Astrophysical Implications of the Bumblebee Model of Spontaneous Lorentz Symmetry Breaking

Astrophysical implications of the Bumblebee model of Spontaneous Lorentz Symmetry Breaking

Gon¸calo Guiomar Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001, Lisboa, Portugal.∗

Jorge Páramos Departamento de F´ısica e Astronomia and Centro de F´ısica do Porto, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal.† (Dated: October 9, 2014) In this work the bumblebee model for spontaneous Lorentz symmetry breaking is considered in the context of spherically symmetric astrophysical bodies. A discussion of the modified equations of motion is presented and constraints on the parameters of the model are perturbatively obtained.

PACS numbers: 04.80.Cc, 97.10.Cv, 11.30.Cp

I. INTRODUCTION post-Newtonian formalism [8]) show very little lenience in allowing the breaking of this symmetry [9]. The study One of the basic assumptions of general relativity of the implications of LSB in gravity through Einstein- is that of Lorentz invariance, a fundamental symmetry aether theories has only recently been explored [10, 11], which to the present day has been verified to a very high and the impact of the bumblebee model on Solar System precision. However, the possibility of this symmetry be- dynamics was assessed in Ref. [12]. ing broken is an ongoing topic of debate [1, 2]: a relevant The purpose of this work is to apply this model to the branch of this discussion is centered on the consequences study of astrophysical bodies such as stars, in order to that Lorentz symmetry breaking (LSB) would have in obtain a lower bond on the parameters of the model. gravitation. The exploration of these consequences can The action for the bumblebee model is given by be made through Kostelecký’s standard model extension [3] which, as the name implies, extends the scope of the 1 S = d4x + (R + ξBµBν R ) (1) standard model by adding a gravitational sector along L 16πG µν − with Lorentz-violating terms [4]. Z 1 The spontaneous Lorentz-breaking mechanism is sim- Bµν B V (BµB b2) , 4 µν − µ ± ilar to the Higgs mechanism, where the system sponta- neously collapses onto the referred vacuum expectation value (VEV), achieving thus a particular four-vector ori- where is the Lagrangian density of matter, Bµν is the entation and creating preferred frame effects as a conse- field strength,L quence.

This can modeled through the introduction of a vector Bµν = µBν ν Bµ, (2) field dynamically driven by a potential which acquires ∇ − ∇ a nonvanishing VEV: notice that, in the context of the and ξ the coupling constant between curvature and the pioneering work developed in Refs. [3, 4], this vector field Bumblebee field; the potential V has a nonvanishing is not simply a classical addition to the matter content of VEV b = 0 signalling the spontaneous Lorentz symmetry the model, but instead is assumed to arise dynamically breaking.6 from the LSB terms in the standard model extension, the underlying quantum field theory. The focus of this work will be mostly on the LSB in the context of gravity, using a subset of the so called Einstein- II. THE MODEL aether theories[5, 6] - the bumblebee models - where a vector field with a nonvanishing VEV is added to the The variation of Eq. (1) with respect to the metric Einstein-Hilbert action of the system. The experimen- yields the modified equations of motion [3], tal constraints obtained in both high-energy cosmic rays [7] and gravitational experiments (the latter through the 1 R Rg = 8πG T M + T B , (3) µν − 2 µν µν µν M B ∗Electronic address: [email protected] where T µν is the matter stress-energy tensor and T µν †Electronic address: [email protected] is the bumblebee stress-energy tensor, defined as 2

B α 1 αβ T µν BµαB ν BαβB gµν V gµν + The only nonvanishing component of Eq. (7) is for µ = r, ≡ − − 4 − yielding 0 ξ 1 α β α 2V BµBν + B B Rαβgµν BµB Rαν 8πG 2 − 16πGV 0g ξR = 0 (12) (4) rr − rr → 1 α 1 α ξ + α µ(B Bν ) + α ν (B Bµ) 2A(g B2 b2) = Gr2[m0ν0 + 2m(ν00 + ν02)] + 2∇ ∇ 2∇ ∇ rr − 16πGr3 1 1 2(B B ) g (BαBβ) . −2∇ µ ν − 2 µν ∇α∇β Gr(2m0 mν0) 2mG r3(ν00 + ν0) . − − − No separate conservation laws are assumed for matter and the bumblebee vector field. The covariant This gives us, after some algebraic manipulation (non)conservation law (which is not used) can be ob- µ 2Gm ξ 0 0 tained directly from the Bianchi identities Gµν ap- B2 = 1 b2 + (Gr2[m ν ∇ − r 32πGAr3 plied to both sides of the modified field equations (3): this leads to µT M = µT B = 0, which may be inter- ∇ µν −∇ µν 6 +2m(ν00 + ν02)] + Gr(2m0 mν0) (13) preted as an energy transfer between the bumblebee and − − matter. 00 0 The equations for the bumblebee field are 2Gm r3[ν + ν ]) . − µν 0 ν ξ µν µB = 2V B BµR , (5) In order to obtain the pressure and density equations, ∇ − 8πG we resort to the trace-reversed field equations, given by, where a prime represents differentiation with respect to the argument. 1 E R 8πG T M + T B g (T M + T B) = 0, A potential of the form µν ≡ µν − µν µν − 2 µν µ 2 n (14) V = A(B B b ) , (6) M B µ ± where T and T are traces of the stress-energy tensors is assumed, so that the Bumblebee field (5) becomes for normal matter and the bumblebee field, respectively. The stress-energy tensor for normal matter is given by Bµ[16πGV 0g ξR ] = 0. (7) the perfect fluid form, µν − µν M Tµν = (ρ + p)uµuν + pgµν , (15) III. STATIC, SPHERICALLY SYMMETRIC SCENARIO where uµ is the four-velocity; in the static scenario and µ ν(r) given that uµu = 1, we have uµ = (e ,~0), so that Given that the relevant quantities such as the density, − pressure and scalar curvature inside a spherical symmet- p ric body such as the Sun have a strong radial variation T M = diag e2ν ρ, , pr2, pr2 sin2(θ) , (16) µν 1 2Gm when compared with very slow temporal changes, one − r ! assumes that the bumblebee field is given by with trace T = 3p ρ. − Bµ = (0,B(r), 0, 0). (8) Using

Accordingly, one resorts to the static Birkhoff metric, gttE grrE = gθθE = 0, (17) tt − rr θθ −1 2Gm one may derive the equations that will allow us to obtain g = diag e2ν(r), 1 , r2, r2 sin2 θ , (9) µν − − r p(r), ρ(r) and ν(r). Without the bumblebee field, these " # quantities (denoted with the subscript 0) are given by where m(r) is the mass profile as a function of the ra- r(r 2Gm )ν0 Gm dial coordinate, and we assume that the potential takes p (r) = − 0 0 − 0 , (18) a quadratic form, for simplicity, 0 4πGr3 0 m0 V = A(B Bµ b2)2, (10) ρ0(r) = , (19) µ − 4πr2 3 with the adopted sign reflecting the spacelike nature of 0 m0 + 4πp0r ν0(r) = G , (20) the bumblebee field. r(r 2Gm0) − For the radial case µ = r, the Ricci tensor is given by, which, along with a state equation that relates p0 and ρ0, G(m0r m)(2 + rν0) yields a closed set of four differential equations with four R = − (ν0)2 ν00. (11) rr r2(r 2Gm) − − unknowns. − 3

In the presence of the bumblebee field, one must also with polytropic index n = 3, and where ρc and pc are the include the related field (7); solving Eq. (14), the pres- central density and pressure, respectively. Although this sure and density are then given by is a crude model, surpassed by state of the art numerical models of the several layers and processes occurring 1 p(r) = rξB(r 2Gm)2B0(2 + rν0) inside the Sun, it is well suited for analytical studies and 8πGr4 − serves the purpose of our study: obtaining bounds for +r(8πGV r3 2Gm + 2r(r 2Gm)ν0) the parameters of the model under scrutiny compatible − − − B2(r 2Gm)( 2ξGm( 1 + r(ν0( 2 + rν0) (21) with a perturbative impact on the interior structure of − − − − our star (as shown for scalar field- [13] and ungravity- 00 0 2 +rν )) + r[16πGV r inspired models [14]). ξ + rξ(ν0( 2 + rν0) + rν00)])] , With the above EOS, the hydrostatic equilibrium con- − − dition gives rise to the Lane-Emden (LE) equation, and 1 d dθ 1 χ2 = θn. (27) ρ(r) = [ r2(8πGV r2 + ξ(r 2Gm)2B02 χ2 dχ dχ − 8πGr4 − − 2Gm0) + rξB(r 2Gm)(B0( 4r + 3Gm Here θ(χ) is a dimensionless function which gives us the − − − +5Grm0) r(r 2Gm)B00) + ξB2(3G2m2. (22) density and pressure profiles of the system through − − 2 0 00 n n+1 2G rm(3m + rm ) ρ = ρcθ (χ) , p = pcθ (χ), (28) − +r2[ 1 + G(m0(4 Gm0) + rm00)])]. − − where χ = r/rn is a dimensionless coordinate, and Although we have a complete set of equations that de- (n + 1)p r2 = c . (29) scribe the behavior of our system, the solution of that n 4πGρ2 set of equations implies very intensive numerical compu- c tations. Instead, since the stellar structure of the Sun The boundary of the spherical body occurs at χ = χf , so is known to be well described by general relativity, one that θ(χf ) = p(χf ) = ρ(χf ) = 0; this is related with its shall adopt a perturbative approach. physical radius R by χf = R/rn. The solution θ(χ) of the LE equation (27) for n 3 ∼ is plotted in Fig. 1, showing e.g. that χf 6.9 for A. Lane-Emden solution n = 3. Notice that this quantity depends solely≈ on the polytropic index n, as does θ: the physical quantities ρc, In order to obtain the perturbations to the pressure pc and R affect only the value of rn. This reflects the and density arising from the effect of the Bumblebee field, homology symmetry of the LE equation, so that all stars one first describes how these quantities are obtained in with the same polytropic index n share a common density the standard scenario of General Relativity. To do so, (and pressure or temperature) profile, scaled only by its one first writes the Tolman-Oppenheimer-Volkov equa- central value. tion, derivable from Eq. (18), One may read the (unperturbed) mass of the spherical 0 2 body from the relation ρ0(r) = m /4πr , obtaining dp m + 4πpr3 = G(ρ + p) . (23) r dr − r(r 2Gm) m(r) = 4π ρ(r)r2dr = − 0 χ Z (30) In the Newtonian regime (invalid for relativistic neutron 3 2 n 3 2 0 stars but a good approximation for main sequence stars 4πrnρc χ θ dχ = 4πρcrnχ θ , 0 − such as the Sun), specified by the following conditions, Z where the LE equation (27) was used; the total mass 3 p(r) ρ(r) , 4πp(r)r m(r) , 2Gm(r) r, (24) of the star is given by replacing ξ = ξf , M = 3 2 0 4πρcrnχf θ (χf ). the above equation may be approximated by the hydro- − static equilibrium condition, IV. PERTURBATIVE EFFECT OF THE dp Gmρ = . (25) BUMBLEBEE FIELD dr − r2 The first model for the internal structure of the Sun The linearization of this system of equations (13,21,22) was put forward by Eddington, assuming that solar mat- brings with it a very large complexity that makes it very ter is described by a polytrope equation of state (EOS), computationally demanding to solve. With that in mind, we consider the perturbation to be of zeroth order, i.e. we 1+1/n pc replace the quantities on the r.h.s. of the equations men- p = Kρ ,K = 1+1/n , (26) ρc tioned above by the unperturbed expressions for m0(r) 4

1 and ν0(r) and the bumblebee ﬁeld. Regarding the latter, n = 2.9 it is more straightforward to resort instead to Eq. (12), n = 3 since at zeroth order one has 0.8 n = 3.1 1 R = 8πG T M g T M = 4πG(ρ p )g , (31) rr rr − 2 rr 0 − 0 rr 0.6 ) χ

( which leads to θ 0.4 2Gm ρ p B2(r) = 1 b2 + 0 − 0 g . (32) − r 8A rr 0.2 Since the unperturbed solutions ρ0(χ) and p0(χ) vanish at the boundary of the spherical body, the above shows 0 that the bumblebee ﬁeld collapses onto its VEV as it 0 1 2 3 4 5 6 7 8 M µ 2 crosses to its outer solution (where Tµν = 0), BµB = b . χ This is consistent with the approach followed in Ref. [12], where the latter condition was also assumed. FIG. 1: Lane-Emden solution for three polytropic Following the above procedure, the expressions for the indices. pressure and density may be obtained, yielding

ξ2(ρ0 p0) 2Gm 3 p(r) = p + [ξ(p ρ)]2 + − 1 (2 + ν0r)+ 0 − 2AGπr − r ξ 2Gm 2 1 [8Ab2 + ξ(ρ p)] (2 + ν0r)(m m0r)+ (33) Aπr3 − r − × − r 2Gm 2Gm 1 1 + r(4Gπpr + 2ν0 r[4Gπρ + (ν0)2 + ν00]) [1 + r(ν0[2 ν0r] rν00)] , G − r − − r − −

2 2 2 3 ξ ξ 2Gm m 0 0 0 2Gm 00 00 ρ(r) = ρ0 (ρ p) + 1 4 + G 9m (p ρ ) + 1 (p ρ )r + − 8 − 128AGπr − r " r − − − r − # h i ξ 2Gm Gm 2 2Gm 1 8Ab2 + ξ(ρ p) 2 + 6Gm0(1 Gm0) + 2Gm00r 1 (1 + 2Gm00r) , 64AGπr2 − r − r − − − r " # (34)

1+n 0 The advantage of considering the admittedly simplistic 0 3γ(χωcθ θ ) ν0(χ) = 2 − 0 , (37) model provided by the polytropic EOS (26) lies in the 2φχf + 6γχθ possibility of rewriting the rather convoluted expressions above in terms of the LE solution θ(χ) only. For this, we 0 and the form factor becomes φ = 3θ (χf )/χf — again now introduce the dimensionless parameters displaying the homology invariance− of the LE Eq. (27). 2 3 2 We may rewrite the above expressions for the pres- ξ ξ b Rs α , β , γ , (35) sure and density in terms of the LE solution θ(χ) and ≡ R2G ≡ R2G ≡ R its derivatives only, obtaining a more manageable form: where Rs 2GM is the Schwarzschild radius of the star, separating the contributions to the pressure and density together with≡ the form factor arising from the nonvanishing VEV b and the potential strength A as 3M φ , (36) ≡ 4πρ R3 c p(χ) = p0(χ) + pb(χ) + pV (χ) + δ(χ), (38) ρ(χ) = p (χ) + ρ (χ) + ρ (χ) δ(χ), and the EOS parameter ωc pc/ρc. 0 b V ≡ 0 − Using the relations (28) and the expression for ν0(r) from equation (18), one obtains we have 5

β 2 0 2 pb(χ) = 3 2 4 (φχf + 3γχθ ) 16πφ ξ χf χ × (39) [3γχ2θn( 1 + ω θ) 2( 1 + χ[ν0( 2 + χν0) + χν00])(φχ2 + 3γχθ0) + 3γχ(2 + χν0)(θ0 + χθ00)], − c − − − f

β 2 0 ρb(χ) = 3 2 4 (φχf + 3γχθ ) 16πφ ξ χf χ × (40) 2 4 02 2 00 2 002 2 000 0 2 2 00 000 [2φ χf + 3γχ(45γχθ + χ[14φχf θ + 9γχ θ + 2φχf χθ ] + 2θ [7φχf + 3γχ (10θ + χθ )])],

2 n 3γα θ 2 0 2 pV (χ) = 2 4 2 4 2 (φχf + 3γχθ ) 1024π Aφ ξ χf χ ×

(ω 1)(3γχ2θn[1 ω θ] + 2[ 1 + χ(ν0[χν0 2] + χν0)](φχ2 + 3γχθ0)) (41) c − − c − − f − χ (2 + χν0)(θ0[θ([1 + n]φχ2 ω + 3γ[ω θ 1]) + 3γχ([1 + n]ω θ n)θ0 nφχ2 ] + 3γχθ[ω θ 1]θ00) , θ f c c − c − − f c − 2 n 3γα θ 2 0 ρV (χ) = 2 4 2 4 2 (φχf + 3γχθ ) 2048π Aφ ξ χf χ × χ (φχ2 + 3γχθ0)(2[1 n]nχθ02[φχ2 + 3γχθ0] + nθ[θ0(χθ0[ 51γ + 2(1 + n)φχ2 ω + 6γ(1 + n)χω θ0] 8φχ2 ) θ2 f − f − f c c − f − 2 0 00 2 0 2 0 2 0 00 (42) χ(2φχf + 33γχθ )θ ] + [1 + n]ωcθ [θ (8φχf + 51γχθ ) + χ(2φχf + 33γχθ )θ ]) 2(2φ2χ4 + 3γχ[45γχθ02 + χ(14φχ2 θ00 + 9γχ2θ002 + 2φχ2 χθ000) + 2θ0(7φχ2 + 3γχ2[10θ00 + χθ000])])+ − f f f f 2 4 02 2 00 2 002 2 000 0 2 2 00 000 2ωcθ(2φ χf + 3γ + χ[45γχθ + χ(14φχf θ + 9γχ θ + 2φχf χθ ) + 2θ (7φχf + 3γχ [10θ + χθ ])]) , and space can be obtained, as depicted in Fig. 3. Notice that 2 2 2n 2 the allowed values for A (seen in the right panel of Fig. 9γ α θ (ωcθ 1) δ(χ) = − . (43) 3) are bounded from below, since this quantity appears 4096π2φ2ξ2 in the denominator of pV and ρV ; conversely, the region The latter appears in both the pressure and density per- allowed for ξb2 is bounded from above. turbations and, as shall be shown, has a negligible impact One thus obtains the bounds when compared with the remaining contributions. ξ ξ ξb2 < 10−23 , GeV2 < 10−3 < 1034. (44) ∼ √A ∼ → √AG ∼ A. Numerical analysis It is also worth noting that for the considered sets of In Fig. 2, the profile of the contributions to Eq. (38) is parameters, the term δ(χ) is negligible in comparison shown for masses and radiuses between 0.1 and 10 times with the other terms in both the pressure and density those of the Sun. The values of the parameters (ξ, b, A) equations, as mentioned after Eq. (43): indeed, one nu- < −34 are chosen so that the maximum of the relative pertur- merically finds that δ 10 pV (plot not shown) for bations is 1%, the order of magnitude of the current ac- the considered masses and∼ radiuses. curacy of the central temperature of the Sun [13–15]. A small variation in the polytropic index does not in- duce significant changes on the obtained bounds: in par- V. DISCUSSION AND OUTLOOK ticular, n does not impact the value of ρb, as can be seen directly in Eq. (40). Increasing the radius (thus lower- The bumblebee field was treated as a zeroth-order per- ing γ and α) raises the impact of the nonvanishing VEV, turbation on a set of stars of varying radius and mass, while leading to a lower contribution from the potential assuming that it follows the underlying symmetry of the term. A greater mass, however, leads to smaller effects problem so that it acquires only a radial component. Be- on all quantities except ρV , which is rather insensitive to cause the impact of the field is considered as a pertur- variations of M. bation, an attempt was made in order to constrain the By fixing M = M and R = R and finding the val- parameters of the model in such a way as to only cause ues for the model parameters (ξ, b, A) that lead to rela- a variation on the system of roughly 1%, following the tive perturbations of less than 1%, the allowed parameter accuracy of our present modeling of the Sun. 6

2 27 11 2 22 6 ξb = 10− ,A = 10− ξb = 10− ,A = 10−

3 n = 2.9 n = 2.9 3 − n = 3 1000R n = 3 −

8 n = 3.1 n = 3.1 8 − 1000R − b b 0 R 0 ρ

p ρ p 13 13 − R − log log 18 18 0.1R − 0.1R − 23 23 − − 28 28 − 3 n = 2.9 n = 2.9 −3 − n = 3 n = 3 − 8 n = 3.1 n = 3.1 8

− − 0.1R 0 0 V 0.1R V ρ p ρ p 13 13

− − log log 18 R 18 − R − 23 1000R 23 − − 1000R 28 28 − 0 1 2 3 4 5 6 0 1 2 3 4 5 6 −

χ χ

2 27 11 2 22 6 ξb = 10− ,A = 10− ξb = 10− ,A = 10−

3 n = 2.9 n = 2.9 3 − n = 3 n = 3 − 8 8 − n = 3.1 0.1M n = 3.1 − M b b 0 M 0.1M 0 ρ p ρ p 13 13 − − log log 18 18 − 10M − 10M 23 23 − − 28 28 − 3 n = 2.9 n = 2.9 −3 − n = 3 n = 3 − 8 n = 3.1 n = 3.1 8

− − 0 0 V M V

0.1M ρ p ρ p 13 13 M

− − log log 18 18 − 10M − 23 23 − − 28 28 − 0 1 2 3 4 5 6 0 1 2 3 4 5 6 −

χ χ

FIG. 2: Proﬁle of the relative perturbations pb/p0, pV /p0, ρb/ρ0 and ρV /ρ0 induced by the bumblebee. The parameters ξb2 and A were chosen so that the maximum of the perturbations reaches the adopted 1% limit. 7

5 0 − pb ρb 10 − 10

) 15 − )

b − A GeV log(

log( 20 − 20 − 25 − pV ρV 30 30 − 20 10 0 10 20 − 20 18 16 14 12 10 8 6 − − − − − − − − − − log(ξ[GeV 2]) log(ξ[GeV 2])

FIG. 3: Allowed region (in grey) for a relative perturbation of less than 1% for pb and ρb on the left side, and pV along with ρV on the right side

The obtained constraint for the value of the nonvan- The application of this same methodology to the study ishing VEV of the potential driving the bumblebee field, of galaxies is also possible, in order to gain further knowl- ξb2 < 10−23, is many orders of magnitude more strin- edge of the constraints to the parameters of our model, as gent∼ than the previously available bound ξb2 < 10−9, well as the possibility of describing galactic dark matter obtained by resorting to tests of Kepler’s law using∼ the as a manifestation of the bumblebee dynamics — follow- orbit of Venus [12]; by assuming that, in the presence of ing analog efforts in both scalar field [13] and vectorial matter, the Bumblebee field is not relaxed at its VEV, aether models [11, 16, 17]. In doing so, a nonvanishing this study has also yielded a constraint on the strength temporal component for a time-evolving Bumblebee field of the corresponding potential, ξ < 1034√AG. Although could also be considered, in order to provide a smooth only a quadratic potential was considered in this study, matching at cosmological scales. the change of the power n would not dramatically change the perturbative treatment followed here. Future refinements of this method could clearly include the use of a more accurate model for stellar structure, Acknowledgments as well as following a more thorough numerical analysis procedure, effectively solving the (differential) modified The authors thank O. Bertolami for fruitful discus- field equations to first order in the model’s parameters: sions, and the referee for his/her useful remarks. J.P. this, however, should only refine the obtained bounds, is partially supported by Funda¸cãopara a Ciência e Tec- with no significant change of their order of magnitude. nologia under the project PTDC/FIS/111362/2009.

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