4,055,354 research outputs found
Normal families and fixed points of iterates
Let F be a family of holomorphic functions and let K be a constant less than
4. Suppose that for all f in F the second iterate of f does not have fixed
points for which the modulus of the multiplier is greater than K. We show that
then F is normal. This is deduced from a result about the multipliers of
iterated polynomials.Comment: 5 page
Forming a constant density medium close to long gamma-ray bursts
The progenitor stars of long Gamma-Ray Bursts (GRBs) are thought to be
Wolf-Rayet stars, which generate a massive and energetic wind. Nevertheless,
about 25 percent of all GRB afterglows light curves indicate a constant density
medium close to the exploding star. We explore various ways to produce this, by
creating situations where the wind termination shock arrives very close to the
star, as the shocked wind material has a nearly constant density. Typically,
the distance between a Wolf-Rayet star and the wind termination shock is too
large to allow afterglow formation in the shocked wind material. Here, we
investigate possible causes allowing for a smaller distance: A high density or
a high pressure in the surrounding interstellar medium (ISM), a weak Wolf-Rayet
star wind, the presence of a binary companion, and fast motion of the
Wolf-Rayet star relative to the ISM. We find that all four scenarios are
possible in a limited parameter space, but that none of them is by itself
likely to explain the large fraction of constant density afterglows. A low GRB
progenitor metallicity, and a high GRB energy make the occurrence of a GRB
afterglow in a constant density medium more likely. This may be consistent with
constant densities beingpreferentially found for energetic, high redshift GRBs.Comment: 13 pages, 13 figures, new version: as accepted by Astronomy &
Astrophysic
The Hubble Constant
I review the current state of determinations of the Hubble constant, which
gives the length scale of the Universe by relating the expansion velocity of
objects to their distance. There are two broad categories of measurements. The
first uses individual astrophysical objects which have some property that
allows their intrinsic luminosity or size to be determined, or allows the
determination of their distance by geometric means. The second category
comprises the use of all-sky cosmic microwave background, or correlations
between large samples of galaxies, to determine information about the geometry
of the Universe and hence the Hubble constant, typically in a combination with
other cosmological parameters. Many, but not all, object-based measurements
give values of around 72-74km/s/Mpc , with typical errors of 2-3km/s/Mpc.
This is in mild discrepancy with CMB-based measurements, in particular those
from the Planck satellite, which give values of 67-68km/s/Mpc and typical
errors of 1-2km/s/Mpc. The size of the remaining systematics indicate that
accuracy rather than precision is the remaining problem in a good determination
of the Hubble constant. Whether a discrepancy exists, and whether new physics
is needed to resolve it, depends on details of the systematics of the
object-based methods, and also on the assumptions about other cosmological
parameters and which datasets are combined in the case of the all-sky methods.Comment: Extensively revised and updated since the 2007 version: accepted by
Living Reviews in Relativity as a major (2014) update of LRR 10, 4, 200
Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity
Quantum gravity is studied nonperturbatively in the case in which space has a
boundary with finite area. A natural set of boundary conditions is studied in
the Euclidean signature theory, in which the pullback of the curvature to the
boundary is self-dual (with a cosmological constant). A Hilbert space which
describes all the information accessible by measuring the metric and connection
induced in the boundary is constructed and is found to be the direct sum of the
state spaces of all Chern-Simon theories defined by all choices of
punctures and representations on the spatial boundary . The integer
level of Chern-Simons theory is found to be given by , where is the cosmological constant and is a
breaking phase. Using these results, expectation values of observables which
are functions of fields on the boundary may be evaluated in closed form. The
Beckenstein bound and 't Hooft-Susskind holographic hypothesis are confirmed,
(in the limit of large area and small cosmological constant) in the sense that
once the two metric of the boundary has been measured, the subspace of the
physical state space that describes the further information that the observer
on the boundary may obtain about the interior has finite dimension equal to the
exponent of the area of the boundary, in Planck units, times a fixed constant.
Finally,the construction of the state space for quantum gravity in a region
from that of all Chern-Simon theories defined on its boundary confirms the
categorical-theoretic ``ladder of dimensions picture" of Crane.Comment: TEX File, Minor Changes Made, 59 page
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