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
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 by HP2SPH (blue) and Absolute error for the angular power spectrum of f for 0 ` 2000 (left) and 0 ` 100 (right) 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 by
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 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.
AAACwXicZVFda9swFJW9r877aLY97kUsCUuhBNtjLBsdNJRBHsbI6JIW6pDJ8nUsKktGul4aQv/g/sHe9lOmOGFs7QXB0bk6917dk1ZSWAzDX55/5+69+w/2HgaPHj95ut969nxqdW04TLiW2pynzIIUCiYoUMJ5ZYCVqYSz9PJkkz/7AcYKrb7hqoJZyRZK5IIzdNS89TPoJkoLlYFCOkytljUCBWO0obk7WABlalFLZmill2CorYCjqUuqc9rJO82rTniUgJRHcRiGHdqTkOOBk2V/E1HDG7EoXIIzyV1BhCxJaNBNV3Q0jk/HI9pLZQ2NMPgyPB2+tnT0afh5LK6o1TkumQFXA7KDbc8ofjuI30dxhyqdge3PW+2wHzZBb4NoB9pkF+N563eSaV6X7utcMmsvorDC2ZoZFFzCdZDUFirGL9kCLhxUrAQ7Wzc7v6Zdx2TNILl2q2vYfxVrVlq7KtND6kDJsGgAFuUh3dxQa2kPt7IbnTAfzNZCVc4IxbeN8lpS1HRjIM2EcQbIlQOMG+FmpbxghnF0NgcJwhUuRYbFx4Fw/rajmwu4DaZxP3rTj7/G7eMPu+XskZfkFemRiLwjx2RExmRCuBd6U2/uffdPfOFXvtk+9b2d5gX5L/z1H2cg0JA=
first light to travel during the early stages of the universe's development 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 first using a low-order approximation to calculate the spherical harmonic coe cients: and gives the strongest evidence for the Big Bang theory. Since the Npix m 4⇡ m b` Y ` ( i, ✓i)f( i, ✓i), 0 ` `max, `max m `max discovery of the CMBR in 1964, scientists have worked to measure it in ⇡ Npix
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 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 )
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 pix scheme on the sphere. While these “HEALPix” points allow for a Step 2: Iterative Refinement O
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 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. 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 ` 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 coe cients): b = Y ( , ✓)f( , ✓) 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 b = Af 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 ` ` ()
transforms for uniform and non-uniform point distributions in order to 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 `=0 m=0 X `Xmax ` to efficiently compute the bivariate Fourier series coefficients. We then m m
Synthesis (coe cients to function): 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 f( , ✓)= Y ( , ✓)b f = Sb utilize a state-of-the-art method which computes spherical harmonic ` ` ()
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 operators are used in the iterativee refinement scheme, which convergese to the enables us to quickly and more accurately compute the angular power (0) (k) (k 1) (k 1)
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 squares solution to the coe cients: b = Af, b = b + A f Sb
spectrum of CMBR data sampled at the HEALPix points. 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 Quadrature Weights ⇣ ⌘
Image credit: NASA AAACEHicZZDLSgMxFIYzXmu9VV26CRbRRSkzVVB3BTcuFWwr2FIymTNtMJMMyRmxDH0EN76KGxeKuHXpzrcxvSy0Hgh8+ZM/J+cPUyks+v63Nze/sLi0XFgprq6tb2yWtrabVmeGQ4Nrqc1NyCxIoaCBAiXcpAZYEkpohXfno/PWPRgrtLrGQQqdhPWUiAVn6KRu6aCN8IBhnLczFYEZPZNfZSwyDDMDtAWi10c7HHZLZb/qj4v+h2AKZTKty27pqx1pniWgkEtm7W3gp9jJmUHBJQyL7cxCyvgd68GtQ8USsJ18PNCQ7jslorE2bimkY/W3I2eJtYMkrFAHCcP+GLCfVOhoh1pLW5nYZjphfNrJhUozBMUnjeJMUtR0lA6NhAGOcuCAcSPcXynvM8M4ugyLLoRgduT/0KxVg6Nq7eq4XD+bxlEgu2SPHJKAnJA6uSCXpEE4eSTP5JW8eU/ei/fufUyuznlTzw75U97nD0Njnbk= 3
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 improve equal weight quadrature above, HEALPix o↵ers quadrature weights: 2 Npix TestAAACAHicZZDLSgMxFIYz3q23qgsXboJFcFHKTBUUcSG4cVmhF8GWkknP2GAmGZIzYhlm46u4caGIWx/DnW9j2s7Cy4HAd/7k5+T8YSKFRd//8mZm5+YXFpeWSyura+sb5c2tttWp4dDiWmpzHTILUihooUAJ14kBFocSOuHdxfi+cw/GCq2aOEqgF7NbJSLBGTqpX97pIjxgGGVNsEijVPGxnp/2yxW/5k+K/oeggAopqtEvf3YHmqcxKOSSWXsT+An2MmZQcAl5qZtaSBi/Y7dw41CxGGwvmyyQ032nDGikjTsK6UT96chYbO0oDqvUQcxwOAEcxlU67lBraatT259JGJ30MqGSFEHx6aAolRQ1HadBB8IARzlywLgR7q+UD5lhHF1mJRdC8Hfl/9Cu14LDWv3qqHJ+VsSxRHbJHjkgATkm5+SSNEiLcJKTJ/JCXr1H79l7896nT2e8wrNNfpX38Q3Mh5ZJ function :f( , ✓)=5g( , ✓,l ,t ) 3g( , ✓,l ,t )+8g( , ✓,l ,t ), 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 g( , ✓,l,t)=(2 2(x x(l, t)+y y(l, t)+z z(l, t)))
1 1 2 2 3 3 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sha1_base64="FEJVx4lNxgyhaG2PsHoPtV+Vzvg=">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sha1_base64="ellaVdlr4l/x4IlgUuW7/tjKSnQ=">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 m m · · · b` wiY ` ( i, ✓i)f( i, ✓i), 0 ` `max, `max m `max x = cos( )sin(✓),y=sin( )sin(✓),z= cos(✓), 0