Table Of ContentPHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September 1
Statistical Challenges of Cosmic Microwave Background Analysis
BenjaminD.Wandelt∗
UniversityofIllinoisatUrbana-Champaign,IL61801,USA
TheCosmicMicrowave Background (CMB) isan abundant source of cosmological information. However, this
information is encoded in non-trivial ways in a signal that is difficult to observe. The resulting challenges in
extracting this information from CMB data sets have created a new frontier. In this talk I will discuss the
challengesofCMBdataanalysis. IreviewwhatcosmologicalinformationiscontainedintheCMBdataandthe
problemofextractingit. CMBanalysescanbedividedintotwotypes: “canonical”parameterextractionwhich
seeks to obtain the best possible estimates of cosmological parameters within a pre-defined theory space and
“hypothesistesting”whichseekstotesttheassumptiononwhichthecanonicaltestsrest. Bothoftheseactivities
arefundamentallyimportant. InadditiontominingtheCMBforcosmologicalinformationcosmologistswould
liketostrengthen the analysiswithdatafromother cosmologicallyinterestingobservations as wellasphysical
4
constraints. This gives an opportunity 1) to test the results from these separate probes for concordance and
0
2)ifconcordance isestablishedtosharpentheconstraints ontheoryspacebycombiningtheinformationfrom
0 theseseparatesources.
2
n
a 1. Overview in several ’colors’. The analysis task is two-fold: in-
J
fer the covariance structure of the field s. Infer the
0
parametersΘ.”Thisiswhatcouldbetermed“canon-
3 What is cosmic microwave background (CMB)
ical” CMB analysis.
1
statistics and what is challenging about it? It in-
1 In this talk I will mainly describe challenges pre-
volvesestimatingthe covariancestructureofofaspa-
v 6 8 sented by this canonical CMB analysis. After a brief
tialrandomfieldwith10 –10 pixels,givenonlyONE
2 review of the scientific motivation for studying the
2 realizationofthisfield. Thecovariancematrixofthese CMB in section 2 I will describe the form of CMB
6 pixelsisacomplicatednon-linearfunctionofthephys-
data as well as the current status and prospects of
1 ical parameters of interest. Of these physical param-
obtaining it in section 3. Section 4 then outlines a
0 etersthere arebetween10and20,soevenfinding the
4 framework for extracting cosmologically useful infor-
maximum likelihood point is hard—determining and
0 mation from the data and section 5 illuminates some
summarizing confidence intervals around the maxi-
/ examples of challenges that arise when implementing
h mum likelihood point is very non-trivial. Cosmolo-
this framework. I will touch on statistical questions
p gistswanttodoallthisandhavetheoptionofbuilding
- concerning “non-canonical” CMB analysis in section
in exact or approximate physical constraints on rela-
o 6 and then conclude in section 7.
r tionships between parameters. In addition, since col-
t So why are we interested in facing the statistical
s lecting cosmologicaldata is so difficult and expensive
challenges of CMB analysis?
a we want to combine all available data sets—both to
:
v testthemformutualdisagreementwhichmightsignal
i newphysics,andtoimprovetheparameterinferences.
X
In all of this the quantification of the uncertainties 2. What can we learn from the CMB?
r in the results is extremely important—after all the
a
stated significance of our results will either drive or
Cosmologists are interested in studying the origins
stop theoretical investigations and the design of new
ofthephysicalUniverse. Inordertodosotheyhaveto
observational campaigns.
relyondata. Forcosmologists,oneofthegreatpracti-
Before I get on to CMB specifics in section 2
caladvantagesofEinstein’srelativityoverNewtonian
let me give you the short version of (most) of this
physics is the fact that we cannot help but look into
talk for statisticians: “The CMB is an isotropic
the past. Therefore, by observing light that reaches
(homoschedastic) Gaussian random field s on the
us from farther and farther away, we can study the
sphere. The desired set of cosmological parameters
Universe directly at earlier and earlier times, at least
Θ = {θ i = 1,...,n}, are related in a non-linear way
i to the extent to which the Universe is transparent to
to the spatial covariance structure S ≡hs s i of the
ij i j light. Since the early Universe was a hot and opaque
field. Observers present us with a sampled, noisy, fil-
plasma we can only see back to the time when the
teredandcensored/pollutedmeasurementofthisfield
plasma cooled sufficiently (due to the Hubble expan-
sion) to combine into neutral atoms and the mean
free time between photons collisions became of order
∗NCSAFacultyFellow of the present age of the Universe. Photons that we
1Foronlinematerialrelatingtothistalkpleasereferto observetoday which scatteredfor the lasttime in the
www-conf.slac.stanford.edu/phystat2003/talks/wandelt/invited/ primordial plasma are the CMB.
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2 PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September
Thechangefromplasmatogas,happenedwhenthe from last scattering to hitting our detectors. This in
Universe was approximately 380,000 years old. CMB turn can tell us about the history of star formation.
photons emitted at this time are therefore the most Averyexcitingprospectisthatbystudyingthede-
directmessengersthatwecandetecttodayofthecon- tailsofCMBpolarizationwecaninferthepresenceor
ditions present in the Universe shortly after the Big absenceofgravitationalwavesatthetimeoflastscat-
Bang.2 They constitute a pristine snapshot of the in- tering. Adetectionwouldofferanindirectviewofone
fantUniversewhichprovideus withdirectcosmologi- oftheelusivemessengersthatstartedtheirjourneyat
calinformationunclutteredbythecomplexnon-linear an even earlier epoch, adding a nearly independent
physicswhichledto the formationofstarsandgalax- constraint on the properties of the Universe at the
ies. One of the main attractions of the CMB is the Planck scale.
conceptual simplicity with which it can be linked to All this information is not encoded in actual fea-
the global properties of the Universe and the physics tures in the CMB map of (temperature or polariza-
which shaped it at or near the Planck scale. tion) anisotropies. In fact in a globally isotropic uni-
verse the absolute placement of individual hot and
As a simple example, the serendipitous discovery
cold spots is devoid of useful information. Informa-
of the CMB by Penzias and Wilson in 1965 showed
tion can, however, be stored in the invariants of the
thatthe CMB is isotropicto averyhighdegree. This
photon brightness fluctuations under the groupof ro-
wasoneofthekeymotivationsforthedevelopmentof
tations SO(3). These are the properties of the field
the inflationaryparadigm[4, 5, 6]. Inflationdescribes
that only depend on the relative angular distance be-
generically the emergence (from the era of quantum
tween two points of the field. For a Gaussian field,
gravity) of a large homogeneous and isotropic Uni-
where 2-point statistics specify all higher order mo-
verse. Inflation also predicted the spectrum of small
ments, this means that the angular power spectrum
metric perturbations from which later structure de-
coefficientsofthe anisotropiescontainalloftheinfor-
veloped through the gravitational instability (though
mation.
it was not the only mechanism to do so). The corre-
The challengefortheoreticians wasthen to develop
sponding anisotropies in the CMB were first convinc-
a detailed theory of the angular power spectrum C ,
ingly detected by the DMR instrument on the COBE ℓ
asa function of angularwavenumberℓ,giventhe cos-
satellite in 1992. The rejection of alternative mecha-
mological parameters Θ. While conceptually simple,
nisms for the generation of the primordial spectrum
it required a decade-long intellectual effort to model
ofmetricperturbationsinfavorofinflationwasama-
therelevantphysicalprocessesatthe requiredlevelof
jor advance driven by the measurement of the large
precision. As a result, there now exist several Boltz-
angle CMB anisotropy. These developments have led
manncodes(e.g. CMBFAST[2]orCAMB[3]),which
toinflationbecomingpartofthe currentcosmological
numerically compute C (Θ) to 1% precision or bet-
standard model. A very robust prediction of generic ℓ
ter. The power spectra C (Θ) are sensitive functions
modelimplementations ofinflationis the Gaussianity ℓ
of certain combinations of the parameters and weak
and homogeneity of the resulting perturbations.
functions of others (degeneracies). These weaklycon-
Quantitatively, the properties of our Universe are
strainedparametercombinationsarereferredtoasde-
encodedinasetofn∼10−20cosmologicalparameters
generacies. Within the context of the standard cos-
Θ where n depends on the level of detail of the mod-
mologicalmodel,thetheoryofthisdependenceiswell-
eling or,commonly, onthe specificationoftheoretical
understood.
priors which fix some of these parameters to “reason-
It is clear from the preceding discussion that the
able” values. These parameters specify the geometry
CMBisanextremelyvaluablesourceofwhatamounts
andaverageenergydensity ofthe Universe,aswellas
to“cosmologicalgold”: informationaboutthephysics
the relativeamountsofenergydensity contributedby
and the global properties of the early Universe. So
the ingredients of the primordial soup (dark matter,
what are the observationalprospects?
ordinarybaryons,neutrinosanddarkenergyandpho-
tons). In addition, the anisotropy carries information
about the spectrum of primordial (inflationary) per-
3. Data from CMB Observations
turbations as well as their type (adiabatic or isocur-
vature). By combining observationsof the anisotropy
A major international effort is underway to make
ofboththeeffectivetemperatureandthepolarization
high quality observations of the microwave sky using
oftheCMBphotonswecaninferhowtransparentthe
ground-based, balloon borne and space missions [19].
UniversereallywasfortheCMBphotonsontheirway
Space missions have the advantage of being able to
scan the whole microwave sky. NASA’s “Wilkinson
MicrowaveAnisotropyProbe”(WMAP)waslaunched
2Thereareothermessengers,namelyneutrinosandgravita- in 2001 with great success and is currently in opera-
tion. It reported its first year of data earlier this year
tionalwaves,reachingusfromevenearliertimesbutwedonot
(yet)havethetechnologytodetecttheminrelevantquantities. [16]. WMAP will continue to collect data for at least
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PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September 3
Figure 1: A schematic of how the cosmological parameters Θ (top left) are linked to thetime ordered data CMB
experimentsactually observe(bottom right). Please see thediscussion in thetext.
another three years. In the medium term (ie. in late of a CMB anisotropy map. This is most simply done
2007) we anticipate the launch of “Planck,” a joint working in the Fourier representation: we draw the
ESA/NASA space mission.3 Additional space mis- sphericalharmonic coefficients a fromuncorrelated
ℓm
sions may follow that will focus on measuring the po- Gaussians with zero mean and variance given by C .
ℓ
larization anisotropies in the CMB. In the meantime Then we compute the spherical harmonic transform
ground and balloon-based missions are jostling to ac- to obtain
celerateour learningcurveby providingmaps athigh
ℓ=∞ m=ℓ
angular resolution on small patches of the sky (up to
T(nˆ)= T Y (nˆ), (1)
ℓm ℓm
a few degrees large). As a result, by the end of this Xℓ=2 mX=−ℓ
decade we will have a mountain of CMB data.
where Y (nˆ) are the spherical harmonics, an or-
Fromthismountain(andofcoursefromthepartof ℓm
thonormal and complete basis for functions on the
it which we have already available today) we would
sphere. The unit vector nˆ points in a direction on
like to extract the cosmological gold. In order to do
the sky.
so we need to understand how the data and the in-
Unfortunately,the T or T(nˆ)cannotbe observed
formation are related. In Figure 1 I show a simpli- ℓm
directly. We are tied to our location in the Galaxy.
fied schematic of this relationship and I will now go
There are various sources of copious amounts of mi-
through the various steps.
crowaveradiationintheGalaxy,suchasdustandelec-
Starting with a set of cosmological parameters Θ tronsfromwindswhichareacceleratedintheGalactic
we can use a Boltzmann code to compute the power magnetic field. These foregrounds add to the CMB
spectrum C . Given this power spectrum and the as- signal to make our sky.
ℓ
sumption of Gaussianity and isotropy we have all the Thisskyisthenobservedinvariousfrequencybands
information we need to create a statistical realization (indicated by the red, green and blue arrows) by a
CMBinstrument(the figureshowsanartist’sconcep-
tion of the WMAP satellite [33]). This instrument it-
self has a complicated transfer function: since weight
3http://www.rssd.esa.int/index.php?project=PLANCK and size constraints force satellite optics to be built
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4 PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September
compactlytheopticsarenotfreefromdistortion. Mi- involvedtoactuallycarryoutthe optimization. Eval-
crowaveshavemacroscopicwavelengthsandtherefore uatinganyoneproposeddesignrequiresmanysimula-
diffract around the edges of the instrument. This tionsofthefullprocessfromobservationtoanalysis—
leads to sidelobes in the beam maps. The instrument theveryprocessthatpresentsthechallengesIamdis-
scans the sky in a certain pattern (the “scan strat- cussing in this talk. Doing this repeatedly to search
egy”) and internal instrument systematics are added thespaceofdesignparametersforanoptimalsolution
to the scanned signal. The microwave detectors (ei- would be an immense task. Asymptotic techniques
therradiometersorbolometers)addnoisetogenerate havebeenimplementedonthebasisoftheFisherma-
the time ordered data (TOD). trix minimum variance predictions, but it is impor-
TherearedifferentlevelsofdetailoftheCMBdata. tant to keep in mind that these are lower bounds on
To give an idea for the orders of magnitude involved, the expected variance, assuming a unimodal, Gaus-
let us go through the sizes of the various objects in sian likelihood shape.
Figure 1 for the Planck experiment.
• ThecompleteTODforPlanckwilltakeupofor-
4. Inference From the Data
12
der1Terabyte(=10 bytes)ofstorage(without
counting house-keeping and pointing data).
Astochasticmodelofthe dataandthe information
• Eachofthe 100detectors(channels)resultsina contained in it can be summarized in the following
map which is of order 10-100 Megabytes. equations. For simplicity we will limit the discussion
to a single channel. The TOD, d say, is modeled as
• The channels are grouped into of order 10 fre- the result of the action of a linear operator A which
quency bands. encodestheopticsandscanningstrategyoftheinstru-
ment,onthesky,madeupofsignalsandforegrounds
• Thecombinedmapsatthesedifferentfrequency f:
bandswillbecombinedintomapsofthephysical
components (such as dust, synchrotron, CMB). d=A(s+f)+n. (2)
• The CMB power spectrum has a few thousand Our assumptions is that s is an isotropic Gaussian
coefficients C . randomfield withzero meanandpowerspectrumC .
ℓ ℓ
We will take the noise correlations N = hnnTi and
• Thesepowerspectrumcoefficientsareafunction A as given—though one of the statistical challenges
of 10−20 cosmologicalparameters. of CMB analysis is to relax this assumption to some
degree. We will not discuss this further here.
Notethatthereisatrade-offbetweenthelevelofcom- The task is then to extract as much information as
pression and what assumptions are implemented in possible about the cosmological parameters Θ from
the data analysis. Note also, that except for the raw the TOD. To set up this inverse problem we write
data each of these data products by themselves mean down Bayes’ theorem
little unless some means of assessing their statistical
uncertainty is provided. P(s,C ,f,Θ|d)P(d)=P(d|f,s,C ,Θ)P(f,s,C ,Θ).
ℓ ℓ ℓ
We immediately run into practical problems. For (3)
example,ifwewouldliketospecifyanoisecovariance In the Bayesian context, solving the inverse problem
matrix for each combined map at each frequency we means exploring and summarizing the posterior den-
would have to specify 10 times ∼ (106)2/2 elements sity P(s,C ,f,Θ|d). On the right hand side we can
ℓ
of a matrix. The necessary storage space of order use the conditional independence of the C and the
ℓ
10,000 Gigabyte basically precludes practical public datagivens,andtheplausibleindependence off and
distribution ofthe data. Other waysofspecifying the s to simplify
uncertainty must be found.
Before we discuss in detail how cosmological infor- P(f,s,C ,Θ)=P(f)P(s|C )P(C |Θ)P(Θ). (4)
ℓ ℓ ℓ
mation is encoded in the data, let us comment about
an aspect of CMB statistics that is challenging even Traditionally, inference is performed in a linear se-
beforethereisanydatainhand. Thisaspectisexper- quenceofstepswhichareconcatenatedintoapipeline.
imentaldesign. Whileourexperimentalcolleaguesare At each individual step a likelihood is written for the
putting a greatdeal of valuable thoughtinto their in- data in its current representation (e.g. TOD, maps,
struments and observationalstrategies,it is currently C ) in terms of the parameters describing the next
ℓ
stilldoneinaninformalway,byingenuityratherthan stage of compression. Due to the complexity of eval-
by formal method. This is partially so because it is uating the likelihoods, often the likelihood approach
hardtodefineoptimalitycriteriathateveryonewould is abandoned and approximate,suboptimal but unbi-
agreewithandpartiallybecauseoftheimmenseeffort ased estimators are constructed.
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PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September 5
Within the above framework we can understand generateaccurateresults. Forlocaloperationssuchas
each of the steps as a limit of Eq. 3. For example nearest neighbor searches and multi-resolution work,
the “map-making” step is signal estimation with pixelizations of the sphere that allow hierarchical re-
finement are useful.
P(f,s,C ,Θ)=P(f)P(s|C )P(C |Θ)P(Θ)=const. Various pixelizations of the sphere have been pro-
ℓ ℓ ℓ
(5) posed [23, 24, 25]. Of these, HEALPix features
Similarly, “Power spectrum estimation” summa- pixels with exactly equal areas throughout, approxi-
rizes the marginal posterior P(C |d), where the d is mateequidistributionofpixelcentersoverthesphere,
ℓ
takenincompressedformastheestimateoftheCMB simple analytical equations for the pixel boundaries,
component. The summary proceeds through a maxi- fast spherical harmonic transforms and very favor-
mumlikelihoodestimate(MLE)andtheevaluationof able quadrature properties. Due to these features,
the curvature of P(C |d) at the MLE. In fact in most HEALPix has developed into the standard pixeliza-
ℓ
practicalcasesanon-optimalestimatorisuseddueto tion for astrophysical all sky maps.
the computational complexity of evaluating P(C |d) Beam Deconvolution Map-making. It is easy
ℓ
or its derivatives and hence of computing the MLE. toshowthattheMLEmˆ forthemapm=s+f isthe
Finally, “parameter estimation” signifies the step result of maximizing Eq. 3 using Eq. 5. This results
from the MLE of C to a density P(Θ|d). The in
ℓ
posterior P(Θ|d) is usually summarized in terms of m≡(ATN−1A)−1ATN−1d (7)
the marginalized means and variances of the param-
eters or in terms of two-dimensional projections or with the associated noise covariance matrix CN =<
marginalizations through the n-dimensional parame- mmT >=(ATN−1A)−1.
ter space. These equations are generally valid whatever the
Recently, we developed MAGIC, a method that formsof A andN. However,the non-trivialstructure
showspromise forglobalinference fromthe joint pos- intheobservationmatrixAwhichisaconsequenceof
terior [15]. The method, based on iterative sampling theunavoidableimperfectionsinsatelliteopticsmakes
from the joint posterior, exploits the full dependence solvingforthemapchallenging. Itisimportanttode-
structureofthedifferentscienceproductsfromaCMB convolve the beam functions, since the beam distor-
mission. For example it is useful for the separationof tionsandsidelobesleadtospuriousshadowimagesof
signal and foregrounds if the covariance of the signal brightforegrounds. Theseimagesarespreadthrough-
is known. The information obtained on C can be out the map in a complicated way that depends on
ℓ
fed backinto the CMB mapestimate, whichwouldin the scanningstrategy. Inaddition, notaccountingfor
turn lead to a better estimate of C , etc. But since I beam imperfections results in distorting the signal it-
ℓ
discussed this technique in my contributed talk [14] I self. Both of these effects can bias not just the map
will not go into details here. but also the result of the covariance estimation.
Forpolarizationmap-makingthisdeconvolutionbe-
I will now discuss a selection of the challenges we
comes even more important, since the polarization
face in the “map-making,” “power spectrum estima-
signal is weak and beam asymmetries can introduce
tion” and “parameter estimation” steps on the way
spurious polarization signals into the data. Further,
from the TOD to Θ.
if bright foregrounds are significantly polarized, they
may induce a significant polarization through their
shadow images if beam convolution effects are ne-
5. Challenges in CMB Analysis
glected in map-making. Making high-quality maps
ofthe polarizationof the CMB anisotropyis the next
5.1. Map-making Challenges frontier in CMB map-making.
Tosimulatetheeffectsofrealisticbeamswehaveto
have a generalconvolutiontechnique for a beam map
Pixelizations of the sphere. A very basic chal-
with a sky map along the scan path. If implemented
lenge in CMB analysis is the fact that the CMB is
as discrete sums over two pixelized maps, rotated to
a random field on the sphere. Convenient numerical
all possible relative orientations, such a technique re-
techniques for storing and manipulating functions on quires of order n2.5 operations, where n is the num-
the sphere needed to be developed. For CMB analy- p p
ber of pixels in the maps. By doing the convolution
sis in particular, fast methods for spherical harmonic
in spherical harmonic space it was shown in [21] that
transforms had to be developed and implemented. this can be reduced to n2 in the general case and to
Point sets were needed that had good generalized p
quadraturepropertiessuchthatdiscretenumericalap- n3p/2 in interesting limit cases. This fast convolution
proximations to Eq. 1 and its inverse method was generalized to polarization in [22].
Component separation. Once a map of the sky
is made from each channel, these maps can be com-
2
a = d nˆY (nˆ)T(nˆ) (6)
ℓm Z ℓm pressedlosslesslyintomapsateachfrequency. Agreat
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6 PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September
6000
WMAP
C_l 68% HPR, single-l
4000 ACBAR
2) Boomerang02
K
µ DASI
(
π CBI
2 MAT98
C/l MAXIMA
)
1 VSA
+
(l
l 2000
500 1000 1500 2000 2500 3000
l
Figure 2: Ourcompilation of all recent CMB power spectrum (Cℓ) data (points), includingthe WMAPdata (gray
band). Pre-WMAPdata which provideredundant information about the power spectrum at low ℓ are omitted. Also
shown (as two solid lines) is the68% constraint on thepower spectrum after implementing a prior which restricts the
range of theories to a 10-parameter space of adiabatic inflationary theories (from [18]).
deal of work in the field has gone into devising meth- timation would be easy. Just compute the spherical
ods for then obtaining an estimate of the CMB sky harmonic transform, Eq. 6. Then the estimator
fromtheseforegroundcontaminatedmapsateachfre-
|a |2
quency band, both for temperature and for polariza- Cˆ = m ℓ (8)
tion. We can either choose to model the foregrounds ℓ P2ℓ+1
physically (e.g. [30, 31]) or we can attempt “blind is the MLE for C . It is also easy to evaluate and
ℓ
separation,” by defining an algorithm for automatic explore the perfect data posterior to quantify the un-
detection of different components in the maps, e.g. certainty in the estimates. However, in the general
basedonthe statisticalindependence ofthese compo- casewewouldliketoevaluateP(S(C )|d,N)whichis
ℓ
nents (e.g. [29]). the result of integrating out (“marginalizing over”) s
Lensing. The CMB has traveled past cosmolog- in the joint posterior. We obtain4
ical mass concentrations which perturb the photon
P(S(C )|d,N)=G(m,S(C )+C ). (10)
geodesics. This distortioncontainsvaluable andcom- ℓ ℓ N
plementary information about cosmological parame-
HereS is the signalcovariancematrix,parameterized
ters. Atthesametimeitmixesthepolarizationmodes
by the C . Since S +C is not a sparse matrix and
ℓ N
in the CMB data, contaminating the primary signal
since the determinant in the Gaussian depends on S,
for the detection of the primordialgravitationalwave
evaluating P(S(C )|d,N) as function of the C costs
ℓ ℓ
background. Methods need to be devised that can of order n3 operations.
p
measure this distortion and extract the information For Planck, n ∼ 107 so if the constant factor in
p
contained in it. It is intriguing that [32] find that
the scaling law was 1 (an unrealistic underestimate)
exact techniques have significant advantages over ap-
proximate,quadraticestimatorsforthereconstruction
of B polarization maps from lensed CMB.
4WeuseG(x,X)asashorthand forthemultivariateGaus-
5.2. Power Spectrum Estimation siandensity
G(x,X)= 1 exp −1xTX−1x . (9)
The Computational Problem. For perfect (all- |2πX| (cid:16) 2 (cid:17)
sky, pure signal, no noise) data, power spectrum es- p
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PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September 7
wellwewillbeabletoconstraincosmologicalparame-
terestimates fromthe CMB.The usualwayofphras-
ing this limitation is in terms of cosmic variance.
Essentially, power spectrum estimation is variance
estimation. For perfect data the solution that maxi-
mizes Eq. 10 is
|a |2
Cˆ = m ℓ . (11)
ℓ P2ℓ+1
Soweareestimatingthevarianceofthesphericalhar-
monic coefficients. But, for example, for ℓ = 2 there
are only 5 such coefficients and a variance estimate
from 5 numbers is statistically uncertain. So “cosmic
variance”isexpressionofthefundamentallimittothe
precisionofanymeasurementofthe C causedbythe
ℓ
fact that the sphere is a bounded space and the fact
thatcausalitywillnotallowustoobserveindependent
patches of the universe.
This fundamental limit to our knowledge provides
a powerful motivation to do the best possible job in
analyzing cosmologicaldata.
5.3. Parameter Estimation
Figure 3: An example result of an online exploration of
themarginal posterior of thedark matter density (Ωm) Techniques. Theproblemofparameterestimation
and dark energy density (ΩΛ) in our Universe. All recent is challenging since we need to explore the posterior
CMB data, including theWMAP data, as well as the density P(Θ|d) which varies over ∼ 10−20 dimen-
HubbleSpaceTelescope key project results and the sions. Various techniques have been used to do this.
Supernovacosmology project were included to obtain For smaller number of dimensions (up to about 5)
these constraints. Points in thefigureare colored (from gridding techniques have worked well. However, the
red to blue) according to how well theparameter
currentstateofthe artistouseMarkovChainMonte
combination at that point agrees with thedata.
Carlo methods [20] such as the Metropolis Hastings
algorithm to sample from P(Θ|d) and to then base
inferencesonsummariesofthe posteriordensitycom-
this would mean 1021 operations for one likelihood
puted from the sampled representation.
evaluation. For a 10 GFLOP CPU this means 1000s
Thequestionhowtoimplementphysicalpriorsand
of CPU years of computation.
constraints was one theme that was discussed at this
Various approaches have been suggested to to conference. Within the Bayesianframeworkthere ex-
counter this challenge. They can be broadly divided ists a unique prescription for applying physical con-
into two classes: 1) specialized exact (maximum like- straints through the specification of informative pri-
lihood estimation) algorithms exploit advantageous ors. The first example of a CMB parameter estima-
symmetries in the observational strategy of a CMB tion which can be explored interactively online is the
mission [9, 10] to reduce the computational scaling CosmicConcordanceProject[17]. Thiscompilesdata
from order n3p to order n2p, and 2) approximate algo- from several recent CMB observations and combines
rithms which filter unwanted properties of the data themwiththeuser’schoiceofother,non-CMBexper-
and simply compute the power spectra on the fil- iments and physical priors. The result is displayed as
tered and incomplete data, and then de-bias the re- the 2D marginal posterior density for any 2 parame-
sultsusingMonteCarlosimulationsafterthefact[12]. terschosenbytheuser. Anexampleisdisplayedinfig-
Thissecondclassofalgorithmsisknownaspseudo-Cℓ ure3. Iinviteyoutohavealookatourprototypeim-
algorithms[11] and has gained a great deal of popu- plementation at http://galadriel.astro.uiuc.edu/ccp.
larity in recent analyses of CMB data, including the Afirstscientificresultfromthis projectwasamea-
WMAP data [13]. surement of the fraction of 4He in the Universe, both
Cosmic Variance. Aside from computational fromCMBdataaloneandinthecombinationofstan-
problems there is an interesting conceptual problem dard Big Bang Nucleosynthesis with the CMB data.
with CMB power spectrum analysis: the fact that we Since standard BBN links the primordial 4He abun-
only have one sky. dance to the baryon to photon ratio which is deter-
This fact induces a fundamental limitation to how mined exquisitely well by CMB data, we obtain the
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8 PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September
mostprecisemeasurementoftheprimordial4Heabun- mological data. Ultimately the goal of the Cosmic
dance to date [18]. Concordance Project is to allow users to select var-
Several open questions remain to be addressed. ious data sets and explore interactively whether the
What number n of parameters is needed to fit the parameterconstraintsfromvariousdatasetsarecom-
data? Which selection of parameters from the full patible with each other. If this agreement is estab-
set should we use in the analysis? These questions lishedthetheconstraintscanbecombinedtogenerate
are currently handled on the basis of personal prefer- yetstrongerconstraints. Ifdisagreementis foundthis
ence of the authors or computational convenience—a is motivation for observationalgroups to collect more
statistically motivated procedure for letting the data data or for theoretical groups to work out new mech-
selecting certain parameters has not yet been imple- anismsthatcanreconcilethediscrepantobservations.
mented.
TestingforstatisticalisotropyintheCMBisawell-
Operationally, it is very computationally expen-
definedoperation,sinceitiseasytospecifyalternative
sive to sample from a posterior with 10 parameters
models. It requires checking whether a model of the
or more, because each likelihood evaluation requires
correlations in the fluctuations in terms of rotation-
running a Boltzmann code which computes C(Θ) .
ℓ allyinvariantquantities(suchasthepowerspectrum)
In many dimensions the Metropolis sampler produces
is better or worse than a model that contains quan-
correlated samples (regardless of whether the target
tities that do not transform as scalars under SO(3).
density is correlated or not—the correlations come
A frequentisttest statistic has been suggestedin [34],
fromthesamplertakingsmallstepsthroughthemany
butaBayesiantreatmenthasnotyetbeenattempted.
dimensional space). New numerical techniques for
The detection of a significant deviation from statisti-
evaluating and sampling from the likelihood in high-
cal isotropy would be a very important result, since
dimensional spaces are needed.
isotropy is a fundamental prediction of inflation.
Beating cosmic variance. In Figure 2 we show
Testing for non-Gaussianity (NG) is similarly im-
another application of parameter estimation tech-
portant,butmuchlesswell-defined. Inthe absenceof
niques whichis relevantto severalthemes wetouched
physical models for NG a Bayesian treatment is not
oninthistalkandatthisconference. Thefigureshows
possible. ThestandardapproachistodefinesomeNG
ourcompilationofCMBdataandthemeanand±68%
statistic, e.g. the skewness of the one-point function
errors on the range of power spectra which are con-
oftheCMBfluctuations. Thenthisstatisticisapplied
tained in our 10 parameter fits. In other words, we
to the data andto a sample of GaussianMonte Carlo
haveimplementedtheoptimalnon-linearfilterforthe
samples (which are usually constrained to match the
C if the Universe is really described by our 10 pa-
ℓ
two-pointstatisticsoftheobserveddata). ThisMonte
rameter model. The physical prior has reduced the
Carlo sample represents the null-hypothesis. A way
cosmic variance error bars far below the limit set by
is defined to assess discrepancy of the data and the
Eq. 11. Since we are using the physical prior that
MonteCarlosample andif the discrepancyis statisti-
the CMB power spectrum is the result of plasma os-
cally significant a detection is claimed.
cillations in the primordial photon baryon fluid the
resultingsmoothnessofthepowerspectrumisusedin Even though this is a straightforward frequentist
constructing the estimate. procedure there is a great deal of arbitrariness in
choosingtheteststatistic. Usuallythe choiceismade
based on a vague notion of genericity, in the sense
6. Testing the Assumptions: Challenges that, for example, the skewness of the data is proba-
of Non-canonical CMB Analysis bly a moregeneric NG statistic thanthe temperature
in pixel number 2,437,549, say. However, it is clearly
not trivial to define a proper measure on the space of
Inadditiontothecanonicalanalysesoutlinedabove
all possible statistics. The arbitrariness in the selec-
itisfundamentallyimportantto testthe assumptions
tion of the statistic (topological features of the map,
on which these analyses rest. I will very briefly men-
n-point functions in pixel and in spherical harmonic
tion two of them here.
space, wavelets, excursion sets, etc.) makes the test
Is the CMB signal really Gaussian? Is it isotropic?
results difficult to interpret. For example, is clearly
These questions touch simultaneously on the issue of
truethattherewillalwaysbesomestatisticthatgives
hypothesis testing, as well as model selection. What
an n-sigma result for any n!
is the evidence in the data for the assumptions of
isotropyandGaussianityonwhichcanonicalanalyses To circumvent this arbitrariness, we could test for
of the CMB are built? Can a goodness of fit criterion the NG predicted in certain variants of inflationary
be defined which allows assessing whether the stan- models ([26, 27, 28] and references therein). These
dard model of cosmology is a complete description of physicalmodelsallowaBayesiananalysisinprinciple,
the CMB data? and it would provide the best possible constraints on
The idea of testing the goodness of fit can be ex- the presence of NG, but implementing Bayesian NG
tended to cross tests of CMB data with other cos- inference these models is computationally tedious.
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PHYSTAT2003, SLAC, Stanford, California :: 8th - 11th September 9
7. Conclusion [14] Wandelt, B. D., in these proceedings.
[15] Wandelt, B.D., Larson, D., and Lakshmi-
In this talk I reviewedsome of the more severe sta- narayanan,A., astro-ph/0310080, PRD in press.
tistical and computational challenges of CMB analy- [16] C. L. Bennett et al., Ap.J.Suppl. 148, 1 (2003)
sis. CMB data are fundamentally important to cos- [17] Huey, G., Wandelt, B. D., and Cyburt, R., in
mology,theproblemsIoutlinedareintellectuallyrich, preparation.
and the knowledge that we need to make the most of [18] Huey, G., Cyburt, R., and Wandelt, B. D.,
theoneCMBskywecanobserveandanalyze,present astro-ph/0307080, PRD in press.
apowerfulmotivationtosolvetheCMBanalysisprob- [19] A. D. Milleret al. 1999 Astrophys.J. 524
lem. (1999) L1-L4; N. W. Halverson et al. 2001,
astro-ph/0104489; S. Hanany et al. Astrophys.J.
545 (2000) L5; Keith Grainge et al. Mon. Not.
Acknowledgments R. Astron. Soc. 000, 15 (2002); Kuo, C. L. et al.
2002, ApJ, astro-ph/0212289; J. E. Ruhl et al.
I thank the organizers of PHYSTAT2003 for mak- 2002, astro-ph/0212229; S., Padin et. al., ApJ
ingthisconferencesoenjoyableandstimulating. This 549, L1, (2001); for a comprehensive list see
work has been partially supported by the National http://lambda.gsfc.nasa.gov
Computational Science Alliance under grant num- [20] Christensen,N.,et al., Class.QuantumGrav.18,
ber AST020003N and by the University of Illinois at 2677 (2001)
Urbana-Champaign through an NCSA Faculty Fel- [21] Wandelt, B.D. and Gorski, K., Physical Review
lowship. D63, 123002 (2001).
[22] Challinor, A. et al Physical Review D62, 123002
(2000).
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