Approximation of Gaussian curvature by the angular defect: an error analysis

It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases

Tilt-To-Length coupling in LISA Pathfinder: model, data analysis and take-away messages for LISA

In the mid-2030s, the laser interferometer space antenna (LISA) is planned to be launched and will be the first gravitational wave detector in space. One of the major noise sources in LISA will be tilt-to-length (TTL) coupling, i.e. the coupling of lateral and angular spacecraft motion into the interferometric readout. Likewise, TTL coupling was a significant noise in LISA’s technology demonstrator mission LISA Pathfinder (LPF), operating from March 2016 until the end of June 2017 with a performance that exceeded the expectations. During this mission, TTL coupling was the main noise contributor between 20 and 200 mHz. It was successfully subtracted via a fit model in post-processing. However, each analytical model existing at that time failed to describe the TTL coupling consistently. The lack of such a physical model limited the ability to minimise the TTL noise via realignments of the optical system a priori. Therefore, I present in this thesis, on the one hand, a detailed description of the TTL mechanisms in general cases and a derivation of the corresponding analytical equations if possible. On the other hand, I interpret these findings for the LPF case, followed by an analysis of the TTL noise during this mission. The analytical models introduced in the first part describe TTL coupling in general interferometric detectors. This analysis covers geometric and non-geometric TTL coupling contributions from lateral and angular jitter of either a mirror or a receiving system. The models can be applied to different interferometric setups modelling the found TTL noise and developing strategies for the TTL coupling suppression. The model predictions have been verified in simulation and, in the case of LPF, by the comparison to data taken during the mission. In particular, the new LPF TTL model successfully describes how the measured coupling depends on the alignment of the test masses hosted by the LPF satellite and, therefore, how they could have been realigned for an optimal TTL suppression. Also, the TTL coupling coefficients match the results from the fit used during the LPF mission in a TTL coupling experiment. Based on this data, I present how a comparable physical model can be derived using the fit results. In addition, the long-term analysis shows that TTL coupling is not stable over the entire mission duration. The individual TTL contributors are affected by a small distortion of the optical bench and its components, mainly due to temperature changes. However, the TTL coupling provides an additional measure for the actual alignment of the full optical system. In summary, I demonstrate within this thesis that modelling TTL noise in space interferometers while complex is possible. Likewise, this result boosts our confidence that the suppression techniques planned for LISA will enable the successful gravitational wave measurement

Tilt-to-length coupling in LISA Pathfinder: Analytical modeling

Tilt-to-length coupling was the limiting noise source in LISA Pathfinder between 20 and 200 mHz before subtraction in postprocessing. To prevent the adding of sensing noise to the data by the subtraction process, the success of this strategy depended on a previous direct noise reduction by test mass alignment. The exact dependency of the level of tilt-to-length coupling on the set points of LISA Pathfinder's test masses was not understood until the end of the mission. Here, we present, for the first time, an analytical tilt-to-length coupling model that describes the coupling noise changes due to the realignments. We report on the different mechanisms, namely the lever arm and piston effect as well as the coupling due to transmissive components, and how they contribute to the full coupling. Further, we show that a pure geometric model would not have been sufficient to describe the coupling in LISA Pathfinder. Therefore, we model also the nongeometric tilt-to-length noise contributions. For the resulting coupling coefficients of the full model, we compute the expected error bars based on the known individual error sources. Also, we validated the analytical model against numerical simulations. A detailed study and thorough understanding of this noise are the basis for a successful analysis of the LISA Pathfinder data with respect to tilt-to-length coupling

LION: laser interferometer on the moon

Gravitational wave astronomy has now left its infancy and has become an important tool for probing the most violent phenomena in our Universe. The LIGO/Virgo-KAGRA collaboration operates ground based detectors which cover the frequency band from 10 Hz to the kHz regime. Meanwhile, the pulsar timing array and the soon to launch LISA mission will cover frequencies below 0.1 Hz, leaving a gap in detectable gravitational wave frequencies. Here we show how a laser interferometer on the moon (LION) gravitational wave detector would be sensitive to frequencies from sub Hz to kHz. We find that the sensitivity curve is such that LION can measure compact binaries with masses between 10 and 100M ⊙ at cosmological distances, with redshifts as high as z = 100 and beyond, depending on the spin and the mass ratio of the binaries. LION can detect binaries of compact objects with higher-masses, with very large signal-to-noise ratios (SNRs), help us to understand how supermassive black holes got their colossal masses on the cosmological landscape, and it can observe in detail intermediate-mass ratio inspirals at distances as large as at least 100 Gpc. Compact binaries that never reach the LIGO/Virgo sensitivity band can spend significant amounts of time in the LION band, while sources present in the LISA band can be picked up by the detector and observed until their final merger. Since LION covers the deci-Hertz regime with such large SNRs, it truly achieves the dream of multi messenger astronomy.DFG, 239994235, SFB 1128: Relativistische Geodäsie und Gravimetrie mit Quantensensoren - Modellierung, Geo-Metrologie und zukünftige Technologie (geo-Q)BMWi, 50OQ1801, LISA Phase A und Vorbereitung der Phase B

Approximation of Gaussian Curvature by the Angular Defect: An Error Analysis

It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases

Tilt-to-length coupling in LISA Pathfinder: analytical modelling

Tilt-to-length coupling was the limiting noise source in LISA Pathfinder between 20 and 200 mHz before subtraction in post-processing. To prevent the adding of sensing noise to the data by the subtraction process, the success of this strategy depended on a previous direct noise reduction by test mass alignment. The exact dependency of the level of tilt-to-length coupling on the set-points of LISA Pathfinder's test masses was not understood until the end of the mission. Here, we present, for the first time, an analytical tilt-to-length coupling model that describes the coupling noise changes due to the realignments. We report on the different mechanisms, namely the lever arm and piston effect as well as the coupling due to transmissive components, and how they contribute to the full coupling. Further, we show that a pure geometric model would not have been sufficient to describe the coupling in LISA Pathfinder. Therefore, we model also the non-geometric tilt-to-length noise contributions. For the resulting coupling coefficients of the full model, we compute the expected error bars based on the known individual error sources. Also, we validated the analytical model against numerical simulations. A detailed study and thorough understanding of this noise are the basis for a successful analysis of the LISA Pathfinder data with respect to tilt-to-length coupling

Geometric tilt-to-length coupling in precision interferometry: mechanisms and analytical descriptions

Tilt-to-length coupling is a technical term for the cross-coupling of angular or lateral jitter into an interferometric phase signal. It is an important noise source in precision interferometers and originates either from changes in the optical path lengths or from wavefront and clipping effects. Within this paper, we focus on geometric TTL coupling and categorize it into a number of different mechanisms for which we give analytic expressions. We then show that this geometric description is not always sufficient to predict the TTL coupling noise within an interferometer. We, therefore, discuss how understanding the geometric effects allows TTL noise reduction already by smart design choices. Additionally, they can be used to counteract the total measured TTL noise in a system. The presented content applies to a large variety of precision interferometers, including space gravitational wave detectors like LISA

Where Brain, Body and World Collide

The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pt < 8 Gev/c with the ALICE experiment at the CERN LHC in pp collisions at a center of mass energy sqrt{s} = 7 TeV using an integrated luminosity of 2.2 nb^{-1}. Electrons from beauty hadron decays were selected based on the displacement of the decay vertex from the collision vertex. A perturbative QCD calculation agrees with the measurement within uncertainties. The data were extrapolated to the full phase space to determine the total cross section for the production of beauty quark-antiquark pairs