Non-linear Data Analysis on the Sphere: The Quest for Anomalies in the Cosmic Microwave Background

134 Pages·2013·5.381 MB·English

by  Gregor Rossmanith (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Gregor Rossmanith

Non-linear Data
Analysis on the Sphere
The Quest for Anomalies in the
Cosmic Microwave Background
Springer Theses
Recognizing Outstanding Ph.D. Research
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Gregor Rossmanith
Non-linear Data Analysis
on the Sphere
The Quest for Anomalies in the Cosmic
Microwave Background
Doctoral Thesis accepted by
the Ludwig-Maximilian University of Munich, Germany
123
Author Supervisor
Dr. GregorRossmanith Prof.Dr. Dr.h.c. Gregor Morfill
Max-Planck-Institute Max-Planck-Institute
forExtraterrestrial Physics forExtraterrestrial Physics
Garching Garching
Germany Germany
ISSN 2190-5053 ISSN 2190-5061 (electronic)
ISBN 978-3-319-00308-5 ISBN 978-3-319-00309-2 (eBook)
DOI 10.1007/978-3-319-00309-2
SpringerChamHeidelbergNewYorkDordrechtLondon
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Supervisor’s Foreword
The analysis of the cosmic microwave background (CMB) radiation has led to
major breakthrough discoveries in Cosmology. The mere discovery of this relic
radiationin1964byPenziasandWilson(NobelPrizein1978)originatingfroma
hot and dense era of the Universe in the very distant past became one of the first
andmajorproofsforthestandardBigBangcosmologicalmodel.Thefirstfull-sky
observations of the CMB with the COBE satellite inthe 1990sof the lastcentury
(NobelPrizeforMatherandSmootin2006)revealedtheamazinghomogeneityof
thisradiation—thoughtheshapeofdetectedfluctuationsinthetemperature(being
as small as of the order of O(10-5) (cid:3)C) could corroborate the assumption of an
inflationary period right after the Big Bang.
HigherprecisionmeasurementsoftheCMBasperformednowadayswith,e.g.,
the WMAP and PLANCK satellite have helped to make Cosmology a precision
science and to further constrain inflationary scenarios by investigating possible
higherordercorrelations,i.e.,non-Gaussianities(NGs),inthefieldoftemperature
and polarization fluctuations.
Testingforhigherorder(nonlinear)correlationshasalsobecomearichfieldof
research in the analysis of complex (chaotic) systems, where it became immedi-
atelyobviousthatalineardescriptionofthecomplexdynamicsofstillcomparably
simpledeterministicchaoticsystemsisalreadyinsufficient.Rather,onehadtofind
new ways to describe the behavior of the system. One very successful approach
turned out to be the quantification of the phase space portrait, e.g., the strange
attractor, formed by the trajectory of the system.
The work presented by Dr. Rossmanith deals with the adaptation and appli-
cationoftwokeyconceptsofcomplexsystemstheory,namelyestimatorsforlocal
scaling properties and the method of surrogates, for the search of non-Gaussia-
nities in the CMB.
Forcalculatingso-calledscalingindicesasestimatorsforthescalingproperties
ofaCMBmap,theideaofembeddingadatasetinahigherdimensionalartificial
representation space has been employed. This technique has its origins in Whit-
neys,Takens,andSauersembeddingtheorems,whichstatethatthetopologyofthe
phase space structure of a dynamical system is preserved when its dynamics (e.g.
v
vi Supervisor’sForeword
theattractor)isreconstructedinanartificialembeddingspacebyusingthemethod
ofdelay-coordinates.Applyingtheseideaslessformallytoimageprocessingtasks
allowsforthecombinationofthespatialandintensity(orcolor)informationofthe
image pixels by representing these physically diverse quantities as points in a
commonembeddingspace.Thedefinitionofhighlysensitivenonlinearlocalfilters
relyingonthecalculationofthelocalscalingofthepointsetisthusmadepossible.
Mostly, tests for NG in Cosmology are performed using models for either
Gaussian density fluctuations or special types of non-Gaussianity like NGs of the
local type, which can be deduced from special shapes of the potential of the
inflaton field.
Todevelopcomplementaryandmodel-independenttestsforNGsthatareableto
investigateanydeviationfromGaussianity,themethodofsurrogateswasadapted
andappliedtothecaseofCMBdataanalysis.Thebasicideaofthisapproach—first
havingbeenpresentedintheseminalpaperbyTheileretal.,1992—istowipeout
the higher order correlations by shuffling or replacing the Fourier phases, while
exactlypreservingthelinearproperties,i.e.,thepowerspectrum.Acomparisonof
theoriginaldatawithitssurrogates—hereusingscalingindicesasteststatistics—
then reveals whether higher order correlations have led to phase correlations that
were then destroyed or not. Since the phase shuffling can also be restricted to
previouslyselectedrangesofscales,thedevelopmentfordedicatedtestsforscale-
dependent NGs is made possiblefor the first time.
Dr.Rossmanithperformedforthefirsttimeadedicatedband-wiseanalysiswith
scalingindicesusingthefive-yearWMAPdata.Hefoundsignificantsignaturesfor
NGs and asymmetries. Further, no dependences on the chosen frequency band
were detected, which suggests that the signatures cannot be attributed to yet not
understood foreground effects.
The applicability of the method of surrogates for CMB analyses was demon-
stratedforthelargestscalesusingforeground-reducedfull-skymapsderivedfrom
the WMAP five-year data release. Highly significant signatures for both non-
Gaussianities and asymmetries were once again found. In fact, these detected
scale-dependent NGs represent the most significant detection of NGs in the
WMAP data to date.
Itisalsodesirabletobeabletousethesurrogateapproachforacutsky,where
the Galactic plane with possible foreground residuals is masked. However, when
applying a sky cut, orthonormality of the spherical harmonics no longer holds on
thisnewincompletesky—makinganaivephaseshufflingimpossible.Ontheother
hand, one can transform the spherical harmonics into a new set of harmonics,
which forms an orthonormal basis on the incomplete sky, where phase manipu-
lation can then take place again.
This thesis combines the transformation of the spherical harmonics into a new
set of harmonics with the idea of phase shuffling, thus enabling investigations by
means of surrogates on an incomplete sky. The feasibility of this approach was
once again first demonstrated for the case of large-scale NGs, which also allows
for a direct comparison of the full sky and cut sky results.
Supervisor’sForeword vii
Ascalingindexanalysisthenshowedstrongnon-Gaussianitiesandpronounced
asymmetries, which are consistent with the full sky results and persist even when
removing larger parts of the sky. This result confirms that the influence of the
Galactic plane is not responsible for these deviations from Gaussianity and isot-
ropy. Intheabsenceofanexplanationinterms ofGalactic foregroundsorknown
systematicartifacts,thesignaturesmustsofarbetakentobecosmologicalathigh
significance. These findings would strongly disagree with predictions of isotropic
cosmologies with single-field slow-roll inflation and might even point to a vio-
lation of the Cosmological principle.
Of course, those results need further confirmation. Currently, the much more
precise CMB data taken with the PLANCK satellite are analyzed with the meth-
odologies outlined in this thesis and results about the large-scale anomalies are
expected in the very near future.
Amoredetailedinvestigationofthepresenceandabsenceofphasecorrelations
as identified in observational CMB data may, on the other hand, also shed more
light on the meaning of Fourier phases for complex structures in general. Those
findingsareofgreatinterestinmany(interdisciplinary)fieldsofresearch—namely
whenever the Fourier representation of sufficiently complex data sets plays a
crucial role in the representation and analysis of the data.
Madrid, February 2013 Prof. Dr. Dr. h.c. Gregor Morfill
Reference
1. J.Theiler,S.Eubank,A.Longtin,B.Galdrikian,J.D.Farmer,PhysicaD58,77(1992)
Preface
OneofthekeychallengesinCosmologytodayistoprobebothstatisticalisotropy
andGaussianityoftheprimordialdensityperturbations,whichareimprintedinthe
cosmic microwave background (CMB) radiation. While single-field slow-roll
inflation predicts the CMB to fulfill these two characteristics, more complex
models may give rise to anisotropy and/or non-Gaussianity. A detection or non-
detection allows therefore to discriminate between different models of inflation
and significantly improves the understanding of the basic conditions of the very
early Universe.
Inthiswork,adetailedCMBnon-Gaussianityandisotropyanalysisofthefive-
andseven-yearobservationsoftheWMAPsatelliteispresented.Ontheonehand,
these investigations are performed by comparing the data set with simulations,
which istheusualapproachforthesekindsofanalyses.Ontheotherhand,anew
model-independentapproach isdeveloped andappliedinthiswork.Starting from
therandomphasehypothesis,so-calledsurrogatemapsarecreatedbyshufflingthe
Fourierphasesoftheoriginalmapsforachosenscale interval.Anydisagreement
between the data and these surrogates points towards phase correlations in the
original map, and therefore—if systematics and foregrounds can be ruled out—
towards a violation of single-field slow-roll inflation.
The construction of surrogate maps only works for an orthonormal set of
Fourier functions on the sphere, which is provided by the spherical harmonics
exclusively on a complete sky. For this reason, the surrogate approach is for the
first time combined with a transformation of the full sky spherical harmonics toa
cut sky version. Both the single surrogate approach as well as the combination
with the cut sky transformation are tested thoroughly to assess and then rule out
the effects of systematics. Thus, this work not only represents a detailed CMB
analysis, but also provides a completely new method to test for scale-dependent
higher order correlations in complete or partial spherical data sets, which can be
applied in different fields of research.
Indetail,theapplicationsoftheabovemethodsinvolvethefollowinganalyses:
First, a detailed study of several frequency bands of the WMAP five-year data
releaseisaccomplishedbymeansofascalingindexanalysis,wherebythedataare
comparedtosimulations.Specialattentionispaidtoanomalouslocalfeatures,and
ix
x Preface
ways to overcome the problem of boundary effects when excluding foreground-
influenced parts of the sky. After this, the surrogate approach is for the first time
applied to real CMB data sets. In doing so, several foreground-reduced full sky
maps from both the five- and seven-year WMAP observations are used. The
analysisincludesdifferentscaleintervalsandahugeamountofchecksonpossible
systematics. Then, another step forward is taken by applying the surrogate
approachforthefirsttimetoincompletedatasets,againfromtheWMAPfive-and
seven-year releases. The Galactic Plane, which is responsible for the largest
amount of foreground contribution, is removed by means of several cuts of dif-
ferentsizes.Inaddition,differenttechniquesforthebasistransformationareused.
Inalloftheseinvestigations,remarkablenon-Gaussianitiesanddeviationsfrom
statistical isotropy are identified. In fact, the surrogate approach shows by far the
most significant detection of non-Gaussianity to date. The band-wise analysis
shows consistent results for all frequency bands. Despite a thorough search, no
candidate for foreground or systematic influences could be found. Therefore, the
findingsoftheseanalyseshavesofartobetakenascosmological,andpointonthe
onehandtowardsastrongviolationofsingle-fieldslow-rollinflation,andquestion
on the other hand the concept of statistical isotropy in general.
Future analyses of the more precise measurements of the forthcoming
PLANCK satellite will yield more information about the origin of the detected
anomalies.