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The Atacama Cosmology Telescope: likelihood for small-scale CMB data

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ANGULAR POWER SPECTRA OF THE mm-WAVELENGTH BACKGROUND LIGHT N. R. Hall, R. Keisler, L. Knox et al. JCAP07(2013)025 i m c n hysics r,k P M.J. Devlin, u le l,m,b c,a D.N. Spergel, ic g,h M. Halpern, t t f ar J.P. Hughes, M. Niemack, q S. Das, m,b doi:10.1088/1475-7516/2013/07/025 f strop N. Sehgal, G.E. Addison, A. Hajian, s T.A. Marriage, and E. Wollack A b l a d R. Hlozek, K. Moodley, J.R. Bond, f e n H. Trac T. Louis, J. Sievers, f M. Gralla, o a B. Partridge, k b osmology and and osmology A.D. Hincks, C F. Menanteau, m,c E.S. Battistelli, E.R. Switzer, L.A. Page, d A. Kosowsky, E. Calabrese, J.W. Fowler, b i,p f a k j rnal of rnal ou An IOP and SISSA journal An IOP and 2013 IOP Publishing Ltd and Sissa Medialab srl Departamento de Astronom´ıay Pontific´ıaUniversidad Cat´olicade Astrof´ısica, Chile, Casilla 306, Santiago 22, Chile NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder,Dept. CO, of U.S.A. Physics 80305 and3400 Astronomy, N. The Charles Johns St., Hopkins Baltimore,Department University, MD of 21218-2686, Astrophysical Sciences, U.S.A. Princeton, Peyton NJ Hall, U.S.A. Princeton 08544 University, High Energy Physics Division,9700 Argonne S National Cass Laboratory, Avenue, LemontBerkeley IL Center 60439, for U.S.A. CosmologicalUniversity Physics, of LBL California, and Berkeley, Department CA, ofDepartment U.S.A. Physics, of 94720 Physics and209 Astronomy, University South of 33rd Pennsylvania, Street, Philadelphia, PA, U.S.A. 19104 Canadian Institute for TheoreticalToronto, Astrophysics, ON, University Canada of M5S Toronto, 3H8 Sub-department of Astrophysics, University ofKeble Oxford, Road, Oxford OX1 3RH,Joseph U.K. Henry Laboratories ofPrinceton, Physics, NJ, Jadwin U.S.A. Hall, 08544 PrincetonDepartment University, of Physics andVancouver, Astronomy, BC, University of Canada British V6T Columbia, 1Z4 McWilliams Center for Cosmology, Wean5000 Hall, Forbes Carnegie Ave., Mellon Pittsburgh University, PADepartment 15213, of U.S.A. Physics, University ofPiazzale Rome Aldo ‘La Moro Sapienza’, 5, I-00185 Rome, Italy l i b c j e g d a k h f c

S.T. Staggs, D. Marsden, M.R. Nolta, M. Hasselfield, K.D. Irwin, R. D¨unner, The Atacama Cosmology Telescope: likelihood for small-scale CMBJ. Dunkley, data J N. Battaglia, m JCAP07(2013)025 697 for / 3500 for use in 10000. We extend /dof= 675 2 χ < ` < < ` < 1301.0776 107 for SPT. We then use the multi-frequency likelihood to estimate the CMB / cosmological parameters from CMBR, CMBR experiments, Sunyaev-Zeldovich The Atacama Cosmology Telescope has measured the angular power spectra of Department of Physics andPittsburgh, Astronomy, University PA, of U.S.A. Pittsburgh, 15260 Department of Physics, University ofCA California 93106, Santa U.S.A. Barbara, Astrophysics and Cosmology Researchof Unit, KwaZulu-Natal, School ofDurban, Mathematical 4041, Sciences, South University Africa Department of Physics, CornellIthaca, University, NY, U.S.A. 14853 Department of Physics andHaverford, Astronomy, Haverford PA, U.S.A. College, 19041 Stony Brook University, Physics andStony Astronomy Brook, Department, NY, U.S.A. 11794 NASA/Goddard Space Flight Center, Greenbelt, MD, U.S.A. 20771 E-mail: [email protected] Department of Physics andPiscataway, Astronomy, NJ Rutgers, U.S.A. The 08854-8019 State University of New Jersey, t s r q o p u n Received January 14, 2013 Accepted June 10, 2013 Published July 16, 2013 Abstract. microwave fluctuations to arcminuteseasons scales of at data. frequencies Atground of small (CMB) scales 148 the and becomeCMB fluctuations 218 increasingly in signals. GHz, the obscured from We primordialeffects, by present three Cosmic including results Microwave extragalactic Back- the from foregounds thermal aclustered and and and nine-parameter Poisson-like kinematic secondary model power Sunyaev-Zel’dovich from describingfrequency (tSZ Cosmic these scaling; and Infrared the secondary Background kSZ) tSZ-CIB (CIB)and power; sources, correlation thermal the and dust coefficient; emission their the fromextract extragalactic Galactic cosmological cirrus radio parameters, in source two we power; differentthis describe regions model a of to the likelihood the sky. function multi-frequencythe spectra In for likelihood in order the to the to ACT multipole include data, range spectra220 500 fitting from GHz. the Accounting South for Polemodel Telescope different at radio provides frequencies source an of levels 95, excellentACT, and 150, and fit Galactic and 96 cirrus to emission, bothpower the datasets same spectrum simultaneously, from with This ACT provides in a bandpowers,cosmological simplified marginalizing parameter ‘CMB-only’ estimation. over likelihood the in secondary the parameters. rangeKeywords: 500 effect ArXiv ePrint: JCAP07(2013)025 13 WMAP – 1 – WMAP 5.1.1 Combining5.1.2 spectra from different Calibration regions factors 22 22 3.3.1 Combining with 3.1.1 Calibration3.1.2 and beam uncertainty Secondary model parameters 11 11 5.2 Marginalized5.3 CMB bandpowers The CMB-only likelihood 22 23 4.1 ACT4.2 data Combination4.3 with SPT Tests of the likelihood 5.1 Method: bandpowers via Gibbs sampling 20 16 17 15 3.2 Combining3.3 with SPT data Multi-frequency likelihood prescription 13 11 3.1 Likelihood from the ACT data9 2.1 Thermal2.25 Sunyaev-Zel’dovich Kinematic2.35 Sunyaev-Zel’dovich Cosmic2.4 infrared6 background tSZ-CIB2.5 cross-correlation 7 Radio2.6 point8 sources Residual Galactic cirrus9 have provided evidenceThe for measurement a of flatpeaks, the universe have Sachs-Wolfe described led plateau byThe to in just Silk constraints the six damping on power cosmologicalabout tail ΛCDM spectrum, parameters. of the parameters and the physics toheights three power of of percent-level spectrum acoustic the the accuracy provides higher-order ([38, earlyscales acoustic a peaks is40 ]). universe, wealth ([64]). complicated of encoded by Extracting additional information in the information from presence its these of angular shape, additional and power from in extragalactic the point positions sources, and 1 Introduction Measurements of the Cosmicconstraining Microwave cosmological Background models. (CMB) have Anisotropies played a measured central over role the in whole sky by 6 Summary A Tests of the CMB likelihood 25 24 4 Tests of the multi-frequency likelihood 5 CMB-only likelihood 13 19 3 Full likelihood from small-scale data9 Contents 11 Introduction 2 Model for the mm-wave sky2 JCAP07(2013)025 ◦ 55 − CIB ν Rad ν 3 tSZ ν 2 c S max ), and the ACT-S region at dec = ` 2 min 500 10000 15 146.9 147.6 149.7 ` (300 deg ◦ – 2 – 4 . Small-scale CMB datasets 1500) the secondary signal, which we define as the sum of ' ` Table 1 Reference Area 1 218150220 1500 10000 220.2 217.6 2000 219.6 9400 2000 9400 6.4 152.9 150.2 153.8 218.1 214.1 219.6 GHz sq degrees mJy GHz GHz GHz 7-year data. In section4 we show the small-scale model fit to the multi-frequency ). 2 The Atacama Cosmology Telescope (ACT) mapped the mm-wave sky with arcminute In this paper we describe a method to fit multi-frequency power spectra from the ACT This is one of a set of papers on the 3-year ACT data; [14] present the angular power ACT 14814] [ 590 All cross-spectra between channelsFlux are cut used imposed in on theEffective map likelihood. band-centers by from its ACT pointThis are source from area mask. [68], includes given the for tSZ, ACT-E radio region sources, at and CIB dec sources. = 0 Dataset Frequency 1 2 3 4 SPT-low 150 [36] 790 650 2000 50 152.9 150.2 153.8 SPT-high 95 [54] 800 2000 9400 97.6 95.3 97.9 (290 deg resolution from 2007to to compute 2010 the in angular1-year two power data, spectrum. distinct from areas. Powerroughly the spectra the 2008 About and same observing cosmological 600presented season, results period, square cosmological using were results the degrees the presented in South were [36, in Pole used 43, [13, , 54 Telescope22]60] alsodata and and simultaneously mapped [66]. [16]. for the CMB,to microwave During foreground, sky, and analyses and SZ in parameters,dataset, [16] following using a and data similar]. [54 from approach combination the with 2008-2010 We data observing describe fromwe seasons, SPT then the construct in and likelihood a a show simpler constructedover self-consistent how CMB-only the likelihood, way. it for SZ estimating Using can the CMB and this be bandpowers foreground 3-year marginalized likelihood used parameters. ACT from in ACT, spectra, and [61] use thebegin likelihoods in presented section2 hereby tolikelihood estimate describing of cosmological the the parameters. ACT model We WMAP data for is the described, mm-wave including emission.data. the combination In In with section5 3 section datawethe from describe SPT full the and compressed CMB-only likelihood,2 concluding in6. section Model for theSky mm-wave maps sky of mm-wavelow fluctuations redshift, at in arcminute addition resolution to22, include the36, components primordial63]. emitting CMB Thein signal at power and figure1, spectra secondary focusing CMB from onsmaller effects, the than angular e.g., complete a [16, scales ACT few dataset, of arcminutes ( reported interest in for14], [ the are primordial shown CMB signal. At scales emission from theSunyaev-Zel’dovich Galaxy, effects]). ([67 and secondary anisotropies due to the thermal and kinematic JCAP07(2013)025 . The WMAP used for SPT power 2 ). The 800 deg 2 250 GHz probed by ACT, SPT, and < ∼ ν < ∼ to highlight the features at these angular scales. satellite ([50]). We follow a similar approach ` overlap with ACT-S. The color scales with Galactic C 2 2 ` – 3 – Planck declination (ACT-S, 292 deg ◦ instead of the conventional ` ), and at -55 2 C 4 ` . Summary of small-scale mm-wave data measured by the Atacama Cosmology Tele- . Regions of the sky used for ACT power spectra (dashed, [14]) in the Equatorial plane cirrus intensity ([21]). foregrounds and SZ effects,the becomes primary significant CMB compared signal,channels, to but, information the since about CMB. there both Werequired want are the to to more separate frequency the extract foreground and signals.of components scale In these than dependence this fluctuations section frequency of over we the the describe a foregrounds frequency model range is to 90 fit the power spectrum spectra (dotted, [36, 54]) is indicated, with 54 deg Figure 2 (ACT-E, 300 deg other CMB experiments including the Figure 1 scope ([14]) anddamping the tail South of Pole the Telescope CMB. ([36, The54]), ACT in and the SPT angular data range are used independently for calibrated measuring to the to [16, 54 , 63]. vertical axis is The primary CMB signaltogether corresponding to with the the best-fitting totalcurve), ΛCDM signal including model at secondary ([61]) is 148 effectsfrom indicated GHz from SZ (dashed), SZ (red, and foregrounds and lower is solid foregrounds. vital curve) Modeling to and allow the 217 extraction secondary GHz of contributions (black, the upper primordial signal solid at small scales. JCAP07(2013)025 (2.5) (2.1) (2.2) (2.3) (2.4) , the Poisson 2 − S ∝ = 15 mJy removes a are calculated as j ij , ν C max − S dN/dS , ) and CIB ` ˆn i , ν B ν, ij , 90 GHz frequency range at these ( + ), is dominated by the sum of tSZ > ij , ∼ sec Gal ` , , ˆn ij P , T B ν E − dS, ) ν, j sec ` + ( ν B ij ( CIB ` dS , dN sec ` ) + ∆ ˜ B + 2 T T ˆn rad ) ` S i ( + B ν ( ij , ∗ CMB ` + max ` – 4 – ˜ S CMB B T ) in the flat-sky approximation. The theoretical ij 0 , kSZ ` D ˆn T is modeled as Z = ( B = CIB π ij T = , we model the signal in the maps as + 2 − ij ` ` / th ` ij ˆn , ij C ) = ∆ C , B tSZ ` ˆn th B ` tSZ ` ν, C B + ( T = is the flux of the brightest sources in the map at a given frequency. ∆ + 1) ij , ` ( sec ` ` max B S and direction ≡ ν ij , and other CMB experiments show that other Galactic emission, including th ` ) are the lensed CMB fluctuations, which are independent of frequency in B ˆn ( , where is the lensed primary CMB power spectrum. In this analysis we model the max CMB is the Fourier transform of WMAP S will be lower, leading to a lower residual power. Radio sources have a shallower T CMB ` ` slope than CIB galaxies, so imposing a flux cut of, e.g., ˜ ∝ B T ` For frequency The Galactic foreground power is also expected to vary between regions on the sky. The cross-correlation power spectra between frequency The majority of these secondary spectra will be common to all regions of the sky and max C S i.e., Deeper surveys, with lower noiseso per pixel, aredN/dS able to detect and mask out dimmer sources, The two regions mapped by ACT (Equatorial, ACT-E, and South, ACT-S) are shown in significant amount of radio power, but little CIB power. thermodynamic units. The secondary signal, ∆ where ∆ and kSZ components,emission emission from from Galactic cirrus, dusty all infrared of galaxies which are and functions radio of galaxies, frequency. and dust where cross-spectrum where secondary spectra as to different experiments. Theto power in vary the among residual data radiodistribution point sets of sources sources due is with to expected, differential the however, number removal counts scaling of as bright sources. For example, for a Poisson with contributions from the tSZInfrared and Background, kSZ both effects; Poisson-like dusty (CIB-P)between galaxies and the that clustered tSZ form (CIB-C); part effect the of cross-correlation Galactic and the the cirrus Cosmic CIB (Gal). (tSZ-CIB);ments radio We by galaxies assume (rad); that and all dust emission other from cross-spectrasmall can scales be and neglected.and locations, CIB Measure- e.g., sources [25]. isper also cent The expected in to cross-correlation simulations. bedepending of small, Since on radio e.g., the the sources [55], kSZ line-of-sightcorrelation who and signal motion function find both consists of with a of the tSZ other correlation positive electrons signals of should and that only average negative source a to fluctuations, the few zero. signal, the two-point synchrotron and free-free emission, is negligible in the power will be JCAP07(2013)025 0 ν (2.6) (2.7) 8, the . = 3000, 0 ` = 0 . 8 8 at σ σ 2 K µ 9, with standard deviation describes its normalization. . 0 ± normalization differs from that kSZ a , 6 . ,` tSZ tSZ 0 a 8, normalized to 1 . B = 5 , ) , scales the expected tSZ emission to j regardless of template, and a constraint = 3000. This differs slightly from the ,` ) ν = 0 kSZ 0 tSZ ( 0 0 8 a ` B ν f tSZ CMB ( σ ) a i 2 T ν kSZ f B ( a f – 5 – = tSZ ij hν/k , a = = kSZ ` B ij x , = 3000. The parameter 0 ` tSZ ` B at 4, for 2 = 150 GHz and scale − K , the effective band-center for the tSZ, given in table1. We ignore µ 0 ν 2) ν x/ coth( x is a template power spectrum corresponding to the predicted tSZ emission at is a template spectrum for the predicted blackbody kSZ emission for a model with ) = is a free parameter describing its amplitude. An example is shown in3. figure The ν ,` ,` ( tSZ 0 kSZ 0 8, normalized to 1 f B . B tSZ The template we adopt is derived from recent hydrodynamic simulations described in [4]. Numerous other authors have also predicted the tSZ spectrum from independent sim- In the rest of this section we describe how each of the components in eq. (2.4) are a = 0 8 The simulations include theAGN and effects supernovae. of The radiativeand predictions cooling, are ACT, consistent star e.g., with formation, [43], SZ and and measurements from feedback the both from shape SPT is shown in3. figure For the model with used in [16]. Theof present choice template, has since themeans the advantage one of main expects reducing to difference the find dependenceon the between same the on various constraint the SZ on templates choice power is can their be converted amplitude. back into This a model-dependent constraint on predicted spectrum reported in [4] has amplitude ulations and analytical models,for e.g., fixed [4, cosmological model19, However, varies37, the depending55, template on57 , shape thesensitive59, is astrophysical69], to broadly modeling and consistent shape of the among theclusters, difference, expected models, clusters. and so and amplitude expect we the the do data total SZ are not power not2.2 include to yet a be the shape Kinematic same uncertainty. Sunyaev-Zel’dovich forThe ACT We kSZ and do power SPT. not isdensity mask ([49]), expected and to in havegalaxy the clusters contributions ionization at arising fraction, later e.g., from times. [27, fluctuations We33, in model45], the the as power electron well as as from the motion of and factor thermodynamic units at estimated from ten simulations. relativistic corrections, e.g., [34],well since approximated by the the low-mass non-relativistic clusters formula. that This dominate the spectra are where for a model with amplitude of matter fluctuations where σ convention used in previous analysesparameterization of used the in ACT the data fiducial ([16, or model;22]). modifications. in In a each later case section we we describe consider the possible2.1 extensions Thermal Sunyaev-Zel’dovich Our model for the power from thermal SZ fluctuations is given by modeled. To allowat for a comparisons pivot between frequency experiments, of we normalize the power spectra figure2 and summarized inof the1, table Galactic together cirrus withexpected map in data from some from] [21 regions SPT. is of The the shown temperature ACT-E for region. comparison. scale A higher level of emission is JCAP07(2013)025 (2.8) 5 for . = 3000 = 3000, ` ` = 1 kSZ a 2 K µ  ) p , β ) j p ν ( , β µ 0 ) ν p 8 cosmology. This is a quarter of the ( . 2 , β µ for simple reionization models at i ν 2 ( = 0 (CIB, [2], the cross-correlation between tSZ and K µ 8 8 µ  ([58, ]). 69 . σ 0 – 6 – 2 5 `  − 5 0 . ` ` 4 8 3 to 5 σ  p ∼ a . 2 = = 10 in a ` ij , z P − CIB ` B . Template power spectra for the thermal and kinetic Sunyaev-Zel’dovich effects (tSZ and Reionization of the universe is not expected to be instantaneous, as was assumed by [5]. comparable to the tSZ atspectrum 150 at scales GHz. probed The byand dominant ACT effect is duration to of of alter patchy reionization, the reionizationinclude amplitude, on e.g., additional depending the shape [6]. on power uncertainty both We in the test midpoint the template a in modified the shape2.3 basic in model. section ,4.3 Cosmic but infrared do background Thermal not dust emission fromBackground high (CIB), redshift is star-forming emitted galaxies, inrange, part the e.g., of rest-frame [31, far the53]. infrared Cosmicmaps and Infrared Clustering redshifted at of into CMB the these frequency mm-wave Following galaxies the ([16, has analyses been, 29 in detected30, [2, a statistically16, 52, Poisson in54], and54, mm-wave the clustered]),60 power component, as from given well these by galaxies as is in modeled the as sub-mm, the e.g., sum of [39, 71]. The shape and amplitudesimulations of predicting the a kSZ power signalThis from at gives patchy least a reionization total as is expected large far signal as less of certain, the with homogeneous signal, e.g., [6, 46, 73]. expected tSZ power. Thehas corresponding a kSZ similar template amplitudepower for is and the expected shape, ‘nonthermal20’ to as model scale does in as [69] roughly the [58] ‘CSF’ model, and the [7] model. The Figure 3 kSZ, [4,5]), clusteredCIB CIB (tSZ-CIB, sources negative scaling at 150 as GHz, [3], and Galactic cirrus ([47]). They are normalized at We use a templateThis that is assumes derived a fromand model the is with same shown instantaneous hydrodynamic in reionization, simulations figure3. described as in the The [5]. tSZ predicted spectra amplitude in from section the2.1, simulations is and 150 GHz,source and power the (not tSZ-CIB shown) scale is as shown for a perfectly correlated signal. Poisson CIB and radio homogeneous reionization at JCAP07(2013)025 (2.9) , and 10 to (2.10) d T ∼ power at ` C 2 ` 7 K. We also assume 2 in the fiducial case . . provides a good fit to 2 = 9 = 1 K ` , and different frequencies µ d n 0 T  ν ) c 2000, a correlation of , β 06 in [2]. Both are in agreement ) and j . c ν 0 ` > 0 , ( ` ) , β µ ± 0 ν ) ν ( for effective dust temperature c ( g 25 2 ) ν . , β d µ i T ν ( ( = 1 ν µ n B  β n ν – 7 – − converts from flux to thermodynamic units. The 2  ) = 0 ` and the Balloon-borne Large-Aperture Submillimeter CMB ` T |  ν, β 1 ( c a − µ is a power law index, and the frequency scaling of each for the Poisson and clustered dust terms respectively. The ) c = n Planck β ij , /∂T ) C − T and ( p ν CIB ` β normalize the two components at B ∂B c a ) = ( ν ) is the Planck function at frequency and ( d p g T a ( ν B in the basic model. c β The frequency dependence we adopt is an approximation to a sum of modified black- The power-law angular scaling of the clustered term, with increasing The tSZ-CIB power is negative at 150 GHz, and can partially cancel power from the We do not expect the CIB power to vary significantly between the ACT and SPT maps, ), in close agreement with the estimate of 8 = . 0 p ` with emissivity indices function the function small scales, is showntrum, in which figure3, includes and contributions approximates frombetween the pairs galaxies shape of of in galaxies the in different non-linear the halos. power same spec- [2] dark matter find halo, that and a power law in parameters bodies at different redshifts,of so the this dust. emissivity Following and] [2 we temperature fix are the only dust effective effective temperature properties to for the clusteredcomponent part. is given by Here, a modified blackbody, channels are assumed to be perfectly correlated. β small-scale power spectra from Telescope (BLAST), and fromwith cross-correlating observations ACT of and BLASTwell the as maps. correlation local function galaxies, This e.g., is from [12, consistent high-redshift23 ]. Lyman We fix break the power-law galaxies, index as to kSZ effect as it does not vary significantly with frequency over the range probed by ACT and ( 30% in power is predicted,redshift with distribution uncertainty dominated of by the uncertaintiesdue in CIB. to the pairs A halo mass of significanthalo and galaxies fraction model). occupying of group the Thesein and CIB same units cluster-mass power halos of halos on power are (theoverall small of responsible ‘one-halo’ fraction tens scales of term for of is CIB of the per emission the tSZ cent associated is power. with therefore massive possible A halos on is tSZ-CIB small small. correlation angular scales even if the despite the different flux cutseffect applied to of remove source sources. masking Using on the model the in CIB [1], power the2.4 is predicted only at the tSZ-CIB per cross-correlation centSome level. spatial correlation isgalaxies, since expected both between trace the clustersthat matter that make density field. contribute an to The importantdusty higher the contribution redshift galaxies. tSZ, and to [3] lower and mass the modelof CIB groups tSZ its this signal angular correlation, ([6, power and spectrum.37, predict69]) the For are mm-wave scale spectra also and at likely frequency to dependence host for the Poisson part, and with galaxy correlation functions. JCAP07(2013)025 ξ 5, . 0 and 0 (2.13) (2.11) (2.12) ` ξ < at 2 5 ([44, 70]). . K 0 µ − = s α 2 , K , µ ) CIB c  − ) j , β i ) ,` ν tSZ 0 ν 0 ( by extrapolating to fainter fluxes ( ν g B ( ) µ ) s i 2 ) ) a ν j ij g 0 ( ν ν ν g ( ( ( 0 0 f  5 in the fiducial model. f . f s 2 0 α c ) + − a 2 ([54]), which we also impose as a Gaussian  c . j 0 = 2 0 ), the frequency scaling of the cross-spectra in ν , β i ν tSZ and the kSZ power, broadening the limit on ν j s ± – 8 – ν is the correlation coefficient. The Poisson CIB ( a ν α 3 ( ξ f  . ξ . Measurements of bright sources from ACT and √ µ 0 2 ξ ) ` i  − 4 after masking sources brighter than 15 mJy in both = 1 ν . 0 ( ` s ` 0 = f a ) converts from flux units as for the CIB sources. The and  ij ± ν , s ( 0 9 a g ) = ν . CIB ij = ν − ( = 2 ij 0 , s f tSZ ` a rad B ` B 2 in the basic model; the effect of widening the range to e.g., . 0 . is the predicted correlation spectrum shape, normalized to 1 0 ν is normalized at is not included in eq. (2.11), unlike in60, [ 73], as the sources that dominate the CIB s p a − a < ξ < ,` tSZ 0 B Since the correlation coefficient is poorly constrained by ACT, we impose a uniform Bright source counts can also be used to predict in thermodyamic units, where SPT give an estimateAssuming for it the holds at spectral fainter index fluxes, in we flux fix unitsusing a of model typically for theand number for of SPT sources in aspredict a [36, a function].54 of residual flux. Using power point This sources was done measured in from [44] the for full ACT, ACT dataset, [26] now amplitude prior of 0 where for pivot scale corresponding to the maximum allowedin correlation in [61]. the models explored Due by to [3], is the discussed correlation between ACT regions (the levelcatalog used to is construct estimated the to150 mask GHz. be for For our complete. comparison, mapsflux the in We estimated level this impose power of analysis), in 6 this where the mJy as SPT the for a power SPT-high spectra Gaussian is after prior masking on to a the power at prior. parameter thermodynamic units is then increases the upper limit on the kSZ power, but2.5 does not affect Radio cosmological point results. sources The radio sourcese.g., at [30, ACT56 ], and frequenciesneighbouring to are good frequencies, not approximation consistent their with expectedsame power simulations spectral should to in indices. be be [55] perfectly Assources and significantly correlated in as valid between [16] clustered, Poisson for and scale-free see [54], sources power, we with with model the the residual power after masking bright CIB Poisson power inwith the the mm-wave tSZ bands clustersin are ([3]). section unlikely2.3, Assuming to and the have tSZ same significant frequency modified redshift-overlap scaling blackbody scaling for the CIB as SPT. As a result,kSZ neglecting power this ([46, component73]). can Following lead [2] to we artificially model tight the constraints spectrum on as the shown in figure3. The free parameter JCAP07(2013)025 . , 2 2. = . . ij 1 0 and ij , − (3.1) They ` (2.14) ± th b th b 21, and 4 5 C ∝ C . . = 0 16, 0 g . 7. This angu- a . 0 = 0 − = , 3000 for cross-spectrum g , Σ B 2 n b K µ  ) , j ] ) ν 8, and a prior of . 0 ( ) + ln det ν g 218 b , ( ) i 2 C = 3 ν g 218 b ( g − g β ,C  th b g C β 218 ( , and 1  − 7 j . 148 b 2 0 ν 0 Σ i 8, and angular scaling ν − . ν T ,C – 9 – ` 2 in the ACT-E and ACT-S spectra respectively. IRIS dust maps ([47]). The frequency scaling is ) .  b 0 148 g , = 3 C n ± µm g −  148 4 b β . 0 ` C th ` b C  = 0 = [ g , we compute bandpower theoretical spectra using from IRIS]). ([47 b a ij gs , = ( C a th ` = µm L C ij , 6 between 150-220 GHz). For consistency we therefore test the effect . 2 ln Gal ` , of the data for each ACT region separately is given by 2, and . B is the bandpower window function in band − ([51]). Using the correlation coefficients estimated in [14], we impose L 0 = 3 ij b` ± , frequency index β g w 8 a . Planck = 0 ge in thermodynamic units at 95, 150, and 220 GHz, with scale dependence a , where 2 ij , K We then marginalize over a residual Galactic cirrus component using a power-law For the SPT data, a small Galactic cirrus residual is also expected. In our basic model The likelihood, th µ ` In this paper we use the original version ‘v1’ of the spectra, unless stated. Since original release, the contain three sets of spectra, C 5 19 ij b` b . we follow the treatment2 in [54], fixingHowever, the this model Galactic has cirrus a power steeperfrequency scaling angular to ( power law than in our ACT model, and a shallower of adopting the ACT model instead, using We find no effect on parameters and no change in3 the goodness of Full fit. likelihood fromIn small-scale this data section weshow describe how the we multi-frequency extend likelihood it used to to include model other3.1 the small-scale ACT datasets, data, in and Likelihood particular from dataThe from the data SPT. from ACT [14] data describe theof two ACT multi-season regions and separately (ACT-E multi-frequencyare and ACT-S); spectra, derived and consist with from an ACTis associated Gaussian-distributed maps to covariance obtained good matrix. usinggiven approximation. some the To model method construct spectra the describedw likelihood in for [18]. each region, The likelihood where Σ is theC bandpower covariance matrix. Each of the model and data vectors described in [14]. binning and the beamsand the have ‘v2’ been spectra slightly are refined, released generating publicly; cosmological ‘v2’ effects of are the negligible spectra. as described This in is [61]. described in [14], lar scaling is estimatedestimated by from correlating the the 100 results IRIS from dust mapspriors with of ACT ([14]), and is consistent with early with amplitude template 2.6 Residual GalacticThe cirrus Galactic emission issignificantly spatially to varying, the and power14] [ spectra,we show particularly apply that in a our dustusing Equatorial mask emission measurements region. can to at As contribute regions 100 reported of in [14], high dust emission before computing the power spectrum, JCAP07(2013)025 . S − itself contains spectra for each ACT L ij b C 2 ln − E − ACT L – 10 – 2 ln − = ACT L 2 ln − . (Top) Power spectra measured by ACT ([14]) at 148 and 218 GHz, and their cross- cross-season. There are twoACT-S seasons (6 used cross-season for spectra). ACT-E The (3 total cross-season likelihood spectra), is and given three by for Figure 4 for ACT-E and ACT-S separately, and each spectra set spectrum, coadded over ACT-E andand ACT-S. secondary We contributions show (dotted the lines)ACT primary cross-frequency to (lensed spectra, the CMB after best-fitting in subtractingand model. dotted the 218 black (Bottom) best-fitting line) GHz Residual model, (right). poweris at in The 148 a the (left), good errors 148x218 at fit (center), small simultaneously to scales ACT-E are and correlated ACT-S, due with to no beam sigificant uncertainty. residual The features. model JCAP07(2013)025 gs a 9400 2. In . 0 and ± ge for ACT-E 4 < ` < a . e 2 y , = 0 e 1 gs y a 2, and describing the CIB power, . 0 c β ± 8 . and c 02. The 218 GHz maps are calibrated = 0 . . 0 a , following the method in [28], at an 0 . Within each frequency, the individual ij bb , ge , s p ± ij a 2 b Σ a y 2 C ) 00 j 2, . j . y y 0 i i WMAP and y y = 1 ( e s 2 – 11 – → 1 y → ij b , y 0 < ξ < for ACT-S. e C ij bb 1 s y = 1500. The 218 GHz calibration is constrained by the 2 , that scales the estimated data power spectra as Σ i y ` 4, 0 , . describing the tSZ-CIB cross-correlation, and s 0 1 ξ y ± 9 . 3000 at 150 GHz, and in [54] for angular scales 2000 = 2 obtained for the model used in [54] using the SPT data. To combine s 2 a describing the SZ emission, , for each map χ i < ` < y kSZ a and = 700, resulting in a 2% map calibration error in CMB temperature units. We ` tSZ a Before fitting the SPT data with the ACT secondary model, we confirm that we recover [14] calibrate the 148 GHz maps using Uncertainties in the measured beam window functions for ACT at 148 GHz are between 2000 (SPT-low) and the [54] data at smaller scales (SPT-high). More radio source power describing the radio power, s describing the Galactic cirrusscribed emission. in section2: The latter four have strong priors imposed, as de- a at 95, 150 andof our 220 work GHz. hereframework is These to to describe observations test the are SZ for and summarizedrefined consistency foreground in spectra between components. the table1. from As two this SPTdata experiments, article One at in was by of being using 150 our prepared, the a comparison. GHz common goals were reported inthe [66]; parameters we and do not include these latest has been removed from the SPT-high spectra due to masking at a deeper flux level, so the the data over the` < full angular range, we follow the method in [54], using the [ 36] data at seasons are calibrated tosingle each overall calibration other; uncertainty. the inter-season calibration error0.7 is and absorbed 0.4%, into the andmeasured beams at by including 218GHz them directly betweenin in 1.5 the [14]. covariance matrix and This for 0.7%. theits technique spectra, exact assumes described We form. a incorporate fiducial uncertainties model in for3.1.2 the the power spectra Secondary but modelOur is parameters model insensitive described to incase: section2 has nine free secondary parameters for ACT in the basic addition to the ninesection 4.3 modelwe parameters, investigatethe how there additional, data. are or To four fewer, compute calibrationthe parameters the band-centers affect parameters model for the for requires SZ, fit an ACT. radio, of effective In and the frequency dusty3.2 model for sources to each given component; in we Combining table use with 1The SPT ([68]). South data Pole Telescopeangular observed scales the 650 sky from 2007-10. Spectra are reported in [36] for To account for both ACT regions, we include four calibration parameters: and the elements of the bandpower covariance matrix as at 148 GHz and 218 GHz, and effective impose this as a Gaussian prior, with cross-spectrum, so no prior is imposed on 3.1.1 Calibration andThe beam data power uncertainty spectrabration are parameter calibrated, but have uncertainties. We therefore include a cali- relative to 148 GHz, at an effective JCAP07(2013)025 2 2 . K 0 7 K. µ . (3.2) ± 2 . 3 = 3000 . ` = 9 ` at = 3000. The B 2 ` K µ ] (3.3) 220 , , compared to 1 2 220 b , to calibrate the 95, 150, power in K 3 y µ . ,C π , 2 4 2 . / 220 ` y , 2 is also defined at SPT , C ξ 1 ± 150 b L y 5 . ,C + 1) 2. . 2 ln ` 0 ( 150 , ` − = 10 ± 150 b 3 . ACT 3000 ,C B L = 1 are the 220 0 , – 12 – s } 95 a b 2 ln s a − ,C , = ge 150 a , L , 95 b gs a ,C 2 ln , . 95 c , } − ). In addition, to fit the SPT data we require a separate radio a s c 95 b a , β C p , a gs is in flux units, for a modified blackbody with effective temperature 9 a , = [ , and three calibration parameters, c and 0 , b β s c ge kSZ a C a a a , , , ξ p { a tSZ , a ξ { , . Distributions for secondary parameters from ACT and SPT, for best-fitting ΛCDM model. kSZ a The likelihood for ACT and SPT together is given by We then extend the ACT secondary model to fit the SPT power spectra. Six of the ACT , tSZ and 220 GHz maps respectively.as We the impose SPT a covariance uniform matrices prior include on the these calibration calibration parameters, uncertainty. for SPT-high. We account for this by first subtracting a radio Poisson power of The SPT likelihood is constructed as in eq. (3.1), with model and data vectors and frequency 150dust GHz. emissivity index TheConversions tSZ-CIB to power correlation at parameter eachare frequency imposed are on given in2. table Strong priors, described in section expected3.1.2 , residual radio power in SPT-low is from the SPT-low data, followingthe the overall approach residual in radio]. [54 level in A Gaussian SPT prior of is then imposed on source parameter, model parameters are expected toa be common for the SPT data (the SZ and CIB parameters: Figure 5 Parameters JCAP07(2013)025 , , 0 at s gs a 186, a . 2 { , K seven- ge = µ . a } θ ] = 3 c , s s 12 β A a , . { 10 1000 angular c ij ∼ a , WMAP = , π sec p ` ` < θ 2 a B / , ξ + , 1122, ratio of the acoustic 1000 . 2000. To compute the C kSZ for all the required cross- CMB ` a = 0 800); the addition of noise ij B , , 2 ` < + 1) × h = sec ` c ` tSZ data measure ij ( B , a ` { 590 th ` √ B = / 040, scalar amplitude ln[10 θ . WMAP analysis using the multi-frequency data, for SPT-low at 8. . 898. 150 . , WMAP – 13 – = 0 150 = 0 b 8 τ C σ = b 02226, cold dark matter density Ω C . = 0 2 WMAP h b data, as it is expected to be small: , and/or SPT-specific secondary and calibration parameters: using the CAMB numerical Boltzmann code ([42]). 2000, and } We then investigate a set of possible extensions or modifications to the s 2 9707, and optical depth 6 y . CMB ` > ` , , is estimated at a level of 8% (using 54 s . B 1 = 0 } y WMAP 3 sky s , y n e /f , 2 2 y y , , e 1 1 Select primary cosmological parameters, andspectrum compute a theoretical lensed CMB power Select values for common secondary parameters: Compute the total theoretical secondary power spectra Compute the total model power at each frequency, y spectra with eq. (2.4), using the effective frequencies for each experiment. Select values for ACT-specific secondary and calibration parameters: Compute the bandpower theoretical powerEquatorial spectra regions for for each ACT dataset (and for for both SPT), South and and compute the likelihood using eq. (3.1). y overlapping region of sky. Covariance between the two spectra due to cosmic variance, There is some degree of covariance between the ACT and SPT spectra, due to the 54 Physical baryon density Ω • • • • 2 • • 6 spectral index horizon to the angular diameter distance at decoupling Θ = 1 61 GHz from the [4] model, assuming for SPT-high at lowers this level, so we neglect the3.3 correlation in our combined Multi-frequency analysis. likelihoodTo prescription return the ACTthis (or approach: ACT+SPT) multi-frequency likelihood for a given model we follow scaling as 1 e.g., [48]. Finally,power in in light the of observations by both ACT and SPT, we4 also neglect the Tests SZ of theIn multi-frequency this likelihood section wethe test ΛCDM the goodness cosmological model. ofparameters fit We using of estimate the the the modelbest-fitting probability MCMC to values. distributions the method of ACT described the power spectra, secondary in assuming [16], fixing the ΛCDM parameters at deg model, we use the band-centers for SZ, radio and dusty sources given in table 1, from [54]. year data to estimate cosmologicalscales, parameters. and The solikelihood have estimates minimal the contamination temperature fromAt spectrum these SZ from frequencies and V and point angular andbe scales, W sources. negligible, the bands consistent infrared (61 The with pointestimated and public source ACT and 94 contribution and 7-year GHz, subtracted is SPT [40]). internally expected measurements. to to the The radio point source level is 3.3.1 Combining with In [61] we use the ACT and SPT likelihood in combination with data from JCAP07(2013)025 2000, with contributions ∼ ` – 14 – 1000, and is dominated by the Poisson and clustered CIB. ∼ ` . Power spectra measured by ACT at 148-218 GHz, with the best-fitting individual SZ and Figure 6 foreground components from table2. The(top) Galactic cirrus the component secondary has been components subtracted. are At 148 significant GHz at scales smaller than secondary model. Weforeground include model the between the SPT two power datasets. spectra We and use examine the ACT the ‘v1’ consistency spectra of for the this analysis; from tSZ, kSZ, radio galaxies,are the non-zero CIB, in and this the model,radio tSZ-CIB cross-correlation. but power are The is not tSZ, individuallysignificant kSZ constrained significantly and by by detected tSZ-CIB from bright the source ACT spectra. counts. The At 218 GHz (bottom) the secondary signal is JCAP07(2013)025 . y 5). 98, . . and 0 p < ξ < a = 0 ξ < 2 χ ), so do not σ 2 . 0 9 [61]. . < 6 < and standard deviation 2400 at 218 GHz, and x kSZ > ∼ a 1 consistent with [2] who ` . over the full angular range. 0 is unconstrained so the prior upper 2 ± ξ χ for mean at 95, 150, and 220 GHz, as measured 2 . y 2 ± K µ x = 2 = 675 for 697 dof (reduced 19 . β 2 χ 21, and 2 . – 15 – . The kSZ power peaks at a non-zero value, but 5, the upper limit increases to . kSZ 16, 0 . 0 a = 0 and < ξ < 3000 B tSZ a is broadened to 0 ξ 72, for 710 data points and 13 parameters). This indicates a good overall fit, 07. The tSZ and kSZ power are individually seen at low significance, with . . 0 ± 3200 at 148 GHz. The goodness of fit is 20 . In figure6 we show the individual components that contribute to the 148 GHz and The marginalized distributions for the secondary parameters fitting the data are shown = 3000. The derived constraints on the CIB and radio source components, and the > ∼ Gal-E and Gal-S are the Galactic cirrus powers in the ACT-E and ACT-S spectra. The levels are close Secondary parameters marginalized over the 6 ΛCDM model parameters are reported in table 1 of [61], A flat prior is imposed, unless indicated as a Gaussian with The SPT cirrus level we use is Results are reported as 68% confidence levels or 95% upper limits; If the prior on ` ` 7 8 9 , are detected at high significance, with index 12 11 10 2, and is also unconstrained by ACT if allowed to vary over a broader range (e.g., c . to the priors imposed from the measured cross-correlations with IRIS ([14]). and are consistentincreasing with errors these by results. at most The 10%. marginalization has little effect on these secondary parameters, in [54]. 0 limit is reported. the distribution is broadsignificance. and The consistent tSZ-CIB with correlation zero. coefficient is The unconstrained total in the SZ prior power range is 0 detected at high Galactic cirrus emission, atcomparison the with ACT other effective models. frequencies are also given in3 table to allow 218 GHz power spectrathere after are removal of contributions thedominated from best-fitting by all Galactic dusty the point cirrusfigure7, sources, power. components. both which At clustered shows At 148 and the GHz 218at Poisson. frequency GHz This dependence the is of illustrated secondary the further dominant spectrum in components is in our model as described earlier, acomplete. refined estimate We of have the checked beams that became available effects after on the parameters analysis was are negligible ( The parameters for the powerprior from distributions. radio [61] sources present and a fromare physical Galactic consistent interpretation cirrus with of are those these driven parameters; found by their the in constraints the 1-year ACT analysis in [16], with reduced errors. in figure5 and summarized in table2. The Poisson-like andan clustered anti-correlation CIB between power, update the parameter constraints or plots in this section. 4.1 ACT data We find that thescales model and provides frequencies. aS, good Figure4 with fitshows the to the the best-fittingsecondary total ACT Galactic contributions. spectra data cirrus over (coadded The component the over SZ removed) full ACT-E decomposed and range and into foregrounds ofwith ACT- primary dominate angular PTE= and 0 at Figure4 therefore shows thenot residual observe power any afterfrequency subtracting significant dependence the features, of best-fitting indicating the model;residuals data that at we in the do the smallest both model scalesaccounted regions. at for fits 218 There in GHz, both is the but a covariance the this matrix. positive is angular consistent excess with and in correlated the beam ACT-E error, a find 2 but localised deviations can be hard to identify using the at JCAP07(2013)025 008 3 1 01 01 02 02 6 07 006 007 2 9 3 11 ...... 9 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 70 . 810 . . 82 0 / 5 0 ± ± ± . ± ± ± ± ± ± ± ± ± ± ± 0 7 4 9 0 2 < < ...... < 99 02 10 01 03 . . . . . 008 006 010 . . . 1 . 10 12 0 89, PTE=0.77). . ). ± ± ± 2 1 1 008 1 471 7 5 2 02 1 02 1 9 4 7 ...... = 0 . K 2 2 0 0 0 0 0 0 0 0 . . µ 2 107 773 ( 77 0 4 0 χ ± ± ± ± ± ± ± . ± / SPT ACT+SPT 4 0 0 0 1 < < . . . . . 58 69 59 2 0 01 02 . . . . . 3000 007 0 0 0 . B ± ± ± 0 8 4 . . . SPT 00701007 —01 — — 1 — 0 1 1 491 7 6 2 2 — 0 4 4 4 — 3 ...... 8 6 2 0 0 0 0 0 0 0 0 1 0 73 — . . 697 96 . 72 0 8 0 / 0 ± ± ± ± . ± ± ± ± ± ± 0202 8 6 8 1 . . . 9 9 2 9 3 1 < < ACT 0 0 0 ...... < 99 03 . . 010 011 ± ± ± . . . The radio Poisson level is lower due to the 2 . c 90 76 95 GHz 150 GHz 220GHz . . 7 β 2 0 0 . 02 1 02 1 222 — 0 1 4 3 0 ...... = 96 for 107 dof (reduced 7 – 16 – 0 0 0 0 0 0 and 0 000 6 4 2 0 3 2 3 2 3 . . . ± ± ± ± ± ± —— 0 — 1 —— — — — 1 1 1 χ 16 0 2 12 2 > > > > > 3 8 4 9 . . . . kSZ ± 00 00 ± ± ± < ξ < ± 1 0 0 . . 2 a 1 1 4 0 0 . . 49 78 54 4 1 22 11 5 ACT . . . . . /dof 675 0 0 0 0 0 2 shifts in 0 χ s s e e c c s p 1 2 3 s gs ge 1 2 1 2 ± ± ± ± ± σ ξ tSZ kSZ a y y y a β a a y y y y a a 1 a 8 8 2 9 4 a PTE 0 . Derived constraints on foreground power, . . . . . 0 148 GHz 218 GHz ∼ Parameter Prior best fit 12 Table 3 10 SZ CIB CIB-PCIB-C 6 4 Radio 3 Gal-E Gal-S 0 Radio . Likelihood parameters, assuming best-fit 6-parameter ΛCDM for the lensed CMB. tSZ-CIB Calibration Table 2 Galactic cirrus greater number of radio sourcesspectra masked in in figure8; the the SPT goodness maps. of The fit model is is shown with the SPT 4.2 Combination with SPT The same modelfrequency also range provides to 95 a GHz,the good adding parameters three fit estimated additional from to cross-spectra SPTthose to the alone from the in ACT, SPT likelihood. figure5 with spectra.and We in show 2; table The they SPT are consistent data with extend the JCAP07(2013)025 = 3000 ` uncertainties from σ = 675 + 96 = 771 for the independent 2 χ – 17 – for the Poisson radio sources in SPT. The goodness of fit of the 0 s a to the nearest integer. 2 χ = 773, which can be compared to 220 GHz the power from fluctuations in the CIB dominates; at lower frequencies the 2 χ − . Frequency dependence of the dominant components of the foreground power at Given the consistency of the two datasets, we combine them to generate a joint likeli- We report the derived constraints on the CIB and radio source components at the ACT Does not include calibrationWe parameters. report the ∆ 13 14 Figure 7 measured by the combined ACT and SPT data sets. The bands show the 1 hood; figure5 includes theare secondary ten parameters foreground derived parameters, fromrameter and a (not seven joint plotted) calibration fit. is parameters.joint In The model this tenth is case foreground there pa- thermal SZ and radiointegration. source The power kSZ is and more tSZ-CIB significant. components The are SPT not radio shown. power is lower due to deeper and SPT effective frequencies forexpected each between the band CIB in power table3.effective at bandpass A 148 frequencies GHz difference and for of the ACTbands. approximately strong and 15% CIB 150 frequency is GHz dependence for across SPT, the due mm-wave to different 4.3 Tests of theThis likelihood model fits thecomponents, ACT with and priors SPT describinga data, our simplified and knowledge parameterization includes from ofmodifications our other the to uncertainties emission. observations. about the the However, model, We therefore physical it and consider is test a how set the of goodness extensions of or fit to the ACT data is affected table2. At 150 fit to each data set. This supports their consistency. JCAP07(2013)025 c ∝ a ` B and p a to vary, we find n 13 2 χ ∆ 14 10 0 parameters 8 9 1 . 8 9 5 – 18 – . 5 8 3 . = 0 8 2 ξ = 1 = 0 9 1 s 6 K 9 1 free 10 -4 kSZ α . . Modifications to secondary model. 06 in this scaling. If we allow the index a . n = 13 p d β T Table 4 6= c Altered kSZ shapeNo tSZ-CIB corr, No SZRadio index 9No Galactic residual 1 7 6 6 21 ModelFiducialCIB index β CIB Poisson corr =CIB 0 Clustered corr = NumberCIB 0 of ACT Fixed kSZ, 9 0 . Power spectra at 95, 150 and 220 GHz, and their cross-spectra, measured by SPT ([54]), The CIB appears to be well-fit currently by a power-law in angular scale, with .[2] find an uncertainty of 0 8 . 0 are broadened, as theyCIB are emission correlated is with perfectlyfor the correlated power-law imperfect among scaling. frequencies, correlation in We was also the reported assume range in that 95-220 the GHz. [52]. Evidence The effect of this assumption is tested by no improvement in the fit, but parameter distributions for the CIB parameters ` by an increasetests, or summarized decrease in in table4, weparameters. parameters, hold or the A cosmological a subset modelon change of fixed the at these in primary the extensions the cosmological best-fitting are prior parameters. ΛCDM considered assumptions. further in In [61], these testing their effect Figure 8 fit with the sameincluded, model with as the excess ACT radioapplied data power to in subtracted the figure4. for ACT Atconsistent. and comparison. 150 SPT GHz Accounting maps, the for and SPT-low the spectrum the different from different [ 36] flux bandpass is effective cuts frequencies, the spectra are JCAP07(2013)025 = 0, finding s α = 21. 2 χ = 3, indicating that the data cannot 2 χ 8, assuming homogeneous reionization. This . – 19 – = 0 , corresponding to a different model for the bright 1 in the model, choosing 0.8 for either the Poisson 8 σ σ < assumption on the frequency scaling of the extragalactic = 0 and keeping only the CIB and Galactic components, the a priori kSZ 15, is consistent with the joint index. Changing the effective dust a . 0 = ± 7 K to 14 K, consistent with the value obtained by [24] from a fit to tSZ . 14 = 6 and the clustered CIB level increases. This indicates a preference for . a 2 χ = 2 p β . c β = 5 compared to the perfectly correlated case. We also assume a common frequency 2 , and removes the detection of the CIB at 148 GHz. The index for the Poisson sources Exploring the SZ assumptions, we consider fixing the kSZ contribution to the linear Our model imposes an Finally, we test the effect of removing the Galactic cirrus components; the goodness of We construct a CMB-only likelihood from ACT data as follows. Instead of using the χ c β yet distinguish between homogenousalso and leads patchy to reionization. tighter constraintsadding Limiting on a the the patchy kSZ tSZ reionization power. in templateaffect this from Modifying the [6] way instead other to the secondary the shape, parameters.data kSZ, we equally which Neglecting find changes the that well its tSZ-CIB formodel shape, cross-correlation for one does also the not fits fewer tSZ-CIB correlation the parameter.altogether, is explored setting The further in dependence [61]. of If we the neglect kSZ the SZ constraints components on the the FIRAS CIB frequency spectrum,change in has no effect on the model, aparttheory from a estimate corresponding for a universe with temperature from 9 goodness of fit significantly worsens, with an increaseradio of sources. ∆ We test the effect of changing the radio spectral index to fit worsens by ∆ ACT likelihood to estimatemating cosmological the parameters, CMB we powerination. take spectrum the intermediate in This step bandpowers,adopted is of marginalizing in a esti- over earlier natural the analyses,of extension possible e.g., subsequent contam- to papers [10]. to formsover Such a combine of variety ‘grand the of CMB unified results nuisance parameters, spectra’ data from e.g., various were compression in CMB used [8,9, that experiments,62]. in have marginalizing At a large been number scales, where contamination setting the correlation coefficientor clustered to components, roughlyreported corresponding in to [52]. theby For degree ∆ the of Poisson correlation componentscaling between we of maps find the clustered thatif, and this Poisson for degrades terms. instance, the A thea goodness different redshift scaling of common in dependence fit index frequency ofsignificantly is may the improve be not the expected clustered necessarily goodness andfor expected. of Poisson fit, power but Allowing is doesin it different, this lead so to case, to vary a independently poorly does constrained distribution not negligible effect on parametersthe and prior goodness on ofradio the fit. sources power that There from lie is above [26]degeneracies also the with by little detection other threshold. 1 effect parameters, from Removing but the changing does prior not altogether opens significantly up improve the goodness ofGalactic fit. cirrus at the 95% confidence level. 5 CMB-only likelihood Understanding the contribution ofvital the for secondary extracting components themary to cosmological and the secondary information, parameters. ACT due powerinteresting. Values to of spectra possible the However, is secondary if degeneracies parameters we betweenlikelihood are pri- are also is astrophysically only desirable. interested in the cosmological parameters, a simplified degrades the goodness of fit of the model by only ∆ JCAP07(2013)025 2000 (5.1) (5.2) > ∼ ` are the bandpower w 20% level at scales ∼ , where ij , th ` , , C ) ) θ ij b` θ ( ( w ij CMB bandpowers, marginalizing over , b sec b = sec ` n C ij , C + th b + C CMB b – 20 – CMB ` C C A = = ij , th b th ` C C 218 GHz, and 218 GHz) are also shown to indicate the significant level of × , is written as ij , th ` C ) is the secondary signal as in eq. (2.4), and is a function of secondary param- θ ( ij , . Estimated CMB bandpowers from ACT, marginalized over extragalactic source and SZ sec ` . Writing the spectrum in bandpowers, C θ By marginalizing over nuisance parameters in the spectrum-estimation step, we can We recall that the model for the theoretical power for a single cross-frequency, cross- effectively decouple the primaryparameters CMB are from then non-CMB needed information. when estimating No cosmological5.1 additional parameters. nuisance Method: bandpowersTo implement via this Gibbs method sampling in practice, we estimate season spectrum, Figure 9 components. Bandpowers are estimatedvariance for weighted ACT-E combination. and The ACT-Sdue bandpowers separately; are here to correlated we at covariance show(dashed, the the with at inverse- the 148 GHz, secondarySZ 148 and parameters. foreground power The at small total scales. multi-frequency spectrafrom for SZ ACT-E and point sourcesaverage is of negligible, the the multi-frequency estimatedhas spectra CMB additional as is uncertainty simply in due an e.g., to optimally [32]. combined secondary At contamination. smaller scales the CMB spectrum the secondary parameters.this, We but use estimate the CMB full bandpowers multi-frequency instead likelihood of cosmological from parameters. section3 to do where eters window functions, we write the model for the bandpowers in vector form as JCAP07(2013)025 . b θ 2 n K (5.6) (5.3) (5.4) (5.5) (5.7) µ , can b C . θ , where ). . The poste- b θ spec C n | ) b × C ) b b CMB b n − ˆ C C ( sec − b p ), and restrict the CMB C , + CMB )] b CMB dθ. b . This gives mean C ) , with elements that are either sec b ( C θ 1 ( CMB ( b C A p . − p C CMB ) b − G b Q C A b L T C ( ), is then also a Gaussian. It has a | ) ), which is the same at all frequencies C 1 b b ( + . − ˆ , θ 1 , n C b A − Σ ˆ CMB b Q C 1 − , and drawing a vector of Gaussian random T Σ C − ) T CMB b ( = b T Σ p C C A ( T LL [ CMB b p 1 − A C – 21 – CMB = − b + ln det ] Z is the number of cross-season and cross-frequency = C sec b Q A C 1 )-length data vector. Q ) = ( ) = − , spec b b + n Σ spec Σ C | T n ). We write the multi-frequency likelihood for a single ACT , of this conditional distribution are obtained by taking the θ, C | b A , marginalized over the secondary parameters, CMB b × Q b CMB C b ,C 4500, where the CMB power is expected to be less than 1 = [ n C CMB A b CMB b b ( + ln det ˆ C p , given the observed multi-frequency, multi-season spectra C C ` > ( CMB b = ( p C | CMB θ b is held fixed, the conditional distribution for the secondary parameters, L ( 2 ln are multi-frequency, multi-season vectors of length C p − 2 ln sec b CMB b C − C = 9 for ACT-E, and 18 for ACT-S). The secondary spectra differ between is held fixed, the conditional distribution for the CMB bandpowers, , and covariance, ), is not a Gaussian, but can be sampled with the Metropolis algorithm that ), and to extract the desired marginalized distribution ), and ), assuming a uniform prior for b b b b b ˆ we alternate a Gibbs sampling step, drawing a new vector of CMB bandpowers, C and C spec . The sample is then given by | sec b ,C n θ, C θ, C G C th b , θ | | C CMB b , with a Metropolis step, drawing a trial vector of the secondary parameters CMB b We want to estimate Gibbs sampling can be used in the special case that at least one conditional slice through We choose a uniform positive prior distribution for If instead If C CMB CMB CMB C b b b | θ C C C CMB b ( ( ( ( is used inand the MCMC sampling in section4. To map out the full joint distribution for C p and in all seasons, onto the ( region, from eq. (3.1), as distribution given by and covariance bandpowers to be zero at spectra ( frequencies but not between seasons. The mapping matrix a multi-dimensional distributionestimate has the large-scale a CMB power knowne.g., spectrum, [17, form, and20, , 35 to and40, marginalize].72 p over has Galactic Here, foregrounds, we been split the used, joint distribution for into two conditional example, distributions: to is the number of bandpowers, and 1 or 0, maps the CMB bandpower vector (of length rior distribution for be written as We find thatp Gibbs sampling provides an efficient way to map out the joint distribution which is a multivariate Gaussian. p variates where The mean, derivatives of the likelihood in eq. (5.4) with respect to We can drawdecomposition a of the random covariance matrix, sample from this Gaussian distribution by taking the Cholesky JCAP07(2013)025 (5.8) ), with its = 3540 for 20% (15%) b ` ∼ C 2000 there is | S − ` < CMB b ,C E − , and relative 218/148 GHz s 1 CMB b y C , ( for the ACT-E bandpowers, and e 3500. At scales p . 1 e , , ) 2 y 1 ) ) < ∼ b b b ` /y ,C E S − − , θ, C , θ, C S E − − CMB b CMB b C , for example. CMB b CMB ,C b = E – 22 – C C | | E − E S − − − Planck CMB b CMB C 0 b CMB CMB b b | , as part of the secondary parameter set. s θ C C C 1 ( ( ( p p p /y s 2 y . The uncertainty on the bandpowers rises at scales smaller than , e ) by taking sequential sampling steps from the conditional distri- for ACT-S. 1 b s C 2 1 /y | e 2 /y , θ WMAP y S S − − CMB b CMB b C ,C = E − S − , are then estimated following the standard MCMC prescription, e.g., [41, 65]. To estimate the joint distribution for the ACT-E and ACT-S bandpowers, we map out We then estimate the 148 GHz calibration factors, Figure 10 shows the effect of marginalization on the bandpower errors, using the ratio = 3200 for the ACT-E (ACT-S) spectra. The marginalized distributions for the CMB 3000, and the correlations between bandpowers increases. CMB b ` CMB C CMB 0 b b ∼ ( 5.1.1 Combining spectraThere is from only different one regions underlyinga CMB single power spectrum, spectrum, or so set thiswindow of method functions bandpowers, could are from be different the used for twoserve to each ACT this estimate region regions. information, due However, we to the estimate theirSince bandpower the distinct the geometries. CMB secondary bandpowers To parameters for easilybe ACT-E con- are correlated and common between ACT-S the to separately. regions both, at the small estimated scales. p CMB bandpowers will butions: About 100,000 stepsthe are [15] required spectral for test.C convergence The of mean the and joint covariance of distribution, the assessed resulting with marginalized bandpowers, 5.1.2 Calibration factors There are four ACT calibrationCMB factors. bandpowers To due minimize to bin-to-bin calibrationthe correlations uncertainty, two in we ACT the divide spectra, estimated out estimating the 148 GHz calibrations for associated covariance matrix, isto then include computed the SPT from data, the or samples. data from This could be extended C The marginalized distribution for the CMB bandpowers, calibration factors, little error inflation due toby foreground uncertainty, and the errorsbandpowers are inflated are by well approximatedACT, by as Gaussians shown for in multipoles the to appendix. band-center 5.2 Marginalized CMBFigure9 bandpowers shows the estimatedadded CMB together bandpowers and fromregion compared the are to ACT-E reported and the inupdated ACT-S table5. ‘v2’ multi-frequency spectra, multi-frequency spectra. In spectra. co- bandpowers this Without over the The table assuming full any we bandpowers angular cosmological range reportmodel for model, are predicted the remarkably each the by consistent CMB CMB with spectra the` theoretical derived ΛCDM using the between the marginalized errors and theis unmarginalized clear errors that for by a measuring fixedseparated secondary the model. from spectrum at secondary It multiple contamination frequencies, out the to CMB can scales be successfully JCAP07(2013)025 4500 (5.9) < ` < 8 8 7 4 1 6 5 7 8 . . . . . 15 . 96 74 42 42 35 25 26 21 14 11 10 78 3 2 2 1 1 ± ± ± ± 113 ˜ Σ 16 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0 2 6 3 1 . . . . . ) 2 + ln det K µ x ( 6 133 15 91 2 57 8 43 8 21 9 1 . . . . . 6158 1124 48 1176 38 1026 36 717 29 802 21 655 17 462 16 372 12 345 1111 244 223 201 − 5 4 4 2 2 π 126102 2370 1923 ± 160 2250 107 1737 2 ˜ ± ± ± ± ± ± ± ± ± ± ± ± Σ ± ± ± ± ± / ± ± ± ± T b 8 4 7 3 0 x . . . . . C 3500. We do not use the 3500 ) = – 23 – + 1) th ` ` 95 204 93 127 87 2 56 2 44 3 19 9 ( C ` < . . . . . | 5861 1187 51 1149 33 1016 39 766 31 787 19 650 16 474 14 355 10 347 10 246 214 ` 3 3 3 2 2 146 2274 109 1903 ± ± 159 2343 115 1744 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± CMB b 9 6 0 5 2 ACT-E ACT-S Coadd ...... Lensed CMB anisotropy power. ˜ C shift in the estimated kSZ power. We find that the CMB ( σ L 2500; a dominant correlation is then with the kSZ power due to its b ` 2 ln Table 5 > ∼ 790 2499 890 1945 990 1068 590 2157 690 1729 − 19902090 230 22402440 199 2640 137 92 2840 57 3140 43 3540 22 9 10901190 1206 1290 1036 1390 679 1490 819 1590 661 1690 452 1790 387 1890 344 242 ` We compare the secondary parameters recovered in this model-independent sampling to This coadds the ACT-ETo and compute ACT-S a CMB likelihood bandpowers using for these plotting data, purposes. ACT-E and ACT-S should be used with the covariance matrix 15 16 and bandpower window functions provided on LAMBDA. blackbody frequency dependence, and aCMB smaller kSZ power power — — compensated iscovariance by matrix larger allowed primary conserves when this the correlation information. ΛCDM assumption is5.3 relaxed. The The CMB CMB-only bandpower likelihood We construct the CMB-onlyare likelihood from Gaussian, the conservatively angularbandpowers range choosing as where they the are CMBand bandpowers increasingly are more non-Gaussian, strongly due correlated to with the foreground foreground parameters. marginalization, The likelihood is given by bandpowers are not stronglythe correlated Silk with damping tail the at secondary parameters until scales well into the case where ΛCDM isare assumed. consistent, This comparison with is about shown in a the appendix; 1 the parameters JCAP07(2013)025 , and (5.10) σ 1 . , and the variation in L A ,  th ` th ` C C s E − − ACT 3500. A single calibration parameter for ACT b`, b`, 2000, the errors are increased due to foreground w w ∼ , the lensing amplitude − ` − – 24 – S eff < ` < E − − N CMB b CMB b ˜ ˜ C C  = x . Parameter constraints using both likelihoods agree to 0 α are the marginalized mean and covariance matrix for the bandpowers, ˜ Σ and . Inflation of errors due to foreground marginalization, relative to the errors for a best- is the lensed CMB spectrum generated from e.g., CAMB. We use 21 bandpowers CMB b ˜ C th ` We have quantified these components using seven power spectra, splitting the CIB To test the performance of this compressed likelihood, results are compared using the C into a Poisson andtSZ clustered emission part, from and clusters,structure. including and power emission Rather from from than the CIB a cross-correlation galaxies between minimal that also model trace with the the large fewest scale parameters, we have sought a fine structure constant, are reported in [61].the We conclude typical that extensions this consideredgive is in more an cosmological efficient optimal analyses, alternative results to although for the the unusual full models full likelihood with likelihood for features may 6 far into the damping Summary tail. In this paper wepower have presented spectra a that likelihood includes formalismto to contributions the describe from lensed the SZ CMB. ACTthermal and We multi-frequency and model foreground kinetic the components SZ, data in emission from including addition CIB four galaxies, late-time and astrophysical emission components: from radio galaxies. full multi-frequency likelihood, andestimated the for CMB-only the likelihood. restrictedthat ΛCDM probe Cosmological 6-parameter the parameters damping model, are tailnumber and and of peak a relativistic shapes, set degrees including of of the freedom more running of extended the models spectral index, the and for ACT-E andeach ACT-S region in is the marginalizedlikelihood range over is 500 analytically, simple, following as [11]. no The extra nuisance prescription parameters for are using needed. this uncertainty. Here where Figure 10 fitting foreground model: at scales smaller than JCAP07(2013)025 , is < ` < kSZ a ) and the ACT website 3900 are significantly non-Gaussian, but ` > ). – 25 – http://lambda.gsfc.nasa.gov/ lower in the model-independent case, as it is anti-correlated with the CMB bandpowers We then compare the secondary parameters estimated in two ways: (1) estimating Modeling these astrophysical components allows us to probe the primordial CMB fluc- We find that data observed by the South Pole Telescope give results consistent with σ 1 http://www.physics.princeton.edu/act/ ( 4500 range. Distributions fordistributions the (dashed ACT-E and curves); ACT-S bandpowers bandpowers are at compared to Gaussian ∼ CMB bandpowers, andin (2) estimating figure 6,12 parameters ΛCDM are and not parameters. strongly are affected by The consistent. the CMB distributions model are The assumptions. shown The tSZ, kSZ power, point source parameters, and Galactic cirrus A Tests of theIn CMB this likelihood appendix, weshow perform a additional selection of tests the on distributions the of CMB-only the likelihood. CMB bandpowers In from figure theare11 estimatedwe well 600 fitbandpowers. by Gaussians at larger scales. The same behaviour is found for the ACT-S tuations down to anan angular estimate resolution of of 4’ theover using primordial the ACT. CMB foreground We spectrum uncertainty. have used well Thiscosmological into the produces parameter the model a estimation. Silk to simplified extract damping compressed likelihood tail, for marginalizing useACT, in accounting for the differenttamination removal by of Galactic radio cirrus. point sources,strategies, SPT and and and different ACT their have degree very observations ofbetween different on con- instrument the the design datasets sky and is scan havecosmic not limited homogeneity. only overlap. an The important excellent cross-check agreement but is anotherAcknowledgments demonstration of This work was supported0408698 and by AST-0965625 the for1214379. U.S. the Funding National ACT was project, also Science provided asand Foundation by well a Princeton through as Canada University, the awards Foundation awards University for PHY-0855887 AST- Astron´omicoAtacama of Innovation Pennsylvania, and in (CFI) PHY- award northern to Tecnol´ogicadeInvestigaci´onCient´ıficay Chile Chile UBC. (CONICYT). ACT under Computations operates the were inon performed the auspices the Parque of GPC the supercomputerunder Comisi´onNacional at the de the auspices of SciNetFund Compute HPC — Canada, Consortium. Research the259505 SciNet Excellence; Government of is supports and Ontario, funded JD, the the byuseful EC, Ontario discussions, the University and Research and CFI of Christian TL. Reichardtuse Toronto. for of We help Funding the with thank the from Legacyfor George SPT LAMBDA Archive ERC data. Efstathiou is for We grant provided and Microwave acknowledge bymade the Background the Steven public Data NASA Gratton Office Analysis through of for (LAMBDA). LAMBDA Space Support ( Science. The likelihood codes will be model that includes ourobservations. uncertainties with For priors example,correlation, describing we our while are knowledge the motivated from tokSZ data additional include power. it do to not avoid placing demand unphysically that strong limits we on include the the tSZ-CIB JCAP07(2013)025 ]. ]. SPIRE SPIRE IN IN [ ][ Power-Law Template for IR arXiv:1210.6697 arXiv:1108.4614 , (2012) 120[ Constraining thermal dust emission in distant 3000 scales, so the preference for a smaller kSZ – 26 – ∼ 752 ` in the latter case. 12 at 95% confidence, more consistent with the limits when < kSZ 3700. The same behavior is seen for the ACT-S bandpowers. Astrophys. J. a , > ∼ kSZ ` a . Secondary SZ and foreground parameters estimated assuming the ΛCDM model . Probability distributions of a selection of CMB bandpowers for ACT-E in the range 4500, marginalized over secondary parameters (solid). The bin-center for each bandpower 2000 due to the common blackbody dependence. 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