Hamiltonian cosmological perturbation theory

Abstract

The Hamiltonian approach to cosmological perturbations in general relativity in finite space-time is developed, where a cosmological scale factor is identified with spatial averaging the metric determinant logarithm. This identification preserves the number of variables and leads to a cosmological perturbation theory with the scalar potential perturbations in contrast to the kinetic perturbations in the Lifshitz version which are responsible for the "primordial power spectrum" of CMB in the inflationary model. The Hamiltonian approach enables to explain this "spectrum" in terms of scale-invariant variables and to consider other topical problem of modern cosmology in the context of quantum cosmological creation of both universes and particles from the stable Bogoliubov vacuum.

The Hamiltonian Approach to General Relativity and CMB Primordial Spectrum

“…We see that the Hubble law in the exact GR appears as spatial averaging the energy constraint (4). Thus, in the contrast with the generally accepted Lifshits theory [15] its Hamiltonian version [9] distinguishes the time-coordinate x 0 as an object of reparametrizations from the reparametrization-invariant time interval (5).…”

“…(here the general coordinate ones). Just this separation is the main difference of the Hamiltonian approach to GR considered here in finite space-time [8,9] from the Dirac -ADM one [10]. This separation can be fulfilled by using the gauge-invariant components of Fock's symplex ω (α) defined as…”

“…This finite volume generalization of the Dirac-ADM approach [10] was called the "Hamiltonian cosmological perturbation theory" [9]. c This approach demonstrates that the naive perturbation theory g µν = g Minkowski µν + h µν , where the coordinate evolution parameter x 0 is identified with measurable quantities, is incorrect because this theory contradicts to the gauge symmetry of GR.…”

“…However, in this theory, the scale factor is an additional variable without any constraint for the deviation Ψ, so that R d 3 xΨ = 0, and there are two zero Fourier harmonics of the determinant logarithm instead of one. This doubling does not allow to express the velocities of both variables log a and R d 3 xΨ through their momenta and to construct the Hamiltonian approach to GR [8,9]. Just this distinction helps us to separate the local part of the energy constraint (4) and determines unambiguously the gauge-invariant Dirac lapse function…”

The Role of Dark Matter and Dark Energy in Cosmological Models: Theoretical Overview

“…The model was successful to explain an acceleration of the Universe, but it has problems to fit Solar system data. Conformal theories are also not excluded (Behnke et al 2002;Barbashov et al 2006aBarbashov et al , 2006b. Recently, a generalization of the approach has been proposed in the framework of higher order theories of gravityalso referred to as f (R) (or fourth order) theories-a modification of the gravity action of the form (Capozziello 2007;Capozziello and Garattini 2007;Capozziello and Francaviglia 2007;Capozziello et al , 2006aCapozziello et al , 2006bCapozziello et al , 2006dCapozziello et al , 2006e, 2006f, 2006g, 2007bCapozziello et al , 2007cCapozziello et al , 2007dCarloni et al 2005;Borowiec et al 2006Borowiec et al , 2007 …”