Cosmic Microwave Background Data Analysis : From Time-Ordered Data To Power SpectraCMB Science - ICMB Science - IICMB Science - IIIPowerPoint PresentationOverview (I)Overview (II)GoalsData CompressionData ParametersComputational ConstraintsSlide 12Map-Making - Formalism (I)Map-Making - Formalism (II)Aside : Noise EstimationMap-Making - Formalism (III)Map-Making - Algorithms (I)Map-Making - Algorithms (II)Map-Making - Algorithms (III)Map-Making - Algorithms (IV)Map-Making - Extensions (I)Map-Making - Extensions (II)Map-Making - Extensions (III)Map-Making - Implementation (I)Map-Making - Implementation (II)Map-Making - Implementation (III)Map-Making - Implementation (IV)Map-Making - Implementation (V)Map-Making - ConclusionsSlide 30Power Spectrum EstimationPower Spectrum - Formalism (I)Power Spectrum - Formalism (II)Power Spectrum - Formalism (III)Power Spectrum - Formalism (IV)Power Spectrum - Algorithms (I)Power Spectrum - Algorithms (II)Power Spectrum - Algorithms (III)Power Spectrum - Algorithms (IV)Power Spectrum - Algorithms (VI)Power Spectrum - Implementation (I)Power Spectrum - Implementation (II)Power Spectrum - Implementation (III)Power Spectrum - Implementation (IV)Aside : Fisher Matrix (I)Aside : Fisher Matrix (II)Power Spectrum - Implementation (V)Aside : Parameterizing ArchitecturesConclusionsCosmic Microwave Background Data Analysis : From Time-Ordered Data To Power SpectraJulian BorrillComputational Research Division, Berkeley Lab& Space Sciences Laboratory, UC BerkeleyCMB Science - IThe CMB is a snapshot of the Universe when it first became neutral 400,000 years after the Big Bang.Cosmic - filling all of space.Microwave - redshifted by the expansion of the Universe from 3000K to 3K.Background - primordial photons coming from “behind” all astrophysical sources.CMB Science - IIThe CMB is a unique probe of the very early Universe.Its tiny (1:105-8) fluctuations carry information about - the fundamental parameters of cosmology - ultra-high energy physics beyond the Standard ModelCMB Science - IIIThe new frontier in CMB research is polarization :• consistency check of temperature results• re-ionization history of the Universe• gravity wave production during inflationBut polarization fluctuations are up to 3 orders of magnitude fainter than temperature (we think) requiring :• many more detectors• much longer observation times• very careful analysis of very large datasetsThe Planck Satellite•A joint ESA/NASA mission due to launch in fall 2007.•An 18+ month all-sky survey at 9 microwave frequencies from 30 to 857 GHz.•O(1012) observations, O(108) sky pixels, O(104) spectral multipoles.CMB analysis moves from the time domain - observations to the pixel domain - maps to the multipole domain - power spectracalculating the compressed data and their reduced error bars at each step.Overview (I)Overview (II)CMB data analysis typically proceeds in 4 steps:- Pre-processing (deglitching, pointing, calibrating).- Estimating the time-domain noise statistics.- Estimating the map (and its errors).- Estimating the power spectra (and their errors).iterating & looping as we learn more about the data. Then we can ask about the likelihoods of the parameters of any particular class of cosmologies.GoalsTo cover the(a)basic mathematical formalism(b)algorithms & their scaling behaviours(c) example implementation issuesfor map-making and power spectrum estimation. To consider how to extract the maximum amount of information from the data, subject to practical computational constraints. To illustrate some of the computational issues faced when analyzing very large datasets (eg. Planck).Data CompressionCMB data analysis is an exercise in data compression:1. Time-ordered data: #samples = #detectors x sampling rate x duration ~ 70 x 200Hz x 18 months for Planck2. (HEALPixelized) Maps: #pixels = #components x sky fraction x 12 nside2 ~ (3 - 6) x 1 x 12 x 40962 for Planck3. Power Spectra: #bins = #spectra x #multipoles / bin resolution ~ 6 x (3 x 103) / 1 for PlanckData ParametersSymbol Description PlanckNumber of samples 5 x 1011Noise bandwidth O(104)Number of pixels 6 x 108Number of spectra 6Maximum multipole 3 x 103Number of spectral bins2 x 104Number of iterations -Number of realizations -Computational Constraints•1 GHz processor running at 100% efficiency for 1 day performs O(1014) operations.•1 Gbyte of memory can hold O(108) element vector, or O(104 x 104) matrix, in 64-bit precision.•Parallel (multiprocessor) computing increases the operation count and memory limits.•Challenges to computational efficiency & scaling:- load balancing (work & memory)- data-delivery, including communication & I/OMap-Making - Formalism (I)Consider data consisting of noise and sky-signalwhere the pointing matrix A encodes the weight of eachpixel p in each sample t - for a total power temperatureobservationand the sky-signal sp is both beam & pixel smoothed.Map-Making - Formalism (II)Assume Gaussian noise with probability distributionand a time-time noise correlation matrixwhose inverse is (piecewise) stationary & band-limitedAside : Noise EstimationTo make the map we need the inverse time-time noise correlations. Approximate:which requires the pure noise timestream. i) Assume nt = dt ii) Solve for the map: dp ~ spiii) Subtract the map from the data: nt = dt - Atp dpiv) Iterate.Map-Making - Formalism (III)Writing the noise in terms of the data and signal & maximizing its likelihood over all possible signals gives the minimum variance mapwith pixel-pixel noise correlationsTaken together, these are a complete description of the data.Map-Making - Algorithms (I)We want to solve the system: Eg. (5 x 1011)2 x (6 x 108) ~ 2 x 1032 for Planck !Equation Naive OpCountMap-Making - Algorithms (II)a) Exploit the structure of the matrices–Pointing matrix is sparse–Inverse noise correlation matrix is band-ToeplitzAssociated matrix-matrix & -vector multiplication scalings reduced from & to .Eg. (5 x 1011) x 104 ~ 5 x 1015 for Planck.Map-Making - Algorithms (III)b) Replace explicit matrix inversion with an iterative solver (eg. preconditioned conjugate gradient) using repeated matrix-vector multiplicationsreducing the scaling from to . depends on the required solution accuracy and the quality of the preconditioner (white noise works well).Eg. 30 x (6 x 108)2 ~ 1019 for Planck.Map-Making - Algorithms (IV)c) Leave the inverse pixel-pixel noise matrix
View Full Document
Unlocking...