26 research outputs found
Complex evaluation of angular power spectra: Going beyond the Limber approximation
Angular power spectra are central to the study of our Universe. In this
paper, I develop a new method for the numeric evaluation and analytic
estimation of the angular cross-power spectrum of two random fields using
complex analysis and Picard- Lefschetz theory. The proposed continuous
deformation of the integration domain resums the highly oscillatory integral
into a convex integral whose integrand decays exponentially. This deformed
integral can be quickly evaluated with conventional integration techniques.
These methods can be used to quickly evaluate and estimate the angular power
spectrum from the three-dimensional power spectrum for all angles (or multipole
moments). This method is especially useful for narrow redshift bins, or samples
with small redshift overlap, for which the Limber approximation has a large
error
Multi-plane lensing in wave optics
Wave effects in lensing form a rich phenomenon at the intersection of
classical caustic singularities and quantum interference, yet are notoriously
difficult to model. A large number of recently observed pulsars and fast radio
bursts in radio astronomy and the prospected increase in sensitivity of
gravitational wave detectors suggest that wave effects will likely be observed
in the near future. The interference fringes are sensitive to physical
parameters which cannot be inferred from geometric optics. In particular, for
multi-plane lensing, the pattern depends on the redshifts of the lens planes. I
present a new method to define and efficiently evaluate multi-plane lensing of
coherent electromagnetic waves by plasmas and gravitational lenses in
polynomial time. This method will allow the use of radio and gravitational wave
sources to probe our universe in novel ways.Comment: 6 pages, 8 figure
Caustic Skeleton & Cosmic Web
We present a general formalism for identifying the caustic structure of an
evolving mass distribution in an arbitrary dimensional space. For the class of
Hamiltonian fluids the identification corresponds to the classification of
singularities in Lagrangian catastrophe theory. Based on this we develop a
theoretical framework for the formation of the cosmic web, and specifically
those aspects that characterize its unique nature: its complex topological
connectivity and multiscale spinal structure of sheetlike membranes, elongated
filaments and compact cluster nodes. The present work represents an extension
of the work by Arnol'd et al., who classified the caustics for the 1- and
2-dimensional Zel'dovich approximation. His seminal work established the role
of emerging singularities in the formation of nonlinear structures in the
universe. At the transition from the linear to nonlinear structure evolution,
the first complex features emerge at locations where different fluid elements
cross to establish multistream regions. The classification and characterization
of these mass element foldings can be encapsulated in caustic conditions on the
eigenvalue and eigenvector fields of the deformation tensor field. We introduce
an alternative and transparent proof for Lagrangian catastrophe theory, and
derive the caustic conditions for general Lagrangian fluids, with arbitrary
dynamics, including dissipative terms and vorticity. The new proof allows us to
describe the full 3-dimensional complexity of the gravitationally evolving
cosmic matter field. One of our key findings is the significance of the
eigenvector field of the deformation field for outlining the spatial structure
of the caustic skeleton. We consider the caustic conditions for the
3-dimensional Zel'dovich approximation, extending earlier work on those for 1-
and 2-dimensional fluids towards the full spatial richness of the cosmic web
Path Integrals in the Sky: Classical and Quantum Problems with Minimal Assumptions
Cosmology has, after the formulation of general relativity, been transformed from a branch of philosophy into an active field in physics. Notwithstanding the significant improvements in our understanding of our Universe, there are still many open questions on both its early and late time evolution. In this thesis, we investigate a range of problems in classical and quantum cosmology, using advanced mathematical tools, and making only minimal assumptions. In particular, we apply Picard-Lefschetz theory, catastrophe theory, infinite dimensional measure theory, and weak-value theory.
To study the beginning of the Universe in quantum cosmology, we apply Picard-Lefschetz theory to the Lorentzian path integral for gravity. We analyze both the Hartle-Hawking no-boundary proposal and Vilenkin's tunneling proposal, and demonstrate that the Lorentzian path integral corresponding to the mini-superspace formulation of the two proposals is well-defined. However, when including fluctuations, we show that the path integral predicts the existence of large fluctuations. This indicates that the Universe cannot have had a smooth beginning in Euclidean de Sitter space. In response to these conclusions, the scientific community has made a series of adapted formulations of the no-boundary and tunneling proposals. We show that these new proposals suffer from similar issues.
Second, we generalize the weak-value interpretation of quantum mechanics to relativistic systems. We apply this formalism to a relativistic quantum particle in a constant electric field. We analyze the evolution of the relativistic particle in both the classical and the quantum regime and evaluate the back-reaction of the Schwinger effect on the electric field in -dimensional spacetime, using analytical methods. In addition, we develop a numerical method to evaluate both the wavefunction and the corresponding weak-values in more general electric and magnetic fields.
We conclude the quantum part of this thesis with a chapter on Lorentzian path integrals. We propose a new definition of the real-time path integral in terms of Brownian motion on the Lefschetz thimble of the theory. We prove the existence of a -measure for the path integral of the non-relativistic free particle, the (inverted) harmonic oscillator, and the relativistic particle in a range of potentials. We also describe how this proposal extends to more general path integrals.
In the classical part of this thesis, we analyze two problems in late-time cosmology. Multi-dimensional oscillatory integrals are prevalent in physics, but notoriously difficult to evaluate. We develop a new numerical method, based on multi-dimensional Picard-Lefschetz theory, for the evaluation of these integrals. The virtue of this method is that its efficiency increases when integrals become more oscillatory. The method is applied to interference patterns of lensed images near caustics described by catastrophe theory. This analysis can help us understand the lensing of astrophysical sources by plasma lenses, which is especially relevant in light of the proposed lensing mechanism for fast radio bursts.
Finally, we analyze large-scale structure formation in terms of catastrophe theory. We show that the geometric structure of the three-dimensional cosmic-web is determined by both the eigenvalue and the eigenvector fields of the deformation tensor. We formulate caustic conditions, classifying caustics using properties of these fields. When applied to the Zel'dovich approximation of structure formation, the caustic conditions enable us to construct a caustic skeleton of the three-dimensional cosmic-web in terms of the initial conditions
Orthogonality relations for conical functions of imaginary order
Orthogonality relations for conical or Mehler functions of imaginary order
are derived and expressed in terms of the Dirac delta function. This work
extends recently derived orthogonality relations of associated Legendre
functions
Cosmic Web & Caustic Skeleton: non-linear Constrained Realizations - 2D case studies
The cosmic web consists of a complex configuration of voids, walls, filaments, and clusters, which formed under the gravitational collapse of Gaussian fluctuations. Understanding under what conditions these different structures emerge from simple initial conditions, and how different cosmological models influence their evolution, is central to the study of the large-scale structure. Here, we present a general formalism for setting up initial random density and velocity fields satisfying non-linear constraints for specialized N-body simulations. These allow us to link the non-linear conditions on the eigenvalue and eigenvector fields of the deformation tensor, as specified by caustic skeleton theory, to the current-day cosmic web. By extending constrained Gaussian random field theory, and the corresponding Hoffman-Ribak algorithm, to non-linear constraints, we probe the statistical properties of the progenitors of the walls, filaments, and clusters of the cosmic web. Applied to cosmological N-body simulations, the proposed techniques pave the way towards a systematic investigation of the evolution of the progenitors of the present-day walls, filaments, and clusters, and the embedded galaxies, putting flesh on the bones of the caustic skeleton. The developed non-linear constrained random field theory is valid for generic cosmological conditions. For ease of visualization, the case study presented here probes the two-dimensional caustic skeleton.</p
Complex classical paths in quantum reflections and tunneling
The real-time propagator of the symmetric Rosen-Morse, also known as the
symmetric modified P\"oschl-Teller, barrier is expressed in the
Picard-Lefschetz path integral formalism using real and complex classical
paths. We explain how the interference pattern in the real-time propagator and
energy propagator is organized by caustics and Stoke's phenomena, and list the
relevant real and complex classical paths as a function of the initial and
final position. We discover the occurrence of singularity crossings, where the
analytic continuation of the complex classical path no longer satisfies the
boundary value problem and needs to be analytically continued. Moreover, we
demonstrate how these singularity crossings play a central role in the
real-time description of quantum tunneling