An Application of Wiener Hermite Expansion to Non-linear Evolution of Dark Matter

Naonori S. Sugiyama; Toshifumi Futamase

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An Application of Wiener Hermite Expansion to Non-linear Evolution of Dark Matter

Cosmology and Nongalactic Astrophysics · Submitted 5 Oct 2012

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The authors' main claim The proposed power spectrum agrees with the result of $N$-body simulation with the accuracy better than 1% or 2% in a range of the BAO scales, where the wave number is about $k= 0.2 \sim 0.4$ $h{\rm Mpc^{-1}}$ at $z=0.5-3.0$.

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arXiv ID1210.1663 (version 2)

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CitationNaonori S. Sugiyama, Toshifumi Futamase. "An Application of Wiener Hermite Expansion to Non-linear Evolution of Dark Matter." arXiv:1210.1663 (2012). https://pith.science/paper/IQEX2ERT

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We apply the Wiener Hermite (WH) expansion to the non-linear evolution of Large-Scale Structure, and obtain an approximate expression for the matter power spectrum in full order of the expansion. This method allows us to expand any random function in terms of an orthonormal bases in space of random functions in such a way that the first order of the expansion expresses the Gaussian distribution, and others are the deviation from the Gaussianity. It is proved that the Wiener Hermite expansion is mathematically equivalent with the $\Gamma$-expansion approach in the Renormalized Perturbation Theory (RPT). While an exponential behavior in the high-$k$ limit has been proved for the mass density and velocity fluctuations of Dark Matter in the RPT, we reprove the behavior in the context of the Wiener Hermite expansion using the result of Standard Perturbation Theory (SPT). We propose a new approximate expression for the matter power spectrum which interpolates the low-$k$ expression corresponding to 1-loop level in SPT and the high-$k$ expression obtained by taking a high-$k$ limit of the WH expansion. The validity of our prescription is specifically verified by comparing with the 2-loop solutions of the SPT. The proposed power spectrum agrees with the result of $N$-body simulation with the accuracy better than 1% or 2% in a range of the BAO scales, where the wave number is about $k= 0.2 \sim 0.4$ $h{\rm Mpc^{-1}}$ at $z=0.5-3.0$. This accuracy is comparable to or slightly less than the ones in the Closure Theory whose the fractional difference from N body result is within 1%. A merit of our method is that the computational time is very short because only single and double integrals are involved in our solution.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply the Wiener Hermite (WH) expansion to the non-linear evolution of Large-Scale Structure, and obtain an approximate expression for the matter power spectrum in full order of the expansion.	contribution 0.97	0.96	0.24	asserted 1.00
s1	This method allows us to expand any random function in terms of an orthonormal bases in space of random functions in such a way that the first order of the expansion expresses the Gaussian distribution, and others are the deviation from the Gaussianity.	method 0.59	0.18	0.04	asserted 0.98
s2	It is proved that the Wiener Hermite expansion is mathematically equivalent with the $\Gamma$-expansion approach in the Renormalized Perturbation Theory (RPT).	result 0.75	0.69	0.09	asserted 1.00
s3	While an exponential behavior in the high-$k$ limit has been proved for the mass density and velocity fluctuations of Dark Matter in the RPT, we reprove the behavior in the context of the Wiener Hermite expansion using the result of Standard Perturbation Theory (SPT).	contribution 0.53	0.82	0.07	asserted 1.00
s4	We propose a new approximate expression for the matter power spectrum which interpolates the low-$k$ expression corresponding to 1-loop level in SPT and the high-$k$ expression obtained by taking a high-$k$ limit of the WH expansion.	contribution 0.90	0.94	0.37	asserted 0.92
s5	The validity of our prescription is specifically verified by comparing with the 2-loop solutions of the SPT.	method 0.73	0.92	0.09	asserted 1.00
s6	The proposed power spectrum agrees with the result of $N$-body simulation with the accuracy better than 1% or 2% in a range of the BAO scales, where the wave number is about $k= 0.2 \sim 0.4$ $h{\rm Mpc^{-1}}$ at $z=0.5-3.0$.	result 1.00	0.93	0.47	asserted 1.00
s7	This accuracy is comparable to or slightly less than the ones in the Closure Theory whose the fractional difference from N body result is within 1%.	result 0.57	0.48	0.12	asserted 0.95
s8	A merit of our method is that the computational time is very short because only single and double integrals are involved in our solution.	interpretation 0.54	0.84	0.06	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.54 · evidence from proof or derivation 0.80.

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Original abstract

We apply the Wiener Hermite (WH) expansion to the non-linear evolution of Large-Scale Structure, and obtain an approximate expression for the matter power spectrum in full order of the expansion. This method allows us to expand any random function in terms of an orthonormal bases in space of random functions in such a way that the first order of the expansion expresses the Gaussian distribution, and others are the deviation from the Gaussianity. It is proved that the Wiener Hermite expansion is mathematically equivalent with the $\Gamma$-expansion approach in the Renormalized Perturbation Theory (RPT). While an exponential behavior in the high-$k$ limit has been proved for the mass density and velocity fluctuations of Dark Matter in the RPT, we reprove the behavior in the context of the Wiener Hermite expansion using the result of Standard Perturbation Theory (SPT). We propose a new approximate expression for the matter power spectrum which interpolates the low-$k$ expression corresponding to 1-loop level in SPT and the high-$k$ expression obtained by taking a high-$k$ limit of the WH expansion. The validity of our prescription is specifically verified by comparing with the 2-loop solutions of the SPT. The proposed power spectrum agrees with the result of $N$-body simulation with the accuracy better than 1% or 2% in a range of the BAO scales, where the wave number is about $k= 0.2 \sim 0.4$ $h{\rm Mpc^{-1}}$ at $z=0.5-3.0$. This accuracy is comparable to or slightly less than the ones in the Closure Theory whose the fractional difference from N body result is within 1%. A merit of our method is that the computational time is very short because only single and double integrals are involved in our solution.

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7 arXiv papers cite this in 10 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Many different approaches have been developed over the past decade for understanding this evolution: standard perturbation theory (SPT; [ref] ), Lagrangian resummation theory (LRT; [ref] ), renormalized perturbation theory (RPT; [ref] ), closure theory [ref] , multi-point propagator method (the $Γ$-expansion method; [ref] ), regularized multi-point propagator method (Reg PT; [ref] ), the Wiener Hermite expansion method [ref] , as well as other techniques [ref] .

How does non-linear dynamics affect the baryon acoustic oscillation? · Introduction · 0.99
Recently there has been a lot of work toward improving perturbative predictions for dark matter clustering [ref] .

What the "simple renormalization group" approach to dark matter clustering really was · Introduction · 1.00
Also, a number of alternative methods have been suggested (e.g. [ref] ) that use different levels of approximation.

Lagrangian perturbation theory at one loop order: successes, failures, and improvements · Introduction · 0.96

What it cites · 23 sentences without a confident label

Uses data or values from 1 paper

It is plotted for the cosmological models with the Wilkinson Microwave Anisotropy Probe ($WMAP$) five year [ref] . cosmological parameters: $Ω_m = 0.279$, $Ω_Λ = 0.721$, $Ω_b = 0.046$, $h = 0.701$, $n_s = 0.96$, $σ_8 = 0.817$.

Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation · Comparison with Other Analytic Predictions and $N$-body Simulations · 0.93

Compares with 2 papers

[The power spectra of SPT and LRT [ref] are given by [equation] [equation] where we denote as $P_13 = 2 δ_L δ_3^(1)$.

Resumming Cosmological Perturbations via the Lagrangian Picture: One-loop Results in Real Space and in Redshift Space · Comparison with Other Analytical Predictions: 1-loop Level · 0.94
Finally, we compare with the closure theory (second Born) in [ref] , which is one of the best predictions at the moment.

Non-linear Evolution of Baryon Acoustic Oscillations from Improved Perturbation Theory in Real and Redshift Spaces · Comparison with closure theory · 1.00

Cites as background 12 papers

One of the main approach is the “Renormalized Perturbation Theory” (RPT; [ref] ), where the basic equations for fluid describing matter perturbation are rewritten in a convenient compact form in order to use a diagrammatic technique developed in quantum field theory [ref] .

Nonlinear Evolution of Baryon Acoustic Oscillations · Introduction · 0.91
On the other hand, there is also an approach to the large-scale structure in the framework of the Lagrangian picture, called “Lagrangian Resummation Theory” (LRT; [ref] ).

Nonlinear Perturbation Theory Integrated with Nonlocal Bias, Redshift-space Distortions, and Primordial Non-Gaussianity · Introduction · 0.94
On the other hand, there is also an approach to the large-scale structure in the framework of the Lagrangian picture, called “Lagrangian Resummation Theory” (LRT; [ref] ).

Next-to-leading resummation of cosmological perturbations via the Lagrangian picture: 2-loop correction in real and redshift spaces · Introduction · 0.94

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