36 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
Loop quantum cosmology of k=1 FRW models
The closed, k=1, FRW cosmology coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k=0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the big-bang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semi-classical at some late time, the big-bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the `difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k=0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k=0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists
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
Semi-classical States, Effective Dynamics and Classical Emergence in Loop Quantum Cosmology
We construct physical semi-classical states annihilated by the Hamiltonian
constraint operator in the framework of loop quantum cosmology as a method of
systematically determining the regime and validity of the semi-classical limit
of the quantum theory. Our results indicate that the evolution can be
effectively described using continuous classical equations of motion with
non-perturbative corrections down to near the Planck scale below which the
universe can only be described by the discrete quantum constraint. These
results, for the first time, provide concrete evidence of the emergence of
classicality in loop quantum cosmology and also clearly demarcate the domain of
validity of different effective theories. We prove the validity of modified
Friedmann dynamics incorporating discrete quanum geometry effects which can
lead to various new phenomenological applications. Furthermore the
understanding of semi-classical states allows for a framework for interpreting
the quantum wavefunctions and understanding questions of a semi-classical
nature within the quantum theory of loop quantum cosmology.Comment: Accepted for publication in Phys Rev D. Updated version to matc
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
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
Cosmological Plebanski theory
We consider the cosmological symmetry reduction of the Plebanski action as a
toy-model to explore, in this simple framework, some issues related to loop
quantum gravity and spin-foam models. We make the classical analysis of the
model and perform both path integral and canonical quantizations. As for the
full theory, the reduced model admits two types of classical solutions:
topological and gravitational ones. The quantization mixes these two solutions,
which prevents the model to be equivalent to standard quantum cosmology.
Furthermore, the topological solution dominates at the classical limit. We also
study the effect of an Immirzi parameter in the model.Comment: 20 page
On the validity of the 5-dimensional Birkhoff theorem: The tale of an exceptional case
The 5-dimensional (5d) Birkhoff theorem gives the class of 5d vacuum
space-times containing spatial hypersurfaces with cosmological symmetries. This
theorem is violated by the 5d vacuum Gergely-Maartens (GM) space-time, which is
not a representant of the above class, but contains the static Einstein brane
as embedded hypersurface. We prove that the 5d Birkhoff theorem is still
satisfied in a weaker sense: the GM space-time is related to the degenerated
horizon metric of certain black-hole space-times of the allowed class. This
result resembles the connection between the Bertotti-Robinson space-time and
the horizon region of the extremal Reissner-Nordstrom space-time in general
relativity.Comment: 13 pages; v2: title amended, to be published in Classical and Quantum
Gravit
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
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