Precision matrix expansion - efficient use of numerical simulations in estimating errors on cosmological parameters

Computing the inverse covariance matrix (or precision matrix) of large data vectors is crucial in weak lensing (and multi-probe) analyses of the large scale structure of the universe. Analytically computed covariances are noise-free and hence straightforward to invert, however the model approximations might be insufficient for the statistical precision of future cosmological data. Estimating covariances from numerical simulations improves on these approximations, but the sample covariance estimator is inherently noisy, which introduces uncertainties in the error bars on cosmological parameters and also additional scatter in their best fit values. For future surveys, reducing both effects to an acceptable level requires an unfeasibly large number of simulations. In this paper we describe a way to expand the true precision matrix around a covariance model and show how to estimate the leading order terms of this expansion from simulations. This is especially powerful if the covariance matrix is the sum of two contributions, $\smash{\mathbf{C} = \mathbf{A}+\mathbf{B}}$, where $\smash{\mathbf{A}}$ is well understood analytically and can be turned off in simulations (e.g. shape-noise for cosmic shear) to yield a direct estimate of $\smash{\mathbf{B}}$. We test our method in mock experiments resembling tomographic weak lensing data vectors from the Dark Energy Survey (DES) and the Large Synoptic Survey Telecope (LSST). For DES we find that $400$ N-body simulations are sufficient to achive negligible statistical uncertainties on parameter constraints. For LSST this is achieved with $2400$ simulations. The standard covariance estimator would require >$10^5$ simulations to reach a similar precision. We extend our analysis to a DES multi-probe case finding a similar performance.Comment: 14 pages, submitted to mnra

Towards generating a new supernova equation of state: A systematic analysis of cold hybrid stars

The hadron-quark phase transition in core-collapse supernovae (CCSNe) has the potential to trigger explosions in otherwise nonexploding models. However, those hybrid supernova equations of state (EOS) shown to trigger an explosion do not support the observational 2 M$_\odot$ neutron star maximum mass constraint. In this work, we analyze cold hybrid stars by the means of a systematic parameter scan for the phase transition properties, with the aim to develop a new hybrid supernova EOS. The hadronic phase is described with the state-of-the-art supernova EOS HS(DD2), and quark matter by an EOS with a constant speed of sound (CSS) of $c_{QM}^2=1/3$. We find promising cases which meet the 2 M$_\odot$ criterion and are interesting for CCSN explosions. We show that the very simple CSS EOS is transferable into the well-known thermodynamic bag model, important for future application in CCSN simulations. In the second part, the occurrence of reconfinement and multiple phase transitions is discussed. In the last part, the influence of hyperons in our parameter scan is studied. Including hyperons no change in the general behavior is found, except for overall lower maximum masses. In both cases (with and without hyperons) we find that quark matter with $c_{QM}^2=1/3$ can increase the maximum mass only if reconfinement is suppressed or if quark matter is absolutely stable.Comment: 14 pages, 11 figures, v2: matches published versio

Vom Treibhausklima der Kreidezeit zum heutigen Eishausklima : auf Spurensuche in Mikrofossilien vom Meeresboden

Wenn Klimaforscher wissen wollen, was die Zukunft bringt, schauen sie gern in die Vergangenheit. Während der Kreidezeit herrschte auf der Erde ein Treibhausklima mit atmosphärischen CO2-Gehalten, die weitaus höher waren als heute. Welche Konsequenzen das für die Meeresströmungen und die marinen Ökosysteme hatte, können Geowissenschaftler heute nicht mehr direkt messen. Bei der Spurensuche helfen ihnen die Fossilien mikroskopisch kleiner Einzeller, deren wunderschöne Kalkschalen als Klimagedächtnis dienen

Self-Organized Synchronization and Voltage Stability in Networks of Synchronous Machines

The integration of renewable energy sources in the course of the energy transition is accompanied by grid decentralization and fluctuating power feed-in characteristics. This raises new challenges for power system stability and design. We intend to investigate power system stability from the viewpoint of self-organized synchronization aspects. In this approach, the power grid is represented by a network of synchronous machines. We supplement the classical Kuramoto-like network model, which assumes constant voltages, with dynamical voltage equations, and thus obtain an extended version, that incorporates the coupled categories voltage stability and rotor angle synchronization. We compare disturbance scenarios in small systems simulated on the basis of both classical and extended model and we discuss resultant implications and possible applications to complex modern power grids.Comment: 9 pages, 9 figure

Statistical properties of the cosmic density field beyond 2-point statistics

Dunkle Materie, dunkle Energie, kosmische Inflation - unser Verständnis der drei Hauptzutaten des kosmologischen Standardmodels ist nach wie vor gering. Das deutet auf allgemeine Lücken in unserem Verständnis der Physik hin. Eine Observable mit dem Potential, diese Lücken zu schließen ist die großskalige Struktur von Dichtefluktuationen im Universum. Beobachtungen der Entwicklung dieser Struktur bei unterschiedlichen Rotverschiebungen können Aufschluss über das genaue Verhalten von dunkler Materie und dunkler Energie geben. Außerdem erlauben solche Beobachtungen Rückschlüsse über die Anfangsbedingungen des Universums, was uns Hinweise auf den genauen Mechanismus der kosmischen Inflation geben kann. Die späten Stadien in der Entwicklung der großskaligen Struktur sind besonders schwer zu analysieren. Ein Grund dafür ist, dass die Differenzialgleichungen, die das Wachs- tum von Dichtefluktuationen beschreiben, im späten Universum nicht mehr gut durch lineare Gleichungen approximiert werden können. Mit dieser Arbeit präsentiere ich in zweierlei Hinsicht Fortschritte in der Behandlung der späten Strukturbildung. Zum einen verbessere ich Techniken zur Schätzung der statistischen Unsicherheiten in Messungen von 2-Punkt Statistiken des kosmischen Dichtefeldes. Das geschieht in einem ersten Schritt, indem ich die Leistung vorhandener Methoden untersuche und verbessere. Und in einem zweiten Schritt, indem ich eine eine völlig neue Methode präsentiere, die den exorbitanten Rechenaufwand einer weit verbreiteten Prozedur umgeht. Zum zweiten entwickle ich ein theoretisches Model für eine neue kosmologische Untersuchungsmethode namens Density Split Statistics. Damit wird es ermöglicht die lokale Wahrscheinlichkeitsdichtefunktion von Fluktuationen des Materiedichtfeldes zu studieren. Das eröffnet ein reiches Spektrum an Informationen über die großskalige Struktur des Universums, die mit rein auf 2-Punkt Statistik basierenden Analysen nicht zu erhalten wäre.Dark matter, dark energy, cosmic inflation - the three main ingredients of the cosmological standard model remain poorly understood. This points to general gaps in our understanding of physics. An observable that has the potential to fill these gaps is the large scale structure of density fluctuations in the universe. Observations of the evolution of that large scale structure over a range of different redshift can be used to study the exact behaviour of dark matter and dark energy. Also, such observations can improve our understanding of the initial conditions of the universe and hence point us to the exact mechanism of inflation. The late stages in the evolution of the large scale structure are particularly difficult to understand. One reason for this is that the differential equations governing the growth of density fluctuations are not well approximated by linear equations in the late time universe. In this work I advance the treatment of late time structure formation in two ways. First, I improve techniques to estimate the statistical uncertainties in measurements of the 2- point statistics of the cosmic density field. This is done in a first step by investigating and refining the performance of existing estimators. And in a second step, by proposing an entirely new method that can bypass the exorbitant computational needs of a prominent existing procedure. Secondly, I develop a theoretical model for a new cosmological probe called density split statistics. This enables the study of the probability density function (PDF) of matter density fluctuations in the universe. This opens up a rich amount of information about the large scale structure of the universe that is otherwise missed if one only analyses 2-point statistics

Statistical properties of the cosmic density field beyond 2-point statistics

Dunkle Materie, dunkle Energie, kosmische Inflation - unser Verständnis der drei Hauptzutaten des kosmologischen Standardmodels ist nach wie vor gering. Das deutet auf allgemeine Lücken in unserem Verständnis der Physik hin. Eine Observable mit dem Potential, diese Lücken zu schließen ist die großskalige Struktur von Dichtefluktuationen im Universum. Beobachtungen der Entwicklung dieser Struktur bei unterschiedlichen Rotverschiebungen können Aufschluss über das genaue Verhalten von dunkler Materie und dunkler Energie geben. Außerdem erlauben solche Beobachtungen Rückschlüsse über die Anfangsbedingungen des Universums, was uns Hinweise auf den genauen Mechanismus der kosmischen Inflation geben kann. Die späten Stadien in der Entwicklung der großskaligen Struktur sind besonders schwer zu analysieren. Ein Grund dafür ist, dass die Differenzialgleichungen, die das Wachs- tum von Dichtefluktuationen beschreiben, im späten Universum nicht mehr gut durch lineare Gleichungen approximiert werden können. Mit dieser Arbeit präsentiere ich in zweierlei Hinsicht Fortschritte in der Behandlung der späten Strukturbildung. Zum einen verbessere ich Techniken zur Schätzung der statistischen Unsicherheiten in Messungen von 2-Punkt Statistiken des kosmischen Dichtefeldes. Das geschieht in einem ersten Schritt, indem ich die Leistung vorhandener Methoden untersuche und verbessere. Und in einem zweiten Schritt, indem ich eine eine völlig neue Methode präsentiere, die den exorbitanten Rechenaufwand einer weit verbreiteten Prozedur umgeht. Zum zweiten entwickle ich ein theoretisches Model für eine neue kosmologische Untersuchungsmethode namens Density Split Statistics. Damit wird es ermöglicht die lokale Wahrscheinlichkeitsdichtefunktion von Fluktuationen des Materiedichtfeldes zu studieren. Das eröffnet ein reiches Spektrum an Informationen über die großskalige Struktur des Universums, die mit rein auf 2-Punkt Statistik basierenden Analysen nicht zu erhalten wäre.Dark matter, dark energy, cosmic inflation - the three main ingredients of the cosmological standard model remain poorly understood. This points to general gaps in our understanding of physics. An observable that has the potential to fill these gaps is the large scale structure of density fluctuations in the universe. Observations of the evolution of that large scale structure over a range of different redshift can be used to study the exact behaviour of dark matter and dark energy. Also, such observations can improve our understanding of the initial conditions of the universe and hence point us to the exact mechanism of inflation. The late stages in the evolution of the large scale structure are particularly difficult to understand. One reason for this is that the differential equations governing the growth of density fluctuations are not well approximated by linear equations in the late time universe. In this work I advance the treatment of late time structure formation in two ways. First, I improve techniques to estimate the statistical uncertainties in measurements of the 2- point statistics of the cosmic density field. This is done in a first step by investigating and refining the performance of existing estimators. And in a second step, by proposing an entirely new method that can bypass the exorbitant computational needs of a prominent existing procedure. Secondly, I develop a theoretical model for a new cosmological probe called density split statistics. This enables the study of the probability density function (PDF) of matter density fluctuations in the universe. This opens up a rich amount of information about the large scale structure of the universe that is otherwise missed if one only analyses 2-point statistics

Emergence of Gravitational Potential and Time Dilation from Non-interacting Systems Coupled to a Global Quantum Clock

We study gravitational back-reaction within relational time formulations of quantum mechanics by considering two versions of time: a time coordinate, modelled as a global quantum degree of freedom, and the proper time of a given physical system, modelled via an internal degree of freedom serving as a local quantum "clock". We show that interactions between coordinate time and mass-energy in a global Wheeler-DeWitt-like constraint lead to gravitational time dilation. In the presence of a massive object this agrees with time dilation in a Schwarzchild metric at leading order in $G$. Furthermore, if two particles couple independently to the time coordinate we show that Newtonian gravitational interaction between those particles emerges in the low energy limit. We also observe features of renormalization of high energy divergences.Comment: Essay written for the Gravity Research Foundation's 2023 Awards for Essays on Gravitatio

Lagrangian Investigation of Two-Dimensional Decaying Turbulence

We present a numerical investigation of two-dimensional decaying turbulence in the Lagrangian framework. Focusing on single particle statistics, we investigate Lagrangian trajectories in a freely evolving turbulent velocity field. The dynamical evolution of the tracer particles is strongly dominated by the emergence and evolution of coherent structures. For a statistical analysis we focus on the Lagrangian acceleration as a central quantity. For more geometrical aspects we investigate the curvature along the trajectories. We find strong signatures for self-similar universal behavior

A multivariate extension of the Lorenz curve based on copulas and a related multivariate Gini coefficient

We propose an extension of the univariate Lorenz curve and of the Gini coefficient to the multivariate case, i.e., to simultaneously measure inequality in more than one variable. Our extensions are based on copulas and measure inequality stemming from inequality in each single variable as well as inequality stemming from the dependence structure of the variables. We derive simple nonparametric estimators for both instruments and exemplary apply them to data of individual income and wealth for various countries