Highly eccentric inspirals into a black hole

We model the inspiral of a compact stellar-mass object into a massive nonrotating black hole including all dissipative and conservative first-order-in-the-mass-ratio effects on the orbital motion. The techniques we develop allow inspirals with initial eccentricities as high as $e\sim0.8$ and initial separations as large as $p\sim 50$ to be evolved through many thousands of orbits up to the onset of the plunge into the black hole. The inspiral is computed using an osculating elements scheme driven by a hybridized self-force model, which combines Lorenz-gauge self-force results with highly accurate flux data from a Regge-Wheeler-Zerilli code. The high accuracy of our hybrid self-force model allows the orbital phase of the inspirals to be tracked to within $\sim0.1$ radians or better. The difference between self-force models and inspirals computed in the radiative approximation is quantified.Comment: Updated to reflect published versio

On sums of three squares

Let $r_3(n)$ be the number of representations of a positive integer $n$ as a sum of three squares of integers. We give two distinct proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of $r_3(n)$.Comment: 11 pages, minor revisions made; to appear in Internat. J. Number Theor

Fast spectral source integration in black hole perturbation calculations

This paper presents a new technique for achieving spectral accuracy and fast computational performance in a class of black hole perturbation and gravitational self-force calculations involving extreme mass ratios and generic orbits. Called \emph{spectral source integration} (SSI), this method should see widespread future use in problems that entail (i) point-particle description of the small compact object, (ii) frequency domain decomposition, and (iii) use of the background eccentric geodesic motion. Frequency domain approaches are widely used in both perturbation theory flux-balance calculations and in local gravitational self-force calculations. Recent self-force calculations in Lorenz gauge, using the frequency domain and method of extended homogeneous solutions, have been able to accurately reach eccentricities as high as $e \simeq 0.7$. We show here SSI successfully applied to Lorenz gauge. In a double precision Lorenz gauge code, SSI enhances the accuracy of results and makes a factor of three improvement in the overall speed. The primary initial application of SSI--for us its \emph{raison d'\^{e}tre}--is in an arbitrary precision \emph{Mathematica} code that computes perturbations of eccentric orbits in the Regge-Wheeler gauge to extraordinarily high accuracy (e.g., 200 decimal places). These high accuracy eccentric orbit calculations would not be possible without the exponential convergence of SSI. We believe the method will extend to work for inspirals on Kerr, and will be the subject of a later publication. SSI borrows concepts from discrete-time signal processing and is used to calculate the mode normalization coefficients in perturbation theory via sums over modest numbers of points around an orbit. A variant of the idea is used to obtain spectral accuracy in solution of the geodesic orbital motion.Comment: 15 pages, 7 figure

Evolution of small-mass-ratio binaries with a spinning secondary

We calculate the evolution and gravitational-wave emission of a spinning compact object inspiraling into a substantially more massive (non-rotating) black hole. We extend our previous model for a non-spinning binary [Phys. Rev. D 93, 064024] to include the Mathisson-Papapetrou-Dixon spin-curvature force. For spin-aligned binaries we calculate the dephasing of the inspiral and associated waveforms relative to models that do not include spin-curvature effects. We find this dephasing can be either positive or negative depending on the initial separation of the binary. For binaries in which the spin and orbital angular momentum are not parallel, the orbital plane precesses and we use a more general osculating element prescription to compute inspirals.Comment: 17 pages, 6 figure

On the p-adic geometry of traces of singular moduli

The aim of this article is to show that p-adic geometry of modular curves is useful in the study of p-adic properties of traces of singular moduli. In order to do so, we partly answer a question by Ono. As our goal is just to illustrate how p-adic geometry can be used in this context, we focus on a relatively simple case, in the hope that others will try to obtain the strongest and most general results. For example, for p=2, a result stronger than Thm.1 is proved in [Boylan], and a result on some modular curves of genus zero can be found in [Osburn] . It should be easy to apply our method, because of its local nature, to modular curves of arbitrary level, as well as to Shimura curves.Comment: 3 pages, Late

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.Comment: 8 pages, submitte

Fractionation of Hydrogen Isotopes by Sulfate- and Nitrate-Reducing Bacteria.

Hydrogen atoms from water and food are incorporated into biomass during cellular metabolism and biosynthesis, fractionating the isotopes of hydrogen-protium and deuterium-that are recorded in biomolecules. While these fractionations are often relatively constant in plants, large variations in the magnitude of fractionation are observed for many heterotrophic microbes utilizing different central metabolic pathways. The correlation between metabolism and lipid δ(2)H provides a potential basis for reconstructing environmental and ecological parameters, but the calibration dataset has thus far been limited mainly to aerobes. Here we report on the hydrogen isotopic fractionations of lipids produced by nitrate-respiring and sulfate-reducing bacteria. We observe only small differences in fractionation between oxygen- and nitrate-respiring growth conditions, with a typical pattern of variation between substrates that is broadly consistent with previously described trends. In contrast, fractionation by sulfate-reducing bacteria does not vary significantly between different substrates, even when autotrophic and heterotrophic growth conditions are compared. This result is in marked contrast to previously published observations and has significant implications for the interpretation of environmental hydrogen isotope data. We evaluate these trends in light of metabolic gene content of each strain, growth rate, and potential flux and reservoir-size effects of cellular hydrogen, but find no single variable that can account for the differences between nitrate- and sulfate-respiring bacteria. The emerging picture of bacterial hydrogen isotope fractionation is therefore more complex than the simple correspondence between δ(2)H and metabolic pathway previously understood from aerobes. Despite the complexity, the large signals and rich variability of observed lipid δ(2)H suggest much potential as an environmental recorder of metabolism

Chemolithotrophy in the continental deep subsurface: Sanford Underground Research Facility (SURF), USA

The deep subsurface is an enormous repository of microbial life. However, the metabolic capabilities of these microorganisms and the degree to which they are dependent on surface processes are largely unknown. Due to the logistical difficulty of sampling and inherent heterogeneity, the microbial populations of the terrestrial subsurface are poorly characterized. In an effort to better understand the biogeochemistry of deep terrestrial habitats, we evaluate the energetic yield of chemolithotrophic metabolisms and microbial diversity in the Sanford Underground Research Facility (SURF) in the former Homestake Gold Mine, SD, USA. Geochemical data, energetic modeling, and DNA sequencing were combined with principle component analysis to describe this deep (down to 8100 ft below surface), terrestrial environment. SURF provides access into an iron-rich Paleoproterozoic metasedimentary deposit that contains deeply circulating groundwater. Geochemical analyses of subsurface fluids reveal enormous geochemical diversity ranging widely in salinity, oxidation state (ORP 330 to −328 mV), and concentrations of redox sensitive species (e.g., Fe(2+) from near 0 to 6.2 mg/L and Σ S(2-) from 7 to 2778μg/L). As a direct result of this compositional buffet, Gibbs energy calculations reveal an abundance of energy for microorganisms from the oxidation of sulfur, iron, nitrogen, methane, and manganese. Pyrotag DNA sequencing reveals diverse communities of chemolithoautotrophs, thermophiles, aerobic and anaerobic heterotrophs, and numerous uncultivated clades. Extrapolated across the mine footprint, these data suggest a complex spatial mosaic of subsurface primary productivity that is in good agreement with predicted energy yields. Notably, we report Gibbs energy normalized both per mole of reaction and per kg fluid (energy density) and find the later to be more consistent with observed physiologies and environmental conditions. Further application of this approach will significantly expand our understanding of the deep terrestrial biosphere

Principal forms X^2 + nY^2 representing many integers

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.Comment: 10 pages, title has been changed, Sections 2 and 3 are new, to appear in Abh. Math. Sem. Univ. Hambur