Moments of zeta and correlations of divisor-sums: III

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^2$ with divisor functions as coefficients

Precision Tests of Parity Violation Over Cosmological Distances

Recent measurements of the Cosmic Microwave Background $B$-mode polarization power spectrum by the BICEP2 and POLARBEAR experiments have demonstrated new precision tools for probing fundamental physics. Regardless of origin, the fact that we can detect sub-$\mu$K CMB polarization represents a tremendous technological breakthrough. Yet more information may be latent in the CMB's polarization pattern. Because of its tensorial nature, CMB polarization may also reveal parity-violating physics via a detection of cosmic polarization rotation. Although current CMB polarimeters are sensitive enough to measure one degree-level polarization rotation with $>5\sigma$ statistical significance, they lack the ability to differentiate this effect from a systematic instrumental polarization rotation. Here, we motivate the search for cosmic polarization rotation from current CMB data as well as independent radio galaxy and quasar polarization measurements. We argue that an improvement in calibration accuracy would allow the precise measurement of parity- and Lorentz-violating effects. We describe the CalSat space-based polarization calibrator that will provide stringent control of systematic polarization angle calibration uncertainties to $0.05^\circ$ -- an order of magnitude improvement over current CMB polarization calibrators. CalSat-based calibration could be used with current CMB polarimeters searching for $B$-mode polarization, effectively turning them into probes of cosmic parity violation, i.e. without the need to build dedicated instruments.Comment: 11 pages, 3 figure

CMB Polarization Systematics Due to Beam Asymmetry: Impact on Inflationary Science

CMB polarization provides a unique window into cosmological inflation; the amplitude of the B-mode polarization from last scattering is uniquely sensitive to the energetics of inflation. However, numerous systematic effects arising from optical imperfections can contaminate the observed B-mode power spectrum. In particular, systematic effects due to the coupling of the underlying temperature and polarization fields with elliptical or otherwise asymmetric beams yield spurious systematic signals. This paper presents a non-perturbative analytic calculation of some of these signals. We show that results previously derived in real space can be generalized, formally, by including infinitely many higher-order corrections to the leading order effects. These corrections can be summed and represented as analytic functions when a fully Fourier-space approach is adopted from the outset. The formalism and results presented in this paper were created to determine the susceptibility of CMB polarization probes of the primary gravitational wave signal but can be easily extended to the analysis of gravitational lensing of the CMB.Comment: 14 pages, 11 figures, 6 tables. Minor corrections included to match published versio

Large Angular Scale Polarization of the Cosmic Microwave Background and the Feasibility of its Detection

In addition to its spectrum and temperature anisotropy, the 2.7K Cosmic Microwave Background is also expected to exhibit a low level of polarization. The spatial power spectrum of the polarization can provide details about the formation of structure in the universe as well as its ionization history. Here we calculate the magnitude of the CMB polarization in various cosmological scenarios, with both an analytic and a numerical method. We then outline the fundemental challenges to measuring these signals and focus on two of them: achieving adequate sensitivity and removing contamination from foreground sources. We then describe the design of a ground based instrument (POLAR) that could detect polarization of the CMB at large angular scales in the new few years.Comment: 40 pages, 7 figures, accepted for publication in the Astrophysical Journa

CMB Polarization Systematics Due to Beam Asymmetry: Impact on Inflationary Science

CMB polarization provides a unique window into cosmological inflation; the amplitude of the B-mode polarization from last scattering is uniquely sensitive to the energetics of inflation. However, numerous systematic effects arising from optical imperfections can contaminate the observed B-mode power spectrum. In particular, systematic effects due to the coupling of the underlying temperature and polarization fields with elliptical or otherwise asymmetric beams yield spurious systematic signals. This paper presents a non-perturbative analytic calculation of some of these signals. We show that results previously derived in real space can be generalized, formally, by including infinitely many higher-order corrections to the leading order effects. These corrections can be summed and represented as analytic functions when a fully Fourier-space approach is adopted from the outset. The formalism and results presented in this paper were created to determine the susceptibility of CMB polarization probes of the primary gravitational wave signal but can be easily extended to the analysis of gravitational lensing of the CMB.Comment: 14 pages, 11 figures, 6 tables. Minor corrections included to match published versio

Methods for stabilizing high Reynolds number Lattice Boltzmann simulations

The Lattice Boltzmann Method (LBM) is a simple and highly efficient method for computing nearly incompressible fluid flow. However, it is well known to suffer from numerical instabilities for low values of the transport coefficients. This dissertation examines a number of methods for increasing the stability of the LBM over a wide range of parameters. First, we consider a simple transformation that renders the standard LB equation implicit. It is found that the stability is largely unchanged. Next, we consider a stabilization method based on introducing a Lyapunov function which is essentially a discrete-time H-function. The uniqueness of an H-function that appears in the literature is proven, and the method is extended to stabilize some of the more popular LB models. We also introduce a new method for implementing boundary conditions in the LBM. The hydrodynamic fields are imposed in a transformed moment space, whereas The non-hydrodynamic fields are shifted over from neighboring nodes. By minimizing population gradients, this method exhibits superior numerical stability over other widely employed schemes when tested on the widely-used benchmark of incompressible flow over a backwards-facing step