Quantifying Selection Bias in Cross-Sectional Studies of Ovarian Hormones

Most studies of ovarian hormones in adult women collect data from a cross-sectional sample of participants meeting various selection criteria including not having been pregnant or breastfeeding for several months. Although this approach is intended to eliminate the effects of these factors on hormonal variation, it introduces a selection bias of unknown magnitude: in a non-contracepting population, those women with the highest fecundity are more likely to be either pregnant or lactating, and so not included in a study sample. Thus a cross-sectional sample disproportionately represents women with the lowest fecundity (and potentially the lowest hormone levels). Here we present a preliminary evaluation of the magnitude of this selection bias, focusing on progesterone ($P$) levels near the luteal peak. We use data from Project REPA, a longitudinal study of reproductive functioning in rural Bolivians, recruited without regard to reproductive status (Vitzthum, Spielvogel, and Thornburg \textit{Proceedings of the U.S. National Academy of Sciences/} 101, 1443 (2004)). Drawing from 542~non-conception cycles in 144~women, we construct simulated cross-sectional samples meeting various inclusion criteria and compare their anovulation rates and progesterone levels.National Science Foundation, University of California, Binghamton University, Indiana Universit

Accurate Evolution of Orbiting Binary Black Holes

We present a detailed analysis of binary black hole evolutions in the last orbit, and demonstrate consistent and convergent results for the trajectories of the individual bodies. The gauge choice can significantly affect the overall accuracy of the evolution. It is possible to reconcile certain gauge dependent discrepancies by examining the convergence limit. We illustrate these results using an initial data set recently evolved by Bruegmann (Phys. Rev. Lett. 92, 211101). For our highest resolution and most accurate gauge, we estimate the duration of this data set's last orbit to be approximately $59 M_{ADM}$.Comment: 4 pages, 3 figure

Study of multi black hole and ring singularity apparent horizons

We study critical black hole separations for the formation of a common apparent horizon in systems of $N$ - black holes in a time symmetric configuration. We study in detail the aligned equal mass cases for $N=2,3,4,5$, and relate them to the unequal mass binary black hole case. We then study the apparent horizon of the time symmetric initial geometry of a ring singularity of different radii. The apparent horizon is used as indicative of the location of the event horizon in an effort to predict a critical ring radius that would generate an event horizon of toroidal topology. We found that a good estimate for this ring critical radius is $20/(3\pi) M$. We briefly discuss the connection of this two cases through a discrete black hole 'necklace' configuration.Comment: 31 pages, 21 figure

Generic effective source for scalar self-force calculations

A leading approach to the modelling of extreme mass ratio inspirals involves the treatment of the smaller mass as a point particle and the computation of a regularized self-force acting on that particle. In turn, this computation requires knowledge of the regularized retarded field generated by the particle. A direct calculation of this regularized field may be achieved by replacing the point particle with an effective source and solving directly a wave equation for the regularized field. This has the advantage that all quantities are finite and require no further regularization. In this work, we present a method for computing an effective source which is finite and continuous everywhere, and which is valid for a scalar point particle in arbitrary geodesic motion in an arbitrary background spacetime. We explain in detail various technical and practical considerations that underlie its use in several numerical self-force calculations. We consider as examples the cases of a particle in a circular orbit about Schwarzschild and Kerr black holes, and also the case of a particle following a generic time-like geodesic about a highly spinning Kerr black hole. We provide numerical C code for computing an effective source for various orbital configurations about Schwarzschild and Kerr black holes.Comment: 24 pages, 7 figures, final published versio

Numerical relativity with characteristic evolution, using six angular patches

The characteristic approach to numerical relativity is a useful tool in evolving gravitational systems. In the past this has been implemented using two patches of stereographic angular coordinates. In other applications, a six-patch angular coordinate system has proved effective. Here we investigate the use of a six-patch system in characteristic numerical relativity, by comparing an existing two-patch implementation (using second-order finite differencing throughout) with a new six-patch implementation (using either second- or fourth-order finite differencing for the angular derivatives). We compare these different codes by monitoring the Einstein constraint equations, numerically evaluated independently from the evolution. We find that, compared to the (second-order) two-patch code at equivalent resolutions, the errors of the second-order six-patch code are smaller by a factor of about 2, and the errors of the fourth-order six-patch code are smaller by a factor of nearly 50.Comment: 12 pages, 5 figures, submitted to CQG (special NFNR issue