Including birefringence into time evolution of CMB: current and future constraints
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Including birefringence into timeevolution of CMB: current andfuture constraints
G. Gubitosi , , M. Martinelli , L. Pagano Physics Department and INFN, Università di Roma “La Sapienza”, Ple Aldo Moro 2, 00185,Rome, Italy SISSA, Via Bonomea 265, Trieste, 34136, Italy Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, U.K.
Abstract.
We introduce birefringence effects within the propagation history of CMB, con-sidering the two cases of a constant effect and of an effect that increases linearly in time,as the rotation of polarization induced by birefringence accumulates during photon propa-gation. Both cases result into a mixing of E and B modes before lensing effects take place,thus leading to the fact that lensing is acting on spectra that are already mixed becauseof birefringence. Moreover, if the polarization rotation angle increases during propagation,birefringence affects more the large scales that the small scales. We put constraints on thetwo cases using data from WMAP 9yr and BICEP 2013 and compare these results with theconstraints obtained when the usual procedure of rotating the final power spectra is adopted,finding that this dataset combination is unable to distinguish between effects, but it never-theless hints for a non vanishing value of the polarization rotation angle. We also forecastthe sensitivity that will be obtained using data from Planck and PolarBear, highlighting howthis combination is capable to rule out a vanishing birefringence angle, but still unable todistinguish the different scenarios. Nevertheless, we find that the combination of Planck andPolarBear is sensitive enough to highlight the existence of degeneracies between birefringencerotation and gravitational lensing of CMB photons, possibly leading to false detection of nonstandard lensing effects if birefringence is neglected. a r X i v : . [ a s t r o - ph . C O ] S e p ontents Recent Cosmic Microwave Background (CMB) observations brought to more and more pre-cise measurements of temperature anisotropies reaching the almost cosmic variance-limitedsensitivity of Planck [1, 2]. While other surveys are focusing on reaching a similar sensitivityon smaller angular scales, e.g. ACT [3] and SPT[4], other CMB experiment were designed inorder to measure the CMB photons polarization properties. After the first detection of thepolarization E modes (parity-even modes) by the DASI interferometer [5], a higher sensitivitywas achieved by following experiments, such as WMAP [6], QUIET [7] and BICEP[8–10]. Up-coming surveys are now designed to achieve even more precise measurements of E-modes andto finally detect the parity-odd modes (B modes) of CMB polarization (see e.g. ACTpol[11],SPTpol[12], PolarBear[13] and EBEX[14]). The lensing B-modes have been already detectedcross-correlating a lensing template with CMB polarization maps, see e.g. [15, 16], whileBICEP team claimed primordial B modes detection [10], although at such scales polarizeddust signal must be taken into account, as pointed out in [17].These observations are crucial to detect signatures of the current standard cosmological modelsuch as the B modes induced by primordial gravitational waves and the leakage of power be-tween E and B modes due to weak gravitational lensing of CMB photons.Moreover, the precise measurement of CMB polarization allows also for tests of new physics,such as the search for CPT and Lorentz violations in the photons sector of particle physics[18]. In particular, some attention has been gained in the last few years by the search forsignals of birefringence, i.e. rotation of the photons polarization direction during in vacuo propagation (see e.g. [9, 19–31] and references therein), whose main effect on CMB photonsconsists in a mixing between E and B polarization modes. To investigate this phenomenon iscrucial also because of the possible contamination that birefringence can have on primordialgravitational wave detection [32, 33].A similar mixing is produced on CMB polarization by weak gravitational lensing and as up-coming surveys will improve our knowledge of this effect, considering these phenomena inthe right order is crucial. In fact, while CMB lensing performs its mixing at “recent” times,birefringence starts to take place right after recombination and it is expected to accumulateduring photons propagation. Therefore the CMB spectra which are modified by lensing do– 1 –ot encode only the effect of primordial anisotropies, but already contain the rotation effectdue to birefringence. However, most of previous works [20–22, 24–27, 29] apply the rotationdue to birefringence on the lensed CMB spectra. This procedure is correct only when oneexpects the polarization rotation not to be a genuine physical effect but to be due to miscal-ibration of the polarimeters [34, 35].In this paper we address this issue comparing the results obtained with currently avail-able datasets in both the early and late time rotation cases, using WMAP and the morepolarization-oriented survey BICEP. We also inquire about the possibility of future CMBsurveys to detect a non-zero birefringence effect or rather to rule it out exploiting forecasteddatasets and we investigate the possibility of future surveys to distinguish among differenttypes of polarization rotation.Furthermore, as birefringence modifies also the power spectrum of B modes, giving it anadditional contribution due to the leakage from the E modes, we also investigate the possibledegeneracies between birefringence parameters and CMB lensing. Indeed lensing effect onCMB spectra, parametrized by the lensing amplitude A L [36], also leads to a leakage from E to B modes, so neglecting the presence of birefringence can in principle produce a misleadingdetection of a non standard lensing effect ( A L (cid:54) = 1 ).The paper is organized as follows. In Section 2 we briefly review birefringence theory andits motivations, also describing how it affects CMB power spectra. In Section 3 we describethe performed analysis and the datasets used to constrain birefringence parameters. Generalresults are presented in Section 4 while we discuss them in the concluding Section 5. Birefringence is the rotation of linear polarization direction during the propagation of radia-tion in vacuo.The standard way birefringence is formalized in the literature is through a sudden ro-tation of the polarization after photon propagation from the last scattering surface to now.There are however exceptions to this, see for example [23, 41], where the amount of birefrin-gence depends on the evolution of a cosmological scalar field (see also [25] for a discussion onthe accuracy of this "sudden rotation" approximation).If the polarization direction rotates counterclockwise (looking at the source) of an angle β > , then the Stokes parameters Q and U get mixed in the following way : Q = ˜ Q cos 2 β + ˜ U sin 2 βU = ˜ U cos 2 β − ˜ Q sin 2 β (2.1) The Stokes parameters are defined in the standard frame used for CMB, see [38, 39], so that a counter-clockwise rotation of the polarization direction (looking at the photons coming toward us) corresponds to arotation of the reference frame from the ˆ x axis to the ˆ y axis. – 2 –nd as a consequence the power spectra become : C EE(cid:96) = ˜ C EE(cid:96) cos (2 β ) + ˜ C BB(cid:96) sin (2 β ) − ˜ C EB(cid:96) sin (4 β ) C BB(cid:96) = ˜ C EE(cid:96) sin (2 β ) + ˜ C BB(cid:96) cos (2 β ) + ˜ C EB(cid:96) sin (4 β ) C EB(cid:96) = 12 (cid:16) ˜ C EE(cid:96) − ˜ C BB(cid:96) (cid:17) sin (4 β ) + ˜ C EB(cid:96) (cid:0) cos (2 β ) − sin (2 β ) (cid:1) C T E(cid:96) = ˜ C T E(cid:96) cos (2 β ) − ˜ C T B(cid:96) sin (2 β ) C T B(cid:96) = ˜ C T E(cid:96) sin (2 β ) + ˜ C T B(cid:96) cos (2 β ) (2.2)The ˜ C (cid:96) are the spectra in absence of polarization rotation (no birefringence), while the C (cid:96) are the observed spectra and we allowed for the presence of non zero parity-violating cross-correlation spectra before the rotation occurs, i.e. ˜ C EB(cid:96) and ˜ C T B(cid:96) .This way of treating polarization rotation is exact when considering the modificationof the spectra that one would expect from a systematic miscalibration of the polarimeters[34, 35], as in this case one would have a genuine effect on the final spectra. However,when dealing with birefringence as the effect of some new physics phenomenon, eqs. (2.20)can only be considered as an approximation. In fact birefringence is a phenomenon due toanomalous photon propagation [23–25, 40, 41, 43], that accumulates during propagation fromlast scattering to now. In this case the amount of rotation is time dependent, given by α ( η ) as a function of conformal time from last scattering η . Depending on the model considered,the actual form of the time dependence of the amount of rotation can vary.For a time-dependent rotation of polarization direction, equation (2.1) is easily general-ized: Q ( η ) = ˜ Q ( η ) cos 2 α ( η ) + ˜ U ( η ) sin 2 α ( η ) U ( η ) = ˜ U ( η ) cos 2 α ( η ) − ˜ Q ( η ) sin 2 α ( η ) (2.3)This induces a modification of the Boltzmann equation for the evolution of polarizationperturbations, ∆ Q ± iU ( (cid:126)k, η ) . In Fourier space [42]: ˙∆ Q ± iU ( (cid:126)k, η ) + ikµ ∆ Q ± iU ( (cid:126)k, η ) = ˙ τ ( η ) (cid:34) − ∆ Q ± iU ( (cid:126)k, η ) − (cid:88) m (cid:114) π ± Y m (ˆ n ) S mp ( (cid:126)k, η ) (cid:35) ∓ i α ( η )∆ Q ± iU ( (cid:126)k, η ) (2.4)where derivatives are taken with respect to conformal time, µ is the cosine of the anglebetween the photon propagation direction and the Fourier mode (cid:126)k . ˙ τ ( η ) is the differentialoptical depth, ˙ τ ( η ) ≡ n e σ T a ( η ) , where n e and σ T are, respectively, the free electron numberdensity and the Thomson cross section, and a is the scale factor. ± Y m are spin-weightedspherical harmonics with spin ± and S ( m ) P is the polarization source ( m = 0 , ± , ± indicates,respectively, scalar, vector and tensor perturbations ). The last term in the equation is theone due to birefringence, and its form can be easily deduced by taking a time derivative ofthe appropriate combination of eqs. (2.3).To formally integrate over the line of sight one observes that ˙∆ Q ± iU + ( ikµ + ˙ τ ± i α )∆ Q ± iU = e − ikµη e τ ( η ) e ∓ i α ( η ) ddη (cid:104) e ikµη e − τ ( η ) e ± i α ( η ) ∆ Q ± iU (cid:105) . (2.5) For reasons that will be clear later one usually rotates only the spectra at multipole (cid:96) (cid:38) . The lowermultipoles are not rotated [37]. – 3 –here we have defined the total optical depth τ ( η ) ≡ (cid:82) η ∗ η ˙ τ ( η (cid:48) ) dη (cid:48) , with η ∗ the time atrecombination, such that dτ /dη = − ˙ τ . The total amount of polarization rotation afterpropagation for a time η from recombination is α ( η ) = (cid:82) ηη ∗ ˙ α ( η (cid:48) ) dη (cid:48) . The integration alongthe line of sight then gives: ∆ Q ± iU ( η ) = (cid:90) η dη e ikµ ( η − η ) e − [ τ ( η ) − τ ( η )] e ± i α ( η ) − α ( η )] ˙ τ ( η ) (cid:88) m (cid:114) π ± Y m (ˆ n ) S mp ( (cid:126)k, η ) (2.6)To go to E, B space one exploits the relations [39]: ∆ E = − (cid:16) ¯ ð ∆ Q + iU + ð ∆ Q − iU (cid:17) (2.7) ∆ B = − i (cid:16) ¯ ð ∆ Q + iU − ð ∆ Q − iU (cid:17) (2.8)Comparing with [39], eqs. (12)-(15) for scalar perturbations and eqs. (26)-(30) for tensorperturbations, it is easy to follow the same procedure outlined there and get: ∆ ( S ) E,(cid:96) ( k, η ) = (cid:114) (cid:96) + 2 (cid:96) − (cid:90) η dηS ( S ) E ( k, η ) cos (2 δα ( η )) j (cid:96) ( k ( η − η )) (2.9) ∆ ( S ) B,(cid:96) ( k, η ) = (cid:114) (cid:96) + 2 (cid:96) − (cid:90) η dηS ( S ) E ( k, η ) sin (2 δα ( η )) j (cid:96) ( k ( η − η )) (2.10) ∆ ( T ) E,(cid:96) ( k, η ) = (cid:90) η dη (cid:104) S ( T ) E ( k, η ) cos (2 δα ( η )) − S ( T ) B ( k, η ) sin (2 δα ( η )) (cid:105) j (cid:96) ( k ( η − η )) (2.11) ∆ ( T ) B,(cid:96) ( k, η ) = (cid:90) η dη (cid:104) S ( T ) B ( k, η ) cos (2 δα ( η )) + S ( T ) E ( k, η ) sin (2 δα ( η )) (cid:105) j (cid:96) ( k ( η − η )) (2.12)where S ( S,T ) E and S ( T ) B are the sources for E and B modes respectively as they appear in eqs.(18) and (30) of [39] and δα ( η ) ≡ α ( η ) − α ( η ) = (cid:82) ηη ˙ α ( η (cid:48) ) dη (cid:48) .The power spectra are computed in the standard way as: C XY(cid:96) = (4 π ) (cid:90) dkk P φ ( k )∆ X,(cid:96) ( k, η )∆ ∗ Y,(cid:96) ( k, η ) (2.13)where P φ ( k ) is the initial power spectrum and ∆ X,(cid:96) ( k, η ) is the perturbation of the mode X = { T, E, B } in Fourier space at time η (the one for temperature is standard and can befound in [39], the ones for E and B modes are given above).In the following we will focus on a linear time dependence, parameterized as α ( η ) = α ηη . (2.14)In this case δα ( η ) = α ηη − α . Motivations for studying this particular time dependencecome from some quantum-gravity-motivated studies [24, 25, 29, 43], where the amount ofrotation is quadratically dependent on the energy of the photons and linearly dependent onthe propagation time. One might of course consider more complicated functional forms forthe time dependence, for example one could link birefringence to a coupling of photons to– 4 –uintessence fields [41] or to pseudo-scalar fields [23]. In that case the time evolution of thebirefringence effect is linked to the time evolution of the fields.In Figure 1 we compare the polarization power spectra that one expects in standard Λ CDM model, with the ones expected if a polarization rotation is present, taking into accountdifferent possibilities: rotation that acts on the time-evolved spectra ( i.e. the one describedby the parameter β in eqs. (2.20)), rotation that evolves with photon propagation with alinear time dependence (described by eqs. (2.9)-(2.13), with relevant parameter α given in(2.14)), a constant rotation that acts on the time evolved spectra, before lensing (see nextsubsection for details) described by the parameter α . Note that in all of the three cases theeffects of lensing are present, even though they are treated differently, as is explained in thefollowing subsection. In particular, in the time-evolving case (parameter α ), lensing acts onthe spectra that were rotated by the time-evolving birefringence effect. ‘ -3 -2 -1 D ‘ [ µ K ] EE Λ CDMα = − ◦ α = +2 ◦ β = − ◦ ‘ -6 -5 -4 -3 -2 -1 D ‘ [ µ K ] BB ‘ D ‘ [ µ K ] TB ‘ -4 -3 -2 -1 | D ‘ | [ µ K ] EB Figure 1 . CMB power spectra obtained in standard Λ CDM (red solid line), late time constantrotation (pink dotted line), early time constant rotation (green dashed line) and time evolving rotation(blue dashed line) cosmologies.
Figure 1 shows how the introduction of the polarization rotation angle transfers power– 5 –rom the EE spectrum to the other polarization spectra both producing C T B(cid:96) and C EB(cid:96) , whichare vanishing in a standard Λ CDM framework where no rotation is present, and contributingto the C BB(cid:96) spectrum.The latter effect is of particular interests as another transfer of power from C EE(cid:96) to C BB(cid:96) ,acting on the same range of angular scales, is produced by gravitational lensing of CMBphotons. The impact of the lensing effect on B mode spectrum can be parametrized throughthe lensing amplitude A L which is equal to in a standard Λ CDM cosmology; this leads, ascan be seen in Figure 2, to possible degeneracies between lensing and polarization rotation,as the effects brought by birefringence can be (partially) mimicked by an enhanced lensingamplitude. Therefore it is crucial to treat in a proper way the combination of the two physicaleffects on CMB spectra, as it is done in the following section. ‘ -6 -5 -4 -3 -2 -1 D ‘ [ µ K ] BB α =0 ◦ α =0 ◦ A L =1 α = − ◦ α =0 ◦ A L =1 α =0 ◦ α = +2 ◦ A L =1 α =0 ◦ α =0 ◦ A L =1 . α =0 ◦ α =0 ◦ A L =2 Figure 2 . BB spectra for Λ CDM (black line), rotation cosmologies with a non zero α (orange line)and α (red line), compared with the spectra produced by cosmologies without rotation, but with A L = 1 . (green line) and A L = 2 (blue line). As we already mentioned, the way birefringence is usually treated is through the rotationof the ’final’ power spectra, resulting from propagation of radiation from the last scatteringsurface until now. This means that birefringence rotates spectra that have already been mixedby lensing . Exceptions to this treatment of lensing are found in [23, 41], where the amount of birefringence dependson the evolution of a cosmological scalar field. However, a detailed study of the degeneracy between lensingand birefringence is missing in those papers. – 6 –t should be clear now that this procedure is consistent only if one assumes that thespectra rotation is due to some systematic effect, which acts at the level of the detectors.If birefringence is due to genuinely physical effects acting along photon propagation, thenone should take into account the fact that lensing will mix spectra that have already beenrotated by birefringence. In particular birefringence will have already generated B modes fromE modes, so that, for example, C T B(cid:96) and C EB(cid:96) are non zero. This means that the formulae forthe lensed spectra have to be derived in the most general case when all the power spectra arenon zero. This is shown in the following, concentrating only on the quantities that involvethe polarization field, since the temperature field is not affected by birefringence.The construction of lensed spectra, which we will call ¯ C (cid:96) , relies on the real-space corre-lation functions [44] ξ X ( γ ) ≡ < T (ˆ n ) P (ˆ n ) >ξ + ( γ ) ≡ < P ∗ (ˆ n ) P (ˆ n ) >ξ − ( γ ) ≡ < P (ˆ n ) P (ˆ n ) > (2.15)where γ is the angle between directions ˆ n and ˆ n , P = Q + iU is the polarization field definedin the local basis with ˆ x direction along the geodesics between ˆ n and ˆ n .It can be shown (see [44]) that in absence of lensing these correlation functions arerelated to the power spectra and to the geometrical factors d (cid:96)ss (cid:48) ( γ ) ≡ (cid:80) m s Y ∗ (cid:96)m (ˆ n ) s (cid:48) Y (cid:96)m (ˆ n ) in the following way: ξ X ( γ ) = (cid:88) (cid:96) (cid:96) + 14 π ( C T E(cid:96) − iC T B(cid:96) ) d (cid:96) ( γ ) ξ + ( γ ) = (cid:88) (cid:96) (cid:96) + 14 π ( C EE(cid:96) + C BB(cid:96) ) d (cid:96) ( γ ) ξ − ( γ ) = (cid:88) (cid:96) (cid:96) + 14 π ( C EE(cid:96) − C BB(cid:96) − iC EB(cid:96) ) d (cid:96) − ( γ ) (2.16)Note that we already are considering the possibility of having the parity-violating spectra C EB(cid:96) and C T B(cid:96) .When lensing is introduced the above formulas have to be modified introducing termsrelated to the power spectrum of the lensing potential. A detailed computation which doesnot consider the parity violating spectra can be found in [45]. When including the parityviolating spectra C EB(cid:96) and C T B(cid:96) the relevant formulas found in [45] generalize to: ¯ ξ X ( γ ) = (cid:88) (cid:96) (cid:96) + 14 π ( C T E(cid:96) − iC T B(cid:96) ) (cid:110) d (cid:96) X X + C gl, X (cid:48) (cid:112) (cid:96) ( (cid:96) + 1) ( X d (cid:96) + X d (cid:96) − )+ 12 C gl, [ d (cid:96) (2 X (cid:48) X (cid:48) + X ) + d (cid:96) − X X ] (cid:111) ¯ ξ + ( γ ) = (cid:88) (cid:96) (cid:96) + 14 π ( C EE(cid:96) + C BB(cid:96) ) (cid:110) d (cid:96) X + 2 C gl, X X d (cid:96) + C gl, [ d (cid:96) ( X (cid:48) ) + d (cid:96) X X ] (cid:111) ¯ ξ − ( γ ) = (cid:88) (cid:96) (cid:96) + 14 π ( C EE(cid:96) − C BB(cid:96) − iC EB(cid:96) ) (cid:110) d (cid:96) − X + C gl, [ X d (cid:96) − + X d (cid:96) − ]+ 12 C gl, [2 d (cid:96) − ( X (cid:48) ) + d (cid:96) X + d (cid:96) − X ] (cid:111) (2.17)– 7 –he ¯ ξ { X, + , −} are the lensed correlation functions, X ijk and C gl, depend on the lensing poten-tial and geometrical factors and are defined in [45]. Note that the power spectra appearinghere are the non-lensed ones, but include already the effect of birefringence if one is notconsidering the late-time birefringence case.Once the lensed correlation functions have been computed as in the above equations,the lensed spectra ¯ C (cid:96) can be derived by using the relations (2.16) and the orthogonality ofthe d (cid:96)ss (cid:48) : ¯ C T E(cid:96) − i ¯ C T B(cid:96) = 2 π (cid:90) − ¯ ξ X d (cid:96) d cos γ ¯ C EE(cid:96) + ¯ C BB(cid:96) = 2 π (cid:90) − ¯ ξ + d (cid:96) d cos γ ¯ C EE(cid:96) − ¯ C BB(cid:96) − i ¯ C EB(cid:96) = 2 π (cid:90) − ¯ ξ − d (cid:96) − d cos γ (2.18)Comparing equations (2.17) and (2.18) one sees that a common effect of lensing andbirefringence is the generation of B modes from E modes. So it is interesting to investigatethe degeneracy between the two effects.A simple way to do this is by studying the difference between a ’late-time’ birefringenceand an ’early-type’ birefringence. In both cases one considers a constant polarization rotationangle, but while in the first case rotation acts on lensed spectra, which are evaluated inthe standard way, in the second one birefringence acts on the spectra propagated from lastscattering surface, but before lensing is applied.More in detail, in the late-time birefringence case the observed spectra are given by C EE(cid:96) = ¯ C EE(cid:96) cos (2 β ) + ¯ C BB(cid:96) sin (2 β ) − ¯ C EB(cid:96) sin (4 β ) C BB(cid:96) = ¯ C EE(cid:96) sin (2 β ) + ¯ C BB(cid:96) cos (2 β ) + ¯ C EB(cid:96) sin (4 β ) C EB(cid:96) = 12 (cid:0) ¯ C EE(cid:96) − ¯ C BB(cid:96) (cid:1) sin (4 β ) + ¯ C EB(cid:96) (cid:0) cos (2 β ) − sin (2 β ) (cid:1) C T E(cid:96) = ¯ C T E(cid:96) cos (2 β ) − ¯ C T B(cid:96) sin (2 β ) C T B(cid:96) = ¯ C T E(cid:96) sin (2 β ) + ¯ C T B(cid:96) cos (2 β ) (2.19)using as ¯ C (cid:96) the non-rotated, but lensed, spectra, computed in the standard way (see e.g. [45]).In the early-time birefringence case, one applies the procedure described in this subsection,using in eq. (2.17) the spectra obtained through: C EE(cid:96) = ˜ C EE(cid:96) cos (2 α ) + ˜ C BB(cid:96) sin (2 α ) − ˜ C EB(cid:96) sin (4 α ) C BB(cid:96) = ˜ C EE(cid:96) sin (2 α ) + ˜ C BB(cid:96) cos (2 α ) + ˜ C EB(cid:96) sin (4 α ) C EB(cid:96) = 12 (cid:16) ˜ C EE(cid:96) − ˜ C BB(cid:96) (cid:17) sin (4 α ) + ˜ C EB(cid:96) (cid:0) cos (2 α ) − sin (2 α ) (cid:1) C T E(cid:96) = ˜ C T E(cid:96) cos (2 α ) − ˜ C T B(cid:96) sin (2 α ) C T B(cid:96) = ˜ C T E(cid:96) sin (2 α ) + ˜ C T B(cid:96) cos (2 α ) (2.20)where ˜ C (cid:96) are now the spectra one obtains after propagation of photons from last scatteringto now, without the inclusion of lensing effect.The difference between the spectra obtained with a late-time birefringence and an early-time birefringence can be seen in Fig.1. – 8 – Analysis
The baseline set of cosmological parameters we evaluate is composed by the standard ones,namely the baryon and CDM physical matter density Ω b h and Ω c h , the scalar spectral index n s , the optical depth τ , the scalar amplitude as evaluated at a pivot scale k = 0 . M pc − A s the angular size of the sound horizon at last scattering surface θ . To this set of parameters,we add the effect of the two constant rotation models discussed above, parametrized by α (early times rotation) and β (late times rotation) and the parameter describing the time-varying rotation α . We fit the C (cid:96) obtained through the theory discussed above combiningdatasets from WMAP latest release [6] and BICEP 2013 release [8]. We do not exploit thelatest release of BICEP [10] experiment as in that case the TB and EB spectra are used tocalibrate the survey to achieve a vanishing rotation angle. As done in [37] in the late timesrotation (i.e. β ) we do not rotate multipoles below (cid:96) = 23 , because the polarization signal,at those multipoles, was generated during reionization and so we would measure only the po-larization rotation between the reionization and present epoch [34, 41]. Therefore, one wouldin principle need to separately analyze the two multipole regions (below and above (cid:96) = 23 ),but the low- (cid:96) one has a much poorer constraining power [37].We assume flat priors on the sampled parameters and we exploit MCMC technique throughthe publicly available package cosmomc [46] with a convergence diagnostic using the Gelmanand Rubin statistics.Furthermore, once the best fit values for the WMAP+BICEP are obtained, we use these asthe fiducial cosmologies to build forecasted datasets for Planck [47] and PolarBear [13] data,in order to investigate how the upcoming data from these surveys will tighten the previouslyobtained constraints and if they would be able to distinguish the effects of the different rota-tion parameters. We produce two sets of forecasted datasets, one assuming a cosmology withthe WMAP+BICEP best fit for α and one with the best fit for α . The first set is analyzedwith a varying α and with a varying β in order to understand if the combination of Planckand PolarBear will be able to distinguish between an early constant rotation and a late con-stant one. The second set is instead analyzed once with a varying α and once with a varying α allowing to inquire about the different effects of a time evolving and a constant rotation.Both these analysis are performed by probing the standard set of parameters alongside thoserelated to rotation effects and using the same MCMC technique employed for WMAP andBICEP. The two datasets are also analyzed with the assumption of no rotation (thus allowingonly standard Λ CDM parameters to vary) in order to investigate whether the assumptionof no rotation would lead to a bias in the recovered best fit of standard parameters in theeventuality of a non vanishing birefringence effect.Finally, we also use the two combinations of simulated datasets to inquire about the possibledegeneracy between weak gravitational lensing of the CMB and rotation of the spectra in-troduced by the birefringence effect. This is done by analyzing the datasets not varying anyrotation parameter, but allowing the lensing amplitude A L to vary. As stated in Section 2this parameter enhances the effect of lensing on CMB spectra if raised above the standard Λ CDM value A L = 1 , thus if the model Λ CDM + A L is used to fit the simulated dataset, anon standard value of A L could be detected in order to mimic the enhancement of BB modesproduced by birefringence rotation angles. – 9 – Results
As stated in the previous section, we exploit currently available data from WMAP and BICEPin order to constrain the amplitude of the birefringence parameter, both in the time evolving( α ) and constant ( α ) case, and of the late time rotation angle ( β ). In Table 1 the resultsobtained with this experimental configuration are reported and it is possible to notice howthe standard cosmological parameters are not affected by the change of theoretical frameworkconsidered, showing how no degeneracies between standard and birefringence parameters aredetected by WMAP+BICEP. Moreover the results obtained for α , α and β are comparableexcept for the time evolving angle which exhibits a sign opposite to the other two, due to thedefinition given in Section 2. Nevertheless, as already found in [8], it must be noted that thecombination WMAP+BICEP favors a non vanishing birefringence angle (see Fig.3, whetherit produces time-varying, constant or late time rotation, at ≈ . σ . β rotation α rotation α rotationParameter Ω b h . ± . . ± . . ± . c h . ± .
004 0 . ± .
004 0 . ± . θ . ± .
002 1 . ± .
002 1 . ± . τ . ± .
01 0 . ± .
01 0 . ± . n s . ± .
01 0 . ± .
01 0 . ± . A s ) 3 . ± .
04 3 . ± .
04 3 . ± . α − − . ± . α − − . ± . − β − . ± . − − H . ± . . ± . . ± . Table 1 . Marginalized mean values and c.l. errors on cosmological parameters usingWMAP+BICEP data. β [deg] R e l a t i v e P r o b a b ili t y varying β WMAP9 WMAP9+BICEP1 4 2 0 2 4 α [deg] R e l a t i v e P r o b a b ili t y varying α α [deg] R e l a t i v e P r o b a b ili t y varying α Figure 3 . Posterior distributions for (from right to left) α , α and β using WMAP (red continuouslines) and WMAP+BICEP (blue dashed lines). – 10 – .1 Forecasted Results Once the best fit values for WMAP+BICEP are obtained in the three different cosmologies,we use them as the fiducial cosmology to forecast future constraints, achievable by the com-bination of Planck and PolarBear surveys. We investigate if the combination of these twoexperiments will be able to significantly distinguish the assumed cosmology from one whereno rotation, of any kind, is present.The first datasets we produce assume as fiducial cosmology the best fit for the early-timerotation with a constant angle α . This is fitted by three different cosmologies: a varying α cosmology, to quantify the improvement brought by Planck+PolarBear on α constraints, avarying late time rotation angle β cosmology, to investigate the possibility to distinguish thetwo rotation mechanisms with this combination of surveys, and a standard Λ CDM cosmologyto quantify the possible bias brought on standard parameters by wrongly assuming a vanish-ing rotation angle.In Table 2 we report the results obtained by combining Planck and PolarBear forecasteddatasets and it is possible to notice how the improvement in sensitivity due to these surveyswill allow to tighten the constraints on α and β and therefore, if WMAP+BICEP best fitvalues are confirmed, to rule out the possibility of a vanishing birefringence angle, both forthe early and late time rotations. α rotation β rotation no rotationParameter Ω b h . ± . . ± . . ± . c h . ± . . ± . . ± . θ . ± . . ± . . ± . τ . ± .
003 0 . ± .
003 0 . ± . n s . ± .
002 0 . ± .
002 0 . ± . A s ) 3 . ± .
01 3 . ± .
01 3 . ± . α − . ± . − − β − − . ± . − H . ± . . ± . . ± . Table 2 . Marginalized mean values and c.l. errors on cosmological parameters usingPlanck+PolarBear forecasted data in the α , β and no rotation analysis. The fiducial model used tobuild the simulated dataset is based on the WMAP+BICEP analysis including α (i.e. early timerotation), second column of Table 1 When fitting the considered datasets with a varying β no shift is detected in the recov-ered values of the standard cosmological parameters, as can be seen in Fig.5, highlighting thefact that the varying β cosmology is able to reproduce the fiducial one. However the recoveredamount of rotation is significantly different if different theoretical models for the rotation areused in the fit, see Figure 4. As the β rotation angle can be thought, due to its properties,as a systematic effect arising from calibration errors [34], the results we show highlight oncemore the necessity to minimize these errors as a poor calibration could lead to false detectionof physical rotation effects.When the datasets are instead analyzed with a standard Λ CDM cosmology without any po-larization rotation we obtain a shift of the order of several standard deviations in standard– 11 –osmological parameters; this effect, clearly visible in Fig.6, arises from the fact that thetheoretical spectra produced assuming a vanishing rotation angle are not able to reproducethe simulated datasets polarization spectra (see Fig. 1).
Angle [deg] R e l a t i v e P r o b a b ili t y Planck+BICEP1 α Best-Fit
Planck varying α Pl+PB varying α Planck varying β Pl+PB varying β Figure 4 . Posterior distribution for early-type and late-time birefringence parameters α (black lines)and β (red lines) using Planck (solid lines) and Planck+PolarBear (dashed lines) forecasted datasets,with a fiducial cosmology equivalent to the second column of Table 1. As a further step we inquire whether Planck+PolarBear can distinguish between anearly times constant rotation and a time evolving one; in order to do this, we use theWMAP+BICEP best fit value, obtained assuming a time evolving rotation, as fiducial cos-mology and we analyze the obtained datasets again with three different cosmology, i.e. with α or α free to vary and with standard Λ CDM.In Table 3 we report the results obtained combining Planck and PolarBear forecasted datasetsand again we notice how the constraining power of Planck+PolarBear allows to rule out thevanishing birefringence angle cosmology.As in the previous analysis, we notice also in this case how the combination of Planckand PolarBear is unable to distinguish the two rotation mechanisms while we again find ashift of a few σ order of magnitude when the datasets are analyzed assuming a vanishingrotation angle (see Fig. 8). As already stated in Section 2 and shown in Fig. 1, one of the effects of a non vanishingbirefringence angle is to shift power from the EE to the BB modes. Another source of mixingbetween EE and BB modes is brought by CMB lensing which produces a non vanishing BBspectrum even if a birefringence angle is not present (see Fig. 2). In order to understandif degeneracies between the two effects exist, we analyze the simulated Planck+PolarBeardatasets, obtained with time evolving rotation as fiducial cosmology, assuming no rotation ispresent, but allowing for a varying lensing amplitude A L .– 12 – .0226 0.0228 0.0230 Ω b h R e l a t i v e P r o b a b ili t y Ω c h R e l a t i v e P r o b a b ili t y θ s R e l a t i v e P r o b a b ili t y τ R e l a t i v e P r o b a b ili t y n S R e l a t i v e P r o b a b ili t y log[10 A S ] R e l a t i v e P r o b a b ili t y Figure 5 . Posterior distribution for several standard cosmological parameters using Planck (solidlines) and Planck+PolarBear (dashed lines) forecasted datasets, with a fiducial cosmology equivalentto the second column of Table 1, analyzed with a varying α (black lines) and with a varying β (red lines). We see that even with Planck+PolarBear sensitivity there is no detectable effect on thestandard cosmological parameters α rotation α rotation no rotationParameter Ω b h . ± . . ± . . ± . c h . ± . . ± . . ± . θ . ± . . ± . . ± . τ . ± .
003 0 . ± .
003 0 . ± . n s . ± .
002 0 . ± .
002 0 . ± . A s ) 3 . ± .
01 3 . ± .
01 3 . ± . α . ± . − − α − − . ± . − H . ± . . ± . . ± . Table 3 . Marginalized mean values and c.l. errors on cosmological parameters usingPlanck+PolarBear forecasted data in the α , α and no rotation analysis. The fiducial model used tobuild the simulated dataset is based on the WMAP+BICEP analysis including α (i.e. time evolvingrotation), third column of Table 1 In Table 4 we report the obtained parameters and it is possible to notice (see also inFigs. 9-10) how the presence of a varying A L mitigates the shift produced assuming the wrongcosmological model. This is due to the fact that raising the value of A L increases the powertransfer from EE to BB modes and therefore can partially account for enhancement in the– 13 – .0225 0.0230 Ω b h R e l a t i v e P r o b a b ili t y Ω c h R e l a t i v e P r o b a b ili t y θ s R e l a t i v e P r o b a b ili t y τ R e l a t i v e P r o b a b ili t y n S R e l a t i v e P r o b a b ili t y log[10 A S ] R e l a t i v e P r o b a b ili t y Figure 6 . Posterior distribution for several standard cosmological parameters using Planck (solidlines) and Planck+PolarBear (dashed lines) forecasted datasets, with a fiducial cosmology equivalentto the second column of Table 1, analyzed with a varying α (black lines) and with no rotation cosmol-ogy (red lines). We see that with Planck sensitivity the effect of disregarding the rotation is significanton the standard cosmological parameters, and it becomes catastrophic with Planck+PolarBear sensi-tivity. α rotation no rotation ∆ /σ no rotation + A L ∆ /σ Parameter Ω b h . ± . . ± . . . ± . . c h . ± . . ± . . . ± . . θ . ± . . ± . . . ± . . τ . ± .
003 0 . ± .
004 1 . . ± .
003 0 . n s . ± .
002 0 . ± .
002 6 . . ± .
002 2 . A s ) 3 . ± .
01 3 . ± .
01 7 . . ± .
01 1 . α − . ± . − − − − A L − − − . ± . − H . ± . . ± . . ± . . Table 4 . Marginalized best fit values and c.l. errors on cosmological parameters usingPlanck+PolarBear simulated data (assumed fiducial cosmology obtained from third column of 1)analyzed with α rotation, no rotation and vanishing rotation angle when a variation of the A L pa-rameter is allowed. Third and fifth columns report the shift with respect to the varying α best fit inunity of σ , for the no rotation and free A L cases respectively. BB spectrum produced by birefringence when the datasets are analyzed assuming vanishingrotation angles. It is interesting to point out how the shift in the standard parameters is– 14 – .0226 0.0228 0.0230 Ω b h R e l a t i v e P r o b a b ili t y Ω c h R e l a t i v e P r o b a b ili t y θ s R e l a t i v e P r o b a b ili t y τ R e l a t i v e P r o b a b ili t y n S R e l a t i v e P r o b a b ili t y log[10 A S ] R e l a t i v e P r o b a b ili t y Figure 7 . Posterior probability distribution for the standard cosmological parameters using Planck(solid lines) and Planck+PolarBear (dashed lines) forecasted datasets, with a fiducial cosmologyequivalent to the third column of Table 1, analyzed with a varying α (black lines) and with a varying α (red lines). We see that even with Planck+PolarBear sensitivity there is no detectable effect onthe standard cosmological parameters α rotation no rotation ∆ /σ no rotation + A L ∆ /σ Parameter Ω b h . ± . . ± . . . ± . . c h . ± . . ± . . . ± . . θ . ± . . ± . . . ± . . τ . ± .
003 0 . ± .
003 0 . . ± .
003 0 . n s . ± .
002 0 . ± .
002 1 . . ± .
002 0 . A s ) 3 . ± .
01 3 . ± .
01 1 . . ± .
01 0 . α − . ± . − − − − A L − − − . ± . − H . ± . . ± . . . ± . . Table 5 . Marginalized best fit values and c.l. errors on cosmological parameters using Plancksimulated data (assumed fiducial cosmology obtained from third column of 1) analyzed with α rotation, no rotation and vanishing rotation angle when a variation of the A L parameter is allowed.Third and fifth columns report the shift with respect to the varying α best fit in unity of σ , for theno rotation and free A L cases respectively. mitigated at the price of a non standard value of A L ; this means that neglecting the presenceof a birefringence rotation can possibly lead to a false detection of A L > when analyzingdata with Planck+PolarBear sensitivity. – 15 – .0225 0.0230 Ω b h R e l a t i v e P r o b a b ili t y Ω c h R e l a t i v e P r o b a b ili t y θ s R e l a t i v e P r o b a b ili t y τ R e l a t i v e P r o b a b ili t y n S R e l a t i v e P r o b a b ili t y log[10 A S ] R e l a t i v e P r o b a b ili t y Figure 8 . Posterior probability distribution for the standard cosmological parameters using Planck(solid lines) and Planck+PolarBear (dashed lines) forecasted datasets, with a fiducial cosmologyequivalent to the third column of Table 1, analyzed with a varying α (black lines) and with norotation cosmology (red lines). We see that with Planck sensitivity the effect of disregarding therotation is significant on the standard cosmological parameters, and it becomes catastrophic withPlanck+PolarBear sensitivity. We found similar, but less significant, results for Planck alone also. The recovered value of A L is . σ away from the standard value and the other parameters are recovered within σ .However, we point out that, should this effect being observed in upcoming data, a practicalway to distinguish between a non vanishing α cosmology and an enhanced lensing amplitudewould lie in the analysis of the χ values obtained when fitting the data; while partially de-generate, in fact, the two cosmologies produce different effect on the CMB spectra (as canbe seen in Fig.2) and therefore will produce different values of the χ when used to analyzedatasets. This implies that Bayesian model selection techniques can be used to quantitativelyunderstand which of the two theoretical models would be preferred by observations. In this paper we investigated the possibility for current and upcoming data to measure theeffect of a rotation of CMB photons polarization direction. This kind of effect can be ascribedeither to calibration issues or to departures from the standard Λ CDM theory and it producesa peculiar mixing of E and B modes. When the effect is due to new physics, it is called bire-fringence, and its effects are different from the ones of a calibration-driven mixing because therotation of polarization accumulates starting from early times, thus it affects CMB spectrabefore the E-B mixing due to gravitational lensing takes place.– 16 – igure 9 . Posterior distribution for the parameters using Planck forecasted datasets, with a fiducialcosmology equivalent to the first column of Table 1, analyzed with a varying α (black lines), with avarying A L (blue lines) and with only standard cosmological parameters (red lines). One sees that A L can compensate for the effects of birefringence quite well in experiments with Planck sensitivity We have considered three scenarios. The first one is a late-time E-B mixing (possibly due tocalibration issues), parameterized by the polarization rotation angle β . This is the mixingmore widely considered in the literature and is applied on the final CMB spectra, alreadyaffected by lensing . The second scenario is an early-time E-B mixing, which is a constantmixing happening before lensing, parameterized by the polarization rotation angle α . Thisis a first approximation of a more physically sound model in which the polarization rotation isdue to non-standard physics and accumulates during photon propagation. This was actuallythe last scenario we considered: a mixing of E and B modes that accumulates during CMBphoton propagation with a linear time dependence. It is parameterized by the dimensionlessquantity α . We analyzed WMAP CMB data alone and combined with BICEP data, findingthat the sensitivity of these survey is not enough to distinguish among the three scenarios,but it is sufficient to show a hint for a non-vanishing value of either α , α or β , pointing outhow the combination of these datasets is able to detect the effect of polarization mixing inCMB spectra.Furthermore, we used the parameters’ values obtained with the latter analysis to forecastupcoming CMB data from Planck and PolarBear, in order to investigate the impact of thesesurveys on the possible detection of the three rotation scenarios and the possibility to dis-tinguish between early and late time rotations and between constant and time-evolving ro-tation angles. We found, as expected, that the Planck+PolarBear analysis will narrow theconstraints on rotation parameters with respect to WMAP+BICEP, possibly ruling out thevanishing rotation scenario, while it will not be able to distinguish among the three rotation– 17 – .0225 0.0230 0.0235 Ω b h R e l a t i v e P r o b a b ili t y var α standardvar A L Ω c h R e l a t i v e P r o b a b ili t y θ s R e l a t i v e P r o b a b ili t y τ R e l a t i v e P r o b a b ili t y n S R e l a t i v e P r o b a b ili t y log[10 A S ] R e l a t i v e P r o b a b ili t y Figure 10 . Posterior distribution for the parameters using Planck+PolarBear forecasted datasets,with a fiducial cosmology equivalent to the first column of Table 1, analyzed with a varying α (blacklines), with a varying A L (blue lines) and with only standard cosmological parameters (red lines).One sees that with the increased sensitivity provided by PolarBear one gets wrong values for thecosmological parameters if birefringence is disregarded. Moreover some parameters are sensitive tothe different effects of A L and birefringence, so that A L can not always compensate for the effects ofbirefringence. mechanisms as, even assuming the wrong scenario, the theoretical spectra produced are stillable to fit the fiducial cosmology.Finally, we inquired about the possible degeneracy between birefringence and CMB lensing,as both physical phenomena produce a leaking from E to B modes; we found that the similareffect on CMB spectra leads to the possibility of partially mimicking the birefringence drivenenhancement of standard Λ CDM B modes with a cosmology where a vanishing rotation isassumed, but a non standard amplitude of CMB lensing is allowed; this result shows howthe sensitivity that will be achieved by Planck+PolarBear will be high enough to prompt theneed for very accurate analysis of these effects as a false detection of a non standard lensingcould be obtained if the birefringence effect is neglected.
AKNOWLEDGMENTS
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