Efficient Wiener filtering without preconditioning

We present a new approach to calculate the Wiener filter solution of general data sets. It is trivial to implement, flexible, numerically absolutely stable, and guaranteed to converge. Most importantly, it does not require an ingenious choice of preconditioner to work well. The method is capable of taking into account inhomogeneous noise distributions and arbitrary mask geometries. It iteratively builds up the signal reconstruction by means of a messenger field, introduced to mediate between the different preferred bases in which signal and noise properties can be specified most conveniently. Using cosmic microwave background (CMB) radiation data as a showcase, we demonstrate the capabilities of our scheme by computing Wiener filtered WMAP7 temperature and polarization maps at full resolution for the first time. We show how the algorithm can be modified to synthesize fluctuation maps, which, combined with the Wiener filter solution, result in unbiased constrained signal realizations, consistent with the observations. The algorithm performs well even on simulated CMB maps with Planck resolution and dynamic range.Comment: 5 pages, 2 figures. Submitted to Astronomy and Astrophysics. Replaced to match published versio

ARKCoS: Artifact-Suppressed Accelerated Radial Kernel Convolution on the Sphere

We describe a hybrid Fourier/direct space convolution algorithm for compact radial (azimuthally symmetric) kernels on the sphere. For high resolution maps covering a large fraction of the sky, our implementation takes advantage of the inexpensive massive parallelism afforded by consumer graphics processing units (GPUs). Applications involve modeling of instrumental beam shapes in terms of compact kernels, computation of fine-scale wavelet transformations, and optimal filtering for the detection of point sources. Our algorithm works for any pixelization where pixels are grouped into isolatitude rings. Even for kernels that are not bandwidth limited, ringing features are completely absent on an ECP grid. We demonstrate that they can be highly suppressed on the popular HEALPix pixelization, for which we develop a freely available implementation of the algorithm. As an example application, we show that running on a high-end consumer graphics card our method speeds up beam convolution for simulations of a characteristic Planck high frequency instrument channel by two orders of magnitude compared to the commonly used HEALPix implementation on one CPU core while maintaining at typical a fractional RMS accuracy of about 1 part in 10^5.Comment: 10 pages, 6 figures. Submitted to Astronomy and Astrophysics. Replaced to match published version. Code can be downloaded at https://github.com/elsner/arkco

Fast calculation of the Fisher matrix for cosmic microwave background experiments

The Fisher information matrix of the cosmic microwave background (CMB) radiation power spectrum coefficients is a fundamental quantity that specifies the information content of a CMB experiment. In the most general case, its exact calculation scales with the third power of the number of data points N and is therefore computationally prohibitive for state-of-the-art surveys. Applicable to a very large class of CMB experiments without special symmetries, we show how to compute the Fisher matrix in only O(N^2 log N) operations as long as the inverse noise covariance matrix can be applied to a data vector in time O(l_max^3 log l_max). This assumption is true to a good approximation for all CMB data sets taken so far. The method takes into account common systematics such as arbitrary sky coverage and realistic noise correlations. As a consequence, optimal quadratic power spectrum estimation also becomes feasible in O(N^2 log N) operations for this large group of experiments. We discuss the relevance of our findings to other areas of cosmology where optimal power spectrum estimation plays a role.Comment: 4 pages, 1 figures. Accepted for publication in Astronomy and Astrophysics Letters. Replaced to match published versio

Using hybrid GPU/CPU kernel splitting to accelerate spherical convolutions

We present a general method for accelerating by more than an order of magnitude the convolution of pixelated functions on the sphere with a radially-symmetric kernel. Our method splits the kernel into a compact real-space component and a compact spherical harmonic space component. These components can then be convolved in parallel using an inexpensive commodity GPU and a CPU. We provide models for the computational cost of both real-space and Fourier space convolutions and an estimate for the approximation error. Using these models we can determine the optimum split that minimizes the wall clock time for the convolution while satisfying the desired error bounds. We apply this technique to the problem of simulating a cosmic microwave background (CMB) anisotropy sky map at the resolution typical of the high resolution maps produced by the Planck mission. For the main Planck CMB science channels we achieve a speedup of over a factor of ten, assuming an acceptable fractional rms error of order 1.e-5 in the power spectrum of the output map.Comment: 9 pages, 11 figures, 1 table, accepted by Astronomy & Computing w/ minor revisions. arXiv admin note: substantial text overlap with arXiv:1211.355

Application of the equipartition theorem to the thermal excitation of quartz tuning forks

The deflection signal of a thermally excited force sensor of an atomic force microscope can be analyzed to gain important information about the detector noise and about the validity of the equipartion theorem of thermodynamics. Here, we measured the temperature dependence of the thermal amplitude of a tuning fork and compared it to the expected values based on the equipartition theorem. In doing so, we prove the validity of these assumptions in the temperature range from 140K to 300K. Furthermore, the application of the equipartition theorem to quartz tuning forks at liquid helium temperatures is discussed.Comment: 8 pages, 3 figures, published in Applied Physics Letter

Forecasting constraints from the cosmic microwave background on eternal inflation

We forecast the ability of cosmic microwave background (CMB) temperature and polarization datasets to constrain theories of eternal inflation using cosmic bubble collisions. Using the Fisher matrix formalism, we determine both the overall detectability of bubble collisions and the constraints achievable on the fundamental parameters describing the underlying theory. The CMB signatures considered are based on state-of-the-art numerical relativistic simulations of the bubble collision spacetime, evolved using the full temperature and polarization transfer functions. Comparing a theoretical cosmic-variance-limited experiment to the WMAP and Planck satellites, we find that there is no improvement to be gained from future temperature data, that adding polarization improves detectability by approximately 30%, and that cosmic-variance-limited polarization data offer only marginal improvements over Planck. The fundamental parameter constraints achievable depend on the precise values of the tensor-to-scalar ratio and energy density in (negative) spatial curvature. For a tensor-to-scalar ratio of $0.1$ and spatial curvature at the level of $10^{-4}$, using cosmic-variance-limited data it is possible to measure the width of the potential barrier separating the inflating false vacuum from the true vacuum down to $M_{\rm Pl}/500$, and the initial proper distance between colliding bubbles to a factor $\pi/2$ of the false vacuum horizon size (at three sigma). We conclude that very near-future data will have the final word on bubble collisions in the CMB.Comment: 14 pages, 6 figure

Search for non-Gaussian Signatures in the Cosmic Microwave Background Radiation

The tremendous impact of Cosmic Microwave Background (CMB) radiation experiments on our understanding of the history and evolution of the universe is based on a tight connection between the observed fluctuations and the physical processes taking place in the very early universe. According to the prevalent paradigm, the anisotropies were generated during the era of inflation. The simplest inflationary models predict almost perfectly Gaussian primordial perturbations, but competitive theories can naturally be constructed, allowing for a wide range in primordial non-Gaussianity. For this reason, the test for non-Gaussianity becomes a fundamental means to probe the physical processes of inflation. The aim of the project is to develop a Bayesian formalism to infer the level of non-Gaussianity of local type. Bayesian statistics attaches great importance to a consistent formulation of the problem and properly calculates the error bounds of the measurements on the basis of the actual data. As a first step, we develop an exact algorithm to generate simulated temperature and polarization CMB maps containing arbitrary levels of local non-Gaussianity. We derive an optimization scheme that allows us to predict and actively control the simulation accuracy. Implementing this strategy, the code outperforms existing algorithms in computational efficiency by an order of magnitude. Then, we develop the formalism to extend the Bayesian approach to the calculation of the amplitude of non-Gaussianity. We implement an exact Hamiltonian Monte Carlo sampling algorithm to generate samples from the target probability distribution. These samples allow to construct the full posterior distribution of the level of non-Gaussianity given the data. The applicability of the scheme is demonstrated by means of a simplified data model. Finally, we fully implement the necessary equations considering a realistic CMB experiment dealing with partial sky coverage and anisotropic noise. A direct comparison between the traditional frequentist estimator and the exact Bayesian approach shows the advantage of the newly developed method. For a significant detection of non-Gaussianity, the former suffers from excess variance whereas the Bayesian scheme always provides optimal error bounds

Improved simulation of non-Gaussian temperature and polarization CMB maps

We describe an algorithm to generate temperature and polarization maps of the cosmic microwave background radiation containing non-Gaussianity of arbitrary local type. We apply an optimized quadrature scheme that allows us to predict and control integration accuracy, speed up the calculations, and reduce memory consumption by an order of magnitude. We generate 1000 non-Gaussian CMB temperature and polarization maps up to a multipole moment of l_max = 1024. We validate the method and code using the power spectrum and the fast cubic (bispectrum) estimator and find consistent results. The simulations are provided to the community.Comment: 18 pages, 19 figures. Accepted for publication in ApJS. Simulations can be obtained at http://planck.mpa-garching.mpg.de/cmb/fnl-simulation