A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy

Off-Diagonal Dark-Matter Phenomenology: Exploring Enhanced Complementarity Relations in Non-Minimal Dark Sectors

In most multi-component dark-matter scenarios, two classes of processes generically contribute to event rates at experiments capable of probing the nature of the dark sector. The first class consists of "diagonal" processes involving only a single species of dark-matter particle -- processes analogous to those which arise in single-component dark-matter scenarios. By contrast, the second class consists of "off-diagonal" processes involving dark-matter particles of different species. Such processes include inelastic scattering at direct-detection experiments, asymmetric production at colliders, dark-matter co-annihilation, and certain kinds of dark-matter decay. In typical multi-component scenarios, the contributions from diagonal processes dominate over those from off-diagonal processes. Unfortunately, this tends to mask those features which are most sensitive to the multi-component nature of the dark sector. In this paper, by contrast, we point out that there exist natural, multi-component dark-sector scenarios in which the off-diagonal contributions actually dominate over the diagonal. This then gives rise to a new, enhanced picture of dark-matter complementarity. In this paper, we introduce a scenario in which this situation arises and examine the enhanced picture of dark-matter complementarity which emerges.Comment: 17 pages, LaTeX, 2 figures. Replaced to match published versio

Cosmological Constraints on Unstable Particles: Numerical Bounds and Analytic Approximations

Many extensions of the Standard Model predict large numbers of additional unstable particles whose decays in the early universe are tightly constrained by observational data. For example, the decays of such particles can alter the ratios of light-element abundances, give rise to distortions in the cosmic microwave background, alter the ionization history of the universe, and contribute to the diffuse photon flux. Constraints on new physics from such considerations are typically derived for a single unstable particle species with a single well-defined mass and characteristic lifetime. In this paper, by contrast, we investigate the cosmological constraints on theories involving entire ensembles of decaying particles --- ensembles which span potentially broad ranges of masses and lifetimes. In addition to providing a detailed numerical analysis of these constraints, we also formulate a set of simple analytic approximations for these constraints which may be applied to generic ensembles of unstable particles which decay into electromagnetically-interacting final states. We then illustrate how these analytic approximations can be used to constrain a variety of toy scenarios for physics beyond the Standard Model. For ease of reference, we also compile our results in the form of a table which can be consulted independently of the rest of the paper. It is thus our hope that this work might serve as a useful reference for future model-builders concerned with cosmological constraints on decaying particles, regardless of the particular model under study.Comment: 41 pages, LaTeX, 12 figures, 4 table

Exact nonadditive kinetic potentials for embedded density functional theory

We describe an embedded density functional theory (DFT) protocol in which the nonadditive kinetic energy component of the embedding potential is treated exactly. At each iteration of the Kohn–Sham equations for constrained electron density, the Zhao–Morrison–Parr constrained search method for constructing Kohn–Sham orbitals is combined with the King-Handy expression for the exact kinetic potential. We use this formally exact embedding protocol to calculate ionization energies for a series of three- and four-electron atomic systems, and the results are compared to embedded DFT calculations that utilize the Thomas–Fermi (TF) and the Thomas–Fermi–von Weisacker approximations to the kinetic energy functional. These calculations illustrate the expected breakdown due to the TF approximation for the nonadditive kinetic potential, with errors of 30%–80% in the calculated ionization energies; by contrast, the exact protocol is found to be accurate and stable. To significantly improve the convergence of the new protocol, we introduce a density-based switching function to map between the exact nonadditive kinetic potential and the TF approximation in the region of the nuclear cusp, and we demonstrate that this approximation has little effect on the accuracy of the calculated ionization energies. Finally, we describe possible extensions of the exact protocol to perform accurate embedded DFT calculations in large systems with strongly overlapping subsystem densities