A note on the birefringence angle estimation in CMB data analysis
Abstract
Parity violating physics beyond the standard model of particle physics induces a rotation of the linear polarization of photons. This effect, also known as cosmological birefringence (CB), can be tested with the observations of the cosmic microwave background (CMB) anisotropies which are linearly polarized at the level of . In particular CB produces nonnull CMB cross correlations between temperature and B modepolarization, and between E and Bmode polarization. Here we study the properties of the so called Destimators, often used to constrain such an effect. After deriving the framework of both frequentist and Bayesian analysis, we discuss the interplay between birefringence and weaklensing, which, albeit parity conserving, modifies preexisting TB and EB cross correlation.
a,b]A. Gruppuso, c]G. Maggio, d,a]D. Molinari, d]P. Natoli
Prepared for submission to JCAP
A note on the birefringence angle estimation in CMB data analysis

INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna,
via P. Gobetti 101, I40129 Bologna, Italy 
INFN, Sezione di Bologna, Via Irnerio 46, I40126 Bologna, Italy

INAF, Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy

Dipartimento di Fisica e Scienze della Terra and INFN, Università degli Studi di Ferrara, Via Saragat 1, I44100 Ferrara, Italy
Contents
1 Introduction
Polarized cosmic microwave background (CMB) data can be used to probe the cosmic birefringence effect [1, 2, 3], i.e. the in vacuo rotation of the photon polarization direction during propagation [4, 5]. Such an effect is naturally parameterized by an angle and results in a mixing between Q and U Stokes parameters. From an observational point of view, the latter mixing produces nonnull CMB cross correlations between temperature and B modepolarization, and between E and Bmode polarization^{1}^{1}1We use the so called “cosmo” convention for the polarization angle. This is typically adopted in CMB analysis, see e.g. http://wiki.cosmos.esa.int/planckpla2015/index.php/Sky_temperature_maps.. Since these correlations are expected to be null under the parity conserving assumptions that are beneath the standard cosmological model, cosmic birefringence represents a well suited tracer of parity violating physics.
The assumption of constant defines the case of “isotropic birefringence”. In this case the effect on the CMB spectra is that of a rotation [6, 7]:
(1.1)  
(1.2)  
(1.3)  
(1.4)  
(1.5) 
with being the observed spectra whereas represent the primordial spectra, i.e. the spectra that would arise in absence of birefringence. Note that the primordial spectra are supposed to be parity conserving, namely and are vanishing. The symbol stands for ensemble average. Generalizations to nonconstant can be found e.g. in [8, 9, 10].
Current constraints from CMB experiments or other astrophysical observations are all compatible with a null effect, see for instance [11, 12, 13]. In CMB data analysis there are at least three ways of constraining :

Monte Carlo Markov Chain (MCMC) approach. In this case MCMC sampling is performed simultaneously over an extended CDM model to include the birefringence angle , see e.g. [1, 14]. This approach assumes that a CMB likelihood is available and coupled to MCMC sampler. Its usefulness lies in the fact that possible correlations among the parameters can be studied. However, the TB and EB spectra, which are the most sensitive to parity violating mechanism, are not always included in the likelihood function because they do not add much information to the standard CDM parameters or its parity conserving extensions. It is possible to estimate without and (see equations (1.1),(1.3),(1.4)), but constraints will be poorer. This is evident once Equations (1.1)(1.5) are Taylorexpanded for small : it turns out that only Equations (1.2) and (1.5) show a linear dependence on which is instead quadratic, and therefore weaker, for Equations (1.1),(1.3),(1.4). Note also that the sign of the angle cannot be determined without TB and EB.

Destimators. These are harmonic based estimators defined through the CMB spectra by the following equations [2, 18, 19, 20]:
(1.6) (1.7) where is an estimate for the birefringence angle . One of their most important features is that they depend explicitly on the multipole . This makes them also suitable to search for scale dependence of the birefringence effect. Isotropic birefringence as described by Equations (1.1)(1.5) is completely degenerate with a systematic, unknown mismatch of the global orientation of the polarimeters. This systematics would clearly generate a scale independent effect. On the contrary most of the birefringence models predict some form of angular dependence of . For example if the birefringence angle is proportional to the distance travelled by photons [4], then one expects it to differ between small and large angular scales, since in the latter case CMB polarization is sourced at the reionization epoch, as opposed to the former in which polarization is generated at recombination. Therefore scale dependence of might be considered as a way to disentangle a pure systematic effect from a fundamental physical mechanism. The properties of the Destimators will be described in detail in Section 2.
Stacking and Destimators analyses are sensitive to the TB and EB spectra and are typically employed fixing other cosmological parameters to the best fit model. In the present article we focus on the Destimators exploring in detail several remarkable properties they exhibit and and providing mathematical expressions that can be fruitfully used both in frequentist and Bayesian analysis. We observe that it is convenient (from the data analysis point of view) to express all the relevant equations in terms of primordial spectra, i.e. unrotated spectra, since in that case the covariance matrices of the Destimators do not depend on if the E and Bmodes noise are equivalent^{2}^{2}2This is typically a good assumption.. Finally we show that weaklensing which obviously impact CMB polarization spectra does not modify a built with the Destimators.
This paper is organized as follows: in Section 2 we describe the properties of the Destimators and provide the relevant expressions that can be used to perform the birefringence angle analysis. Section 3 is devoted to the interplay between Destimators and weaklensing effect. Conclusions are drawn in Section 4.
2 Destimators
2.1 Expectation values
Taking the ensamble average of (1.6) and (1.7), one finds
(2.1)  
(2.2) 
where we have explicitly written the dependence on for sake of clarity. Replacing Equations (1.11.5) in equations (2.1) and (2.2), after some algebra one obtains:
(2.3) 
(2.4) 
where primordial and are set to zero^{3}^{3}3This is a work hypothesis that holds throughout the paper.. It is easy to see from Equations (2.3) and (2.4) that the functions and vanish when
(2.5) 
Therefore, looking for that nulls the expectation values of the Destimators, is equivalent to looking for the birefringence angle that has rotated the primordial CMB spectra. From a data analysis point of view such a search of is conveniently performed with a standard technique which is presented in Section 2.4. From now on, unless otherwise specified, we always consider that Equation (2.5) holds.
2.2 Single realization results
Equations (1.11.5) follow from
(2.6)  
(2.7)  
(2.8) 
where we have explicitly written the contribution of the primordial signal (i.e. , and ) and of the instrumental noise (i.e. , and ) present on the maps^{4}^{4}4We are neglecting the contribution of spurious astrophysical emissions.. From Equations (2.62.8) one recovers (1.11.5) as follows:

define the angular power spectra (APS), , as
(2.9) where stand for or , with . Sometimes we also use the following notation: . Notice that Equation (2.9) is unbiased.

set to zero the primordial spectra and ;

perform an average over a set of realizations.
Replacing in Equations (1.6, 1.7) what found in step 1 above, one obtains the following expressions (for ):
(2.10)  
(2.11)  
As a sanity test it is easy to check that taking the ensemble average of Equations (2.10), (2.11) one finds
(2.12)  
(2.13) 
whatever the value of , which confirms the result of Section 2.1.
2.3 Covariances
Given the null means, the covariances of the Destimators are
(2.14) 
(2.15) 
(2.16) 
(2.17) 
and can be computed starting from Equations (1.6)(1.7). In this case one recovers what already provided in [19] where the covariances are given in terms of observed APS. Plugging instead Equations (2.10)(2.11) in (2.14)(2.17) allows one to connect the covariances with the primordial (i.e. unrotated) APS. Employing Wick’s theorem (all are Gaussian variables) one finds
(2.18)  
(2.19)  
(2.20)  
(2.21) 
These are very useful expressions for data analysis as we will see in Section 2.4. Note that if (typically a good assumption) they simplify in
(2.22)  
(2.23)  
(2.24)  
(2.25) 
and the dependence on drops out. The latter Equations are formally identical to (2.18)(2.21) with vanishing . Also note that the and are uncorrelated only in the ideal case in which the primordial Bmodes are zero and noise can be assumed to vanish, see Equations (2.20) and (2.24).
2.4 minimization
The best fit angle can be obtained minimizing the following expressions:
(2.26)  
(2.27) 
where . One can also minimize the combination
(2.28) 
with being implicitly defined by
(2.29) 
Since if and for vanishing noise, under the latter condition
(2.30) 
Equations (2.26)(2.28) can be minimized following either a Bayesian or frequentist approach. In the latter case a specific model is chosen and the matrices are built for such a reference model (the natural choice is , at least as long as there is no clear evidence of an detection). The idea is then to test whether the observed value of is compatible with model assumption. Through Monte Carlo simulations an empirical posterior distribution for can be obtained and used to derive confidence intervals. On the contrary, in the Bayesian approach the matrices keep their dependence on . The idea here is to find the angle which best describes the observations. Within the latter approach Monte Carlo simulations are in principle unneeded to evaluate statistical uncertainties. However in both approaches the matrices must be explicitly known. Failing analytical approximations they have to be derived using Monte Carlo simulations. To this extent the computation for the Bayesian case is readily simplified by the following argument: in principle for each fixed angle one should simulate a large number of primordial maps, rotate them by a quantity , add a random noise realization compatible with the data, apply an (unbiased) APS estimator to each of the simulated maps, build the matrix , invert it and evaluate the corresponding . This pipeline should be repeated for every in order to sample the as a function of and find its minimum. In practice one would perform sets of simulations where , with being the adopted step in sampling . By noticing that the dependence on can be factorized as in Eqs (2.10), (2.11) it is possible to perform just one set of simulations (i.e. ) for every and build empirically the matrix (or ) through Eqs. (2.14), (2.15), (2.16) and (2.17).
2.5 minimization in subintervals of .
Equations (2.26), (2.27) and (2.28) can be minimized in subintervals of the multipole range to investigate possible scale dependences of the birefringence effect. Note that such an investigation using the other methods listed in Section 1 is troublesome. Therefore the Destimators turn out to be the most suitable for such a kind of analysis.
Let us consider now the most ideal case in which there is no correlation between  and in which the dependence of the covariance on is dropped out, see Equations (2.22) and (2.23). Mathematically it would seem possible to minimize for each single ,
(2.31)  
(2.32) 
where are defined by Equations (2.22),(2.23). In this case, expanding Equations (2.31), (2.32) for small up to second order, one finds
(2.33)  
(2.34) 
where
(2.35)  
(2.36)  
(2.37) 
with . Of course, in the considered regime of , Equations (2.33),(2.34) are simply parabolic functions. Some considerations are in order:

Equations (2.33),(2.34) are convex (as it must be for a statistics) parabolic functions since the coefficients of ( and ) are always non negative. Only in case of noise dominated regime, these coefficients tend to zero and consequently the estimators stop being capable to constrain the birefringence angle.

the vertices are different from zero only if and .

for both the Destimators the uncertainties are driven by the coefficient of the term, namely and . The larger is that coefficient, the narrower is the parabola and the smaller is the uncertainty. Conversely the smaller is that coefficient, the wider is the parabola and therefore the larger is the uncertainty. In practice, considering Eq. (2.33) one sees that when is far from zero (even negative) then the corresponding statistical uncertainty of is small. Similarly Eq. (2.34) shows that when is far from zero (i.e. is large) then the corresponding statistical uncertainty of is small. These simple considerations represent an handle to understand the behavior of the uncertainties at different angular scale.
3 Interplay between birefringence and weaklensing effect
3.1 Weak lensing
The impact of weaklensing on CMB spectra is computed in [21] and extended to , in [22]. The lensed twopoint correlation functions, i.e. , are given by:
(3.1)  
(3.2)  
(3.3)  
with being the angle between two given points of the sphere, and where and are objects that depend on the lensing potential and geometrical factors, see [21, 22] for further details. Following [22], the power spectra appearing in the right hand side of Equations (3.1),(3.2),(3.3) are unlensed, but they include already the effect of birefringence, if any. Once the lensed twopoint correlation functions are computed, the lensed spectra can be obtained through Equations (3.1),(3.2),(3.3) and employing the orthogonality properties of :
(3.4)  
(3.5)  
(3.6) 
Note that since the functions between curl brackets in Equations (3.1),(3.2),(3.3) are real, weaklensing modifies ,, and only separately. This shows that the weaklensing effect is a parity conserving phenomenon. Considering Equations (3.4) and (3.6), and separating the real and imaginary part, one can write
(3.7)  
(3.8)  
(3.9)  
(3.10) 
with
(3.11)  
(3.12) 
where
(3.13)  
3.2 Minimization of the built with Destimators
We consider now again the Destimators, defined in Equations (1.6) and (1.7). As done in [22] we suppose that weaklensing is a late time effect. Therefore after the weaklensing takes place, the Destimators, denoted as , can be written as
(3.14)  
(3.15) 
where the sum over is understood and where Equations (3.7)(3.10) have been used. For sake of simplicity we omit now the label , and adopt a matrixvector formalism. The built with , for both and , is then given by
(3.16)  
Equation (3.16) shows that the function is invariant under the weaklensing effect. We stress that not only the minimum but the whole shape does not depend on the fact that weaklensing took place or not. Notice how this follows from the structure of the weaklensing kernel and in particular its parity conserving nature as highlighted above.
We can follow two ways to estimate : (a) we can take the observed APS (that do include the weaklensing) and build a covariance matrix with the inclusion of the weaklensing or (b) we can delens the observed spectra and build the covariance matrix with APS that do not include the weaklensing effect. Of course both approaches assume that the lensing potential is known.
3.3 Mismatch in the weaklensing matrix
In fact the lensing potential would be known with a certain degree of uncertainty. We suppose that the weaklensing matrix adopted, , differ from the real weaklensing matrix for a term proportional to a quantity
(3.17) 
In this case the “wrong” statistics will be given by
(3.18) 
Replacing Equation (3.17) in Equation (3.18), one finds
(3.19)  
where higher order terms (i.e. ) have been neglected. Equation (3.19) shows that a mismatch of order in the weaklensing matrix as described in Eq. (3.17) provides no bias in the estimate of but increases its uncertainty by . This conclusion of course might change if the mismatch between and is more complicated and cannot be modeled as in Eq. (3.17).
4 Discussion and conclusion
We have studied in detail the Destimators, which are used to constrain the birefringence angle from CMB observations. We have discussed the mathematical framework and pointed out that the close relationship between Destimators and primordial APS can be fruitfully exploited to speed up the analysis. In particular if a sufficient large set of simulations are available, then Equations (2.10)(2.11) can be used to build the covariance matrix through Eqs. (2.14), (2.15), (2.16) and (2.17). The latter are obtained in the signal plus noise case, but can be generalized to account for residual systematic effects. We have also computed the covariance matrices in the ideal case, see in Equations (2.18)(2.21) noting that in the case in which the noise of E and B modes are equal, the covariance matrix for frequentist and Bayesian approach are formally equivalent and do not depend on . We have used this ideal case to describe the behavior of the uncertainties of when estimated in subinterval of the multipoles range in search of a possible scale dependence of the birefringence effect which can be exploited to disentangle instrumental systematics from physical effects. Moreover we have shown that the formalism built with the Destimators and the statistics derived from it, is not influenced from weaklensing effect provided that the weaklensing kernel is exactly known. When the lensing matrix is misestimated by a term proportional to the identity as described in Eq. (3.17), then is not biased, but its uncertainty is increased by . However, it is not guaranteed that such a yields unbiased estimates for when the lensing matrix deviates from the real one in a more complicated way with respect to Eq. (3.17).
Acknowledgments
A.G, D.M. and P.N. acknowledge support by ASI/INAF Agreement 2014024R.1 for the Planck LFI Activity of Phase E2.
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