Gauge invariant cosmological perturbations: theory and application

A thermodynamically oriented gauge-invariant description of scalar cosmological perturbations around a flat Friedman universe is given. All basic variables have their physical meaning on comoving hypersurfaces, i.e. on hypersurfaces orthogonal to the matter worldlines. Several well known growth and decay rates of cosmological perturbation modes are rediscovered in a transparent way. The formalism is extended to multicomponent systems and applied to a mixture of radiation and non-relativistic particles for perturbations both smaller and larger than the particle horizon. Thermodynamical perturbation relations in the Minkowski spacetime turn out to be valid in the expanding universe, provided all perturbed quantities are replaced by their gauge-invariant counterparts.

We present a summary of the results, in the large scale, that were derived from the gauge-invariant (GI) cosmological density perturbation analysis. Comparisons are made with the synchronous gauge (SG) results, pointing out past incorrect analyses concerning evolution during the inflationary era and the decaying mode during the matter dominated era. The GI method, using the covariant equations, is recommended for cosmological perturbation analysis, not only because of its gauge-independence, but because of its simplicity and similarity to the Newtonian analysis. Solutions in other specific gauges can simply be recovered from the known results of the GI analysis.

We study the growth of energy-density perturbations in inflationary-universe models, applying an extension of Bardeen's gauge-invariant framework derived in a previous paper. The complete analysis is exemplified in the case of the ''new inflationary universe'' of Linde and Albrecht and Steinhardt. For this model we obtain the following result: the amplitude of energy-density fluctuations at horizon crossing is of order 50, far too large to match with the usual pictures of galaxy formation. Our result agrees with other published analyses. We conclude that the amplification of energy-density perturbations is determined by the change in the equation of state between initial and final Hubble radius crossing. It is, however, independent of the phase structure between the two crossings. In particular, it is independent of the reheating mechanism. We also derive a simple formula which describes the complete evolution of perturbations outside the horizon whenever entropy and anisotropic stress perturbations are unimportant. Finally, we comment on previous published methods and discuss their limitations.

We use gauge-invariant cosmological perturbation theory to calculate the displacement field that sets the initial conditions for $N$-body simulations. Using first and second-order fully relativistic perturbation theory in the synchronous-comoving gauge, allows us to go beyond the Newtonian predictions and to calculate relativistic corrections to it. We use an Einstein--de Sitter model, including both growing and decaying modes in our solutions. The impact of our results should be assessed through the implementation of the featured displacement in cosmological $N$-body simulations.

The translationally invariant diagrammatic quantum perturbation theory (TPT) is applied to the polaron problem on the 1D lattice, modeled through the Holstein Hamiltonian with the phonon frequency omega0, the electron hopping t and the electron-phonon coupling constant g. The self-energy diagrams of the fourth-order in g are calculated exactly for an intermittently added electron, in addition to the previously known second-order term. The corresponding quadratic and quartic corrections to the polaron ground state energy become comparable at t/omega0>1 for g/omega0~(t/omega0)^{1/4} when the electron self-trapping and translation become adiabatic. The corresponding non adiabatic/adiabatic crossover occurs while the polaron width is large, i.e. the lattice coarsening negligible. This result is extended to the range (t/omega0)^{1/2}>g/omega0>(t/omega0)^{1/4}>1 by considering the scaling properties of the high-order self-energy diagrams. It is shown that the polaron ground state energy, its width and the effective mass agree with the results found traditionally from the broken symmetry side, kinematic corrections included. The Landau self trapping of the electron in the classic self-consistent, localized displacement potential, the restoration of the translational symmetry by the classic translational Goldstone mode and the quantization of the polaronic translational coordinate are thus all encompassed by a quantum theory which is translationally invariant from the outset.

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

I briefly review the main features of inflationary universe models. Inflation provides a mechanism which produces energy density fluctuations on cosmological scales. In the original models it was not possible to obtain the correct magnitude of these fluctuations without fine tuning the particle physics models. I discuss two mechanisms, “chaotic inflation”, and a “dynamical relaxation” process, by which inflation may be realized in models which give the right magnitude of fluctuations.

Evolution equation for the expectation value of a scalar field in spatially flat RW universes

We consider an inflationary universe model in the context of generalized cosmic Chaplygin gas by taking matter field as standard and tachyon scalar fields. We evaluate the corresponding scalar fields and scalar potentials during intermediate and logamediate inflationary regimes by modifying the first Friedmann equation. In each case, we evaluate the number of e-folds, scalar as well as tensor power spectra, scalar spectral index and important observational parameter, i.e., tensor-scalar ratio in terms of inflatons. The graphical behavior of this parameter shows that the model remains incompatible with WMAP7 and Planck observational data in each case.

We study the production of large-scale (\ensuremath{\sim}Mpc) magnetic fields in inflationary Universe models. The magnetic fields produced are uninterestingly small unless the conformal invariance of the electromagnetic field is broken. Once the conformal invariance is broken, a mechanism akin to ``superadiabatic amplification'' can operate and lead to sizable primeval magnetic fields. We consider three ways of breaking the conformal invariance: through gravitational couplings of the photon; through the coupling of the photon to a charged massless, nonconformally invariant scalar field; and through the anomalous coupling of the photon to axions. The primeval magnetic fields which result can have astrophysically interesting strengths, but are very model dependent.

The authors propose a new covariant and gauge-invariant (GI) treatment of perturbations in a Robertson-Walker universe dominated by a classical scalar field phi . They first set up the formalism, based on the natural slicing of the problem by the surfaces phi =constant, and introduce a set of covariantly defined GI variables. In their approach the whole inhomogeneity of the matter field is incorporated in the GI spatial fluctuations of the momentum psi of phi ; then the GI density perturbations are simply proportional to the momentum perturbations. The inhomogeneity of the geometry is characterized by GI fluctuations of the 3-curvature scalar of the surfaces phi =constant. The time evolution of the matter and curvature perturbations are coupled by a pair of first-order linear differential equations. Correspondingly, each GI variable satisfies a second-order linear homogeneous differential equation. When the background curvature vanishes, k=0, the curvature variable is conserved for perturbation scales larger than the horizon, but this is no longer true in general if k not=0. They discuss simple examples, including the case when more than one scalar field is present, recovering standard results for inflationary universe models. They also demonstrate that in coasting solutions with k=-1, inhomogeneities are damped out on all scales.

We derive a set of equations monitoring the evolution of covariant and gauge-invariant linear scalar perturbations of Friedman-Lema\^itre-Robertson-Walker models with multiple interacting non-linear scalar fields. We use a dynamical systems' approach in order to perform a stability analysis for some classes of scalar field potentials. In particular, using a recent approximation for the inflationary dynamics of the background solution, we derive conditions under which homogenization occurs for chaotic (quadratic and quartic potentials) and new inflation. We also prove a cosmic no-hair result for power-law inflation and its generalisation for two scalar fields with independent exponential potentials (assisted power-law inflation).

We explore the possible cosmological consequences of a running Newton’s constant, G(⎕), as suggested by the non-trivial ultraviolet fixed point scenario for Einstein gravity with a cosmological constant term. Here, we examine what possible effects a scale-dependent coupling might have on large-scale cosmological density perturbations. Starting from a set of manifestly covariant effective field equations, we develop the linear theory of density perturbations for a non-relativistic perfect fluid. The result is a modified equation for the matter density contrast, which can be solved and thus provides an estimate for the corrections to the growth index parameter, ɤ.

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.

The possibility of nontrivial representations of the gauge group on wave functionals of a gauge invariant quantum field theory leads to a generation of mass for intermediate vector and tensor bosons. The mass parameters m show up as central charges in the algebra of constraints, which then become of second-class nature. The gauge group coordinates acquire dynamics outside the null-mass shell and provide the longitudinal field degrees of freedom that massless bosons need to form massive bosons. This is a non-Higgs mechanism that could provide new clues for the best understanding of the symmetry breaking mechanism in unified field theories. A unified quantization of massless and massive non-Abelian Yang–Mills, linear gravity and Abelian two-form gauge field theories are fully developed from this new approach, where a cohomological origin of mass is pointed out.