Scalar-tensor mode mixing in higher-order cosmological perturbations

openThe theory of cosmological perturbations has become a very important subject of modern Cosmology because it allows to link the models of the very early Universe, such as the inflationary scenario, with the massive high-precision data on the Cosmic Mi- crowave Background radiation, on large-scale structures and future data on the primordial Gravitational-Wave stochastic background. When working with perturbations within General Relativity a difficulty arises: we have After an introduction, I start recovering the first-order results: scalar and tensor perturbations evolve independently, thus I can easily study and write Einstein’s equations for scalars and then for tensors. First order vectors are neglected due to the fact that, if generated, they are fast redshifted away with the expansion of the Universe. So, the equations governing the evolution of these quantities are obtained, recovering the results in the literature. When one goes to second order, computations start to be more complex, revealing the underlying non-linearity of Einstein’s equations. For the first time, the second-order perturbed metric is obtained directly in the Poisson gauge, with scalars at first and second order, vectors at second order, tensors at first and second order. With this choice, I show that we have second-order mixed terms which source scalar and tensor perturbations. Namely, second-order scalar modes are sourced by first-order scalars coupled with first-order tensors, by two coupled first-order scalar and two first- order tensors. The same problem has been discussed in other papers but in a different gauge. Now, Einstein’s equations start to be more complicated and finding solutions is not so easy. Starting from computing the second-order perturbed quantities such as the Ricci tensor and the Ricci scalar, the second order energy-momentum tensor, from the traceless i − j component of the Einstein field equation I can find the difference between the second order scalar perturbations, which as a first result is non zero even for a perfect fluid in absence of an anisotropic stress tensor, in contrast with the first order result, which gives ψ = φ for a perfect fluid. With this equation in hand, I derived the other Einstein equations and the continuity equation to close the system for the scalar quantities. The following part is focused on writing the Boltzmann equations at second order in order to study the evolution of particle species, such as photons, baryons and cold dark matter. In this section I write the Boltzmann equation accounting for first and second order scalar perturbation, second order vector perturbation, first and second order tensor perturbation. The same formalism can be later applied to derive the Boltzmann equation for other particle species, such as baryons and cold dark matter (CDM). Studying the evolution of cold dark matter is a very important topic because it plays a fundamental role in structure formation. In conclusion the main goal of this project is to add an original contribution to the second-order evolution of scalar quantities, in Poisson gauge, such as the gravitational potentials, the density contrast and the velocity of baryons and CDM, starting from the perturbed expression of the metric through the Einstein and Boltzmann equations, con- sidering as non-negligible the contributions from first-order tensor modes which can be coupled to themselves and to other first-order scalar modesThe theory of cosmological perturbations has become a very important subject of modern Cosmology because it allows to link the models of the very early Universe, such as the inflationary scenario, with the massive high-precision data on the Cosmic Mi- crowave Background radiation, on large-scale structures and future data on the primordial Gravitational-Wave stochastic background. When working with perturbations within General Relativity a difficulty arises: we have After an introduction, I start recovering the first-order results: scalar and tensor perturbations evolve independently, thus I can easily study and write Einstein’s equations for scalars and then for tensors. First order vectors are neglected due to the fact that, if generated, they are fast redshifted away with the expansion of the Universe. So, the equations governing the evolution of these quantities are obtained, recovering the results in the literature. When one goes to second order, computations start to be more complex, revealing the underlying non-linearity of Einstein’s equations. For the first time, the second-order perturbed metric is obtained directly in the Poisson gauge, with scalars at first and second order, vectors at second order, tensors at first and second order. With this choice, I show that we have second-order mixed terms which source scalar and tensor perturbations. Namely, second-order scalar modes are sourced by first-order scalars coupled with first-order tensors, by two coupled first-order scalar and two first- order tensors. The same problem has been discussed in other papers but in a different gauge. Now, Einstein’s equations start to be more complicated and finding solutions is not so easy. Starting from computing the second-order perturbed quantities such as the Ricci tensor and the Ricci scalar, the second order energy-momentum tensor, from the traceless i − j component of the Einstein field equation I can find the difference between the second order scalar perturbations, which as a first result is non zero even for a perfect fluid in absence of an anisotropic stress tensor, in contrast with the first order result, which gives ψ = φ for a perfect fluid. With this equation in hand, I derived the other Einstein equations and the continuity equation to close the system for the scalar quantities. The following part is focused on writing the Boltzmann equations at second order in order to study the evolution of particle species, such as photons, baryons and cold dark matter. In this section I write the Boltzmann equation accounting for first and second order scalar perturbation, second order vector perturbation, first and second order tensor perturbation. The same formalism can be later applied to derive the Boltzmann equation for other particle species, such as baryons and cold dark matter (CDM). Studying the evolution of cold dark matter is a very important topic because it plays a fundamental role in structure formation. In conclusion the main goal of this project is to add an original contribution to the second-order evolution of scalar quantities, in Poisson gauge, such as the gravitational potentials, the density contrast and the velocity of baryons and CDM, starting from the perturbed expression of the metric through the Einstein and Boltzmann equations, con- sidering as non-negligible the contributions from first-order tensor modes which can be coupled to themselves and to other first-order scalar mode

A certified RB method for PDE-constrained parametric optimization problems

Abstract We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an "optimize-then-reduce" approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows

Non equilibrium optical properties in semiconductors from first--principles: a combined theoretical and experimental study of bulk silicon

The calculation of the equilibrium optical properties of bulk silicon by using the Bethe--Salpeter equation solved in the Kohn--Sham basis represents a cornerstone in the development of an ab--initio approach to the optical and electronic properties of materials. Nevertheless calculations of the {\em transient} optical spectrum using the same efficient and successful scheme are scarce. We report, here, a joint theoretical and experimental study of the transient reflectivity spectrum of bulk silicon. Femtosecond transient reflectivity is compared to a parameter--free calculation based on the non--equilibrium Bethe--Salpeter equation. By providing an accurate description of the experimental results we disclose the different phenomena that determine the transient optical response of a semiconductor. We give a parameter--free interpretation of concepts like bleaching, photo--induced absorption and stimulated emission, beyond the Fermi golden rule. We also introduce the concept of optical gap renormalization, as a generalization of the known mechanism of band gap renormalization. The present scheme successfully describes the case of bulk silicon, showing its universality and accuracy.Comment: 14 pages, 13 figure

Optimal plant water use strategies explain soil moisture variability

Plant responses to water stress influence water and carbon cycles and can lead to feedbacks on climate yet characterizing these responses at ecosystem levels remains uncertain. Quantifying ecosystem-level water use strategies is complex due to challenges of upscaling plant traits and disentangling confounding environmental factors, ultimately limiting our ability to understand and anticipate global change in ecosystem dynamics and ecohydrological fluxes. We reduce the dimensionality of this problem and quantify plant water use strategies by combining plant traits with soil and climate variables into parameter groups that synthesize key eco-physiological tradeoffs. Using a parsimonious soil water balance framework, we explore variations in plant water uptake capacity, water stress responses, and water use performance via these non-dimensional parameter groups. The group characterizing the synchronization of plant water transport and atmospheric water demand emerges as the primary axis of variation in water use strategies and interacts with the group representing plant hydraulic risk tolerance, especially in arid conditions when plant water transport is limiting. Next, we show that specific plant water use strategies maximize plant water uptake (leading to carbon gain benefits) weighted by risks of water stress (leading to higher costs of water use). A model-data comparison demonstrates that these ecohydrologically optimal parameter groups capture observed soil moisture variability in 40 ecosystems and beyond aridity, rainfall frequency is an important environmental control for plant water use strategies. The emerging parsimonious link between ecohydrological performance and non-dimensional parameters provides a tractable representation of plant water use strategies, relevant to parameterize global models while accounting for ecological and evolutionary constraints on the water cycle

The coordination of green-brown food webs and their disruption by anthropogenic nutrient inputs

Aim Our goal was to quantify nitrogen flows and stocks in green-brown food webs in different ecosystems, how they differ across ecosystems and how they respond to nutrient enrichment. Location Global. Time period Contemporary. Major taxa studied Plants, phytoplankton, macroalgae, invertebrates, vertebrates and zooplankton. Methods Data from >500 studies were combined to estimate nitrogen stocks and fluxes in green-brown food webs in forests, grasslands, brackish environments, seagrass meadows, lakes and oceans. We compared the stocks, fluxes and metabolic rates of different functional groups within each food web. We also used these estimates to build a dynamical model to test the response of the ecosystems to nutrient enrichment. Results We found surprising symmetries between the green and brown channels across ecosystems, in their stocks, fluxes and consumption coefficients and mortality rates. We also found that nitrogen enrichment, either organic or inorganic, can disrupt this balance between the green and brown channels. Main conclusions Linking green and brown food webs reveals a previously hidden symmetry between herbivory and detritivory, which appears to be a widespread property of natural ecosystems but can be disrupted by anthropogenic nitrogen additions

Damping behavior of 316L lattice structures produced by Selective Laser Melting

Selective Laser Melting is a powder-bed additive manufacturing technology that allows producing fully-dense metal objects with complex shapes and high mechanical properties. In this work, Selective Laser Melting was used to produce 316L specimens including lattice structures with the aim of exploring the possibility given by additive manufacturing technologies to produce parts with increased damping capacity, especially in relation to their weight. The internal friction of bulk and lattice specimens was measured in terms of delay between stress and deformation (i.e. tanδ) for different applied loads and frequencies. A finite element model was used to design the test and microstructure investigations were performed to support the results obtained by dynamo-mechanical tests. Keywords: Additive manufacturing, Selective Laser Melting, 316L, Lattice structure, Damping, Internal frictio

Enhanced vibration damping by means of a negative capacitance

The use of shunted piezoelectric transducers to damp mechanical vibrations is an interesting approach thanks to its low cost and the light weight of the actuators used. Among the different ways to build the shunt impedance, the use of negative capacitances is very attractive because it allows for high damping performances with low power required by the control system. Negative capacitances do not exist as physical components but they can be designed and built using circuits based on operational amplifiers. The use of shunt circuits based on a negative capacitance coupled to a resistance allows to have a broadband control. This paper explains how to increase the bandwidth of this controller by adding to such a shunt circuit an inductance. The dynamics of the controlled system is solved analytically and the reason why the introduction of the inductance is able to give the mentioned improvement is made clear also using numerical simulations. Furthermore, this improvement also allows to increase the attenuation performance in a certain frequency band. The conditions necessary to assure the stability of the electro-mechanical system are found and explained

Interactions of nutrient and water availability control growth and diversity effects in a Salix two-species mixture

Plant growth is constrained by resource availability and interactions among limiting resources—abundance in one resource (e.g., nutrients) might promote growth, thereby causing the depletion of other resources (e.g., water), potentially inducing stress or mortality. In a diverse plant community, complementary resource use has been hypothesized to increase the overall productivity, but how diversity effects vary with interacting water and nutrient limitation and through time is not known. Here, we address this knowledge gap in a controlled pot experiment where species composition (two Salix species in monoculture or mixture), nutrient addition, and watering frequency (for fixed total water inputs) were varied during two growing seasons. High nutrient availability promoted plant growth and nitrogen accumulation at the pot scale, as well as increased allocation aboveground, but also triggered more intense water stress and mortality, as larger plants depleted soil water during warm periods. Supplying water more frequently slightly alleviated water stress under high nutrient availability, thus promoting growth and nitrogen accumulation. The species mixtures performed better than the average of the mixture constituents (positive net diversity effects) and increasingly so through time. The complementarity and selection effects, respectively, increased and decreased under both high nutrient availability and high watering frequency. Overall, these results suggest that as plants grow larger, plant interactions and resource partitioning intensify, causing the positive diversity effects, but also that drought consequences might be exacerbated in plant communities rapidly growing thanks to high nutrient supply

Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains

In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model