Cosmic Microwave Background Mini-Review

0.22 MB·English

by  Douglas Scott

1 Cosmic Microwave Background Scott & Smoot
COSMIC MICROWAVE BACKGROUND MINI-REVIEW
Revised August 2005 by Douglas Scott (University of British
Columbia) and George F. Smoot (UCB/LBNL) for ‘The Review
ofParticlePhysics’,S.Eidelmanet al., PhysicsLetters,B.592,1 Ateverypointintheskyoneobservesablackbodyspectrum,with
(2004) and 2005 partial update for the 2006 edition available on temperature T(θ). The spectrum of the dipole is the differential
thePDGWWWpages (URL:http://pdg.lbl.gov/) ofablackbodyspectrum,asconfirmedbyRef.[8].
The implied velocity [9] [5] for the solar system barycenter is
1. Introduction v = 368±2kms−1, assuming a value T0 =Tγ, towards (ℓ,b)=
◦ ◦ ◦ ◦
(263.85 ±0.10 ,48.25 ±0.04 ). Suchasolarsystemvelocityim-
The energy content in radiation from beyond our Galaxy is
plies a velocity for the Galaxy and the Local Group of galaxies
deroemdininat1e9d6b5y[1t]h.eTChoesmspicecMtriucmrowoafvtehBeaCcMkgBrouisndwe(lClMdeBsc)r,ibdiesdcobvy- relativetotheCMB◦.Th◦ede◦rived◦valueisvLG=627±22kms−1
towards(ℓ,b)=(276 ±3 ,30 ±3 ),wheremostoftheerrorcomes
ablackbody function with T =2.725K.Thisspectral formisone
6 fromuncertaintyinthevelocityofthesolarsystemrelativetothe
of the main pillars of the hot Big Bang model for the early Uni-
0 LocalGroup.
verse. Thelackofanyobserveddeviationsfromablackbodyspec-
0 The dipole is a frame dependent quantity, and one can thus
trumconstrainsphysicalprocessesovercosmichistoryatredshifts
2 z∼<107(seepreviousversionsofthismini-review[2]). However,at dtheeteCrmMinBedtihpeo‘laebwsooluultdebreestzefrroa.mOe’uorfvtheleoUcitnyivreerlsaetiavsethtoatthinewLhoiccahl
n the moment, all viablecosmological models predict avery nearly
Group,aswellasthevelocityoftheEartharoundtheSun,andany
a Planckianspectrum,andsoarenotstringentlylimited.
J AnotherobservablequantityinherentintheCMBisthevaria- velocityofthereceiverrelativetotheEarth,isnormallyremoved
forthepurposesofCMBanisotropystudy.
4 tionintemperature(orintensity)fromonepartofthemicrowave
1 sky to another [3]. Since the first detection of these anisotropies
bytheCOBE satellite[4],therehas beenintenseactivitytomap 2.3. Higher-OrderMultipoles:

1 the sky at increasing levels of sensitivity and angular resolution. Excess variance in CMB maps at higher multipoles (ℓ≥2) is
v Aseriesofground-andballoon-basedmeasurementswasjoinedin interpreted as being the result of perturbations in the density of
7 2003 by the first results from NASA’s Wilkinson Microwave An- the early Universe, manifesting themselves at the epoch of the
0 isotropy Probe (WMAP) [5]. These observations have led to a lastscatteringoftheCMBphotons. InthehotBigBangpicture,
3 stunning confirmation of the ‘Standard Model of Cosmology.’ In this happens at a redshift z ≃ 1100, with little dependence on
1 combination with other astrophysical data, the CMB anisotropy thedetailsofthemodel. Theprocess bywhichthehydrogenand
0 measurements placequitepreciseconstraints onanumberofcos- helium nuclei can hold onto their electrons is usually referred to
6 mologicalparameters,andhavelaunchedusintoaneraofprecision as recombination [10]. Before this epoch, the CMB photons are
0 cosmology. tightly coupled to the baryons, while afterwards they can freely
/ streamtowardsus.
h 2. Description of CMB Anisotropies
p Theoretical models generally predict that the aℓm modes are
Observations show that the CMB contains anisotropies at the Gaussian random fields. Tests show that this is an extremely
o- 10−5level,overawiderangeofangularscales. Theseanisotropies good simplifying approximation [11], with only some relatively
r areusually expressed byusing aspherical harmonic expansion of weak indications of non-Gaussianity or statistical anisotropy at
st theCMBsky: large scales. Although non-Gaussianity of various forms is possi-
a T(θ,φ)= aℓmYℓm(θ,φ). bleinearlyUniversemodels[12],thesignaturesfoundinexisting
: Xℓm WMAP data are generally considered to be subtle foreground or
v
i Thevastmajorityofthecosmologicalinformationiscontainedin instrumentalartefacts [13][14].
X the temperature 2 point function, i.e., the variance as a func- With the assumption of Gaussian statistics, and if there is no
r tion of separation θ. Equivalently, the power per unit lnℓ is preferredaxis,thenitisthevarianceofthetemperaturefieldwhich
a ℓ m|aℓm|2/4π. carriesthecosmologicalinformation,ratherthanthevaluesofthe
2P.1. TheMonopole: individualaℓms;inotherwordsthepowerspectruminℓfullychar-
The CMB has a mean temperature of Tγ = 2.725±0.001K acterizestheanisotropies. Thepowerateachℓis(2ℓ+1)Cℓ/(4π),
(1σ) [6], which can be considered as the monopole component whereCℓ≡ |aℓm|2 ,andastatisticallyisotropicskymeansthat
D E
of CMB maps, a00. Since all mapping experiments involve dif- all ms are equivalent. We use our estimators of the Cℓs to con-
ference measurements, they are insensitive to this average level. straintheirexpectationvalues,whicharethequantitiespredicted
Monopole measurements can only be made with absolute temp- by a theoretical model. For an idealized full-skyobservation, the
erature devices, such as the FIRAS instrument on the COBE varianceofeachmeasuredCℓ (i.e.,thevarianceofthevariance)is
satellite [6]. Such measurements of the spectrum are consis- [2/(2ℓ+1)]C2. Thissamplinguncertainty(knownas‘cosmicvari-
ℓ
tent with a blackbody distribution over more than three decades ance’)comesaboutbecauseeachCℓisχ2distributedwith(2ℓ+1)
in frequency. A blackbody of the measured temperature corre- degreesoffreedomforourobservablevolumeoftheUniverse. For
spondstonγ=(2ζ(3)/π2)Tγ3≃411cm−3 andργ=(π2/15)Tγ4≃ fractionalskycoverage, fsky,thisvarianceisincreasedby1/fsky
4.64×10−34gcm−3≃0.260eVcm−3. andthemodesbecomepartiallycorrelated.
2.2. TheDipole: It is important to understand that theories predict the expec-
tation value of the power spectrum, whereas our sky is a single
The largest anisotropy is in the ℓ = 1 (dipole) first spherical
realization. Hence the cosmic variance is an unavoidable source
harmonic, withamplitude 3.346±0.017mK [5]. The dipoleis in-
ofuncertaintywhenconstrainingmodels;itdominatesthescatter
terpretedtobetheresultoftheDopplershiftcausedbythesolar
at lower ℓs, whilethe effects of instrumental noise and resolution
system motionrelative to the nearlyisotropicblackbody field, as
dominateathigherℓs[15].
confirmedbymeasurementsoftheradialvelocities oflocalgalax-
ies[7]. Themotionofanobserverwithvelocityβ=v/crelativeto
anisotropicPlanckianradiationfieldoftemperature T0 produces 2.4. AngularResolutionandBinning:
aDoppler-shiftedtemperaturepattern
Thereisnoone-to-oneconversionbetween multipoleℓandthe
T(θ)=T0(1−β2)1/2/(1−βcosθ) aHnogwleevseurb,taensdinegdlebyspahpearirctaicluhlaarrmwoanviecveYcℓtmorcporrorjeescptoenddosnttootahnegusklayr.
≈T0 1+βcosθ+(β2/2)cos2θ+O(β3) . variations ofθ∼π/ℓ. CMBmapscontain anisotropyinformation
(cid:16) (cid:17)
2 Cosmic Microwave Background Scott & Smoot
fromthesizeofthemap(orinpracticesomefractionofthatsize) somethingelse,the‘initialconditions’giverisetoaminimumof3
downtothebeam-sizeoftheinstrument,σ. Onecanthinkofthe parameters: A,n,andr.
effect of a Gaussian beam as rollingoff the power spectrum with
thefunctione−ℓ(ℓ+1)σ2.
Forlessthanfullskycoverage, theℓmodesbecomecorrelated.
Hence,experimentalresultsareusuallyquotedasaseriesof‘band
powers’,definedasestimatorsofℓ(ℓ+1)Cℓ/2πoverdifferentranges
of ℓ. Because of the strong foreground signals in the Galactic
Plane, even ‘all-sky’surveys, such as COBE and WMAP involve
acutsky. Theamountofbinningrequiredtoobtainuncorrelated
estimates ofpoweralsodepends onthemapsize.
3. Cosmological Parameters
Thecurrent‘StandardModel’ofcosmologycontainsaround10
freeparameters(see‘CosmologicalParameters’mini-review[16]).
The basic framework is the Friedmann-Robertson-Walker metric
(i.e.,auniversethatisapproximatelyhomogeneous andisotropic
on large scales), with density perturbations laid down at early
timesandevolvingintotoday’sstructures(see‘Big-BangCosmol-
ogy’ mini-review [17]). These perturbations can be either ‘adia-
batic’(meaningthatthereisnochangetotheentropyperparticle
foreachspecies,i.e.,δρ/ρformatteris(3/4)δρ/ρforradiation)or
‘isocurvature’ (meaning that, for example, matter perturbations
compensate radiationperturbations sothat thetotal energyden- Figure 1: The theoretical CMB anisotropy power spec-
sityremainsunperturbed,i.e.,δρformatteris−δρforradiation). trum,usingastandardΛCDMmodel fromCMBFAST. Thex-
These different modes give riseto distinct phases during growth, axis islogarithmichere. The regions, each covering roughly
with those of the adiabatic scenario being strongly preferred by adecadeinℓ,arelabeledasinthetext: theISWRise;Sachs-
the data. Models that generate mainly isocurvature type pertur- Wolfe Plateau; Acoustic Peaks; and Damping Tail. Also
bations (such as most topological defect scenarios) are no longer shownistheshapeofthetensor(gravitywave)contribution,
consideredtobeviable. withanarbitrarynormalization.
Within the adiabatic family of models, there is, in principle,
The background cosmology requires an expansion parameter
a free function describing how the comoving curvature perturba-
(the Hubble Constant, H0, often represented through H0 =
tions, R, vary with scale. In inflationary models [18], the Taylor 100hkms−1Mpc−1)andseveralparameterstodescribethemat-
seriesexpansionoflnR(lnk)hastermsofsteadilydecreasingsize.
ter and energy content of the Universe. These are usually given
For the simplest models, there are thus 2 parameters describing
tshloepieniotfiatlhceopnodwiteiornsspfeocrtrduemns,ity|Rp|e2rtu∝rbkanti.onTsh:itshceaanmbpelietxupdleicaitnldy wianrheteerwremhρasctroigftot=vhee3rnHcr02tiht/ie8cπaplGhdy.esSnicisnsitcyoe,fpit.hhey.e,siCfcoaMrlsBdpeeancnsieiitssioe‘tsxr’oρ,pxΩie∝xs,=Ωixtρhxis2/ρt≡hcreωistxe,
D E
defined, forexample,through: ωs that are best constrained by CMB data. In particular CMB
observations constrain ΩBh2 for baryons and ΩMh2 for baryons
∆2R≡(k3/2π2) |R|2 , plusColdDarkMatter.
D E The contribution of a cosmological constant Λ (or other form
of Dark Energy) is usually included through a parameter which
and using A2 ≡∆2R(k0) with k0 =0.05Mpc−1, say. There are quantifies the curvature, ΩK≡1−Ωtot, whereΩtot=ΩM+ΩΛ.
many other equally valid definitions of the amplitude parameter Theradiationcontent, whileinprincipleafreeparameter, ispre-
(seealsoRefs.[16]and[17]),andwecautionthattherelationships ciselyenoughdeterminedbythemeasurementofTγ,andmakesa
betweensomeofthemcanbecosmologydependent. In‘slowroll’ negligiblecontributiontoΩtot today.
inflationarymodelsthisnormalizationisproportionaltothecom- The main effect of astrophysical processes on the Cℓs comes
binationV3/(V′)2,fortheinflationarypotentialV(φ). Theslope throughreionization. TheUniversebecamereionizedatsomered-
′′
n also involves V , and so the combination of A and n can, in shiftzi,longafterrecombination, affecting theCMBthroughthe
principle,constrainpotentials. integratedThomsonscatteringoptical depth:
Inflationgeneratestensor(gravitywave)modesaswellasscalar
(density perturbation) modes. This fact introduces another pa- zi dt
rametermeasuringtheamplitudeofapossibletensorcomponent, τ=Z0 σTne(z)dzdz,
orequivalentlytheratioofthetensortoscalarcontributions. The
tensor amplitude AT∝V, and thus one expects a larger gravity whereσT istheThomsoncross-section,ne(z)isthenumberden-
wavecontributioninmodelswhereinflationhappensathigheren- sityoffreeelectrons(whichdependsonastrophysics)anddt/dz is
ergies. Thetensorpowerspectrumalsohasaslope,oftendenoted fixed bythe background cosmology. In principle, τ can be deter-
nT,butsincethisseemslikelytobeextremelyhardtomeasure,it minedfromthesmallscalematter power spectrum together with
issufficientfornow tofocus onlyontheamplitudeofthegravity the physics of structure formation and feedback processes. How-
wavecomponent. Itismostcommontodefinethetensorcontribu- ever, this isasufficiently complicated calculation that τ needs to
tionthroughr,theratiooftensortoscalarperturbationspectraat beconsideredasafreeparameter.
largescales (sayk=0.002Mpc−1);however, thereareother defi- Thus we have 8 basic cosmological parameters: A, n, r, h,
nitions in terms of the ratio of contributions to C2, for example. ΩBh2, ΩMh2, Ωtot, and τ. One can add additional parameters
Different inflationary potentials will lead to different predictions, tothislist,particularlywhenusingtheCMBincombinationwith
e.g. for λφ4 inflation with 50 e-folds, r=0.32, while other mod- other data sets. The next most relevant ones might be: Ωνh2,
els can have arbitrarily small values of r. In any case, whatever the massive neutrino contribution; w (≡ p/ρ), the equation of
the specific definition, and whether they come from inflation or state parameter for the Dark Energy; and dn/dlnk, measuring
3 Cosmic Microwave Background Scott & Smoot
deviationsfromaconstantspectralindex. Tothese11onecouldof Universe became neutral the proton-electron plasma was tightly
courseaddfurtherparametersdescribingadditionalphysics,such coupledtothephotons,andthesecomponentsbehavedasasingle
asdetailsofthereionizationprocess,featuresintheinitialpower ‘photon-baryonfluid’. Perturbationsinthegravitationalpotential,
spectrum,asub-dominantcontributionofisocurvaturemodes,etc. dominatedbythedarkmattercomponent,weresteadilyevolving.
As well as these underlying parameters, there are other quan- They drove oscillations in the photon-baryon fluid, with photon
tities that can be obtained from them. Such derived parameters pressureprovidingmostoftherestoringforceandbaryons giving
include the actual Ωs of the various components (e.g., ΩM), the some additional inertia. The perturbations were quite small in
variance of density perturbations at particular scales (e.g., σ8), amplitude, O(10−5), and so evolved linearly. That means each
the age of the Universe today (t0), the age of the Universe at Fourier mode evolved independently and hence can be described
recombination,reionization,etc. byadrivenharmonicoscillator,withfrequencydeterminedbythe
sound speed inthe fluid. Thus the fluid density oscillates, which
4. Physics of Anisotropies gives oscillations in temperature, together with a velocity effect
which is π/2 out of phase and has its amplitude reduced by the
Thecosmologicalparametersaffecttheanisotropiesthroughthe
soundspeed.
well understood physics of the evolution of linear perturbations
After the Universe recombined the baryons and radiation de-
within a background FRW cosmology. There are very effective,
coupled,andtheradiationcouldtravelfreelytowardsus. Atthat
fast,andpublicly-availablesoftwarecodesforcomputingtheCMB
point the phases of the oscillations were frozen-in, and projected
anisotropy, polarization, and matter power spectra, e.g.,CMBFAST
on the sky as a harmonic series of peaks. The main peak is the
[19] and CAMB [20]. CMBFAST is the most extensively used code; it
mode that went through 1/4 of a period, reaching maximal com-
hasbeentestedoverawiderangeofcosmologicalparametersand
pression. Theevenpeaks aremaximalunder-densities, whichare
isconsideredtobeaccuratetobetter thanthe1%level[21].
generally of smaller amplitude because the rebound has to fight
AdescriptionofthephysicsunderlyingtheCℓscanbeseparated against the baryon inertia. The troughs, which do not extend to
into3mainregions,asshowninFig.1.
zeropower,arepartiallyfilledbytheDopplereffect because they
4.1. TheSachs-Wolfeplateau: ℓ∼<100: areatthevelocitymaxima.
The horizon scale (or more precisely, the angle subtended by An additional ingredient comes from geometrical projection.
theHubbleradius)atlastscattering correspondsto ℓ≃100. An- The scale associated with the peaks is the sound horizon at last
isotropiesatlargerscaleshavenotevolvedsignificantly,andhence scattering,whichcanbestraightforwardlycalculatedasaphysical
directly reflect the ‘initial conditions.’ The combination of grav- length scale. This length is projected onto the sky, leading to an
itational redshift and intrinsic temperature fluctuations leads to angular scale that depends on the background cosmology. Hence
δT/T ≃(1/3)δφ/c2, where δφ is the perturbation to the gravita- the angular position of the peaks is a sensitive probe of the spa-
tional potential. This is usually referred to as the ‘Sachs-Wolfe’ tialcurvatureoftheUniverse(i.e.,Ωtot),withthepeakslyingat
effect[22]. higherℓinopenuniversesandlowerℓinclosedgeometry.
Assumingthatanearlyscale-invariantspectrumofdensityper- Onelasteffectarisesfromreionizationatredshiftzi. Afraction
turbationswaslaiddownatearlytimes(i.e.,n≃1,meaningequal of photons (τ) will be isotropically scattered at z <zi, partially
powerperdecadeink),thenℓ(ℓ+1)Cℓ≃constantatlowℓs. This erasingthe anisotropies atangular scales smallerthan thosesub-
effect is hard to see unless the multipole axis is plotted logarith- tendedbytheHubbleradiusatzi,whichcorrespondstypicallyto
mically(asinFig.1,butnotFig.2). ℓs above about a few 10s, depending on the specific reionization
Time variation of the potentials (i.e., time-dependent metric model. Theacousticpeaksarethereforereducedbyafactore−2τ
perturbations)leadstoanupturnintheCℓsinthelowestseveral relativetotheplateau.
multipoles;anydeviationfromatotalequationofstatew=0has These peaks were a clear theoretical prediction going back to
such aneffect. Sothe dominance of the DarkEnergy at low red- about1970[25]. Onecanthinkofthemasasnapshotofstochastic
shiftmakesthelowestℓsriseabovetheplateau. Thisissometimes standing waves. Since the physics governing them is simple and
called the ‘integrated Sachs-Wolfe effect’ (or ISW Rise), since it theirstructurerich,thenonecanseehowtheyencodeextractable
comes from the line integral of φ˙; it has been confirmed through information about the cosmological parameters. Their empirical
correlations between the large-angle anisotropies and large-scale existencestartedtobecomecleararound1994[26],andtheemer-
structure [23]. Specific models can also give additional contribu- gence, over the following decade, of a coherent series of acoustic
tionsatlowℓ(e.g.,perturbations intheDarkEnergycomponent peaks and troughs is a triumph of modern cosmology. This pic-
itself[24]),buttypicallytheseareburiedinthecosmicvariance. turehasreceivedfurtherconfirmationwiththerecentdetectionin
In principle, the mechanism that produces primordial pertur- the power spectrum of galaxies (at redshifts close to zero) of the
bationscouldgeneratescalar,vector,andtensormodes. However, imprintoftheacousticoscillationsinthebaryoncomponent[27].
thevector(vorticity)modesdecaywiththeexpansionoftheUni- 4.3. Thedampingtail: ℓ∼>1000:
verse. Tensors also decay when they enter the horizon, and so
◦ Therecombinationprocessisnotinstantaneous,givingathick-
theycontributeonlytoangularscalesaboveabout1 (seeFig.1).
ness to the last scattering surface. This leads to a damping of
Hence somefractionofthelow ℓsignal couldbedueto agravity
theanisotropiesatthehighestℓs,correspondingtoscalessmaller
wave contribution, although small amounts of tensors are essen-
than that subtended bythis thickness. Onecanalsothink ofthe
tiallyimpossibletodiscriminatefromothereffectsthatmightraise
photon-baryon fluid as having imperfect coupling, so that there
theleveloftheplateau. However,thetensorscanbedistinguished
is diffusion between the two components, and hence the ampli-
usingpolarizationinformation(seeSec.6).
tudes oftheoscillations decrease withtime. Theseeffects leadto
4.2. Theacousticpeaks: 100∼<ℓ∼<1000: adamping of the Cℓs,sometimes calledSilkdamping[28], which
Onsub-degreescales,therichstructureintheanisotropyspec- cuts offtheanisotropiesatmultipolesaboveabout2000.
trumistheconsequenceofgravity-drivenacousticoscillationsoc- An extra effect at high ℓs comes from gravitational lensing,
curring before the atoms in the Universe became neutral. Per- causedmainlybynon-linearstructuresatlowredshift. TheCℓsare
turbations inside the horizon at last scattering have been able convolvedwithasmoothingfunctioninacalculableway,partially
to evolve causally and produce anisotropy at the last scattering flattening the peaks, generating a power-law tail at the highest
epoch,whichreflectsthatevolution. Thefrozen-inphasesofthese multipoles, and complicating the polarization signal [29]. This is
soundwavesimprintadependenceonthecosmologicalparameters, anexampleofa‘secondaryeffect’,i.e.,theprocessingofanisotro-
whichgivesCMBanisotropiestheirgreatconstrainingpower. pies due to relatively nearby structures. Galaxies and clusters of
Theunderlyingphysicscanbeunderstoodasfollows. Beforethe galaxiesgiveseveralsucheffects,butallareexpectedtobeoflow
4 Cosmic Microwave Background Scott & Smoot
amplitudeandaretypicallyonlyimportantforthehighestℓs. 6. CMB Polarization
SinceThomsonscatteringofananisotropicradiationfieldalso
5. Current Anisotropy Data generateslinearpolarization,theCMBispredictedtobepolarized
at the roughly 5% level [39]. Polarization is a spin2 field on the
There has been a steady improvement in the quality of CMB sky,andthealgebraofthemodesinℓ-spaceisstronglyanalogous
datathathasledtothedevelopmentofthepresent-daycosmolog- to spin-orbitcouplinginquantum mechanics [40]. Thelinearpo-
icalmodel. Probablythemostrobustconstraints currentlyavail- larization pattern can be decomposed in a number of ways, with
ablecomefromthecombinationofthe WMAP firstyeardata[5] two quantities required for each pixel in a map, often given as
withsmallerscaleresultsfromtheCBI[30]andACBAR[31]ex- theQandU Stokes parameters. However,themostintuitiveand
periments. Newer data from BOOMERANG [32] and the VSA physical decomposition is a geometrical one, splitting the polar-
[33] are also useful at smaller angular scales and give consistent ization pattern into a part that comes from a divergence (often
results for parameters. We plot power spectrum estimates from referred to as the ‘E-mode’) and a part with a curl (called the
these five experiments in Fig. 2. Other recent experiments, such ‘B-mode’)[41]. Moreexplicitly,themodesaredefinedintermsof
asARCHEOPS[34],DASI[35]andMAXIMA[36]alsogivepow- secondderivativesofthepolarizationamplitude,withtheHessian
erfulconstraints,whicharequiteconsistentwithwhatwedescribe fortheE-modeshavingprincipleaxesinthesamesenseasthepo-
below. There have been some comparisons among data-sets [37], larization,whilethe B-modepattern canbethought ofsimplyas
whichindicateverygoodagreement,bothinmapsandinderived a45◦ rotationofthe E-modepattern. Globallyoneseesthat the
power spectra (up to systematic uncertainties in the overall cali- E-modes have (−1)ℓ parity (like the spherical harmonics), while
brationforsomeexperiments). Thismakesitclearthatsystematic theB-modeshave(−1)ℓ+1 parity.
effects are largely under control. However, a fully self-consistent The existence of this linear polarization allows for 6 different
jointanalysisofallthecurrentdatasetshasnotbeenattempted, crosspowerspectratobedeterminedfromdatathatmeasurethe
oneofthereasonsbeingthatitrequiresacarefultreatmentofthe full temperature and polarization anisotropy information. Parity
overlappingskycoverage. considerations make 2 of these zero, and we are left with 4 po-
tential observables: CTT, CTE, CEE, and CBB. Since scalar
ℓ ℓ ℓ ℓ
perturbations have no handedness, the B-mode power spectrum
canonlybegeneratedbyvectorsortensors. Hence,inthecontext
of inflationary models, the determination of a non-zero B-mode
signal is a way to measure the gravity wave contribution (and
thus potentially derive the energy scale of inflation), even if it is
rather weak. However, one must first eliminate the foreground
contributionsandothersystematiceffectsdowntoverylowlevels.
The oscillating photon-baryon fluid also results in a series of
acoustic peaks in the polarization Cℓs. The main ‘EE’ power
spectrum has peaks that are out of phase with those inthe ‘TT’
spectrum, because the polarization anisotropies are sourced by
the fluid velocity. The ‘TE’ part of the polarization and temp-
eraturepatterns comesfromcorrelationsbetweendensityandve-
locity perturbations on the last scattering surface, which can be
bothpositiveandnegative,andisoflargeramplitudethantheEE
signal. Thereisnopolarization‘Sachs-Wolfe’effect,andhenceno
large-angle plateau. However, scattering during a recent period
of reionization can create a polarization ‘bump’ at large angular
scales.
The strongest upper limits on polarization are at the roughly
10µK level from the POLAR [42] experiment at large angular
scales and the PIQUE [43], COMPASS [44] and CBI [45] experi-
ments at smaller scales. The first measurement of a polarization
Figure 2: Band-power estimates from the WMAP, signal came in 2002 from the DASI experiment [46], which pro-
BOOMERANG, VSA, CBI, and ACBAR experiments. We vided a convincing detection, confirming the general paradigm,
have suppressed some of the low-ℓ and high-ℓ band-powers butoflowenoughsignificancethatitlentlittleconstrainttomod-
which have large error bars. Note also that the widths of els. As well as the E-mode signal, DASI also made a statistical
theℓ-bandsvariesbetweenexperiments. Thisplotrepresent detection oftheTEcorrelation.
onlyaselectionofavailableexperimental results,withsome In2003theWMAP experimentdemonstrated thatiswasable
other data-sets being of similar quality. The multipole axis to measure the TE cross-correlation power spectrum with high
hereislinear,sotheSachs-Wolfeplateauishardtosee. How- precision [47]. Other recent experimental results include a weak
ever,theacousticpeaksanddampingregionareveryclearly detectionoftheEEsignalfromCAPMAP[48]andmoresignificant
observed, with no need for a theoretical curve to guide the detectionsfromCBI[49],DASI[50]andBOOMERANG[51]. The
eye. TE signal has also been detected in several multipole bands by
Fig. 2 shows band-powers from the first year WMAP data BOOMERANG [52], and there are statistical detections by CBI
[38], together with BOOMERANG [32], VSA [33], CBI [30] and [49]andDASI[50]. SomeupperlimitsonCℓBB alsoexist,butare
ACBAR[31]dataathigherℓ. Thepointsareinverygoodagree- currentlynotveryconstraining.
mentwitha‘ΛCDM’typemodel,asdescribedearlier,withseveral The results of WMAP CℓTE are shown in Fig. 3, along with
ofthepeaksandtroughsquiteapparent. Fordetailsofhowthese estimates from the DASI and BOOMERANG experiments. The
estimateswerearrivedat,thestrengthofanycorrelationsbetween measured shape of the cross-correlationpower spectrum provides
band-powersandotherinformationrequiredtoproperlyinterpret supporting evidence of the adiabatic nature of theperturbations,
them,turntotheoriginalpapers. aswellasdirectlyconstrainingthethicknessofthelastscattering
surface. Since the polarization anisotropies are generated in this
scattering surface, the existence of correlations at angles above
5 Cosmic Microwave Background Scott & Smoot
aboutadegreedemonstratethatthereweresuper-Hubblefluctua- However, for the time being, foreground contamination is not a
tionsattherecombinationepoch. majorissueforCMBexperiments.
7.2. SecondaryAnisotropies:
With increasingly precise measurements of the primary aniso-
tropies, there is growing theoretical and experimental interest in
‘secondary anisotropies.’ Effects which happen at z ≪1000 be-
come moreimportant as experiments pushto higher angular res-
olutionandsensitivity.
These secondary effects include gravitational lensing, patchy
reionization and the Sunyaev-Zel’dovich (SZ) effect [54]. This is
′ ′
Compton scattering (γe → γ e) of the CMB photons by a hot
electrongas,whichcreatesspectraldistortionsbytransferringen-
ergy fromthe electrons to the photons. The effect isparticularly
important for clusters of galaxies, through which one observes a
partiallyComptonizedspectrum,resultinginadecrementatradio
wavelengths and an increment in the submillimeter. This can be
usedtofindandstudyindividualclustersandtoobtainestimates
oftheHubbleconstant. Thereisalsothepotentialtoconstrainthe
equation of state of the Dark Energy through counts of detected
clustersasafunctionofredshift[55].
7.3. Higher-orderStatistics:
AlthoughmostoftheCMBanisotropyinformationiscontained
in the power spectra, there will also be weak signals present in
higher-order statistics. These statistics can measure any primor-
Figure 3: Cross power spectrum of the temperature an-
dial non-Gaussianity in the perturbations, as well as non-linear
isotropies and E-mode polarization signal from WMAP, to-
growth of the fluctuations on small scales and other secondary
getherwithestimatesfromDASIandBOOMERANGwhich
effects (plus residual foreground contamination). Although there
extendtohigherℓ. NotethattheBOOMERANGbandsare
are an infinite variety of ways in which the CMB could be non-
wider in ℓ than those of WMAP, while those of DASI are
Gaussian,thereisagenericformtoconsiderfortheinitialcondi-
almost as wide as the features inthe power spectrum. Also
tions, where a quadratic contribution to the curvature perturba-
note thatthey-axishereisnotmultipliedbytheadditional
ℓ,whichhelpstoshowboththelargeandsmallangularscale tionsisparameterizedthroughadimensionlessnumberfNL. This
weaklynon-linearcomponentcanbeconstrainedthroughmeasure-
features.
ments of the bispectrum or Minkowski functionals for example,
Themostintriguingresultfromthepolarizationmeasurements andtheresultfromWMAP is−58<fNL<134(95% confidence
is at the largest angular scales (ℓ<10), where there is an excess region)[11].
signal compared to that expected from the temperature power
spectrum alone. This is precisely the signal expected from an
early period of reionization, arising from Doppler shifts during 8. Constraints on Cosmologies
the partial scattering at z < zi. It seems to indicate that the Theclearestoutcome ofthenewerexperimental resultsisthat
firststars,presumablythesourceoftheionizingradiation,formed
the standard cosmological paradigm is in good shape. A large
aroundz=20(although theuncertainty isstillquitelarge).
amountofhighprecisiondataonthepowerspectrumisadequately
7. Complications fit with fewer than 10 free parameters. The frameworkis that of
Friedmann-Robertson-Walker models, which have nearly flat ge-
There are a number of issues which complicate the interpreta- ometry, containing Dark Matter and Dark Energy, and with adi-
tionofCMBanisotropydata,someofwhichwesketchoutbelow. abatic perturbations having close to scale invariant initial condi-
7.1. Foregrounds: tions.
The microwave sky contains significant emission from our Within this framework,bounds can be placed on the values of
Galaxyandfromextra-galactic sources[53]. Fortunately, thefre- thecosmologicalparameters. Ofcourse,muchmorestringentcon-
quencydependence ofthesevarioussourcesisingeneral substan- straintscanbeplacedonmodelswhichcoverarestrictednumber
tially different from that of the CMB anisotropy signals. The of parameters, e.g.assumingthat Ωtot=1, n=1orr=0. More
combination of Galactic synchrotron, bremsstrahlung and dust generally,theconstraintsdependupontheadoptedpriors,evenif
emissionreachesaminimumatawavelengthofroughly3mm(or theyareimplicit,forexamplebyrestrictingtheparameterfreedom
about 100GHz). As one moves to greater angular resolution, the or the ranges of parameters (particularly where likelihoods peak
minimummovestoslightlyhigherfrequencies,butbecomes more near the boundaries), or by using different choices of other data
sensitivetounresolved(point-like)sources. incombinationwiththeCMB.Whenthedatabecomeevenmore
Atfrequenciesaround100GHzandforportionsoftheskyaway precise, these considerations will be less important, but for now
fromtheGalacticPlanetheforegroundsaretypically1to10%of we caution that restrictions on model space and choice of priors
the CMB anisotropies. By making observations at multiple fre- needtobekept inmindwhenadoptingspecific parameter values
quencies, it is relatively straightforward to separate the various anduncertainties.
components and determine the CMB signal to the few per cent There are some combinations of parameters that fit the CMB
level. For greater sensitivity it is necessary to improve the sep- anisotropiesalmostequivalently. Forexample,thereisanearlyex-
aration techniques by adding spatial information and statistical act geometricdegeneracy, whereany combinationof ΩM and ΩΛ
propertiesoftheforegroundscomparedtotheCMB. that gives the same angular diameter distance to last scattering
TheforegroundsforCMBpolarizationareexpectedtofollowa will give nearly identical Cℓs. There are also other near degen-
similarpattern,butarelesswellstudied,andareintrinsicallymore eracies among the parameters. Such degeneracies can be broken
complicated. Whether it is possible to achieve sufficient separa- whenusingtheCMBdataincombinationwithothercosmological
tiontodetectB-modeCMBpolarizationisstillanopenquestion. datasets. Particularlyusefularecomplementaryconstraintsfrom
6 Cosmic Microwave Background Scott & Smoot
galaxyclustering,theabundanceofgalaxyclusters,weakgravita- tween modes (as mentioned in Sec. 2.3), etc., the significance of
tionallensingmeasurements,TypeIasupernovadistancesandthe thisfeatureisstillamatter ofdebate [13][64].
distributionofLymanαforestclouds. Foranoverviewofsomeof In addition it is also possible to put limits on other pieces of
theseothercosmologicalconstraints, seeRef.[16]. physics[65],forexampletheneutrinochemical potentials,contri-
The combination of WMAP, CBI and ACBAR, together with butionofwarmdarkmatter, decayingparticles,timevariationof
weakpriors(onhandΩBh2forexample),andwithinthecontext the fine-structure constant, or physics beyond general relativity.
ofa6parameterfamilyofmodels(whichfixesΩtot=1andr=0), Further particle physics constraints will follow as the anisotropy
yieldsthefollowingresults[56]: A=2.7(±0.3)×10−9,n=0.97± measurements increaseinprecision.
0.03, h=0.73±0.05, ΩBh2=0.023±0.001, ΩMh2=0.13±0.01 Careful measurement of the CMB power spectra and non-
and τ =0.17±0.07. Other combinations of data, e.g. including Gaussianity can in principle put constraints on physics at the
BOOMERANGandothernewerCMBmeasurements,orimproved highestenergies,includingideasofstringtheory,extradimensions,
large-scale structure data, lead to consistent results, sometimes collidingbranes,etc. Atthemomentanycalculationofpredictions
withsmallererrorbars,andwiththeprecisevaluesdependingon appears to be far from definitive. However, there is a great deal
data selection [57]. Notethat forh, theCMBdata alone provide ofactivityonimplicationsofstringtheoryfortheearlyUniverse,
onlyaveryweak constraint, unless spatial flatness orsomeother and hence a very real chance that there might be observational
cosmological data are used. For ΩBh2 the precise value depends implicationsforspecificscenarios.
sensitively on how much freedom is allowed in the shape of the
primordialpowerspectrum(see‘Big-BangNucleosynthesis’mini-
10. Fundamental Lessons
review[58]).
The best constraint on Ωtot is 1.02±0.02. This comes from More important than the precise values of parameters is what
including priors from h and supernova data. However, slightly we have learned about the general features which describe our
different, but consistent results come fromusing other data com- observable Universe. Beyond the basichot BigBang picture, the
binations. CMBhastaughtusthat:
The 95% confidence upper limit on r is 0.53 (including some • The Universe recombined at z ≃1100 and started to become
extra constraint from galaxy clustering). This limitis stronger if ionizedagainatz≃10–30.
werestrictourselveston<1andweakerifweallowdn/dlnk6=0. • ThegeometryoftheUniverseisclosetoflat.
There are also constraints on parameters over and above the • BothDarkMatterandDarkEnergyarerequired.
basic 8 that we have described. But for such constraints it is • Gravitationalinstabilityissufficienttogrowalloftheobserved
necessary to include additional data inorder to break the degen-
largestructuresintheUniverse.
eracies. ForexampletheadditionoftheDarkEnergyequationof
• Topologicaldefectswerenotimportantforstructureformation.
state, w adds the partial degeneracy of being able to fit a ridge
• Thereare‘synchronized’ super-Hubblemodes generated inthe
in(w,h)space, extending tolowvalues ofbothparameters. This
earlyUniverse.
degeneracy isbrokenwhenthe CMBisusedincombination with
• Theinitialperturbations wereadiabaticinnature.
independent H0 limits,forexample [59], giving w<−0.5at 95% • The perturbations had close to Gaussian (i.e., maximallyran-
confidence. Tighter limits can be placed using restricted model-
dom)initialconditions.
spaces and/oradditionaldata.
It is very tempting to make an analogy between the status of
For the optical depth τ, the errorbar is largeenough that ap-
the cosmological ‘Standard Model’ and that of particle physics.
parentlyquitedifferentresultscancomefromothercombinations
In cosmology there are about 10 free parameters, each of which
ofdata. TheconstraintfromthecombinedWMAPCTTandCTE
ℓ ℓ isbecomingwelldetermined,andwithagreatdealofconsistency
dataisτ=0.17±0.04,whichcorresponds(withinreasonablemod-
between differentmeasurements. However,noneoftheseparame-
els) to a reionization redshift 9<zi<30 (95% CL) [47]. This is terscanbecalculated fromafundamental theory,andsohintsof
a little higher than some theoretical predictions and some indi-
thebiggerpicture,‘physicsbeyondtheStandardModel’arebeing
cations from studies of absorption in high-z quasar spectra [60]. searchedforwithevermoreambitiousexperiments.
TheexcitementhereisthatwehavedirectinformationfromCMB Despite this analogy, there aresome basicdifferences. For one
polarizationwhichcanbecombinedwithotherastrophysicalmea- thing, many of the cosmological parameters change with cosmic
surementstounderstandwhenthefirststarsformedandbrought epoch,andsothemeasuredvaluesaresimplytheonesdetermined
abouttheendofthecosmicdarkages. today, andhence theyarenot‘constants’, likeparticlemassesfor
example (although they are deterministic, so that if one knows
9. Particle Physics Constraints
their values at one epoch, they can be calculated at another).
CMBdataarebeginningtoputlimitsonparameterswhichare Moreover, the number of parameters is not as fixed as it is in
directlyrelevantforparticlephysicsmodels. Forexamplethereis theparticlephysicsStandardModel;differentresearcherswillnot
a limit on the neutrino contribution Ωνh2 <0.0076 (95% confi- necessarily agree on what the free parameters are, and new ones
dence) fromacombinationofWMAP andgalaxyclusteringdata can be added as the quality of the data improves. In addition,
from the 2dFGRS project [61]. This directly implies a limit on parameters likeτ, whichcome fromastrophysics, areinprinciple
neutrino mass, mν <0.7eV, assuming the usual number den- calculable from known physical processes, although this is cur-
sityoffermionsPwhichdecoupledwhentheywererelativistic. rently impractical. Ontop of all this, other parameters might be
A combination of the WMAP data with other data-sets gives ‘stochastic’inthattheymaybefixedonlyinourobservablepatch
somehintofarunningspectral index, i.e.,dn/dlnk6=0[56]. Al- oftheUniverseoramongcertainvacuumstatesinthe‘Landscape’
though this is still far from resolved [62], things will certainly [66].
improve as new data come in. A convincing measurement of a In a more general sense the cosmological ‘Standard Model’ is
non-zerorunningofthe indexwouldbequite constrainingforin- muchfurtherfromtheunderlying‘fundamentaltheory’whichwill
flationarymodels[63]. ultimately provide the values of the parameters from first princi-
Oneother hintofnew physics liesinthefactthat thequadru- ples. Nevertheless, any genuinely complete ‘theory of everything’
pole and some of the other low ℓ modes seem anomalously low must include an explanation for the values of these cosmological
comparedwiththebest-fitΛCDMmodel[38]. Thisiswhatmight parameters as well as the parameters of the Standard Model of
be expected in a universe which has a large scale cut-off to the particlephysics.
powerspectrum,oristopologicallynon-trivial. However,because
ofcosmicvariance,possibleforegrounds,apparentcorrelationsbe-
7 Cosmic Microwave Background Scott & Smoot
11. Future Directions Acknowledgements
With all the observational progress in the CMB and the ty- Wewouldliketothankthenumerouscolleagueswhohelpedin
ing down of cosmological parameters, what can we anticipate for compiling and updating this review. We are also grateful for the
the future? Of course there will be a steady improvement in the diligence ofthe RPP staff. Thisworkwas partiallysupported by
precisionandconfidence withwhichwecandeterminetheappro- theCanadianNSERCandtheUSDOE.
priate cosmological model and its parameters. We can anticipate
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