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Pos(BHCB2018)015 + 2 P 0 to = 2 P , Mapse@Fisica.Ufc.Br , , Mapse Barroso F Modification in Gravitational Waves Production Triggered by Spontaneous Lorentz Violation PoS(BHCB2018)015 Kevin M. Amarilo∗, Mapse Barroso F. Filho and Roberto V. Maluf Universidade Federal do Cearà ˛a,Brazil E-mail: [email protected], [email protected], [email protected] In this work, we search how the production of gravitational waves is corrected by a spontaneous Lorentz violation triggered by the vacuum expectation value (VEV) of the bumblebee vector field (bm ). We consider the modified graviton equations of motion by the dynamics of the bumblebee field. In this formulation the graviton is still non-massive and has two degrees of freedom, but does not has a gauge transformation and the dispersion relation changes from p2 = 0 to p2 + m 2 x(b pm ) = 0. Then, we solve these equations considering a current that acts as a source of the radiation. In this case, we apply the method of the Green’s function to find a modified quadrupole formula for the perturbation. International Conference on Black Holes as Cosmic Batteries: UHECRs and Multimessenger Astronomy - BHCB2018 12-15 September, 2018 Foz du Iguazu, Brasil ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Production of Gravitational Waves with Lorentz Violation Kevin M. Amarilo 1. Introduction The idea that the Lorentz symmetry may not hold in greater energies scales began in the context of the string theory [1]. Since then, the investigation of the Lorentz Symmetry Breaking (LSB) was developed into the effective field theory known as Standard Model Extension (SME), in which the usual fields of the SM and GR are coupled with fixed background fields that trigger the Lorentz and CPT Breaking. Among the fields that break the Lorentz Symmetry, the bumblebee field (Bm ) is one relatively simple. The symmetry is spontaneously broken by the dynamics of Bm that acquires a nonzero PoS(BHCB2018)015 vacuum expectation value (VEV) [2, 3]. This model was first considered with the LSB being m m 2 2 triggered by a Higgs-like potential V(B ) = l(B Bm ∓ b ) =2. In this work, we analyze the repercussion of the LSB triggered by the bumblebee model in the production of Gravitational Waves (GW) scenario. We start modifying the free wave equation for the graviton. Then, a current Jmn is added to the model and a Green’s function is derived for the timelike bm (b0;0) and spacelike bm = (0;b) configurations for the bumblebee VEV. With these results, the Green’s Function for the modified equation will be compared with the usual one, showing the modifications in the theory. 2. Modifying the Graviton’s Wave Equation The extension of the gravitational sector including the Lorentz-violating terms is given by the action S = SEH + SLV + Smatter: (2.1) The first term refers to the usual Einstein-Hilbert action Z p 2 S = d4x −g (R − 2L); (2.2) EH k2 where R is the curvature scalar and L is the cosmological constant, which will not be considered in this analysis. The SLV term consists in the coupling between the Bumblebee field and the curvature of space- time. The leading terms are Z p 2 S = d4x −g (uR + smn R +tab mn R ); (2.3) LV k2 mn ab mn where u, smn and tab mn are dynamical fields with zero mass dimension. Finally, SM takes account the matter-gravity couplings, which in principle should include all fields of the standard mode model as well as the possible interactions with the coefficients u, smn and tab mn . The gravity-bumblebee coupling can be represented by the action (2.3) defining 1 1 u = xBa B ; x Bm Bn − gmn Ba B ; tab mn = 0: (2.4) 4 a 4 a The dynamics of Bm is dictated by the action Z p 1 2x S = d4x −g − Bmn B + Bm Bn R −V(Bm B ∓ b2) ; (2.5) B 4 mn k2 mn m 1 Production of Gravitational Waves with Lorentz Violation Kevin M. Amarilo where we introduced the field strength Bmn Bmn = ¶m Bn − ¶n Bm ; (2.6) in analogy with the electromagnetic field tensor Fmn , in fact, the bumblebee models are not used only as a toy model to investigate the excitation originated from the LSB mechanism, but also as a alternative to U(1) gauge theory for the photon. In this theory, the photon appears not as a fundamental particle, but as a Nambu-Goldstone mode due to spontaneous Lorentz violation [4]. The potential, defined as l PoS(BHCB2018)015 V = (Bm B ∓ b2)2; (2.7) 2 m is responsible for triggering the spontaneous breakdown of diffeomorphism and Lorentz symmetry. Here b2 is a positive constant that stands for the nonzero expectation value of this field. In the pursue of the influence of the gravity-bumblebee coupling on the graviton, we must assess the linearized version [7]. Therefore, we split the dynamic fields into the vacuum expectation values and the nearby quantum fluctuations: gmn = hmn + khmn ; (2.8) Bm = bm + B˜m ; where hmn and Bm represent small perturbations around the Minkowski background and a constant vacuum value bm , respectively. Following the procedure described in Ref. [7], the solution for the linearized bumblebee equa- tion of motion in the momentum space is m ab am m ab m m m 2 k p ba b h 2sb R 2s p ba b R s p R sb R s p b R B˜m = b + a − b + − + ; (2.9) 2(b · p) p2 p2(b · p) 4l(b · p) p2 p2(b · p) m 0 2 with p = (p ;p);s = (2x=k ), while Rmn and R are taken in their linearized form. This solu- tion can be inserted in the Lagrangian (2.3) as specified in [5] leading to the following effective Lagrangian 2 mn a 1 2 a 2 LLV = x p b b h h + (b · p) (h ) m n a 2 a 1 − (b · p)2 hmn h + p2b b hma hn − b b p p + b p b p hmn hab 2 mn m n a m n a b (m n) (a b) 4x 2 p2 p p + −2p2b b − 2b2 p p + 4b · pb p − m n hmn ha k2 m n m n (m n) 4l a 2 ! b pm pn pa p 2(b · p)pm pn b( p ) pm pn pa p + 2b b p p − b p b p + b − a b + b hmn hab m n a b (m n) (a b) p2 p2 4l 4 2 ! # p (b · p) pm pn + b2 p2 − (b · p)2 + (ha )2 + p2b b − 2(b · p)b p + hml hn + O(h3); 4l a m n (m n) p2 l (2.10) 2 Production of Gravitational Waves with Lorentz Violation Kevin M. Amarilo a where h = h a , it is worth noting that this Lagrangian is not invariant under the usual gauge trans- formation of the metric perturbation (hmn ! hmn + ipm zn + ipn zm ) Proceeding with the analysis, we must add the Lorentz-violating terms of Eq. (2.10) to the bilinear terms of the Einstein-Hilbert Lagrangian 1 1 p p hmn ha − p p hmn h + p2hm h − p2hmn h ; (2.11) m a n m n 2 m 2 mn forming the kinetic Lagrangian Lkin = LEH + LLV : (2.12) PoS(BHCB2018)015 Next, we must evaluate the Feynman propagator of the graviton using this Lagrangian, there- fore first we rewrite the Lkin into the bilinear form 1 mn ab Lkin = − h Oˆ h ; (2.13) 2 mn;ab where this operator is symmetric in the indices (mn), (ab) (the same way as hmn and hab ) and under the interchange of the pairs (mn) and (ab). In this notation, the graviton propagator defined as h0jT[hmn (x)hab (y)]j0i = Dmn;ab (x − y); (2.14) where Dmn;ab (x − y) is the operator that satisfies the Feynman’s propagator equation, given as ˆmn; ls;ab mn;ab (4) Ols D (x − y) = iI d (x − y) (2.15) mn;ab 1 ma nb mb na where I = 2 (h h + h h ) is the identity operator. From this point, the problem of determining the Feynman propagator is reduced to the inversion of the Oˆ operator. One method for obtaining the inverse of rank two tensors is based on the Barnes-Rivers spin projection operators. The process, which is considerable tedious, and the propagator is found in Ref. [6]. The propagator has two poles, that represents two dispersion relations for the graviton, p2 + x(b · p)2 = 0; (b · p)2 − b2 p2 = 0: (2.16) The first mode has no problems concerning the stability and causality, but the second, even though preserves the causality, is non-unitary and spoils the physical consistency. The equation of motion obtained from the effective Lagrangian (2.12) is ˆ ab Omn;ab h = 0: (2.17) The Eq. (2.17) is then saturated with pm pn , bm bn , p(m bn) and the Minkowski metric, revealing the following constraints mn mn mn pm pn h = 0; bm bn h = 0; p(m bn)h = 0; h = 0: (2.18) Even more constraints can be found saturating (2.17) with pm and bm alone, which leads to m m pm hn = 0; bm hn = 0: (2.19) 3 Production of Gravitational Waves with Lorentz Violation Kevin M. Amarilo The Eqs. (2.18)-(2.19) consist in a set of 12 constraints that can reduce the 14 degrees of freedom of the theory (ten of the graviton and four of the bumblebee field).
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