arXiv:astro-ph/0601095v3 8 Jun 2006 ucino ic scalar Ricci of function inlbrolpoeei 7 8], [7, baryo/leptogenesis tional causes which h orvector baryo/leptogenesis four quintessential of the scenario the In zero. h unesne1]scalar, quintessence[12] the h ulo h lcrmgei esr h cino (1) of if action invariant The gauge tensor. electromagnetic is the of dual the unesne h iecmoetof component time the quintessence, etro h rdeto clr tvoae oet and Lorentz violates It scalar. T a CP of gradient the or vector where hr-iostr nteeetv arnino the of lagrangian effective 11] the [10, in form introduce term we Phenomenologically, Chern-Simons a sector. photon the in iru 4 ,6 ,8 9]. 8, 7, equi- 6, thermal 5, in [4, produced librium is the asymmetry where number mechanism baryon baryogenesis a provide leptons and evolution. cosmological in example, role For active an played have may o omlg rvdsaohrwyt etti im- this test the to of breaking way Also, symmetry. another portant provides cosmology experimen- precision Now on high tests of number model. of tal standard a the detection been beyond have physics The physics There particle new reveal of symmetry. will tion model exact standard an the is In tion. icin ftemn hoeia oeso e physics. new of models dis- possible, theoretical effective cosmological many and now the the tests of is robust tinctions of have It to precision 2], observations[1, unprecedented of established. the fronts structure been with of observational has model” the “standard formation a and research, theoretical cosmological the both erhn o P ilto ihCsi irwv Backgrou Microwave Cosmic with Violation CPT for Searching nti ae esuytecosmological the study we paper this In osbesgaueo e hsc sthe is physics new of signature possible A in advances many and pursuance of decades After ymtisi h akrudvleof value background the if symmetries d p nttt fHg nryPyis hns cdm fScienc of Academy Chinese Physics, Energy High of Institute µ safu etrand vector four a is a em obndwt h 0 aa efudta ozr rota nonzero a interestin that an found put we could data, ∆ B03 one favored: the alone, with data Combined WMAP term. vector the polarization Using the four-vector, tated. external couple an which to lagrangian, tensor effective the in term Chern-Simons OMRN B3 aa ent hti h oet and Lorentz the if t that using note We by anisotropies data. polarization (B03) BOOMERANG and temperature (CMB) ainlAtooia bevtre,CieeAaeyo S of Academy Chinese Observatories, Astronomical National PT CP PT CP PT CP esac o intrso oet and Lorentz of signatures for search We p c µ oFeng Bo L ntttfu hoeicePyi,Pioohne 6 691 16, Philosophenweg Physik, f¨ur Theoretische Institut voaigitrcin ntebaryons the in interactions -violating ilto.I h cnroo gravita- of scenario the In violation. osraini h aoaoy[3]. laboratory the in conservation si h omo h eiaieof derivative the of form the in is int p rdaeSho fSine h nvriyo oy,Tko1 Tokyo Tokyo, of University The Science, of School Graduate µ α R ∼ sacntn n homogeneous and constant a is = p [13]. − µ a,b ∂ 6 A µ . igh Li Mingzhe , φ 0 ν +4 − uigteeouinof evolution the During . F F e e 4 b p . µν . µν 0 0 µ eerhCne o h al Universe(RESCEU), Early the for Center Research +3 − ∂ , stegain fa of gradient the is 3 (1 = µ . . 9 7 φ e 1 2 (1, deg PT CP osntvanish, not does / PT CP 2) c u-igXia Jun-Qing , PT CP n BOOMERANG and PT CP ǫ µνρσ p symmetry µ violation σ snon- is F ). viola- viola- PT CP ρσ (1) PT CP is d iltosi h omcmcoaebackground microwave cosmic the in violations invco fec htni oae ya nl ∆ angle an by rotated is photon each polariza- of the vector birefringence, tion cosmological of With presence in- harmonics. the the spherical tensor to the due of vanish properties spectra trinsic and power TC correlation in cross the a violated GC not and distribution, is (G) temperature/polarization parity gradient-like If the a [17]. components into (C) decomposed curl-like be can tion h urn M aapoiea neetn indication interesting nonzero an a provide for data CMB current the oigterttdqatte ihapie n es[18] De- gets one surface. prime, scattering a last with the quantities at rotated the zero noting are correlations, they GC if and TC even non-zero observe would one and iltn term violating zto aat esr hstp fLrnz and Lorentz- of type this measure to data 16]. ization 15, 11, [10, used electrodynamics been modified has this constrain and to strength birefringence, the cosmological of the measure of sensitive a galaxies provides radio quasars, distant and from radiation polarized observing and[19] p nl fteplrzto ln ∆ plane polarization position the as the of known in angle change is The effect birefringence”. This “cosmological propagating [10]. vec- are distances polarization they cosmological the when over of waves rotation electromagnetic a of to tor lead can term This ere,a ogas long as metries, nteohrhn,teoiia G GadC spectra CC and GG modified: TG, also original are the hand, other the On uliChen Xuelei , µ h tksprmtr n fteCBpolariza- CMB the of U and Q parameters Stokes The ntepeetpprw s h urn M polar- CMB current the use we paper present the In h neato n()voae lothe also violates (1) in interaction The stegain fatm eedn clrfil)[14]. field) scalar dependent time a of gradient the is ,PO o 1-,Biig104,P .China R. P. 100049, Beijing 918-4, Box P.O. e, fpoaaigCBpooswl e ro- get will photons CMB propagating of s C h uleetoantcfil strength field electromagnetic dual the s l ′ G T ine,Biig101,P .China R. P. 100012, Beijing ciences, eWA n h 03flgtof flight 2003 the and WMAP he C PT CP 0Hiebr,Grayand Germany Heidelberg, 20 inageo h htn smildly is photons the of angle tion osrito h ieo uha such of size the on constraint g l ′ = GC p a µ o h rttime first the for n imnZhang Xinmin and , . C C = ymtisaeboe ya by broken are symmetries l G T l ′ C T p 1 2 303,Japan 13-0033, 0 ( o 2∆ cos C oentvns eg,i h case the in (e.g., vanish not dose = l GG C dDt rmWMAP from Data nd l G T − , α C i 2∆ sin l CC u eut hwthat show results Our . α hc sotie by obtained is which , i 4∆ sin ) d α P . α and CP PT CP sym- (2) (3) (4) α - , 2

C′GG = CGG cos2 2∆α + CCC sin2 2∆α , (5) l l l ′CC CC 2 GG 2 Cl = Cl cos 2∆α + Cl sin 2∆α . (6) WMAP3

WMAP3 + B03 (TT,TG,GG,CC))

WMAP3 + B03 full dataset From the above discussion, we see that even with only 1.0 the TG cross correlation power spectrum (and the TT L autocorrelation power spectrum), the Lorentz- and CPT -

0.5 violating term can still be measured. Of course, direct measurements of the TC and GC power spectra would allow more stringent constraints. Indeed, the GC spec- 0.0

-20 -10 0 10 20

trum will be the most sensitive probe of such Lorentz- (deg) and CPT -violating term [19], this is because the GC power spectrum is generated by the rotation of the GG power spectrum, which is a more sensitive probe of the FIG. 1: (color online). One dimensional constraints on the α primordial fluctuation than the TT and TG spectra [20]. rotation angle ∆ from WMAP data alone (Green or light gray line), WMAP and the 2003 flight of BOOMERANG B03 To break possible degeneracy between this term and TT, TG, GG and CC(Orange or gray line), and from WMAP the variation of other parameters, we make a global fit to and the full B03 observations (TT, TG, GG, CC, TC, GC) the CMB data with the publicly available Markov Chain (Black line). Monte Carlo package cosmomc[21, 22], which has been modified to allow the rotation of the power spectra dis- cussed above, with a new free parameter ∆α. We as- +11.6 +5.9 sume purely adiabatic initial conditions, and impose the ∆α =0.0−11.7 −5.9 deg. The uncertainty is considerable, flatness condition motivated by inflation. The follow- as the error bars of the WMAP TG data are relatively ing set of 8 cosmological parameters are sampled: the large, and TG data are not very sensitive probe. In the Hubble constant h, the physical baryon and cold dark likelihood of Fig. 1 we have gained double peaks, which 2 2 can be easily understood from Eq.(4) due to the symme- matter densities, ωb = Ωbh and ωc = Ωch , the ratio of the sound horizon to the angular diameter distance at try around ∆α = 0. With the inclusion of the B03 data, the measurement decoupling, Θs, the scalar spectral index and the over- could be improved dramatically. In a first step we also all normalization of the spectrum, ns and As, the ten- sor to scalar ratio of the primordial spectrum r, and, consider the indirect measurements only by including the B03 TT, TG, GG and CC data. We find the constraint on finally, the optical depth to reionization, τr. We have imposed the Guassian Hubble Space Telescope prior[23]: ∆α becomes a bit more stringent compared with WMAP h = 0.72 ± 0.08 and also a weak big-bang nucleosynthe- only, a nonzero ∆α is slightly favored and the double 2 peaks are still present. When the B03 TC and GC data sis prior[24]: Ωbh = 0.022 ± 0.002 (1 σ). For the other are also included the degeneracy around ∆α = 0 is bro- parameters we have adopted flat priors, and in the com- +4.0 ken. We get the 1, 2 σ constraints to be ∆α = −6.0−4.0 putation of the CMB spectra we have considered lensing +3.9 contributions. −3.7 deg with WMAP3 and the B03 full data set. When the first version of this paper was being pre-

pared, the available temperature and polarization power

spectra included the results from the first-year observa- TC rotated only

GC rotated only tion of WMAP [1, 25, 26, 27], and those from the Jan- 1.0 uary 2003 Antarctic flight of BOOMERANG (Hereafter

B03)[28, 29, 30]. The WMAP three year data (WMAP3) L

has since been released, with significant improvements on 0.5 the estimate of the TE power at small ℓ [31]. We have repeated our analysis with the new WMAP three year

data. 0.0

-50 -40 -30 -20 -10 0 10 20 30

In Fig. 1 we plot our one dimensional constraints on (deg) ∆α from the WMAP data alone, and from the combined WMAP and B03 data. We have assumed that the cos- mic rotation is not too large and imposed a flat prior FIG. 2: (color online). One dimensional constraints on the rotation angle ∆α from WMAP and the B03 observations, −π/2 ≤ ∆α ≤ π/2. The CMB temperature power spec- assuming only CMB TC is rotated to be nonzero by ∆α(Green trum remains unchanged with the rotation while the TG or gray line) and only GC rotated(Black line). power spectrum gets modified, as given by Eq. (4). Using the data from WMAP alone, for both the first and three year data set, we obtain a null detection within The covariance matrices of the B03 TC and GC data the error limits. For WMAP3 the 1, 2 σ constraints are are correlated. In order to find out what role the TC and 3

GC data play in our fitting respectively, we have made been constrained by looking for correlations between the fits with, in one case, only the TC spectrum rotated as elongation axes and polarization vectors of distant ra- Eq. (2), and in the other case only the GC spectrum dio galaxies and quasars. The most recent searches yield rotated. To make the comparison clear and avoid the null results, with an error on ∆α at the order of 1◦ level problem of convergence we set the flat prior −1.2 ≤ ∆α ≤ [11, 15, 16]. The typical redshifts of the sources in these 0.8. In Fig. 2, we plot the resulting one dimensional searches are of order of unity. Conceivably, for the much constraints. In neither case is the likelihood symmetric greater redshift range between the last scattering surface at ∆α = 0. In the TC rotated only case, the symmetric and the present-day observer, the cumulative effect of points are around ±π/4, as we can see from Eq.(2). Such cosmological birefringence could be stronger. a symmetry is lost for this narrower prior, but in our It was claimed that the individual B03 CC and CG global fittings (Fig. 1) we have allowed a larger range of data are consistent with zero [29], however, we found that ∆α. We find from Fig. 2 the TC data are very weak a negative rotation angle is preferred in our combined in breaking the degeneracy around ∆α = 0, while for analysis. It is noteworthy that our result relies mainly GC the rotation is more eminent, where the likelihood in on the fact that at l ∼ 350, GG power of B03 is positive, Fig. 2 is centered around ∆α = −π/8. In this fit ∆α =0 CC power is (slightly) positive and GC power is (slightly) has an excess of ∆χ2 = 4, which is disfavored compared negative. Given Eq.(3), the GC power spectrum helps to with the best fit case. increase the statistical significance on nonzero ∆α. At present, the only publicly available (polarization) data 0.16 1.04 are the three-year WMAP and the data from a 200-hour 0.14 1.02 0.12 flight of the BOOMERANG balloon. In the coming few 1 0.1 r s years, the quantity and quality of the CMB polarization τ n 0.08 0.98 data are likely to be improved rapidly, with the ongoing 0.96 0.06 WMAP observations and many balloon experiments like 0.04 0.94 0.02 0.92 the BOOMERANG. These would allow better measure- −20 −15 −10 −5 0 5 10 −20 −15 −10 −5 0 5 10 ∆α (deg) ∆α (deg) ments of ∆α. While nonzero TG and CG power can also be in-

3.2 0.5 duced by Faraday rotations[32] and higher dimensional 3.15 Lorentz and CPT violating operators[33], these are of- 0.4 ] 3.1 s

A ten frequency-dependent, while the effects described here 3.05 0.3 10 r 3 are not[34]. This provides, at least in principle, a way to 0.2 log[10 2.95 distinguish between these different effects. The Faraday 2.9 0.1 2.85 rotation induced by magnetic field is given by 0 −20 −15 −10 −5 0 5 10 −20 −15 −10 −5 0 5 10 2 ∆α (deg) ∆α (deg) ∆α λ L B n dL =8.1 × 105 k e . (7) rad m Z  Gs  cm−3 pc FIG. 3: (color online). Joint 2-dimensional posterior proba- 0   bility contour plots in the ∆α − τr (top left), ∆α − nS (top 10 If we assume that reionization occurs at z < 20, then for α − AS α − r right), ∆ log[10 ] (bottom left) and ∆ (bottom −9 right), showing the 68% and 95% contours from the WMAP a global intergalactic magnetic field of 10 Gs, at the + B03 constraints. frequency of 145 GHz where BOOMERANG operates, the Faraday rotation is only of the order of 10−3 deg, which is much smaller than the range of α uncertainty The effect of the polarization rotation is degenerate distribution and hence insignificant. The apparent rota- with variation on the amplitude of the primordial spec- tion might also be due to contamination from foreground trum and the tensor to scalar ratio. These parameters are emission. In some attempts to obtain CMB tempera- also degenerate with the optical depth of reionization. In ture and polarization spectra, including those of WMAP, Fig. 3 we plot the joint two-dimensional posterior proba- foreground-removing procedure has been applied. For bility contours of ∆α with τr, nS, AS and r. More precise the BOOMERANG experiment, which operates at rel- measurements on these four parameters will help to break atively high frequency, it is believed that the primary the degeneracy on the constraints of cosmic Lorentz and CMB polarization signal is dominant, and the contribu- CPT violations discussed here. We have also made fits tion of the polarized galactic synchrotron foreground is with a running spectrum index, but found that it does small [29], but at present a small contamination can not not affect the above results significantly. The inclusion be ruled out completely. Future multi-wavelength polar- of the matter power spectrum obtained from large scale ization observations would help distinguish this possibil- structure measurements also does not change our con- ity. straints on ∆α significantly. We could not yet conclude that CPT is definitely vi- Previously, the cosmological birefringence effect has olated if a non-zero ∆α is detected. However, if such a 4 detection is confirmed, it would certainly raise the possi- Phys. Rev. D 67, 043509 (2003); S. M. Carroll and J. Shu, bility of a Lorentz-violating term like that given in Eq.(1), hep-ph/0510081. or others of the similar form. For example, a term of this [7] H. Davoudiasl et al., Phys. Rev. Lett. 93, 201301 (2004). 70 form could be due to the interaction between dark energy [8] H. Li, M. Li, and X. Zhang, Phys. Rev. D , 047302 (2004). and the electromagnetic sector, if we take pµ as ∂µφ with [9] G. L. Alberghi, R. Casadio, and A. Tronconi, φ being quintessence field. Thus, the results we obtained hep-ph/0310052. can be used to put additional constraints on the behav- [10] S. M. Carroll, G. B. Field, and R. Jackiw, Phys. Rev. D iors of dynamical dark energy between the redshift range 41, 1231 (1990). z ∼ 1 to z ∼ 1000. [11] S. M. Carroll and G. B. Field, Phys. Rev. D 43, 3789 A Lorentz violation also implies the violation of the (1991). equivalence principle. In our case where only a small vi- [12] R. D. Peccei, J. Sola and C. Wetterich, Phys. Lett. B 195, 183 (1987); C. Wetterich, Nucl. Phys. B 302, 668 olation is present, the group velocity of light remains un- (1988); B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, changed, and the weak equivalence principle is satisfied. 3406 (1988). On the other hand, the Einstein equivalence principle is [13] For other relevant studies see e.g. A. Kostelecky, R. Lehn- violated, as there would be a split of photon helicities[35]. ert, and M. Perry, Phys. Rev. D 68, 123511 (2003); O. Furthermore, causality is violated for timelike pµ. How- Bertolami, R. Lehnert, R. Potting, and A. Ribeiro, Phys. ever this violation is significant only in the regions where Rev. D 69, 083513 (2004); A. A. Andrianov, P. Giacconi, the wavelengthes of photons are very large [36]. and R. Soldati, JHEP 0202, 030 (2002). [14] F. R. Klinkhamer, Nucl. Phys. B 578, 277 (2000); In summary, current cosmological observations have hep-ph/0511030, to appear in Acta. Phys. Polon. B. opened a new window for probing new physics. In this [15] S. M. Carroll and G. B. Field, Phys. Rev. Lett. 79, 2394 paper we show that the current data from WMAP and (1997). BOOMERANG might indicate a rotated polarization an- [16] S. M. Carroll, Phys. Rev. Lett. 81, 3067 (1998). gle, which can be resulted from the CPT and Lorentz [17] See e.g. M. Kamionkowski, A. Kosowsky and A. Stebbins, violations. Such a result, if confirmed at greater sig- Phys. Rev. D 55, 7368 (1997). nificance by future observations, would reveal hitherto [18] A. Lue, L. M. Wang, and M. Kamionkowski, Phys. Rev. Lett. 83, 1506 (1999). unknown dynamics of the nature. [19] B. Feng, H. Li, M. Li, and X. Zhang, Phys.Lett. B 620, We are grateful to T. Montroy for making the B03 TC 27 (2005). and GC covariance matrix available [30] and kind com- [20] A. H. Jaffe, M. Kamionkowski and L. Wang, Phys. Rev. ments. We thank G. Hinshaw and L. Page for helpful D 61, 083501 (2000). correspondences. Our fittings were finished in the Shang- [21] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002). hai Supercomputer Center (SSC). We thank K. Ichiki, [22] Available from http://cosmologist.info. 553 A. Lewis, H. Peiris, J. Yokoyama and G. Zhao for help- [23] W. L. Freedman et al., Astrophys. J. , 47 (2001). [24] S. Burles, K. M. Nollett and M. S. Turner, Astrophys. J. ful discussions and F. Klinkhamer and R. Lehnert for 552, L1 (2001). comments. This work is supported in part by the Na- [25] G. Hinshaw et al., Astrophys. J. Suppl.148, 135 (2003). tional Natural Science Foundation of China (NSFC) un- [26] A. Kogut et al., Astrophys. J. Suppl.148, 161 (2003). der Grant Nos. 10533010, 90303004, 19925523 and by [27] L. Verde et al., Astrophys. J. Suppl. 148, 195 (2003). the Ministry of Science and Technology of China under [28] W. C. Jones et al., astro-ph/0507494; F. Piacentini et al., Grant No. NKBRSF G19990754. B. F. is supported by astro-ph/0507507. JSPS and M. L. is by the Alexander von Humboldt Foun- [29] T. E. Montroy et al., astro-ph/0507514. [30] Available from http://cmb.phys.cwru.edu/boomerang/. dation. X.C. is supported by the China National Science [31] D. N. Spergel et al., arXiv:astro-ph/0603449; Fund for Distinguished Scholars (Grant No 10525314). L. Page et al., astro-ph/0603450; G. Hinshaw et al., astro-ph/0603451; N. Jarosik et al., astro-ph/0603452. [32] See e.g. E. S. Scannapieco and P. G. Ferreira, Phys. 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