Constructing Reference Metrics on Multicube Representations of Arbitrary Manifolds

Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).Comment: 37 pages, 16 figures; additional introductory material added in version accepted for publicatio

Scalar, Vector and Tensor Harmonics on the Three-Sphere

Scalar, vector and tensor harmonics on the three-sphere were introduced originally to facilitate the study of various problems in gravitational physics. These harmonics are defined as eigenfunctions of the covariant Laplace operator which satisfy certain divergence and trace identities, and ortho-normality conditions. This paper provides a summary of these properties, along with a new notation that simplifies and clarifies some of the key expressions. Practical methods are described for accurately and efficiently computing these harmonics numerically, and test results are given that illustrate how well the analytical identities are satisfied by the harmonics computed numerically in this way.Comment: 14 pages, 9 figures, to appear in General Relativity and Gravitatio

What does a binary black hole merger look like?

We present a method of calculating the strong-field gravitational lensing caused by many analytic and numerical spacetimes. We use this procedure to calculate the distortion caused by isolated black holes and by numerically evolved black hole binaries. We produce both demonstrative images illustrating details of the spatial distortion and realistic images of collections of stars taking both lensing amplification and redshift into account. On large scales the lensing from inspiraling binaries resembles that of single black holes, but on small scales the resulting images show complex and in some cases self-similar structure across different angular scales.Comment: 10 pages, 12 figures. Supplementary images and movies can be found at http://www.black-holes.org/the-science-numerical-relativity/numerical-relativity/gravitational-lensin

Spectral methods for the wave equation in second-order form

Current spectral simulations of Einstein's equations require writing the equations in first-order form, potentially introducing instabilities and inefficiencies. We present a new penalty method for pseudo-spectral evolutions of second order in space wave equations. The penalties are constructed as functions of Legendre polynomials and are added to the equations of motion everywhere, not only on the boundaries. Using energy methods, we prove semi-discrete stability of the new method for the scalar wave equation in flat space and show how it can be applied to the scalar wave on a curved background. Numerical results demonstrating stability and convergence for multi-domain second-order scalar wave evolutions are also presented. This work provides a foundation for treating Einstein's equations directly in second-order form by spectral methods.Comment: 16 pages, 5 figure

Comparing Gravitational Waveform Extrapolation to Cauchy-Characteristic Extraction in Binary Black Hole Simulations

We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar $\Psi_4$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects---unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-$\Psi_4$ waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-$\Psi_4$ and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for $m=0$ "memory" modes.Comment: 26 pages, 20 figure

Suitability of hybrid gravitational waveforms for unequal-mass binaries

This article studies sufficient accuracy criteria of hybrid post-Newtonian (PN) and numerical relativity (NR) waveforms for parameter estimation of strong binary black-hole sources in second- generation ground-based gravitational-wave detectors. We investigate equal-mass non-spinning binaries with a new 33-orbit NR waveform, as well as unequal-mass binaries with mass ratios 2, 3, 4 and 6. For equal masses, the 33-orbit NR waveform allows us to recover previous results and to extend the analysis toward matching at lower frequencies. For unequal masses, the errors between different PN approximants increase with mass ratio. Thus, at 3.5PN, hybrids for higher-mass-ratio systems would require NR waveforms with many more gravitational-wave (GW) cycles to guarantee no adverse impact on parameter estimation. Furthermore, we investigate the potential improvement in hybrid waveforms that can be expected from 4th order post-Newtonian waveforms, and find that knowledge of this 4th post-Newtonian order would significantly improve the accuracy of hybrid waveforms.Comment: 11 pages, 14 figure

Microlensing of the Lensed Quasar SDSS0924+0219

We analyze V, I and H band HST images and two seasons of R-band monitoring data for the gravitationally lensed quasar SDSS0924+0219. We clearly see that image D is a point-source image of the quasar at the center of its host galaxy. We can easily track the host galaxy of the quasar close to image D because microlensing has provided a natural coronograph that suppresses the flux of the quasar image by roughly an order of magnitude. We observe low amplitude, uncorrelated variability between the four quasar images due to microlensing, but no correlated variations that could be used to measure a time delay. Monte Carlo models of the microlensing variability provide estimates of the mean stellar mass in the lens galaxy (0.02 Msun < M < 1.0 Msun), the accretion disk size (the disk temperature is 5 x 10^4 K at 3.0 x 10^14 cm < rs < 1.4 x 10^15 cm), and the black hole mass (2.0 x 10^7 Msun < MBH \eta_{0.1}^{-1/2} (L/LE)^{1/2} < 3.3 x 10^8 Msun), all at 68% confidence. The black hole mass estimate based on microlensing is consistent with an estimate of MBH = 7.3 +- 2.4 x 10^7 Msun from the MgII emission line width. If we extrapolate the best-fitting light curve models into the future, we expect the the flux of images A and B to remain relatively stable and images C and D to brighten. In particular, we estimate that image D has a roughly 12% probability of brightening by a factor of two during the next year and a 45% probability of brightening by an order of magnitude over the next decade.Comment: v.2 incorporates referee's comments and corrects two errors in the original manuscript. 28 pages, 10 figures, published in Ap

Soluble CD23 Levels are Inversely Associated with Atopy and Parasite-Specific IgE Levels but Not with Polyclonal IgE Levels in People Exposed to Helminth Infection

BACKGROUND: Protective acquired immunity against helminths and allergic sensitisation are both characterised by high IgE antibody levels. Levels of IgE antibodies are naturally tightly regulated by several mechanisms including binding of the CD23 receptor. Following observations that helminth infections and allergic sensitisation may co-present, the current study aims to investigate the relationship between the soluble CD23 (sCD23) receptor, parasite-specific IgE responses and allergic sensitisation in people exposed to the helminth parasite Schistosoma haematobium. METHODS: A cohort of 434 participants was recruited in two villages with different levels of S. haematobium infection in Zimbabwe. Serum levels of the 25-kDa fragment of sCD23 were related to levels of schistosome infection intensity, allergen (house dust mite, HDM) and schistosome-specific IgE, total IgE and skin sensitisation to HDM. RESULTS: sCD23 levels rose significantly with schistosome infection intensity but declined significantly with schistosome-specific IgE levels. Furthermore, sCD23 levels were negatively associated with skin sensitisation and IgE reactivity against HDM, but showed no relationship with total IgE. CONCLUSION: The results are consistent with the suppression of parasite and allergen-specific IgE levels by sCD23. Further mechanistic studies will determine the relevance of this potential regulatory mechanism in the development of helminth-specific immune responses in atopic individuals

Periastron Advance in Spinning Black Hole Binaries: Gravitational Self-Force from Numerical Relativity

We study the general relativistic periastron advance in spinning black hole binaries on quasi-circular orbits, with spins aligned or anti-aligned with the orbital angular momentum, using numerical-relativity simulations, the post-Newtonian approximation, and black hole perturbation theory. By imposing a symmetry by exchange of the bodies' labels, we devise an improved version of the perturbative result, and use it as the leading term of a new type of expansion in powers of the symmetric mass ratio. This allows us to measure, for the first time, the gravitational self-force effect on the periastron advance of a non-spinning particle orbiting a Kerr black hole of mass M and spin S = -0.5 M^2, down to separations of order 9M. Comparing the predictions of our improved perturbative expansion with the exact results from numerical simulations of equal-mass and equal-spin binaries, we find a remarkable agreement over a wide range of spins and orbital separations.Comment: 18 pages, 12 figures; matches version to appear in Phys. Rev.