ONDES MATIÈRE ETUNIVERS- RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS Abstract 5 based onthepublishedorsubmittedworks ofthe and of characteristics of the primordial fluctuations. This overview is entirely temperature andpolarisationdata, bothinterms ofcontenttheuniverse pioneering polarisationdata. Idescriberesults we obtainedsofarfrom these with theirfullstatistical characterisation. alsoincludes The 2015delivery Cosmic Microwave Background (CMB), its angular power spectrum together formid-2017. The is expected 2015 data delivery provides a “definitive” delivery map of the anisotropies of “legacy” the final 2015.A February in set a first set of cosmological data and results in March 21 March in results and data cosmological of set first a campaign thatexceededallexpectations. collaboration deliveredThe Planck launched in 2009, the Planck satellite was shut off in 2013, after a measuring Sketched outin1992, selectedby theEuropean Space Agency in1996, and Institut d’AstrophysiqueInstitut Paris, Université-UPMC, de CNRS &Sorbonne

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131 132 ments provided by two scientific consor tia funded by ESA member states and led by Principal Investigators Investigators 4years. about of duration LFI full the the for while data Principal months, 30 taking kept about by for i.e. led cryogens, the of and exhaust the till scientific a data and survey took ESA HFI the states fridge, between member ESA collaboration a by (USA). NASA from through funded contributions provided additional tia and Denmark, by funded and led reflectors consor consortium telescope Italy, and scientific France two from by provided ments tors of the High Frequency Instrument (HFI; Lamarre et al. 2010; Planck HFI Core Team Core HFI Planck 2010; al. 2011) et Lamarre were bolometers, cooled (HFI; Instrument were 2011) Frequency al. High et the of tors 70 Mennella and 44, 2010; 30, at al. et centred bands covering Bersanelli radiometers, (LFI; pseudo-correlation Instrument Frequency Low the of tors mission The Planck 1. was launched on May 14 May on launched was satellite The evolution. subsequent its and Universe early the studying to 2011) dedicated is so far are discussed in detail in a suite of more than 150 papers by the Planck bythe 150 papers than more of a suite in detail in discussed are far so included in a issue special of Astronomy & Astrophysics (Volume 520). All obtained results 13 in papers described are launch of time the at predicted as performance its and payload, 857 and 545, 353, 1 & tables in provided is mission an array of 74 detectors in nine frequency bands sensitive to frequencies between 25 and and 25 between 1000 frequencies to sensitive bands frequency nine in detectors 74 of array an 2013. Planck October 23 and 2009 12 August between uously of sensitivity, angular resolution, and frequency coverage never before achieved. achieved. before never coverage frequency and resolution, angular sensitivity, of

(2) (1) Planck Agency’s Space European The A summary of the characteristics of the LFI and HFI maps based on the data of the full full the of data the on based maps HFI and LFI the of characteristics the of A summary

GHz, which scanned the sky with angular resolution between 33 between resolution angular with sky the scanned which GHz, Since the HFI operational temperature of 0.1 K was achieved thanks to an open loop dilution dilution loop open an to thanks achieved K was 0.1 of temperature operational HFI the Since Planck COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. Résumé les travaux delacollaboration Planck, déjàpubliésousoumis. sur basée entièrement est revue primordiales. Cette fluctuations les caractérisant concernant les paramètres comme ceux caractérisant le contenu del’univers jusqu’ici,de température desdonnées de polarisation, etdesdonnées àpartir dufondsdiffus. sur lapolarisation originales données lesrésultatsobtenus Jedécris une complète caractérisation statistique. également des Cette fournée 2015 inclut 2017. micro-ondes, milieu son spectre angulairede puissance et en donnemêmetemps pour attendue est finale « La livraison 2015fournitcarte une livraison La 2015. février en complet 2013,cosmologiques etderésultatsle21mars fournir avantd’en un ensemble les attentes. aproduit Lacollaboration unpremier Planck ensemblededonnées fonctionner en 2013, après unecampagnede mesures qui a dépassé toutes Européenne en1996, lancé en2009, s’est arrêté de le satellite Planck ses grandes dans lignes en1992, Ebauché retenu parl’Agence Spatiale cosmique réseau jusqu’au quantique lamousse Cosmologie :depuis avec lesatellitePlanck

(http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instru with (ESA) Agency Space European the of a project is (http://www.esa.int/Planck) GHz. GHz. th Planck 2009 and 2009 scanned the microwave and sub-millimetre contin sky 2 imaged the whole sky twice per year, with a combination acombination year, with per twice sky whole the imaged 2. Note in particular the excellent sensitivity achieved by HFI by HFI achieved sensitivity excellent the particular in Note 2. at 0.1±0.001 K and covering bands centred at 100, 143, 217, 143, 100, at centred bands covering and K 0.1±0.001 at satellite » des anisotropies dufondsdéfinitive diffus 1 (Tauber et al. 2010; Planck Collaboration I I Collaboration Planck 2010; al. et (Tauber ’s scientific payload contained contained payload ’s scientific and 5 and GHz. The detec The GHz. collaboration. . The detec . The Planck

, its , its - - - - RELATIVITÉ, ONDES DE L’UNIVERS 133

Table 1. Main characteristics of LFI full mission maps.

Frequency band Characteristic 30 GHz 44 GHz 70 GHz Centre frequency [GHz] 28.4 44.1 70.4 Effective beam FWHMa [arcmin] 32.29 27.00 13.21 Effective beam ellipticitya 1.32 1.04 1.22

◦ b Temperature noise (1 ) [µKCMB] 2.5 2.7 3.5 ◦ b Polarization noise (1 ) [µKCMB] 3.5 4.0 5.0 Overall calibration uncertaintyc [%] 0.35 0.26 0.20

d Systematic effects uncertainty in Stokes I [µKCMB] . 0.19 0.39 0.40 d Systematic effects uncertainty in Stokes Q [µKCMB] 0.20 0.23 0.45

d Systematic effects uncertainty in Stokes U [µKCMB] 0.40 0.45 0.44

a Calculated from the main beam solid angle of the effective beam, Ωeff = mean(Ω). These values are used in the source extraction pipeline (Planck Collaboration XXVI 2015). b Noise rms computed after smoothing to 1°. c Sum of the error determined from the absolute and relative calibration, see Planck Collaboration IV (2015). d Estimated rms values over the full sky and after full mission integration. Not included here are gain reconstruction uncertainties, estimated to be of order 0.1 % .

in the core CMB channels, on which most cosmology results rely, of 1.29, 0.55, 0.78 µKCMB deg at (respectively) 100, 143, and 217 GHz. These numbers indicate the rms of the fluctuations contributed by detector noise in pixels of 1 degree on a side (leading to 7.5, 4.3, 8.7 µKCMB per (Gaussian) beam solid angle of FWHM equal to 9.66, 7.22, 4.90 arcmin (the rms of the CMB anisotropy is about 100 µKCMB). For white noise (a reasonable approximation at scales below a degree), the detector noise scales inversely proportional to the pixel linear size.

2. CMB maps from Planck

The nine all-sky high-sensitivity high angular resolution Planck maps in intensity (figure 1) and their associated polarisation maps at seven frequencies are a treasure trove for astro- physics, which have already allowed many progresses in the understanding of the various astrophysical sources of emission in the millimetre and sub-millimetre range, e.g., on the diffuse Galactic emission (in particular synchrotron, free-free, CO, spinning and thermal dust), as well as compact sources (radio-sources, Infra-red galaxies, Sunyaev-Zel’dovich clusters) and the unresolved Cosmic Infra-red background, which is the integrated light from all infrared sources along the line of sight.

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 134 one of them (here SMICA). (here them of one difference of) Figure (lack resilient. truly are Their features map. which probe to CMB allows that characterising together map, ( estimate map noise confidence a map, CMB a least at produces specific the method with separation line component Each in best. solve to out set they problem possibilities, stochastic and objectives different have NILC, dif methods Indeed, (Commander, maps. ferent frequency approaches various the differently different combine which four SMICA) used SEVEM, have we galaxies, other and maps. mission full HFI of characteristics Main Table 2. 2.1 CMB map cleaning 2.1 CMBmap CIB monopole prediction [ MJy sr [MJy prediction monopole CIB

Noise per beam solid angle [ angle solid beam per Noise In order to clean the background CMB map from foreground emissions from our from emissions foreground from map CMB background the clean to order In e d c3 c2 c1 b3 b2 b1 a1 uncertainty). at the20%level (alsofor constant According totheBétherminetal. (2012)model, isestimated tobe whoseuncertainty Calibrationaccuracy(at545and857GHz, the5%accountsfor themodel Numberofbolometerswhosedatawere usedinproducing thechannelmap. Estimateofthenoiseinpolarizationscaledto1 Estimateofthenoiseinintensityscaledto1 Estimateofthenoiseperbeamsolidangle, asgiven in Ratioofthemajortominoraxisbest-fitGaussian averaged over thefullsky. FWHMoftheellipticalGaussianfit. FWHMoftheGaussianwhosesolidangleisequivalenttothateffective beams. Effective beam FWHM beam Effective Effective beam FWHM beam Effective Temperature noise [ noise Temperature COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC WEB - Polarization noise [ noise Polarization Characteristic Effective beam ellipticity beam ellipticity Effective Calibration accuracy [%] accuracy Calibration Number of bolometers

[kJy sr [kJy µ µ 1 K K 2 [arcmin] [arcmin] [kJy sr [kJy CMB CMB µ −1 K deg] deg] deg] deg] CMB −1 −1 ϵ ] ] ] νI ν

0.0030 ). .8 1.040 1.186 100 0.09 9.68 9.66 1.29 1.96 ...... 7.5 8 ° 0.0079 assumingthatthenoiseiswhite. 0.55 0.07 143 7.30 7.22 .71.75 1.17 4.3 11 Reference frequency i.e ° assumingthatthenoiseiswhite. ., a mask), an effective beam, and a a and beam, effective an mask), a ., .6 1.166 1.169 0.033 5.02 .04.92 4.90 0.78 217 0.16 b1 8.7 212 12 .

2 illustrates the results of results the 2 illustrates 2.56 353 0.78 4.94 29.7 7.31 0.13 1.1(+5) 1.137 545 4.83 0.78 4.67 0.35 . . . . 9.1 3 ν F. R. F. R. [GHz] BOUCHET 1.4(+5) 1.336 4.22 4.64 857 0.72 0.64 8.8 4

-

Notes b3 b2 b1 c2 c2 c3 c1 c1 a1 d e RELATIVITÉ, ONDES DE L’UNIVERS 135

2.2 CMB map isotropy and general non-Gaussian statistics

The previous CMB maps may be used to study the statistical isotropy and Gaussianity of the CMB. Here I survey null-hypothesis testing: a number of tests are performed, then p-values are calculated. A posteriori correction for “look-elsewhere effect” was ad- dressed whenever possible. However, it is in the very nature of such a model-independent approach to leave the interpretation to further research.

All of the results we obtained are robust with respect to the choice of component sep- arated CMB map. This is important since it demonstrates the high quality and equivalence for that purpose of the Planck component-separated data products rendered by different methodologies under varying assumptions.

It is found that the CMB is largely consistent with statistical isotropy, although there are a few indications of anomalies with respect to the expectations of the ΛCDM cos- mological model. Some of the tests performed in 2015 are the same as those in the 2013 release, in which case the results are consistent. Since many of these anomalies were also observed in the WMAP temperature data, the agreement between the two independent experiments effectively rules out the possibility that the origin of these features can be found in residual systematic artefacts present in either data set (either originating from the instruments or foregrounds).

Aspects of the statistics of the CMB fluctuations were assessed with tests of skewness, kurtosis, multinormality, N-point functions, and Minkowski functionals, and none yielded indications of significant departures from Gaussianity, while the variance of the CMB map was found to be low, in agreement with previous studies. First-order moments of filtered maps also exhibit the low variance anomaly, as well as a kurtosis excess on certain scales associated with the so called “Cold Spot”. A study of peak statistics finds results consis- tent with the expectations for a Gaussian random field, although the Cold Spot is again detected.

The low variance anomaly appears to be associated with the known low- deficit in the angular power spectrum (see §3.3 below). The lack of large-scale angular correlations, the relatively featureless northern ecliptic hemisphere 3- and 4-point functions,퓁 and indi- cations of violations of point- and mirror-parity symmetry are also confirmed (although little attempt was made to correct these for a posteriori effects). Tight constraints on a quadrupolar power modulation were also obtained.

The now well-known large-scale dipolar power asymmetry (at 7 % level) was given particular attention. This asymmetry was detected via pixel-to-pixel variance, as well as by measuring power explicitly or indirectly via to ± 1 mode coupling. The latter approach lends itself to a posteriori correction, which reduces the significance of the asymmetry substantially when no model for the퓁 anomaly퓁 is assumed. In addition, two independent but related tests of directionality were conducted: one finds suggestions of anomalous clustering of directions out to relatively small scales while the other does not, evidently due to being optimized for slightly different forms of directionality.

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 136

30 GHz 44 GHz 70 GHz COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R.

100 GHz 143 GHz 217 GHz

353 GHz 545 GHz 857 GHz

-1 1

-103 -102 -10-101 10 102 103 104 105 106

30-353 GHz: δT [μKCMB]; 545 and 857 GHz: surface brightness [kJy/sr]

Figure 1. The nine Planck frequency maps between 30 and 857 GHz (these are the central frequency of the bands). The color scale, based on an inversion of the function y = 10x − 10−x, is tailored to show the full dynamic range of the maps. RELATIVITÉ, ONDES DE L’UNIVERS 137

-300 µμKK 300

Figure 2. Top: Planck 2015 temperature anisotropies map. A small strip in the direction of the Galactic plane is lled in by a constrained realization that has the same statistical properties as the rest of the sky (this area of the sky is obviously not use for any CMB science analysis). Bottom: Planck 2015 CMB polarisation map zoomed, with 20 arc minute smoothing, revealing smaller scales details (the data is natively at 5 minutes of arc resolution).The coloured background shows the temperature anisotropies, enabling to visually perceive the correlation between the temperature and polarisation elds.

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 138 auto-spectra three the namely spectra, power independent four measure to able be to we expect Thus only with fields, random Gaussian isotropic during an inflationary epoch. The The epoch. an inflationary during for example as produced radiation), the (gravitational modes by tensor times at sourced early observable, polarization other The depth). optical reionization the background cosmology, as well as an improved determination of some parameters ( parameters some of determination improved an as well as cosmology, background independent an allows therefore and intensity, the as physics same so-called the One, 1997). can in turn be decomposed into two coordinate-independent quantities with different different ( with cosmology the quantities on dependence coordinate-independent it two into direction, and decomposed be amplitude an turn in both by can given is polarization linear Because 2014). al. et Collaboration 2012;Polarbear al. et Collaboration QUIET 2009; al. et Pryke 2009; al. et ( data polarization (2015). XX Collaboration in Planck (2014),may found be 2015XV the and analysis Collaboration (2015). XVI Collaboration Planck in detailed are results these of All cosmology. of model standard the beyond us take will far so described anomalies the of that some It possibility is asignificance. tantalizing statistical their in interpreting ambiguity and subjectivity the eliminate or reduce will this Importantly, features. peculiar these of tests independent 2017,enable in should release Planck final the for po- expected larization, large-angular-scale of use The alone. fluctuations temperature with anomalies CMB deficit. power large-scale the of reflection yet be another to seems which profile, temperature unoriented low a is exception The stacking. unoriented and oriented for both simulations, isotropic statistically with consistent ly of balloon and ground-based telescopes ( telescopes ground-based and balloon of COBE by sky the of fractions large over measured been has spectrum power the temperature, In parameters. its and model cosmological underlying the by determined uniquely turn in are spectra power CMB The foregrounds. astrophysical from contamination of and fects, ef systematic processing or instrumental various of noise, instrumental of models with augmented be must spectra empirical these data, realistic For approximation. excellent an is know we now which Gaussian, amultivariate as distributed and isotropic statistically 3. Power spectra from3. Planck et al. 2013). The The 2013). al. et Das 2012; al. et 2011; al. Story 2011;et al. Keisler et Das 2010; al. et Fowler 2009; al. et Reichardt 2006; al. et Jones 2005; al. et Tristram 2003; al. et Pearson 2003; al. et Grainge

Over the last decade, CMB intensity (temperature) has been augmented by linear linear by augmented been has (temperature) intensity CMB decade, last the Over the to probe of our ability limit the near one is probably 2015 Planck the release, With large are They analysed. was peaks polarization and temperature of stacking Finally, The CMB angular power spectra contain all of the information available if the CMB is (Wright et al. 1996) and WMAP and 1996) al. et (Wright COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R.

C l TT Planck , e.g.

C l EE , Kovac et al. 2002; Kogut et al. 2003; Sievers et al. 2007; Dunkley Dunkley 2007; al. et Sievers 2003; al. et Kogut 2002; al. et Kovac , and and 2013 power spectrum and likelihood were discussed in Planck Planck in discussed were likelihood and 2013 spectrum power mode which is the curl-free part, is determined by much the is determined part, E mode is curl-free which the

C l BB , along with the cross-spectrum the with , along and E and e.g , Kamionkowski et al. 1997; Zaldarriaga and Seljak Seljak and Zaldarriaga 1997; al. et Kamionkowski , (Bennett et al. 2013),(Bennett and by in regions smaller a host components are also conventionally taken to be be to taken conventionally also are B components e.g. , Netterfield et al. 1997; Hanany et al. 2000; 2000; al. et Hanany al. 1997; et Netterfield , expected to be correlated with intensity. intensity. with correlated be to E expected

C l TE measurement of the the of measurement . mode, is only only is B mode, e.g ., ., - - RELATIVITÉ, ONDES DE L’UNIVERS 139

6000

5000 ] 2 4000 [ µ K TT ℓ 3000 2000

1000

0 600 60

TT 300 ℓ 30 0 0 ∆ -300 -30 -600 -60 2 10 30 500 1000 1500 2000 2500 ℓ ]

140 2 100 ] 2

µ K 80

70 -5

[ µ K 60 TE 0 [10 ℓ 40 EE ℓ

-70  20 -140 0 TE

10 EE 4 ℓ 0 ℓ 0

∆  -4 ∆ -10 30 500 1000 1500 2000 30 500 1000 1500 2000 ℓ ℓ

Figure 3.Planck 2015 CMB power spectra of TT (top), TE and EE (bottom), compared with the base LCDM t (red line). The upper panels show the binned spectra and the lower pan- els the residuals of the t. For all plots, the horizontal scale changes from logarithmic to linear at the hybridization scale, = 29. For the residuals, the vertical axis scale changes as well, shown by different left and right tick marks. Note that we show = ( +1) C /(2π) 퓁 for TT and TE but C for EE, which also has different vertical scales at low-퓁 and high-퓁 . 퓓 퓁 퓁 퓁 퓁

The distribution of temperature and polarization on the sky is further affected by grav- itational lensing by the inhomogeneous mass distribution along the line of sight between the last scattering surface and the observer. This introduces correlations between large and small scales, which can be gauged by computing the expected contribution of lensing to the 4-point function (i.e., the trispectrum). This can in turn be used to determine the power spectrum of the lensing potential, as is done in Planck Collaboration XV (2015) for the 2015 Planck release, and to further constrain the cosmological parameters via a separate likelihood function (Planck Collaboration XIII 2015). The following paragraphs

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 140 likelihood including both temperature and polarization for 퓁 and multipoles likelihood polarization temperature including both scales. smaller at estimates pseudo-spectral of use the and scales large on likelihood the of calculation direct a between splitting 2006), maps, while for polarization we used the 70 the used we polarization 2008) for 2004, while al. maps, et (Eriksen Commander cleaned the uses formalism the temperature, function based on pseudo- likelihood. the in covariance noise induced the for cases both in accounting respectively, 30 the over arespectra high- at quite a low level of (of the order of a few ( contribution the that shows It release. spectra(figure Tand the modifying to addition in effect, lensing The 4-pt). lensing and (2-pt both to introduction an provide themselves, the main advances over 2013 include the use of high- of use the 2013 include over advances main the themselves, structed our likelihood approximation at high- at approximation likelihood our con- We structed effects. instrumental and foregrounds of models detailed more with along tion ( effects instrumental and emission astrophysical foreground of contribution the describing eters esting comparison of the polarization-based cosmological results with those derived from derived those with results cosmological of the polarization-based comparison esting systematic effect, and we preferred to defer this more powerful but delicate analysis to a later release. a later to analysis delicate but powerful more this defer to preferred we and effect, systematic scales, scales, angular smaller at function likelihood the of behaviour Gaussian increasingly the as well as data. polarization large at reason, this For ( scales angular temperature. in than definitive less is which effects astrophysical systematic and instrumental and noise random the of understanding careful a for require (especially ratio signal-to-noise lower inherently the with along anal- ysis, data subsequent and measurements polarization with involved difficulties technical pose on any possible primordial ones. primordial possible any on pose using these spectra and the corresponding analytical covariance matrix: covariance analytical corresponding the and likelihood spectra these grand using a built and cross-spectra, (pseudo-) mask-corrected into data detector 3.1 CMBpower spectra from Planck C

l TT At low multipoles, the current Planck current the multipoles, low At At high multipoles multipoles high At The The (3) Because of these different sensitivities to systematic errors at different angular scales, scales, angular different at errors systematic to sensitivities different these of Because − alone. Planck

The HFI data, having much less noise, therefore requires a much tighter control of any residual residual any of control tighter amuch requires therefore noise, less much having data, HFI The ln COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. Planck L Planck e.g. ( C ˆ , calibration, beams). Aside from the processing improvements of the data data the of improvements processing the from Aside beams). calibration, , | GHz and 353 and GHz C adopted a hybrid approach to the likelihood calculation (Efstathiou 2004, 2004, (Efstathiou calculation likelihood the to approach hybrid a adopted 3), likelihood functions, and basic cosmological parameters from the 2015 the from parameters cosmological basic and functions, likelihood 3), paper Planck Collaboration XX (2015) obtains the the obtains (2015) XX Collaboration Planck paper (θ i.e )) ., low multipoles presented results for for results presented = modes (from the initial Eones) (from B modes superim initial which the generates E fields, 2 1 ( 퓁 > 29) C ˆ − GHz maps taken as tracers of synchrotron and dust emission, emission, dust and synchrotron of tracers as taken maps GHz C C 퓁 s calculated from (θ , as in Planck Collaboration XV (2014), we use a likelihood likelihood a use (2014),we XV Collaboration Planck in as , )  T

퓁 C ) the 2015 baseline results use only a subset of of subset a only use results 2015 baseline the ) −1 C ˆ release implements a standard joint pixel-based pixel-based joint standard a implements release

C − l TE C and and (θ by compressing all of the individual Planck individual the of all 퓁 bycompressing GHz LFI maps LFI GHz Planck )  systematic errors to the polarization polarization the to errors 퓁 systematic +

C const l EE HFI data, as well as further param further as well as data, HFI µ at high multipoles. However, the However, the multipoles. high at K , )

2 ), therefore allowing an inter 3 and explicitly marginalize and explicitly polarization informa 퓁 polarization

C l TT , modes), thus thus modes), B

C l EE ≤ 29 and and Planck . For For .

C (1) l TE - - - -

RELATIVITÉ, ONDES DE L’UNIVERS 141

where Ĉ is the data vector, C(θ) is the model with parameters θ, and C is the covariance matrix. Note that this formalism allows to separately marginalize over or condition upon different components of the model vector, separately treating cases such as individual frequency-dependent spectra, or temperature and polarization spectra. Obviously, Planck maps at different frequencies have different constraining powers on the underlying CMB, and we used this to impose and assess various cuts to keep only the most relevant data in the data vector. Indeed, we retained only the three best CMB Planck channels, i.e., 100 GHz, 143 GHz, and 217 GHz, in the multipole range where they have significant CMB contributions and low enough foreground contamination after masking. Further, in order to achieve a significant reduction in the covariance matrix size (and computation time), we compressed the data vector (and accordingly the covariance matrix), both by co-adding the individual detectors for each frequency and by binning the combined power spectra.

The construction of a Gaussian approximation to the likelihood function requires building covariance matrices for the pseudo-power spectra. Mathematically exact ex- pressions exist, but they are prohibitively expensive to calculate numerically at Planck resolution (Wandelt et al. 2001); we thus made use of analytical approximations (Hansen et al. 2002; Hinshaw et al. 2003; Efstathiou 2004; Challinor and Chon 2005). For our base- line likelihood, we calculate covariance matrices for all 45 unique detector combinations that can be formed out of the six frequency-averaged half-mission maps at 100, 143, and 217 GHz. To do so, we assume a fiducial power spectrum that includes the data variance induced by the CMB and all identified foreground components (which allow a good de- scription of all relevant Planck data, including at frequencies not used in the likelihood); this variance is computed assuming these components are Gaussian-distributed. The ef- fect of this approximation regarding Galactic foregrounds was shown negligible by means of simulations. The fiducial model is taken from the best-fit cosmological and foreground parameters; since they only become available after a full exploration of the likelihood, we iteratively refined our initial guess.

The resulting spectra are shown in figure 3, with the full statistical description being embodied in a likelihood code (applied to a set of empirical spectra from Planck) which nu- merically returns the likelihood of an input set of theoretical CMB spectra, and accounts for relevant uncertainties both instrumental and astrophysical in nature (as well as the correlations induced by the specific processing of the data). This code is publicly available. With this release, Planck now detects 36 extrema in total, consisting of 19 peaks and 17 troughs. The figure also shows the best-fit base-ΛCDM model obtained from TT data alone (red lines), which is a sufficient description for all spectra.

Detailed checks of consistency and null tests lead us to conclude that there might still be low-level systematics residuals in the E polarisation spectra which are not yet fully unaccounted for, at the (1µK2) , an example of which is shown by the green lines in the bottom panel. We therefore advised against using this polarisation information for test- ing non-minimal models𝓞𝓞 differing from more standard cases by wiggles of that order of magnitude. Nevertheless, the high-multipole polarization spectra from Planck are already good enough to allow a separate high-accuracy determination of the parameters of the ΛCDM model, showing consistency with those established independently from tempera- ture information alone. As a graphical example of this consistency, the bottom panel of

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 142 the the parameters for Λ for parameters Planck the of values numerical The polarisation. in now including data, the fact that high- that fact the for information same the Λ mostly access they that fact the to due model), parameters 6 relatively modest for base is one temperature the to added is information polarisation when constraints the of ing are consistent with current astrophysical knowledge. Of course, we also verified the consis the verified also we course, Of knowledge. experiments. CMB other from results the with tency astrophysical current with consistent are can be trusted for that purpose. 퓁 the to consistent restricted when Wegenerally least at also here, are obtained (2015) checked those with IX pow thatCollaboration the that derivedPlanck note in foreground and above further We propertiesdescribed methods parameters. by component-separation the cosmological obtained maps CMB from derived the of parameters and er spectra determination the affect not do ones. adiabatic dominating the to addition in modes isocurvature primordial of existence the like extensions, potential tightly rather constrains also it but parameters, basic the all constraining experiment single no we had ago long so not when the play at of physics basic confirmation strong a offers this does only Not part. TT corresponding the from Λ most for precision comparable of are which straints figure of CMB anisotropies is to determine the effective number of of number effective the determine to is anisotropies CMB of detailed more by confirmed is which analysis. statistical good, very indeed is fit The panel). TT top the in of the sky used), with with used), sky the of out Planck Carrying the for procedure etc.). this probes, other with cross-correlations CIB, the lensing, from coming information non-Gaussian all ignore (and hence Gaussian purely are anisotropies the that we assume if information cosmological available the all contains that ameasure spectra, 2 the to times of square This estimating S/N is the in total equivalent measured. the power extensions to the base Λ base the to extensions ( parameters particular on constraints tighter Planck fact in and others, than valuable more are formation in of pieces some since story, whole the not is this course Of 2013 the in release. than 2015 the Planck perspective this From 2015 The masking). and variance cosmic noise, TT instrumental of effects the includes (which CDM constraints from3.2 ΛCDMconstraints CMBspectra alone Planck CDM, albeit through different physical mechanisms. Maybe more impressive is actually actually is impressive more Maybe mechanisms. physical different through albeit CDM,

data have increased this value to 1 to value this increased have data The base (minimal) (minimal) base The We verified that degeneracies between foreground and calibration parameters generally generally parameters calibration and foreground between degeneracies that verified We One way of assessing the constraining power contained in a particular measurement measurement a particular in contained power constraining the assessing of way One Λ

CDM model with the best parameters obtained on TT alone (also shown in red red in shown (also alone TT on obtained parameters best the with model CDM 3 shows in red, superimposed to the the predicted data, TE and withinEE spectra COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. CDM are given in table 3 (columns 2 and 5) in both cases. The tighten- The cases. both in 5) and 2 (columns 3 table in given are CDM 퓁 temperature-polarisation correlation in TE is already providing con TE Λ and and CDM model) by including new polarization data. polarization new byincluding model) CDM CDM model with 6 parameters provides excellent fit to the the to fit excellent provides parameters 6 with model CDM Λ CDM ( 2013 TT 2013 EE 60 afurther adding i.e ., without considering any extension of the minimal minimal the of extension any considering without .,

114 power spectrum data yields the number 826 000 000 826 number the yields data spectrum power data constrain approximately 55 000 (in large part due to the increased fraction fraction increased the to due part large (in 000 e.g ., reionization optical depth or certain certain or depth optical reionization ., 96 and 000 CDM parameters than those arising arising those than CDM parameters < is able to place considerably considerably place to able is 2000 range in TT where they they where TT in range 2000 a 퓁 000 modes,000 respectively. m modes that have been been have that modes 2015 cosmological % more modes modes more % - - - - RELATIVITÉ, ONDES DE L’UNIVERS 143

σ Ω Ω ln(10 z k 100 Ω ω n Ω Ω z Ω ω Σ Ω z k Gy r. Age/ Ag z σ Ω τ Ω Pa k n ln(1 10 Ω k r d Y Ω Ω Ω N H H N 0 eq re eq D s s n n P m m 8 e m m c m b m Λ m b m c 0 K ...... 002 rameter aee 68 rameter s s 0   h BAO+JLA+H (“ext”, (“lensing”) data for external and reconstruction lensing with combination in spectra, power con CMB Parameter Table 3. Planck from limits dence that predicted by BBN (with a posterior mean Y mean (with a posterior BBN by predicted that Λ the to gives 95 extensions line parameter double one the below set for some third % limits The Λ base the specify to used parameters the on constraints the from (as obtained parameters derived of gives 68 set a number on % constraints h h h e ν ν / / h h h h ...... θ θ 0 2 2 / 2 2 dl dl 3 2 [eV [eV ]. 3 2 MC MC ...... Gy 10 .. .0 .. .0 .. .0 .. .0 ...... < k n n A r. ]. k s s )3 )3 .1 .1 .0 .0 .0 .0 .0 .< . . .0 .0 . . .0 0.14050 0 0.01035 0 lkT+oTBPi TT+lowTEB Plik TT+lowTEB Plik Plik .0959 7± .04085 .02222 . . .14050 .04085 .02222 0. 13.81 3± 13 0. 0959 0103 .142 .1197 .142 . 0 67 0.829 0 ± 67.31 0. 0. −0 − −0 − 9655 33±49 ± 3393 9655 1197 3393 .315 . .685 .315 .685 . .089 .089 68 0.25 −1 − 078 82 07 81 0 0 3 . < < TT 31 1 .008 .052 .052 .008 9 9.9 .1 .1 %l %l 6 9 8 3± 6 7± 5 .5 .5 0 . 0. 0 0. 0 9 ). Note that TT,TE,EE is actually a shorthand for Plik TT,TE,EE. The  The for Plik TT,TE,EE. rows gives of 68 set rst Λ base a shorthand for the % limits actually is TT,TE,EE that Note ). 3 3 2 2 + ±0 ±0 ±0 ± ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ± ±0 ±0 ± ±0 ± ±0 ±0 ±0 ±0 ±0 ±0 ±0 4 4 .103 .103 − − + + − − + − − 715 71 − + − imit s6 imit lo − + − − − + 1 1 1 1 0 0 0 0 0 0 0. 0.01 0 0 0. 49 0.01 0 . . 0 0 0 . . 0 0 0 0 0.049 0 . . 6 8 64 63 .002 .002 .006 .00047 .038 0050.01027 .00015 .019 .01 30 0030.02226 .00023 .014 .9 .00052 .038 .002 . .9 .00047 .00015 .014 .006 .036 .019 .002 .00023 .036 .00052 042 041 wTEB 5 . . . .049 . . 00045 01 0004 50 62 055 016 016 66 66 s6 3 30 3 0 2 20 0 20 2 5 Plik 0.14024 1.04103 0.0959 1± 0 0 1 0 0 TT 0. 0. 13 0. 0 . . 13 0. .04103 .01027 0. 0. 0 .02226 0959 1402 0.30 0.06 6± .118 3 0 .118 0.30 3.06 2± −0 − −0.003 − 1415 67± 9677 19±0 ± 8149 8149 9677 3365 1415 3365 .799 .692 .692 .799 . . .1±0 ± 7.81 7 0 −1.41 − + 06 06 0 0 3.13 . 8% 8% .251 .251 0 < <0 <0 < 81 1 lo . .005 .005 8 00 0.012 ± 8 8 2± 6± 6 6 1± 4 . .8 .8 41 wTEB 0 ±0 ±0 0 ± ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±4 ±0 ± ± ±0 ± ±0 ± ± ±0 ±0 ±4 .675 .114 .114 .675 3 − − + − + − limits limits − − + − − + − + − − − + 1 1 1 1 0 0 0 0 0 0 0 0 0. 0. 0.0004 0. 0 0 0 0.0004 0 0. 0. 0. . . 0 0 0 0 .6 .6 . 0 0 0 0 0 0 0 0 . . 4 7 61 .001 .00047 .038 .009 .0001 40 .00046 .012 .00023 .9 .0020 . .038 .009 .001 .012 .9 . .00023 . . .00046 43 43 040 039 ...... 0060 016 029 0060 01 0004 02 0020 0001 01 56 64 2 2 015 017 015 016 2 2 +l +l 6 9 2 9 3 3 9 nigPi TT+lowTEB+lensing+ Plik ensing ensing 5 7 40 5 P ≈ 0.2453, and theoretical uncertainties in the BBN predictions dominating over the Planck error on Ω on error Planck over the dominating predictions BBN the in uncertainties theoretical and ≈ 0.2453, Plik TT 1.0410 0.0959 .010258 0.02227 1.0410 0 . 0.0959 0 0. 010258 0. 0 0 0 13 0. . 13 .02227 0. 0 0 0. 0 1402 1402 2± + −0 .968 0. .306 .815 .118 79 ± 67.90 .815 .306 67 .968 3. 0. .118 3. −1.006 − −0.003 − 1413 6935 1413 6935 lo .796 .796 68 68 0.25 067 06 06 064 36 36 1± 1 0 . .0001 3 . wTEB 0001 <0 <0 <0 <0 90 . . 8 .1 .1 00 00 %l %l 1± 5± 4± 4 7 4± 6 3± 1± 4± 5± 1± 4± 6 2± 3± .9 .9 5 5 1 1 ±0 ±0 ±0 ±0 0.02 91 ± ±0 ±0 ±0 ± ±0 ± ±0 ±0 ±0 ±0 ±0 ±0 .234 .114 .234 .114 6 3 − − + − + − − + − imits imits − − + − − + 1 1 1 1 − − + 0 0 0 0 0 0 0 0 0.0044 0.007 20 0.0090 0.0004 0. 0 0 0 0 27 0.0004 0 0.0004 0 0. 27 0.0004 .3 .3 . . . 0 0 0 0 0 0 0 0 . . 0 0.0052 0 0.0052 2 + 41 40 .001 .00041 .013 .000083 .007 .00020 .5 . .007 .5 . . .001 . . .00041 .00020 .02 43 . .007 036 035 . . . . . 0012 02 0090 0044 000083 02 0012 01 091 085 014 015 0054 lensing 5 5 91 43 3 1 2 2 1 20 5 20 5 20 + ex ex t TT,ElwE TT,TE,EE TT,TE,EE+lowTEB t TT,TE,EE 1.04077 0.0960 0.01036 0.0222 0.0960 1 0.0222 0.01036 0.1427 0.9645 0.1198 0.6844 0.1427 0 0.9645 . .14059 0.1198 .14059 04077 0. 35 .010.3121 0.0091 ± .3156 0.831 67.27 0 . . 67 0. 3. 3. −0 −0 6844 3156 3395 35±33 ± 3395 . . .094 68 68 0.250 −1.55 − 079 1 .2 13.807 0.026 ± 813 831 094 813 079 <0 <0 2.99 . 10.0 <0 <0 27 1 .040 . .006 .040 006 %l %l 1± 5± 1± 5± . 55 .0987 .0987 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ± ±0 ± ±0 ±0 ±0 ± ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 .492 − − .492 + 0.027 − + − 0.027 + − − + − − + imits imits − − − + − + 0 0 0.41 0.41 0 0 0. 0.0001 0 0. 0 0 33 1 1 1. 1. lo . 0 0 0 0 . 0 0.041 0 0 0. 0 0. 0.038 026 . 39 01 .42±0.0013 ± 0.1422 .0014 .017 .013 04 .63± 0.9653 .0049 .00032 .0015 0000.01032 .00010 .66 .034 .026 09 0.6879 .0091 .013 . .0091 . .0049 . .00032 .0001 .00032 .0091 . . .00010 0020104± 0.14044 .00032 . . 5 7 7 . . .038 00029 0002 66 017 0014 034 0015 48 58 041 014 014 014 wTEB CDM model. In all cases the helium mass fraction used is is used fraction mass helium the cases all In model. CDM 60 9 60 TT,TE,EE 0 1.04087 0.0959 .01032 0 0.0959 1.04087 0.8150 0.1193 .02226 0 .14044 13 0 .0222 0 0 0 0.1193 0.063 67 .5 ± 3.059 .8150 . . 67 . .9653 3 0 −0 −0.002 − 3382 3382 1422 3121 6879 . . .063 68 68 0.247 −1 807 059 0 2. .5 .5 +l +l <0 <0 <0 <0 .004 .002 .004 8. %l %l 94 94 ± 1 1 ± 6 2± 6 6 2± . .42 o wE+esn TT,TE,EE owTEB+lensing 42 5 5 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±0 ±3 ±0 ±0 ± ± ±0 ±0 ± ±0 ±0 ±0 ±3 ± ± ± ±0 wTEB .112 .589 .112 . − − + − − + − − + 589 − − + imits imits − + − − − + 1 1 1 1 0 0 0 0 0 0 0 0 0.0003 0. 0.000096 0.64 0. 0 0 0. 0.64 0. 0. 0.0003 0. . . 0 0 0 0 .3 .3 . 0.013 0.015 0 0 0 0 0 0 . . 2 4 38 .014 .00032 .0087 .0014 .0087 .026 .0001 .0087 .00009 .026 .0013 .0087 .00032 . . . .0001 . 2 2 026 027 ...... 0048 025 0087 0048 00032 025 014 0014 0087 00032 56 62 8 8 015 015 013 013 + lensing 0 60 0 60 6 TT,TE,EE CDM model, the second second the model, CDM 0.01028 0. 1.04093 0.0959 0.0959 0.14038 0 1.04093 0.14038 0.1188 0.8159 .09±0.0062 ± 0.3089 13.799 .0223 0.6911 13.799 . 0.6911 0.8159 .0223 0 0.9667 0.1188 47 ± 14170 14170 .6 0.012 ± 0.066 .6 ± 3.064 67.7 . 67.7 3 0 0.00080 +l +l −1.019 − −0.002 − 3371 3089 31±23 ± 3371 .066 .064 68 0. . 0008 1 0 3 ow ow < 0.194 < < < 249 249 . . 8. .0 .0 002 019 %l %l 4± 8± 4± 0± 8± 0± 8± 8± CDM model). model). CDM 0.113 0.113 0 8 8 4 4 TEB TEB+lensing ±0 ± ±0 ±0 0.0062 ± ±0 ±0 ±0 ± ±0 ± ±0 ± ± ± ±0 ± ±0 ± .194 − − + − − + − − + − − + imits imits − − + − − + 1. 1. 1 1 0 0 0 0 0 0 0 0 0. 0.0010 0.00029 0.0001 0.023 0.46 0.000071 0 0 0 23 0. 0 0.46 0.0010 0 0.0001 0.023 0 0 0 0 . .3 .3 . 0 0 0.013 0 0 0 0.013 . . 1 1 . . 2 33 .00030 .0086 .021 .0040 .0040 . . . .00029 . .00030 .00029 .012 . .0062 025 026 0039 0040 . . . 00097 0002 00007 021 0086 00097 0062 3 3 080 075 013 + lensing 4 9 4 1 b h 2 +e +e ). xt

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 144 harmonic coefficients L multipole agiven at that tri-spectrum Indeed, noisy. rather is figure map This two). and 1.5 between redshift a at peaks (which kernel dependent distance byabroad weighted potential gravitational the of sight of line the along translational invariance of the correlations). This can then be used to obtain an estimator of of estimator an scales. obtain different at to smoothed B) (T, E, used maps be CMB then bycross-correlating can potential This lensing the correlations). the of invariance requiring by translational expressed is (which homogeneous statistically where fluctuations initial the if 퓁 adjacent couple distortions These ations. B-modes, into E-modes polarisation from contribution tensor primordial fluctu B-modes the pre-existing adding to potentially the of some transform to is one Another model). a on constraints parameter the deriving when codes numerical bythe for accounted fully is (which spectra power CMB the of structure trough and peak the smooth slightly to is One effects. several has This surface. scattering last the at imprinted image the distorts slightly Planck related, large scale anomalies which we large scale anomalies earlier. mentioned related, possibly other, the all of list the in remains it but data, low- processed better more, of inclusion specific this of significance 퓁 low- the that concluding tests, by mined high- Planck the with a tension exhibited spectrum power scale modes around L around modes scale large afew grey, in only combination variance minimum the for Even data. polarisation using reconstruction noisier the curves other the and map, a temperature from deriving map ing bration and improved analysis on a larger portion of the sky. We also found with the same same the with found also We sky. the of portion Planck new the larger that a test on analysis improved and bration cali revised to due spectra power 2013 the 2015 and Commander between changes small weakened has map from Commander the 0.7 low- at spectrum observed the ( physics new its of rendition. 2015 the in data again status checked we tracer so (2008)), al. et Destri potential also a see (2012);as al. et Dudas 2014);(2015, attention significant drawn depending on the Planck the on depending was foundtobe around thisanomaly 99 of significance statistical lensingpower3.4 Planck spectrum 3.3 Thelow-퓁 “anomaly” =

In Planck Collaboration XV (2014) we noted that the the that noted (2014)we XV Collaboration Planck In Figure observer the to path their on structures scale large by photons CMB the of Lensing Using a statistical measure (based on the Hausman test) of the relative bias between between bias relative the of test) Hausman the on (based measure statistical a Using 20

base Λ base

4 shows at right, in grey, the corresponding power spectrum (which is therefore a a therefore is (which spectrum power corresponding the grey, in right, at shows 4 and COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. 4 shows at left the resulting map of the estimated lensing potential, an integral integral an potential, lensing estimated the of map resulting the left at shows 4

30, which happen to be systematically low with respect to the base model. The The model. base the to respect with low systematically be to happen which 30, CDM model, the red curve shows the power spectrum obtained from a red lens- the obtained shows curve powermodel, the CDM spectrum information. In order to quantify such a tension, we performed a series of of series a performed we tension, a such quantify to order In 퓁 information.

a ∼ 퓁 low- m

20−50 have a signal to noise comparable to one! De-biasing from from De-biasing one! to comparable noise to signal a have 20−50 ). The black curve shows the predicted (noiseless) spectrum in the the in spectrum (noiseless) predicted the shows curve black The ). CMB solution or the estimator considered. This anomaly has has anomaly This considered. estimator the or solution CMB polarization maps are anomalous only at the 7.7 the at only anomalous are maps 퓁 polarization power anomaly was mainly driven by multipoles between between bymultipoles driven mainly was anomaly 퓁 power and a model, we found that the significance of that test for for test that of significance the that found we model, a l and “anomaly” has therefore not been strengthened with the the with strengthened been not therefore has 퓁 “anomaly” is obtained form the weighted product of four map map four of product weighted the form obtained is modes which would otherwise be uncorrelated uncorrelated be otherwise would which modes 2013 2.8 in % to best-fit model, which is mostly deter mostly is which model, best-fit Planck % in 2015. This arises from e.g 2013 low- %, with slight variations variations slight with %, ., Kitazawa and Sagnotti Sagnotti and Kitazawa ., 퓁 temperature % level. The - - - RELATIVITÉ, ONDES DE L’UNIVERS 145

]

7 10103 3 ] 7 10

× 2 [ 10102 π 2

/ 2π [× 10 / φφ L

ϕϕ 1 1

L C 10

2 10

C )] ^ ^ ϕφˆMVMV ϕφˆEEEE

2 +1 ^ˆTTTT ^ˆEBEB L 0 0 ϕφ ϕφ ( 1010 ^ TE ^ TB L ˆTE ˆTB [ ϕφ ϕφ

+ 1)]

(L

L

[ 1010 100 100 500 500 1000 1000 2000 2000 L

Figure 4. a) Wiener-ltered lensing potential estimate with minimal masking (using the NILC component separated map), in Galactic coordinates with a Mollweide projection (Planck Collaboration XV 2015). The reconstruction has been bandlimited to 8 ≤ L ≤ 2048 where L is the multipole index in the lensing power spectrum). b) Lens reconstruction noise ΦΦ levels N L (on top of the signal) for the TT, TE, EE, EB, and TB estimators applied to the SMICA full-mission CMB map. The noise level for their minimum-variance combination (MV) is also shown. The ducial ΛCDM theory power spectrum L used in our Monte Carlo simulations (to estimate biases) is plotted as the black solid line.

the noise contribution therefore requires a quite accurate estimate of its value though detailed Monte-Carlo simulation.

Figure 5 shows the final (debiased) result as greyed boxes, compared with all other de- terminations obtained so far. Planck for the first time measured the lensing power spectrum with higher accuracy than it is predicted by the base ΛCDM model that fits the tempera- ture data (when the uncertainties of the fit are propagated forward). The amplitude is constrained to about 2.5 %, a 40 σ detection for this lensing effect. The columns 3 and 6 of table 3 show the improvements with respect to the already discussed columns 2 and 5 when Planck lensing is used in conjunction with Planck CMB power spectra in constraining the base ΛCDM parameters. The improvement is modest for the base model where there is no spatial curvature and the dark energy equation of state is w = −1, two areas where the lower-z origin of lensing actually helps at lifting the degeneracies inherent to CMB only constraints (as can be seen in the bottom part of the table).

3.5 ΛCDM 1-parameter extensions and constraints from additional data

The Planck paper dedicated to cosmological parameters (Planck Collaboration XIII 2015) considered many other astrophysical sources of information, pertaining to Baryonic Acoustic Oscillations (BAO), Type-Ia supernovae (JLA, for Joint Light-curve Analysis), the 4 current Hubble constant (H0), as well as Planck cluster counts, redshift space distortions,

(4) The Planck “ext” analysis relies only on the H0 prior derived in Efstathiou (2014) by reanalysing Cepheid data using the revised geometric maser distance to NGC 4258 of Humphreys et al. (2013).

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 146 table in numerically given are which combinations various in sets data external and lensing, if changes in Ω in changes if magnitude improvement on possible deviation of of deviation possible on improvement magnitude of addition the 1997), al. et 1997; al. et Zaldarriaga (Bond effects or physics beyond base Λ base beyond physics or effects most likely the origin(s)should soon be to suggest available systematic sets of tensions, these data New analyse. to difficult notoriously also are data These tension. some exhibit The at low constraints. fluctuations of matter previous amplitude the the constrain to generically which bring sets, data other they improvements the gives 3 table of 7 and 4 with consistent fully be to tofound were “ext”, referred jointly three, first The lensing. gravitational weak and for comparison. experiments other well as as Planck from reconstructions previous well as as map, CMB SMICA 2015), the on XV based Collaboration (Planck release 2015 the data from estimate spectrum power potential Lensing Figure 5. empirically quite precisely. For precisely. quite empirically cannow be tested which assumption isThis restrictive a very geometry. a spatial flat with last scattering. last Figure A non zero detection of of detection zero A non by including BAO data, yielding data, BAO by including

The base Λ base The (5) Ω Ω Ω

K K K

This degeneracy allows for the small-scale linear CMB spectrum to remain almost unchanged unchanged almost remain to spectrum CMB linear small-scale the for allows degeneracy This 3. While the the While 3. 2 ϕϕ 7

[L ( L + 1)] C /2π[×10 ] COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. 6 illustrate graphically the tightening of the constraint with with constraint the of tightening the graphically 6 illustrate = = = L −0 0. 1. 0.000 −0.005 −0.052 .5 2 0 5 1 5 K are compensated by changes in H in changes by compensated are 11 CDM model assumes an Friedman-Lemaître-Robertson-Walker metric metric Friedman-Lemaître-Robertson-Walker an assumes model CDM ± −0.017 +0.016 −0.055 +0.049 0 (95%, 0.005 Planck 100 0 Planck CMB spectra remain affected by the “geometric degeneracy “geometric the by affected remain spectra CMB Ω (95%, (95%, CMB + lensing information within within information lensing + CMB K would surely have far-reaching implications for cosmology. cosmology. for implications far-reaching have surely would Λ CDM models, the curvature parameter parameter the curvature CDM models, CDM. Planck Planck Planck TT+lowP+lensingBA TT+lowP+lensing) TT+lowP) 0 to obtain the same angular diameter distance to to distance diameter angular same the obtain to L Ω 0 1000 500 Planck Planck K from zero, which is further improved improved further is which zero, from (2013) (2015) Planck O). Λ lensing leads to order of of order to leads lensing CDM, and the columns columns the and CDM,

Planck Ω AC SPT K CMB spectra, spectra, CMB

T ≡ 1 − Ω 2000 m − Ω (2) 5 z Λ ” , . RELATIVITÉ, ONDES DE L’UNIVERS 147

The bottom part of table 3 summarizes further constraints on single parameter exten- sions to the base ΛCDM model on, e.g., neutrinos properties Σmν and Ne, the primordial

Helium fraction, YP (when the standard BBN value is not assumed to hold), or the dark energy equation of state, w = p/ρ (the cosmological constant corresponding to the case w = −1).

Regarding the constraints on initial conditions, it is convenient to expand the power spectra of primordial curvature and tensor perturbations on super-Hubble scales as

 1 ns 1+ dns/dln k ln(k/k )+... k − 2 ∗ (k) = As , (3) PR k ∗

 1 nt+ dnt/dln k ln(k/k )+... k 2 ∗ (k) = A , (4) Pt t k ∗

where As (At) is the scalar (tensor) amplitude and ns (nt), and dns/d ln k (dnt/d ln k) are the scalar (tensor) spectral index, and the running of the scalar (tensor) spectral index, respec- tively. Figure 7 shows the Planck joint constraints on (ns, r0.002), where r0.002 stands for the tensor-to-scalar ratio at the pivot scale k = 0.002 Mpc−1 from Planck alone and with other datasets, including the co-analysis with Bicep/Keck (see below). The plot also shows the expected parameter range for selected inflation models, when the number of e-folds to the end of inflation after the current Hubble scale froze out,N ★ , is in the standard range 50 to 60. This plot shows that the canonical polynomial potential V φ2 is increasingly disfavoured and, more generically, all convex potentials as well (therefore favouring slow-roll models for which both ϕ& and &/ ρϕ increase with time). ∝

The numerical constraints on the tensor-to-scalar ratio r0.002 (and on possible cur- vature of the scalar spectrum, dns/d ln k) using different data combinations are given in table 3. Let us note that the constraint r0.002 < 0.11 at 95 % CL which arises from Planck TT+lowP+lensing (and which is not really improved when adding external data, or high- polarisation) can hardly get tighter by using only T & E data. Indeed, T & E fluctuations are both sourced by scalar and tensor fluctuations, and residual degeneraries cannot be lifted퓁 without further information. Let us further note that the 2013 Planck constraint on r was already quite similar.

3.6 Co-analysis of Bicep2 and Planck data

This is in this context that the Bicep2 collaboration announced worldwide, on the 17th of March 2014, that it had detected the long-sought after microwave B-modes, an extraordinary result from a beautiful, extremely sensitive, experiment. This thus ended a long series of ever tighter upper limits on the B-modes power spectrum. They further claimed that the search for primordial gravitational waves was over, that r = 0.2, and that it was a 5 σ detection!

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 148 The red contours tightly constrain the spatial geometry of our Universe to be nearly at. nearly be to Universe our of geometry spatial the constrain tightly red contours The BAO (solid and red contours). contours) (blue reconstruction lensing Planck the of inclusion the by signicantly are improved limits These spectra. power polarization and temperature bottom), compared to the theoretical predictions of selected inationary models. inationary selected of predictions theoretical the to compared bottom), BAO, BKP, and aka, (Bicep2+Keck sets Planck, cross data other with combination in and 68 Figure 7. joint Marginalized Ω the in Constraints 6. Figure ric degeneracy between Ω between degeneracy ric colour-coded by the value of H of value the by colour-coded

Tensor-to-scalar ration (r0.002) Ω^

0.00 0.05 0.10 0.15 0.20 0.25 COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R.

Concave 0.30 0.45 0.60 0.75 Convex 0.94 0.30 m and Ω and 0.96 Primordial tilt(n 0 ) and Planck TT,TE,EE+lowP (solid contours). The geomet The (solid contours). TT,TE,EE+lowP Planck ) and % and 95 % and Λ m is partially broken because of the effect of lensing on the the on lensing of effect the of because broken partially is -Ω Λ plane from the Planck TT+lowP (samples; Planck data the from plane 0.45 % CL regions for n regions % CL 0.98 s Ω ) m lensing+ BA lensing TE+EE 0.60 1.00 s and r and O N V ∝ϕ N V ∝ϕ V ∝ϕ V ∝ϕ V ∝ϕ R Low scaleSBSUSY Power-law in ation α attractors Hilltop quarticmodel Natural in ation Planck TT+lowP+BKP+B Planck TT+lowP+BKP Planck TT+lowP 0.002 2 * * in ation =60 =50 0.75 from Planck alone alone Planck from 2/3 4/3 2 3

40 44 48 52 56 60 64 68

0 AO H -

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BKxBK 1 BK+P 0.05 B+P ] (BKxBK-αBKxP)/(1-α) 2 K+P 0.04 0.8 0.03 /2π[μK I 0.6 0.02 peak

L/L 0.4 0.01

BB I(I+1)C 0 0.2

-0.01 0 0 50 100 150 200 250 300 00.050.1 0.15 0.20.250.3 Multipole r

Figure 8. Results of the co-analysis of Planck and Bicep/Keck (BK) data. The left panel shows B-modes power spectra, both dust-correct (blue points) and uncorrected (black points), as well as the expected B-modes from lensing of the E-modes (given in red). The right panel shows the likelihood of r values, leading to the quoted 95 % CL upper limit of r0.05 < 0.11.

It was nevertheless pointed out extremely rapidly that a more mundane interpretation was possible (Flauger et al. 2014; Planck Collaboration et al. 2014; Mortonson and Seljak 2014). Indeed, the 5 σ detection of r relied on the assumption that the polarised emission from Galactic dust was not contributing significantly at that frequency in that field, while the analy- sis presented could not exclude that it was actually all coming from such dust emission, even at the 3 σ level (for lack of sufficient spectral information, Ade et al. (2014)). And indeed we published shortly after a paper showing, on the basis of the all-sky 353 GHz Planck data (used as a tracer for polarised dust emission) that about half of the detected B-modes in that field was expected to be due to dust emission, albeit with sizeable uncertainties. It was indeed not possible to be very precise without having access to detailed non-public information like the weights of the BICEP2 pixels in the final results. The two collaborations therefore per- formed a joint analysis (BICEP2/Keck Array and Planck Collaborations 2015), in particular by directly cross-correlating the Planck 353 GHz data and the 145 GHz data from Bicep (and the additional Keck data from the same collaboration, noted below as BK) to assess the dust contamination level and shape.

The main results of the joint analysis are shown in figure 8. The left panel displays B-modes power spectra. The black point correspond to the uncorrected spectrum determination (BKxBK) which shows the basis of the claimed detection as a clear excess with respect to the ex- pected B-modes from lensing given by the red curve. The blue points shows the dust-corrected result with the help of Planck data, which is consistent with the predicted level from the lensing of E-modes and actually offers a beautifully precise determination of the later (at the 0.1 µK level!). The right panel displays the likelihood of r when either Bicep2 (B), or Keck (K), or both datasets (BK) are cleaned with Planck data (P). There is no clear evidence left for primordial B-modes, and we quoted the upper limit r0.05 < 0.11 at 95 % CL, or equivalently r0.005 < 0.12. This direct upper limit on r is now completely compatible with the indirect limit set previously by Planck in 2013, re- moving the need for less minimal model than base ΛCDM. Figure 7 shows the further tightening obtained when Planck and BKP (BKxPlanck) constraints are combined (The combined constraint

Planck TT+lowP+BKP yields r0.002 < 0.08 at 95 % CL). This is the state of the art.

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 150 The upper horizontal axes give the approximate corresponding multipoles via via multipoles corresponding approximate the give axes horizontal upper The variance arevariance large. cosmic from fluctuations the where spectrum the of region a this in is it for since evidence marginal is the feature However, itself. data CMB the in inherent rather but methods, the of This agreement suggests that the reconstruction of this “anomaly” is not an artefact of any outweigh those of the higher models, so no oscillatory features are significant there. significant are features oscillatory no so models, higher the of those outweigh 퓟 spectrum, power primordial the or potential the of features possible d generally of high-energy early universe physics, in a way that is strongly complementary to to complementary strongly is that way a in physics, universe early high-energy of generally the low quadrupole (for (for quadrupole low the a power deficit corresponding to the anomaly of the TT spectrum at spectrum TT the of anomaly the to corresponding deficit a power 4. Primordial Non-Gaussianity 4. where where reconstruction and the colour scale is chosen so that darker regions correspond to lower- to correspond regions darker that so chosen is intervals. confidence scale colour the and each For probabilities. reconstruction equal have colours Equal movable. freely are knots These 10. to ( knots 2 from assumed, recon- such three Figure struction. presents 2015) XX Collaboration (Planck inflation to devoted paper The oroscillations). potential inflaton in the a step is a linear transformation, one can unambiguously reconstruct some part of it. The The it. of part some reconstruct unambiguously can one transformation, a linear is of presence the for allowing when features). these affected significantly not are parameters mological cos- remaining the on constraints the (and considered priors of choice the for spectrum, over a power-law are preferred models features of these none but spectrum, of the tions parametriza motivated theoretically of a number for evidences Bayesian (2015) provides variance reduces our knowledge of 퓟 of knowledge our reduces variance higher about precise anything say to lacking is resolution the but The region evidence. Bayesian cording to respective their construct a power deficit linked to the dip in the TT angular power spectrum at spectrum power angular TT the in dip the to linked deficit a power construct there is no statistically significant evidence for any features departing from a simple power power simple a from ( departing law features any for evidence significant statistically no is there is clearly scale-free, with n with scale-free, clearly is 3.7 Exploring freedom of degrees further n

s These reconstructions can also be averaged over different values of N of values different over averaged be also can reconstructions These Table Primordial non-Gaussianity (NG) is one of the most powerful tests of inflation, and more more and inflation, of tests powerful most the of one is (NG) non-Gaussianity Primordial that conclude to lead and reconstructions consistent broadly yield methods three All Since the mapping from the primordial power spectrum to the angular power spectra spectra power angular the to spectrum power primordial the from mapping the Since /d ln i.e ., ., COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. D 퓟 k

rec , but that single extra parameter would not allow capturing well, or at all, many many all, at or well, capturing allow not would parameter extra single that , but 3 already gave constraints on possible curvature of the scalar spectrum, spectrum, scalar the of curvature possible on constraints gave 3 already 퓡 is the comoving distance to recombination. Apart from a low k alow driven by from deficit Apart recombination. to distance comoving the is ( k ) ∝

k 9 shows one of them, for an increasingly number of reconstruction knots knots reconstruction of number increasingly an for them, of one 9 shows n s -1 ) primordial spectrum. But at low But spectrum. ) primordial 1 and σ and i.e N ., only a single power law is allowed, as in the standard analysis), analysis), ., a as power law only in is standard the single allowed, knot s ≈ 0.96 > 2 2 confidence intervals are also sketched (black curves). curves). (black sketched also are intervals confidence σ ), the main feature which is clearly visible with 8 knots is is 8 knots with visible clearly is which feature main ), the . 퓡 ( k ) . The weights assigned to the lower N lower the to assigned weights . The Planck k ≈ 1.5 − 2.0 × 10 30 < paper Planck Collaboration XX XX Collaboration Planck paper 퓁 < 2300 퓁 . At lower 퓁 lower . At is highly constrained, is constrained, highly −3 퓁 Mpc ∼ 30 퓡 (as for example example for (as knot −1 weighted ac . But the rest rest the . But , all three re three , all 퓁 knot

퓁 ≈ ∼ , cosmic 20 − 30 models models k Planck / D rec σ - - - . ,

RELATIVITÉ, ONDES DE L’UNIVERS 151

ℓ ℓℓ 4.0 10 100 1000 4.0 10 100 1000 4.0 10 100 1000 3σ 3σ 3σ ) 3.5 3.5  3.5

2σ 2σ 2σ 10 3.0 3.0 3.0 1σ 1σ 1σ

log(10 2.5 2.5 2.5 0 0 0 2.0 σ2.0 σ2.0 σ 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 4.0 10 100 1000 4.0 10 100 1000 4.0 10 100 1000 3σ 3σ 3σ

) 3.5  3.5 3.5

2σ 2σ 2σ 10 3.0 3.0 3.0 1σ 1σ 1σ

log(10 2.5 2.5 2.5 0σ 0σ 0σ 2.0 2.0 2.0 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 10 100 1000 10 100 1000 10 100 1000 4.0 4.0 4.0 3σ 3σ 3σ )

 3.5 3.5 3.5 2σ 2 2σ 10 σ 3.0 3.0 3.0 1σ 1σ 1σ

log(10 2.5 2.5 2.5 0σ 0σ 0σ 2.0 2.0 2.0 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 k / Mpc-1 k / Mpc-1 k / Mpc-1

Figure 9. Bayesian movable knot reconstructions of the primordial power spectrum (k) using Planck TT+lowP data. The plots indicate our knowledge of the P( (k)|k, Nknot퓡 for 퓟 a given number of reconstruction knots, Nknot. The number of these knots퓡 Nknot increases (left to right and top to bottom) from 2 to 10. 1 σ and 2 σ condence intervals퓟 are outlined (black curves). The upper horizontal axes give the approximate corresponding multipoles via ≈ k/Drec, where Drec is the comoving distance to recombination.

퓁 the power-spectrum constraints already discussed. The simplest models of inflation, charac- terized by a single scalar field slowly rolling along a smooth potential, predict the generation of primordial fluctuations which are almost Gaussian distributed, with a tiny deviation from Gaussianity of the order of the slow-roll parameters (Acquaviva et al. 2003; Maldacena 2003). On the other hand any departure from these hypothese has the potential to produce distinctive NG signatures at a detectable level in the CMB anisotropies.

We used several analysis methods, e.g., the modal approach, as well as other estima- tors [the optimal “KSW” with its skew-Cl extension (Komatsu et al. 2005; Munshi and Heavens 2010), wavelets, Minkowski functionals...], since these different weighting of the triple-product of modes would be affected differently by residual systematic effects. We have extensively tested for the latter using simulations and the 4 component separation methods of Planck, testing for the effect of masks and max of the analysis, and debiased them when needed, in particular to account for the ISW-lensing part or residual compact sources. 퓁

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 152 the tightest constraints available on extensions to the simplest models of inflation: the stan the inflation: of to-date. test models stringent most its survived simplest has the inflation to slow-roll single-field of scenario dard extensions on available constraints tightest the far by are These NG. primordial of shapes most-studied the of three of amplitudes the for analysis in the context of an extended extended an of context the in analysis (2015). inBouchet morein A depth complete are which commented for inflation, interest par In fluctuations. primordial Gaussian with data, only temperature consistent using ticular, are NG primordial on 2015) XVII models are discussed in other Planck other in discussed are of models set the to extensions Wider (2015). XI Collaboration Planck in given is observations, Planck CMB polarisation. CMB the concern mostly will improvements the 2016, in where release for prepared being The models. inflationary discusses (2015) XX andCollaboration Planck forcomponent models the (2015)darkenergy specific examines 5. Conclusions 5.

This contribution gave a short overview of the latest Planck latest the of overview a short gave contribution This i.e conclusion, same the to lead approaches All f f f NL NL NL regarding the lensing power spectrum results, as well as constraints from other other from constraints as well as results, spectrum power lensing the regarding COSMOLOGY WITH THE PLANCKSATELLITE: FROM QUANTUM FOAM TO THE COSMIC BOUCHET WEB -F.R. Acknowledgements Collaboration. http://www.sciops.esa.int/index.php?project=planck&page=Planck_at found they which have in activities scientific or technical the involved, been be can description of the PlanckCollaboration and a list of its members, including (EU).PRACEA and (Portugal); (Ireland);FCT/MCTES (Norway);SFI RCN (Switzerland);(Denmark);SER/SSO Space (Canada);DTU (Germany);CSA (Spain);RES and MICINN,Tekes,JA MPG and (Finland);DLR CSC and AoF CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, (France);ASI, CNRS/INSU-IN2P3-INP and CNES by:ESA; supported been :-).mine of are development top submitted.The on mistakes All resultsAll presented collaboration, are thoseofthePlanck aspublishedor − − − ortho equil local = =+ = −9.2 − 1.8 20 ± ± ± 33 69 5.6

2015 papers; for example, Planck Collaboration XIV XIV Collaboration Planck example, for 2015 papers; Λ CDM cosmology of these and other results from from results other and these of cosmology CDM Planck ., ., the Planck results Collaboration (Planck final “legacy” data and analyses are are analyses and data “legacy” final data and findings of most most of findings and data has Planck (5) - - RELATIVITÉ, ONDES DE L’UNIVERS 153

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