16 research outputs found
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
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
Effective State Metamorphosis in Semi-Classical Loop Quantum Cosmology
Modification to the behavior of geometrical density at short scales is a key
result of loop quantum cosmology, responsible for an interesting phenomenology
in the very early universe. We demonstrate the way matter with arbitrary scale
factor dependence in Hamiltonian incorporates this change in its effective
dynamics in the loop modified phase. For generic matter, the equation of state
starts varying near a critical scale factor, becomes negative below it and
violates strong energy condition. This opens a new avenue to generalize various
phenomenological applications in loop quantum cosmology. We show that different
ways to define energy density may yield radically different results, especially
for the case corresponding to classical dust. We also discuss implications for
frequency dispersion induced by modification to geometric density at small
scales.Comment: Revised version; includes expanded discussion of natural
trans-Planckian modifications to frequency dispersion and robustness to
quantization ambiguities. To appear in Class. Quant. Gra
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
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
Loop Quantum Cosmology: A Status Report
The goal of this article is to provide an overview of the current state of
the art in loop quantum cosmology for three sets of audiences: young
researchers interested in entering this area; the quantum gravity community in
general; and, cosmologists who wish to apply loop quantum cosmology to probe
modifications in the standard paradigm of the early universe. An effort has
been made to streamline the material so that, as described at the end of
section I, each of these communities can read only the sections they are most
interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical
and Quantum Gravity. Typos corrected, clarifications and references adde
On Energy Conditions and Stability in Effective Loop Quantum Cosmology
In isotropic loop quantum cosmology, non-perturbatively modified dynamics of
a minimally coupled scalar field violates weak, strong and dominant energy
conditions when they are stated in terms of equation of state parameter. The
violation of strong energy condition helps to have non-singular evolution by
evading singularity theorems thus leading to a generic inflationary phase.
However, the violation of weak and dominant energy conditions raises concern,
as in general relativity these conditions ensure causality of the system and
stability of vacuum via Hawking-Ellis conservation theorem. It is shown here
that the non-perturbatively modified kinetic term contributes negative pressure
but positive energy density. This crucial feature leads to violation of energy
conditions but ensures positivity of energy density, as scalar matter
Hamiltonian remains bounded from below. It is also shown that the modified
dynamics restricts group velocity for inhomogeneous modes to remain sub-luminal
thus ensuring causal propagation across spatial distances.Comment: 29 pages, revtex4; few clarifications, references added, to appear in
CQ
Loop Quantum Cosmology
Quantum gravity is expected to be necessary in order to understand situations
where classical general relativity breaks down. In particular in cosmology one
has to deal with initial singularities, i.e. the fact that the backward
evolution of a classical space-time inevitably comes to an end after a finite
amount of proper time. This presents a breakdown of the classical picture and
requires an extended theory for a meaningful description. Since small length
scales and high curvatures are involved, quantum effects must play a role. Not
only the singularity itself but also the surrounding space-time is then
modified. One particular realization is loop quantum cosmology, an application
of loop quantum gravity to homogeneous systems, which removes classical
singularities. Its implications can be studied at different levels. Main
effects are introduced into effective classical equations which allow to avoid
interpretational problems of quantum theory. They give rise to new kinds of
early universe phenomenology with applications to inflation and cyclic models.
To resolve classical singularities and to understand the structure of geometry
around them, the quantum description is necessary. Classical evolution is then
replaced by a difference equation for a wave function which allows to extend
space-time beyond classical singularities. One main question is how these
homogeneous scenarios are related to full loop quantum gravity, which can be
dealt with at the level of distributional symmetric states. Finally, the new
structure of space-time arising in loop quantum gravity and its application to
cosmology sheds new light on more general issues such as time.Comment: 104 pages, 10 figures; online version, containing 6 movies, available
at http://relativity.livingreviews.org/Articles/lrr-2005-11