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Register a new userFast Large Scale Structure Perturbation Theory using 1D FFTs
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Added by
Marcel M. Schmittfull
Publication date
2016
fields
Physics
and research's language is
English
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The usual fluid equations describing the large-scale evolution of mass density in the universe can be written as local in the density, velocity divergence, and velocity potential fields. As a result, the perturbative expansion in small density fluctuations, usually written in terms of convolutions in Fourier space, can be written as a series of products of these fields evaluated at the same location in configuration space. Based on this, we establish a new method to numerically evaluate the 1-loop power spectrum (i.e., Fourier transform of the 2-point correlation function) with one-dimensional Fast Fourier Transforms. This is exact and a few orders of magnitude faster than previously used numerical approaches. Numerical results of the new method are in excellent agreement with the standard quadrature integration method. This fast model evaluation can in principle be extended to higher loop order where existing codes become painfully slow. Our approach follows by writing higher order corrections to the 2-point correlation function as, e.g., the correlation between two second-order fields or the correlation between a linear and a third-order field. These are then decomposed into products of correlations of linear fields and derivatives of linear fields. The method can also be viewed as evaluating three-dimensional Fourier space convolutions using products in configuration space, which may also be useful in other contexts where similar integrals appear.
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We present a simple numerical scheme for perturbation theory (PT) calculations of large-scale structure. Solving the evolution equations for perturbations numerically, we construct the PT kernels as building blocks of statistical calculations, from which the power spectrum and/or correlation function can be systematically computed. The scheme is especially applicable to the generalized structure formation including modified gravity, in which the analytic construction of PT kernels is intractable. As an illustration, we show several examples for power spectrum calculations in $f(R)$ gravity and $Lambda$CDM models.
Perturbation theory (PT) calculation of large-scale structure has been used to interpret the observed non-linear statistics of large-scale structure at the quasi-linear regime. In particular, the so-called standard perturbation theory (SPT) provides a basis for the analytical computation of the higher-order quantities of large-scale structure. Here, we present a novel, grid-based algorithm for the SPT calculation, hence named GridSPT, to generate the higher-order density and velocity fields from a given linear power spectrum. Taking advantage of the Fast Fourier Transform, the GridSPT quickly generates the nonlinear density fields at each order, from which we calculate the statistical quantities such as non-linear power spectrum and bispectrum. Comparing the density fields (to fifth order) from GridSPT with those from the full N-body simulations in the configuration space, we find that GridSPT accurately reproduces the N-body result on large scales. The agreement worsens with smaller smoothing radius, particularly for the under-dense regions where we find that 2LPT (second-order Lagrangian perturbation theory) algorithm performs well.
The linear matter power spectrum is an essential ingredient in all theoretical models for interpreting large-scale-structure observables. Although Boltzmann codes such as CLASS or CAMB are very efficient at computing the linear spectrum, the analysis of data usually requires $10^4$-$10^6$ evaluations, which means this task can be the most computationally expensive aspect of data analysis. Here, we address this problem by building a neural network emulator that provides the linear theory matter power spectrum in about one millisecond with 0.3% accuracy over $10^{-3} le k [h,{rm Mpc}^{-1}] < 30$. We train this emulator with more than 150,000 measurements, spanning a broad cosmological parameter space that includes massive neutrinos and dynamical dark energy. We show that the parameter range and accuracy of our emulator is enough to get unbiased cosmological constraints in the analysis of a Euclid-like weak lensing survey. Complementing this emulator, we train 15 other emulators for the cross-spectra of various linear fields in Eulerian space, as predicted by 2nd-order Lagrangian Perturbation theory, which can be used to accelerate perturbative bias descriptions of galaxy clustering. Our emulators are especially designed to be used in combination with emulators for the nonlinear matter power spectrum and for baryonic effects, all of which are publicly available at http://www.dipc.org/bacco.
The form of the primordial power spectrum (PPS) of cosmological scalar (matter density) perturbations is not yet constrained satisfactorily in spite of the tremendous amount of information from the Cosmic Microwave Background (CMB) data. While a smooth power-law-like form of the PPS is consistent with the CMB data, some PPS with small non-smooth features at large scales can also fit the CMB temperature and polarization data with similar statistical evidence. Future CMB surveys cannot help distinguish all such models due to the cosmic variance at large angular scales. In this paper, we study how well we can differentiate be- tween such featured forms of the PPS not otherwise distinguishable using CMB data. We ran 15 N-body DESI-like simulations of these models to explore this approach. Showing that statistics such as the halo mass function and the two-point correlation function are not able to distinguish these models in a DESI-like survey, we advocate to avoid reducing the dimensionality of the problem by demonstrating that the use of a simple three-dimensional count-in-cell density field can be much more effective for the purpose of model distinction.
Standard cosmological perturbation theory (SPT) for the Large Scale Structure (LSS) of the Universe fails at small scales (UV) due to strong nonlinearities and to multistreaming effects. In Pietroni et al. 2011 a new framework was proposed in which the large scales (IR) are treated perturbatively while the information on the UV, mainly small scale velocity dispersion, is obtained by nonlinear methods like N-body simulations. Here we develop this approach, showing that it is possible to reproduce the fully nonlinear power spectrum (PS) by combining a simple (and fast) 1-loop computation for the IR scales and the measurement of a single, dominant, correlator from N-body simulations for the UV ones. We measure this correlator for a suite of seven different cosmologies, and we show that its inclusion in our perturbation scheme reproduces the fully non-linear PS with percent level accuracy, for wave numbers up to $ksim 0.4, h~{rm Mpc^{-1}}$ down to $z=0$. We then show that, once this correlator has been measured in a given cosmology, there is no need to run a new simulation for a different cosmology in the suite. Indeed, by rescaling this correlator by a proper function computable in SPT, the reconstruction procedure works also for the other cosmologies and for all redshifts, with comparable accuracy. Finally, we clarify the relation of this approach to the Effective Field Theory methods recently proposed in the LSS context.
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