Power Spectrum Estimation in Theory and in Practice Adrian Liu, MIT

What we would like to do Inverse noise and foreground covariance matrix Vector containing measurement

What we would like to do Bandpower at k  “Geometry” -- Fourier transform, binning Noise/residual foreground bias removal

The Essence of the Method For similar methods, see also N. Petrovic & S.P. Oh, MNRAS 413, 2103 (2011) G. Paciga et. al., MNRAS 413, 1174 (2011) Filter Rapidly fluctuating modes retained Smooth modes suppressed High foreground scenario Foregroundless scenario

Why we like this method Lossless Cleaned Data Raw Data Cleaning

Why we like this method Lossless Smaller “vertical” error bars

Why we like this method Lossless Smaller “vertical” error bars mK 1 K 100 mK Log 10 T (in mK) Errors using Line of Sight Method AL, Tegmark, Phys. Rev. D 83, (2011)

Why we like this method Lossless Smaller “vertical” error bars <10 mK 130 mK Log 10 T (in mK) Errors using Inverse Variance Method 30 mK AL, Tegmark, Phys. Rev. D 83, (2011)

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars AL, Tegmark, Phys. Rev. D 83, (2011)

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars AL, Tegmark, Phys. Rev. D 83, (2011)

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars No additive noise/foreground bias

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars No additive noise/foreground bias A systematic framework for evaluating error statistics

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars No additive noise/foreground bias A systematic framework for evaluating error statistics BUT

Why we like this method Lossless Smaller “vertical” error bars Smaller “horizontal” error bars No additive noise/foreground bias A systematic framework for evaluating error statistics BUT Computationally expensive because matrix inverse scales as O(n 3 ). [Recall C -1 x] Error statistics for 16 by 16 by 30 dataset takes CPU-months

Quicker alternatives Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams, AL, Hewitt, Tegmark

Quicker alternatives Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams, AL, Hewitt, Tegmark

O(n log n) version Finding the matrix inverse C -1 is the slowest step.

O(n log n) version Finding the matrix inverse C -1 is the slowest step. Use the conjugate gradient method for finding C -1 x, which only requires being able to multiply by Cx.

O(n log n) version Finding the matrix inverse C -1 is the slowest step. Use the conjugate gradient method for finding C -1, which only requires being able to multiply by C. Multiplication is quick in basis where matrices are diagonal.

O(n log n) version Finding the matrix inverse C -1 is the slowest step. Use the conjugate gradient method for finding C -1, which only requires being able to multiply by C. Multiplication is quick in basis where matrices are diagonal. Need to multiply by C = C noise + C sync + C ps + …

Different components are diagonal in different combinations of Fourier space C = C ps + C sync + C noise + … Real spatial Fourier spectral Fourier spatial Fourier spectral Real spatial Real spectral

Comparison of Foreground Models GSM Our model Eigenvalue AL, Pritchard, Loeb, Tegmark, in prep.

Quicker alternatives Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams, AL, Hewitt, Tegmark

FKP + FFT version Bandpower at k  “Geometry” -- Fourier transform, binning Noise/residual foreground bias removal

FKP + FFT version Foreground avoidance instead of foreground subtraction mK 1 K 100 mK

FKP + FFT version Foreground avoidance instead of foreground subtraction. Use FFTs to get O(n log n) scaling, adjusting for non- cubic geometry using weightings.

FKP + FFT version Foreground avoidance instead of foreground subtraction. Use FFTs to get O(n log n) scaling, adjusting for non- cubic geometry using weightings. Use Feldman-Kaiser-Peacock (FKP) approximation –Power estimates from neighboring k-cells perfectly correlated and therefore redundant. –Power estimates from far away k-cells uncorrelated. –Approximation encapsulated by FKP weighting. –Optimal (same as full inverse variance method) on scales much smaller than survey volume.

FKP + FFT version mK 1 K 100 mK

Summary Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams, AL, Hewitt, Tegmark