5,085 research outputs found
On the degree of Polar Transformations -- An approach through Logarithmic Foliations
We investigate the degree of the polar transformations associated to a
certain class of multi-valued homogeneous functions. In particular we prove
that the degree of the pre-image of generic linear spaces by a polar
transformation associated to a homogeneous polynomial is determined by the
zero locus of . For zero dimensional-dimensional linear spaces this was
conjecture by Dolgachev and proved by Dimca-Papadima using topological
arguments. Our methods are algebro-geometric and rely on the study of the Gauss
map of naturally associated logarithmic foliations
Isotropization of the universe during inflation
A primordial inflationary phase allows one to erase any possible anisotropic
expansion thanks to the cosmic no-hair theorem. If there is no global
anisotropic stress, then the anisotropic expansion rate tends to decrease. What
are the observational consequences of a possible early anisotropic phase? We
first review the dynamics of anisotropic universes and report analytic
approximations. We then discuss the structure of dynamical equations for
perturbations and the statistical properties of observables, as well as the
implication of a primordial anisotropy on the quantization of these
perturbations during inflation. Finally we briefly review models based on
primordial vector field which evade the cosmic no-hair theorem.Comment: 9 pages, 3 figures. Invited review article for the French Academy of
Scienc
Testing gaussianity, homogeneity and isotropy with the cosmic microwave background
We review the basic hypotheses which motivate the statistical framework used
to analyze the cosmic microwave background, and how that framework can be
enlarged as we relax those hypotheses. In particular, we try to separate as
much as possible the questions of gaussianity, homogeneity and isotropy from
each other. We focus both on isotropic estimators of non-gaussianity as well as
statistically anisotropic estimators of gaussianity, giving particular emphasis
on their signatures and the enhanced "cosmic variances" that become
increasingly important as our putative Universe becomes less symmetric. After
reviewing the formalism behind some simple model-independent tests, we discuss
how these tests can be applied to CMB data when searching for large scale
"anomalies"Comment: 52 pages, 22 pdf figures. Revised version of the invited review for
the special issue "Testing the Gaussianity and Statistical Isotropy of the
Universe" for Advances in Astronomy
CMB statistical isotropy confirmation at all scales using multipole vectors
We present an efficient numerical code and conduct, for the first time, a
null and model-independent CMB test of statistical isotropy using Multipole
Vectors (MVs) at all scales. Because MVs are insensitive to the angular power
spectrum , our results are independent from the assumed cosmological
model. We avoid a posteriori choices and use pre-defined ranges of scales
, and in our analyses. We
find that all four masked Planck maps, from both 2015 and 2018 releases, are in
agreement with statistical isotropy for , . For
we detect anisotropies but this is indicative of simply the
anisotropy in the noise: there is no anisotropy for and an
increasing level of anisotropy at higher multipoles. Our findings of no
large-scale anisotropies seem to be a consequence of avoiding \emph{a
posteriori} statistics. We also find that the degree of anisotropy in the full
sky (i.e. unmasked) maps vary enormously (between less than 5 and over 1000
standard deviations) among the different mapmaking procedures and data
releases.Comment: v4: additional analysis which increased statistical sensitivity,
including new plots and tables; extended discussion; 15 pages, 14 figures, 7
tables. Matches published versio
Weak-lensing -modes as a probe of the isotropy of the universe
We compute the angular power spectrum of the -modes of the weak-lensing
shear in a spatially anisotropic spacetime. We find that there must also exist
off-diagonal correlations between the -modes, -modes, and convergence
that allow one to reconstruct the eigendirections of expansion. Focusing on
future surveys such as Euclid and SKA, we show that observations can constrain
the geometrical shear in units of the Hubble rate at the percent level, or even
better, offering a new and powerful method to probe our cosmological model.Comment: 4 pages, 3 figures. This version matches the published on
Sacrifice and Efficiency of the Income Tax Schedule
We investigate the efficiency of equal sacrifice tax schedules in an economywhich primitives are exactly those in Mirrlees (1971): a continuum of individualswith identical preferences defined over consumption and leisure who differ withrespect to their labor market productivity. Using a separable specification forpreferences we derive the minimum equal sacrifice allocation and recover thetax schedule that implements it. The separable specification allows us to usethe methodology developed by Werning (2007b) to check whether the scheduleis efficient, that is, whether there is no alternative tax schedule that raises morerevenue while delivering less utility to no one. We find that inefficiency does notarise for most parametrizations we use to approximate the US economy. For thefew cases for which inefficiency does arise, it does so only for very high levels ofincome and marginal tax rates.
Cosmological Signatures of Anisotropic Spatial Curvature
If one is willing to give up the cherished hypothesis of spatial isotropy,
many interesting cosmological models can be developed beyond the simple
anisotropically expanding scenarios. One interesting possibility is presented
by shear-free models in which the anisotropy emerges at the level of the
curvature of the homogeneous spatial sections, whereas the expansion is
dictated by a single scale factor. We show that such models represent viable
alternatives to describe the large-scale structure of the inflationary
universe, leading to a kinematically equivalent Sachs-Wolfe effect. Through the
definition of a complete set of spatial eigenfunctions we compute the two-point
correlation function of scalar perturbations in these models. In addition, we
show how such scenarios would modify the spectrum of the CMB assuming that the
observations take place in a small patch of a universe with anisotropic
curvature.Comment: 21 pages, 1 figure. To appear in JCA
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