[Paper Review] Integrated perturbation theory for cosmological tensor fields. I. Basic formulation

Takahiko Matsubara|arXiv (Cornell University)|Oct 19, 2022

Stellar, planetary, and galactic studies4 citations

TL;DR

This paper introduces an integrated perturbation theory (iPT) framework for cosmological tensor fields—such as galaxy shapes and spins—by generalizing scalar-bias iPT to tensor-valued bias. It employs irreducible tensor decomposition under rotational symmetry to derive nonlinear correlation statistics, enabling analytical modeling of large-scale structure correlations involving intrinsic alignments and other tensor observables in future large-scale surveys.

ABSTRACT

In order to extract maximal information about cosmology from the large-scale structure of the Universe, one needs to use every bit of signal that can be observed. Beyond the spatial distributions of astronomical objects, the spatial correlations of tensor fields, such as galaxy spins and shapes, are ones of promising sources that can be accessed in the era of large surveys in the near future. The perturbation theory is a powerful tool to analytically describe the behaviors and evolutions of correlation statistics on large scales for a given cosmology. In this paper, we formulate a nonlinear perturbation theory of tensor fields in general, based on the formulation of integrated perturbation theory for the scalar-valued bias, generalizing it to include the tensor-valued bias. To take advantage of rotational symmetry, the formalism is constructed on the basis of the irreducible decomposition of tensors, identifying physical variables which are invariant under the rotation of the coordinates system.

Motivation & Objective

• To extend the integrated perturbation theory (iPT) from scalar-valued bias to tensor-valued bias for cosmological fields.
• To develop a formalism that captures nonlinear statistical correlations of tensor observables like galaxy shapes and spins in large-scale structure.
• To ensure rotational invariance by using irreducible tensor decomposition to identify physical, coordinate-invariant variables.
• To enable analytical prediction of correlation functions involving tensor fields in the quasi-linear regime, crucial for upcoming large imaging surveys.
• To provide a systematic framework that separates large-scale dynamics from small-scale nonlinear physics via renormalized bias functions for tensor fields.

Proposed method

• Generalizes iPT to tensor fields by extending the Lagrangian perturbation theory framework to include tensor-valued bias functions.
• Uses irreducible decomposition of symmetric traceless tensors to identify rotationally invariant physical variables.
• Constructs correlation functions of tensor fields using Wigner 3j, 6j, and 9j symbols to handle angular momentum coupling in multi-field correlation statistics.
• Applies orthogonal decomposition of bias in Lagrangian space to isolate nonlinear structure formation effects into renormalized bias functions.
• Derives evolution equations for correlation statistics by combining perturbative dynamics with tensor bias functions, valid in the quasi-linear regime.
• Employs group-theoretic techniques and angular momentum recoupling to ensure consistency and invariance under coordinate rotations.

Experimental results

Research questions

• RQ1 How can the integrated perturbation theory (iPT) be generalized to describe nonlinear correlations of cosmological tensor fields such as galaxy shapes and spins?
• RQ2 What is the appropriate mathematical framework for handling tensor-valued bias in Lagrangian space while preserving rotational invariance?
• RQ3 How do irreducible tensor components of galaxy shapes and intrinsic alignments couple to the underlying matter density field in the nonlinear regime?
• RQ4 What are the key symmetry properties and invariants that must be preserved in the correlation functions of tensor fields in cosmology?
• RQ5 How can renormalized bias functions for tensor fields be systematically defined and computed within the iPT formalism?

Key findings

• The paper successfully generalizes iPT to tensor fields by formulating a consistent framework for tensor-valued bias in Lagrangian space.
• Irreducible tensor decomposition ensures that all physical variables are invariant under coordinate rotations, preserving gauge invariance.
• Correlation functions of tensor fields are expressed using Wigner 3j, 6j, and 9j symbols, enabling systematic angular momentum coupling in multi-point statistics.
• The formalism separates large-scale dynamical evolution from small-scale nonlinear physics via renormalized bias functions, analogous to scalar iPT.
• The framework allows analytical prediction of correlation statistics involving intrinsic alignments and other tensor observables in the quasi-linear regime.
• The method provides a foundation for modeling cosmological information from galaxy shapes and spins in upcoming large-scale imaging surveys such as LSST and Euclid.