Set Of Gauge-invariant Variables Research Articles

Relating dust reference models to conventional systems in manifestly gauge invariant perturbation theory

Models with dust reference fields in relational formalism have proved useful in understanding the construction of gauge-invariant perturbation theory to arbitrary orders in the canonical framework. These reference fields modify the dynamical equations for perturbation equations. However, important questions remain open on the relation with conventional perturbation theories of inflaton coupled to gravity and of multi-fluid systems, and on understanding modifications in terms of physical degrees of freedom. These gaps are filled in this manuscript for Brown-Kucha\v{r} and Gaussian dust models, both of which involve three scalar physical degrees of freedom. We establish a relationship of these models with conventional inflationary and multi-fluid system of inflation and ordinary dust by introducing a set of gauge invariant variables on the reduced phase space of the dust reference models. We find the modifications due to dust clocks to Bardeen equation in the longitudinal gauge and Mukhanov-Sasaki equation in the spatially-flat gauge, in terms of physical degrees of freedom. This results in a closed system of equations for all the degrees of freedom needed to explore the evolution of the scalar perturbations. Our analysis shows for the first time that even for two-fluid systems, there is a natural choice of the set of gauge invariant variables for each chosen gauge which not only offers a direct physical interpretation but also results in simplifications to the dynamical equations.

Hamiltonian approach to second order gauge invariant cosmological perturbations

In view of growing interest in tensor modes and their possible detection, we clarify the definition of tensor modes up to 2nd order in perturbation theory within the Hamiltonian formalism. Like in gauge theory, in cosmology the Hamiltonian is a suitable and consistent approach to reduce the gauge degrees of freedom. In this paper we employ the Faddeev-Jackiw method of Hamiltonian reduction. An appropriate set of gauge invariant variables that describe the dynamical degrees of freedom may be obtained by suitable canonical transformations in the phase space. We derive a set of gauge invariant variables up to 2nd order in perturbation expansion and for the first time we reduce the 3rd order action without adding gauge fixing terms. In particular, we are able to show the relation between the uniform-$\phi$ and Newtonian slicings, and study the difference in the definition of tensor modes in these two slicings.

Exact multi-matrix correlators

We argue that restricted Schur polynomials provide a useful parameterization of the complete set of gauge invariant variables of multi-matrix models. The two point functions of restricted Schur polynomials are evaluated exactly in the free field theory limit. They have diagonal two point functions.

Theory of cosmological perturbations in an anisotropic universe

This paper describes the theory of cosmological perturbations around a homogeneous andanisotropic universe of the Bianchi I type. Starting from a general parametrization of theperturbed spacetime à la Bardeen, a complete set of gauge invariant variables isconstructed. Three physical degrees of freedom are identified and it is shownthat, in the case where matter is described by a scalar field, they generalize theMukhanov–Sasaki variables. In order to show that they are canonical variables, the actionfor the cosmological perturbations at second order is derived. Two major physicalimprints of the primordial anisotropy are identified: (1) a scalar–tensor ‘seesaw’mechanism arising from the fact that scalar, vector and tensor modes do notdecouple and (2) an explicit dependence of the statistical properties of the densityperturbations and gravity waves on the wavevector instead of its norm. Thisanalysis extends, but also sheds some light on, the quantization procedure that wasdeveloped under the assumption of a Friedmann–Lemaître background spacetime,and allows one to investigate the robustness of the predictions of the standardinflationary scenario with respect to the hypothesis on the symmetries of thebackground spacetime. These effects of a primordial anisotropy may be related to someanomalies of the cosmic microwave background anisotropies on large angularscales.

Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. II. Basic gauge-invariant variables with applications to de sitter space-time

In a companion paper, a systematic treatment of linearized perturbations and a new geometric definition of gauge-invariant variables, based on the theory of vector bundles and applicable to the case of an arbitrary system of covariant field equations, were carefully presented. One of the purposes of the present paper is to specify a necessary and sufficient condition that a given, finite set of gaugeinvariant variables, denoted collectively by ω and referred to as the complete set of basic variables, can be used to extract the equivalence classes of perturbations from ω in a unique way. The above set is complete because it has the following property: a knowledge of ω is all one needs in the sense that ifx represents an arbitrary point of the “space-time” manifoldX andG denotes any gauge-invariant tensor field onX, then the value ofG atx∈X is uniquely specified by giving the germs of basic gauge-invariant variables atx∈X. Arguments are proposed that ω also has a stronger property which is more immediately useful: anyG is obtainable directly from the basic variables through purely algebraic and differential operations. These results are of practical interest, and one concrete setting where one is led to the explicit definition of ω occurs when considering the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time and obeying the linearized empty-space Einstein equations with nonnegative cosmological constant Λ; the case Λ=0 corresponds to linear perturbation theory in Minkowski space-time.

Generalized lines of force as the gauge invariant degrees of freedom for general relativity and Yang-Mills theory.

We present a set of equations that determine (with certain qualifications) what is (i) a complete canonical set of gauge-invariant variables for Yang-Mills (YM) theory that generalize Faraday's electric lines of force and (ii) a complete canonical set of the diffeomorphism and gauge invariant observables for general relativity (GR). Both results come from solving the Hamilton-Jacobi version of the constraint equations: the Gauss law constraints of YM theory and the Gauss law and diffeomorphism constraint of GR in the Ashtekar formalism. These results are local in physical space as well as in phase space.