Quantum-to-classical Transition of Cosmological Perturbations for Non-vacuum Initial States

Transition from quantum to semiclassical behaviour and loss of quantum coherence for inhomogeneous perturbations generated from a non-vacuum initial state in the early Universe is considered in the Heisenberg and the Schr\"odinger representations, as well as using the Wigner function. We show explicitly that these three approaches lead to the same prediction in the limit of large squeezing (i.e. when the squeezing parameter $|r_k|\to \infty$): each two-modes quantum state (k, -k) of these perturbations is equivalent to a classical perturbation that has a stochastic amplitude, obeying a non-gaussian statistics which depends on the initial state, and that belongs to the quasi-isotropic mode (i.e. it possesses a fixed phase). The Wigner function is not everywhere positive for any finite $r_k$, hence its interpretation as a classical distribution function in phase space is impossible without some coarse graining procedure. However, this does not affect the transition to semiclassical behaviour since the Wigner function becomes concentrated near a classical trajectory in phase space when $|r_k|\to \infty$ even without coarse graining. Deviations of the statistics of the perturbations in real space from a Gaussian one lie below the cosmic variance level for the N-particles initial states with N=N(|k|) but may be observable for other initial states without statistical isotropy or with correlations between different k modes. As a way to look for this effect, it is proposed to measure the kurtosis of the angular fluctuations of the cosmic microwave background temperature.Comment: LaTeX (28 pages),+2 eps figure

Inflation, quantum fields, and CMB anisotropies

Inflationary cosmology has proved to be the most successful at predicting the properties of the anisotropies observed in the cosmic microwave background (CMB). In this essay we show that quantum field renormalization significantly influences the generation of primordial perturbations and hence the expected measurable imprint of cosmological inflation on the CMB. However, the new predictions remain in agreement with observation, and in fact favor the simplest forms of inflation. In the near future, observations of the influence of gravitational waves from the early universe on the CMB will test our new predictions.Comment: 11 pages, 1 figure, Awarded with the fourth prize in the Gravity Research Foundation 2009 Essay Competitio

Quasi-Isotropization of the Inhomogeneous Mixmaster Universe Induced by an Inflationary Process

We derive a ``generic'' inhomogeneous ``bridge'' solution for a cosmological model in the presence of a real self-interacting scalar field. This solution connects a Kasner-like regime to an inflationary stage of evolution and therefore provides a dynamical mechanism for the quasi-isotropization of the universe. In the framework of a standard Arnowitt-Deser-Misner Hamiltonian formulation of the dynamics and by adopting Misner-Chitr\`e-like variables, we integrate the Einstein-Hamilton-Jacobi equation corresponding to a ``generic'' inhomogeneous cosmological model whose evolution is influenced by the coupling with a bosonic field, expected to be responsible for a spontaneous symmetry breaking configuration. The dependence of the detailed evolution of the universe on the initial conditions is then appropriately characterized.Comment: 17 pages, no figure, to appear on PR

The stress-energy tensor for trans-Planckian cosmology

This article presents the derivation of the stress-energy tensor of a free scalar field with a general non-linear dispersion relation in curved spacetime. This dispersion relation is used as a phenomelogical description of the short distance structure of spacetime following the conventional approach of trans-Planckian modes in black hole physics and in cosmology. This stress-energy tensor is then used to discuss both the equation of state of trans-Planckian modes in cosmology and the magnitude of their backreaction during inflation. It is shown that gravitational waves of trans-Planckian momenta but subhorizon frequencies cannot account for the form of cosmic vacuum energy density observed at present, contrary to a recent claim. The backreaction effects during inflation are confirmed to be important and generic for those dispersion relations that are liable to induce changes in the power spectrum of metric fluctuations. Finally, it is shown that in pure de Sitter inflation there is no modification of the power spectrum except for a possible magnification of its overall amplitude independently of the dispersion relation.Comment: 18 pages, 2 figures. Version to appear in PRD (minor modifications

Particle production and classical condensates in de Sitter space

The cosmological particle production in a $k=0$ expanding de Sitter universe with a Hubble parameter $H_0$ is considered for various values of mass or conformal coupling of a free, scalar field. One finds that, for a minimally coupled field with mass $0 \leq m^2 < 9 H_0^2/4$ (except for $m^2= 2H_0^2$), the one-mode occupation number grows to unity soon after the physical wavelength of the mode becomes larger than the Hubble radius, and afterwards diverges as $n(t) \sim O(1)(\lambda_{phys}(t)/H_0^{-1})^{2\nu}$, where $\nu \equiv [9/4 - m^2/H_0^2]^{1/2}$. However, for a field with $m^2 > 9H_0^2/4$, the occupation number of a mode outside the Hubble radius is rapidly oscillating and bounded and does not exceed unity. These results, readily generalized for cases of a nonminimal coupling, provide a clear argument that the long-wavelength vacuum fluctuations of low-mass fields in an inflationary universe do show classical behavior, while those of heavy fields do not. The interaction or self-interaction does not appear necessary for the emergence of classical features, which are entirely due to the rapid expansion of the de Sitter background and the upside-down nature of quantum oscillators for modes outside the Hubble radius.Comment: Revtex + 5 postscript figures. Accepted for Phys Rev D15. Revision of Aug 1996 preprint limited to the inclusion and discussion of references suggested by the referee

Improved Estimates of Cosmological Perturbations

We recently derived exact solutions for the scalar, vector and tensor mode functions of a single, minimally coupled scalar plus gravity in an arbitrary homogeneous and isotropic background. These solutions are applied to obtain improved estimates for the primordial scalar and tensor power spectra of anisotropies in the cosmic microwave background.Comment: 31 pages, 4 figures, LaTeX 2epsilon, this version corrects an embarrasing mistake (in the published version) for the parameter q_C. Affected eqns are 105, 109-110, 124, 148-153 and 155-15

A quantum mechanical relation connecting time, temperature, and cosmological constant of the universe: Gamow's relation revisited as a special case

Considering our expanding universe as made up of gravitationally interacting particles which describe particles of luminous matter and dark matter and dark energy which is described by a repulsive harmonic potential among the points in the flat 3-space, we derive a quantum mechanical relation connecting, temperature of the cosmic microwave background radiation, age, and cosmological constant of the universe. When the cosmological constant is zero, we get back the Gamow's relation with a much better coefficient. Otherwise, our theory predicts a value of the cosmological constant $2.0 10^{-56} {\rm {cm^{-2}}}$ when the present values of cosmic microwave background temperature of 2.728 K and age of the universe 14 billion years are taken as input.Comment: 4 pages, 1 figure, Study of the Universe from a condensed matter point of view, section III corrected with a single body potentia

Duality Invariance of Cosmological Perturbation Spectra

I show that cosmological perturbation spectra produced from quantum fluctuations in massless or self-interacting scalar fields during an inflationary era remain invariant under a two parameter family of transformations of the homogeneous background fields. This relates slow-roll inflation models to solutions which may be far from the usual slow-roll limit. For example, a scale-invariant spectrum of perturbations in a minimally coupled, massless field can be produced by an exponential expansion with $a\propto e^{Ht}$, or by a collapsing universe with $a\propto (-t)^{2/3}$.Comment: 5 pages, Latex with Revtex. Hamiltonian formulation added and discussion expanded. Version to appear in Phys Rev

Conditions for Successful Extended Inflation

We investigate, in a model-independent way, the conditions required to obtain a satisfactory model of extended inflation in which inflation is brought to an end by a first-order phase transition. The constraints are that the correct present strength of the gravitational coupling is obtained, that the present theory of gravity is satisfactorily close to general relativity, that the perturbation spectra from inflation are compatible with large scale structure observations and that the bubble spectrum produced at the phase transition doesn't conflict with the observed level of microwave background anisotropies. We demonstrate that these constraints can be summarized in terms of the behaviour in the conformally related Einstein frame, and can be compactly illustrated graphically. We confirm the failure of existing models including the original extended inflation model, and construct models, albeit rather contrived ones, which satisfy all existing constraints.Comment: 8 pages RevTeX file with one figure incorporated (uses RevTeX and epsf). Also available by e-mailing ARL, or by WWW at http://star-www.maps.susx.ac.uk/papers/infcos_papers.html; Revised to include extra references, results unchanged, to appear Phys Rev

de Sitter spacetime: effects of metric perturbations on geodesic motion

Gravitational perturbations of the de Sitter spacetime are investigated using the Regge--Wheeler formalism. The set of perturbation equations is reduced to a single second order differential equation of the Heun-type for both electric and magnetic multipoles. The solution so obtained is used to study the deviation from an initially radial geodesic due to the perturbation. The spectral properties of the perturbed metric are also analyzed. Finally, gauge- and tetrad-invariant first-order massless perturbations of any spin are explored following the approach of Teukolsky. The existence of closed-form, i.e. Liouvillian, solutions to the radial part of the Teukolsky master equation is discussed.Comment: IOP macros, 10 figure