Simultaneous determination of the cosmic birefringence and miscalibrated polarisation angles from CMB experiments
Yuto Minami, Hiroki Ochi, Kiyotomo Ichiki, Nobuhiko Katayama, Eiichiro Komatsu, Tomotake Matsumura
PProg. Theor. Exp. Phys. , 00000 (13 pages)DOI: 10.1093 / ptep/0000000000 Simultaneous determination of the cosmicbirefringence and miscalibrated polarisation angles from CMB experiments
Yuto Minami , Hiroki Ochi , Kiyotomo Ichiki , Nobuhiko Katayama , EiichiroKomatsu , and Tomotake Matsumura High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki305-0801, Japan ∗ E-mail: [email protected] Graduate School of Engineering Science, Yokohama National University, 79-5Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan Graduate School of Science, Division of Particle and Astrophysical Science, NagoyaUniversity, Chikusa-ku, Nagoya 464-8602, Japan Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, NagoyaUniversity, Chikusa-ku, Nagoya 464-8602, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU,WPI), Todai Institutes for Advanced Study, The University of Tokyo, Kashiwa277-8583, Japan Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85748Garching, Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We show that the cosmic birefringence and miscalibrated polarisation angles can bedetermined simultaneously by cosmic microwave background (CMB) experiments usingthe cross-correlation between E - and B -mode polarisation data. This is possible becausepolarisation angles of the CMB are rotated by both the cosmic birefringence and mis-calibration effects, whereas those of the Galactic foreground emission only by the latter.Our method does not require prior knowledge of the E - and B -mode power spectra ofthe foreground emission, but uses only the knowledge of the CMB polarisation spec-tra. Specifically, we relate the observed EB correlation to the difference between the observed E - and B -mode spectra in the sky, and use different multipole dependence ofthe CMB (given by theory) and foreground spectra (with no assumption) to derive thelikelihood for the miscalibration angle α and the birefringence angle β . We show thata future satellite mission similar to LiteBIRD can determine β with a precision of tenarcminutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index xxxx, xxx
1. Introduction
Cross-correlation between E - and B -mode polarisation of the cosmic microwave background(CMB) is sensitive to parity-violating physics in the Universe. One of the physical effects,known as “cosmic birefringence” [1–4], rotates CMB polarisation angles as CMB photons c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. a r X i v : . [ a s t r o - ph . C O ] A p r ropagate to us since last scattering at z ≈ EB correlation to solve for the miscalibration angle α [6], we may run into two issues: (1) welose sensitivity to the cosmic birefringence angle β ; and (2) we would infer a wrong value of α if there were non-vanishing β . In this paper, we mitigate this issue by using the polarisedGalactic foreground emission. Polarisation of the Galactic foreground is insensitive to thecosmic birefringence effect because of a limited propagation length of photons. We can distin-guish between α (affecting both CMB and foreground) and β (affecting only CMB) using thedifferent multipole dependence of the CMB and Galactic foreground polarisation power spec-tra. This is possible because the observed EB correlation is related to the difference betweenthe observed E - and B -mode power spectra of the sky (including CMB and foreground) [7],and we know the CMB power spectra well; thus, we can simultaneously determine α and β without prior knowledge of the foreground polarisation power spectra.Throughout this paper, we shall assume that β is uniform over the sky, as this is the casethat is degenerate with α . There are physical mechanisms to produce spatially-varying β [8–12], which can be estimated from CMB polarisation data with a suitable estimator [13].See refs. [14–16] for the current constraints on spatially-varying β .The rest of the paper is organised as follows. In Sect. 2 we describe our methodology forevaluating the posterior distribution of α and β . In Sect. 3 we validate our method usingsky simulations, and present the main results. We conclude in Sect. 4.
2. Methodology EB to the observed EE − BB When polarisation angles are rotated uniformly over the sky by an angle α , spherical har-monics coefficients of the observed E - and B -mode polarisation, denoted by “o”, are relatedto the intrinsic ones by E o (cid:96),m = E (cid:96),m cos(2 α ) − B (cid:96),m sin(2 α ) ,B o (cid:96),m = E (cid:96),m sin(2 α ) + B (cid:96),m cos(2 α ) . (1)Throughout this paper, we shall adopt the notation that all spherical harmonics coefficientsand power spectra have been multiplied by the appropriate beam transfer functions, unlessnoted otherwise.In this paper, we shall work with full-sky data without a mask, as we do not wish toremove the Galactic foreground. Our method can be adopted straightforwardly to work infractions of the sky (e.g., ground-based experiments), or with masks. In this paper we do not discuss the
T B correlation but focus only on the EB correlation, since alarge cosmic variance in T makes T B less sensitive than EB to α and β when the instrumental noiseis sufficiently low to measure B . In any case, it is straightforward to extend our method to include T B . 2/13 efining the power spectra as C XY(cid:96) = (2 (cid:96) + 1) − (cid:80) (cid:96)m = − (cid:96) X (cid:96),m Y ∗ (cid:96),m , we obtain [2–4] C EE, o (cid:96) = C EE(cid:96) cos (2 α ) + C BB(cid:96) sin (2 α ) − C EB(cid:96) sin(4 α ) , (2) C BB, o (cid:96) = C EE(cid:96) sin (2 α ) + C BB(cid:96) cos (2 α ) + C EB(cid:96) sin(4 α ) , (3) C EB, o (cid:96) = 12 (cid:0) C EE(cid:96) − C BB(cid:96) (cid:1) sin(4 α ) + C EB(cid:96) cos(4 α ) . (4)Using Eq. (2) and (3) in Eq. (4), we find C EB, o (cid:96) = 12 (cid:16) C EE, o (cid:96) − C BB, o (cid:96) (cid:17) tan(4 α ) + C EB(cid:96) cos(4 α ) . (5)This result was first derived in ref. [7] except for the EB term. We can use Eq. (5) to solvefor α with no assumption about the intrinsic C EE(cid:96) − C BB(cid:96) . Then, we no longer have to worryabout a bias in α induced by incorrect modelling of the intrinsic C EE(cid:96) − C BB(cid:96) , which wasstudied in ref. [17].Next, we include the cosmic birefringence angle β and noise (“N”), and write separatelythe foreground (“fg”) and CMB (“CMB”) components. We obtain E o (cid:96),m = E fg (cid:96),m cos(2 α ) − B fg (cid:96),m sin(2 α ) + E CMB (cid:96),m cos(2 α + 2 β ) − B CMB (cid:96),m sin(2 α + 2 β ) + E N (cid:96),m , (6) B o (cid:96),m = E fg (cid:96),m sin(2 α ) + B fg (cid:96),m cos(2 α ) + E CMB (cid:96),m sin(2 α + 2 β ) + B CMB (cid:96),m cos(2 α + 2 β ) + B N (cid:96),m . (7)From these coefficients, we can relate C EB, o (cid:96) to C EE, o (cid:96) − C BB, o (cid:96) as (cid:16) C EB, o (cid:96) − tan(4 α )2 (cid:16) C EE, o (cid:96) − C BB, o (cid:96) (cid:17)(cid:19) cos(4 α ) =( C EE,
CMB (cid:96) − C BB,
CMB (cid:96) ) sin(4 β ) / − ( C EE, N (cid:96) − C BB, N (cid:96) ) sin(4 α ) / C EB, fg (cid:96) + C EB, N (cid:96) cos(4 α ) + C EB,
CMB (cid:96) cos(4 β )+ ( C E fg B CMB (cid:96) + C E CMB B fg (cid:96) ) cos(2 β ) + ( C E fg E CMB (cid:96) − C B fg B CMB (cid:96) ) sin(2 β )+ ( C E fg B N (cid:96) + C E N B fg (cid:96) ) cos(2 α ) − ( C E fg E N (cid:96) − C B fg B N (cid:96) ) sin(2 α )+ ( C E CMB B N (cid:96) + C E N B CMB (cid:96) ) cos(2 α − β ) − ( C E CMB E N (cid:96) − C B CMB B N (cid:96) ) sin(2 α − β ) . (8)If we divide throughout by cos(4 α ), then among these terms the following do not vanishupon ensemble average: (cid:104) C EB, o (cid:96) (cid:105) = tan(4 α )2 (cid:16) (cid:104) C EE, o (cid:96) (cid:105) − (cid:104) C BB, o (cid:96) (cid:105) (cid:17) + sin(4 β )2 cos(4 α ) (cid:16) (cid:104) C EE,
CMB (cid:96) (cid:105) − (cid:104) C BB,
CMB (cid:96) (cid:105) (cid:17) + 1cos(4 α ) (cid:104) C EB, fg (cid:96) (cid:105) + cos(4 β )cos(4 α ) (cid:104) C EB,
CMB (cid:96) (cid:105) . (9)The last term, (cid:104) C EB,
CMB (cid:96) (cid:105) , is the intrinsic EB correlation at the last scattering surface. Thisterm could arise from, e.g., chiral gravitational waves [2, 18, 19], anisotropic inflation [20], tc. We can measure this signal if its multipole dependence is sufficiently different from thatof (cid:104) C EE,
CMB (cid:96) (cid:105) − (cid:104) C BB,
CMB (cid:96) (cid:105) [21]. Therefore, we shall focus on the cosmic birefringence termand ignore (cid:104) C EB,
CMB (cid:96) (cid:105) throughout this paper without loss of generality.For the moment, we shall also assume that the ensemble average of the intrinsic EB cor-relation of the foreground emission vanishes over the full sky, i.e., (cid:104) C EB, fg (cid:96) (cid:105) = 0. Of course, anon-zero EB arises from a statistical fluctuation in one realisation of our sky. The question iswhether a EB correlation is statistically significant compared to the cosmic-variance uncer-tainty. The current data show no evidence for non-zero EB correlation from the foregroundemission [22, 23]; however, it is still possible that more sensitive future experiments may finda statistically-significant EB correlation in the foreground. We shall show how to deal withthis term in Sect. 4. We determine α and β by fitting C EB, o (cid:96) with C EE, o (cid:96) − C BB, o (cid:96) and C EE,
CMB (cid:96) − C BB,
CMB (cid:96) usingEq. (9). To this end, we use a likelihood analysis. The log-likelihood function is given by − L = (cid:96) max (cid:88) (cid:96) =2 (cid:104) C EB, o (cid:96) − tan(4 α )2 (cid:16) C EE, o (cid:96) − C BB, o (cid:96) (cid:17) − sin(4 β )2 cos(4 α ) (cid:16) C EE,
CMB (cid:96) − C BB,
CMB (cid:96) (cid:17)(cid:105) Var (cid:16) C EB, o (cid:96) − tan(4 α )2 (cid:16) C EE, o (cid:96) − C BB, o (cid:96) (cid:17)(cid:17) . (10)The expression of variance in the likelihood is given in Appendix A. As we do notknow the C EE,
CMB (cid:96) and C BB,
CMB (cid:96) realised in our sky, we replace them by the best-fitting ΛCDM theoretical power spectra multiplied by the beam transfer functions,i.e., C EE,
CMB , th (cid:96) b (cid:96) and C BB,
CMB , th (cid:96) b (cid:96) , respectively. In principle, we could marginalisethe likelihood over the difference between C XX,
CMB (cid:96) and C XX,
CMB , th (cid:96) by adding (cid:104) ( C XX,
CMB (cid:96) − C XX,
CMB , th (cid:96) b (cid:96) ) (2 (cid:96) + 1) / (2 C XX,
CMB , th (cid:96) b (cid:96) ) (cid:105) with X = E or B . In practice, wefind that marginalisation has little effect on the uncertainty of α and β .We minimise Eq. (10) with respect to α and β , given C EB, o (cid:96) , (cid:0) C EE, o − C BB, o (cid:1) , C EE,
CMB , th (cid:96) ,and C BB,
CMB , th (cid:96) . Assuming flat priors on α and β , the likelihood gives the posteriordistribution of α and β .
3. Results
To validate our methodology, we use the “PySM” package [24] to produce realistic simula-tions of the microwave sky, with an experimental specification similar to the future satellitemission LiteBIRD [25]. Specifically, we include the polarised Galactic foreground emission(“s1” synchrotron model and “d1” dust emission, as described in ref. [24]), a CMB mapgenerated from the power spectra based on CAMB [26], and white noise with standard devi-ation given by σ N = ( π/ w − / /µ K arcmin) µ K str − / [27] with w − / given in the“Polarisation Sensitivity” column of Table 1.We use the HEALPix package [28] to generate maps with the resolution parameter N side =512. To incorporate beam smearing, the spherical harmonics coefficients of the CMB andforeground maps at each frequency are multiplied by a beam transfer function, b (cid:96) , for whichwe assume a Gaussian beam with full-width-at-half-maximum (FWHM) given in the thirdcolumn of Table 1. We calculate the power spectra up to (cid:96) max = 2 N side = 1024. able 1: Polarisation sensitivity and beam size of the LiteBIRD telescopes [25]
Frequency (GHz) Polarisation Sensitivity ( µ K (cid:48) ) Beam Size in FWHM (arcmin)40 37.5 6950 24.0 5660 19.9 4868 16.2 4378 13.5 3989 11.7 35100 9.2 29119 7.6 25140 5.9 23166 6.5 21195 5.8 20235 7.7 19280 13.2 24337 19.5 20402 37.5 17All of the results reported below will be derived from one realisation of the CMB, which iscommon to all frequencies, and one realisation of noise generated at each frequency. We donot generate many different realisations, since the foreground emission is always common tothose realisations and thus the uncertainty derived from the ensemble will miss the cosmic-variance contribution of the foreground emission. In our likelihood (Eq. 10), this foregroundcosmic-variance contribution is included in the variance term in the denominator, which isderived in Appendix A. We have checked that the foreground EB correlation in the PySMsimulations is consistent with zero within the cosmic variance. α and β First, we report the results with no cosmic birefringence by setting β = 0 in the simulationand fitting only α in the likelihood .In Figure 1, we show that our method recovers correctly the input values of α in from − ◦ to 3 ◦ in all frequency bands to within the uncertainties. We show the 1- σ uncertainties, σ ( α ), in the second column of Table 2 in units of arcminutes. The smallest uncertainty is σ ( α ) = 1 arcmin, which is achieved at 119 and 140 GHz.Next, we determine α and β simultaneously. We show two representative results: (1) withthe input miscalibration angles, α in , varied from − ◦ to 3 ◦ while β in = 0 (Figure 2); and(2) with the input birefringence angles, β in , varied from − ◦ to 3 ◦ while α in = 0 (Figure 3).We find that our method recovers correctly the input values of α in and β in in all frequencybands to within the uncertainties. When the foreground and the angle miscalibration are ignored, we can interpret α as the bire-fringence angle. See, e.g., refs. [29–34] and references therein for forecasts of the capability of futureexperiments to constrain the birefringence angle in this simplest case.5/13 α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
50 GHz α out = α in α out α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
78 GHz α out = α in α out α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
119 GHz α out = α in α out α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
195 GHz α out = α in α out α (deg.)32101 R e c o v e r e d a n g l e ( d e g . )
235 GHz α out = α in α out α (deg.)32101 R e c o v e r e d a n g l e ( d e g . )
337 GHz α out = α in α out Fig. 1:
Recovery of the miscalibration angle α in the absence of the cosmic birefringence (i.e., with β = 0).Each panel shows the recovered values of α out (red dots with error bars, in units of degrees) againstthe input values α in for 6 frequency bands out of 15 specified in Table 1. α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
50 GHz α out = α in β out = β in α out β out α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
78 GHz α out = α in β out = β in α out β out α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
119 GHz α out = α in β out = β in α out β out α (deg.)3210123 R e c o v e r e d a n g l e ( d e g . )
195 GHz α out = α in β out = β in α out β out α (deg.)32101 R e c o v e r e d a n g l e ( d e g . )
235 GHz α out = α in β out = β in α out β out α (deg.)32101 R e c o v e r e d a n g l e ( d e g . )
337 GHz α out = α in β out = β in α out β out Fig. 2:
Simultaneous determination of α and β with the input values α in varied from − ◦ to 3 ◦ and β in = 0.Each panel shows the recovered values of α out (red dots with error bars, in units of degrees) against α in . The blue stars with error bars show the recovered values of β out at each α in . β (deg.)3210123 R e c o v e r e d β ( d e g . )
50 GHz β out = β in α out = α in α out β out β (deg.)3210123 R e c o v e r e d β ( d e g . )
78 GHz β out = β in α out = α in α out β out β (deg.)3210123 R e c o v e r e d β ( d e g . )
119 GHz β out = β in α out = α in α out β out β (deg.)3210123 R e c o v e r e d β ( d e g . )
195 GHz β out = β in α out = α in α out β out β (deg.)32101 R e c o v e r e d β ( d e g . )
235 GHz β out = β in α out = α in α out β out β (deg.)32101 R e c o v e r e d β ( d e g . )
337 GHz β out = β in α out = α in α out β out Fig. 3:
Simultaneous determination of α and β with the input values α in = 0 and β in varied from − ◦ to3 ◦ . Each panel shows the recovered values of β out (blue stars with error bars, in units of degrees)against β in . The red dots with error bars show the recovered values of α out at each β in . able 2: Marginalised 1- σ uncertainties on α and β from the experimental specifications given in Table 1.The input values are α in = 0 and β in = 0 Frequency (GHz) α only case (arcmin) α and β (arcmin) σ ( α ) σ ( α ) σ ( β )40 8.3 16 7050 6.3 17 3160 4.9 20 2568 3.6 22 2478 2.6 21 2289 1.9 19 20100 1.2 17 17119 1.0 15 15140 1.0 12 13166 1.2 11 12195 1.5 9.6 11235 2.4 8.4 12280 4.6 8.3 26337 4.7 7.5 76402 4.4 7.0 4 . × We give the marginalised 1- σ uncertainties, σ ( α ) and σ ( β ), in the third and fourth columnsof Table 2 in units of arcminutes. The smallest σ ( β ) is 11 arcmin at 195 GHz (where σ ( α )is 9 . α and β increases the uncertainties on α significantly at all frequency bands.The angles α and β are expected to be correlated in any fit. When the power spectrumdata are dominated by the CMB, we can only determine a linear combination α + β . On theother hand, when the data are dominated by the Galactic foreground emission, we can onlydetermine α . Therefore, the foreground helps to break degeneracy between α and β , allowingus to determine them simultaneously. We show this in Figure 4. The cosmic birefringenceangle β is poorly constrained in the foreground-dominated frequency bands (lowest andhighest frequencies) relative to the miscalibration angle α , whereas the degeneracy given by α + β (dotted lines) is broken in between. Since α + β is constrained tightly by the CMB,adding the foreground yields similar uncertainties on both α and β . In other words, theaccuracy of β is determined by the accuracy of α provided by the foreground emission.This method can be applied not only to experiments focused on the CMB, but also tocalibrate polarisation angles of other experiments focused on the foreground emission, suchas those observing at lower and higher frequencies. When applied to low frequency data,it is straightforward to extend our method to incorporate the effect of Faraday rotation onaverage by rotating the foreground polarisation angle by an additional angle γ ∝ ν − , i.e., α → α + γ . Accurate subtraction of the Faraday rotation requires estimation of rotationangles per pixel rather than the full-sky average (see, e.g., refs. [34, 35] and referencestherein). ( d e g ) ( d e g ) ( d e g ) ( d e g ) ( d e g ) ( d e g ) Fig. 4:
Joint constraints on α (horizontal axis) and β (vertical axis) in units of degrees, from the experimentalspecifications given in Table 1. We show only 6 frequency bands out of 15. The contours show∆( − L ) = 2 .
30 (68.3% CL) and 6.17 (95.4% CL). The input values are α in = 0 and β in = 0 . ◦ .The black dotted lines show α + β = 0 . ◦ . Finally, it is possible to combine all frequency bands to obtain the best estimate of β withthe smallest uncertainty. To perform such an analysis, however, we must take into accountthe covariance between different frequency bands. While the instrumental noise is expectedto be uncorrelated to a good approximation, both the CMB and foreground emission arehighly correlated across different frequency bands. We leave the computation of the fulllikelihood combining all frequency bands to future work. . Discussion and Conclusions In this paper, we have shown that it is possible to determine the cosmic birefringence andmiscalibrated polarisation angles simultaneously from the observed EB cross-correlationpower spectrum, contrary to what has been usually assumed in the literature. The ideabehind our method is simple: the miscalibration angle α affects both CMB and the Galacticforeground emission, whereas the cosmic birefringence angle β affects only CMB.The key observation is that the EB correlation induced by the angle miscalibration isrelated to the difference between the observed EE and BB power spectra on the sky, asshown by ref. [7]. We can then use accurate knowledge of the CMB polarisation powerspectra from the best-fitting ΛCDM model to separately constrain α and β . To this end wehave derived the likelihood function given in Eq. (10), which yields the posterior distributionof α and β given the observational data and the best-fitting CMB model.Applying our method to simulated maps including realistic foreground emission [24] andCMB as well as instrumental noise and beam smearing similar to the future CMB mis-sion LiteBIRD [25], we find that the method successfully recovers the input values of α and β simultaneously, with the minimum uncertainty on the cosmic birefringence being σ ( β ) = 11 arcmin at 195 GHz. Therefore, CMB experiments are capable of constrainingparity-violating physics even when we use the EB correlation to self-calibrate polarisationangles as proposed by ref. [6].So far we have assumed that the intrinsic EB correlation of the foreground emissionvanishes when measured over the full sky, i.e., (cid:104) C EB, fg (cid:96) (cid:105) = 0 in Eq. (9). Let us now addressthe impact of a possible EB signal from the foreground. To model this, one could write (cid:104) C EB, fg (cid:96) (cid:105) = f c (cid:113) (cid:104) C EE, fg (cid:96) (cid:105)(cid:104) C BB, fg (cid:96) (cid:105) as in ref. [17]. As the foreground EE and BB power spectraare similar [23], let us write the BB spectrum as (cid:104) C BB, fg (cid:96) (cid:105) = ξ (cid:104) C EE, fg (cid:96) (cid:105) , where ξ is for exampleequal to (cid:104) A BB /A EE (cid:105) ≈ . (cid:104) C EB, fg (cid:96) (cid:105) = f c √ ξ − ξ (cid:16) (cid:104) C EE, fg (cid:96) (cid:105) − (cid:104) C BB, fg (cid:96) (cid:105) (cid:17) . (11)This result can be put into the same form as in the angle miscalibration, if we introducea new angle γ and write sin(4 γ ) / f c √ ξ/ (1 − ξ ), with 0 ≤ f c √ ξ/ (1 − ξ ) ≤
1. Therefore,the foreground emission is now rotated by α ( ν ) + γ ( ν ), whereas the CMB is rotated by α ( ν ) + β . Here, ν denotes an observing frequency band. We then use the difference betweenthe multipole dependence of the foreground and the CMB to determine β − γ ( ν ); thus, weneed to give up measuring β if we use only one frequency band. Fortunately, as the cosmicbirefringence effect is independent of frequency, we can distinguish between β and γ ( ν ) usingmulti-frequency data.While we have calculated the expected uncertainties on α and β from the future satellitemission LiteBIRD, we can use the same formalism to calculate those from ground-basedand balloon-borne experiments. This new framework allows us to enhance our ability toconstrain parity-violating physics from the CMB polarisation data. Acknowledgment
We thank Joint Study Group of the LiteBIRD collaboration for useful discussion and feed-back on this project, and A. Gruppuso, M. Hazumi, D. Molinari, P. Natoli, and D. Scottfor comments on the draft. We acknowledge the use of the MINUIT algorithm via the minuit Python interface. This work was supported in part by Japan Society for the Promo-tion of Science (JSPS) KAKENHI Grant Numbers JP15H05896, JP18K03616, JP16H01543,JP15H05891, and JP15H05890, and JSPS Core-to-Core Program, A. Advanced ResearchNetworks.
A. Variance in the likelihood
In this section, we derive the variance of C EB, o (cid:96) − ( C EE, o (cid:96) − C BB, o (cid:96) ) tan(4 α ) / C XY(cid:96) = (2 (cid:96) + 1) − (cid:80) m X (cid:96),m Y ∗ (cid:96),m and assum-ing Gaussianity, i.e., (cid:104) X (cid:96),m Y ∗ (cid:96),m X (cid:48)∗ (cid:96),m (cid:48) Y (cid:48) (cid:96),m (cid:48) (cid:105) = (cid:104) C XY(cid:96) (cid:105)(cid:104) C X (cid:48) Y (cid:48) (cid:96) (cid:105) + (cid:104) C XX (cid:48) (cid:96) (cid:105) δ mm (cid:48) (cid:104) C Y Y (cid:48) (cid:96) (cid:105) δ mm (cid:48) + (cid:104) C XY (cid:48) (cid:96) (cid:105) δ m − m (cid:48) (cid:104) C X (cid:48) Y(cid:96) (cid:105) δ m − m (cid:48) , we obtainVar (cid:104) C EB, o (cid:96) − ( C EE, o (cid:96) − C BB, o (cid:96) ) tan(4 α ) / (cid:105) = (cid:68)(cid:104) C EB, o (cid:96) − ( C EE, o (cid:96) − C BB, o (cid:96) ) tan(4 α ) / (cid:105) (cid:69) − (cid:104) C EB, o (cid:96) − ( C EE, o (cid:96) − C BB, o (cid:96) ) tan(4 α ) / (cid:105) = 12 (cid:96) + 1 (cid:104) C EE(cid:96) (cid:105)(cid:104) C BB(cid:96) (cid:105) + tan (4 α )4 22 (cid:96) + 1 (cid:0) (cid:104) C EE(cid:96) (cid:105) + (cid:104) C BB(cid:96) (cid:105) (cid:1) − tan(4 α ) 22 (cid:96) + 1 (cid:104) C EB(cid:96) (cid:105) (cid:0) (cid:104) C EE(cid:96) (cid:105) − (cid:104) C BB(cid:96) (cid:105) (cid:1) + 12 (cid:96) + 1 (cid:0) − tan (4 α ) (cid:1) (cid:104) C EB(cid:96) (cid:105) . (A1)Here, (cid:104) C XY(cid:96) (cid:105) is the sum of all terms including the CMB, foregrounds, and instrumental noise.Since we make no assumption about the foreground power spectra, we shall approxi-mate (cid:104) C XY(cid:96) (cid:105) with the observed spectra, C XY, o (cid:96) . This is a reasonable approximation at highmultipoles, where we have enough statistics. The fitting results are also dominated by theinformation at high multipoles. However, we find that the last term in Eq. (A1) causes aproblem: as the observed EB spectrum oscillates around zero due to the smallness of thesignal (if any) and the large scatter from cosmic variance, squaring it yields a biased estimateof the variance. Fortunately this term makes only a sub-dominant contribution to the totalvariance; thus, we shall ignore it from now on. The final formula is thenVar (cid:104) C EB, o (cid:96) − ( C EE, o (cid:96) − C BB, o (cid:96) ) tan(4 α ) / (cid:105) ≈ (cid:96) + 1 C EE, o (cid:96) C BB, o (cid:96) + tan (4 α )4 22 (cid:96) + 1 (cid:104) ( C EE, o (cid:96) ) + ( C BB, o (cid:96) ) (cid:105) − tan(4 α ) 22 (cid:96) + 1 C EB, o (cid:96) (cid:16) C EE, o (cid:96) − C BB, o (cid:96) (cid:17) . (A2) References [1] S. M. Carroll, Phys. Rev. Lett., , 3067–3070 (1998), arXiv:astro-ph/9806099.[2] A. Lue, L.-M. Wang, and M. Kamionkowski, Phys. Rev. Lett., , 1506–1509 (1999), arXiv:astro-ph/9812088.[3] B. Feng, H. Li, M. Li, and X. Zhang, Phys. Lett., B620 , 27–32 (2005), arXiv:hep-ph/0406269.[4] B. Feng, M. Li, J.-Q. Xia, X. Chen, and X. Zhang, Phys. Rev. Lett., , 221302 (2006), arXiv:astro-ph/0601095.[5] E. Komatsu et al., Astrophys. J. Suppl., , 18 (2011), arXiv:1001.4538.[6] B. Keating, M. Shimon, and A. Yadav, Astrophys. J., , L23 (2012), arXiv:1211.5734.[7] G.-B. Zhao, Y. Wang, J.-Q. Xia, M. Li, and X. Zhang, JCAP, (07), 032 (2015), arXiv:1504.04507.[8] A. Kosowsky and A. Loeb, Astrophys. J., , 1–6 (1996), arXiv:astro-ph/9601055.[9] S. Gardner, Phys. Rev. Lett., , 041303 (2008), arXiv:astro-ph/0611684.[10] M. Pospelov, A. Ritz, and C. Skordis, Phys. Rev. Lett., , 051302 (2009), arXiv:0808.0673.
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