Probing Lorentz-violating electrodynamics with CMB polarization

We perform a comprehensive study of the signatures of Lorentz violation in electrodynamics on the Cosmic Microwave Background (CMB) anisotropies. In the framework of the minimal Standard Model Extension (SME), we consider effects generated by renormalizable operators, both CPT-odd and CPT-even. These operators are responsible for sourcing, respectively, cosmic birefringence and circular polarization. We propagate jointly the effects of all the relevant Lorentz-violating parameters to CMB observables and provide constraints with the most recent CMB datasets. We bound the CPT-even coefficient to kF,E+B < 2.31 × 10-31 at 95% CL. This improves previous CMB bounds by one order of magnitude. The limits we obtain on the CPT-odd coefficients, i.e. |k (3) (V)00| < 1.54 × 10-44 GeV and |kAF | < 0.74 × 10-44 GeV at 95% CL, are respectively one and two orders of magnitude stronger than previous CMB-based limits, superseding also bounds from non-CMB searches. This analysis provides the strongest constraints to date on CPT-violating coefficients in the minimal SME from CMB searches.


Introduction
Lorentz symmetries are at the foundation of the current description of nature.However, theoretical investigations have suggested that they may only be exact symmetries at low energies [1][2][3].Motivations for this hypothesis are rooted in quantum gravity.Therefore, it is expected that the energy scale at which Lorentz invariance could be violated is the Planck scale.While the magnitude of this scale might discourage searches for Lorentz violations, high-precision experimental tests might be sensitive to their small low-energy residual effects.
There are currently a number of different theoretical frameworks describing departures from Lorentz symmetries [1][2][3].The most conservative approach is that of effective field theory, which incorporates Lorentz violation via the introduction of extra tensors in the Lagrangian of the standard model.The new operators can be ordered according to their mass dimension: operators which introduce Lorentz violations at some high-energy scale have mass dimension higher than four, and therefore are non-renormalizable.Without some custodial symmetries, these operators might also induce Lorentz violations in operators with lower mass dimension.The lower-dimension renormalizable operators produce effects that are not suppressed by the high-energy scale, and could in principle dominate over the non-renormalizable operators, possibly leading to stronger signatures on low-energy physics.The Lagrangian containing such renormalizable terms, known as minimal Standard Model Extension (SME), was first derived in [4].In this work we focus on the radiation sector of the SME Lagrangian [5] and test Lorentz invariance with observations of the CMB.We concentrate on renormalizable operators, leaving the study of the non-renormalizable operators [6] to a future work.
The Cosmic Microwave Background (CMB) is an ideal probe of possible departures from standard electrodynamics.The CMB radiation is linearly polarized due to Compton scattering at the epochs of recombination and reionization [7].A non standard propagation of light might induce distinctive patterns on the CMB polarization.A very well known example is the cosmic birefringence effect, namely the in-vacuo rotation of the linear polarization plane of the CMB radiation 1 .
Since the CMB last scattering surface is the farthest source of electromagnetic radiation available in nature, the cosmic birefringence effect accumulates during propagation of the CMB, increasing the chances of detecting a non-vanishing signal.Evidence for new physics could also come from the observation of a sizable level of circular polarization.In the standard cosmological model, circular polarization is not expected at the time of last scattering, even though a tiny amount can be generated by known physics at a later time [8,[11][12][13][14][15][16] as CMB photons propagate across the Universe.
As we will show in this paper, some combinations of these effects are expected within the SME framework, depending on which operators are considered.Among the two operators analyzed in our work, one violates CPT symmetry and is responsible for the generation of cosmic birefringence.The CPT-even operator, instead, leads to the generation of circular polarization from the conversion of the primordial linear polarization components.While we will focus on the SME framework, we remark that both birefringence and the generation of circular polarization can emerge in other theoretical scenarios.In particular, cosmic birefringence can be generated by Chern-Simons terms in the electrodynamics Lagrangian [17][18][19][20], by the coupling of the electromagnetic field to quintessence [21][22][23] or axion [24] fields, or in quantum-gravity motivated effective theories for electromagnetism [25][26][27].These scenarios might be distinguished because they predict different dependence on the frequency of the CMB signal [28,29] and on the propagation direction [19,20,27,30,31].Production of circular polarization is instead predicted by several scenarios beyond the standard model of particle physics, including a possible coupling between photons and an external vector field via a Chern-Simons term [32], the Cotton-Mouton effect [33], propagation of CMB photons in a non-commutative spacetime [34] and other non-standard effects [35][36][37][38][39].
Previous tests of the minimal SME focussed on one operator at a time [35,40], and neglected the possible interplay between them, which instead might affect theoretical predictions and then observational constraints.In this work, we exploit the large amount of information stored in the CMB polarization spectra [41][42][43] to perform a more complex analysis, accounting for different operators at the same time.In doing so, we employ the formalism recently developed by some of the authors of this work [44].This novel formalism allows to describe in all generality the effects of anomalous propagation of polarized radiation in terms of an effective susceptibility tensor.Any model implying anomalous propagation of radiation can be mapped into the components of this effective susceptibility tensor and the implications for the CMB power spectra can be readily derived.
This work is timely, since it provides updated constraints on Lorentz violating coefficients using a novel mathematical formalism and state-of-the-art CMB data.Moreover, it paves the way to analogous tests with upcoming CMB data.Indeed, CMB polarization is the main observational target of next-generation CMB experiments [45][46][47][48][49][50].
The paper is structured as follows.In Section 2 we map the coefficients of the minimal SME operators which describe Lorentz violation in the radiation sector onto the effective susceptibility tensor.We define a number of phenomenological parameters related to the SME operators.This allows us to propagate the combined effects of the SME operators to the CMB spectra.We discuss the phenomenological impact of individual operators as well as their interplay.We then proceed to constrain the operators using data from a number of CMB experiments, as detailed in Section 3. Results are reported in Section 4. In Section 5, we translate the bounds obtained on the phenomenological parameters into bounds on the actual coefficients appearing in the minimal SME Lagrangian.We compare our results to constraints obtained both with other CMB datasets and with other kinds of observations.

Imprints of Lorentz violation on the CMB spectra
In this section, we introduce the theoretical model that describes Lorentz violating (LV) effects in the electromagnetic sector and propagate the effects to the cosmological observables of interest, namely, the CMB spectra.As we mentioned in the Introduction, we treat LV effects within the SME framework [5], focussing on the so-called minimal SME, which only contains renormalizable operators, with mass dimension d ≤ 4. For the photon sector and in a general spacetime with metric g µν this is characterized by the action where we set ε αβµν = αβµν / √ −g, with αβµν being the completely antisymmetric Levi-Civita symbol and g = det(g µν ).F µν and A µ are the field-strength tensor and the electromagnetic 4-potential, respectively.The first term in Eq. (2.1) is just the standard Maxwell Lagrangian.The couplings k AF account for operators with mass-dimension d = 3 which violate CPT symmetries besides Lorentz symmetries.The vector (k AF ) α has dimensions of a mass and 4 independent components.The couplings k F govern operators with massdimension d = 4 that are invariant under CPT.The tensor (k F ) µαβγ is dimensionless and obeys the following symmetries plus a vanishing double trace, thus implying a total of 19 independent components.Applying the Euler-Lagrange equations to the action in Eq. (2.1) leads to the following modified Maxwell's equations: The usual Maxwell's theory is invariant under conformal transformations of the metric g µν → ag µν .This guarantees that Maxwell's equations in a Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe, described by the metric are the same as in Minkowski spacetime with the metric η µν .Here τ represents the conformal time.In order for this invariance to be preserved by the LV theory in Eq. (2.1), the coefficients (k F ) αβµν must transform according to [51]: such that the a −4 factor cancels out the a 4 coming from √ −g.Instead, the vector (k AF ) α must be invariant under the conformal transformation, since the scaling of √ −g is canceled by that of the Levi-Civita tensor ε αβµν ∝ 1/ √ −g.
A well-known analogy exists between LV electrodynamics in vacuum and the standard Maxwell electrodynamics in an anisotropic medium, as first explored in Ref. [17] (see also Refs.[5,27]).This analogy can be exploited to define an effective susceptibility tensor χ ij from the modified Ampère-Maxwell equation, i.e. the space component (ν = i) of Eq. (2.4). 2his reads where ω and k are the comoving angular frequency and wave-number, respectively: (2.9) Note that the CPT-odd operator introduces in χ ij only terms that are zero-and first-order in the wave-vector, whereas the CPT-even operator produces also a contribution that is quadratic in k.This does not come as a surprise, since it is not possible to construct a quadratic term in the wave-vector by contracting its components with those of the 3D Levi-Civita tensor and the three-vector k AF .
To evaluate the effects of the LV operators on the CMB power spectra, we first need to link the susceptibility tensor to the components of the mixing matrix in the radiative transfer equation for the Stokes parameters Q, U and V of the polarized CMB radiation.Employing the formalism developed in Ref. [44], the components of the susceptibility tensor can be then recast in terms of three quantities, ρ Q , ρ U and ρ V , describing a general mixing between the U and V , Q and V , U and Q Stokes parameters, respectively (see Eq. ( 1) and Eq. ( 12) of Ref. [44]).Using the same conventions as in Ref. [44], we introduce where ρ ±2 = (ρ Q ± i ρ U )/ √ 2 and τ LS is the conformal time at the last scattering surface.As usual in CMB analysis, we expand (τ 0 − τ LS )ρ ±2/V in spherical harmonics 3 , with expansion coefficients b ±2/V, m .Note that the b V, m are only non-vanishing for = {0, 1}, whereas the b ±2, m are non-vanishing only for = 2.
At this stage, it is useful to combine the expansion coefficients b ±2/V, m to define the following dimensionless parameters: ) In fact, these parameters are directly connected to the CMB power spectra, as we will show below, and can be related to the physical parameters appearing in the action in Eq. (2.1) as follows: where the bar denotes quantities averaged along the line of sight, e.g. ( kAF ) 0 ≡ τ 0 τ LS (k AF ) 0 dτ and4 ( kF ) ijkl ≡ τ 0 τ LS ω(k F ) ijkl dτ .To derive these relations, we have assumed that the standard dispersion relation for photons holds true, i.e. ω = ck.In principle, one should take into account the corrections to the dispersion relation, which are of the kind ω When included in Eqs.(2.14)-(2.17),these corrections lead to higherorder contributions in k F and k AF .Since LV effects are constrained to be very small [40], we can work at leading order in the LV coefficients, so that we can take ω ck.
The phenomenological parameters of Eqs.(2.14)-(2.17)relate the observed CMB spectra C XX to those expected if no LV effects are in place, which we denote CXX .Keeping terms up to second order in the β's, which corresponds to working at second order in the parameters appearing in the action (2.1), we find: where 1 (2 + 1) (2.28) By inspecting Eqs.(2.18)-(2.24)we can identify the effects of different classes of LV operators: • the CPT-odd operators, parametrized by β 2 AF,T and β 2 AF,S , lead to the well-known cosmic birefringence effect.In particular, β 2 AF,T , related to the time component of the 4-vector k AF , gives rise to isotropic birefringence [26,[53][54][55][56][57][58], which produces nonvanishing EB and TB spectra and the mixing between EE and BB spectra.Anisotropic birefringence [59][60][61][62][63][64] is induced by the parameter β 2 AF,S , related to the space components of k AF .This mixes the EE and BB spectra by introducing a coupling among different multipoles (i.e.off-diagonal correlations), such that the -th multipole is coupled to both the ( − 1)-th and ( + 1)-th ones; • the VV spectrum is sourced from EE and BB spectra when the CPT-even operators are present.Similarly to what observed for anisotropic birefringence, a coupling between different multipoles is induced.In this case, it affects all the multipoles between the ( − 2)-th and the ( + 2)-th.Note that the VV spectrum is the only one which, if measured, could break the degeneracy between β 2 F,E and β 2 F,B , since in the other spectra only the sum of these two parameters comes into play.In this model, no mixing is predicted between V modes and E-or B-modes; • both the CPT-even and CPT-odd operators rescale the EE, BB and TE spectra via the parameter Z.
The modifications to the CPT-even linear polarization spectra, Eqs.(2.18)-(2.20),and the introduction of the circular polarization spectrum, Eq. (2.23), have been implemented in a customized version of the Boltzmann code CAMB [65,66], hereafter camb-cpt 5 .In the code, we have treated gravitational lensing of the CMB and the modifications induced by the extra terms in the action, Eq. (2.1), as two distinct effects.In principle, these two mechanisms should be propagated simultaneously along the line of sight, see for example Ref. [67].However, the kernel of the lensing effect is peaked at low redshift while the effect of the LV electrodynamics on the CMB is integrated from the last scattering surface and acts as a small correction.Therefore, as far as B-modes are concerned, we can safely rotate the tensor signal and then add the B-mode lensing contribution computed assuming no rotation.Instead, regarding EE and TE, we apply corrections to the lensed spectra.This is justified by considering that, for the noise level of current CMB experiments and even for the noise level of SO and LiteBIRD6 , there is a negligible difference between modifying the lensed spectra (i.e., applying the Lorentz-violating effect after the lensing contribution is included) and acting on the unlensed ones before adding lensing, see [44,67].
Figure 1 shows a comparison between the standard CMB spectra (solid) and those obtained with camb-cpt by setting all the β 2 parameters equal to 0.001 (dashed).The most relevant feature is the leakage of E-into B-modes.Another clear effect is the VV power spectrum mostly sourced by the E modes.The linear-polarization spectra are also rescaled by the Z factor in Eq. (2.25), which depends on all the β 2 parameters.The latter effect is barely visible on the scale of the figure.

Analysis method and dataset
We perform a Monte Carlo Markov Chain (MCMC) analysis to obtain constraints on the Lorentz-violating parameters β 2 AF,T , β 2 AF,S , β 2 F,E , β 2 F,B jointly with other cosmological, foreground and nuisance parameters.To this scope, the code camb-cpt has been interfaced with the MCMC sampler Cobaya [68].Using the Gelman-Rubin convergence statistics [69], we have assumed that our MCMC chains have reached convergence when R − 1 ∼ 0.01.
We analyze the following data: • Planck 2018: Planck temperature and polarization power spectra [70], and lensing reconstruction power spectrum [71], from the Planck 2018 legacy release.
• ACT: Atacama Cosmology Telescope temperature and polarization power spectra as published in the Data Release 4 [43].Since the ACT data are always used in combination The VV spectrum is non-vanishing only in the LV case, sourced by both the standard EE and BB spectra.Note that the EE spectra are almost overlapped and practically indistinguishable with this choice of the LV parameters.
with Planck, following the prescription of the ACT collaboration, we only consider multipoles larger than 1800 in temperature.For more details see section 6.2.3 of [43].
• VV: V modes power spectra as published by CLASS [72] and SPIDER [73] experiments.
For Planck, BICEP/Keck and ACT we employ the official likelihood packages released by the respective collaborations [42,43,70,71].For the V-modes data, a simple custom-made likelihood has been added to the framework.The χ 2 for the V modes is computed as: where D V V b,data and D V V b,theory are the data and the binned theory respectively and σ 2 b is the error on the bandpowers. 7Since CLASS and SPIDER are both completely noise dominated, we can safely add together their respective χ 2 computed as in Eq. (3.1).
The ΛCDM+r model (i.e., allowing for non-vanishing primordial gravitational waves with amplitude set by the tensor-to-scalar ratio r) provides our baseline scenario, unless otherwise stated.See Ref. [74] for details about parametrization, theoretical assumptions and priors used.For the foreground and nuisance parameters, we follow the prescriptions provided by Planck [74] and BICEP [42] collaborations.In addition to the baseline, we consider the β 2 parameters defined in Eqs.(2.14)-(2.17).On those parameters we impose uniform positive priors.Further model extensions are not considered in this work.

Constraints on phenomenological parameters
In this section, we present the constraints derived on β 2 AF,T , β 2 AF,S , β 2 F,E and β 2 F,B using the aforementioned datasets and parametrizations.

Constraints on CPT-odd terms only
As a first step in our analysis, we consider only the CPT-odd term in Eq. (2.1) and fix to zero the parameters related to the CPT-even term.The effect of this term on the CMB spectra is encoded in two parameters β 2 AF,T and β 2 AF,S and leads to isotropic and anisotropic birefringence effects, respectively.In Figure 2, we show the two-dimensional and one-dimensional posterior probability distributions of a subset of cosmological parameters, including β 2 AF,T and β 2 AF,S , explored in the analysis with the combination of Planck + BK18 data.The baseline model is given by the ΛCDM+r cosmology.To better elucidate the effect of β 2 AF,T and β 2

AF,S
on the constraints of the remaining parameters, we also vary them one at the time while fixing the other to zero.We note that varying either β 2 AF,T or β 2 AF,S has equivalent impact on the constraints on other cosmological parameters.This is due to the fact that both β 2 AF,T and β 2 AF,S lead to qualitatively equivalent modifications of the BB spectrum.Indeed, an inspection of Eq. (2.20) and Eq.(2.27) shows that the overall effect produced by non-vanishing β 2 AF,T or β 2 AF,S is an effective rotation of E-modes into B-modes.Such rotation competes with r in increasing the power in B-modes (see Figure 1, where the two β 2 AF and r enhance the reionization and recombination bumps in the BB power spectrum).This explains why the marginalization over β 2 AF,T and β 2 AF,S tightens the constraints on r with respect to those obtained in the ΛCDM+r baseline analysis.
Even though in Figure 2 we report results from Planck +BK18, the two β 2 AF could be also constrained with Planck data only, exploiting their effect on E-mode polarization.However, the resulting bounds on β 2 AF,T and β 2 AF,S are nearly an order-of-magnitude broader than those obtained when adding BK18 to Planck data.This is due to the lack of constraining power from B-modes which are more strongly affected by the two β 2 AF .In Fig. 3a, we show the constraints on a subset of parameters and compare the results obtained with Planck data only in ΛCDM+β 2 AF and ΛCDM+r+β 2 AF with those obtained with the combination of Planck + BICEP/Keck data in ΛCDM+r+β 2 AF .As expected, the bounds on β 2 AF are tightened when r is varied jointly with the CPT-odd parameters, even if using Planck data only.However, the improvement is dramatic when BICEP/Keck data are added to the analysis.In Fig. 3b we show a zoom-in of the lower right triangle of Figure 3a to better appreciate the impact of BICEP/Keck data on the constraints on the β 2 AF .We stress again that no V-modes are sourced by the CPT-odd term of the Lagrangian.

Constraints on CPT-even terms only
We now focus on the CPT-even term of the action in Eq. (2.1).The effects on the CMB spectra are in this case encoded by the two parameters β 2 F,E and β 2 F,B , which are responsible for an overall rescaling of the TE, EE and BB C s via the parameter Z, see Eqs. (2.18), (2.19), (2.20).If we restrict our analysis to consider only linear polarization, the impact of the two β 2 F,E/B is degenerate.However, the CPT-even term sources a degree of circular polarization from a mixing of E-and B-modes appropriately rescaled by β 2 F,E and β 2 F,B , see Eq. (2.23).The sourcing of V-modes could in principle be used to individually constrain the β 2 F,E/B , provided that a V-mode experiment puts statistically significant bounds on the VV signal.However, the signal-to-noise ratio in the SPIDER and CLASS data is insufficient to put significant bounds on the two parameters.This is shown in Figure 9 of the App.A, where the posterior distributions on cosmological parameters, including β 2 F,E , with and without V-mode data are perfectly overlapping.We expect this to be exactly the same for β 2 F,B , since in the absence of sensitive enough V-mode data both β 2 F,E and β 2 F,B parameters are constrained through the rescaling of TE, EE and BB spectra within Z. Therefore, in the following, we neglect the contribution of V modes data and we quote results for the effective parameter β 2 F , defined as In Fig. 4, we show 2D and 1D posterior probabilities of a subset of cosmological parameters explored with the combination of Planck +BK18 and Planck +BK18+ACT data.We compare the results within the ΛCDM+r + β 2 F model and the baseline ΛCDM+r model.Differently from what discussed for the CPT-odd parameters, we do not see any improvement in the bounds on r when β 2 F is varied.In this case, we expect a positive correlation between r and β 2 F , contrarily to what happens with the β 2 AF .Indeed, a non-vanishing β 2 F reduces the amplitude of the BB spectrum, which could be compensated by higher values of r.However, we do not appreciate such a correlation in Fig. 4. The reason is that most of the constraining power on β 2 F comes from TE and EE spectra, making any degeneracy with r undetectable.Indeed, the sensitivity on β 2 F from T-and E-modes only is at the same level as that on β 2 AF,T /S , being driven by the scaling in amplitude of EE and TE spectra.In Fig. 4, we also note a shift in Ω b h 2 and Ω c h 2 with respect to the constraints obtained when β 2 F = 0.The shifts can be easily explained when considering the impact of the parameters on the shape of the TE and EE spectra.The main effect of the non-vanishing β 2 F on the polarization power spectra is to rescale their overall amplitude through Z in Eq. (2.25).A change in Ω b h 2 , instead, affects the amplitude of the TE and EE acoustic oscillations both in the photon density field (by modifying the inertia of the baryon-photon fluid, which is relevant for the temperature transfer function) and in the photon velocity field (as a result of the change in the density), which is relevant for the E-polarization transfer function.From these considerations, we can understand the correlation between β 2 F and Ω b h 2 .At sub-degree scales (high multipoles ), a change in Ω b h 2 modifies the damping angular scale since a different baryon density affects the photon mean free path.As a result, the power at small scales is more or less suppressed depending on the value of Ω b h 2 .This effect goes in the opposite direction of the change in the amplitude of the first peaks: a lower value of Ω b h (in blue) and ΛCDM+r (in orange) using Planck TTTEEE+lensing+BK18 data.Note the tighter limit on r when one of the β 2 AF parameters is allowed to vary with respect to the case in which they are both equal to zero.Opening to both β 2 AF further improves the individual constraints on β 2 AF,T , β 2 AF,S and r, see the main text for a detailed discussion.
the amplitude of the oscillations at intermediate scales and suppresses the power at small scales.A similar effect at intermediate scales is provided by Ω c h 2 .A decrease of the latter delays the onset of matter-radiation equality, thus shifting to larger scales the boosting effect due to radiation driving on the acoustic oscillations.Therefore, we expect Ω c h 2 to decrease when allowing for a non-vanishing β 2 F .The inclusion of ACT data causes the same shift of Ω b h 2 and Ω c h 2 when sampling over β 2 F , as can be seen in Fig. 4.Moreover, the limit on β 2 F is broader.This is likely driven by the known preference of ACT for larger A s and n s [43], which can be compensated by a larger value of β 2 F .

Joint constraints on CPT-odd and CPT-even terms
Finally, we investigate the case in which all the CPT-even and CPT-odd parameters are varied jointly.This allows us to investigate how the interplay between the effects induced by different Figure 3: On the left, one and two-dimensional posterior probability distributions for a subset of parameters varied in the MCMC analysis.We report the constraints obtained when assuming a ΛCDM+β 2 AF,T + β 2 AF,S (in pink) and a ΛCDM+r+β 2 AF,T + β 2 AF,S (in cyan and green) models.The former using only Planck TTTEEE+lensing dataset, while the latter using both Planck TTTEEE+lensing and Planck TTTEEE+lensing+BK18 datasets.Note how much the constraints on the β 2 AF parameters improve when we include BK18 data.On the right, a zoom-in showing the constraints on β 2 AF,T and β 2 AF,S using Planck TTTEEE+lensing+BK18 datasets.
operators affects the constraints on the LV parameters.We have collected the 95% CL on r, β 2 AF,T , β 2 AF,S , β 2 F for the cases analyzed in Tab. 1.The posteriors on all the cosmological parameters, including those not quoted in this Section, can be found in the Appendix A. Figure 5 shows the 2D and 1D posterior probabilities of a subset of cosmological parameters plus the β 2 s assuming a ΛCDM+r+β 2 AF,T +β 2 AF,S +β 2 F model.For comparison, we have also included the posteriors for the ΛCDM+r+β 2 AF,T +β 2 AF,S and ΛCDM+r+β 2 F models.On the one hand, we see that the bounds on β 2 F improve when all the β 2 are allowed to vary.In fact, in absence of V-mode data, the only effect of β 2 F is to contribute to the rescaling of the CMB spectra via Z, in the same way as β 2 AF,T and β 2 AF,S do.On the other hand, the constraints on β 2 AF,T and β 2 AF,S do not improve significantly when the two parameters are varied jointly with β 2 F .In fact, besides rescaling the spectra, they also induce a mixing between E and B modes, which allows to disentangle them from β 2 F .Note again the improved bounds on r when β 2 AF,T and β 2 AF,S are varied.The inclusion of ACT mostly affects the constraint on β 2 F (see Figure 6), as discussed before.F (in purple when using the Planck TTTEEE+lensing+BK18 dataset, in blue when adding ACT) and ΛCDM+r (in dashed orange and dashed cyan respectively).Since not enough constraining power comes from current V-mode data, we are note able to disentangle the effects of β 2 F,E and β 2 F,B , and we can only set a limit on their combination Note the shifts in the posteriors of Ω b h 2 and Ω c h 2 when considering the ΛCDM + r + β 2 F extension, see the main text for a detailed discussion.

Implications for the LV coefficients in the minimal SME action
In this Section, we translate the bounds on the phenomenological parameters β 2 AF,T , β 2 AF,S and β 2 F introduced in Eqs.(2.14), (2.15) and (4.1) into constraints on the LV couplings k AF and k F appearing in the action in Eq. (2.1).We focus on the constraints obtained with the full dataset combination, Planck +BK18+ACT.We report these results in Tab. 2.
Focussing first on the CPT-odd effects, the constraints on the time component of k AF are usually rephrased as bounds on the parameter k [40,55]).This parameter can be linked to the phenomenological parameter β 2 AF,T as follows: where we have assumed that (k AF ) 0 is constant along the line of sight and To account for the fact that we are combining information coming from different experiments, observing the sky in different frequency channels, we can define an effective frequency ν f following the method presented in Ref. [28].Given the frequency dependence in Eq. (5.4), we Figure 6: One and two-dimensional posterior probability distributions for the LV parameters β 2 varied in the MCMC analysis.We report the constraints obtained when assuming a ΛCDM+r+β 2 AF,T +β 2 AF,S +β 2 F model using Planck TTTEEE+lensing+BK18 (in dark blue) and Planck TTTEEE+lensing+BK18+ACT (in red).Including ACT data weakens the constraints mostly on β 2 F , see the main text for a detailed discussion. find where ν i − , ν i + is the frequency interval of the i-th frequency channel and σ i is the noise level.Using Eq. (5.5), we obtain ν f = 158.8GHz, 121.7 GHz and 122.7 GHz for Planck [41,75], BK18 [42,76] and ACT [43,77], respectively.
We now report the 95% CL constraints on the LV coefficients using Planck +BK18+ACT data, in the case where the three parameters β 2 AF,T , β 2 AF,S and β 2 F are all free to vary.For the CPT-odd terms we find ) whereas for the CPT-even operator we obtain  2.28) show how the β 2 parameters affect the CMB spectra.The limits have been expressed in units of 10 −2 .The key "VV" represents the combined CLASS+SPIDER dataset for V-modes.
Note that the bound on k F,E+B in Eq. (5.8) has been obtained by normalizing the effective frequency to 121.7 GHz, which is the value computed for BK18.This choice is motivated by the fact that BK18 data give the highest constraining power on the LV coefficients, see discussion in Sec. 4. We remind the reader that the full set of constraints derived from different data and parameter combinations can be found in Table 2.
The bounds on the LV coefficients derived in previous literature are collected in [40], see Tables D15 and D16.For the CPT-odd case, an upper bound on the parameter |k (3) (V )00 | has been obtained in Ref. [51] using WMAP data, leading to the result |k −43 GeV at 95% CL.We note that the limit derived in our analysis using Planck +BK18+ACT data is stronger by more than one order of magnitude, see Eq. (5.6).Analogously, a limit on the coefficient |k AF | from WMAP data has been obtained in [35,55], yielding |k AF | < 2 × 10 −42 GeV at 95% CL.In this case, the bound derived in our analysis is stronger by two orders of magnitude, see Eq. (5.7).We stress that the bounds on the CPT-odd coefficients derived in this work are the strongest to date, both considering CMB and other sources.See again Ref. [40] for an exhaustive list of current bounds.
For what concerns CPT-even Lorentz violation, our bound on k F,E+B improves previous constraints by roughly one order of magnitude [35].The CMB-based cosmological bounds on the CPT-even coefficients are only overcome by those obtained from optical polarimetry of extragalactic sources, see Refs.[40,78,79].
The bounds presented in the previous paragraphs are obtained in the most general case with all the β 2 parameters jointly varied.This represents a further novelty of our work.However, it is worth mentioning that, since all the parameters compete for the same power, the bounds obtained with a single parameter exploration are slightly weaker, as can be seen

Conclusions
In this paper, we have derived the signatures of Lorentz-violating (LV) electrodynamics on the polarization of the cosmic microwave background (CMB) and provided the most stringent constraints to date on LV coefficients from CMB observations.We computed the modified CMB spectra, employing the full expression of the LV action given in Eq. (2.1), and we performed a likelihood analysis exploiting the most recent CMB datasets.To our knowledge, this is the first time that such an end-to-end analysis has been performed.We considered the minimal Standard Model extension of electrodynamics, including both CPT-odd (mass dimension d = 3) and CPT-even (mass dimension d = 4) operators.The CPT-odd operator, characterized by the 4-vector (k AF ) µ , is responsible for the standard cosmic birefringence effect (isotropic and anisotropic).The CPT-even operator, instead, is characterized by a tensor (k F ) µνρσ and converts linear into circular polarization, giving rise to a non-zero Vmode spectrum.
The expressions for the modified CMB spectra are presented in Eqs.(2.18)-(2.28)and are obtained following the formalism laid down in Ref. [44].The LV effects are encoded in four phenomenological parameters, defined in Eqs.(2.14)-(2.17).The parameters characterizing the CPT-odd term are β 2 AF,T and β 2 AF,S , related to the time and space components of k AF , respectively.The CPT-even terms are β 2 F,E and β 2 F,B , which depend on the components of the tensor k F .The theoretical predictions of the modified CMB spectra in presence of LV effects are computed by using a customized version of the Boltzmann solver camb. 9e derived constraints on the phenomenological LV parameters from state-of-the-art CMB datasets: Planck [41], BK18 [42], ACT [43], CLASS [72] and SPIDER [73].Table 1 shows the 95% confidence intervals of the β 2 parameters, for different combinations of datasets and different choices of the underlying cosmological model.Sampling the LV coefficients does not affect significantly the standard cosmological parameters.The tensor-to-scalar ratio r represents the only relevant exception.Indeed, the constraint on r is ∼ 10% tighter when all β 2 parameters are sampled with respect to the ΛCDM+r model (see the discussion in Section 4 and the full triangle plots in Appendix A).As far as the CPT-even term is concerned, we found that current V-mode datasets have negligible constraining power compared to measurements of linear CMB polarization, in agreement with previous findings (see Ref. [44]).
Finally, we recast the constraints on the phenomenological parameters β 2 into bounds on the coefficients of the CPT-even and -odd operators appearing in the minimal SME action, see Table 2.We compared the constraints derived in this work with previous bounds from astrophysical and laboratory probes available in literature 10 .
Our constraints on the CPT-odd parameters, i.e. |k (3) (V )00 | < 1.54 × 10 −44 GeV and |k AF | < 0.74×10 −44 GeV, are roughly one and two orders of magnitude tighter than previous CMB limits, respectively.Moreover, they are the strongest bounds obtained to date on the CPT-odd LV coefficients considering all other probes.Concerning the CPT-even case, the bounds are currently dominated by the constraint coming from optical polarimetry of extragalactic sources.Nevertheless, we improve previous CMB-based results by one order of magnitude, yielding k F,E+B < 2.31 × 10 −31 .
Forthcoming CMB experiments, such as LiteBIRD [49], Simons Observatory [48] and CMB-Stage 4 [50], will largely improve our sensitivity on such extensions of the standard electrodynamics, thanks to unprecedented sensitivity to linear CMB polarization as well as better sensitivity to V-mode polarization.A rough estimate of the expected improvements can be obtained by conservatively assuming that the constraints on the β 2 parameters will still be dominated by B-mode measurements.Future CMB experiments will increase their sensitivity to the tensor-to-scalar ratio r by more than a factor of twenty compared to current bounds.The improvement on r can be then translated to the same improvement on each β 2 , since both parameters act as a rescaling factor for the BB spectrum (see Eq. (2.20)).From Eqs. (5.1),(5.3),(5.4), it is straightforward to eventually forecast a factor of 5 improvement on the physical coefficients in the LV action.Note that this is a conservative estimate since it does not account for the increased constraining power coming from more accurate measurements of E-mode polarization.Improved V-mode bounds would also allow to disentangle the effects of the phenomenological β 2 F,E and β 2 F,B parameters.This would potentially set individual bounds on these two coefficients, whose effects are indistinguishable when exploiting measurements of linear polarization only, see discussion in Sec. 4. A detailed forecast analysis is left as the subject of a future publication.F,E model using the Planck TTTEEE+lensing+BK18+CLASS+SPIDER dataset (in lime) and assuming ΛCDM+r+β 2 F,E (in purple) and ΛCDM+r (in orange), both using the Planck TTTEEE+lensing+BK18 dataset.The cases ΛCDM+r+β 2 F,E with and without V-modes data are perfectly overlapping, showing the lack of constraining power from the current circular polarization data and justifying the choice of sampling over the combination β 2 F in Eq. (4.1)  F .The addition of ACT data widens the limit on β 2 F , due to the preference of ACT data for higher A s and n s .Figure 12: One and two-dimensional posterior probability distribution for the full set of parameters varied in the MCMC analysis.We report the constraints obtained when assuming the ΛCDM+r+β 2 AF,T +β 2 AF,S +β 2 F model using Planck TTTEEE+lensing+BK18 (in dark blue) and Planck TTTEEE+lensing+BK18+ACT (in red) datasets.For comparison, we also the constrain obtained assuming a ΛCDM+r model using the Planck TT-TEEE+lensing+BK18+ACT dataset(in violet).Including ACT data in the analysis degrades the constraints on β 2 F .The shifts between the blue and red cases in some cosmological parameters (such as n s , θ MC and A s ) are due to the addition of ACT data.They are indeed also present in the violet case.This shift can be also noticed in Figure 17 of Ref. [43].

Figure 4 :
Figure4: One and two-dimensional posterior probability distributions for a subset of parameters varied in the MCMC analysis.We report the constraints obtained when assuming ΛCDM+r+β 2 F (in purple when using the Planck TTTEEE+lensing+BK18 dataset, in blue when adding ACT) and ΛCDM+r (in dashed orange and dashed cyan respectively).Since not enough constraining power comes from current V-mode data, we are note able to disentangle the effects of β 2F,E and β 2 F,B , and we can only set a limit on their combination β 2 F = (β 2 F,E + β 2 F,B )/4.Note the shifts in the posteriors of Ω b h 2 and Ω c h 2 when considering the ΛCDM + r + β 2 F extension, see the main text for a detailed discussion.

Figure 9 :
Figure9: One and two-dimensional posterior probability distribution for the full set of parameters varied in the MCMC analysis.We report the constraints obtained when assuming the ΛCDM+r+β 2F,E model using the Planck TTTEEE+lensing+BK18+CLASS+SPIDER dataset (in lime) and assuming ΛCDM+r+β 2F,E (in purple) and ΛCDM+r (in orange), both using the Planck TTTEEE+lensing+BK18 dataset.The cases ΛCDM+r+β 2F,E with and without V-modes data are perfectly overlapping, showing the lack of constraining power from the current circular polarization data and justifying the choice of sampling over the combination β 2 F in Eq. (4.1)

Figure 10 :
Figure10: One and two-dimensional posterior probability distribution for the full set of parameters varied in the MCMC analysis.We report the constraints obtained when assuming the ΛCDM+r+β 2 F model using the Planck TTTEEE+lensing+BK18 dataset (in orange) and Planck TTTEEE+lensing+BK18+ACT (in blue) and when assuming the ΛCDM+r model using Planck TTTEEE+lensing+BK18 (in dashed orange) and Planck TT-TEEE+lensing+BK18+ACT (in dashed cyan).Notice the shifts in Ω b/c h 2 due to the sampling of β 2 F .The addition of ACT data widens the limit on β 2 F , due to the preference of ACT data for higher A s and n s .
Planck TTTEEE+lensing+BK18+ACT dataset.The posterior in dashed yellow is the reference for the ΛCDM+r case using same dataset.The joint marginalization over all the β 2 parameters improves the constraints on β 2 F , while keeping unchanged those on r and the β 2 AF parameters.For what concerns the CPT-even effects, recasting our constraints on β 2 F into bounds on the components of k F is less trivial, due to the frequency dependence of Eqs.(2.16)-(2.17).
[74]rder to get the estimate in Eq. (5.2), we have used the best-fit values for the cosmological parameters taken from Planck 2018 (TT, TE, EE + lowE constraints for ΛCDM model)[74].F Figure 5: One and two-dimensional posterior probability distributions for the LV parameters β 2 varied in the MCMC analysis.We report the constraints obtained when assum-Analogously, from Eq. (2.15) we find for the space components of k AF |k AF | 2.93 × 10 −43 β 2 AF,S GeV .(5.3)

Table 2 :
Bounds at 95% CL on k |k AF | and k F,E+B for the listed datasets and models The constraints on k F,E+B are derived taking ν f = 158.8GHz for Planck alone and ν f = 121.7 GHz for the combination of Planck, BK18 and ACT.As discussed in the main text, this choice is justified by the highest constraining power on LV coefficients given by BK18 data. in

Table 2 .
One and two-dimensional posterior probability distribution for the full set of parameters varied in the MCMC analysis.We report the constraints obtained when assuming ΛCDM+r+β 2 AF,T +β 2 AF,S (in green), ΛCDM+r+β 2 AF,T (in red), ΛCDM+r+β 2 AF,S (in blue) and ΛCDM+r (in orange) using the Planck TTTEEE+lensing+BK18 dataset.The marginalization over either β 2 AF,T or β 2 AF,S has the same effect on the other parameters.It is worth to underline the tighter limit on r with respect to the case in which each β 2 AF is equal to zero.Opening to both the β 2 AF parameters shrinks even more the constraints on β 2 One and two-dimensional posterior probability distribution for the full set of parameters varied in the MCMC analysis.We report the constraints obtained when assuming ΛCDM+β 2 AF,T + β 2 AF,S (in pink) and ΛCDM+r+β 2 AF,T + β 2 AF,S (in cyan and green).The former using only Planck TTTEEE+lensing dataset, while the latter using both Planck TTTEEE+lensing and Planck TTTEEE+lensing+BK18 datasets.Note how much the constraints on the β 2 AF parameters shrink when marginalizing over r and adding the BK18 dataset.The shift on n s and Ω b h 2 is instead due especially to the change of dataset.
CDM+rFigure 7: One and two-dimensional posterior probability distribution for the full set of parameters varied in the MCMC analysis.We report the constraints obtained when assumingΛCDM+r+β 2 AF,T +β 2 AF,S +β 2 F (in dark blue), ΛCDM+r+β 2 AF,T +β 2 AF,S (in green), ΛCDM+r+β 2F (in purple) and ΛCDM+r (in orange), using the Planck TT-TEEE+lensing+BK18+ACT dataset.Varying all the β 2 parameters together shrinks the constraints on β 2 F , leaving almost unchanged those on r and the β 2 AF parameters.Note the same shifts on Ω b h 2 , Ω c h 2 in both the blue and the purple case, driven by the marginalization over β 2 F .