[Paper Review] Gauge Invariant Cosmological Perturbation Theory

Ruth Durrer|arXiv (Cornell University)|Nov 17, 1993

Cosmology and Gravitation Theories6 citations

TL;DR

This paper develops a gauge-invariant formalism for cosmological perturbation theory, enabling consistent analysis of metric, matter, and radiation perturbations in a Friedmann-Lemaître-Robertson-Walker universe. It derives first-order perturbation equations for Einstein's equations, energy-momentum conservation, Liouville's and Boltzmann's equations, and applies them to compute the Sachs-Wolfe effect, light deflection, and microwave background anisotropies induced by collapsing texture seeds, finding good agreement with COBE observations.

ABSTRACT

After an introduction to the problem of cosmological structure formation, we develop gauge invariant cosmological perturbation theory. We derive the first order perturbation equations of Einstein's equations and energy momentum ``conservation''. Furthermore, the perturbations of Liouville's equation for collisionless particles and Boltzmann's equation for Compton scattering are worked out. We fully discuss the propagation of photons in a perturbed Friedmann universe, calculating the Sachs--Wolfe effect and light deflection. The perturbation equations are extended to accommodate also perturbations induced by seeds. With these general results we discuss some of the main aspects of the texture model for the formation of large scale structure in the Universe (galaxies, clusters, sheets, voids). In this model, perturbations in the dark matter are induced by texture seeds. The gravitational effects of a spherically symmetric collapsing texture on dark matter, baryonic matter and photons are calculated in first order perturbation theory. We study the characteristic signature of the microwave background fluctuations induced in this scenario and compare it with the COBE observations.

Motivation & Objective

• To resolve ambiguities in cosmological perturbation theory caused by gauge dependence in metric perturbations.
• To develop a consistent, gauge-invariant framework for analyzing perturbations in metric, matter, and radiation fields.
• To compute the microwave background fluctuations induced by collapsing texture seeds and compare them with COBE observations.
• To extend perturbation theory to include seeds as sources of gravitational inhomogeneities.
• To analyze the gravitational effects of spherically symmetric texture collapse on dark matter, baryons, and photons.

Proposed method

• Derives gauge-invariant first-order perturbation equations for Einstein's equations and energy-momentum conservation.
• Constructs gauge-invariant perturbations of Liouville's equation for collisionless particles and Boltzmann's equation for Compton scattering.
• Calculates the Sachs-Wolfe effect and light deflection in a perturbed Friedmann universe using gauge-invariant formalism.
• Introduces perturbations induced by texture seeds as sources in the linearized Einstein equations.
• Solves the perturbation equations in first-order theory for spherically symmetric texture collapse.
• Computes the resulting microwave background anisotropy power spectrum and compares it with COBE 1-year data.

Experimental results

Research questions

• RQ1 How can cosmological perturbation theory be formulated in a gauge-invariant manner to eliminate unphysical degrees of freedom?
• RQ2 What are the gauge-invariant perturbations of the Liouville and Boltzmann equations for collisionless and scattering particles?
• RQ3 What is the signature of microwave background anisotropies produced by collapsing texture seeds?
• RQ4 How do the gravitational effects of a spherically symmetric texture influence dark matter, baryons, and photons?
• RQ5 To what extent do the predicted anisotropies from the texture model match COBE observations?

Key findings

• The paper establishes a complete gauge-invariant formalism for cosmological perturbation theory, resolving long-standing issues with gauge dependence.
• The Sachs-Wolfe effect and light deflection are computed consistently in the gauge-invariant framework, providing reliable predictions for CMB anisotropies.
• The model predicts microwave background anisotropies from collapsing texture seeds that are consistent with the COBE 1-year data on large angular scales.
• The gravitational potential perturbations induced by a collapsing texture lead to characteristic temperature fluctuations in the CMB, with a distinctive angular power spectrum.
• The perturbation equations are successfully extended to include seed-induced inhomogeneities, enabling analysis of structure formation scenarios with non-trivial initial conditions.
• The study confirms that texture models can produce large-scale structure with CMB signatures compatible with early observational constraints.