Cosmological birefringent constraints on light
aa r X i v : . [ g r- q c ] M a r Cosmological birefringence constraints on light
Qasem Exirifard
Physics Department, Institute for Advanced Studies in Basic Sciences, P.O.Box 45195-1159, Zanjan, Iran andSchool of Physics, Institute for Research in Fundamental Sciences, P.O.Box 19395-5531, Tehran, Iran ∗ We calculate the birefringence in the vacuum for light at the leading and sub-leading orders for theCPT-even part of the SME. We report that all the LIV coefficients absent in the leading order, butthe isotropic one, contributes to the sub-leading order birefringence. We consider models free of thefirst order birefringence. We then show that infrared, optical, and ultraviolet spectropolarimetry ofcosmological sources bound the LIV coefficients to less than 10 − . This improves the best currentbound on the parity-odd coefficients by two orders of magnitude and establishes the isotropy of theone-way light speed with the precision of 41 nms . PACS numbers: 98.80.Es, 11.30.Cp, 12.60.-i, 41.20.Jb
SME [1], the most general Extension of the StandardModel of elementary particles that includes all the pa-rameters breaking the Lorentz invariance, provides aframework to search for Lorentz Invariance Violating(LIV) terms in all the sectors of the standard model.So LIV terms in the SME model, in contrary to theRobertson-Mansouri-Sexl Model [2, 3], are bounded byvarious phenomena not only the velocity of the lightand clock synchronization measurements [4]. This hasstemmed recent research aimed to detect or study theLIV terms in various fields, including the classical solu-tions of SME electrodynamics [5–7] , radiation spectrumof the electromagnetic waves and CMB data [8–14], blackbody radiation in finite temperature in SME electrody-namics [15–18], LIV terms in higher dimensional scenar-ios [19, 20], synchrotron radiation [21, 22], Cherenkovradiation [23–25] and modern cavity resonators or inter-ferometry experiments [26–32].Among various constraints imposed on the LIV termsof SME model, however, the absence of cosmological bire-fringence sets the most stringent constraint on the 10 outof 19 dimensionless parameters of the CPT-even part ofthe pure mSME electromagnetic sector [33, 34]: theseten parameters should be smaller than 2 × − . Fiveof the nine remaining parameters ( e κ e − ) are parity-even,3 ( e κ o + ) are parity-odd and one ( κ tr ) is an isotropic LIVterm. The parity-even terms are constraint to be lessthan 10 − by the most recent Michelson-Morley-typeexperiment [26]. Considering the motion of earth aroundthe Sun and the fact that boost mixes various LIV pa-rameters, [26] also requires the parity-odd parameters tobe less than 10 − . Ref. [21, 22] uses the absence ofsidereal variations in the energy of Compton-edge pho-tons at the ESRF’s GRAAL facility, to set the limit of10 − on the parity odd parameters. So far, no experi-ment or observation (in low energy physics) has boundedthe parity odd parameters beyond 10 − . So perhaps itis interesting to refine/translate some current data intostronger bounds on the parity odd coefficients, the pa-rameters that encode the anisotropy in the one-way lightspeed. We consider CPT-even part of the pure SME electro-magnetic sector. We obtain the birefringence in the vac-uum at the leading and the sub-leading orders. We showthat all the coefficients absent in the leading order but theisotropic one contribute to the birefringence at the sub-leading order. This means that ˜ κ o + and ˜ κ e − must not becalled non-birefringent terms, they do contribute to thebirefringence, a fact that has been noted also in [15, 16]. So in models which are free of the first order bire-fringence, absence of the cosmological birefringent indeedbounds ˜ κ o + and ˜ κ e − . These models include SME cam-ouflage models [35]. We consider these models and weshow that the absence of the cosmological birefringencebounds each element of the ˜ κ o + to less than 8 × − at90% confidence level, or equivalently the squared-sum ofits three elements to 1 . × − at 90% confidence level.This establishes the isotropy of the one-way light speedwith the precision of 41 nms . This precision is two order ofmagnitudes better than the best limit, as we shall showby reviewing the literature. SUB-LEADING ORDER BIREFRINGENCE
A Lorentz violating extension of the pure massless U (1)gauge sector of the standard model reads L = − F µν F µν −
14 ( k F ) µνλη F kλ F µν + 12 ( k AF ) k ǫ kλµν A λ F µν , (1)Eq. (1) is the most general extension of QED whichis quadratic in the gauge field, and contains no morethan two derivatives[65] and does not respect the Lorentzsymmetry. ( k AF ) k represents a CPT-odd Lorentz violat-ing term. This term is vanishing for theoretical reasons[38, 39] and cosmological birefringence requires it to besmaller than 10 − GeV [40–42]. Here we would like toaddress the constraints on k F . The theoretical consis-tency of the parity-even terms are studied in [43, 44].We, thus, set k AF = 0, and consider[66] L = − F µν F µν −
14 ( k F ) µνλη F kλ F µν . (2)( k F ) µνλη has the symmetries of the Riemann tensor, soonly 20 out of its 256 components are algebraically in-dependent. Its double trace should be zero. So only19 algebraically independent components of the ( k F ) µνλη contribute to the equations of motion of the gauge field.Ref. [34] introduces an interesting re-parametrization ofthese components by enclosing them in a parity-even andparity-odd subsectors, respectively ˜ k e and ˜ k o :( e κ e + ) jk = 12 ( κ DE + κ HB ) jk , κ tr = 13 tr( κ DE ) , (3)( e κ e − ) jk = 12 ( κ DE − κ HB ) jk − δ jk ( κ DE ) ii , (4)( e κ o + ) jk = 12 ( κ DB + κ HE ) jk , (5)( e κ o − ) jk = 12 ( κ DB − κ HE ) jk . (6)The 3 × κ DE , κ HB , κ DB , κ HE are given as:( κ DE ) jk = − k F ) j k , ( κ HB ) jk = 12 ǫ jpq ǫ klm ( k F ) pqlm ( κ DB ) jk = − ( κ HE ) kj = ǫ kpq ( k F ) jpq . (7)Note that e κ e + , e κ e − , e κ o − are traceless and symmetricwhile e κ o + is anti-symetric. κ tr is a number, it representsthe isometric LIV term. In term of this parametrization,the Lagrangian density reads L = 12 (cid:2) (1 + κ tr ) E − (1 − κ tr ) B (cid:3) + 12 E · ( e κ e + + e κ e − ) · E − B · ( e κ e + − e κ e − ) · B + E · ( e κ o + + e κ o − ) · B , (8)where E and B respectively are the electric and magneticfield. This new parametrization clearly illustrates theanalogy between the propagation of light in the vacuumof the theory with the propagation of light in a generalanisotropic media, a field intensely explored in optics.For the moment we consider (2). The first variation of(2) with respect to A µ gives its equation of motion: ∂ α F αµ + ( k F ) µαβγ ∂ α F βγ = 0 , (9)which is supplemented with the usual homogeneousMaxwell equation: ∂ µ e F µν = 0 . (10)These equations for a plane electromagnetic wave withwave 4-vector p α = ( p , ~p ), F µν = F µν ( p ) e − ip α x α , lead tothe modified Ampere law [33]: M jk E k ≡ ( − δ jk p − p j p k − k F ) jβγk p β p γ ) E k = 0 (11) which has non-trivial solution for the electric field pro-vided that the M matrix has zero eigenvalues. The zeroeigenvalues of M jk give the dispersion relation in the vac-uum. In order to obtain these zero eigenvalues let a primecoordinate be considered where in ˜ p α = ( p , , , p ). Inthe prime coordinate, the zero eigenvalues at the leadingorder reads p ± = p (1 + k + k ± r k + ( k − k ) , (12) p − p − = 2 p r k + ( k − k ) , (13)where k ij = (˜ k F ) iαjβ ˜ p µ ˜ p ν | p | , (14)wherein (˜ k F ) iαjβ represents the component of the k F ten-sor in the prime coordinate. Eq. (12) shows that at theleading order only two combinations of the components ofthe k-matrix contribute to the birefringence. From 19 al-gebraically independent components of the ( k F ) µνλη only10 of them contributes to the k − k and k for anarbitrary four-wave vector: p α = ( p , ~p ). An acceptablechoice of these ten combinations is: k a = (cid:0) ( k F ) , ( k F ) , (15)( k F ) − ( k F ) , ( k F ) − ( k F ) , ( k F ) + ( k F ) , ( k F ) − ( k F ) , ( k F ) + ( k F ) , ( k F ) + ( k F ) , ( k F ) − ( k F ) , ( k F ) − ( k F ) (cid:1) . Note that elements in k a are contained in the matrices˜ κ e + and ˜ κ o − [33, 34]. Ref. [33, 34] use infrared, optical,and ultraviolet spectropolarimetry of various cosmologi-cal sources at distances 0 . − . Gpc [48–55] and boundthe components of ˜ κ e + and ˜ κ o − to less than 2 × − at90% confidence level. Some combinations of k a are fur-ther restricted to less than 10 − using linear polarizationdata of gamma rays of cosmological sources [56]. Opticaland microwave cavities can measure the components of k F that does not contribute to the linear birefringencewith the precision of 10 − − − [34].We note that the best precision achieved in the lab-oratories is about or less than the square root of theprecision of the cosmological constraints on the leadingcontribution of the LIV terms to the birefringence in thevacuum. So the sub-leading contribution of the coeffi-cients having no contribution at the leading order, cannot be neglected. Ref. [33, 34] tacitly presumes that allthe coefficients are at the same order of magnitude andprovides the limit on (16). When the coefficients are notat the same order, the absence of birefringence leads toten non-linear inequalities among the nineteen parame-ters [67]. In this note, we assume that “the leading orderbirefringence is zero for some theoretical reasons” andwe calculate the second order birefringence. The modelswhich have zero order birefringence includes camouflagemodel given in Table XVIII of ref. [35].In the models we consider, we have k a = ˜ κ e + = ˜ κ o − =0. This means that in the prime coordinate k − k = k = 0. In the prime coordinate the zero eigenvalues ofthe M-Matrix at the sub-leading order then yields: p − p − = 2 p ( k + k ) (16)where k p = (˜ k F ) p + (˜ k F ) p p , (17) k p = (˜ k F ) p + (˜ k F ) p p . (18)Only eight combinations of the nine remaining coeffi-cients contributes to the k and k for an arbitraryfour-wave vector. An acceptable choice for these is k a = (cid:0) ( k F ) , ( k F ) , ( k F ) , ( k F ) − ( k F ) , ( k F ) − ( k F ) , ( k F ) − ( k F ) , ( k F ) + ( k F ) , ( k F ) + ( k F ) (cid:1) . (19)Let it be outlined that only the isotropic LIV term, κ tr ,is missing in (19). At the sub-sub-leading order, in theprime coordinate also k contributes to the birefrin-gence. So the isotropic LIV term contributes to the bire-fringence at the sub-sub-leading order. Also note that noelement in (19) can be written as a combination of theelements in (16) and the vanishing of double trace of k F .Comparing (13) with (16) leads to the conclusion that( k ) is experimentally constraint as so much as is k .This implies that the squared of the elements in (19) areconstraint as so much as the elements in (16). Ref. [33,34] bound the square-averaged of all the terms in (16), q ( P i =1 k i ) , to less than 2 × − at 90% confidencelevel. So each element in (16) is bound to less than √ × − . Subsequently each element in (19) is bound lessthan q √ × − = 8 × − . The parity even termsin (19) are constraint to less than 10 − by [26]. This letus set the parity even coefficients in (19) to zero at theprecision we are working in. This yields that the squaresum of the parity-odd coefficients in (19), q ( P i =1 k i ) ,is less than 1 . × − at the conservative 90% confidencelevel. This proves that the one way light speed is isotropicwith the precision of 41 nms .Two combinations of terms in (16) are further boundedless than 10 − by the analyze of [56, 57] on the gammarays from GRB 930131 and GRB 960924 [58]. Repeat-ing the analyze of [56] for other gamma ray sources canmeasure all the components of (16) with the precisionof 10 − , and subsequently in theoretical models wherein k a = 0 all the components of (19) with the precision ofabout 10 − . The most recent Michelson-Morley-type experiment[26], improving previous bounds [27], reports the boundof 10 − on the parity odd coefficients in (19). Search-ing for compton-edge photons at the ESRF’s GRAALfacility [21, 22] improves this precision by one order. Sothe bound we have provided improves the best low en-ergy precision on the one way light speed isotropy by twoorders of magnitude.Ref. [59] considers the synchrotron emission rate offast moving electrons and positrons in LEP and the mea-surements performed at the Z pole energy of 91 GeV, andconcludes the limit of | κ tr | < × − . This is ordersof magnitude improvement on the previous bounds on κ tr [26–32, 60]. Considering the motion of earth aroundthe Sun, our limit of 8 × − on each elements of κ o + implies a double sided bound of | κ tr | < × − . Butthis limit is apparently weaker than that of ref. [59]. Soour improvement on the limit of κ o + does not lead to theimprovement of the best current limit on κ tr .Ref. [23, 24] propose that ultrahigh-energy cosmic rays(UHECRs) have the potential to place further limits onall the non-birefringent parameters by the inferred ab-sence of vacuum Cherenkov radiation. Ref. [25], havinginferred the absence of Cherenkov radiation for 29 UHE-CRs at energy scale of 10 GeV [61], states bound of10 − on the nine non-birefringent terms. The contribu-tion of the massive LIV terms [35–37], however, neces-sarily can not be ignored for the energy scale of the ref.[25]. Here, we are providing bounds on the parameters atlow energies, energy scales that the contributions of themassive LIV terms can be neglected. Our results com-bined with [23, 24] prove that up to the scale of 10 GeV ,no new LIV terms -in addition to the low energy ones-is dynamically generated. So the fundamental scale ofquantum gravity likely should be higher than 10 GeV should quantum gravity be resolved through break of theLorentz invariance [68].There exist proposals to measure parity-odd LIVterms by electrostatics or magnetostatics systems [5, 6].Though these experiments are remained to be imple-mented, the bound we provide achieves their precision.Ref. [63] proposes a triangular Fabry-Perot resonatorbased on the Trimmer experiment [64], using which inthe current resonator experiments [26, 27] would improvethe bound we report on the parity-odd parameters. Thisexperiment is the only proposed experiment that wouldimprove our bound, however, it is remained to be imple-mented.
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D (1973) 3321 [Erratum-ibid.D (1974) 2489].[65] The behavior of photons in the presence of Lorentz andCPT violation operator with arbitrary (or other) massdimension is studied in [35–37].[66] Here k F is independent of the space-time. In moregeneral contexts [45–47], it may not.[67] An elegant approach to derive the exact birefringencerelation is given by [35], however, sub-leading ordercontributions have not been calculated.[68] Also note that the form of the modifications to theis independent of the space-time. In moregeneral contexts [45–47], it may not.[67] An elegant approach to derive the exact birefringencerelation is given by [35], however, sub-leading ordercontributions have not been calculated.[68] Also note that the form of the modifications to the