Fast and Accurate Algorithms for Cosmic Microwave Background Radiation Data on HEALPix Points Kathryn P. Drake Grady B. Wright Supported by NASA Idaho Space Grant Consortium Boise State University Boise State University Motivation Current Method: HEALPix Numerical Test #1 Faintly glowing at the edge of the observable universe, the Cosmic Step 1: “Zeroth” Order Approximation <latexit sha1_base64="HnunS9VSEFab1LJstn7fZlV23pA=">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</latexit> by HP2SPH (blue) and Absolute error for the angular power spectrum of f for 0 ` 2000 (left) and 0 ` 100 (right) calculated<latexit sha1_base64="3Ll9kKBMy30sdkbZjWd7jRWuiO0=">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</latexit> by <latexit sha1_base64="nEcKyHbLgppwFC7rCdmQo/oNFzw=">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</latexit> Microwave Background (CMB) is “relic radiation” that represents the HEALPix software estimates the angular power spectrum of a function f on the sphere NASA’s HEALPix software (red) for 12582912 nodes. <latexit sha1_base64="PVHsnbaXS+mUIpN21BOOKD1jGqo=">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</latexit> first light to travel during the early stages of the universe's development by<latexit sha1_base64="7EdS6bq7oGc8pfkX+0tbyOdMcu8=">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</latexit> first using a low-order approximation to calculate the spherical harmonic coefficients: and gives the strongest evidence for the Big Bang theory. Since the Npix m 4⇡ m b` Y ` (λi, ✓i)f(λi, ✓i), 0 ` `max, `max m `max discovery of the CMBR in 1964, scientists have worked to measure it in ⇡ Npix − <latexit sha1_base64="cqL8TuVz0hC/q3z7cVSBcjZOrQg=">AAACoXicbZFtaxQxEMez61M9H3rqS0GCh1DhPHZrwaIIBX2hLyyneG3lcl2yuWwvNNmkyazcEfZ7+Tl857cxu7dYvToQ+M1/MjPJTG6kcJAkv6L42vUbN29t3e7duXvv/nb/wcMjpyvL+IRpqe1JTh2XouQTECD5ibGcqlzy4/z8XRM//s6tE7r8CivDZ4qelaIQjEKQsv6PPCNcylNFqDFWL0lhKfN7xIjaH2beiGVdE1epzIu3aX16qelQtWnqv9VdhR0iQ985zcSQwIJDgOfF/8QhJm9wQiS/wE0m/kOZV3RZt+EXl34bVpu3sv4gGSWt4auQdjBAnY2z/k8y16xSvAQmqXPTNDEw89SCYJLXPVI5big7p2d8GrCkiruZbydc42dBmeNC23BKwK36d4anyrmVyoc4gKKwaAEWaogbD7SWbrhO2+gExf7Mi9JUwEu2blRUEoPGzbrwXFjOQK4CUGZFeCtmCxp2BGGpvTCEdPPLV+Fod5S+HO1+3hscvO7GsYUeo6doB6XoFTpAH9AYTRCLnkTvo0/RYTyIP8bj+Mv6ahx1OY/QPxZPfwPWcM/b</latexit> i=1 full detail using a Hierarchical Equal Area isoLatitude Pixelization 3/2 Due to the isolatitudeX nature of the pixels, this computation is done with (N ) <latexit sha1_base64="nhN0esx2L+dKI6vjwde9QOtC7/k=">AAACaXicZVHda9RAEN/Ej57n11VfRF8GU6HCcU2uoCIIBX3wSSt4baE5j81m0izd7Ibdie0R8n/qm+/+D+Je7h6sHVj47cz85uM3Wa2kozj+GYQ3bt66vTW4M7x77/6Dh6PtR0fONFbgTBhl7EnGHSqpcUaSFJ7UFnmVKTzOzt+v4sff0Tpp9Fda1jiv+JmWhRScvGsxsqk2UueoCT40CGSASgTpjPIJ1OQImlNjEUzRR2p5icqNPZYOhKnqhvpKngK50QgXkkrYSStOpeCq/dztflq0ntV9a/f3pt3LHViMongS9wbXQbIBEdvY4WL0K82NaCo/pFDcudMkrmnecktSKOyGaeOw5uKcn+Gph5pX6OZtr00HL7wnh8JY//ySvfdfRssr55ZVNgYPVlP3gMpqDKsfGbNad13saicq3sxbqb0CqMW6UdGoXkIvNOTSoiC19IALK/2sIEpuuSB/jmFKeEkXMqfy3Svp7xAl/wtwHRxNJ8n+JP4yjQ7ebsQZsGfsOdtlCXvNDthHdshmTLAf7E+wFQyC3+F2+CR8uk4Ngw3nMbtiYfQXUMm5/g==</latexit> pix scheme on the sphere. While these “HEALPix” points allow for a Step 2: Iterative Refinement O <latexit sha1_base64="2PrZIVX/SRA21N1uqK6wv3lULak=">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</latexit> quasiuniform discretization of the sphere, they are not well suited for the In order to improve the estimate of the bms, HEALPix uses an iterative process. <latexit sha1_base64="EEIt1WeuQykzPprqiff9TpoJgZk=">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</latexit> ` fast algorithms necessary for mining the massive CMBR data sets. For `max ` m ¯ m this work, we apply the Double Fourier Sphere method and fast Fourier Analysis (function to coefficients): b = Y (λ, ✓)f(λ, ✓) <latexit sha1_base64="wm5q2gE8roxtAu/VLm1R8v+vQog=">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</latexit> b = Af <latexit sha1_base64="KVpHnQ2tEMccIi2+e9D9nJyf1CY=">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</latexit> ` ` () transforms for uniform and non-uniform point distributions in order to <latexit sha1_base64="EZp3Ti5xaJo+5XYmCseoXO+DFpA=">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</latexit>