Gauge-invariant Perturbation Theory Research Articles

Primordial magnetic seed field amplification by gravitational waves

Using second-order gauge-invariant perturbation theory, a self-consistent framework describing the non-linear coupling between gravitational waves and a large-scale homogeneous magnetic field is presented. It is shown how this coupling may be used to amplify seed magnetic fields to strengths needed to support the galactic dynamo. In situations where the gravitational wave background is described by an `almost' Friedmann-Lema{\^i}tre-Robertson-Walker (FLRW) cosmology we find that the magnitude of the original magnetic field is amplified by an amount proportional to the magnitude of the gravitational wave induced shear anisotropy and the square of the field's initial co-moving scale. We apply this mechanism to the case where the seed field and gravitational wave background are produced during inflation and find that the magnitude of the gravitational boost depends significantly on the manner in which the estimate of the shear anisotropy at the end of inflation is calculated. Assuming a seed field of $10^{-34}$ $\rm{G}$ spanning a comoving scale of about $10 \rm{kpc}$ today, the shear anisotropy at the end of inflation must be at least as large as $10^{-40}$ in order to obtain a generated magnetic field of the same order of magnitude as the original seed. Moreover, contrasting the weak field approximation to our gauge-invariant approach, we find that while both methods agree in the limit of high conductivity, their corresponding solutions are otherwise only compatible in the limit of infinitely long-wavelength gravitational waves.

Gravitational waves from oscillating accretion tori: Comparison between different approaches

Quasiperiodic oscillations of high density thick accretion disks orbiting a Schwarzschild black hole have been recently addressed as interesting sources of gravitational waves. The aim of this paper is to compare the gravitational waveforms emitted from these sources when computed using (variations of) the standard quadrupole formula and gauge-invariant metric perturbation theory. To this goal we evolve representative disk models using an existing general relativistic hydrodynamics code which has been previously employed in investigations of such astrophysical systems. Two are the main results of this work: First, for stable and marginally stable disks, no excitation of the black hole quasinormal modes is found. Second, we provide a simple, relativistic modification of the Newtonian quadrupole formula which, in certain regimes, yields excellent agreement with the perturbative approach. This holds true as long as back-scattering of GWs is negligible. Otherwise, any functional form of the quadrupole formula yields systematic errors $\ensuremath{\sim}10%$.

The general relativistic magnetohydrodynamic dynamo equation

The magnetohydrodynamic dynamo equation is derived within general relativity, using the covariant 1 + 3 approach, for a plasma with finite electrical conductivity. This formalism allows for a clear division and interpretation of plasma and gravitational effects, and we are not restricted to a particular space‐time geometry. The results should be of interest in astrophysics and cosmology, and the formulation is well suited to gauge-invariant perturbation theory. Moreover, the dynamo equation is presented in some specific limits. In particular, we consider the interaction of gravitational waves with magnetic fields, and present results for the evolution of the linearly growing electromagnetic induction field, as well as the diffusive damping of these fields. Ke yw ords: gravitation ‐ gravitational waves ‐ MHD.

Publisher's Note: Cosmological density fluctuations and gravity waves: A covariant approach to gauge-invariant nonlinear perturbation theory [Phys. Rev. D70, 103524 (2004)

We present a new approach to gauge-invariant cosmological perturbations at second order, which is also covariant. We examine two cases in particular for a dust Friedman-Lemaitre-Robertson-Walker model of any curvature: we investigate gravity waves generated from clustering matter, that is, induced tensor modes from scalar modes; and we discuss the generation of density fluctuations induced by gravity waves -- scalar modes from tensor perturbations. We derive a linear system of evolution equations for second-order gauge-invariant variables which characterise fully the induced modes of interest, with a source formed from variables quadratic in first-order quantities; these we transform into fully-fledged second-order gauge-invariant variables. Both the invariantly defined variables and the key evolution equations are considerably simpler than similar gauge-invariant results derived by other methods. By finding analytical solutions, we demonstrate that non-linear effects can significantly amplify or dampen modes present in standard linearised cosmological perturbation theory, thereby providing an important source of potential error in, and refinement of, the standard model. Moreover, these effects can dominate at late times, and on super-Hubble scales.

Cosmological density fluctuations and gravity waves: A covariant approach to gauge-invariant nonlinear perturbation theory

We present a new approach to gauge-invariant cosmological perturbations at second-order, which is also covariant. We examine two cases, in particular, for a dust Friedman-Lema\^{\i}tre-Robertson-Walker model of any curvature: we investigate gravity waves generated from clustering matter, that is, induced tensor modes from scalar modes; and we discuss the generation of density fluctuations induced by gravity waves-scalar modes from tensor perturbations. We derive a linear system of evolution equations for second-order gauge-invariant variables which characterize fully the induced modes of interest, with a source formed from variables quadratic in first-order quantities; these we transform into fully-fledged second-order gauge-invariant variables. Both the invariantly defined variables and the key evolution equations are considerably simpler than similar gauge-invariant results derived by other methods. By finding analytical solutions, we demonstrate that nonlinear effects can significantly amplify or dampen modes present in standard linearized cosmological perturbation theory, thereby providing an important source of potential error in, and refinement of, the standard model. Moreover, these effects can dominate at late times, and on super-Hubble scales.

Second-Order Gauge Invariant Perturbation Theory: -- Perturbative Curvatures in the Two-Parameter Case --

Based on the gauge invariant variables proposed in our previous paper [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], some formulae of the perturbative curvatures of each order are derived. We follow the general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime to derive these formulae. These perturbative curvatures do have the same form as the definitions of gauge invariant variables for arbitrary perturbative fields which are previously proposed. As a result, we explicitly see that any perturbative Einstein equations are given in terms of gauge invarinat form. We briefly discuss physical situations to which this framework should be applied.

Nonlinear impact of perturbation theory on numerical relativity

I discuss the impact of gauge-invariant perturbation theory, as developed originally by Vincent Moncrief, on numerical simulations of Einstein's theory. Far from being replaced by numerical relativity, perturbative approaches remain essential for analysing, interpreting, extending and complementing fully nonlinear approaches. In the last decade, as computers have become ever more powerful tools for studying the full nonlinear equations, the power and application of perturbation theory has also grown. Its impact on numerical relativity is profound (decidedly nonlinear), and will surely continue to be for years to come.

On asymptotic perturbation theory for quantum mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory

Singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory. Under very general conditions, asymptotic estimations on the Rayleigh–Schrödinger expansions of the perturbed spectral projections are obtained. As a consequence almost invariant subspaces of exponential order are constructed. The results cover practically all singular perturbations considered in nonrelativistic quantum mechanics. In the magnetic field case, under the condition that the magnetic field does not increase at infinity, a gauge invariant perturbation theory leading to convergent series with field-dependent coefficients is developed.

Preheating and the Einstein field equations

We inaugurate a framework for studying preheating and parametric resonance after inflation without resorting to any approximations, either in gravitational perturbation theory or in the classical evolution of the field(s). We do this by numerically solving the Einstein field equations in the post-inflationary universe. In this paper we show how to compare our results to those of gauge invariant perturbation theory. We then verify the analysis of Finelli and Brandenberger (hep-ph/9809490) of super-horizon modes in ${m}^{2}{\ensuremath{\varphi}}^{2}$ inflation, showing that they are not amplified by resonant effects. Lastly, we make a preliminary survey of the nonlinear couplings between modes, which will be important in models where the primordial metric perturbations undergo parametric amplification.

Curvature-based gauge-invariant perturbation theory for gravity: A new paradigm

A new approach to gravitational gauge-invariant perturbation theory begins from the fourth-order Einstein-Ricci system, a hyperbolic formulation of gravity for arbitrary lapse and shift whose centerpiece is a wave equation for curvature. In the Minkowski and Schwarzschild backgrounds, an intertwining operator procedure is used to separate physical gauge-invariant curvature perturbations from unphysical ones. In the Schwarzschild case, physical variables are found which satisfy the Regge-Wheeler equation in both odd and even parity. In both cases, the unphysical "gauge'' degrees of freedom are identified with violations of the linearized Hamiltonian and momentum constraints, and they are found to evolve among themselves as a closed subsystem. If the constraints are violated, say by numerical finite-differencing, this system describes the hyperbolic evolution of the constraint violation. It is argued that an underlying raison d'\^etre of causal hyperbolic formulations is to make the evolution of constraint violations well-posed.

Curvature-based gauge-invariant perturbation theory for gravity: A new paradigm

Curvature-based gauge-invariant perturbation theory for gravity: A new paradigm

Einstein and Yang-Mills theories in hyperbolic form without gauge fixing.

The evolution of physical and gauge degrees of freedom in the Einstein and Yang-Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can be written in flux-conservative first-order symmetric hyperbolic form where the only nonzero characteristic speed is that of light. This dynamical form is ideal for global analysis, analytic approximation methods such as gauge-invariant perturbation theory, and numerical solution.

Collisions of boosted black holes: Perturbation theory prediction of gravitational radiation.

We consider general relativistic Cauchy data representing two nonspinning, equal-mass black holes boosted toward each other. When the black holes are close enough to each other and their momentum is sufficiently high, an encompassing apparent horizon is present so the system can be viewed as a single, perturbed black hole. We employ gauge-invariant perturbation theory, and integrate the Zerilli equation to analyze these time-asymmetric data sets and compute gravitational waveforms and emitted energies. When coupled with a simple Newtonian analysis of the infall trajectory, we find striking agreement between the perturbation calculation of emitted energies and the results of fully general relativistic numerical simulations of time-symmetric initial data.

Scalar perturbations in string-dilaton cosmology

Scalar perturbative potentials are constructed on exact background solutions for string-dilaton cosmology. We deal with the string effective Lagrangian under the same standard of an Induced Gravity Theory non-minimally coupled with a scalar field (the dilaton) and apply the theory of gauge-invariant perturbations. The main results are that we are able to obtain density contrasts (δρ/ρ) which can be connected with the observations and to control the horizon crossing for the perturbation scales.

Gauge-invariant cosmological perturbation theory

The linear perturbation theory of Friedman universes is treated in a gauge-invariant manner (i.e., invariant under linearized coordinate transformations). After presenting some mathematical tools in Chapter 1, the perturbation equations are derived with the help of the 3+1 formalism of general relativity in Chapter 2. Bardeens perturbation theory [4] is thereby extended to the treatment of collisionless matter. In Chapter 3 we solve the perturbation equations for perfect fuids explicitely. The results are applied to the discussion of the Sachs-Wolfe effet and of the traditional baryonic scenario of galaxy formation. In Chapter 4 numerical solutions of the gauge invariant perturbation equations are presented for a universe consisting of collisionless particles of mass mx ≠ 0, massless neutrinos and radiation. In this scenarios the masses of the first objects to collapse are of the order of m³pl/m²x. Within the gauge-invariant treatment, the perturbation amplitudes exhibit no growth on scales which are larger than the present horison size.

Gauge-invariant cosmological perturbations and the spectrum of primeval density fluctuations.

Fluctuation-dissipation theorems of the Landau-Lifshitz type are applied to cosmological perturbations about a homogeneous and isotropic background universe. A gauge-invariant perturbation theory is used to describe dissipative effects due to shear viscosity and bulk viscosity and the corresponding random fluctuations. All basic gauge-invariant quantities are physically identified on comoving hypersurfaces. The spectrum of adiabatic energy density fluctuations on superhorizon scales is calculated. A ${k}^{4}$ power spectrum ($k=\mathrm{comoving}\mathrm{wave}\mathrm{number}$) is shown to be a necessary consequence of a nonvanishing shear viscosity in the early universe with an equation of state $p=\frac{\ensuremath{\varepsilon}}{3}$. Bulk viscosity is connected with a $k$-independent spectrum. The possible relevance of this spectrum for the theory of galaxy formation is discussed and its relation to the Harrison-Zel'dovich spectrum is clarified.

Inhomogeneous cosmological models which are homogeneous and isotropic on average.

A scheme to construct inhomogeneous universes which are nevertheless homogeneous and isotropic on average is presented in the framework of general relativity for models containing dust and a cosmological constant. A class of exact is shown in the case that the collapse is locally one dimensional. They represent relativistic pancake solutions analogous to Zel'dovich's solution in Newtonian cosmology. The relation to gauge-invariant perturbation theory is also discussed.

On the quantization of perturbations in inflation

The theory of gauge-invariant scalar perturbations in chaotic inflation is briefly reviewed. The authors clarify a problem related to their quantization, pertinent to any inflationary model.

Gauge-invariant cosmological perturbation theory with seeds.

Gauge-invariant cosmological perturbation theory is extended to handle perturbations induced by seeds. A calculation of the Sachs-Wolfe effect is presented. A second-order differential equation for the growth of density perturbations is derived and the perturbation of Liouville's equation for collisionless particles is also given. The results are illustrated by a simple analytic example of a single texture knot, where we calculate the induced perturbations of the energy of microwave photons, of baryonic matter, and of collisionless particles.

Gauge-invariant treatment of gravitational radiation near the source: Analysis and numerical simulations.

We discuss a procedure based on the use of multipole-moment expansions for matching numerical solutions for the gravitational-radiation field around a compact source to linear analytic solutions. Gauge-invariant perturbation theory is used to generate even- and odd-parity matching equations for each spherical harmonic order $l$, $m$. This technique determines asymptotic wave forms, valid in the local wave zone, from the numerically evolved fields in a weak-field annular region surrounding the isolated source. The separation of the wave form from near-zone and residual gauge effects is demonstrated using fully general-relativistic simulations of relativistic stars undergoing nonradial pulsation.