27 research outputs found
Loop Quantum Cosmology and Boundary Proposals
For many years, the most active area of quantum cosmology has been the issue of choosing boundary conditions for the wave function of a universe. Recently, loop quantum cosmology, which is obtained from loop quantum gravity, has shed new light on this question. In this case, boundary conditions are not chosen by hand with some particular physical intuition in mind, but they are part of the dynamical law. It is then natural to ask if there are any relations between these boundary conditions and the ones provided before. After discussing the technical foundation of loop quantum cosmology which leads to crucial ifferences to the WheelerâDeWitt quantization, we compare the dynamical initial conditions of loop quantum cosmology with the tunneling and the no-boundary proposal and explain why they are closer to the no-boundary condition. We end with a discussion of recent developments and several open problems of loop quantum cosmology
Lattice Refining Loop Quantum Cosmology from an Isotropic Embedding of Anisotropic Cosmology
We demonstrate that it is possible to produce different isotropic embeddings
of anisotropic Loop Quantum Cosmology, resulting to "lattice refinement" in the
isotropic system. To introduce the general approach, we first use a simple
model with only two anisotropic directions. We then employ the specific case of
a Bianchi I model, to show how the method extends to three-dimensional systems.
To concisely calculate the step-size of the resulting isotropic state, we
define the "symmetric dual" of states and operators, for the two- and
three-dimensional systems, respectively.Comment: 19 pages, 1 figure; slightly amended version to appear in Classical
and Quantum Gravit
Hubble operator in isotropic loop quantum cosmology
We present a construction of the Hubble operator for the spatially flat
isotropic loop quantum cosmology. This operator is a Dirac observable on a
subspace of the space of physical solutions. This subspace gets selected
dynamically, requiring that its action be invariant on the physical solution
space. As a simple illustrative application of the expectation value of the
operator, we do find a generic phase of (super)inflation, a feature shown by
Bojowald from the analysis of effective Friedmann equation of loop quantum
cosmology.Comment: 20 pages, 3 eps figures, few comments and clarifications added to
match with the published versio
Consistency Conditions for Fundamentally Discrete Theories
The dynamics of physical theories is usually described by differential
equations. Difference equations then appear mainly as an approximation which
can be used for a numerical analysis. As such, they have to fulfill certain
conditions to ensure that the numerical solutions can reliably be used as
approximations to solutions of the differential equation. There are, however,
also systems where a difference equation is deemed to be fundamental, mainly in
the context of quantum gravity. Since difference equations in general are
harder to solve analytically than differential equations, it can be helpful to
introduce an approximating differential equation as a continuum approximation.
In this paper implications of this change in view point are analyzed to derive
the conditions that the difference equation should satisfy. The difference
equation in such a situation cannot be chosen freely but must be derived from a
fundamental theory. Thus, the conditions for a discrete formulation can be
translated into conditions for acceptable quantizations. In the main example,
loop quantum cosmology, we show that the conditions are restrictive and serve
as a selection criterion among possible quantization choices.Comment: 33 page
The Early Universe in Loop Quantum Cosmology
Loop quantum cosmology applies techniques derived for a background
independent quantization of general relativity to cosmological situations and
draws conclusions for the very early universe. Direct implications for the
singularity problem as well as phenomenology in the context of inflation or
bouncing universes result, which will be reviewed here. The discussion focuses
on recent new results for structure formation and generalizations of the
methods.Comment: 10 pages, 3 figures, plenary talk at VI Mexican School on Gravitation
and Mathematical Physics, Nov 21-27, 200
Closed FRW model in Loop Quantum Cosmology
The basic idea of the LQC applies to every spatially homogeneous cosmological
model, however only the spatially flat (so called ) case has been
understood in detail in the literature thus far. In the closed (so called: k=1)
case certain technical difficulties have been the obstacle that stopped the
development. In this work the difficulties are overcome, and a new LQC model of
the spatially closed, homogeneous, isotropic universe is constructed. The
topology of the spacelike section of the universe is assumed to be that of
SU(2) or SO(3). Surprisingly, according to the results achieved in this work,
the two cases can be distinguished from each other just by the local properties
of the quantum geometry of the universe. The quantum hamiltonian operator of
the gravitational field takes the form of a difference operator, where the
elementary step is the quantum of the 3-volume derived in the flat case by
Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are
studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself
is not an eigenvalue, the eigenvectors form a basis. An estimate on the
dimension of the spectral projection on any finite interval is provided.Comment: 19 pages, latex, no figures, high quality, nea
Quantum Nature of the Big Bang: Improved dynamics
An improved Hamiltonian constraint operator is introduced in loop quantum
cosmology. Quantum dynamics of the spatially flat, isotropic model with a
massless scalar field is then studied in detail using analytical and numerical
methods. The scalar field continues to serve as `emergent time', the big bang
is again replaced by a quantum bounce, and quantum evolution remains
deterministic across the deep Planck regime. However, while with the
Hamiltonian constraint used so far in loop quantum cosmology the quantum bounce
can occur even at low matter densities, with the new Hamiltonian constraint it
occurs only at a Planck-scale density. Thus, the new quantum dynamics retains
the attractive features of current evolutions in loop quantum cosmology but, at
the same time, cures their main weakness.Comment: Typos corrected. Revised version to appear in Physical Review
Inflationary scalar spectrum in loop quantum cosmology
In the context of loop quantum cosmology, we consider an inflationary era
driven by a canonical scalar field and occurring in the semiclassical regime,
where spacetime is a continuum but quantum gravitational effects are important.
The spectral amplitude and index of scalar perturbations on an unperturbed de
Sitter background are computed at lowest order in the slow-roll parameters. The
scalar spectrum can be blue-tilted and far from scale invariance, and tuning of
the quantization ambiguities is necessary for agreement with observations. The
results are extended to a generalized quantization scheme including those
proposed in the literature. Quantization of the matter field at sub-horizon
scales can provide a consistency check of such schemes.Comment: 29 pages, 2 figures. v2: typos corrected, discussion improved and
extended, new section added. Conclusions are unchange
Effects of the quantisation ambiguities on the Big Bounce dynamics
In this paper we investigate dynamics of the modified loop quantum cosmology
models using dynamical systems methods. Modifications considered come from the
choice of the different field strength operator and result in
different forms of the effective Hamiltonian. Such an ambiguity of the choice
of this expression from some class of functions is allowed in the framework of
loop quantisation. Our main goal is to show how such modifications can
influence the bouncing universe scenario in the loop quantum cosmology. In
effective models considered we classify all evolutional paths for all
admissible initial conditions. The dynamics is reduced to the form of a
dynamical system of the Newtonian type on a 2-dimensional phase plane. These
models are equivalent dynamically to the FRW models with the decaying effective
cosmological term parametrised by the canonical variable (or by the scale
factor ). We find that for the positive cosmological constant there is a
class of oscillating models without the initial and final singularities. The
new phenomenon is the appearance of curvature singularities for the finite
values of the scale factor, but we find that for the positive cosmological
constant these singularities can be avoided. For the positive cosmological
constant the evolution begins at the asymptotic state in the past represented
by the deSitter contracting (deS) spacetime or the static Einstein
universe H=0 or state and reaches the deSitter expanding state
(deS), the state H=0 or state. In the case of the negative
cosmological constant we obtain the past and future asymptotic states as the
Einstein static universes.Comment: RevTeX4, 28 pages, 11 figs; rev.2 new section on exact solutions;
(v3) published versio
Lattice refinement in loop quantum cosmology
Lattice refinement in LQC, its meaning and its necessity are discussed. The
r\^ole of lattice refinement for the realisation of a successful inflationary
model is explicitly shown. A simple and effective numerical technique to solve
the constraint equation for any choice of lattice refinement model is briefly
illustrated. Phenomenological and consistency requirements leading to a
particular choice of lattice refinement model are presented, while it is
subsequently proved that only this choice of lattice refinement leads to a
unique factor ordering in the Wheeler-De Witt equation, which is the continuum
limit of LQC.Comment: 17 pages, 1 figure, to appear in the Proceedings of "Recent
Developments in Gravity-NEB XIII"; Thessaloniki (Greece), June 200