This document discusses backreaction in the standard cosmological model. It notes that while the universe appears homogeneous and isotropic on large scales, structure formation leads to inhomogeneities on small scales that may affect the background dynamics. There are challenges in properly averaging quantities over domains to account for these effects. Different approaches to studying backreaction effects include analyzing the variance of local expansion rates and using correlation tensors. It is unclear whether any backreaction corrections would be equivalent to dark energy or distinguishable from it. More work is needed to fully compute and model backreaction quantitatively.
Large scale coherent structures and turbulence in quasi-2D hydrodynamic modelsColm Connaughton
This document discusses turbulence in two-dimensional systems and the inverse energy cascade phenomenon. It begins with an overview of turbulence in 3D and 2D, describing the inverse energy cascade in 2D systems whereby energy is transferred to larger scales rather than smaller scales. It then discusses how finite size effects can generate large-scale coherent structures by blocking the inverse cascade. The document concludes by noting that extracting coherent flow from turbulent fluctuations is challenging and that diagnostics like the third-order structure function may not be reliable indicators of the energy cascade direction due to the presence of coherent structures.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
Decomposition and Denoising for moment sequences using convex optimizationBadri Narayan Bhaskar
This document summarizes research on using convex optimization techniques like atomic norm minimization to solve problems involving decomposing signals into sparse representations using atoms from predefined dictionaries. It discusses how atomic norm regularization provides a unified framework for problems like sparse recovery, low-rank matrix recovery, and line spectral estimation. It presents theoretical guarantees on exact recovery and convergence rates for atomic norm denoising and shows how to implement it using alternating direction methods and semidefinite programming. Experimental results demonstrate state-of-the-art performance of atomic norm techniques on line spectral estimation tasks.
This document provides an introduction to the concept of wave turbulence. It discusses how waves interact nonlinearly at finite amplitudes to produce a statistical, non-equilibrium dynamics. Key points:
- Wave turbulence involves dispersive waves that are excited and damped by external processes, leading to interactions between many degrees of freedom.
- The nonlinear interactions can be modeled using a Hamiltonian approach by including higher-order terms that couple different Fourier modes.
- A central goal is developing a statistical description of the system using correlation functions and obtaining a closed kinetic equation for the wave spectrum.
- In the weak turbulence regime, this kinetic equation can be solved perturbatively to obtain scaling laws for the wave spectrum in both physical and
3D gravity inversion by planting anomalous densitiesLeonardo Uieda
Paper presented at the 2011 SBGf International Congress in Rio de Janeiro, Brazil.
Abstract:
This paper presents a novel gravity inversion method for estimating a 3D density-contrast distribution defined on a grid of prisms. Our method consists of an iterative algorithm that does not require the solution of a large equation system. Instead, the solution
grows systematically around user-specified prismatic
elements called “seeds”. Each seed can have a different density contrast, allowing the interpretation of multiple bodies with different density contrasts and interfering gravitational effects. The compactness of the solution around the seeds is imposed by
means of a regularizing function. The solution
grows by the accretion of neighboring prisms of the
current solution. The prisms for the accretion are chosen by systematically searching the set of current neighboring prisms. Therefore, this approach allows that the columns of the Jacobian matrix be calculated on demand. This is a known technique from computer science called “lazy evaluation”, which greatly reduces the demand of computer memory and processing time. Test on synthetic data and on real data collected over the ultramafic Cana Brava complex, central Brazil, confirmed the ability of our method in detecting sharp
and compact bodies.
The renyi entropy and the uncertainty relations in quantum mechanicswtyru1989
The document discusses uncertainty relations in quantum mechanics using information-theoretic measures like Rènyi entropy. It summarizes that:
1) Standard deviations are not always good measures of uncertainty, as shown by examples where distributions have multiple "humps".
2) Shannon entropy provides a better measure of uncertainty as the average number of questions needed to obtain information.
3) Uncertainty relations can be formulated using Rènyi entropy, a generalization of Shannon entropy, in terms of probabilities from "histograms" of position and momentum measurements.
4) Mathematical inequalities are used to derive an uncertainty relation between the Rènyi entropies of position and momentum that reduces to the known
Primer for ordinary differential equationsTarun Gehlot
This document provides an introduction and overview of ordinary differential equations (ODEs). It defines an ODE, distinguishes ODEs from partial differential equations, and outlines techniques for solving linear ODEs with constant coefficients using classical and Laplace transform methods. The document also discusses how engineers formulate and approach solving ODEs based on physical systems and provides examples of solving first and second order linear ODEs.
Large scale coherent structures and turbulence in quasi-2D hydrodynamic modelsColm Connaughton
This document discusses turbulence in two-dimensional systems and the inverse energy cascade phenomenon. It begins with an overview of turbulence in 3D and 2D, describing the inverse energy cascade in 2D systems whereby energy is transferred to larger scales rather than smaller scales. It then discusses how finite size effects can generate large-scale coherent structures by blocking the inverse cascade. The document concludes by noting that extracting coherent flow from turbulent fluctuations is challenging and that diagnostics like the third-order structure function may not be reliable indicators of the energy cascade direction due to the presence of coherent structures.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
Decomposition and Denoising for moment sequences using convex optimizationBadri Narayan Bhaskar
This document summarizes research on using convex optimization techniques like atomic norm minimization to solve problems involving decomposing signals into sparse representations using atoms from predefined dictionaries. It discusses how atomic norm regularization provides a unified framework for problems like sparse recovery, low-rank matrix recovery, and line spectral estimation. It presents theoretical guarantees on exact recovery and convergence rates for atomic norm denoising and shows how to implement it using alternating direction methods and semidefinite programming. Experimental results demonstrate state-of-the-art performance of atomic norm techniques on line spectral estimation tasks.
This document provides an introduction to the concept of wave turbulence. It discusses how waves interact nonlinearly at finite amplitudes to produce a statistical, non-equilibrium dynamics. Key points:
- Wave turbulence involves dispersive waves that are excited and damped by external processes, leading to interactions between many degrees of freedom.
- The nonlinear interactions can be modeled using a Hamiltonian approach by including higher-order terms that couple different Fourier modes.
- A central goal is developing a statistical description of the system using correlation functions and obtaining a closed kinetic equation for the wave spectrum.
- In the weak turbulence regime, this kinetic equation can be solved perturbatively to obtain scaling laws for the wave spectrum in both physical and
3D gravity inversion by planting anomalous densitiesLeonardo Uieda
Paper presented at the 2011 SBGf International Congress in Rio de Janeiro, Brazil.
Abstract:
This paper presents a novel gravity inversion method for estimating a 3D density-contrast distribution defined on a grid of prisms. Our method consists of an iterative algorithm that does not require the solution of a large equation system. Instead, the solution
grows systematically around user-specified prismatic
elements called “seeds”. Each seed can have a different density contrast, allowing the interpretation of multiple bodies with different density contrasts and interfering gravitational effects. The compactness of the solution around the seeds is imposed by
means of a regularizing function. The solution
grows by the accretion of neighboring prisms of the
current solution. The prisms for the accretion are chosen by systematically searching the set of current neighboring prisms. Therefore, this approach allows that the columns of the Jacobian matrix be calculated on demand. This is a known technique from computer science called “lazy evaluation”, which greatly reduces the demand of computer memory and processing time. Test on synthetic data and on real data collected over the ultramafic Cana Brava complex, central Brazil, confirmed the ability of our method in detecting sharp
and compact bodies.
The renyi entropy and the uncertainty relations in quantum mechanicswtyru1989
The document discusses uncertainty relations in quantum mechanics using information-theoretic measures like Rènyi entropy. It summarizes that:
1) Standard deviations are not always good measures of uncertainty, as shown by examples where distributions have multiple "humps".
2) Shannon entropy provides a better measure of uncertainty as the average number of questions needed to obtain information.
3) Uncertainty relations can be formulated using Rènyi entropy, a generalization of Shannon entropy, in terms of probabilities from "histograms" of position and momentum measurements.
4) Mathematical inequalities are used to derive an uncertainty relation between the Rènyi entropies of position and momentum that reduces to the known
Primer for ordinary differential equationsTarun Gehlot
This document provides an introduction and overview of ordinary differential equations (ODEs). It defines an ODE, distinguishes ODEs from partial differential equations, and outlines techniques for solving linear ODEs with constant coefficients using classical and Laplace transform methods. The document also discusses how engineers formulate and approach solving ODEs based on physical systems and provides examples of solving first and second order linear ODEs.
This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include:
- Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades.
- Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments.
- Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales.
- The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.
Fractal dimensions of 2d quantum gravityTimothy Budd
After introducing 2d quantum gravity, both in its discretized form in
terms of random triangulations and its continuum description as
Quantum Liouville theory, I will give a (non-exhaustive) review of the
current understanding of its fractal dimensions. In particular, I will
discuss recent analytic and numerical results relating to the
Hausdorff dimension and spectral dimension of 2d gravity coupled to
conformal matter fields.
This document outlines a presentation on formulating QCD coupled with QED (quantum electrodynamics) on the lattice for the purpose of studying isospin breaking effects. It discusses challenges in putting QED on the lattice due to the zero charge constraint with periodic boundary conditions. Several proposed approaches are mentioned, including QEDL, twist averaging, massive QED, and using charge conjugation boundary conditions. The document contains sections on isospin in QCD, challenges of QED on the lattice, previous QED+QCD simulations, and proposed new approaches.
Current knowledge of the transversity quark distribution function, or how transversely polarized quarks are distributed in a transversely polarized proton?
Virtual Nodes: Rethinking Topology in CassandraEric Evans
The document discusses Cassandra's topology and how it is moving from a single token per node model to a virtual node model where each node is assigned multiple tokens. This improves load balancing and data distribution in the cluster. Specifically, it addresses problems with the single token approach like poor load distribution when nodes fail and inefficient data movement when adding or replacing nodes. The virtual node model with random token assignment provides better scaling properties as the number of nodes and data size increases.
Recovering vital physiological signals from ambulatory devicesPraveen Pankajakshan
This document proposes a Bayesian framework to recover physiological signals from ambulatory devices using sparse regularization. It suggests minimizing l2-l2 cost functions on devices for initial processing and l2-TV cost functions on servers for improved accuracy. The approach models signal gradients as sparse and estimates signals iteratively using majorization-minimization. It applies these techniques to recover electrocardiogram signals from ambulatory recordings.
This document discusses Born reciprocity in string theory and how it relates to the nature of spacetime. It argues that while string theory is formulated in terms of maps into spacetime, this breaks Born reciprocity. The document suggests that quasi-periodicity of strings without assuming a periodic target spacetime better respects Born reciprocity. This leads to a phase space formulation of string theory without assuming locality of spacetime at short distances.
The document summarizes key points about equality constrained minimization problems and Newton's method for solving them. It discusses:
1) Equality constrained minimization problems and their equivalent forms via eliminating constraints or using the dual problem.
2) Newton's method extended to include equality constraints, where the Newton step is defined to satisfy the linearized optimality conditions and ensures feasible descent.
3) An infeasible start Newton method that computes steps to reduce the primal-dual residual norm, ensuring iterates become feasible within a finite number of steps.
Nonequilibrium statistical mechanics of cluster-cluster aggregation, School o...Colm Connaughton
Colm Connaughton presented on nonequilibrium statistical mechanics models of cluster-cluster aggregation. He discussed simple models where particles move randomly and merge upon contact. More sophisticated models track the size distribution of clusters as they aggregate. The Smoluchowski equation describes this process. For certain collision kernels, clusters of arbitrarily large size can form in finite time, known as gelation. While some kernels mathematically describe instantaneous gelation, physical models avoid this with a cluster size cutoff. Stationary states can be reached with a particle source.
Neural network precept diagnosis on petrochemical pipelines for quality maint...Alexander Decker
This document describes a proposed neural network model for predicting degradation in petrochemical pipelines. It begins with background on pipelines and fatigue crack propagation based on Paris' law. It then discusses stresses in cylindrical pipelines under internal pressure. The model represents crack growth as a function of stress intensity factor and uses a recurrent formula to calculate cumulative damage over time. The goal is to develop a prognostic tool for quality maintenance in pipeline systems.
This document reviews research on the convergence of perturbation series in quantum field theory. It discusses Dyson's argument that perturbation series in quantum electrodynamics (QED) have zero radius of convergence due to vacuum instability when the coupling constant is negative. Large-order estimates show that perturbation series coefficients grow factorially fast in quantum mechanics and field theories. Finally, it describes the method of Borel summation, which may allow extracting the exact physical quantity from a divergent perturbation series through a unique mapping.
Conformal Field Theory and the Holographic S-Matrixliam613
This document discusses how conformal field theories (CFTs) can describe gravitational scattering and provide an effective field theory (EFT) description of gravity in anti-de Sitter space (AdS). It introduces CFTs and issues with describing gravity at high energies. It then explains how the holographic duality between CFTs and gravity theories can be used to calculate scattering matrices and understand gravitational dynamics. In particular, it outlines how calculations in Mellin space allow CFT correlation functions to describe scattering in AdS space. The document also discusses when and why CFTs exhibit an EFT structure in AdS based on the structure of EFTs with a mass gap between light and heavy states.
This short document discusses enjoying life each day. It recommends living every day with enjoyment by focusing on the present moment and appreciating life's simple pleasures instead of worrying about the future or dwelling on the past. Make the most of each day by living in the now.
Este documento fornece dicas para identificar agências de emprego fraudulentas, incluindo fazer perguntas sobre como sua candidatura foi recebida, se há custos envolvidos e se é realmente uma vaga de emprego ou um serviço sendo oferecido. Alerta sobre sinais como recepções lotadas, falta de detalhes sobre a vaga e pressa para assinar contratos. Aconselha o leitor a manter o foco em sua busca por emprego de forma planejada.
O documento discute vários condimentos e suas potenciais vantagens para a saúde, incluindo o ketchup que contém antioxidantes que podem reduzir o risco de doenças cardíacas, mel escuro que contém antioxidantes que podem reduzir o risco de várias doenças, e azeite que contém um composto que pode melhorar a memória de longo prazo.
The document discusses falsifying cosmological models like LCDM and quintessence using galaxy cluster number counts. It summarizes three potential "pink elephant" galaxy clusters at z > 1 that have masses much larger than expected in LCDM. However, there is significant statistical uncertainty from both sample variance and parameter variance given current cosmological constraints. Future surveys could provide tighter constraints and potentially rule out LCDM if more such massive high-z clusters are found. Formulas are proposed to evaluate the expected cluster counts needed to rule out LCDM at given confidence levels accounting for these sources of uncertainty.
O documento fornece instruções sobre como realizar autoexame das mamas. Ele explica a anatomia dos seios, os fatores de risco para câncer de mama, quando e como realizar o exame, o que procurar e a importância de realizá-lo mensalmente para detecção precoce da doença.
Prof. Rob Leight (University of Illinois) TITLE: Born Reciprocity and the Nat...Rene Kotze
This document discusses Born reciprocity in string theory and how it relates to the nature of spacetime. It argues that while string theory is formulated in terms of maps into spacetime, this breaks Born reciprocity. The document suggests that quasi-periodicity of strings without assuming a periodic target spacetime better respects Born reciprocity. This leads to a phase space formulation of string theory without assuming locality of spacetime at short distances.
This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include:
- Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades.
- Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments.
- Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales.
- The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.
Fractal dimensions of 2d quantum gravityTimothy Budd
After introducing 2d quantum gravity, both in its discretized form in
terms of random triangulations and its continuum description as
Quantum Liouville theory, I will give a (non-exhaustive) review of the
current understanding of its fractal dimensions. In particular, I will
discuss recent analytic and numerical results relating to the
Hausdorff dimension and spectral dimension of 2d gravity coupled to
conformal matter fields.
This document outlines a presentation on formulating QCD coupled with QED (quantum electrodynamics) on the lattice for the purpose of studying isospin breaking effects. It discusses challenges in putting QED on the lattice due to the zero charge constraint with periodic boundary conditions. Several proposed approaches are mentioned, including QEDL, twist averaging, massive QED, and using charge conjugation boundary conditions. The document contains sections on isospin in QCD, challenges of QED on the lattice, previous QED+QCD simulations, and proposed new approaches.
Current knowledge of the transversity quark distribution function, or how transversely polarized quarks are distributed in a transversely polarized proton?
Virtual Nodes: Rethinking Topology in CassandraEric Evans
The document discusses Cassandra's topology and how it is moving from a single token per node model to a virtual node model where each node is assigned multiple tokens. This improves load balancing and data distribution in the cluster. Specifically, it addresses problems with the single token approach like poor load distribution when nodes fail and inefficient data movement when adding or replacing nodes. The virtual node model with random token assignment provides better scaling properties as the number of nodes and data size increases.
Recovering vital physiological signals from ambulatory devicesPraveen Pankajakshan
This document proposes a Bayesian framework to recover physiological signals from ambulatory devices using sparse regularization. It suggests minimizing l2-l2 cost functions on devices for initial processing and l2-TV cost functions on servers for improved accuracy. The approach models signal gradients as sparse and estimates signals iteratively using majorization-minimization. It applies these techniques to recover electrocardiogram signals from ambulatory recordings.
This document discusses Born reciprocity in string theory and how it relates to the nature of spacetime. It argues that while string theory is formulated in terms of maps into spacetime, this breaks Born reciprocity. The document suggests that quasi-periodicity of strings without assuming a periodic target spacetime better respects Born reciprocity. This leads to a phase space formulation of string theory without assuming locality of spacetime at short distances.
The document summarizes key points about equality constrained minimization problems and Newton's method for solving them. It discusses:
1) Equality constrained minimization problems and their equivalent forms via eliminating constraints or using the dual problem.
2) Newton's method extended to include equality constraints, where the Newton step is defined to satisfy the linearized optimality conditions and ensures feasible descent.
3) An infeasible start Newton method that computes steps to reduce the primal-dual residual norm, ensuring iterates become feasible within a finite number of steps.
Nonequilibrium statistical mechanics of cluster-cluster aggregation, School o...Colm Connaughton
Colm Connaughton presented on nonequilibrium statistical mechanics models of cluster-cluster aggregation. He discussed simple models where particles move randomly and merge upon contact. More sophisticated models track the size distribution of clusters as they aggregate. The Smoluchowski equation describes this process. For certain collision kernels, clusters of arbitrarily large size can form in finite time, known as gelation. While some kernels mathematically describe instantaneous gelation, physical models avoid this with a cluster size cutoff. Stationary states can be reached with a particle source.
Neural network precept diagnosis on petrochemical pipelines for quality maint...Alexander Decker
This document describes a proposed neural network model for predicting degradation in petrochemical pipelines. It begins with background on pipelines and fatigue crack propagation based on Paris' law. It then discusses stresses in cylindrical pipelines under internal pressure. The model represents crack growth as a function of stress intensity factor and uses a recurrent formula to calculate cumulative damage over time. The goal is to develop a prognostic tool for quality maintenance in pipeline systems.
This document reviews research on the convergence of perturbation series in quantum field theory. It discusses Dyson's argument that perturbation series in quantum electrodynamics (QED) have zero radius of convergence due to vacuum instability when the coupling constant is negative. Large-order estimates show that perturbation series coefficients grow factorially fast in quantum mechanics and field theories. Finally, it describes the method of Borel summation, which may allow extracting the exact physical quantity from a divergent perturbation series through a unique mapping.
Conformal Field Theory and the Holographic S-Matrixliam613
This document discusses how conformal field theories (CFTs) can describe gravitational scattering and provide an effective field theory (EFT) description of gravity in anti-de Sitter space (AdS). It introduces CFTs and issues with describing gravity at high energies. It then explains how the holographic duality between CFTs and gravity theories can be used to calculate scattering matrices and understand gravitational dynamics. In particular, it outlines how calculations in Mellin space allow CFT correlation functions to describe scattering in AdS space. The document also discusses when and why CFTs exhibit an EFT structure in AdS based on the structure of EFTs with a mass gap between light and heavy states.
This short document discusses enjoying life each day. It recommends living every day with enjoyment by focusing on the present moment and appreciating life's simple pleasures instead of worrying about the future or dwelling on the past. Make the most of each day by living in the now.
Este documento fornece dicas para identificar agências de emprego fraudulentas, incluindo fazer perguntas sobre como sua candidatura foi recebida, se há custos envolvidos e se é realmente uma vaga de emprego ou um serviço sendo oferecido. Alerta sobre sinais como recepções lotadas, falta de detalhes sobre a vaga e pressa para assinar contratos. Aconselha o leitor a manter o foco em sua busca por emprego de forma planejada.
O documento discute vários condimentos e suas potenciais vantagens para a saúde, incluindo o ketchup que contém antioxidantes que podem reduzir o risco de doenças cardíacas, mel escuro que contém antioxidantes que podem reduzir o risco de várias doenças, e azeite que contém um composto que pode melhorar a memória de longo prazo.
The document discusses falsifying cosmological models like LCDM and quintessence using galaxy cluster number counts. It summarizes three potential "pink elephant" galaxy clusters at z > 1 that have masses much larger than expected in LCDM. However, there is significant statistical uncertainty from both sample variance and parameter variance given current cosmological constraints. Future surveys could provide tighter constraints and potentially rule out LCDM if more such massive high-z clusters are found. Formulas are proposed to evaluate the expected cluster counts needed to rule out LCDM at given confidence levels accounting for these sources of uncertainty.
O documento fornece instruções sobre como realizar autoexame das mamas. Ele explica a anatomia dos seios, os fatores de risco para câncer de mama, quando e como realizar o exame, o que procurar e a importância de realizá-lo mensalmente para detecção precoce da doença.
Prof. Rob Leight (University of Illinois) TITLE: Born Reciprocity and the Nat...Rene Kotze
This document discusses Born reciprocity in string theory and how it relates to the nature of spacetime. It argues that while string theory is formulated in terms of maps into spacetime, this breaks Born reciprocity. The document suggests that quasi-periodicity of strings without assuming a periodic target spacetime better respects Born reciprocity. This leads to a phase space formulation of string theory without assuming locality of spacetime at short distances.
The document summarizes research on threshold network models, which generate scale-free networks without growth by assigning intrinsic weights to nodes based on a given distribution and connecting nodes based on whether their total weight exceeds a threshold. The model has been extended to spatial networks by incorporating distance between nodes and to include homophily. Analytical results show the degree distribution and other properties depend on the weight distribution and thresholding function used. Several open problems are also discussed.
Further discriminatory signature of inflationLaila A
These are the slides of the talk I gave on discriminating between models of inflation using space based gravitational wave detectors, at KEK in Tskuba University, Japan.
Geometric properties for parabolic and elliptic pdeSpringer
This document discusses recent advances in fractional Laplacian operators and related problems in partial differential equations and geometric measure theory. Specifically, it addresses three key topics:
1. Symmetry problems for solutions of the fractional Allen-Cahn equation and whether solutions only depend on one variable like in the classical case. The answer is known to be positive for some dimensions and fractional exponents but remains open in general.
2. The Γ-convergence of functionals involving the fractional Laplacian as the small parameter ε approaches zero. This characterizes the asymptotic behavior and relates to fractional notions of perimeter.
3. Regularity of interfaces as the fractional exponent s approaches 1/2 from above, which corresponds to a critical threshold
This document describes a Godunov smoothed particle hydrodynamics (SPH) method for simulating geophysical flows over natural terrain. Classical SPH struggles to model flows with steep slopes and resolve shocks accurately. The proposed method overcomes these issues by setting up Riemann problems between interacting particles and using corrected derivative formulas. Boundary conditions are handled by approximating boundaries as piecewise polynomials and using ghost particles. A background mesh is employed for neighbor searching and dynamic load balancing in parallel simulations. Results are presented for cliff collapse and granular jump scenarios.
Galaxies are collisionless systems that can be modeled using continuum methods. The evolution of a collisionless system is governed by the collisionless Boltzmann equation (CBE). N-body simulations solve the CBE by tracking the trajectories of particles in phase space over time. Poisson solvers are used to estimate gravitational forces, with grid-based methods like particle-mesh being fast but providing only approximate forces below the grid scale.
This document discusses vector fields and equipotentials in physics, using examples of gravitational fields near the Earth's surface and in binary star systems. It shows how to calculate vector fields from gradient of potentials, and plots of fields and equipotentials for these examples. Key relationships discussed are that equipotentials are perpendicular to field vectors, and vector fields point from high to low potential.
This document discusses vector fields and equipotentials in physics, using examples of gravitational fields near the Earth's surface and in binary star systems. It shows how to calculate vector fields from gradient of potentials, and plots of fields and equipotentials for these examples. Key relationships discussed are that equipotentials are perpendicular to field vectors, and vector fields point from high to low potential.
This document summarizes a presentation on distributed subgradient methods for saddle-point problems. It begins with an overview of distributed convex optimization and consensus-based algorithms. It then discusses using Lagrangian decomposition to distribute constraints, which allows agents to agree on Lagrange multipliers through communication. A more general saddle-point framework is presented, along with an algorithm using projected subgradients and Laplacian averaging. The algorithm is proven to converge to a saddle-point evaluation error of O(1/√t). Applications to distributed constrained optimization and low-rank matrix completion are discussed.
The document discusses pseudospectra as an alternative to eigenvalues for analyzing non-normal matrices and operators. It defines three equivalent definitions of pseudospectra: (1) the set of points where the resolvent is larger than ε-1, (2) the set of points that are eigenvalues of a perturbed matrix with perturbation smaller than ε, and (3) the set of points where the resolvent applied to a unit vector is larger than ε. It also shows that pseudospectra are nested sets and their intersection is the spectrum. The definitions extend to operators on Hilbert spaces using singular values.
Spectral clustering works by creating an affinity matrix from a similarity matrix and then applying dimensionality reduction before clustering in the reduced space. It represents the data as an undirected graph and uses the graph Laplacian matrix to perform the dimensionality reduction. The number of clusters can be determined using the eigengap heuristic or by setting k equal to the logarithm of the number of data points. The Gaussian kernel is commonly used to create the affinity matrix from the similarity matrix.
Galadriel's Mirror uses transformation optics to mimic a curved spacetime that allows for closed timelike curves, enabling time travel. The presentation discusses:
1) Using a curved spacetime metric from general relativity that allows time travel, even if not physically realistic. Transformation optics can then create an equivalent material.
2) A curved spacetime example that tips light cones, making a path that circles the angular direction both null and closed, enabling light to travel in time.
3) The proposed mirror material would use this curved spacetime, curving light along a closed null curve that takes it into the past, allowing users to see into the future or past through the mirror.
Similar to Chris Clarkson - Dark Energy and Backreaction (14)
This document discusses machine learning concepts including supervised vs. unsupervised learning, clustering algorithms, and specific clustering methods like k-means and k-nearest neighbors. It provides examples of how clustering can be used for applications such as market segmentation and astronomical data analysis. Key clustering algorithms covered are hierarchy methods, partitioning methods, k-means which groups data by assigning objects to the closest cluster center, and k-nearest neighbors which classifies new data based on its closest training examples.
- The document discusses methods for characterizing dark energy and modified gravity models in a model-independent way using cosmological observations.
- Due to the "dark degeneracy" between dark matter and dark energy, it is not possible to separately measure the properties of dark matter and dark energy without assuming a specific model class.
- Observables like the Hubble parameter H(z) and gravitational potentials can be reconstructed from the data, but this does not break the degeneracy between dark matter and dark energy contributions.
- The scale-dependence of quantities like the gravitational potentials and growth rate can be used to test and constrain broad classes of dark energy and modified gravity models in a more model-independent way.
Seminar by Prof Bruce Bassett at IAP, Paris, October 2013CosmoAIMS Bassett
This document discusses the rise of machine learning and artificial intelligence in astronomy due to a massive increase in data from upcoming surveys. It will produce around an exabyte of data per day, far more than has been produced throughout human history. This raises issues around preparing students, and how science may be done. The document discusses using machine learning for tasks like supernova identification and classification. It also discusses challenges like ensuring machine learning results are trustworthy, and whether this can truly replace human genius. It explores the idea of a universal language for scientific theories that could be searched algorithmically.
The 21cm line from neutral hydrogen can be used to study cosmology during the first billion years of the universe. This includes the Dark Ages when no structures formed, the Cosmic Dawn when the first luminous objects formed, and the Epoch of Reionization when these objects reionized the intergalactic medium. Current and future 21cm experiments like LOFAR, MWA, PAPER, and HERA aim to detect the signal from these eras but face challenges in calibrating the instruments and subtracting bright foreground sources. Some progress has been made in placing upper limits on the signal and constraining the heating of the intergalactic medium by X-rays, but a clear detection of the signal is still needed
The document discusses the cosmic dawn and reionization period in the early universe. It describes the evolution from the dark ages after recombination to the epoch of reionization around z=6-20. Key aspects discussed include understanding the sources and sinks of ionizing photons that drove reionization, and challenges in modeling this period due to the large parameter space and scales involved, from single stars to the entire universe. Seminumerical simulations are presented as an efficient method to model reionization and predict 21cm signals.
A short introduction to massive gravity... or ... Can one give a mass to the ...CosmoAIMS Bassett
1. The document discusses massive gravity and proposes that giving the graviton a small mass could potentially explain dark matter and dark energy without needing to introduce those concepts.
2. It reviews several models of massive gravity, including the Dvali-Gabadadze-Porrati model, which produces cosmic acceleration similar to dark energy. Kaluza-Klein theory is also discussed as producing massive gravitons.
3. Nonlinear extensions of the Pauli-Fierz theory are examined, finding solutions only with singularities. The "Goldstone" description of massive gravity is introduced as a way to better understand nonlinear effects like the Vainshtein mechanism.
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Chris Clarkson - Dark Energy and Backreaction
1. backreaction in the concordance model
Chris Clarkson
Astrophysics, Cosmology & Gravitation Centre
University of Cape Town
Thursday, 26 January 12
2. key ingredients for cosmology
• universe homogeneous and isotropic on large scales (>100 Mpc)
• background dynamics determined by amount of matter + curvature present
(+ a theory of gravity)
• inflation lays down the seeds for structure formation from quantum
fluctuations
• works ... if we include a cosmological constant or ‘dark energy’
• matter + curvature not enough
Thursday, 26 January 12
6. 3 issues
•Averaging
•Coarse-graining of structure, such that small-scale effects are hidden
to reveal large scale geometry and dynamics.
•Backreaction
•Gravity gravitates, so local gravitational inhomogeneities may affect
the cosmological dynamics.
•Fitting
•How do we fit an idealized model to observations made from one
location in a lumpy universe, given that the ‘background’ does not in
fact exist? (Averaging observables is not same as spatial average)
Thursday, 26 January 12
7. scale of homogeneity of the dist
1 1 it seems 3natural to define the e
HD ⇤ ↵N ⇧ D = N ⇧Jd x , As pointed out in
matter flow.
3 3VD D
Averaging is difficult one introduces an ambiguity in
VD ⇤ Jd x is the Riemannian volume of D. The avereaging as m
3 expansions: the expansion oper
measured by observers comoving
•Define Riemannian averaging on ⌥. As mentionned before, the co
er the domain D: domain D
1 the matter distribution, so that
↵D = ↵↵ D ⇤ ↵(t, xi )Jd3 x ,the space-time metri
write down
VD D In the following we will then ret
scalar function ↵. One can then define the e⌥ective scale factor for
:
Riemannian volume ✏ element
⇣t aD where J ⇤ det(hij ) and VD ⇤
HD = . the Riemannian average over the
spatial average implies wrt
aD
some foliation of spacetime
that is well defined for any scala
how can average of a tensor preserve Lorentz invariance?
the function aD (t) obeying:
Thursday, 26 January 12
8. scale of homogeneity of the dist
1 1 it seems 3natural to define the e
HD ⇤ ↵N ⇧ D = N ⇧Jd x , As pointed out in
matter flow.
3 3VD D
Averaging is difficult one introduces an ambiguity in
VD ⇤ Jd x is the Riemannian volume of D. The avereaging as m
3 expansions: the expansion oper
measured by observers comoving
•Define Riemannian averaging on ⌥. As mentionned before, the co
er the domain D: domain D
1 the matter distribution, so that
↵D = ↵↵ D ⇤ ↵(t, xi )Jd3 x ,the space-time metri
write down
VD D In the following we will then ret
scalar function ↵. One can then define the e⌥ective scale factor for
: eg, to specify average
Riemannian volume ✏ element
energy density need ⇣t aD where J ⇤ det(hij ) and VD ⇤
full solution of the HD = . the Riemannian average over the
spatial average implies wrt
aD
field equations some foliation of spacetime
that is well defined for any scala
how can average of a tensor preserve Lorentz invariance?
the function aD (t) obeying:
Thursday, 26 January 12
13. Buchert backreaction
from non-local
variance of
local expansion
rate
Zalaletdinov
Thursday, 26 January 12
14. Buchert backreaction
from non-local
variance of
local expansion
rate
Zalaletdinov
Thursday, 26 January 12
15. Buchert backreaction
from non-local
variance of
local expansion
rate
Zalaletdinov
macroscopic Ricci
correlation tensor
Thursday, 26 January 12
16. Another view of the averaging problem
on l ity d ay
e nd ati ua to Hubble radius
i nfl eq
averaging gives corrections here
Thursday, 26 January 12
17. Another view of the averaging problem
on l ity d ay
e nd ati ua to Hubble radius
i nfl eq
model = flat FLRW + averaging gives corrections here
perturbations
Thursday, 26 January 12
18. Another view of the averaging problem
on l ity d ay
nd ati
howfl do
e weua to
remove backreaction bits
Hubble radius
in eq
to get to ‘real’ background?
smoothed background today is not
same background as at end of inflation
?
model = flat FLRW + averaging gives corrections here
perturbations
Thursday, 26 January 12
19. Fitting isn’t obvious either
a single line of
sight gives D(z) {
averaged on the
Hubble rate along the
sky to give
line of sight
smooth ‘model’
Thursday, 26 January 12
21. Aren’t the corrections just ~10-5?
•No. Of course not. Λ is bs [Buchert, Kolb ...]
Thursday, 26 January 12
22. Aren’t the corrections just ~10-5?
•No. Of course not. Λ is bs [Buchert, Kolb ...]
•Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...]
Thursday, 26 January 12
23. Aren’t the corrections just ~10-5?
•No. Of course not. Λ is bs [Buchert, Kolb ...]
•Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...]
•Well, maybe. I’ve no idea what’s going on. [everyone else ... ?]
Thursday, 26 January 12
24. Aren’t the corrections just ~10-5?
•No. Of course not. Λ is bs [Buchert, Kolb ...]
•Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...]
•Well, maybe. I’ve no idea what’s going on. [everyone else ... ?]
•Corrections from averaging enter Friedmann and Raychaudhuri
equations
•is this degenerate with ‘dark energy’?
•can we separate the effects [if there are any]?
•or ... is it dark energy? neat solution to the coincidence
problem
Thursday, 26 January 12
25. Could it be dark energy?
?
‘average’ expansion can
accelerate while local one
decelerates everywhere
regions with faster expansion dominate the volume over time
so, the average expansion will increase
Thursday, 26 January 12
26. Could it be dark energy?
?
‘average’ expansion can
accelerate while local one
decelerates everywhere
regions with faster expansion dominate the volume over time
so, the average expansion will increase
Thursday, 26 January 12
30. What to compute?
•general formalisms lack quantitative computability
•perturbative methods don’t give coherent picture
•but they do give predictions
Thursday, 26 January 12
32. Different aspects
•Non-linear perturbations [Newtonian vs non-Newtonian]
Thursday, 26 January 12
33. Different aspects
•Non-linear perturbations [Newtonian vs non-Newtonian]
•Relativistic corrections
•linear and non-linear
Thursday, 26 January 12
34. Different aspects
•Non-linear perturbations [Newtonian vs non-Newtonian]
•Relativistic corrections
•linear and non-linear
•backreaction from averaging
Thursday, 26 January 12
35. Different aspects
•Non-linear perturbations [Newtonian vs non-Newtonian]
•Relativistic corrections
•linear and non-linear
•backreaction from averaging
•corrections to observables
Thursday, 26 January 12
36. Canonical Cosmology
•compute everything as power series in small parameter ε
gµ⇥ = gµ⇥ + ⇥
¯ (1)
gµ⇥ + ⇥
2 (2)
gµ⇥ + · · ·
‘real’ spacetime second-order
first-order
‘background’ spacetime perturbation correction
FLRW
fit to ‘background’ observables - SNIa etc
Thursday, 26 January 12
37. Canonical Cosmology
•compute everything as power series in small parameter ε
gµ⇥ = gµ⇥ + ⇥
¯ (1)
gµ⇥ + ⇥
2 (2)
gµ⇥ + · · ·
‘real’ spacetime second-order
first-order
‘background’ spacetime perturbation correction
FLRW
how do these
?
fit in?
fit to ‘background’ observables - SNIa etc
Thursday, 26 January 12
39. so, ....
is it big or is it small ... ?
Thursday, 26 January 12
40. so, ....
is it big or is it small ... ?
(I don’t know either)
Thursday, 26 January 12
41. Simulations can’t see it
periodic BC’s and no
horizon give
Olbers paradox -
cancels backreaction
Thursday, 26 January 12
42. Simulations can’t see it
periodic BC’s and no
horizon give
Olbers paradox -
cancels backreaction
Thursday, 26 January 12
43. Perturbation theory
metric to second-order
Bardeen equation
no real backreaction from first-order perturbations
- average of perturbations vanish by assumption
Thursday, 26 January 12
44. Perturbation theory
metric to second-order
second-order potentials induced by first-order scalars
vectors and tensors give measure of relativistic corrections
Thursday, 26 January 12
45. Perturbation theory
metric to second-order
second-order potentials induced by first-order scalars
vectors and tensors give measure of relativistic corrections
Thursday, 26 January 12
48. induced tensors
bigger than
primordial [today]
Thursday, 26 January 12
49. backreaction as correction to the background
•second-order modes give non-trivial ‘backreaction’
•Hubble rate depends on
•What is it?
Thursday, 26 January 12
50. backreaction as correction to the background
•second-order modes give non-trivial ‘backreaction’
•Hubble rate depends on
•What is it?
Thursday, 26 January 12
51. backreaction as correction to the background
•second-order modes give non-trivial ‘backreaction’
•Hubble rate depends on
•What is it?
Thursday, 26 January 12
52. backreaction as correction to the background
•second-order modes give non-trivial ‘backreaction’
•Hubble rate depends on
•What is it?
determines amplitude
of backreaction
Thursday, 26 January 12
53. backreaction is concerned with the
homogeneous, average contributions
what does this depend on?
Thursday, 26 January 12
54. scaling behaviour d lity y
en ua da
to
eq Hubble radius
at first-order
scale increasing
Thursday, 26 January 12
58. amplitude of second-order contributions
large equality scale suppresses
backreaction - but overcomes
factor(s) of Delta
Thursday, 26 January 12
59. backreaction - expansion rate
•second-order modes give non-trivial backreaction
•Hubble rate depends on
•UV divergent terms don’t contribute on average [Newtonain]
•well defined and well behaved backreaction
•but, this is only well behaved because of the long radiation era
•what would we do if the equality scale were smaller?
Thursday, 26 January 12
60. on we wish to probe, and some subtleties arise because we have to
hen we examineto the Hubble we areat second-order
Change averagedHubble as a function of redshift as a fractional in twothe backgroundthe rate.
the rate rate interested things:
ed Friedmann equation. In the Friedmann Friedmann equation, and theinterested
FIG. 1: The
˘
Hubble rate
the Hubble rate H calculated from the ensemble-averaged
D
equationchange are left shows the Hubble rat
we to Hubble
eld equations asBothresult of and EdS models are considered, and the di erentcomponents indicated
directly, H . a concordance averaging, which are three new averaging schemes,
D
if we put R = 0, H still doesn’t have a UV divergence.
of dark energy, it is common to think of these as e⇥ective fluid or
S D
hese two agree up to rate
Normalised Hubble perturbative order, when we take the ensemble
as function of redshift
e given by the generalised Friedmann equation. For a given domain
from averaging Friedmann
veraged Hubble rate we might expect to find. When we present HD
equation
⇧
˘ 2
HD ⌅ HD (98)
Friedmann equation and then taken the square-root. This does not
(55) directly (but note that if we square Eq (55), take the ensemble
equality scale domain
he Hubble rate as calculated directly from the Friedmann equation).
ce’ of the Hubble rate, which may be defined by
Clarkson, Ananda & Larena, 0907.3377
2
[HD ] = HD 2 HD . (99)
⌥⌃ ⌥⌃ ˘
FIG. 2: Plots for H
D with RD = 1/kequality , RS = 0, with the variance included, as a function of redshif
Thursday, 26 January 12
62. effective fluid
•amplitudes apply to effective fluid approach [Baumann etal]
and Green and Wald [PPN]
•they claim backreaction is small from this
Thursday, 26 January 12
64. great!
backreaction is small ...
Thursday, 26 January 12
65. backreaction - acceleration rate
•other quantities are much stranger
•time derivative of the Hubble rate represented in the
deceleration parameter
•UV divergent terms do not cancel out
Thursday, 26 January 12
66. backreaction - acceleration rate
•other quantities are much stranger
•time derivative of the Hubble rate represented in the
deceleration parameter
•UV divergent terms do not cancel out
Thursday, 26 January 12
67. backreaction - acceleration rate
•other quantities are much stranger
•time derivative of the Hubble rate represented in the
deceleration parameter
•UV divergent terms do not cancel out
Thursday, 26 January 12
68. backreaction - acceleration rate
•other quantities are much stranger
•time derivative of the Hubble rate represented in the
deceleration parameter
•UV divergent terms do not cancel out
dominates backreaction
Thursday, 26 January 12
75. oh no!
backreaction is huge ...
but wait!
Thursday, 26 January 12
76. use observables - spatial averaging is meaningless!
•we fit our cosmology to an all-sky average of observed
quantities
•distance-redshift, number-count redshift, etc.
•if averaging/backreaction is significant then these should
fit to the wrong model (should use LTB metric!)
•can be computed using Kristian-Sachs approach
Thursday, 26 January 12
77. expectation value
observed deceleration governed by cut-off
Thursday, 26 January 12
78. expectation value
observed deceleration governed by cut-off
Thursday, 26 January 12
79. expectation value
observed deceleration governed by cut-off
Thursday, 26 January 12
81. UV cutoff
•modes below equality scale dominate effect - not physical
Thursday, 26 January 12
82. UV cutoff
•modes below equality scale dominate effect - not physical
•where should they be cut off?
Thursday, 26 January 12
83. UV cutoff
•modes below equality scale dominate effect - not physical
•where should they be cut off?
•inflation scale [last mode to leave the Hubble scale]?
backreaction >>>1
Thursday, 26 January 12
84. UV cutoff
•modes below equality scale dominate effect - not physical
•where should they be cut off?
•inflation scale [last mode to leave the Hubble scale]?
backreaction >>>1
•dark matter free-streaming scale? [~ pc] backreaction >>1
Thursday, 26 January 12
85. UV cutoff
•modes below equality scale dominate effect - not physical
•where should they be cut off?
•inflation scale [last mode to leave the Hubble scale]?
backreaction >>>1
•dark matter free-streaming scale? [~ pc] backreaction >>1
•‘spherical scale’ [Kolb]
Thursday, 26 January 12
86. UV cutoff
•modes below equality scale dominate effect - not physical
•where should they be cut off?
•inflation scale [last mode to leave the Hubble scale]?
backreaction >>>1
•dark matter free-streaming scale? [~ pc] backreaction >>1
•‘spherical scale’ [Kolb]
•virial ‘scale’ ~ Mpc’s [Baumann etal] - gives O(1) backreaction
Thursday, 26 January 12
87. wtf?
amplitude of backreaction determined purely from cut-off
virial scales only sensible cutoff O(1) backreaction
ignore them - maybe gauge or unphysical ?
Thursday, 26 January 12
94. ok ... so,
backreaction could be ...
anything ...
Thursday, 26 January 12
95. key questions
•convergence of perturbation theory function of equality
scale
•why are we so lucky?
•with less radiation, scales up to ‘virial scale’ must
contribute to backreaction - how would we compute this?
•model with no radiation era might be best model to
explore backreaction
Thursday, 26 January 12
96. What is the background?
Do observations measure the background?
What is ‘precision cosmology’?
Thursday, 26 January 12
97. Interfering with dark energy
•until we understand
backreaction precision
cosmology not secure
•what are we measuring?
•Zalaletdinov’s gravity
predicts decoupled
geometric and dynamical
curvature
•removes constraints on
dynamical DE
•evidence for acceleration
only provided by SNIa
Thursday, 26 January 12
98. a single line of
sight gives D(z) {
•We average this over the sky, and (re)construct model
- alternative aspect to ‘backreaction’
•average depends on z, so ‘looks’ spherically symmetric
and inhomogeneous
•would remove copernican constraints on void models!
•no need for LTB solution, kSZ constraints removed...
Thursday, 26 January 12
99. Conclusions Confusions
•Why are second-order perturbations so large?
•
•tells us that perturbation theory must be relativistic, not
Newtonian
•role of UV divergence must be understood to decide whether
backreaction is small - higher order or resummation methods
needed? must include tensors!
•do we need relativistic N-body replacement?
•what is the background? what is ‘precision cosmology’?
•‘void models’ may be mis-interpretation of backreaction ...
Thursday, 26 January 12
103. Curvature test for the Copernican Principle
• in FLRW we can combine Hubble rate and distance data to find curvature
2
[H(z)D (z)] 1
k =
[H0 D(z)]2
⇥
dL = (1 + z)D = (1 + z) dA
2
• independent of all other cosmological parameters, including dark energy
model, and theory of gravity
• tests the Copernican principle and the basis of FLRW (‘on-lightcone’ test)
⇥
C (z) = 1 + H 2
DD D 2
+ HH DD = 0
Clarkson, Basset & Lu, PRL 100 191303
Thursday, 26 January 12
104. Using age data to reconstruct H(z)
Shafieloo & Clarkson, PRD
Thursday, 26 January 12
105. Consistency Test for ΛCDM For flat ΛCDM mod-
d for all curvatures, where H(z) is given by the Friedmann equation,
els the slope of the distance data curve must satisfy
onsistency Test for ΛCDM For flat ΛCDM mod- lies outside the n-σ er
H(z) 2
H 2
m
1/ + Ωk (1 + z)3 + (1 − Ωm ). + Rearranging for
D=(z)of= (1 + z)3Ωm (1 +z)2 + ΩDE exp 3satisfy w(zdeviations from Λ, as
the A litmus test for flat ΛCDM
slope 0 Ωthe distance data curve must
z
1 )
dz ,
z) = 1/ wemhavez)3 + (1 − Ωm ). Rearranging for + z below for the general
Ωm Ω (1 + 0 1
Ωk The 2 If, on the other hand
we .have usual procedure is to postulate a several parameter form for w(z) and calculate
ernative method is to reconstruct w(z) by1 − D reconstructingfit for all suitable para
directly (z) the luminosity-distance
Ωm = w(z). Writing D(z) = (H ./c)(1 + z)−1d (z), we hav
1 − D (z) 2
[9? ] dL (z) Ωm = inverted to yield + z)
(5)
3 − 1]D (5) 20 that is good evidence
may be
[(1 + z)
[(1 .
3 − 1]D (z)2 (z) L
is that only one choic
2(1 + z)(1 + Ωk D )D − (1 + z) Ωk D + 2(1 + z)Ωk DD − 3(1 + Ωk D2 ) D
2 2 2
hin the flat ΛCDMflat ΛCDMwe measure D if we2 measure D to provide e
Within the paradigm, if paradigm, (z) at L (z) = 0 (z) at
2 [Ω + (1 + z)Ω ] D 2 − (1 + Ω D )} D
.
3 {(1 + z)
e z and calculate the rhs kof this equation, we should
m k
some z and calculate the rhs of this equation,parameterisation
ery we should
in the same answer independently of the for flat LCDM w(z) that ].could ne
this is constant redshift equation of state [? of
e-redshift curve D(z), we can reconstruct the dark energyof of the redshift Typicall
in
obtain the same answer independently
surement. Differentiating Eq. (5)ansatz used in [5],
ed ansatz for D(z), such as the Pade we then find that through various ensur
measurement. Differentiating Eq. (5) we then find that nated.
L (z) = ζD (z) + 3(1(1 + z)D α (1 − z) − 1 2 ] α
+ z)2 − (z)[1 + D (z)+
DL (z) = = ζD (z) √ 3(1 + z)2 D (z)[1 − D (z)2 ]the test To
L (z) flat ΛCDMγmodels. 2 − α − β −(6) .
+ +z+ Testing
= 0 for all β(1 + z) + 1 γ
strate that it will wo
have used then shorthandw(z) from Eq ΛCDM models.in Eq. (4) toisthe lum
e data, and the calculatesfor = 2[(1 + z)3 − 1]. a reconstruction method (6) g
= 0 ζ all flat (3). Such Note more
w(z) directly because we are fitting directly to the observable, and 3 can spot small The
this is completely independent of the value of Ωm . simulated data. vari
so
slate to dramatic a test for Λ as Zunckel & Clarkson, PRL, arXiv:0807.4304;Note err
mayWe this as variations in w(z). Unfortunately,param- + z) to larger and substit
use have used the shorthand ζ a this also leads − 1]. reported
follows: take = 2[(1 calculated
form this is directly. It is also Sahni L (z) which errors = 0 within .the e
see data. If etal 0807.3548 value of Ω
sedparameterising it completely not clear, however,= 0 theL (z)on our understand
by that for D(z) and fit to the independent of m
e taken most seriously. What is very nice about this method is that, if done in small re
f w(z) in a given useis independent of bins atΛ as z; this is not the case param-
We may bin this as a test for lower follows: take a for parameteri
grateeterised form for D(z)dand Bothto the data. If strong degeneracies
over redshift when calculating (z). fit methods suffer from L (z) = 0
Thursday, 26 January 12
106. Consistency Test for ΛCDM For flat ΛCDM mod-
d for all curvatures, where H(z) is given by the Friedmann equation,
els the slope of the distance data curve must satisfy
onsistency Test for ΛCDM For flat ΛCDM mod- lies outside the n-σ er
H(z) 2
H 2
m
1/ + Ωk (1 + z)3 + (1 − Ωm ). + Rearranging for
D=(z)of= (1 + z)3Ωm (1 +z)2 + ΩDE exp 3satisfy w(zdeviations from Λ, as
the A litmus test for flat ΛCDM
slope 0 Ωthe distance data curve must
z
1 )
dz ,
z) = 1/ wemhavez)3 + (1 − Ωm ). Rearranging for + z below for the general
Ωm Ω (1 + 0 1
Ωk The 2 If, on the other hand
we .have usual procedure is to postulate a several parameter form for w(z) and calculate
ernative method is to reconstruct w(z) by1 − D reconstructingfit for all suitable para
directly (z) the luminosity-distance
Ωm = w(z). Writing D(z) = (H ./c)(1 + z)−1d (z), we hav
1 − D (z) 2
[9? ] dL (z) Ωm = inverted to yield + z)
(5)
3 − 1]D (5) 20 that is good evidence
may be
[(1 + z)
[(1 .
3 − 1]D (z)2 (z) L
is that only one choic
2(1 + z)(1 + Ωk D )D − (1 + z) Ωk D + 2(1 + z)Ωk DD − 3(1 + Ωk D2 ) D
2 2 2
hin the flat ΛCDMflat ΛCDMwe measure D if we2 measure D to provide e
Within the paradigm, if paradigm, (z) at L (z) = 0 (z) at
2 [Ω + (1 + z)Ω ] D 2 − (1 + Ω D )} D
.
3 {(1 + z)
e z and calculate the rhs kof this equation, we should
m k
some z and calculate the rhs of this equation,parameterisation
ery we should
in the same answer independently of the for flat LCDM w(z) that ].could ne
this is constant redshift equation of state [? of
e-redshift curve D(z), we can reconstruct the dark energyof of the redshift Typicall
in
obtain the same answer independently
surement. Differentiating Eq. (5)ansatz used in [5],
ed ansatz for D(z), such as the Pade we then find that through various ensur
measurement. Differentiating Eq. (5) we then find that nated.
L (z) = ζD (z) + 3(1(1 + z)D α (1 − z) − 1 2 ] α
+ z)2 − (z)[1 + D (z)+
DL (z) = = ζD (z) √ 3(1 + z)2 D (z)[1 − D (z)2 ]the test To
L (z) flat ΛCDMγmodels. 2 − α − β −(6) .
+ +z+ Testing
= 0 for all β(1 + z) + 1 γ
strate that it will wo
have used then shorthandw(z) from Eq ΛCDM models.in Eq. (4) toisthe lum
e data, and the calculatesfor = 2[(1 + z)3 − 1]. a reconstruction method (6) g
= 0 ζ all flat (3). Such Note more
w(z) directly because we are fitting directly to the observable, and 3 can spot small The
this is completely independent of the value of Ωm . simulated data. vari
so
slate to dramatic a test for Λ as Zunckel & Clarkson, PRL, arXiv:0807.4304;Note err
mayWe this as variations in w(z). Unfortunately,param- + z) to larger and substit
use have used the shorthand ζ a this also leads − 1]. reported
follows: take = 2[(1 calculated
form this is directly. It is also Sahni L (z) which errors = 0 within .the e
see data. If etal 0807.3548 value of Ω
sedparameterising it completely not clear, however,= 0 theL (z)on our understand
by that for D(z) and fit to the independent of m
e taken most seriously. What is very nice about this method is that, if done in small re
f w(z) in a given useis independent of bins atΛ as z; this is not the case param-
We may bin this as a test for lower follows: take a for parameteri
grateeterised form for D(z)dand Bothto the data. If strong degeneracies
over redshift when calculating (z). fit methods suffer from L (z) = 0
Thursday, 26 January 12
107. A litmus test for flat ΛCDM
these are better fits to constitution data than LCDM
with Arman Shafieloo, PRD
Thursday, 26 January 12
108. A litmus test for flat ΛCDM
these are better fits to constitution data than LCDM
should be zero!
with Arman Shafieloo, PRD
Thursday, 26 January 12
109. A litmus testno dependence on Omega_m
for flat ΛCDM
these are better fits to constitution data than LCDM
should be zero!
with Arman Shafieloo, PRD
Thursday, 26 January 12