Breaking the degeneracy between polarization efficiency and cosmological parameters in CMB experiments
Silvia Galli, W.L. Kimmy Wu, Karim Benabed, François Bouchet, Thomas M. Crawford, Eric Hivon
BBreaking the degeneracy between polarization e ffi ciency and cosmological parameters in CMBexperiments Silvia Galli,
1, *
W. L. Kimmy Wu,
2, 3, †
Karim Benabed, Franc¸ois Bouchet, Thomas M. Crawford,
3, 4 and Eric Hivon Sorbonne Universit´e, CNRS, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014 Paris, France SLAC National Accelerator Laboratory & KIPAC, 2575 Sand Hill Road, Menlo Park, CA 94025 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, U.S.A Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA (Dated: February 9, 2021)Accurate cosmological parameter estimates using polarization data of the cosmic microwave background(CMB) put stringent requirements on map calibration, as highlighted in the recent results from the
Planck satel-lite. In this paper, we point out that a model-dependent determination of polarization calibration can be achievedby the joint fit of the TE and EE CMB power spectra. This provides a valuable cross-check to band-averagedpolarization e ffi ciency measurements determined using other approaches. We demonstrate that, in Λ CDM, thecombination of the TE and EE constrain polarization calibration with sub-percent uncertainty with
Planck dataand 2% uncertainty with SPT pol data. We arrive at similar conclusions when extending Λ CDM to includethe amplitude of lensing A L , the number of relativistic species N e ff , or the sum of the neutrino masses (cid:80) m ν .The uncertainties on cosmological parameters are minimally impacted when marginalizing over polarizationcalibration, except, as can be expected, for the uncertainty on the amplitude of the primordial scalar powerspectrum ln(10 A s ), which increases by 20 − ∼ ∼ Planck polarization maps, whose statistical uncertainty is limited to ∼ . ffi cient for upcoming exper-iments. I. INTRODUCTION
The Λ cold dark matter ( Λ CDM) model has emerged tobe the leading model in describing our universe since theadvent of precision measurements of the anisotropies in thecosmic microwave background (CMB). On the largest angu-lar scales, we have satellite measurements from WMAP and
Planck that reach cosmic-variance limits in the temperatureanisotropy spectrum up to multipoles (cid:96) ∼
500 and (cid:96) ∼ Planck and recent results from ground-based telescopes, po-larization measurements of the CMB are increasingly domi-nating over the temperature measurements in terms of theirstatistical constraining power on cosmological parameters.However, in order to fully take advantage of these upcom-ing data sets, systematic errors that could bias the polariza-tion measurements must be su ffi ciently mitigated and con-trolled. Specifically, recent Planck results show that cosmo-logical parameters can be biased by one of the main polar-ization systematics—errors in the estimates of the polariza-tion e ffi ciencies of the detectors [7, 8]. For Planck , the polar-ization e ffi ciencies of its detectors as measured in-flight were * [email protected] † [email protected] discrepant from what were expected from laboratory measure-ments by up to 5 times the statistical uncertainties of the lab-oratory measurements [9]. To account for this discrepancy,the Planck polarization calibrations at di ff erent frequencieswere then re-evaluated by requiring the polarization spectrato recover the Λ CDM cosmology inferred by the temperaturespectrum measurements, e ff ectively modeling the detector po-larization e ffi ciencies as overall calibration of the polarizationmaps per frequency, P cal .In this work, we propose an alternative method to extractpolarization calibration as a potential cross-check for directapproaches. Typically, polarization calibration parametersare included in cosmological parameter estimation as nui-sance parameters with priors informed by external calibrationsteps [e.g. 10–12]. Here, we jointly fit the Λ CDM and ex-tension models to the CMB TE and EE spectra allowing thepolarization calibration parameters to float, i.e., we let the datato self-calibrate P cal given a model. We show that the combi-nation of just TE and EE is su ffi cient in providing a tight P cal constraint, and the P cal uncertainty can be further improved byincluding the temperature power spectrum TT. Atmosphericnoise degrades the ground-based TT measurement more thansatellite TT or ground-based TE and EE measurements. Forthis reason, the ability to self-calibrate P cal with only TE andEE as demonstrated by this work is of particular interest tocurrent and upcoming ground-based experiments [e.g. 13–17].The inferred polarization calibration from our proposedmethod can produce tight constraints because of the di ff erentdependence on P cal of TE and EE, which breaks parameter de-generacies with other cosmological parameters. While this in-ferred polarization calibration is admittedly model-dependent, a r X i v : . [ a s t r o - ph . C O ] F e b it is nevertheless useful as a consistency check against po-larization calibration estimated through other methods. Fur-thermore, we show that most Λ CDM parameter constraintsare only mildly to negligibly degraded when marginalizingover P cal , and common extensions to Λ CDM are insensitiveto marginalizing over this extra parameter.In the following, we apply this method to SPT pol and
Planck data and show that P cal are constrained to percentlevel precision for these experiments across Λ CDM and itsextensions, including the lensing amplitude A L , the e ff ectivenumber of relativistic species N e ff , and the sum of neutrinomasses (cid:80) m ν . We take inputs from a recent SPT pol powerspectrum analysis [18, hereafter H18] and Planck ’s latest datarelease [19] and sample parameter spaces without imposingpriors on their respective polarization calibration parameters.With the recent release of the ACTpol DR4 data, we applythis method to the publicly available
ACTPollite likeli-hood [20, 27] to demonstrate the ease of application of this ap-proach. We use C osmo
MC [21] for sampling the posterior dis-tributions of SPT pol and
Planck , and C obaya [22] for ACTpol.To check the relevance of this method for upcoming and fu-ture data sets, we forecast the P cal uncertainty and the changesin cosmological parameter uncertainties when marginalizingover P cal for SPT-3G and CMB-S4. While this paper was inits final stages of preparation, the results from the first seasonof the SPT-3G experiment were released [23]. We leave theapplication of our method to this data set to future work.This paper is organized as follows. In Sec. II, we summa-rize polarization calibration as defined in SPT pol and Planck .We present results for SPT pol , ACTpol, and
Planck in Sec-tions III to V. Our forecasts for SPT-3G and CMB-S4 are de-tailed in Sec. VI. We conclude in Sec. VII.
II. POLARIZATION EFFICIENCY AND EFFECTIVECALIBRATION
The power absorbed by a polarized detector in an experi-ment such as
Planck or SPT pol at time t can be modeled as: P ( t ) = G (cid:8) I + ρ (cid:2) Q cos 2( ψ ( t )) + U sin 2( ψ ( t )) (cid:3)(cid:9) + n ( t ) , (1)where I , Q , and U are the Stokes parameters that characterizethe intensity and polarization fields, G is the e ff ective gain(setting the absolute calibration), ρ is the detector polarizatione ffi ciency, ψ ( t ) is the angle of the detector with respect to thesky and n ( t ) is the detector noise. Here we have omitted e ff ectsfrom beams and bandpasses without loss of generality.Intensity and polarization I , Q , and U maps per frequencyare then produced via map-making [e.g., 9] by co-adding ob-servations at di ff erent times and from di ff erent detectors. Rel-ative calibration corrections are applied across detectors andthe co-addition is weighted given the noise of the time-ordereddata over some observing period. In the following, we focuson the impact of errors in the estimate of detector polarizatione ffi ciency at the coadded map level, which can be e ff ectivelycaptured at each frequency by a polarization calibration cor- rection parameter P cal . For the SPT pol
TE and EE analysis in H18, polarizationmaps are first made incorporating detector polarization e ffi -ciencies and angles measured on ground. Then, before form-ing data power spectra, the temperature and polarization mapsare calibrated against Planck maps. The calibration factors (cid:15) are formed by first taking the ratio of the cross-spectrumbetween two halves of SPT pol maps and the cross-spectrumbetween
Planck maps and SPT pol maps. The
Planck mapsare masked and filtered identically as the SPT pol maps andthus have the same filter transfer function and mode-coupling.The remaining di ff erences from the beams B b and the pixel-window function √ F b of the input Planck maps are accountedfor as follows: (cid:15) b = (cid:113) F Planckb B Planckb B SPT b C SPT i × SPT j b C SPT × Planckb , (2)where subscript b denotes binned multipole, and i , j denotedi ff erent halves of the SPT pol data. The calibration fac-tors are extracted by averaging across the multipole ranges600 < (cid:96) < < (cid:96) < Planck
DR2
Commander polarization mapsare used to obtain the polarization calibration factor, and pro-vide a ∼
6% correction to the Q and U maps (see sections4.5.2 and 7.3 in H18 for further details). The uncertainties ofthe calibration factors are incorporated when sampling cosmo-logical and nuisance parameters. Specifically, the theoreticalspectra to which the data are compared are scaled by 1 / ( T cal2 P cal ) for TE and 1 / ( T cal P cal ) for EE, where T cal denotes theoverall residual calibration of the maps and P cal denotes thepolarization calibration correction. Gaussian priors with meanof unity and uncertainties of 0.34% and 1% are applied to T cal and P cal respectively, based on the uncertainties of the ratioestimates in Eq. 2. It is the prior on P cal that we remove in thiswork.For Planck , the modeling of polarization calibration is dif-ferent from the one used in H18 in two ways. First, the
Planck likelihood at high- (cid:96) includes maps from 3 frequencies, 100,143, and 217 GHz, in constrast to the single-frequency analy-sis done in H18 at 150 GHz. Second, while the SPT pol P cal is defined at the map level, the Planck e ff ective polarizationcalibration parameters c EE ν are defined at the power spectrumlevel for each frequency spectrum ν × ν used in the high- (cid:96) likelihood. Thus, P cal = √ c EE for each frequency. The polarization calibration correction parameter, P cal , are sometimes calledpolarization e ffi ciency corrections in Planck papers. Unless specifically refer-ring to detector polarization e ffi ciencies, we use polarization calibration P cal as applied at the map level to refer to this correction. In this paper, we wouldoften shorten “polarization calibration correction parameter” to polarizationcalibration. The high- (cid:96) likelihood covers (cid:96) >
30. We assume here that polarization e ffi -ciency corrections have a negligible impact on the low- (cid:96) polarization likeli-hood due to the large uncertainties in this regime due a combination of cosmicvariance, noise, and systematic uncertainties. Thus, the polarization e ffi ciency for a cross-frequency spectrum ν × ν (cid:48) in, e.g., EE is (cid:113) c EE ν × c EE ν (cid:48) . Specifically, the theory power spectra to which the data iscompared are multiplied by a calibration factor g defined as g XY ν × ν (cid:48) = y (cid:113) c XX ν c YY ν (cid:48) + (cid:113) c XX ν (cid:48) c YY ν . (3)Here, ν × ν (cid:48) indicate the frequency spectra with ν, ν (cid:48) = , ,
217 GHz; the spectra are then either for XY = T E or XY = EE . c TT ν denotes temperature calibration parameters,which are separately determined and on which priors are set. c TT is set to unity so that the 143 GHz temperature map istaken as a reference. Finally, y P is the overall Planck calibra-tion parameter defined at the map level, on which a Gaussianprior of y P = (1 , . ) is set (see Section 3.3.4 of [7] forfurther details). As detailed in Sec. V, in the baseline Planck analysis, c EE ν are fixed to the values obtained by comparingthe EE data spectra to the theory spectra computed given thebest-fit cosmology to the TT spectra. In this work, c EE ν arenuisance parameters to be constrained by the data themselves.Given the di ff erent definitions of the polarization calibrationin these SPT pol and Planck works, in the rest of this paperwe will always specify whether the quoted uncertainties referto the map-level ( P cal ) or power-spectrum level ( c EE ν ) correc-tions. In Sec. V, we will provide results for the Planck datausing both definitions.
III.
SPTPOL
A. Data and model description
We use the SPT pol
TE and EE power spectrum measure-ments from H18. The generation of these measurements isdescribed in detail in H18 and here we highlight relevant as-pects of that work. Data in H18 came from the 150 GHz bandobservations made by the SPT pol camera on the South PoleTelescope over an e ff ective area of 490 deg . The power spec-tra cover angular multipoles (cid:96) between 50 and 8000. The po-larization noise level measured in the range 1000 < (cid:96) < . µ K arcmin.For the Λ CDM baseline case, we sample the identicalmodel space as in H18 using the same covariance matrix withC osmo
MC [21]. The model parameter space is composed of Λ CDM, foreground, and nuisance parameters. The Λ CDMparameters are the cold dark matter density Ω c h ; the baryondensity Ω b h ; the amplitude and tilt of the primordial scalarpower spectrum ln(10 A s ) and n s ; the optical depth to reion-ization τ ; C osmo MC’s internal proxy to the angular scale ofthe sound horizon at decoupling, θ MC . A Gaussian prior is seton τ : (0 . , . ) given the Planck results [24]. Thesum of neutrino mass (cid:80) m ν , when not sampled, is fixed to0.06 eV. On the other cosmological parameters, we set largeuniform priors. We denote Gaussian priors with mean µ and standard deviation σ as ( µ, σ ),and uniform priors between v min and v max as [ v min , v max ]. We consider Galactic dust foregrounds and the extragalac-tic foregrounds from polarized point sources. We model andset priors for them identically as in H18. The priors on theamplitudes of dust at (cid:96) of 80, A TE and A EE , are set to be uni-form with [0 , µ K ]; the priors on the spatial spectral indices, α TE and α EE , are set to ( − . , . ). Finally, the prior onthe amplitude of polarized sources D PS EE is set to [0 , . µ K ].As in H18, the nuisance parameters are beam uncertainties,super-sample lensing [25], and temperature and polarizationcalibrations. We include e ff ects from super-sample lensingwith the prior on κ to be (0 . , . ). We model beam un-certainties using two eigenmodes with prior (0 . , ) on eachmode. The overall residual calibration parameter T cal has prior(1 . , . ). Finally, as for the focus of this paper P cal , weeither set a prior of (1 . , . ), which is the baseline of H18,or no prior, which is the method we propose to let P cal be de-termined by the data.In the following, we will report results obtained either fromTE and EE separately, or from the combination of the two,which we will refer to as TE,EE. B. Main results
To illustrate the idea, in Fig. 1, we show the 2D posteriorof ln(10 A s ) and P cal from TE, EE, and TE,EE without im-posing a P cal prior. We see that without a P cal prior, the con-straints on A s from TE alone and EE alone are very degener-ate with P cal . However, since the P cal dependence from TEand EE are di ff erent (linear versus quadratic in P cal respec-tively), the combined TE,EE constraint on A s and P cal withouta prior are significantly reduced. This illustrates the poten-tial of combining the TE and EE spectra in constraining P cal without significantly degrading constraints on Λ CDM param-eters. Furthermore, we find that the P cal parameter as sam-pled is consistent with unity. This serves as cross-check to thepolarization calibration determined by the comparison to the Planck
Commander polarization maps. In the following, wefirst show that the constraints on P cal are su ffi ciently preciseand stable across di ff erent models to be used as a cross-checkfor other sources of measurements. We then discuss e ff ectson cosmological parameter uncertainties when marginalizingover P cal .For this SPT pol data set, we obtain a ∼
2% constraint on P cal in Λ CDM and three extensions— A L , N e ff , and (cid:80) m ν , aslisted in Tab. I and shown in Fig. 2. This level of precision issu ffi cient to cross-check the baseline approach used in H18 inwhich the SPT pol polarization maps are calibrated against the Planck
Commander maps. In other words, without applyingthe polarization calibration correction from comparing against
Planck , one would arrive at a similar conclusion that a 6% cor-rection should be applied to the calibration of the polarizationmaps if one lets P cal float while sampling the Λ CDM and ex-tension model spaces with the TE,EE data set. We note that inall three extension scenarios, the P cal constraint does not de-grade significantly, which shows that this approach is usefulas cross-checks beyond just the Λ CDM model.The stable uncertainties on P cal across Λ CDM and the few . . . . . ln(10 A s ) . . . . . . P c a l TEEETE,EE
FIG. 1. ln(10 A s ) vs P cal in Λ CDM for SPT pol
TE, EE, and TE,EE,with no P cal priors. We exploit the di ff erent degeneracy directionsbetween ln(10 A s ) and P cal from TE and EE to constrain P cal .TABLE I. Polarization calibration parameters obtained from SPT pol data assuming di ff erent models. For reference, using the baseline P cal prior in H18 of 1%, we find P cal = ± Λ CDMmodel. Model SPT pol
TE,EE (no P cal prior) Λ CDM 1.0061 ± Λ CDM + A L ± Λ CDM + N e ff ± Λ CDM + (cid:80) m ν ± extensions suggest that P cal has little degeneracy with otherparameters. Indeed, most cosmological parameter constraintsare only negligibly to mildly degraded when we relax the P cal prior for the SPT pol TE,EE data set. We show in Fig. 3 theratios of cosmological parameter uncertainties between the no P cal prior and the baseline P cal prior case for the models con-sidered. The constraints on A s degrade most, by 40 −
60% de-pending on the model. This is expected given the correlation Λ C D M A L N e ff Σ m ν P cal,pol SPTpol TE,EE
FIG. 2. Marginal mean and 68% confidence level error bars on P cal obtained from SPT pol TE,EE data assuming the Λ CDM model anda few of its extensions. The determination of P cal is only slightlya ff ected by the choice of cosmological model. Λ C D M A L N e ff Σ m ν Ω b h Λ C D M A L N e ff Σ m ν Ω c h θ MC n s ln(10 A s ) H σ A L N eff m ν , 95% SPTpol TE,EE ( σ f r ee P c a l / σ b a s e li n e − ) [ % ] FIG. 3. Impact of freeing P cal on the error bars of cosmologicalparameters for the SPT pol TE,EE data . We show the ratio of theerror bars obtained letting the P cal parameter free to vary, over theones obtained using the baseline SPT pol settings, in units of percent, σ freePcal /σ baseline − Λ CDM modeland a few of its extensions. Only the constraints on ln(10 A s ) aresignificantly weakened by letting P cal free to vary. between ln(10 A s ) and P cal . The correlation is 84% for the Λ CDM case, as suggested in Fig. 1. All of the rest of the pa-rameter uncertainties increase by (cid:46)
10% when marginalizingover the broadened P cal posterior space. We show in Sec. VIthat the degradation in A s disappears if we include the tem-perature spectrum measurement TT as part of the input. Thisis because TT tightly constrains A s independent of P cal . Fordata sets similar to SPT pol , not only are the constraints on P cal precise enough for cross-checks with other approaches,most cosmological parameter constraints are also minimallydegraded when no P cal priors are imposed.As one way of demonstrating consistency, we comparethe inferred P cal values from the TE-only and EE-only datasets when the rest of the parameters are fixed to the best-fit from the TE,EE joint fit in Λ CDM with the baseline P cal prior. The marginalized P cal are P cal = . ± .
020 and P cal = . ± .
005 for the TE and the EE data set respec-tively. This shows that the individual data set does not prefera statistically di ff erent P cal ; there is no significant systematicresiduals that project onto P cal . We define the correlation between two parameters x , y as ρ x , y = cov( x , y ) / (cid:112) cov( x , x )cov( y , y ), with cov( x , y ) the elements of the parametercovariance matrix. .
95 1 .
00 1 . P cal . . . A L . . . l n ( A s ) . . . ln(10 A s ) . . . A L LCDM+ A L (no P cal prior)LCDM+ A L (with P cal prior)LCDM (no P cal prior) FIG. 4. A s , P cal , and A L posteriors with and without A L free for theSPT pol TE,EE data set. The uncertainty on A L is unchanged withand without the P cal prior, ensuring the strong A L constraint frompolarization-only spectra even when freeing P cal . C. The A L case We now turn to one particularly interesting parameter ex-tension, A L , a non-physical parameter that tunes the e ff ectof gravitational lensing on the CMB primary spectra, chang-ing the amount of smoothing of its peaks and troughs [26].In Planck , an excess peak smoothing was observed in theirtemperature power spectrum at the 2.8 σ level compared withthe Λ CDM expectation [24]. One key way of di ff erentiatingwhether the excess smoothing is a statistical fluctuation, theresult of unmodeled systematic errors, or new physics is totest if this trend persists in polarization [e.g. 20]. One concernof our method would be that marginalizing over the no-prior P cal would degrade the A L constraint enough that one can nolonger tell if the polarization data show similar trends. Asshown in Fig. 4, the constraints on A L for TE,EE with andwithout P cal priors are almost identical, retiring related con-cerns.As an aside, we note that the lensing information from peaksmoothing reduces what would otherwise be almost completedegeneracy between P cal and A s in the TE-only and EE-onlycases. Specifically, because peak smoothing provides a sec-ond handle for measuring A s , the Λ CDM TE-only and EE-only constraints on P cal are 21% and 13% respectively (asshown in the red and yellow contours in Fig. 1). By contrast,when the peak-smoothing information is absorbed by the ad-ditional parameter A L , the P cal constraints degrade by morethan a factor of two in both TE-only and EE-only cases. IV. ACTPOL
We apply this method to the recent ACTpol DR4 dataset on the frequency-combined CMB-only spectra, using the
ACTPollite likelihood [20, 27]. We note that the flat priorapplied on y P , the ACTpol polarization calibration parameter,is su ffi ciently broad ([0.9, 1.1]) that it is already allowing y P to float to that extent. Here we estimate how well this ACT-pol data set can constrain polarization calibration using justthe TE and EE spectra, with the prior on y P further widened.We also check if the TE,EE y P result is consistent with theTT,TE,EE y P result.We use only the TE and EE frequency-combined spectrawithout TT on both the wide and the deep patch. We thentransform the y P samples by applying an inverse to match the P cal definition, P cal = / y P . With this setup, we find P cal = . ± . Λ CDM model. It is consistent with the y P result from [20], which includes the TT spectra, of y P = . ± . V. PLANCK
A. Data and model description
In this section, we test whether jointly fitting the
Planck
TE and EE spectra with no prior on P cal would produce suf-ficiently precise P cal measurements to serve as useful cross-checks for other approaches. We also test the level of impactof this approach on the uncertainties on cosmological param-eters.In Planck , polarization e ffi ciencies, as well as polarizationangles, were measured on the ground in [28] and taken intoaccount in the map-making algorithm SRoll [9]. At thefrequencies used in the high-multipole likelihood (100, 143,217 GHz), polarization e ffi ciencies per detector were found tobe between 83% and 96%, with estimated uncertainties be-tween 0 . .
3% at the map level. However, tests per-formed on the maps, which compared strongly emitting po-larized galactic dust regions as observed by di ff erent detec-tors, suggested that residual polarization e ffi ciency errors areseveral times larger than the expected uncertainties reportedin [28], as shown in [9]. Left uncorrected, these residuals inthe polarization e ffi ciencies can impact cosmological parame-ters up to fractions of a sigma by biasing the overall amplitudeof the TE and EE spectra used in the high-multipole likeli-hood.In order to correct for this e ff ect, e ff ective polarizationcalibrations were estimated by the Planck collaboration bycomparing the TE and EE power spectra at 100, 143, and217 GHz to fiducial TE and EE spectra computed from the Λ CDM best-fit to the TT data. Polarized galactic contamina-tion was cleaned using information from the 353 GHz chan-nel [7]. The fits were performed on a limited range of mul-tipoles ( (cid:96) = − ff ected by fore-ground cleaning or noise uncertainties and over about ∼ ffi ciency correctionsfound in this way depend on the cosmological model fitted tothe temperature data (although this was tested to have a smallimpact). This method enabled determinations of the polariza-tion calibration for EE with uncertainties below (cid:46) .
5% at themap level ( (cid:46)
1% at power spectrum level) and for TE withuncertainties below (cid:46)
1% ( (cid:46)
Planck likelihood. Upto a global polarization calibration, the derived c EE ν s werefound to be consistent with the results of the component sep-aration algorithm SMICA [9], which measures relative inter-frequency calibration ratios between foreground-cleaned po-larization maps. Furthermore, in [7], it was noted that theestimates obtained separately from EE and TE should agreegiven the same polarization maps. However, the two measure-ments were found to di ff er by up to 1 . ±
1% at the map levelat 143 GHz (see Section 3.3.4 of [7]). As we will show below,this di ff erence cannot be reconciled by the approach we pro-pose in this work—leaving polarization e ffi ciencies to freelyvary. Since the di ff erence in polarization calibration from TEand EE is small enough that it could be caused by statisticalfluctuations, we leave the investigation of potential biases toparameters to future work and focus on the constraints on P cal given the Planck data set and impact on cosmological param-eters.We consider the 2018 final release of the
Planck data [7].We use the low-multipole likelihood in polarization
SimAll ( (cid:96) = −
29 in EE only), which we will refer to as “lowE.” Forhigh multipoles, we use the
Plik likelihood ( (cid:96) = − Commander likelihood at low- (cid:96) ( (cid:96) = − Plik at high- (cid:96) ( (cid:96) = − (cid:96) likelihoods, because their impact on low- (cid:96) spectra are negligible compared to cosmic variance, noise, andsystematic uncertainties in this multipole range. In the base-line Planck results using the
Plik likelihood, the polariza-tion calibration to the TE and EE spectra are fixed to the onesobtained from comparing the EE spectra at di ff erent frequen-cies to the Λ CDM best-fit of the TT + lowE data combination.These baseline parameters are listed in Table II. B. Main results and robustness assessment
We first discuss the uncertainties on P cal for the Planck dataset when it is free to vary. Using TE,EE + lowE, we find onecan determine the polarization calibrations with uncertaintiessmaller than ∼
1% at the map level. More specifically we finduncertainties of 0 . .
6% and 0 .
8% at the map level for ν = , ,
217 GHz respectively (corresponding to 1 . .
2% and 1 .
7% at the power spectrum level). Furthermore,we compare these uncertainties to the ones obtained with theTT power spectra included. We find that the error bars shrinkby almost a factor of 2 to 0 . .
31% and 0 .
51% at themap level for the three frequencies and similarly at the power-spectrum level. The measurements and uncertainties are re-
TABLE II. Polarization calibrations at power spectrum level obtainedfrom
Planck data assuming di ff erent cosmological models. We alsoreport the corresponding polarization calibrations at map level ( P cal = √ c EE , σ ( P cal ) ∼ ( c EE ) − . σ ( c EE ) / . ), to ease the comparison withthose obtained for SPT in Section III. The column ”baseline” liststhe fixed values used in the baseline Planck likelihood, which weredetermined with an uncertainty of ∼
1% at the power-spectrum level( ∼ .
5% at the map level).Parameter
Planck
TE, EE + lowE Planck
TT,TE, EE + lowE baseline Λ CDM c EE ± ± c EE ± ± c EE ± ± P cal EE ± ± P cal EE ± ± P cal EE ± ± Λ CDM + A L c EE ± ± c EE ± ± c EE ± ± P cal EE ± ± P cal EE ± ± P cal EE ± ± Λ CDM + N e ff c EE ± ± c EE ± ± c EE ± ± P cal EE ± ± P cal EE ± ± P cal EE ± ± ported in Tab. II and shown in Fig. 5. With and without TT,the uncertainties on the P cal factors are comparable to onesused in the Plik likelihood. This demonstrates that this ap-proach yields relevant constraints on P cal for cross-checks ofother approaches.In Tab. II, we observe shifts in the mean values of the po-larization calibrations when TT are added to TE and EE. Tocheck that the shifts are consistent with statistical fluctuations,we employ the formalism described in [29], which is appli-cable for comparing two data sets in which one is a subsetof the other. We find that the observed shifts are consistentwith statistical fluctuations at better than the 2 σ exp level, with σ exp = (cid:113) σ , EE − σ , TE , EE . Finally, we note that the meanvalues recovered from the TT,TE,EE combination are slightlydi ff erent from the ones used in the baseline because of statis-tical fluctuations due to the di ff erent multipole range and skymask used in the two cases (see also the discussion in section3.7 of [7]).We further check how much the constraints degrade whenwe exclude the cross-frequency spectra and only use the com-bination of the TE and EE frequency auto-spectra 100 × × ×
217 GHz. We find in this case compa-rable constraints on polarization calibrations to our baselineresults. Furthermore, if we include TE and EE from only onefrequency instead of all three as in our previous cases, i.e.,we use only the 100 × × ×
217 GHz Λ C D M A L N e ff Σ m nu Pcal EE Pcal EE Pcal EE Planck TE,EE Λ C D M A L N e ff Σ m nu Pcal EE Pcal EE Pcal EE Planck TT,TE,EE
FIG. 5. Marginal mean and 68% confidence level error bars on thethree
Planck P cal frequency parameters when they are let free tovary assuming di ff erent cosmological models. The top plot showsthe results for Planck
TE,EE, while the bottom one shows
Planck
TT,TE,EE. Estimates on the P cal parameters do not change signifi-cantly when varying the cosmological model. power spectra, the uncertainties of the polarization calibra-tions worsen to 1 . , .
75% and 2 .
1% at the map level (2 . , .
5% and 4 .
1% at the power spectrum level) respectively. Thelarge increase in uncertainty for the 217 ×
217 GHz case is be-cause of the more restrictive (cid:96) range of 500 − P cal parametersat di ff erent frequencies and the most degenerate cosmologicalparameter, ln(10 A s ). When using TE,EE + lowE, ln(10 A s )has a ∼
40% correlation with each of the three P cal parameters.The second most degenerate parameter is Ω b h ( ∼
30% cor-relation), while all other parameters have smaller correlations.As can be expected, we also find the degeneracies amongst the P cal parameters to be large: ρ c EE , c EE = ρ c EE , c EE = ρ c EE , c EE = P cal parameters to float, we showthe fractional di ff erence in Λ CDM parameter uncertaintiesin Fig. 7 for TE,EE and Fig. 8 for TT,TE,EE. Similar towhat we see in SPT pol , we observe negligible to mild degra-dation in Λ CDM parameter uncertainties besides those forln(10 A s ), given the correlations between the P cal parame-ters and ln(10 A s ). For the TE,EE data set, the uncertainty ofln(10 A s ) increases by ∼
20% when the P cal parameters areallowed to float. Once TT is included, which independentlyconstrains ln(10 A s ), we see that floating P cal has negligibleimpact on all Λ CDM parameters. We will see similar trendsin our forecasts in Sec. VI.
C. Extended models
We now turn to extensions to the Λ CDM model. Sim-ilar to Sec. III, we check the constraints on P cal for threeextensions, A L , (cid:80) m ν , and N e ff . The P cal uncertainties areshown in Fig. 5 for TE,EE and TT,TE,EE. We see that in allcases, the uncertainties of the P cal parameters are similar tothose in Λ CDM. As for the cosmological parameter uncer-tainties, Figures 7 and 8 show the increase in their error barswhen marginalizing over polarization calibration parametersfor
Planck
TE,EE and TT,TE,EE respectively.The parameter uncertainties in Λ CDM + N e ff are little af-fected, with increases in the error bars by less than 15%. Onthe contrary, we find a somewhat larger e ff ect on parameteruncertainties in the Λ CDM + (cid:80) m ν model for the TE,EE data.In this case, marginalizing over P cal increases the upper limiton (cid:80) m ν by almost 40%, while degrading the uncertainties on H and σ by almost 30%. We note that the main source caus-ing the degradation in (cid:80) m ν does not come from a drastic in-crease in posterior uncertainty given the degeneracy between (cid:80) m ν and P cal . The main e ff ect rather comes from a shift in thebest-fit values of correlated parameters (cid:80) m ν , ln(10 A s ), and P cal . For this data set, TE dominates the fit and causes (cid:80) m ν and ln(10 A s ) to be anti-correlated. With P cal free, the best fitfor ln(10 A s ) shifts to lower values by about 0 . σ . Thus, alower value of ln(10 A s ) induces a shift of the (cid:80) m ν posteriordistribution to higher values. Since this distribution is single-tailed with (cid:80) m ν >
0, this shift is perceived as a change in theupper bounds. These degradations disappear once the TT datais included, because TT strongly constrains ln(10 A s ). While c EE c EE c EE ln(10 A s ) c EE c EE c EE Planck TE,EE, free P cal Planck TT,TE,EE, free P cal A L c EE c EE c EE ln(10 A s ) A L c EE c EE c EE Planck TE,EE, free P cal Planck TE,EEPlanck TT,TE,EE, free P cal Planck TT,TE,EE
FIG. 6. One- and two-dimensional posterior distributions of the polarization e ffi ciency parameters and cosmological parameters for Planck
TE,EE. The left panel shows the results for the Λ CDM model, while the right panel shows results for the Λ CDM + A L model. this shift could be due to either a statistical fluctuation or asystematic error, it highlights the impact of P cal on constrain-ing (cid:80) m ν .For the Λ CDM + A L model, it was noted in Planck that the A L parameter is high compared to the Λ CDM expectation—at the 2.8 σ or 2.1 σ levels for polarization calibrations es-timated using Plik ’s baseline or estimated using separatefits of TE and EE respectively, as already described abovein Sec. V A. Here we show that leaving the polarization cal-ibrations free to vary cannot alleviate the di ff erence betweenthese two results. This is due to the fact that the di ff erencebetween the two Planck estimates of polarization e ffi ciencyfrom TE alone or from EE alone ( ∆ P cal ∼ .
017 at 143 Ghzat map level) is larger than the P cal posterior width when P cal is free to vary when fitting the TE,EE or TT,TE,EE data( σ ( P cal ) (cid:46) . P cal mean values measuredfrom these fits are in good agreement with those of the base-line estimates. Therefore, leaving P cal free to vary providesresults which are similar to the baseline case. Specifically,using the TE,EE + lowE data set, the A L parameter best fit is A L = . ± .
13, which is within 0 . σ exp of the value ob-tained when fixing P cal in the baseline case, A L = . ± . These results refer to the baseline data combination TT,TE,EE + lowE + CMBlensing. Note that the A L parameter only impacts the amplitude of lensing inthe TT,TE,EE power spectra, while it leaves the Planck
CMB lensing recon-struction power spectrum unaltered. This is not surprising since the P cal fits obtained from the TE,EE or TT,TE,EEdata are dominated by EE, which is also the data set used for the baselineestimates. also including TT, varying the polarization calibrations leadsto A L = . ± . P cal fixed A L = . ± .
068 (see also the dis-cussion in section 3.7 of [7]). Thus, leaving the polarizationcalibrations free to vary has a very small impact on the valueand error bar of the A L parameter, which remains higher thanunity at the 2 . σ level, due to the tight constraint provided bythe TE,EE or TT,TE,EE data combinations which agree withthe baseline estimate. For the same reason, the other cosmo-logical parameters are little a ff ected as well. VI. FORECASTS
In this section, we forecast how well P cal could be measuredwith our method and the impact on cosmological parameteruncertainties when marginalizing over P cal for ongoing andfuture experiments. We consider two experiment configura-tions: SPT-3G, the third-generation camera currently installedon the South Pole Telescope [13, 30], and CMB-S4, a next-generation ground-based CMB experiment [17]. A. SPT-3G
The SPT-3G receiver observes in three frequency bands95, 150, and 220 GHz in both intensity and polarization with ∼ ∼ of the sky in its mainsurvey field. The full-width half-maximum of the beams areapproximately 1.7, 1.2, and 1.1 arcminutes at 95, 150, and220 GHz respectively. The first science results from SPT- Λ C D M A L N e ff Σ m nu Ω b h Λ C D M A L N e ff Σ m nu Ω c h θ MC n s ln(10 A s ) H σ A L N eff m ν , 95% Planck TE,EE ( σ f r ee P c a l / σ b a s e li n e − ) [ % ] FIG. 7. Same as Figure 3, but for the
Planck
TE,EE data. Freeing the
Planck P cal parameters for this data combination has a large impactonly in the Λ CDM +Σ m ν case, where the 95% confidence level upperlimit on the sum of neutrino masses Σ m ν and the error bars on derivedparameters H and σ are increased by 30 − A s ), rather than an increase indegeneracies between parameters, see Sec. V C.
3G using TE and EE spectra measured using data collectedin 2018 have recently been released [23]. However, the datawere only collected for half of the observing season with partof the focal plane operable. Therefore, for this forecast, weuse noise level projections starting from 2019 when the activedetector count nearly doubled. With five seasons of obser-vations on the main survey field (2019–2023 inclusive), thenoise levels in the final coadded temperature maps are pro-jected to be 3.0, 2.2, and 8.8 µ K arcmin in the three frequencybands, and those in the polarization maps are a factor of √ P cal constraints along with constraints on Λ CDM and extension parameters for SPT-3G for two scenar-ios. First, we use data from only one of the three frequencybands, 150GHz, for more direct comparison with SPT pol , de-scribed in Sec. III, and to verify the impact of using only onefrequency channel. Second, we report the constraints whencombining maps from all three bands.We use the Fisher Matrix formalism and code describedin [6] for extracting the 1- σ parameter uncertainties. As in-puts, we use lensed power spectra of TT, TE, and EE; wedo not include the lensing reconstruction spectrum C φφ L . Wepresent constraints from the combination of TE and EE as abaseline and also those including all three spectra to study thee ff ect of including TT. We restrict the power spectrum angular Λ C D M A L N e ff Σ m nu Ω b h Λ C D M A L N e ff Σ m nu Ω c h θ MC n s ln(10 A s ) H σ A L N eff m ν , 95% Planck TT,TE,EE ( σ f r ee P c a l / σ b a s e li n e − ) [ % ] FIG. 8. Same as Figure 3, but for the
Planck
TT,TE,EE data. Freeingthe three
Planck P cal frequency parameters has a very minor impacton the cosmological parameter error bars, smaller than 15%, in allthe cosmological models considered here. multipole range to (cid:96) = − σ ( τ ) = . / f noise or marginalizing over foregrounds do not changethese results substantially.Table III shows results for the Λ CDM case. The SPT-3GTE and EE combination is projected to constrain P cal at thelevel of ∼ . P cal ,the constraint on ln(10 A s ) is degraded by about 50% whilethe rest of the Λ CDM parameters are mildly a ff ected (belowthe 15% level). Similar to what is seen in the Planck case, thedegraded constraints can be recovered by adding the TT data.In this case, marginalizing over P cal has negligible impact oncosmological parameters and the constraint on P cal tightens to0 . P cal are obtainedin extensions of the Λ CDM model, such as Λ CDM + N e ff , Λ CDM + A L or Λ CDM + (cid:80) m ν , for both the TE + EE and theTT + TE + EE data combination. As for the cosmological pa-rameters, we highlight here the ones with constraints de-graded when marginalizing over P cal . In Λ CDM + (cid:80) m ν , theln(10 A s ) uncertainty increases by 40% for the TE + EE datacombination. In Λ CDM + N e ff , the ln(10 A s ) uncertainty in-creases by 70% and the uncertainties on Ω b h and H increaseby ∼ Λ CDM case, when includ-ing the TT data, the marginalization over P cal has minimal im-pact on the constraints on cosmological parameters.0 TABLE III. Fisher matrix forecast on cosmological parameters and P cal for SPT-3G, using the 150 GHz channel alone or all of the threechannels. As a comparison, we also show constraints when fixing P cal . Ω b h Ω c h H τ ns ln [10 As ] P cal [ × − ] [ × − ] [ × − ] [ × − ] [ × − ] [ × − ] [ × − ] Λ CDMSPT-3G TE + EE 150GHz 1.4 2.0 7.5 6.6 8.0 1.3SPT-3G TE + EE 1.3 1.9 7.1 6.6 7.7 1.3SPT-3G TT + TE + EE 1.4 1.7 6.5 6.4 7.4 1.2 Λ CDM + P cal SPT-3G TE + EE 150GHz 1.6 2.1 8.0 6.6 8.2 2.0 7.6SPT-3G TE + EE 1.5 2.0 7.7 6.6 7.9 1.9 7.4SPT-3G TT + TE + EE 1.4 1.8 6.8 6.4 7.4 1.2 2.1
B. CMB-S4
CMB-S4 is a next-generation ground-based CMB experi-ment aiming to observe ∼
70% of the sky. It is planned tohave a frequency coverage from 20 to 270 GHz and the full-width half-maximum of its beam at 150 GHz is (cid:46) P cal given thewide survey from Chile. We use noise curves from [31],which combine information from all frequencies using an in-ternal linear combination method. The per-frequency noiseinput includes atmospheric noise; the output noise curvesinclude residuals from component separation. We assume f sky = .
42, which excludes the area covering the galaxy inthe wide survey. As in the forecast for SPT-3G, we use lensedpower spectra in the multipole range of (cid:96) = − C φφ L .Table IV shows results for the Λ CDM case. We find thatwith just TE and EE, CMB-S4 data could constrain P cal at thelevel of ∼ . . P cal , constraints on cosmological pa-rameters are mildly degraded without TT, and negligibly de-graded with TT. As in the previous sections, we verify thatextending the Λ CDM model with (cid:80) m ν , N e ff , and A L does notsignificantly change the constraints on P cal . Conversely, leav-ing the P cal parameter free has the largest impact on the con-straints on Ω b h , H and ln(10 A s ) in the Λ CDM + N e ff modelfor TE + EE, at the level of 30%. Similarly to previous cases,including the TT data allows us to marginalize over P cal withno loss of precision on cosmological parameters. VII. CONCLUSIONS
In this paper, we demonstrate that e ff ective polarization cal-ibrations P cal could be precisely determined by fitting CMBTE and EE spectra to the Λ CDM model and its common ex-tensions with P cal as a free parameter. This is possible thanksto the di ff erent dependence of the TE and EE spectra on P cal .While allowing P cal to float does increase the posterior vol-ume and therefore degrades some constraints on cosmological parameters, we show that the degradation becomes negligibleonce TT is included.We apply the method to SPT pol and Planck . For the SPT- pol
150 GHz TE and EE data set presented in H18, we ex-tract P cal with an uncertainty of ∼
2% at the map level, inde-pendent of the considered models. For the data set from the
Planck P cal at 100, 143, and 217 GHz with uncertainties of0 . .
6% and 0 .
8% at the map level. We highlight how thismethod can be useful for detecting inconsistencies in the data.In particular, P cal determined using TE and EE should agreewith the ones determined with TT included or the ones mea-sured from external data sets. If not, this could suggest theexistence of unaccounted for systematics which project intothese multiplicative factors.Finally, we forecast the capabilities of current and futureexperiments to constrain P cal . We find that using its 3 fre-quency channels, SPT-3G will be able to measure P cal withan uncertainty of 0.7% from TE and EE, and the uncertaintycan be improved to 0.2% when including TT. We find thatleaving P cal free to vary will degrade the constraints on A s from TE and EE by 30%, while constraints from TT,TE,EEare not a ff ected. Furthermore, we find that CMB-S4 couldfurther tighten the uncertainty of P cal to 0.2% with its TE andEE measurements and to 0.06% with TT,TE,EE. Similarly toSPT-3G, while constraints on A s are a ff ected by the variationof P cal by about 20% when using TE,EE, the constraints fromTT,TE,EE are una ff ected.We highlight that these uncertainties on P cal are compara-ble to or tighter than those derived for the Planck baseline( ∼ . Planck to calibratepolarization maps will ultimately limit the accuracy of theseexperiments, provided that the
Planck uncertainty is folded inthe power spectrum covariance matrix. Furthermore, if theexternal P cal determination is biased and the systematic un-certainties are not properly included, cosmological parame-ters constraints would be biased. We observe a possible hintof this in the Planck
TE,EE (cid:80) m ν upper limits between the(baseline) fixed P cal case and the free P cal case. For Planck however, the di ff erence of (cid:80) m ν upper limits due to a shift inthe P cal values is still compatible with a statistical fluctuation.For future experiments, we emphasize that stringent controlon P cal is important for accurate and precise inference on cos-mological parameters, such as the sum of neutrino masses.1 TABLE IV. Fisher matrix forecast on cosmological parameters and P cal for CMB-S4. As a comparison, we also show constraints when notvarying the P cal . Ω b h Ω c h H τ ns ln [10 As ] P cal [ × − ] [ × − ] [ × − ] [ × − ] [ × − ] [ × − ] [ × − ] Λ CDMCMB-S4 TE + EE 0.36 0.71 2.7 5.1 2.5 0.88CMB-S4 TT + TE + EE 0.36 0.67 2.5 4.9 2.3 0.85 Λ CDM + P cal CMB-S4 TE + EE 0.42 0.75 2.9 5.1 2.5 1.0 2.0CMB-S4 TT + TE + EE 0.37 0.70 2.6 4.9 2.3 0.86 0.56
Beyond using the primary CMB spectra TT, TE, and EE, weacknowledge the possibility of adding lensing potential powerspectrum measurements to further tighten constraints on P cal and reduce degradations in cosmological parameters. Weleave this for future work. In conclusion, this paper demon-strates that a significant source of systematic error for futureCMB polarization experiments can be self-calibrated withoutmajor consequences on the constraints on cosmological pa-rameters. ACKNOWLEDGMENTS
We thank the participants of the CMB systematics and cal-ibration focus workshop hosted virtually by Kavli IPMU for helpful comments and feedback. This work was completed inpart with resources provided by the University of Chicago’sResearch Computing Center. This work has received fundingfrom the French Centre National d’Etudes Spatiales (CNES).This research used resources of the IN2P3 Computer Center( http://cc.in2p3.fr ). WLKW is supported in part bythe Kavli Institute for Cosmological Physics at the Univer-sity of Chicago through grant NSF PHY-1125897, an endow-ment from the Kavli Foundation and its founder Fred Kavli,and by the Department of Energy, Laboratory Directed Re-search and Development program and as part of the PanofskyFellowship program at SLAC National Accelerator Labora-tory, under contract DE-AC02-76SF00515. TC is supportedby the the National Science Foundation through South PoleTelescope grant OPP-1852617. [1] C. Bennett et al. (WMAP), Astrophys. J. Suppl. , 20 (2013),arXiv:1212.5225 [astro-ph.CO].[2] N. Aghanim et al. (Planck), Astron. Astrophys. , A95(2017), arXiv:1608.02487 [astro-ph.CO].[3] Y. Akrami et al. (Planck), (2018), arXiv:1807.06205 [astro-ph.CO].[4] T. Louis et al. (ACTPol), JCAP , 031 (2017),arXiv:1610.02360 [astro-ph.CO].[5] K. Story et al. (SPT), Astrophys. J. , 86 (2013),arXiv:1210.7231 [astro-ph.CO].[6] S. Galli, K. Benabed, F. Bouchet, J.-F. Cardoso, F. Elsner,E. Hivon, A. Mangilli, S. Prunet, and B. Wandelt, Phys. Rev. D , 063504 (2014), arXiv:1403.5271 [astro-ph.CO].[7] N. Aghanim et al. (Planck), Astron. Astrophys. (2020),10.1051 / / (2020),10.1051 / / , 1141(2010), arXiv:0906.4069 [astro-ph.CO].[11] B. Koopman et al., Proc. SPIE Int. Soc. Opt. Eng. , 99142T (2016), arXiv:1607.01825 [astro-ph.IM].[12] F. Nati, M. J. Devlin, M. Gerbino, B. R. Johnson, B. Keating,L. Pagano, and G. Teply, J. Astron. Inst. , 1740008 (2017),arXiv:1704.02704 [astro-ph.IM].[13] A. N. Bender et al., Proc. SPIE Int. Soc. Opt. Eng. ,1070803 (2018), arXiv:1809.00036 [astro-ph.IM].[14] Y. Inoue et al. (POLARBEAR), Proc. SPIE Int. Soc. Opt. Eng. , 99141I (2016), arXiv:1608.03025 [astro-ph.IM].[15] S. M. Simon et al., Journal of Low Temperature Physics ,1041 (2018).[16] P. Ade et al. (Simons Observatory), JCAP , 056 (2019),arXiv:1808.07445 [astro-ph.CO].[17] K. Abazajian et al., (2019), arXiv:1907.04473 [astro-ph.IM].[18] J. Henning et al. (SPT), Astrophys. J. , 97 (2018),arXiv:1707.09353 [astro-ph.CO].[19] N. Aghanim et al. (Planck), Astron. Astrophys. (2020),10.1051 / / , 103511 (2002),arXiv:astro-ph / ,023003 (2014), arXiv:1401.7992 [astro-ph.CO].[26] E. Calabrese, A. Slosar, A. r. Melchiorri, G. F. Smoot, and O. Zahn, Phys. Rev. D , 123531 (2008), arXiv:0803.2309[astro-ph].[27] S. K. Choi et al. (ACT), JCAP , 045 (2020),arXiv:2007.07289 [astro-ph.CO].[28] C. Rosset et al., A&A , A13 (2010), arXiv:1004.2595[astro-ph.CO]. [29] S. Gratton and A. Challinor, arXiv e-prints , arXiv:1911.07754(2019), arXiv:1911.07754 [astro-ph.IM].[30] B. Benson et al. (SPT-3G), Proc. SPIE Int. Soc. Opt. Eng. ,91531P (2014), arXiv:1407.2973 [astro-ph.IM].[31] https://cmb-s4.org/wiki/index.php/Survey_Performance_Expectationshttps://cmb-s4.org/wiki/index.php/Survey_Performance_Expectations