[Paper Review] Integrated perturbation theory for cosmological tensor fields. II. Loop corrections

Takahiko Matsubara|arXiv (Cornell University)|Oct 20, 2022

Galaxies: Formation, Evolution, Phenomena4 citations

TL;DR

This paper extends integrated perturbation theory to cosmological tensor fields, enabling efficient computation of loop corrections for nonlinear power spectra and correlation functions in real and redshift space. By leveraging spherical harmonics and Hankel transforms via FFTLog, it reduces multidimensional integrals to fast one-dimensional transforms, significantly accelerating numerical evaluation for tensor-biased observables such as galaxy shapes and spins.

ABSTRACT

In the previous paper [arXiv:2210.10435], the nonlinear perturbation theory of cosmological density field is generalized to include the tensor-valued bias of astronomical objects, such as spins and shapes of galaxies and any other tensors of arbitrary ranks which are associated with objects that we can observe. We apply this newly developed method to explicitly calculate nonlinear power spectra and correlation functions both in real space and in redshift space. Multi-dimensional integrals that appear in loop corrections are reduced to combinations of one-dimensional Hankel transforms, thanks to the spherical basis of the formalism, and the final expressions are numerically evaluated in a very short time using an algorithm of the fast Fourier transforms such as extsc{FFTLog}. As an illustrative example, numerical evaluations of loop corrections of the power spectrum and correlation function of the rank-2 tensor field are demonstrated with a simple model of tensor bias.

Motivation & Objective

• To generalize nonlinear perturbation theory to include tensor-valued bias in cosmological observables such as galaxy shapes and spins.
• To develop a formalism for computing loop corrections to power spectra and correlation functions of tensor fields in both real and redshift space.
• To enable fast numerical evaluation of these corrections using one-dimensional Hankel transforms via FFTLog.
• To demonstrate the method with a simple model of rank-2 tensor bias, showing its feasibility for precision cosmology.

Proposed method

• The formalism employs a spherical basis expansion of the n-point correlation functions and propagators, enabling decomposition of multidimensional integrals into products of angular and radial parts.
• The radial integrals are expressed as Hankel transforms of the linear power spectrum and bias propagators, which are efficiently computed using FFTLog.
• The method applies to both real-space and redshift-space configurations by incorporating line-of-sight dependence into the propagators via angular momentum coupling.
• For semi-local bias models, the propagators allow separable dependence on wavevectors, making the Hankel transform approach viable at one-loop and two-loop orders.
• The formalism uses Wigner 3j and 6j symbols to couple angular momenta in the spherical harmonic expansions, ensuring rotational invariance.
• The final expressions for the power spectrum are written in terms of coupled tensorial and radial components, enabling direct numerical evaluation.

Experimental results

Research questions

• RQ1 How can nonlinear perturbation theory be extended to include tensor-valued bias in cosmological observables such as galaxy shapes and spins?
• RQ2 What is the structure of loop corrections to the power spectrum and correlation function for tensor fields in real and redshift space?
• RQ3 Can multidimensional integrals arising in loop corrections be reduced to computationally tractable one-dimensional Hankel transforms?
• RQ4 How does the inclusion of tensor bias affect the nonlinear evolution of large-scale structure?
• RQ5 What is the numerical efficiency of the proposed method compared to standard approaches?

Key findings

• The method reduces n-loop corrections for tensor fields to a series of one-dimensional Hankel transforms, enabling fast numerical evaluation using FFTLog.
• The formalism is applicable to both real and redshift space by incorporating line-of-sight dependence into the bias propagators via angular momentum coupling.
• For semi-local bias models, the propagators allow separable dependence on wavevectors, making the Hankel transform approach feasible at one-loop and two-loop orders.
• The derived expressions for the power spectrum in redshift space reduce to the real-space case when the line-of-sight angular momentum is set to zero.
• Numerical demonstrations confirm the method's efficiency and accuracy for a simple rank-2 tensor bias model, with computation times significantly reduced compared to brute-force integration.
• The approach preserves rotational invariance and handles arbitrary tensor ranks through systematic use of Wigner symbols and spherical harmonic expansions.