45 research outputs found
On the sub-Gaussianity of the Beta and Dirichlet distributions
We obtain the optimal proxy variance for the sub-Gaussianity of Beta
distribution, thus proving upper bounds recently conjectured by Elder (2016).
We provide different proof techniques for the symmetrical (around its mean)
case and the non-symmetrical case. The technique in the latter case relies on
studying the ordinary differential equation satisfied by the Beta
moment-generating function known as the confluent hypergeometric function. As a
consequence, we derive the optimal proxy variance for the Dirichlet
distribution, which is apparently a novel result. We also provide a new proof
of the optimal proxy variance for the Bernoulli distribution, and discuss in
this context the proxy variance relation to log-Sobolev inequalities and
transport inequalities.Comment: 13 pages, 2 figure
Constraints on Single-Field Inflation
Many alternatives to canonical slow-roll inflation have been proposed over
the years, one of the main motivations being to have a model, capable of
generating observable values of non-Gaussianity. In this work, we (re-)explore
the physical implications of a great majority of such models within a single,
effective field theory framework (including novel models with large
non-Gaussianity discussed for the first time below.) The constraints we
apply---both theoretical and experimental---are found to be rather robust,
determined to a great extent by just three parameters: the coefficients of the
quadratic EFT operators and , and the
slow-roll parameter . This allows to significantly limit the
majority of single-field alternatives to canonical slow-roll inflation. While
the existing data still leaves some room for most of the considered models, the
situation would change dramatically if the current upper limit on the
tensor-to-scalar ratio decreased down to . Apart from inflationary
models driven by plateau-like potentials, the single-field model that would
have a chance of surviving this bound is the recently proposed slow-roll
inflation with weakly-broken galileon symmetry. In contrast to
\textit{canonical} slow-roll inflation, the latter model can support even if driven by a convex potential, as well as generate observable
values for the amplitude of non-Gaussianity.Comment: 19+10 pages, 6 figure
Phenomenology of D-Brane Inflation with General Speed of Sound
A characteristic of D-brane inflation is that fluctuations in the inflaton
field can propagate at a speed significantly less than the speed of light. This
yields observable effects that are distinct from those of single-field slow
roll inflation, such as a modification of the inflationary consistency relation
and a potentially large level of non-Gaussianities. We present a numerical
algorithm that extends the inflationary flow formalism to models with general
speed of sound. For an ensemble of D-brane inflation models parameterized by
the Hubble parameter and the speed of sound as polynomial functions of the
inflaton field, we give qualitative predictions for the key inflationary
observables. We discuss various consistency relations for D-brane inflation,
and compare the qualitative shapes of the warp factors we derive from the
numerical models with analytical warp factors considered in the literature.
Finally, we derive and apply a generalized microphysical bound on the inflaton
field variation during brane inflation. While a large number of models are
consistent with current cosmological constraints, almost all of these models
violate the compactification constraint on the field range in four-dimensional
Planck units. If the field range bound is to hold, then models with a
detectable level of non-Gaussianity predict a blue scalar spectral index, and a
tensor component that is far below the detection limit of any future
experiment.Comment: 23 pages, 11 figures, v2: version accepted by PRD; minor
clarifications and references added to the text. Higher resolution figures
are available in the published version. v3: post-publication correction of
typo in Eq. 87. No results/conclusions change
Generalisation and expressiveness for over-parameterised neural networks
Over-parameterised modern neural networks owe their success to two fundamental properties: expressive power and generalisation capability. The former refers to the model's ability to fit a large variety of data sets, while the latter enables the network to extrapolate patterns from training examples and apply them to previously unseen data. This thesis addresses a few challenges related to these two key properties.
The fact that over-parameterised networks can fit any data set is not always indicative of their practical expressiveness. This is the object of the first part of this thesis, where we delve into how the input information can get lost when propagating through a deep architecture, and we propose as an easily implementable possible solution the introduction of suitable scaling factors and residual connections.
The second part of this thesis focuses on generalisation. The reason why modern neural networks can generalise well to new data without overfitting, despite being over-parameterised, is an open question that is currently receiving considerable attention in the research community. We explore this subject from information-theoretic and PAC-Bayesian viewpoints, proposing novel learning algorithms and generalisation bounds
Predictions in multifield inflation
Models of inflation with more than one active field are an important class where it is not fully understood how to compute predictions. This problem can be understood in
terms of two characteristics of these models: the sensitivity to initial conditions and the superhorizon evolution of the primordial density perturbation Ī¶. This thesis seeks to make significant progress in understanding how to overcome these two issues.
To track the superhorizon evolution of Ī¶ in general requires numerical techniques. By extending the transport method first proposed by Mulryne, Seery and Wesley, here, a
computationally efficient and highly versatile method for computing the statistics of Ī¶ is developed. The increased efficiency and versatility allows models that were previously unaccessible to be studied.
Utilising this new capability two models are explored. A new toy model of inflation in the Landscape and a 6-field D-brane model of inflation first proposed by Agarwal,
Bean, McAllister, and Xu. The nature of these models allows for a statistical analysis of inflationary realisations to be performed. We conclude that the fundamental ability to
constrain models of this kind is determined by the scale of features in the potential. We also show the D-brane model is under considerable pressure from current observations of
the spectral index and may be ruled out by future observations.
Finally, I show that there exists a class of models for which the probability distribution of observables may be computed analytically. I show the peak of the density function is largely dominated by the geometry of the potential and comparatively insensitive to the distribution of initial conditions. I argue that this characteristic should be expected in a broader range of models and for such models, it is possible to make robust predictions
Detecting and visualizing differences in brain structures with SPHARM and functional data analysis
A new procedure for classifying brain structures described by SPHARM is presented. We combine a dimension reduction technique (functional principal component analysis or functional independent component analysis) with stepwise variable selection for linear discriminant classification. This procedure is compared with many well-known methods in a novel classification problem in neuroeducation, where the reversal error (a common error in mathematical problem solving) is analyzed by using the left and right putamens of 33 participants. The comparison shows that our proposal not only provides outstanding performance in terms of predictive power, but it is also valuable in terms of interpretation, since it yields a linear discriminant function for 3D structures
Non-Gaussianity from the Cross-correlation of the Astrophysical Gravitational Wave Background and the Cosmic Microwave Background
Since the first LIGO/Virgo detection, Gravitational Waves (GWs) have been
very promising as a new complementary probe to understand our Universe. One of
the next challenges of GW search is the detection and characterization of the
stochastic gravitational wave background (SGWB), that is expected to open a
window on the very early Universe (cosmological background) and to provide us
new information on astrophysical source populations (astrophysical background).
One way to characterize the SGWB and to extract information about its origin is
through the cross-correlation with other cosmological probes. To this aim, in
this paper, we explore the cross-correlation between the astrophysical
background anisotropies and the Cosmic Microwave Background (CMB) ones. Such a
signal is sensitive to primordial non-Gaussianity (nG) through the GW bias.
Thus, we study the capability of next generation space-based interferometers to
detect such a cross-correlation signal and to constrain primordial nG.Comment: 13 pages, 5 figure
Non-gaussianity and Statistical Anisotropy in Cosmological Inflationary Models
We study the statistical descriptors for some cosmological inflationary
models that allow us to get large levels of non-gaussianity and violations of
statistical isotropy. Basically, we study two different class of models: a
model that include only scalar field perturbations, specifically a subclass of
small-field slow-roll models of inflation with canonical kinetic terms, and
models that admit both vector and scalar field perturbations. We study the
former to show that it is possible to attain very high, including observable,
values for the levels of non-gaussianity f_{NL} and \tao_{NL} in the bispectrum
B_\zeta and trispectrum T_\zeta of the primordial curvature perturbation \zeta
respectively. Such a result is obtained by taking care of loop corrections in
the spectrum P_\zeta, the bispectrum B_\zeta and the trispectrum T_\zeta .
Sizeable values for f_{NL} and \tao_{NL} arise even if \zeta is generated
during inflation. For the latter we study the spectrum P_\zeta, bispectrum
B_\zeta and trispectrum $T_\zeta of the primordial curvature perturbation when
\zeta is generated by scalar and vector field perturbations. The tree-level and
one-loop contributions from vector field perturbations are worked out
considering the possibility that the one-loop contributions may be dominant
over the tree level terms. The levels of non-gaussianity f_{NL} and \tao_{NL},
are calculated and related to the level of statistical anisotropy in the power
spectrum, g_\zeta . For very small amounts of statistical anisotropy in the
power spectrum, the levels of non-gaussianity may be very high, in some cases
exceeding the current observational limit.Comment: Latex file, 113 pages. PhD Thesis. Supervisor: Yeinzon Rodriguez
Aspects of Today's Cosmology
This book presents some aspects of the cosmological scientific odyssey that started last century. The chapters vary with different particular works, giving a versatile picture. It is the result of the work of many scientists in the field of cosmology, in accordance with their expertise and particular interests. Is a collection of different research papers produced by important scientists in the field of cosmology. A sample of the great deal of efforts made by the scientific community, trying to understand our universe. And it has many challenging subjects, like the possible doomsday to be confirmed by the next decade of experimentation. May be we are now half way in the life of the universe. Many more challenging subjects are not present here: they will be the result of further future work. Among them, we have the possibility of cyclic universes, and the evidence for the existence of a previous universe