cosmic magnetism with information field theorysemikoz/egmf/ensslin.pdf · 2010. 12. 17. · cosmic...
TRANSCRIPT
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Max-Planck-Institut für Astrophysik
Cosmic Magnetism withInformation Field TheoryTorsten Enßlin, Mona Frommert, Henrick Junklewitz,
Niels Oppermann, Georg Robbers, Petr Kuchar
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data: NVSS, Taylor et al. 2009map: Oppermann et al., arXiv:1008.1243
data: WMAP, WMAP-team map: Page et al.
magnetic feld observablesmagnetic feld observables
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magnetic helicitymagnetic helicity
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Probing Helicity by LITMUSProbing Helicity by LITMUS
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Taylor,Stil,Sunstrum 2009
37,534 RM extragalactic RM-sources in the northern sky
The Faraday rotation skyThe Faraday rotation sky
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Information Field TheoryInformation Field Theory
{
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inverse problem => Information Theory spatially distributed quantity => Field Theory
Why Information Field Theory ?
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Information fields of interest:
● magnetic field in 3d● Faraday rotation map● polarized emissivity per Faraday depth● cosmic matter distribution● magnetic power spectra, ...
Information sources:
● radio polarimetry: RM● radio polarimetry: total I, PI● galaxy surveys● X-rays, ...
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What is information theory ?
What is information?knowledge about
the set of possibilities Ω their individual probabilities P:Ω ➝ [0,1]
knowledge state (Ω, P)
information theory is the math of knowledge statesit is simply probability theory applied to reasoning
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How to obtain information ?
physical signal s
1) prior knowledge: P(s)
2) measurement: d = R(s) + n, P(d|s)
3) inference: P(s|d)=P(d|s) P(s)/P(d), d ➙ s
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When the signal is a field, Bayesian methods are unavoidable sincethe number of signal space dimensions
exceed the data space dimensions by a large – if not infinite – factor!
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priorlikelihoodposterior
evidence
s = signald = data
Basic idea: expand posterior around Gaussian Basic idea: expand posterior around Gaussian
Information HamiltonianInformation Hamiltonian
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Free TheoryFree TheoryGaussian signal & noise, linear responseGaussian signal & noise, linear response
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Free TheoryFree TheoryGaussian signal & noise, linear responseGaussian signal & noise, linear response
Wiener flter theoryWiener flter theoryknown for 60 years information
sourceinformation
source
informationpropagator
informationpropagator
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Interacting TheoryInteracting Theorynon-Gaussian signal, noise, or non-linear responsenon-Gaussian signal, noise, or non-linear response
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Optimal algorithms for non-linear inverse problems can be build on linear operations like Wiener filtering !!!
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Optimal algorithms for non-linear inverse problems can be build on linear operations like Wiener filtering !!!
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Optimal algorithms for non-linear inverse problems can be build on linear operations like Wiener filtering !!!
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dictionary
Translation:inference problem → statistical field theory
Dictionary:log-Posterior = negative HamiltonianEvidence = partition function ZWiener variance = information propagatornoise weighted data → information sourceinference algorithms ← Feynman diagramsmaximum a Posteriori = classical solutionuncertainty corrections = loop correctionsShannon information = negative entropy
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Reconstruction of signals with unknown spectra
Enßlin & Frommert arXiv:1002.2928
Image credits: Planck team
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Joint posterior
signal space
parameter space
P (s,Sp|d)
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Gaussian data model
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Wiener filter
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Spectral prior
P (pi)
pi
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Generic filter formula
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Jeffreys prior
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Classical map
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Filter parameters
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Taylor,Stil,Sunstrum 2009
37,534 RM extragalactic RM-sources in the northern sky
The Faraday rotation skyThe Faraday rotation sky
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Oppermann et al. arXiv:1008.1243
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The Faraday rotation skyThe Faraday rotation sky
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Information Field TheoryInformation Field Theory
}
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LITMUSLITMUSLLocalocal I Inference nference TTest forest for M Magnetic feldsagnetic felds U Uncovering Helicencovering HeliceSS
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LITMUSLITMUSLLocalocal I Inference nference TTest forest for M Magnetic feldsagnetic felds U Uncovering Helicencovering HeliceSS
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Observational Setup
Hydra A cluster
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Kuchar & Enßlin (2009)
Magnetic power spectra in theMagnetic power spectra in thecool core of the Hydra A clustercool core of the Hydra A cluster
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Conclusions
● magnetic field observables allow deduction of helicity and energy spectrum
● inference problems require information theory
● information field theory is a statistical field theory
● information Hamiltonian = - log P(data, signal)
● information propagator = Wiener variance, ...
● perturbative, renormalization, thermodynamical, & sampling methods can be used
● providing us with the mean map, signal uncertainty, ...
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Outlook
1.4 GHz: Reich & Wolleben22 GHz: WMAP team
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Thank you !