Primordial Non-Gaussianity from a Joint Analysis of Cosmic Microwave Background Temperature and Polarization

We explore a systematic approach to the analysis of primordial non-Gaussianity using fluctuations in temperature and polarization of the Cosmic Microwave Background (CMB). Following Munshi & Heavens (2009), we define a set of power-spectra as compressed forms of the bispectrum and trispectrum derived from CMB temperature and polarization maps; these spectra compress the information content of the corresponding full multispectra and can be useful in constraining early Universe theories. We generalize the standard pseudo-C_l estimators in such a way that they apply to these spectra involving both spin-0 and spin-2 fields, developing explicit expressions which can be used in the practical implementation of these estimators. While these estimators are suboptimal, they are nevertheless unbiased and robust hence can provide useful diagnostic tests at a relatively small computational cost. We next consider approximate inverse-covariance weighting of the data and construct a set of near-optimal estimators based on that approach. Instead of combining all available information from the entire set of mixed bi- or trispectra, i.e multispectra describing both temperature and polarization information, we provide analytical constructions for individual estimators, associated with particular multispectra. The bias and scatter of these estimators can be computed using Monte-Carlo techniques. Finally, we provide estimators which are completely optimal for arbitrary scan strategies and involve inverse covariance weighting; we present the results of an error analysis performed using a Fisher-matrix formalism at both the one-point and two-point level.Comment: 25 Pages, 4 Figure

Optimal Estimation of Several Linear Parameters in the Presence of Lorentzian Thermal Noise

In a previous article we developed an approach to the optimal (minimum variance, unbiased) statistical estimation technique for the equilibrium displacement of a damped, harmonic oscillator in the presence of thermal noise. Here, we expand that work to include the optimal estimation of several linear parameters from a continuous time series. We show that working in the basis of the thermal driving force both simplifies the calculations and provides additional insight to why various approximate (not optimal) estimation techniques perform as they do. To illustrate this point, we compare the variance in the optimal estimator that we derive for thermal noise with those of two approximate methods which, like the optimal estimator, suppress the contribution to the variance that would come from the irrelevant, resonant motion of the oscillator. We discuss how these methods fare when the dominant noise process is either white displacement noise or noise with power spectral density that is inversely proportional to the frequency ($1/f$ noise). We also construct, in the basis of the driving force, an estimator that performs well for a mixture of white noise and thermal noise. To find the optimal multi-parameter estimators for thermal noise, we derive and illustrate a generalization of traditional matrix methods for parameter estimation that can accommodate continuous data. We discuss how this approach may help refine the design of experiments as they allow an exact, quantitative comparison of the precision of estimated parameters under various data acquisition and data analysis strategies.Comment: 16 pages, 10 figures. Accepted for publication in Classical and Quantum Gravit

Heterogeneous multireference alignment: a single pass approach

Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where $K$ signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the $K$ signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the $K$ signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals $K$ that can be resolved as a function of the signal length $L$ is on the order of $\sqrt{L}$.Comment: 6 pages, 3 figure