PoS(BHCB2018)015 + 2 p 0 to = https://pos.sissa.it/ 2 p , [email protected] , , Mapse Barroso F. Filho and Roberto V. Maluf ∗ 0. Then, we solve these equations considering a current that acts as a source of the = 2 ) µ p µ ). We consider the modified equations of motion by the dynamics of the bumblebee [email protected] b µ ( b radiation. In this case, we apply theformula method for of the the perturbation. Green’s function to find a modified quadrupole ξ Speaker. In this work, we search howLorentz the violation production triggered of by the gravitational vacuum waves expectation is( value corrected (VEV) by of the a bumblebee spontaneous vector field field. In this formulation thedoes graviton not is has still a non-massive gauge and transformation has and two degrees the of dispersion freedom, relation but changes from ∗ Copyright owned by the author(s) under the terms of the Creative Commons c International Conference on Black Holes asBHCB2018 Cosmic Batteries: UHECRs and Multimessenger12-15 Astronomy September, - 2018 Foz du Iguazu, Brasil Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). [email protected] Kevin M. Amarilo Universidade Federal do Cearà ˛a,Brazil E-mail: Modification in Gravitational Waves Production Triggered by Spontaneous Lorentz Violation PoS(BHCB2018)015 µν s (2.5) (2.2) (2.4) (2.3) (2.1) Kevin M. Amarilo , ) is one relatively  µ ) B 2 . b 0 ) defining , that acquires a nonzero ∓ ) = µ µ 2.3 B B µ αβ µν B αβ µν ( R t V , ) , − αβ µν Λ 2.  t . 2 µν / α 2 + R B − ) ν 2 α R µν B b matter ( B µ R S 2 ∓ 2 B configurations for the bumblebee VEV. With µν κ µν + µ 2 g ξ ) s g B κ 1 4 2 b LV − µ + , 1 S 0 − + B √ ( ν + uR x ( λ µν B 4 = ( 2 B µ d EH 2 µ κ B S b ) = µν g ]. This model was first considered with the LSB being Z is added to the model and a Green’s function is derived  µ 3 B = − = ξ B is the cosmological constant, which will not be considered in 4 1 , S ( µν √ 2 J Λ − V x EH ,  4 S α g d B − ]. Since then, the investigation of the Lorentz Symmetry Breaking α Z 1 √ B = ξ x and spacelike 4 is dictated by the action 4 1 ) d LV are dynamical fields with zero mass dimension. µ 0 S = , B Z 0 u b = ( αβ µν µ B t takes account the matter- couplings, which in principle should include all b S M and term consists in the coupling between the Bumblebee field and the curvature of space- S LV µν . s S , u αβ µν The The gravity-bumblebee coupling can be represented by the action ( The dynamics of Finally, The idea that the Lorentz symmetry may not hold in greater energies scales began in the Among the fields that break the Lorentz Symmetry, the bumblebee field ( In this work, we analyze the repercussion of the LSB triggered by the bumblebee model in The extension of the gravitational sector including the Lorentz-violating terms is given by the The first term refers to the usual Einstein-Hilbert action t where R is the curvature scalar and this analysis. time. The leading terms are simple. The symmetry isvacuum spontaneously expectation broken value by (VEV) the [ dynamics of where fields of the standard modeand model as well as the possible interactions with the coefficients u, triggered by a Higgs-like potential context of the [ Production of Gravitational Waves with Lorentz Violation 1. Introduction (LSB) was developed into the effective fieldwhich theory the known usual as Standard fields Model of Extension theLorentz (SME), and SM in CPT and Breaking. GR are coupled with fixed background fields that trigger the for the timelike the production of Gravitational Waves (GW)for scenario. the We start graviton. modifying the Then, free wave a equation current action these results, the Green’s Function forshowing the the modified modifications equation in will the be theory. compared with the usual one, 2. Modifying the Graviton’s Wave Equation PoS(BHCB2018)015 , ) 3 h ( αβ h O ]. µν + 4 (2.7) (2.9) (2.6) (2.8) h (2.10) # ! λ ν β h , p ) R µλ α p 2 p h · λ b ν 4 b µ ! p ( p Kevin M. Amarilo ν  2 µ p p σ p αβ µ h p + + 2 2 ) p R µν ) β 2 µ h p p p b ·  ) α b α σ ( β α ( b p h ν − α + p 2 ( ) µν ) p µ b p h ν ) R p · p ν µ  ) ] leading to the following effective b p µ p p ν ( 5 ( µ · p b σ ( λ ) , , b µ b 4 λ , 2 p ( µ p 4 ) · 2 + 2 2 B + µν b p ν b β h , ( − ∂ p , in fact, the bumblebee models are not used µ 2 κ αβ ∓ − β are taken in their linearized form. This solu- ˜ α B − R ) ) p µ p − µν + ν β R p ν ], the solution for the linearized bumblebee equa- α B F ν + ν p b · 2 B 7 p b µ b µν µ µ α 2 b µ µ ν µ ( B η b ( b p ∂ ( p and b b 2 µ 2 pb µ p = = 2 ) as specified in [ λ p = p · p 2 µν − µ ) σ 2 b

µν = µν R 2.3 B α 2 b µν α 4 ν α g B B + h V h + − + ( 2 ) ) ν 2 µα β ) p α h αµ α p p µ ν h R , while · α 2 p b ( ) ( α p b 2 2 µ ]. Therefore, we split the dynamic fields into the vacuum expectation b b (  ) b b 7 κ ν σ 2 2 4 1 2 / λ p 2 p p 4 ξ − µ + 2 ( + + ν + α b b α 2 h µ µν ) = ( − αβ b h p h ) β 2 µν · σ represent small perturbations around the Minkowski background and a constant β p p p h , µν b b · ) , respectively. ν 2 α µ h ( α b µ p b p 2 B − , ( b ) b µ − ν 0 2 µ p b b 2 p p 2 ·  p µ and p 2 b κ b 2 2 (  b 2 = ( ξ is a positive constant that stands for the nonzero expectation value of this field. µν κ 1 2 = ξ  4

µ 2 h µ b p = − + + + ˜ In the pursue of the influence of the gravity-bumblebee coupling on the graviton, we must The potential, defined as Following the procedure described in Ref. [ B LV L is responsible for triggering the spontaneous breakdownHere of diffeomorphism and Lorentz symmetry. assess the linearized version [ with tion can be inserted inLagrangian the Lagrangian ( values and the nearby quantum fluctuations: only as a toyas model a to alternative investigate to the U(1)fundamental excitation gauge particle, originated but theory as from for a the the Nambu-Goldstone mode . LSB due mechanism, to In spontaneous but this Lorentz also violation theory, [ the photon appears not as a tion of motion in the momentum space is vacuum value Production of Gravitational Waves with Lorentz Violation where we introduced the field strength in analogy with the electromagnetic field tensor where PoS(BHCB2018)015 ) and (2.15) (2.14) (2.16) (2.19) (2.11) (2.13) (2.17) (2.12) (2.18) ) to the αβ h 2.10 and µν Kevin M. Amarilo ]. The propagator h 6 . 0 alone, which leads to , = ) is operator. One method for µ h µν ˆ b O h 2.12 ) , . ) y µν 0 0; y and h ) − 2 = µ = µ − x p p 2 ζ ( x 1 2 ) p ( ν , and the Minkowski metric, revealing (the same way as µν 4 2 ( ) h . ip ) − b ) αβ αβ ν δ , 0 . ν h h b + − b µ µ ) with µ µν αβ . αβ = LV 2 ν , ( µ h ( αβ 0 ) ( ζ , D p 2 µ ν L , p p µν µ p , h ) = · µν 2.17 = ν µ + ˆ 1 2 ip I b b i O b i ( µν αβ 0 µ . In this notation, the graviton propagator defined + + 0; | ( EH 3 h ) µν b h , h )] ) = L = µν 0; y ν αβ 1 2 0; y , µν h ( αβ p = ( h µν = − − µ = µν ν αβ h ˆ → is the identity operator. From this point, the problem of p x µ ν 2 p kin O h ν = ( ) ) h ) µ b and p µν L µ x p µ ) αβ ( · kin να h p , b ( b − η L µν ( λσ µν α ν h ( ξ [ µβ D h , 0; T η + | µν µν 2 λσ = 0 + h into the bilinear form ˆ p h O α µν p νβ kin h µ η ν L p p µα µ is the operator that satisfies the Feynman’s propagator equation, given as η p ) ( y ) is then saturated with 1 2 − = , it is worth noting that this Lagrangian is not invariant under the usual gauge trans- x 2.17 ( α α αβ , αβ h , µν µν = I D h The equation of motion obtained from the effective Lagrangian ( The Eq. ( Proceeding with the analysis, we must add the Lorentz-violating terms of Eq. ( Next, we must evaluate the Feynman propagator of the graviton using this Lagrangian, there- Even more constraints can be found saturating ( where where The first mode has no problemspreserves concerning the the causality, stability is and non-unitary causality, and but spoils the the second, physical even consistency. though under the interchange of the pairs as determining the Feynman propagator is reduced toobtaining the the inversion inverse of of the rankThe two process, tensors which is is based considerable tedious, on andhas the the two Barnes-Rivers propagator poles, spin is that found projection represents in operators. Ref. two dispersion [ relations for the graviton, bilinear terms of the Einstein-Hilbert Lagrangian forming the kinetic Lagrangian where this operator is symmetric in the indices formation of the metric perturbation fore first we rewrite the Production of Gravitational Waves with Lorentz Violation where the following constraints PoS(BHCB2018)015 = → µ 0 (3.3) (3.2) (3.5) (3.1) (3.4) x (2.20) = 2 p Kevin M. Amarilo that acts as a source to the µν J has not produced massive modes . , is the Retarded Green’s function ) µν ) y ) τ ( . R is the Heaviside step function. The , y ( ) , ν , ) y µν 0 µν Θ 2 − B x ] J J ) ( µ x r ) − = p ( y B Θ x = · − ( considering a non-relativistic, isolated and ) b − µν EH τ 4 ), and it simplifies to ( [ µν h 1 x ( G ] ( h ξ µν and δ 2 δ ] h r ) 2 | + 4 ) 2.17 p 1 y π y G 2 p · 4 4 ) = p · − b y d ( b x ( | − ξ Z ) = ξ ) = x y + ( = p + 2 ( − r G ) = ˜ 2 p x x G ˆ , [ p ( ( O 0 [ ) to the Eq. ( R y µν G h − 2.19 0 ) resulting in the equation x )-( ) consist in a set of 12 constraints that can reduce the 14 degrees of = 2.20 ) we get , which is the dispersion relation in the configuration space. Defining ). Therefore, we can conclude that the spontaneous Lorentz violation τ 2 2.18 ) 3.2 2.19 µ ∂ 2.16 )-( µ b ( 2.18 0. Moreover, the non-minimal coupling ξ is determined by the convolution integration = − 2 2 ) µν ∂ p h · , = ) b y ( ) in this scenario, ˆ ) O ξ In the usual Einstein-Hilbert linearized theory, For the modified graviton’s equation, we can represent the Green’s function in the momentum The method of the Green’s function is commonly used to solve this kind of differential equa- The Eqs. ( We apply the Eqs. ( For the analysis of the production of GW we add a current − x , x + 0 ( 2 x p space, solving the Eq. ( here convolution leads to thefar-away quadrupole object. formula of G here it was considered that( the four-vectors are divided into a temporal and a spatial part ( tion. The Green’s function must satisfy that is the correct energy-momentummined dispersion in relation the associated Eqs. with the physical ( pole deter- triggered by the bumblebee field modified the Einstein-Hilbert’s dispersion relation as freedom of the theory (ten ofwith the two graviton physical and degrees four of ofgraviton. the freedom, bumblebee which field). is Therefore, the we same are number left of the usual Einstein-Hilbert’s Production of Gravitational Waves with Lorentz Violation for the graviton. 3. The Production of Gravitational Waves in the Presence of the Bumblebee Field dispersion relation in Eq. ( PoS(BHCB2018)015 − = b µ φ (3.7) (3.8) (3.9) (3.6) b ( (3.10) (3.12) (3.13) (3.11) must be cos 0 ) . b b to θ o ) ( ) r p is ( | sin Kevin M. Amarilo ˜ − b ) G | τ θ ( ( δ , respectively. ) sin , b b θ # 2 2 ) + ( ) b and 0 θ p b ( cos . Considering that ( 2 2 , ξ , ) τ | cos ) selects a preferential direction for 0 has a timelike configuration rb p ) y + b , | τ b µ ( | ) θ 1 − ( b p , which will reduce ξ + , , Ψ , x ) p ) ) i ( 2 0 b + r r ) 00 2 − | cos t t , r − 1 ( ( 2 0 T 0 G j cos r = p i j i j 2 2 − ] p I I 2 y " − i | 2 2 2 τ = ( Ψ dt dt ) ) 1 1 δ b ( d d 0 | . Therefore the solution for the perturbation is 2 y 0 r µ y y and since the bumblebee field does not couples b ξ 2 3 b G G r r ( − µν ) ]. The vector τδ 5 d ) 2 2 0 − x ξ 8 0 b ( , ) expanded to the second order of 1 GT Z ( R + 0 ) + ( π ξ r ) = ) = b 1 G is the quadrupole moment tensor defined as [ 3.8 r r 1 − 16 , , ) = + − 2 0 i j t t t I = ( 1 p ( ( ( τ ) = = ( i j i j ) = i j y µ p I δ h h p b , therefore cos µν ( π −  b ) = J ˜ 4 x 1 G p ( ( ) only by the factors − ˜ G G and ) = r τ y 3.4 p h − × ) the angular coordinates of the trivectors 2 x ) b ) ( τ φ r ( · , G b Θ b θ ( 3 2 r n b π ξ × 8 is the retarded time and r ) and ( in the configuration space, we may use the inverse Fourier Transform. The inversion ) = φ − y , G t θ − = is the angle between x r ( , in this case the Green’s function resumes to t 2  Ψ The inverse Fourier transform of Eq. ( This term of the Green’s function highlights an interesting property of the bumblebee models, For the modified equation with The solution for the perturbation, considering that the source has a slow motion, is the well- The second case is a spacelike configuration 0 G , , with ( 0 ) b φ the existence of an anisotropy in the solutions [ with the matter fields, we may that known quadrupole formula which differs from the Eq. ( here, the second term is defined as the propagation of the wave. where is made for two particular cases, the first is considering that Production of Gravitational Waves with Lorentz Violation then to get and the inversion results to small in comparison to the componentsthe of unit. the perturbation, this factor will be slightly greater than PoS(BHCB2018)015 (3.14) . Kevin M. Amarilo  ) r t . In the second case, ( 2 ], modifications in the ) i j 8 3 0 I b 3 dt ( d , in this case, only the the ξ µ 2 known as bumblebee vector ) b + r µ r · 1 B 2 b ( p ξ , 065020 (2008). ) + 77 r t ( i j 2 I 2 dt d , n. 2, p. 683, 1989 , 065008 (2005).  6 b 39 71 θ that appeared as consequence of the expansion of the i j I cos2 2 b 2 ξ − , 124036 (2014). 1 90 is the modified retarded time. Therefore, all the parameters of the  2 ) G r 0 2 b ( configuration, the solution is ξ ) ) = + r b , , 1 ` t y and S. Samuel, Phys. Rev. D 0 ( p i j r h = ( − µ 0 b x = 0 r t Analyzing this new relation we determine the solution for this model in two special configu- For the In this work, we searched the effects of the spontaneous LSB on the production of the GW. We In this equation, we can observe once more the existence of anisotropy, the amplitude of the [2] R. Bluhm and V. A. Kostelecký, Phys.[3] Rev. D R. Bluhm, S-H. Fung, and V. A. Kostelecký, Phys. Rev. D [4] C. A. Hernaski, Phys. Rev. D [5] R. V. Maluf, et al, Physical Review[6] D, v.R. 88, V. n. Maluf, 2, et p. al, 025005, Physical 2013. Review[7] D, v.Q. 90, G. n. Bailey, 2, V. A. p. Kostelecký, 025007, Physical 2014. [8] Review D,V. v. A. 74, Kostelecký, n. and 4, M. p. Matthew, 045001, Physical 2006. Review D 66.5 (2002): 056005. [1] V. A. Kosteleck amplitude and presence of a third derivative of the quadrupole moment. References rations. In the first one, we considered a timelike configuration for field that has a nonzero VEV. This leads to a modified wave equation for the graviton propagation. velocity of propagation of the wavea were spacelike smaller configuration by was a considered factortence for of of the anisotropy bumblebee in VEV. the The solution, solution which showed is the well exis- known in the literature [ introduced the LSB by the coupling of the graviton with a vector wave, besides the slower propagation velocity, must be the same. Green function to the second order. 4. Conclusion wave is modified, therefore, theSince existence the of quadrupole the moment bumblebee remained fieldOther unmodified, can peculiarity be the is attested frequency the in of third derivative a the of experiment. wave must not change. Production of Gravitational Waves with Lorentz Violation where