21 research outputs found
WKB approximation for inflationary cosmological perturbations
A new method for predicting inflationary cosmological perturbations, based on
the Wentzel-Kramers-Brillouin (WKB) approximation, is presented. A general
expression for the WKB scalar and tensor power spectra is derived. The main
advantage of the new scheme of approximation is that it is valid even if the
slow-roll conditions are violated. The method is applied to power-law
inflation, which allows a comparison with an exact result. It is demonstrated
that the WKB approximation predicts the spectral indices exactly and the
amplitude with an error lower than 10%, even in regimes far from
scale-invariance. The new method of approximation is also applied to a
situation where the slow-roll conditions hold. It is shown that the result
obtained bears close resemblance with the standard slow-roll calculation.
Finally, some possible improvements are briefly mentioned.Comment: 11 pages, 1 figure, RevTeX; minor changes, reference added (v2);
typos corrected (v3
Canonical Quantum Statistics of an Isolated Schwarzschild Black Hole with a Spectrum E_n = sigma sqrt{n} E_P
Many authors - beginning with Bekenstein - have suggested that the energy
levels E_n of a quantized isolated Schwarzschild black hole have the form E_n =
sigma sqrt{n} E_P, n=1,2,..., sigma =O(1), with degeneracies g^n. In the
present paper properties of a system with such a spectrum, considered as a
quantum canonical ensemble, are discussed: Its canonical partition function
Z(g,beta=1/kT), defined as a series for g<1, obeys the 1-dimensional heat
equation. It may be extended to values g>1 by means of an integral
representation which reveals a cut of Z(g,beta) in the complex g-plane from g=1
to infinity. Approaching the cut from above yields a real and an imaginary part
of Z. Very surprisingly, it is the (explicitly known) imaginary part which
gives the expected thermodynamical properties of Schwarzschild black holes:
Identifying the internal energy U with the rest energy Mc^2 requires beta to
have the value (in natural units) beta = 2M(lng/sigma^2)[1+O(1/M^2)], (4pi
sigma^2=lng gives Hawking's beta_H), and yields the entropy S=[lng/(4pi
sigma^2)] A/4 + O(lnA), where A is the area of the horizon.Comment: 14 pages, LaTeX A brief note added which refers to previous work
where the imaginary part of the partition function is related to metastable
states of the syste
Looking Beyond Inflationary Cosmology
In spite of the phenomenological successes of the inflationary universe
scenario, the current realizations of inflation making use of scalar fields
lead to serious conceptual problems which are reviewed in this lecture. String
theory may provide an avenue towards addressing these problems. One particular
approach to combining string theory and cosmology is String Gas Cosmology. The
basic principles of this approach are summarized.Comment: invited talk at "Theory Canada 1" (Univ. of British Columbia,
Vancouver, Canada, June 2 - 4, 2005) (references updated
Area spectrum and quasinormal modes of black holes
We demonstrate that an equidistant area spectrum for the link variables in
loop quantum gravity can reproduce both the thermodynamics and the quasinormal
mode properties of black holes.Comment: 11 pages, no figures; references adde
Algebraic approach to quantum black holes: logarithmic corrections to black hole entropy
The algebraic approach to black hole quantization requires the horizon area
eigenvalues to be equally spaced. As shown previously, for a neutral
non-rotating black hole, such eigenvalues must be -fold degenerate if
one constructs the black hole stationary states by means of a pair of creation
operators subject to a specific algebra. We show that the algebra of these two
building blocks exhibits symmetry, where the area
operator generates the U(1) symmetry. The three generators of the SU(2)
symmetry represent a {\it global} quantum number (hyperspin) of the black hole,
and we show that this hyperspin must be zero. As a result, the degeneracy of
the -th area eigenvalue is reduced to for large , and
therefore, the logarithmic correction term should be added to the
Bekenstein-Hawking entropy. We also provide a heuristic approach explaining
this result, and an evidence for the existence of {\it two} building blocks.Comment: 15 pages, Revtex, to appear in Phys. Rev.
The Coherent State Representation of Quantum Fluctuations in the Early Universe
Using the squeezed state formalism the coherent state representation of
quantum fluctuations in an expanding universe is derived. It is shown that this
provides a useful alternative to the Wigner function as a phase space
representation of quantum fluctuations. The quantum to classical transition of
fluctuations is naturally implemented by decohering the density matrix in this
representation. The entropy of the decohered vacua is derived. It is shown that
the decoherence process breaks the physical equivalence between vacua that
differ by a coordinate dependent phase generated by a surface term in the
Lagrangian. In particular, scale invariant power spectra are only obtained for
a special choice of surface term.Comment: 25 pages in revtex 3. This version is completely revised with
corrections and significant new calculation
Brane cosmological perturbations
We address the question of cosmological perturbations in the context of brane
cosmology, where our Universe is a three-brane where matter is confined,
whereas gravity lives in a higher dimensional spacetime. The equations
governing the bulk perturbations are computed in the case of a general warped
universe. The results are then specialized to the case of a five-dimensional
spacetime, scenario which has recently attracted a lot of attention. In this
context, we decompose the perturbations into `scalar', `vector' and `tensor'
modes, which are familiar in the standard theory of cosmological perturbations.
The junction conditions, which relate the metric perturbations to the matter
perturbations in the brane, are then computed.Comment: 14 pages, Latex; no figur
Bulk Gravitational Field and Cosmological Perturbations on the Brane
We investigate the effect of the bulk gravitational field on the cosmological
perturbations on a brane embedded in the 5D Anti-de Sitter (AdS) spacetime. The
effective 4D Einstein equations for the scalar cosmological perturbations on
the brane are obtained by solving the perturbations in the bulk. Then the
behaviour of the corrections induced by the bulk gravitational field to the
conventional 4D Einstein equation are determined. Two types of the corrections
are found. First we investigate the corrections which become significant at
scales below the AdS curvature scales and in the high energy universe with the
energy density larger than the tension of the brane. The evolution equation for
the perturbations on the brane is found and solved. Another type of the
corrections is induced on the brane if we consider the bulk perturbations which
do not contribute to the metric perturbations but do contribute to the matter
perturbations. At low energies, they have imaginary mass m^2=-(2/3) \k^2 in
the bulk where \k is the 3D comoving wave number of the perturbations. They
diverge at the horizon of the AdS spacetime. The induced density perturbations
behave as sound waves with sound velocity in the low energy
universe. At large scales, they are homogeneous perturbations that depend only
on time and decay like radiation. They can be identified as the perturbations
of the dark radiation. They produce isocurvature perturbations in the matter
dominated era. Their effects can be observed as the shifts of the location and
the height of the acoustic peak in the CMB spectrum.Comment: 35 pages, 1 figur
Inflationary Perturbations: the Cosmological Schwinger Effect
This pedagogical review aims at presenting the fundamental aspects of the
theory of inflationary cosmological perturbations of quantum-mechanical origin.
The analogy with the well-known Schwinger effect is discussed in detail and a
systematic comparison of the two physical phenomena is carried out. In
particular, it is demonstrated that the two underlying formalisms differ only
up to an irrelevant canonical transformation. Hence, the basic physical
mechanisms at play are similar in both cases and can be reduced to the
quantization of a parametric oscillator leading to particle creation due to the
interaction with a classical source: pair production in vacuum is therefore
equivalent to the appearance of a growing mode for the cosmological
fluctuations. The only difference lies in the nature of the source: an electric
field in the case of the Schwinger effect and the gravitational field in the
case of inflationary perturbations. Although, in the laboratory, it is
notoriously difficult to produce an electric field such that pairs extracted
from the vacuum can be detected, the gravitational field in the early universe
can be strong enough to lead to observable effects that ultimately reveal
themselves as temperature fluctuations in the Cosmic Microwave Background.
Finally, the question of how quantum cosmological perturbations can be
considered as classical is discussed at the end of the article.Comment: 49 pages, 6 figures, to appear in a LNP volume "Inflationary
Cosmology
Conformal Transformations in Cosmology of Modified Gravity: the Covariant Approach Perspective
The 1+3 covariant approach and the covariant gauge-invariant approach to
perturbations are used to analyze in depth conformal transformations in
cosmology. Such techniques allow us to obtain very interesting insights on the
physical content of these transformations, when applied to non-standard
gravity. The results obtained lead to a number of general conclusions on the
change of some key quantities describing any two conformally related
cosmological models. In particular, it is shown that the physics in the
Einstein frame has characteristics which are completely different from those in
the Jordan frame. Even if some of the geometrical properties of the cosmology
are preserved (homogeneous and isotropic Universes are mapped into homogeneous
and isotropic universes), it can happen that decelerating cosmologies are
mapped into accelerated ones. Differences become even more pronounced when
first-order perturbations are considered: from the 1+3 equations it is seen
that first-order vector and tensor perturbations are left unchanged in their
structure by the conformal transformation, but this cannot be said of the
scalar perturbations, which include the matter density fluctuations. Behavior
in the two frames of the growth rate, as well as other evolutionary features,
like the presence or absence of oscillations, etc., appear to be different too.
The results obtained are then explicitly interpreted and verified with the help
of some clarifying examples based on -gravity cosmologies.Comment: 26 pages, 8 figure