Linear theory results (blue lines) evolve due to nonlinearities and yield results given by N-body simulations (black points). Wiggle damping for all these models is well described by the 1LPT (Zel'dovich) model (green dashed line). Note that all the initial wiggles are highly dampened at lower scales, $k \lesssim 0.5 h/$Mpc. All results are shown at redshift $z=0.0$.
• Figure 5: Residual wiggles, relative to the 1LPT (Zel'dovich). In top two lines iLPT models, as well as CLPTs model (see 2015PhRvD..91b3508V) are shown using definitions in Eq. \ref{['eq:ilpttf']}. In panels in lines three and four we show residuals of hybrid models defined in Eq. \ref{['eq:Hymodels']}, as well as LEFT theory developed recently in 2015JCAP...09..014V. In a bottom line we show SPT-EFT one loop and two loop models (IR resummation included), given by Eq. \ref{['eq:eft1loop']} and \ref{['eq:eft2loop']} respectively. All results are shown at redshift $z=0.0$.