Astrophysical implications of the Bumblebee model of Spontaneous Lorentz Symmetry Breaking
Gonçalo Dias Pereira Guiomar
Thesis to obtain the Master of Science Degree in Engineering Physics
Supervisor(s): Prof. Dr. Vítor Manuel dos Santos Cardoso Prof. Dr. Jorge Tiago Almeida Páramos
Examination Committee
Chairperson: Prof. Dr. Ana Maria Vergueiro Monteiro Cidade Mourão Supervisor: Prof. Dr. Jorge Tiago Almeida Páramos Members of the Committee: Prof. Dr. Amaro José Rica da Silva
November 2014
Acknowledgments
This thesis could not have been possible without the help and guidance of my supervisor, Professor Jorge Páramos, with whom I rediscovered the joy of doing Physics. His immense patience when dealing with my incompetence, along with his immense knowledge of unorthodox working places made this work a fun and rewarding experience. Also, I would like to thank Professor Vítor Cardoso for his help and availability in the process of realizing this thesis. For my family, I am truly grateful for your continuous support and for providing me the opportunity of realizing my goals, no matter how uncertain they might have seem in the past. For my friends, who accompanied me throughout this journey, thank you for joining me in my culinary digressions. This last paragraph I dedicate to Geisa, for helping me collapse onto a better state of being.
i
Abstract
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 modi ed equations of motion is presented and constraints on the parameters of the model are perturbatively obtained. Along with this, a detailed review of this model is given, ranging from the questioning of the basic assumptions of General Relativity, to the role of symmetries in Physics and the Dark Matter problem.
Keywords
General Relativity, Bumblebee Model, Lorentz Symmetry Breaking, Stellar Equilibrium (English)
iii
Resumo
Neste trabalho, consideramos o modelo Bumblebee para a quebra espontânea da simetria de Lorentz no contexto de corpos celestes com simetria esférica. Uma discussão das equações de movimento mod- i cadas é apresentada, juntamente com os constragimentos do modelo obtidos de modo perturbativo. De modo a contextualizar o modelo e o problema em questão, uma revisão é apresentada, onde se abordam temas tais como os fundamentos da Relatividade Geral, o papel das simetrias na física e o problema da matéria escura.
Palavras Chave
Relatividade Geral, Modelo Bumblebee, Quebra da simetria de Lorentz, Equilíbrio Estelar
v
Contents
1 Introduction 2 1.1 Thesis Outline...... 3 1.2 Essential concepts...... 3 1.3 General Relativity...... 4 1.4 The limits of Einstein's Relativity...... 5
2 Astrophysics 9 2.1 Introduction...... 10 2.2 Tolman-Openheimer-Volkov equation...... 10 2.3 Polytropes and the Lane-Emden Equation...... 12
3 Symmetry Breaking 15 3.1 Introduction...... 16 3.2 Explicit and Spontaneous Breaking of Symmetries...... 17 3.3 Observer and Particle LSB...... 20 3.4 Standard Model Extension and LSB...... 21
4 Vector Theories 25 4.1 Introduction...... 26 4.2 Aether Theories...... 26 4.3 The Bumblebee Model...... 34
5 Astrophysical constraints on the Bumblebee 37 5.1 Introduction...... 38 5.2 Static, spherically symmetric scenario...... 38 5.3 Perturbative Eect of the Bumblebee Field...... 39 5.4 Numerical analysis...... 41
6 Conclusions 45
Bibliography 47
vii
List of Figures
2.1 Lane-Emden solution for a spherical body in hydrostatic equilibrium...... 14
3.1 Snow akes generated through a Linenmayer rule set for the rst, second and third iterations...... 16 3.2 The Mexican Hat potential...... 19
4.1 The Bullet Cluster...... 28
5.1 Pro le of the relative perturbations pb/p0, pV /p0, ρb/ρ0 and ρV /ρ0 induced by the Bumblebee...... 42
5.2 Allowed region (in grey) for a relative perturbation of less than 1% for pb and ρb..... 43
5.3 Allowed region (in grey) for a relative perturbation of less than 1% for pV and ρV .... 44
ix
List of Tables
5.1 Selected stars that used as models for the numerical analysis of the Bumblebee pertur- bation...... 43 5.2 Table of non-dimensional parameters...... 44
xi
Abbreviations
GR - General Relativity TOV - Tolman-Oppenheimer-Volko LE - Lane-Emden EP - Equivalence Principle LLI - Local Lorentz Invariance LPI - Local Position Invariance EP - Equivalence Principle WEP - Weak Equivalence Principle SEP - Strong Equivalence Principle LSB - Lorentz Symmetry Breaking PPN - Parametrized Post Newtonian CPT - Charge Parity Time Symmetry GZK - Greisen-Zatsepin-Kuzmin CMBR - Cosmic Microwave Background Radiation CMBR - Cosmic Microwave Background Radiation HI-RES - High Resolution Fly's Eye SME - Standard Model Extension SM - Standard Model SSB - Spontaneous Symmetry Breaking QED - Quantum Electrodynamics MOND - Modi ed Newtonian Dynamics FLRW - Friedmann-Lemaitre-Roberston-Walker VEV - Vacuum Expectation Value
1 1 Introduction
Contents 1.1 Thesis Outline...... 3 1.2 Essential concepts...... 3 1.3 General Relativity...... 4 1.4 The limits of Einstein's Relativity...... 5
2 1.1 Thesis Outline
1.1 Thesis Outline
This work will begin by a small introduction of the relevant concepts needed from General Relativity in the subsection 1.2 and the current observed limitations of the theory in subsection 1.4. The Tolman- Oppenheimer-Volko equations are then introduced in Chapter2 along with the Lane-Emden (LE) model for spherical bodies, which will be needed in order to understand Chapter5. In Chapter3, the concept of Lorentz Symmetry Breaking (LSB), in both its explicit and spontaneous form, is introduced as a way to portray one of the fundamental properties of the model being used, the Bumblebee Model. This will later be introduced in4 as a particular case of the more general Aether Theories. The two nal chapters (5,6) contain the kernel of this work: The application of the models presented as a way of constraining their parameters using a spherical astrophysical body modeled by a polytrope. This thesis closely follows the work done in [1].
1.2 Essential concepts
Presently, Einstein's general theory of relativity serves as a tool to understand a wide range of phenomena. From the dynamics of compact astrophysical bodies, such as stars and black holes, to cosmology, its striking predictions have sustained this theoretical framework and made it one of the most relevant scienti c achievements in the history of science [2]. The experimental con rmation of the existence of gravitational lensing, time dilation and gravi- tational redshift have solidi ed this fact but this by no means imply that the theory is completely correct; in fact, such consistence is a motivating factor for testing its limits even further. Questioning the basic assumptions of General Relativity is thus a valid way to achieve this goal, as they are the fundamental rule set from which it emerges. These basic assumptions can be expressed in the following manner [3]:
Weak Equivalence Principle - Bodies in free fall have the same acceleration independently of • their compositions.
Local position invariance - The rate at which a clock ticks is independent of its position. • Local Lorentz invariance - The rate at which a clock ticks is independent of its velocity. • This work shall focus on the second and third basic assumptions presented above. A more detailed discussion will be presented in3 in how this principle can be used to test the validity of physical theories. As a short introduction to the review that will be made in the following section, a small appetizer is given in the context of special relativity, showing how one can test the limits of such theory. The tests are based on testing the principles of relativity (physical laws are independent of the inertial frame of reference used to describe them) and the constancy of the speed of light. In the case of inertial frames of reference (which can always be found, at least in the vicinity of any given point in spacetime), this translates into invariance under Lorentz transformations, a tenet of Special Relativity: deviations from these transformations would also imply deviations from the underlying principles.
3 1. Introduction
This can be approached through the Robertson-Sexl-Mansouri formalism, which consists on the known Lorentz transformations,
vx t 2 x vt t0 = − c , x0 = − , y0 = y, z0 = z, (1.1) 1 v2 1 v2 − c2 − c2 q q altered in a way as to express a preferred frame of reference Σ(T, X~ ). The transformations now take the form,
t ~.~x ~x 1 1 ~v~x T = − , X~ = ~v + ~vT. (1.2) a d − d − b v2 With these transformations, it can be shown [3] that through an expansion of a, b, d and ~ around v/c2 one can obtain the following expression for the relative shift in the two way speed of light,
c (θ, v) v 2 3δ2 β2 β 3 v 4 2 1 = sin2 θ (δ β) + − β (1 + δ) δ + (β δ)2 cos 2θ , (1.3) c (0, v) − − c 4 − 2 − 2 − 2 4 − c 2 with θ the angle between the velocity ~v of the frame of reference and the path of light, and αn, βn, δn, n the expansion terms of a(v), b(v), d(v) and respectively with n = 1, 2 the order of the expansion. If any of the referred expansion terms are veri ed to be non-vanishing in some measurement, this would imply the violation of Lorentz invariance.
1.3 General Relativity
Having introduced the general method, we now move onto de ne the basic formalism of General Relativity, along with the conventions that will be used in the remainder of this work. General Relativity proceeds by claiming that the principle of relativity indeed applies to all frames of reference, and not just inertial ones. Thus, instead of considering only invariance under Lorentz transformations, it imposes general (dieomorphism) invariance, i.e. the laws of physics are invariant under general (dierentiable) coordinate transformations. The need for having a priori coordinate invariance implies that we need to use a scalar Lagrangian density which, depends on the elds and their derivatives (up to second order in order to avoid Ostro- gradsky's instabilities [6]). In Riemannian spacetime, this eld is the metric tensor and its derivatives are embodied in the Ricci curvature scalar, leading to the standard Einstein-Hilbert action = R. If, L however, we promote R to a fundamental variable, then more general forms are admissible
= g [Λ + bR + c Rµν + ... + f(R)] , (1.4) L | | ∇µ∇ν p with f(R) a possible function of the metric from which we can build modi ed versions of the basic Einstein-Hilbert (which is obtained by selecting the linear terms alone).Variation of the Einstein-Hilbert action leads to
1 1 1 δS = dx4√ g Rµν gµνR + T µν δg , (1.5) − −16πG − 2 2 µν Z with
2 δ(√ g ) T = − L . (1.6) µν −√ g δgµν −
4 1.4 The limits of Einstein's Relativity
Imposing a null variation yields the Einstein's equations of motion,
1 Gµν Rµν gµν R = 8πGT µν (1.7) ≡ − 2 where G is the gravitational constant. This identity tells us one of the most important results of this theory: The distribution of energy in spacetime dictates its curvature.
The doubly contracted Bianchi identities Gµν = 0 then imply the conservation of the energy- ∇ν momentum tensor T µν = 0. In the absence of matter, one would obtain R = 0 for the Ricci scalar; ∇ν in a static and spherically symmetric spacetime, this leads to the Schwarzschild solution. The following chapters will use these results to derive the Tolman-Oppenheimer-Volko equation in chapter2 or the results presented in the nal chapter.
1.4 The limits of Einstein's Relativity
Having introduced the basic formalism that will be used throughout this work, we shall now discuss some of the known experimental limits that constrain General Relativity as well as the methods used in obtaining them. These observational bounds can be used both to test the validity of the foundations of GR and alternative theories of gravity [2].
Varying the action S = √ g µ ν one then gets the geodesic equation of motion − µνdx dx R d2xµ dxα dxβ = Γµ . (1.8) dτ 2 αβ dτ dτ In the static weak- eld limit, the metric can be written as
2Φ 2Φ g = g = 0, g = 1 N , g = 1 + N δ (1.9) 0i i0 00 − − c2 ij c2 ij with ΦN the Newtonian gravitational potential. Inserting this metric into the geodesic equation we get Newton's second law of motion for the (0, 0) component
d2xi ∂Φ = Γi = N (1.10) dt2 − 00 − dxi and, from Einstein's equations of motion 1.7, we get the Poisson equation
2Φ = 4πGρ. (1.11) ∇ N This simply means that the metric plays the role of the Newtonian gravitational potential, with the Christoel symbol behaving analogously to an acceleration. Thus, by changing the metric one could explore how certain aspects of a theory cascade into the equations of motion of the system.
1.4.1 Parametrized Post Newtonian Formalism
We now introduce the Parametrized Post Newtonian Formalism, or PPN for short. Behind every metric theory of gravity lies the underlying principle that the eponymous tensor directly aects the way in which the gravitational eld interacts with matter. As such, this formalism serves as a generalization of the metric to include parameters which express certain symmetries and laws of invariance of the
5 1. Introduction system, which then allows us to measure the deviation of these GR parameters in relation to Newtonian gravity. Assuming Local Lorentz and Position Invariance, along with conservation of momentum, it can be shown that the metric tensor in PPN is given by,
g = 1 + 2U 2βU 2 2ξΦ + (2γ + 2α + ζ 2ξ)Φ + 2(3γ 2β + 1 + ζ + ξ)Φ (1.12) 00 − − − W 3 1 − 1 − 2 2 +2(1 + ζ )Φ + 2(3γ + 3ζ 2ξ)Φ (ζ 2ξ) (α α α )ω2U 3 3 4 − 4 − 1 − A − 1 − 2 − 3 α ωiωjU + 2(2α α )ωiV + O(3) − 2 ij 3 − 1 i 1 1 g = (4γ + 3 + α α + ζ 2ξ)V (1 + α ζ + 2ξ)W (1.13) 0i −2 1 − 2 1 − i − 2 2 − 1 i 1 (α 2α )ωiU α ωjU + O(5/2) −2 1 − 2 − 2 ij 2 gij = (1 + 2γU)δij + O( ), (1.14) where β is a measure of the non-linearity of the law of superposition of gravitational elds, γ mea- sures the curvature of the spacetime created per unit rest mass, α1, α2, α3 measure deviations from
Lorentz invariance, ζ the violation of Local Position invariance and α3, ζ1, ζ2, ζ3, ζ4 measure the pos- sible violation of the conservation of momentum. The expressions for the gravitational potentials
U, U , Φ , ,V ,W are given in [3]. GR is characterized by β = γ = 1, and all other PPN parameters ij i A i i vanish.
Current experimental tests on these parameters, particularly the couple γ, β, are the result of measurements made by the Cassini 2003 spacecraft and helioseismology show that γ 1 2.3 10−5 − ≈ × and β 1 3 10−4, respectively [7]. − ≈ × 1.4.2 Equivalence Principle, Lorentz and Position Invariance
We now move to the testing of the postulates presented in the previous section; the equivalence principle and both the Local invariance principles (position and Lorentz), which are the fundamental groundwork from which GR is built from. Tests for the Equivalence Principle can be divided into two groups: those that test the weak version (WEP) and those that test the strong equivalence principle (SEP). The WEP states that the Equivalence Principle (all non-gravitational laws should behave in free- falling frames as if there was no gravity) is satis ed by all interactions except that of gravity. One could test its validity by simply measuring the dierence in the free-fall accelerations between two test bodies a1 and a2 [3],
∆a 2(a a ) M M M = 1 − 2 = G G = ∆ G . (1.15) a a + a M − M M 1 2 I 1 I 2 I where MG and MI represent the gravitational and inertial masses, or by directly measuring the ratio
MG/MI . For the latter, various experiments have been made where the most recent and strongest constraint being given by Adelberger in Ref. [8] of 1 M /M 1.4 10−13. | − G I | ≈ × Another consequence of the WEP is the existence of a gravitational Doppler eect in bodies which travel trough a changing gravitational potential. One of the most historically relevant experiments
6 1.4 The limits of Einstein's Relativity was made by Robert Pound and Glen Rebka [9], where photons where emitted by a moving source and later absorbed by a stationary target located at the top of a tower. The absorption was only possible as the relativistic doppler shift of the moving source cancelled the gravitational doppler eect of the graviational eld of the earth. The measured change in frequency ∆ν/ν = 2.57 0.26 10−15 ± × con rmed the existence of this eect, as predicted by GR. The SEP, on the other hand, states that every measurement is independent of the velocity and position of the laboratory, even accounting for the self-energy of massive bodies such as stars and black holes. Testing the SEP implies measuring the contributions of this gravitational energy which were not considered in the WEP. In order to accomplish this we resort to the PPN formalism introduced above, introducing the quantity
M Ω ∆ G = η (1.16) M Mc2 I where Mc2 is the total mass energy of the body and Ω its negative gravitational self-energy. Here η = 4β γ 3 is a combination of PPN parameters. This combination implies that in GR one should − − have η = 0, since γ = β = 1. We now move on to the two nal assumptions presented in the previous section: Local Lorentz and Position Invariance, or LLI and LPI respectively. The phenomenological eect of moving in a reference frame relatively to a stationary one can be probed by assuming a cosmological vector eld which collapses onto a non-vanishing minimum via spontaneous symmetry breaking. These types of models have been proposed by Kostelecký [10] and its impact on solar system observables was discussed in [11]. The mechanism underlying the breaking of the symmetry is explained in chapter3. Experimental searches for the breaking of this symmetry have been made through diverse physical phenomena. As a violation of this symmetry could imply the breaking of the CPT symmetry, numerous proposals have been made for the possible testing of this possibility [12]. As referred above, the PPN formalism is also a good way in which to infer the departure of an experimental phenomenon from what would be expected from GR. In what regards the parameters that relate to preferred frame eects, the more relevant of these is α2, for which the observational limit is α < 4 10−7[13] and re ects the existence of spin precession anomalies, along with α , which | 2| × 3 re ects self-acceleration eects. For the latter, a measurement was made via pulsar statistics in order to measure its deviation if it is non-vanishing, yielding the constraint α < 2.2 1020 [14]. 3 × Another way to measure possible Lorentz violation is through the study of the Greisen and Zatsepin & Kuzmin (GZK) cut-o. This stems from the interaction between protons with energies of the order of 1020eV and Cosmic Microwave Background Radiation (CMBR) photons in nuclear reactions of the type
p + γ p + π0. (1.17) CMB → Due to these reactions, the primary protons would have their energy decreased, suering a type of
7 1. Introduction friction from the cosmic background radiation. The threshold for these reactions is given by
2 mπ + 2mπmp 19 Ef = 10 eV, (1.18) 4EγCMB ≈ which implies that above this energy value, we should not observe cosmic rays with these energies on Earth. Con rmation of the GZK limit through experiments such as the Pierre Auger Observatory [15] would impose strong constraints [16] on the possible observation of LV in the QED as discussed above.
As discussed in Ref. [17], the detection of those 1019eV photons would imply,
a 10−25, a = a 10−7 (1.19) | 1| ≤ | 2| | 3| ≤ which would indicate a very weak presence of LV at the high energy scale. However, recent results from both HI-RES (High Resolution Fly's Eye) and the Pierre Auger collaborations have shown evidence that the cut-o has not been statistically broken, which implies that Lorentz invariance has also been maintained [18]. In what regards LPI, the already mentioned Pound-Rebka experiment can serve as a test for this eect, with the measurement of the change in frequency given by
∆ν (1 + µ)U = (1.20) ν c2 were µ = 0 in GR, would give an idea of how far the assumption holds. Measurements made with hydrogen-maser frequencies on earth and on altitudes of ten thousand kilometres yield µ of µ < | | 2 10−4 [19]. × Although LI has these strong constraints, the fact is that there is still room for obtaining deviations in energy scales that are unreachable today [13, 20]. It is through this window of opportunity that we shall peer through in the following sections.
8 2 Astrophysics
Contents 2.1 Introduction...... 10 2.2 Tolman-Openheimer-Volkov equation...... 10 2.3 Polytropes and the Lane-Emden Equation...... 12
9 2. Astrophysics
2.1 Introduction
In this section, the hydrostatic equilibrium equation is obtained via an approximation of the known Tolmann-Oppenheimer-Volko equation which, is derived directly from Einstein's equations of motion. Assuming a polytropic equation of state, these equations are the so called Lane-Emden dierential equations, which have a solution depending on the polytropic index n alone. This model will be later used in chapter5 as a description of a non-perturbed star from which the subsequent analysis of the Bumblebee model will follow.
2.2 Tolman-Openheimer-Volkov equation
The description of a star's interior when in hydrostatic equilibrium can be obtained in General Relativity through the choice of an appropriate energy-momentum tensor along with a static spherically symmetric metric. Assuming that the uid inside the star behaves like a perfect uid, then the appropriate energy- momentum tensor is given by,
Tµν = ρuµuν + p(gµν + uµuν ), (2.1) where the signature ( , +, +, +) is used both in this case and throughout the remainder of this work. − Given the static, spherically symmetric geometry, the Birkho metric is chosen, as it is described by the line element,
ds2 = e2ν(r)dt2 + e2λ(r)dr2 + r2dθ2 + r2 sin2(θ)dφ2 (2.2) − and so the energy-momentum tensor is given by
2ν(r) 2λ(r) 2 2 2 Tµν = diag(ρe , pe , pr , pr sin (θ)) (2.3)
and the trace by
T = gµν T = ρ + 3p. (2.4) µν − Using the trace-reversed form of the Einstein eld equations referred in Chapter1
1 R 8π T g T = 0 (2.5) µν − µν − 2 µν tt rr θθ the equations g Rtt,g Rrr and g Rθθ are then, respectively: e2(ν−λ) r(λ0 ν0)ν0 rν00 2ν0 = 4πGe2ν (3p + ρ) (2.6) r − − − − 1 2λ0 + r(λ0 ν0)ν0 rν00 = 4πGe2λ(p ρ) (2.7) r − − − − e−2λ 1 + e2λ rλ0 + rν0 = 4πGr2(p ρ) (2.8) − − − Subtracting 2.6 and 2.7 then gives,
2e−2λ(λ0 + ν0) = 8πG(p + ρ). (2.9) − r
10 2.2 Tolman-Openheimer-Volkov equation
For that we need to nd a relation between the density and the metric function λ(r). By using the combination,
gttR + grrR 2gθθR = 0 (2.10) tt rr − θθ we nd that
2[1 8πGρr2 + e−2λ( 1 + 2λ0r)] − − = 0 (2.11) r2 which gives us, through some manipulation, the equation
1 e−2λ + 2rλ0e−2λ = 8πGρr2 (2.12) − d r re−2λ = 8πGρr2. (2.13) dr − We can now obtain a relation pertaining λ
2Gm e−2λ = 1 (2.14) − r which can be obtained by calculating the mass through
r m(r) = 4πρ(r0)r02dr0. (2.15) Z0 After dierentiating λ one gets
2m0r 2m 2Gm 2λ0e−2λ = G − = 8πGρr , (2.16) − r2 − r2 which, by substitution into equation 2.9, gives us a relation between the remainder metric function
ν(r) and the pressure,
Gm + 4πGpr3 ν0 = . (2.17) r(r 2Gm) − We know from the twice contracted Bianchi identities that
Gµν = 0, (2.18) ∇ν which implies that the conservation of the energy-momentum tensor T µν = 0, from the Einstein ∇ν eld equations presented in Chapter1.
By selecting the radial component µ = r, we arrive at the conservation equation for a perfect and static uid,
p0 + ν0(p + ρ) = 0, (2.19)
which allows us to reach a dierential equation for the pressure inside the star:
dp G(4πGpr3 + Gm)(p + ρ) 2Gm −1 = 1 (2.20) dr r2 − r which is the so called Tolmann-Oppenheimer-Volko equation for hydrostatic equilibrium. In order to solve the set of Eqs. 2.15 2.19 2.20 , which consists of a system of 4 unknown functions
λ, ν, p, ρ and three independent equations, we need another equation that relates the pressure and the
11 2. Astrophysics
density, i.e. an equation of state p = p(ρ). This motivates the next section, where we will be describing the polytrope equation of state and the Lane-Emden dierential equation for a star in hydrostatic equilibrium, as well as an introduction to the methodology applied in Chapter5 for obtaining the equations for the unperturbed star. In order to proceed into the next section, we will assume the conditions of hydrostatic equilibrium, i.e.:
p(r) << ρ(r), 4πp(r)r3 << m(r), 2Gm(r) << r, (2.21)
then the equation 2.20 can be approximated by its hydrostatic equilibrium version,
dp Gm = ρ (2.22) dr − r2 dm = 4πr2ρ (2.23) dr
which when combined give us the Poisson equation φ = 4πGρ in spherical coordinates ∇ 1 d r2 dp = 4πGρ. (2.24) r2 dr ρ dr − 2.3 Polytropes and the Lane-Emden Equation
With the previous conditions in mind, we can now assume that the pressure is given by the poly- tropic relation in equation 2.25 for a certain polytropic index n
p = Kρ1+1/n, (2.25)
where K is the polytropic constant and ρ0 the baryonic mass density. The polytropic index describes the basic thermodynamical processes. Taking n = 1 we get an isobaric sphere, n = 0 an isometric one and n = gives the isothermal condition for that same sphere. The adiabatic processes are ∞ related by n = 1/(γ 1) within this framework, where γ represents the adiabatic coe cient γ = c /c . − p V These indexes can provide crude approximations to known astrophysical bodies [21]. Degenerate star cores found in giant gaseous planets can be studied using a polytropic index of n = 3/2, boundless systems which were rst use in the description of stellar systems by Arthur Schuster in 1883 with n = 5 and the rst solar models, which were rst proposed by Arthur Eddington (known as the Eddington standard model of stellar structure), circa 1916 for a polytropic index of n = 3. Although these models are rather simplistic, they are still useful as rst order approximations, as they allow for an easier algebraic manipulation. Dierentiation of 2.25 gives
dp n + 1 1 dρ = Kρ n (2.26) dr n dr which, when inserted into equation 2.24 changes into
2 1 d n + 1 r 1 dρ Kρ n = 4πGρ. (2.27) r2 dr n ρ dr −
12 2.3 Polytropes and the Lane-Emden Equation
We can now rewrite this dierential equation in its dimensionless form, by selecting the transfor- mations
n n+1 ρ = ρcθ (χ) , p = pcθ (χ), (2.28)
The dierential equation is now given by
1+ 1 Kρ n 1 d dθ (n + 1) c r2 = θn. (2.29) 4πGρ2 r2 dr dr − c
where the underscript c refers to the central values of the quantities.
We can now de ne χ = r/rn, where rn is de ned through the star's radius and the terms present in equation 2.29,
1 (n + 1)K 2 1 n R = r χ = ρ 2−n χ . (2.30) n f 4πG c f
where χf signals the boundary of the spherical body (θ(χf ) = p(χf ) = ρ(χf ) = 0). With all these components, we can now assemble the dimensionless Lane-Emden dierential equa- tion:
1 d dθ χ2 = θn. (2.31) χ dχ dχ − The analytical solution for this equation was found by Chandrasekar [22] for a set of polytropic indexes n = (0, 1, 5). These are given by, respectively
1 sin(χ) 1 θ(χ) = 1 χ2, θ(χ) = , θ(χ) = (2.32) − 6 χ χ2 1 + 3 q where the nal solution is in nite in radial extent.
As we want to calculate the LE solution with n values around the Eddington solution with n = 3, these exact solutions will not be useful. We thus proceed with the analysis of the system via a numerical method. We know that the boundary conditions for this system are given by
θ(0) = 1 (2.33)
θ0(0) = 0 (2.34) and so by making a series expansion of our equation at the point χ = 0, we can then calculate the initial conditions of our system for some point χ = χi, and solve numerically the equation for any n up to 5, where the analytical solution is radially in nite as referred above. Throughout this work, three values for the polytropic index where chosen, with the intent of showing how its variation aects the model. The values are n = 2.8, 3, 3.2 and so the need to obtain a general expression for the expansion of θ(χ) around χ = 0 arises. Assuming that the solution θ is symmetric under the transformation χ χ and that the solution → − θ is analytic we can expand θ as,
χ2 nχ4 (5n 8n2)χ6 θ(χ) 1 + + − + ... (2.35) ≈ − 6 120 15120
13 2. Astrophysics
where the terms of order between n = 6 and n = nf where not shown for simplicity. This is useful in a numerical computation where it might be necessary the computation of a non-zero initial value close to the real null initial condition.
Using these conditions one then obtains the following pro les for the Lane-Emden function θ(χ), as can be seen in Figure 2.1
1 n = 2.9 n = 3 0.8 n = 3.1
0.6 ) χ ( θ 0.4
0.2
0 0 1 2 3 4 5 6 7 8 χ
Figure 2.1: Lane-Emden solution for the a spherical body in hydrostatic equilibrium. The roots of the three functions are 2.8 3 and 3.2 . χf = 6.191, χf = 6.896 χf = 6.768
Besides obtaining the pro le θ(χ), we can also relate the stars fundamental observed properties, its mass and radius, and relate them with the central density through the LE solution.
In order to go from the set (M,R) to the set (ρc, pc) from which the LE solution takes its scaling 0 2 parameters, we start by integrating the relation ρ0(r) = m /4πr in order to obtain
R χf 3 2 n (2.36) M = 4πρ(r)dr = 4πrnρc χ θ dχ. Z0 Z0 By inserting the LE equation into the integral one then gets,
3/2 3 n (n + 1)K 2−n 2 0 M = 4π ρc χ θ (χ ) (2.37) 4πG − f f which allows us to obtain K in terms of the mass and radius of the star and the polytropic index n
n 1 n n 3 n 1 π 2 0 − − −1 −n K = G 4 [ χ θ (χ )] n χ n MR . (2.38) n + 1 − f f f The central pressure of the system is then given by inverting rn,
2n −1/2 n− 1 n R K(n + 1) − ρ = (4π) n 1 (2.39) c − χ G " f # completing the parameters needed to completely describe the star.
14 3 Symmetry Breaking
Contents 3.1 Introduction...... 16 3.2 Explicit and Spontaneous Breaking of Symmetries...... 17 3.3 Observer and Particle LSB...... 20 3.4 Standard Model Extension and LSB...... 21
15 3. Symmetry Breaking
3.1 Introduction
The concept of symmetry can be stated as an intrinsic property of a system which does not change under certain transformations. Its usage throughout history can be traced back to both aesthetic and technical means of describing apparent order and beauty, such as the apparent mirror symmetry of the human body, or the more complex six-fold symmetry present in snow akes. One of the most interesting aspects of this idea is the fact that when applied to physical systems, the existence of symmetries restricts the possible outcomes that certain phenomena may have under that system. This idea was stated by Pierre Curie in his Sur la symètrie dans les phénoménes physiques [23], where he found that the thermal and electric properties of crystals varied with the underlying crystal structure.
Figure 3.1: Snow akes generated through a Linenmayer rule set for the rst, second and third iterations.
A simple example can be given, as represented in gure 3.1. The snow ake was generated through a formal rule set rst created by Aristid Lindenmayer, a Hungarian botanist whose work consisted in trying to describe the evolution of the visual patterns present in the growth of yeast. The Linden- mayer system takes an alphabet of symbols, an initial state for the system and a set of production rules and applies the rules iteratively. In the case of our snow ake, for each 60◦ angle a branch is generated with two smaller branches, with each iteration adding onto the already growing pattern. We have thus created a model that describes, up to a certain degree of certainty, a snow ake which one could nd in nature, only through the rules that describe its underlying symmetry. But what if there were only two snow akes in the Universe, and one snow ake was slightly dierent than the one described by our model? Well, this would imply that one of the symmetries that described the "good" snow ake, would be broken by the "bad" one, and thus our model would have to be updated in order to incorporate this fact. This idea of looking for the underlying symmetries of a certain system and checking for their existence in the Universe portrays the motivation behind the model used in this work, although it is inserted in the larger context of both Einstein's Relativity and the Standard Model of Particle Physics. The Principle of Relativity, as rst adopted by Einstein in his special theory of relativity, states that the laws that govern the change of a physical system are independent of the chosen coordinate systems, even when those are moving relatively to each other in uniform translational motion. This principle, when coupled with the light postulate which guarantees a xed speed of light in all
16 3.2 Explicit and Spontaneous Breaking of Symmetries inertial frames of reference, leads to the known Lorentz transformations between coordinate systems which are moving relatively to each other. This can be written succinctly, trough the matrix equation
x0 = Λ(v)x (3.1) which, for a boost in the x direction for example, would be described by ct0 γ βγ 0 0 c t x0 βγ− γ 0 0 x = , (3.2) y0 −0 0 1 0 y z0 0 0 0 1 z representing thus the underlying symmetry of boosts in the x direction. In General Relativity, this symmetry holds locally i.e. in distances small enough that the variations in the gravitational eld are unnoticed. This is but a statement of the weak equivalence principle which, as stated by Einstein, takes the form [24]
The outcome of any local non-gravitational experiment in a freely falling laboratory is in- dependent of the velocity of the laboratory and its location in spacetime.
In the following sections, when speaking of Lorentz invariance, the concept behind the name will follow closely what was described in the introduction. The goal is to provide a brief introduction to the concept of Lorentz Symmetry Breaking (LSB) and its usage as a probe for studying fundamental physical phenomena in both large and small scales.
3.2 Explicit and Spontaneous Breaking of Symmetries
The breaking of symmetries is a great source of richness in both physics and everyday life. Consider then a hungry donkey who is placed between two stacks of hay and assume he always goes to the nearest one. By the Principle of Su cient Reason put forward by Leibniz, the donkey cannot justify going for either stack of hay and so it dies of hunger as it is unable to decide. Unless, of course, by some unknown force of will, he breaks the symmetry of his dire situation and goes for one of those stacks of hay! Thus it is justi ed that breaking a symmetry brings with it a lot more to life than simply perturbing a perfect, albeit immutable situation. In the case of physics, things are not so simple. We shall start by de ning the two ways in which these symmetries can be broken: explicitly and spontaneously, and provide some examples of these kinds of phenomena. For explicit symmetry breaking, the underlying symmetry that is broken appears in the description of the physical laws themselves. This can take the form of certain terms in the Lagrangian density that describes the system, as the dynamical equations are not invariant under that transformation. One example of this is the parity violation in the weak interaction, which was initially proposed by Yang & Lee circa 1950 [25] by studying the beta decay of cobalt-60. Up until this point the laws of nature were assumed to be invariant under mirror re ection transformations (as observed in gravity, electromagnetism and the strong force), i.e. observing a certain experiment and its mirror re ected
17 3. Symmetry Breaking copy should yield the same results; however, in the case of beta decay, a preferred direction of the emitted electrons was always observed. In relativistic quantum mechanics, this symmetry portrays an invariance in the change between particles and antiparticles within a system. Along with parity, charge conservation is also seems to be broken in the weak interactions. The conjugation of both these symmetries (CP), was once thought to be a valid symmetry of nature, but was later veri ed by Cronin & Fitch to be broken in the decay of neutral kaons [26], which gave them the Nobel Prize in Physics in 1980. Although the individual symmetries appear to not to be fundamental properties of our universe, a combination of them does: a theorem proved by Schwinger, Pauli, Bell and Lüders around 1950 shows that the combination of charge conjugation, parity and time-reversal symmetries is conserved in quantum eld theories which have Lorentz invariance, local causality and positive energy [27]. This is called the CPT theorem and up to this day this symmetry seems to hold up experimentally [28]. Another example is the occurrence of symmetry breaking through non-renormalizable eects. Ef- fective eld theories appear as low-energy approximations to a more overreaching theory, as they only accurately describe the particles that fall within the energy range considered. Although the eects of heavier particles do not appear on the low energy regimes, when moving on to higher energies, symmetries which were assumed on the low end of the energy scale could be broken on the higher energy theories. As for the case of spontaneous breaking of symmetries, things are a bit dierent. Instead of the asymmetry existing in the equations of motion themselves, it occurs instead in one of the possible solutions to those equations, arising dynamically from the system. A simple example can be given in order to illustrate this. Consider a cylindrical rod held hori- zontally. If we let the rod fall, it will spontaneously choose a direction, breaking the initial rotational symmetry. The state in which the symmetry of the system is broken is one of the in nite solutions of the dynamical equations governing the system. This same phenomenon can be observed when a ferromagnet is cooled below its critical temperature
(Curie temperature Tc). Initially the system presents no magnetization, as it is has T > Tc, but a net magnetization emerges as soon as T < Tc, where the spins align spontaneously in a given direction, breaking the initial symmetry of the system. In the context of particle physics, the occurrence of spontaneous symmetry breaking is fundamental in explaining certain phenomena. One example of the breaking of a discrete symmetry is the Yukawa interaction, which can be explained via the Lagrangian density
1 = (∂φ)2 µ2φ2 λφ4. (3.3) L 2 − − where φ is a real scalar eld. This Lagrangian density represents a system consisting of a self-interacting scalar eld φ, with the potential of the system consisting of the V (φ) = µ2φ2 λφ4 terms. It is − − invariant under the global Z symmetry φ φ or, if we instead adopt a complex scalar eld, the → − global U(I) symmetry φ φeiα, with α a constant. → If our potential has a minimum value at some point φ = 0, then its symmetry would be eectively 0 6
18 3.2 Explicit and Spontaneous Breaking of Symmetries
2 broken. In this case, by taking µ = 0 the potential would have two possible minimum values φ0 = 1/(2λ)µ, which when selected by the system would spontaneously break the underlined global ± symmetry.p In the case of continuous SSB, the discovery of the Goldstone's Theorem [29] is fundamental in explaining several of results in particle physics. As a result of the continuous symmetry being broken, massless bosons appear (Goldstone bosons), with the number of bosons being equal to the number of generators of the broken symmetry. These massless bosons are crucial in the understanding the Higgs mechanism. In this case, the picture changes as the symmetry being broken is local, instead of the global one described by the last example. A similar model to the one given by equation 3.3 can be given in order to explain this mechanism.
Consider the Lagrangian of the complex scalar eld φ, φ∗ coupled with the electromagnetic eld 1 1 = (Dµφ)∗(D φ) + µ2φ∗φ λ(φ∗φ)2 F µν F (3.4) L − µ − 4 − 4 µν where D φ = ∂φ iqA φ is the covariant derivative, and the potential can be identi ed similarly as µ − µ V (φ, φ∗) = µ2φ∗φ 1 λ(φ∗φ)2. This potential is what is called the "Mexican Hat Potential", due to − 4 its shape, which can be seen on Figure 3.2.
Figure 3.2: The famous Mexican Hat potential which describes how a system spontaneously breaks its initial symmetry by rolling onto the minimum of the potential, where a circle of in nite possible solutions exist.
The minima is now given by a point in a circle on the complex plane, i.e. φ 2 = µ2a2/2 with | | a = 4µ2/λ, and so the "rolling" of the eld to one of these values would eectively break the symmetry.p If we de ne the scalar eld φ as, 1 φ(x) = (α + β(x) + iγ(x)) (3.5) √2 then our kinetic terms are changed into
1 1 1 (Dµφ)∗(D φ) = (∂β)2 (∂γ)2 q2α2A2 + qαA ∂µβ + ... (3.6) − µ −2 − 2 − 2 µ
19 3. Symmetry Breaking
where we end up having a quadratic coupling between the gauge eld A and the scalar γ. With an −1 appropriate gauge change, given by Aµ(x) = Vµ(x) + (qα) ∂µγ(x), this can be corrected, as now the kinetic terms are give by
1 1 (Dµφ)∗(D φ) = (∂β)2 q2α2V 2 + .... (3.7) − µ −2 − 2
This is the Higgs mechanism: The gauge change brought with it a new mass term with m = qα, with the added advantage of absorbing the scalar eld γ and collapsing the potential onto a non-zero minimum, thus breaking its symmetry. The goal of this work is not to explain the Higgs mechanism in detail, but rather bring some light into the relevance that the breaking of spontaneous symmetries has in explain certain physical phenomena. The gravitational models that will be presented will have some aspects of these SSB in some sense, which though not representing the same mechanism, are easier to understand having a more complete conceptual baggage.
3.3 Observer and Particle LSB
Due to the extensive application of this symmetry in physical theories, discussing it without clarify- ing which exact symmetry is broken brings with it a lot of confusion, as not all Lorentz Transformations (LT's) are the same. We can group the dierent types into two groups: as Observer and Particle Lorentz transformations. Following this labeling, we can characterize them in the following manner.
Observer Lorentz transformations: These are the transformations one usually thinks of • when working in Special Relativity. They relate the observations of two inertial observers which can have dierent velocities or can be rotated in some manner in relation to each other. Not having this kind of invariance would mean that the choice of reference frame would alter the results obtained when measuring particular phenomena, i.e. a particle's mass would vary from one point to another. Due to this, theories must preserve this symmetry, something which is accomplished by writing the laws of physics in terms of covariant equations. If one takes an
action S de ned on a manifold with a metric g and which depends on some eld Φ(x), then M
S[g, Φ(x)] = (g, Φ(x), Φ(x)) (3.8) L ∇ ZM must be a Lorentz scalar in order to be invariant. The equations of motion derived from this action will be invariant to boosts and rotations between reference frames of the form
xµ x0µ = Λµxν (3.9) → ν with the elds also being transformed accordingly.
Particle Lorentz transformations: If in the rst case the dierence between dierent exper- • iments was only a change of coordinates, in this case two identical experiments can be boosted or rotated relative to each other by the same observer. This happens because as we rotate our
20 3.4 Standard Model Extension and LSB
experiment in relation to a certain eld, the relations between them are also altered. In order to clarify this fact, a couple of examples can be given.
Consider a system composed of a charge of mass m and charge q being aected by a perpendicular magnetic eld B~ . Its motion can be described by
d~v m = q~v B,~ (3.10) dt × an equation that we know is valid in all reference frames. The particle in this situation would move in a circular motion in a plane perpendicular to the eld. Suppose we made a particle LT by means of a boost; the consequence of this would be a larger radius in the trajectory of the particle, as its momentum would be increased. On the other hand, if we made an observer boost along the particle's trajectory, the result would be dierent, as a drift in the particle would arise due to now there being an electric eld.
The other example is also straightforward. Considering a mass on an inclined plane on the surface of the earth [30], if one simply changes the system of coordinates that describe the system, the acceleration that the mass obtains is the same in both reference frames, diering only from a change of coordinates. However, if we rotate the inclined plane, that acceleration would change perceptively as the direction of the gravitational eld in relation to the xed background would be dierent (it would be as if we created a new ramp with a more drastic incline).
In conclusion, the type of Lorentz symmetry that is eectively broken in the following sections will be of the second type, i.e. particle Lorentz transformations.
3.4 Standard Model Extension and LSB
The Standard Model gives us an accurate description of the myriad of phenomena which occur between the basic particles and forces at very small scales. On larger scales, this burden falls on the classical description provided to us by General Relativity which, as far as experimental con rmation goes, has proven itself within the class of physical eects it tries to eectively describe [10].
As these two eld theories are expected to merge at the Planck level, with energies around mP = 1019GeV, into a single uni ed and consistent description of nature, search for possible signals at this scale is paramount to achieve a deeper understanding how this merger might theoretically occur. One possible candidate for these signals is the breaking of Lorentz symmetry. This work will be centered on a particular type of alternative gravitational theories which are based on the introduction of vector elds in the Lagrangian density of our system, the so called Aether theories. The main motivation for this type of approach emerged from the work done initially by Alan Kostelecký circa 1988, which consisted in the study of natural Lorentz symmetry breaking (LSB) mechanisms in bosonic string theory, with the goal of trying to explain the relationship between the 26-dimensional spacetime needed for that theory, and the four at dimensions that we know of, with
21 3. Symmetry Breaking the assumption that the breaking the symmetry could bring with it the compacti cation of extra dimensions [31]. The raise in interest regarding the breaking of fundamental symmetries (such as Lorentz and CPT violation) in eld theories then culminated on the creation of the Standard-Model extension. A framework built from the core elements of the Standard Model and General Relativity, as Kostelecký refers to in Ref. [10]
(...) suppressed eects emerging from the underlying uni ed quantum gravity theory might be observable in sensitive experiments performed at our presently attainable low-energy scales. (...) Any observable signals of Lorentz violation can be described using eective eld theory. To ensure that known physics is reproduced, a realistic theory of this type must contain both general relativity and the SM, perhaps together with suppressed higher-order terms in the gravitational and SM sectors.
To give an idea, the Standard-Model Extension can be represented by an action consisting of a partial sum of terms given by,
SSME = SSM + SLV + Sgravity + ... (3.11) where SSM is the SM action (although with some gravitational couplings [10]) with a corresponding Lagrangian density given by
= + + + + (3.12) LSM Llepton Lquark LY ukawa LHiggs Lgauge
SLV corresponds to the SM Lorentz and CPT-violating terms and Sgravity is the gravity sector of the Lagrangian. The Lorentz-violating terms on the SME take the form of Lorentz-violating operators coupled to coe cients which will be de ned through Lorentz indexes. The existence of non-zero LV coe cients could appear through various mechanisms, one of which (and the one which will be crucial for the models referred to herein) is spontaneous Lorentz violation (SLV). The classi cation of the Lorentz-violating terms can be done through the observed properties under CPT [10], as the breaking of this symmetry in Minkowski-spacetime implies Lorentz violation. The pure gravity action can be written as,
1 S = d4x (3.13) gravity 2k Lgravity Z with the Lagrangian density consisting of a Lorentz invariant and a Lorentz violating part = Lgravity LI + LV + ... the latter being considered in the limit in which the torsion vanishes [10]. L L Although it is in this Lorentz violating Lagrangian density that we shall focus our attention, a brief example follows of how the insertion of Lorentz breaking terms into our Lagrangian density can change some known physical properties of a system. Consider the following QED Lagrangian density with isotropic LV [17] which allows us to study LV in the case of the photons and electron/positrons:
22 3.4 Standard Model Extension and LSB
1 ia a = ψ¯(iγµD m)ψ F F µν + ia ψγ¯ iD ψ + 2 D ψγ¯ iD D ψ + 3 F ∂2F kj. (3.14) L µ − − 4 µν 1 i M 2 j i j 4M 2 kj i The Lorentz violating terms are coupled to parameters which allow to test the constraints on some observable quantities. The rst two terms belong to the usual QED Lagrangian (the Dirac term along with the electromagnetic energy). The remainder terms are the Lorentz violating ones, which are coupled to the scale constant M and the dimensionless parameters a1, a2, a3. The u, j indexes represent the spatial components of the respective terms. From this Lagrangian one can then derive the dispersion relations, which will naturally depend on the LV parameters, as follows,
a k4 E2 = k2 + 3 (3.15) γ M 2 2a p4 E2 = m2 + p2(1 + 2a ) + 2 . (3.16) e 1 M 2 which are clearly a deviation from the relativistic dispersion relations. The work done in [17] focused on obtaining constraints for the ai parameters above by studying the interaction of high-energy photons with the Earth's atmosphere and magnetic eld and how the detection of photon-induced showers with energies above 1019eV would constrain those parameters to values of the order of a 10−25, a = | 1| ≤ | 2| a 10−7, as referred to in chapter1. The small values for these parameters are in agreement with | 3| ≤ the fact that no compelling evidence exists for Lorentz violation. In the following section a group of models which where created with the goal of trying to obtain similar constraints will be presented. Focusing on the gravitational sector of the Standard Model Extension, the aether models present themselves as phenomenological probes to test the existence of LSB within astrophysical bodies and cosmologies.
23 3. Symmetry Breaking
24 4 Vector Theories
Contents 4.1 Introduction...... 26 4.2 Aether Theories...... 26 4.3 The Bumblebee Model...... 34
25 4. Vector Theories
4.1 Introduction
In the following section, a brief introduction to Einstein-Aether theories will be made. Starting from the de ning Lagrangian density of these theories, a brief historical review of how they came into existence is made. As these theories are relevant in the context of dark matter, particularly as a result of the emergence of the MOND theories (Modi ed Newtonian Dynamics) and later on the TeVeS models which were used as an attempt to explain the phenomenon, the dark matter and energy problem will also be described. Within the Aether models, two applications will be presented: one regarding the change in the gravitational constant and the other regarding in ation. Finally, the Bumblebee model will be introduced. This model will later on be used in the work presented in chapter5 as a way to obtain astrophysical constraints on the existence of a vector eld with a non-vanishing expectation value inside spherical star-like bodies.
4.2 Aether Theories
Although some of the more direct references [3235] mention the introduction of Lorentz breaking symmetries in gravitational theories to the work of Kostelecký [10, 12, 31], the study of gravitationally coupled vector eld theories dates back to a decade earlier, with the work of Will and Nordtvedt. Their work, circa 1972, was based on using the Post-Newtonian Formalism (PPN) (see chapter1 for more details) to show the eects of a preferred frame of reference in the tides of the earth, the perihelion-shift of the planets and the variation of the earth's rotation rate [36]. Aether models are based on the introduction of a vector eld in the Lagrangian density of the system with a non-vanishing vacuum expectation value. Due to that property, the vector eld will dynamically select a preferred frame at each point in space-time, spontaneously breaking Lorentz invariance. This is a mechanism reminiscent of the breaking of local gauge symmetry in the Higgs mechanism as explained in chapter3 and in [37] and it serves as phenomenological representation of the LSB terms in the gravitation sector of the SME, as discussed in Ref. [10]. Aether theories have been used, in parallel with the work of Kostelecký [10], as phenomenological probes of LSB in quantum gravity and as models for in ation and dark energy [32] and the corre- sponding action consists in a 4-vector Aµ coupled to gravity, which can be written as, R S = d4x√ g + (gµν ,Aα) + S (gµν , Φ) (4.1) − 16πG LAE M Z N µν where SM stands for the matter action, g the metric and Φ the matter elds which couple only to the metric [33, 35] and not to Aα. For the Einstein-aether Lagrangean we can start by writing it in the most general way given LAE by Ref. [33] (and the references found within),
= Kαβ V (AµA ) (4.2) LAE µν − µ µ with V (A Aµ) being a general potential which depends on our vector Aµ. The coe cients αβ can be given by Kµν
αβ αβ αβ αβ (4.3) Kµν = K(1)µν + K(2)µν + K(3)µν
26 4.2 Aether Theories with
K(1)αβ = (c gαβg + c δαδβ + c δαδβ + c AαAβg )( Aµ Aν ) (4.4) µν 1 µν 2 µ ν 3 ν µ 4 µν ∇α ∇β
K(2)αβ = (c δαAβA + c gαβA A + c δαAβA + c AαAβA A )( Aµ Aν ) (4.5) µν 5 ν µ 6 µ ν 7 µ ν 8 µ ν ∇α ∇β 1 K(3)αβ = c F F µν + c δαδβR AµAν + c RδαA Aβ (4.6) µν 9 2 µν 10 µ ν αβ 11 µ α in which the c and c gauge the relevance of the covariant derivatives along Aµ and F = ∂ A ∂ A . 4 8 µν µ ν − ν µ The K(3) is the most relevant of the three, as it contains terms that represent the coupling be- tween the vector eld and the geometry of the system given by the Riemann tensor (particularly the
α β µ and α β terms) and the and are simpli ed to a single kinetic c10δµ δν RαβA Aν c11Rδµ AαA K(1) K(2) term β( Aµ)2. ∇µ As stated previously these models are a subclass of Lagrangian densities where torsion is not considered, in contrast to the more general case considered by Kostelecký in Ref. [10] and following work starts with a Lagrangian density consisting of a terms mainly pertaining to the K(3) tensor, as can be seen in Ref. [35]:
R β S = d4x√ g 1 F F µν β( Aµ)2 + c R AµAν + c RA Aµ V (A Aµ) (4.7) − 16πG − 2 µν − ∇µ 10 µν 11 µ − µ Z These models did not evolve from a vaccum, but rather from the complete opposite: the existence of dark matter. Hints for the existence of dark matter emerged in the early 1930s with the work of J. Oort and F. Zwicky. Oort, by studying the Doppler shifts of stars in the Milky Way and thus obtaining their velocities, observed that those stars had enough velocity to escape the gravitational pull provided by the luminous mass of the galaxy i.e. the mass of the bodies that were directly visible [38]. The easiest way of studying stellar bodies is by measuring how much light they emit per unit of time, i.e. their luminosity. Because we can accurately measure the mass of the Sun, we calculate its mass-to-luminosity ratio M and use it as a standard, which allows us to obtain estimates of the masses L of other astronomical bodies by comparison.
Zwicky used the M/L ratios of the nebulae in the Coma cluster in order to obtain their mass, and discovered that this was only 2% of the average mass of the nebulae. This value was obtained by dividing the total mass of the cluster by the number of observed nebulae (about 1000). [39] Another way in which this mass discrepancy was observed was by assuming that galaxies in galaxy clusters behaved like planets in the Solar System, with a velocity dispersion given by the Newtonian expression
G m(r) v(r) = N . (4.8) r r where m(r) is the mass enclosed within a sphere of radius r. Vera Rubin et al. [40] studied the rotation curves of 60 galaxies and compared the results with what would be obtained by the expression above. She found that the rotation curves of those galaxies where " at", meaning that for larger radii the velocities of the galaxies increase until a threshold is reached and remain relatively constant for larger
27 4. Vector Theories radii. This is unexpected because if one considers that only the luminous mass in the center of the cluster as the only source of gravitational pull m(r) = M, the velocity should be higher closer to that center and lower in the periphery, where the gravitational pull is smaller (as can be immediately seen by the expression above). This then implied that that there is some sort of extended non-luminous matter in the periphery of the galaxy that exerts a gravity pull large enough to keep the galaxies rotating at those speeds. Pieces of evidence from other sources also exist: X-ray radiation from the hot gas surrounding galaxies was used by Vikhlinin et al. [41] to determine the mass distribution of the galaxies; Ratios of baryonic to total mass in the order of 13% where found, again indicating the existence of extra mass in those galaxies. Gravitational lensing also provides some con rmation for these results, and some consider it as a direct observation of dark matter, as it does not depend on the dynamics of the clusters [42]. Gravitational lensing consists in measuring masses through the de ection of light presents when passing through a gravitational eld, creating one of more images of the original object in a dierent location where the object would be directly observed. Weak gravitational lensing (where a single image is formed) has been used to infer the existence of dark matter in the Bullet Cluster [42], where the matter distribution was found to be dislocated from the cluster's luminous center of mass. This discrepancy can be seen in Figure 4.1, where the two sets of contour lines (blue and magenta as seen in color) which represent the observed gravitational mass are located away from the larger physical bodies (in black) that are part of the cluster.
Figure 4.1: F606W-band image of the Bullet Cluster. Image taken from [43]
As a nal example, Cosmic Microwave Background (CMB) radiation can also be used as a test for inferring the existence of dark matter, through the measurement of the density parameters present in the Friedman-Robertson-Walker model. Starting with the Friedman equation,
8πG R˙ 2 ρR2 = kc2 (4.9) − 3 −
28 4.2 Aether Theories
which describes a at, closed or open universe for k = 0, +1, 1, we can rewrite it by considering the − densitiy parameter given by ρ 8πGρ (4.10) Ω = = 2 ρc 3H with H the Hubble parameter. This allows us to obtain the relation
kc2 H2(Ω 1) = , (4.11) − R2 which directly shows a relation between the matter density and the resulting curvature. The limits Ω = 1, Ω > 1, Ω < 1 make the correspondence between a at, closed and open Universe. CMB anisotropy measurements allow us to obtain values for the total density parameter Ωtot = Ωm +Ωrad +ΩΛ and for the isolated Ωm = ΩBaryonic +Ωnon−Baryonic parameter. Various measurements have been made, with the most recent belonging to the Planck 2013 experiment [44]. Its results where Ω h2 = 0.1423 0.0029, m ± Ω = 0.02207 0.00033 and for non-baryonic cold dark matter Ω h2 = 0.1196 0.0031 (with baryonic ± c ± h = 6.2606957(29) 10−34). These results are in agreement with previous experiments and show a × great disparity in the relative percentages of baryonic to non-baryonic matter, again indicating a strong presence of dark matter in our Universe. Numerous candidates have been proposed to ll the role of dark matter. Standard model neutrinos along with sterile neutrinos [45] (same as Standard Model neutrinos but without weak interactions) have been considered as candidates due to their weak interaction with baryonic matter. One of the main problems in this assumption is that their abundance in the Universe does not allow them to be the dominant component of dark matter. [45] Axions, a result of CP violation physics postulated by Peccei and Quinn [46], have also been considered, with searches currently being done by the Axion Dark Matter Experiment [47]. Another class of candidates encompasses the super-symmetric candidates emerging from SUSY. Neutralinos have been widely studied as cold dark matter candidates due to being heavy stable particles with coupling strengths in the order of the weak interaction [45]. Experiments such as the Cryogenic Dark Matter Search [48] and more recently the Large Underground Xenon (LUX) [49] seek to detect these kinds of weakly interacting massive particles (WIMPs) but the results don't seem to strengthen the dark matter hypothesis. For more comprehensive reviews of particle candidates to dark matter please check [45, 50, 51]. Besides particle candidates, other alternatives exist to the dark matter hypothesis. Modi ed New- tonian Dynamics is one of such alternatives and it consists in a modi cation of Newton's laws in order to explain the at rotation curves discussed earlier. The main assumption in MOND theories is that Newton's law is modi ed with a dependence in the acceleration of the system. For low accelerations we have, instead of the usual second law of motion:
F~ = m~aµ(x) (4.12) where µ(x) is any function exhibiting the asymptotic behaviors
x if x 1 µ (x) = (4.13) 1 if x 1. (
29 4. Vector Theories
For small values of the ratio a we can compare the acceleration to what we would normally have, a0 i.e., a F~ = m~g = m~a a = √ga0. (4.14) a0 → So, for the simple case of a body in an orbital motion around a central mass M, where GM we g = r2 would have a centripetal acceleration given by v2 so our velocity would be a = r
4 v = GMa0, (4.15) which would be dependent from the central mass M. The parameter a0 can now be tted to the results obtained in the velocity curves yielding a = 1.2 10−10m/s2 [52]. One of the most interesting aspects 0 × of this theory is that the value of a0 obtained from observations is in the same order of magnitude of the Hubble constant, a cH [53]. 0 ∼ 0 Although this framework allows us to explain some of the velocity curves of individual galaxies, some problems do exist. As Sanders and McGaugh explain [53],
There have also been several contributions attempting to formulate MOND either as a covariant theory in the spirit of General Relativity, or as a modi ed particle action (modi ed inertia). Whereas none of these attempts has, so far, led to anything like a satisfactory or complete theory, they provide some insight into the required properties of generalized theories of gravity and inertia.
One example of an attempted merger of both MOND and General Relativity is what is called the Tensorial-Vector-Scalar theory, proposed by Bekenstein in 2004 [54]. The assumptions of the theory are also its main source of criticism [55], as it assumes the existence, besides the metric gαβ and the α matter elds φi, one extra vector eld U and two extra scalar elds σ and φ, the rst of which lacks of a convincing justi cation due to not having dynamic terms in the action. The action of this theory also goes in the inverse direction of one of the main aspects of MOND that made it so appealing, as its simplicity is completely lost, as one can see by the action S = SG + SV + SS + SM with the corresponding terms given by
S = R( g)1/2d4x (4.16) G − Z K 2λ S = gαβgµν U U (gµν U U + 1) ( g)1/2d4x (4.17) V −32πG [α,µ] [β,ν] − K µ ν − Z 1 G S = σ2hαβφ φ + σ4F (kGσ2) ( g)1/2d4x (4.18) S 2 ,α ,β 2l2 − Z S = (φ )( g˜)1/2d4x. (4.19) M L i − Z Another point in which MOND theories fall short is the inability to solve some discrepancies in particular matter dispositions of galaxy clusters, as discussed in [5658]. The matter distributions of visible matter do not correlate exactly with the non-visible part, an eect which can be seen directly in the Bullet Cluster, where two clusters of galaxies collided to form a matter distribution with two distinct peaks of visible matter density, and two other zones where the gravitational lensing eect is stronger, these being located both far away from each other and far relatively far away from the visible density peaks.
30 4.2 Aether Theories
4.2.1 Eects on the Gravitational Constant
In this section the goal is to give an overview of some interesting aspects and results obtained by applying these types of models to the study of cosmology. Starting by selecting the relevant terms for our Lagrangian, we have
= K(1)µν λ(uµu + m2). (4.20) L αβ − µ where all but the c4 coe cients are non-zero and our potential is a Lagrange multiplier eld associ- ated with the variable λ, for which we will later on deduce its equations of motion. Following the methodology presented by Ref. [59] and [60], we de ne a new tensor given by,
J µ = Kµν uβ (4.21) α αβ∇ν which allow us to obtain the equation of motion from the action above, with respect to the vector eld uµ
J µν = λuν . (4.22) ∇µ The constraint condition for this equation is
uµu = m2 (4.23) µ − which basically says that we are solving our equations of motion in the vacuum expectation value of our eld.
Given the adopted metric signature ( , +, +, +), we require that the vector uµ is timelike, which − implies that m2 > 0.
Multiplying the λ equation by uν on both sides gives 1 λ = u J µν . (4.24) −m2 ν ∇µ The metric that we will be using is the FLRW metric, which describes a spatially and isotropic universe. 1 ds2 = dt2 + a2(t) dr2 + r2dΩ2 (4.25) − 1 kr2 − The condition of spatial isotropy implies that our Aether vector eld must have only a temporal component. Because of the condition g uµuν = m2, the vector must be something like νµ − uµ = (m, 0, 0, 0). (4.26)
Inserting this in equation 4.22 we get,
a¨ λ(t) = 3(c + c + c )H2 + 3c (4.27) − 1 2 3 2 a which depends directly on the aether coe cients de ned in K(1) term of the previous section. Besides this, another useful expression is that of the stress energy tensor which can be obtained by varying the action with respect to the metric gµν . The calculations, albeit cumbersome, are spread out in the literature related to these models [61], and its expression can be simpli ed to [59]:
31 4. Vector Theories
T = 2c ( uβ u βu u ) [ (u J β )+ µν 1 ∇ ∇ν β − ∇ µ∇β ν − ∇β (µ ν) 2 (4.28) (uβJ ) (u J β )] u J βαu u + g . ∇β (µν) − ∇β (µ ν) − m2 α∇β µ ν µν L The next step is to analyze how the introduction of the aether vector eld aects gravity. The rst assumption is to consider that we have a system composed of both matter and the vector eld, which can be expressed by the Einstein eld equations by the sum of the two stress-energy tensors
1 R Rg = 8πG (T matter + T aether). (4.29) µν − 2 µν N µν µν
The gravitational constant GN is the one de ned in the Aether Lagrangian from equation 4.1. Matter is assumed to behave as a perfect uid, with a matter-energy tensor:
matter (4.30) Tµν = (ρm + pm)ηµην + pmgµν
The Aether eld can be described in the same fashion, de ning the corresponding pressure and density
a¨ ρ = 3αH2 , p = α H2 + 2 , (4.31) ae − ae a with a˙ the Hubble parameter and 2 [59]. H = a α = (c1 + 3c2 + c3)m The Einstein equations lead, through the Bianchi identities, to the energy conservation equation for a general density ρ and pressure p
ρ˙ + 3H(ρ + p) = 0 (4.32)
If we use the de nitions above for the aether density and pressure and insert it in equation (31), we have,
ρae˙ + 3H(ρae + pae) = dH a¨ 3α2H + 3H( 3αH2 + α + 2α ) = − dt − a a¨ a¨ (4.33) 3α2H( H2) + 3H( 2αH2 + 2α ) = − a − − a a¨ a¨ 6αH( H2) + 6αH( H2) = 0, − a − a − which proves that the selected terms are a good choice. For the FLRW metric we have
1 3 k R Rg = H2 + = T = ρ + ρ . (4.34) 00 − 2 00 8πG a2 00 ae m N The spatial components are given by
1 1 a¨ k R Rg = H2 + 2 + = T = p + p (4.35) ik − 2 ik 8πG a a2 ik ae m N Rewriting these equations in a form similar to the Friedmann equations, substituting the Aether terms for pressure and density and assuming a at space k = 0, we get 8πG H2 = c ρ , (4.36) 3 m a¨ 4πG = c (ρ + 3p ) (4.37) a − 3 m m
32 4.2 Aether Theories with the de nition of a new gravitational constant given by
GN Gc = . (4.38) 1 + 8πGN α The aether energy density suggests that for a positive energy density, we must have α < 0; positivity of H2 further implies that 1/(8πG ) M 2 < α < 0 (where M 1019GeV is the Planck mass). This N ∼ p p ∼ would imply that our gravitational constant would increase in relation to the original G∗. Because the acceleration equation (36) depends linearly on Gc, this implies that the net eect would be that the acceleration rate of the expansion of the universe would become larger.
4.2.2 In ation
Based of the Lagrangian density used in the study above, one can also study the role that these models play on in ation. For this case the potential chosen by Kanno & Soda [62] has a Lagrange
2 multiplier potential similar to the one used above, where m = 1. The ci terms adopted are also dierent, along with the expression for the timelike vector eld uµ
uµ = (1, 0, 0, 0) . (4.39)
The inital action is of a Scalar-Vector-Tensor theory, where besides the vector eld, we have a scalar eld φ coupled to it. This new Lagrangian can be understood as a perturbation of the scalar eld in ation model, where a Lorentz violating vector eld is introduced. The inital action is given by
1 S = d4x√ g R β (φ) µuν u β (φ) µuν u β (φ)( uµ)2 − 16πG − 1 ∇ ∇µ ν − 2 ∇ ∇ν µ − 3 ∇µ Z (4.40) 1 β (φ)uµuν uα u + λ(uµu + 1) ( φ)2 V (φ) − 4 ∇µ ∇ν α µ − 2 ∇ − and one can immediately see that the terms pertaining solely to the scalar eld, are those used in the most simple models that describe in ation. The metric used is
ds2 = dt2 + a2(t)δ dxidxj. (4.41) − ij where a = eα. Inserting the vector eld and the metric in the action gives[62],
1 3 1 S = dt e3α 1 + 8πGβ H2 + φ˙2 V (φ) . (4.42) N −8πG 2 − Z where β = β1 + 3β2 + β3. From this action the procedure is similar to what was done in the previous section, as we obtain the relevant equations of motion for the variables of our system β, φ, H. They are, respectively
1 H2φ0 V γH2 = + (4.43) 3 2β β H0 φ02 β0 γ + + = 0 (4.44) H 2β β H0 V φ00 + + 3φ0 + ,φ + 3β = 0 (4.45) H H2 ,φ
33 4. Vector Theories
where ˙ dQ dη with 0 dQ and 1 . Q = dη dt Q = dη γ = 1 + 8πgβ The existence of Lorentz violation implies that the terms relating to it in our model are compar- atively big to the scalar eld ones. This assumption, i.e. β >> 1&β >> f(φ, φ00, β0) will imply that γ 1 and the rst and third equations will now be → V H2 = (4.46) 3β V φ0 + ,φ + β = 0 (4.47) 3H2 ,φ The choice of potential is that of a parabola, in order to nd the similarities between these conditions and the slow roll conditions one forces when studying independent scalar eld in ation. Besides that, the authors suggest a quadratic coupling of the scalar eld to the Aether parameters β, so we have
1 β = ηφ2 ; V = m2φ2. (4.48) 2 These conditions, along with equations (45) and (46) allow us to obtain the solutions
φ φ(α) = 0 (4.49) a4η m2 H2 = (4.50) 6η which represents an in ationary model as a consequence of Lorentz violating parameters in our model. This can be seen directly if we consider the way in which these parameters where choosen.
Because , we de ned 2α(t) then a(˙t) which is the de nition of the Hubble H =α ˙ a(t) = e α˙ = a(t) parameter. The deceleration parameter q can be related to H by
H˙ H˙ = (1 + q) q = 1 . (4.51) H2 − ⇔ − − H2 Because H is a constant we have q = 1, which is the necessary condition for our theory to have in ation.
4.3 The Bumblebee Model
A model that contains a vector eld which dynamically break Lorentz symmetry is called a Bumble- bee model. These models, although with a simpler form, contain interesting features such as rotations, boosts and CPT violations. This subclass of aether models posits the following action functional,
1 1 S = d4x (R + ξBµBν R ) Bµν B V (BµB b2) , (4.52) B 16πG µν −4 µν − µ ± Z where the Bµ is the same as the vector eld Aµ (but now following the KS convention), and Bµν being the Bumblebee eld strenght given by
B = B B . (4.53) µν ∇µ ν − ∇ν µ The parameter ξ, with dimensions of M −2, represents the coupling between the Ricci tensor and the Bumblebee eld Bµ and V the potential of the Bumblebee eld which, as in the case of the aether
34 4.3 The Bumblebee Model theories, is the term that drives the breaking of the Lorentz symmetry of our Lagrangian by collapsing onto a non-zero minimum at B Bµ = b2. Here, Bµ is one of the Lorentz breaking coe cients µ ∓ referred in chapter3. The presence of this coe cient implies a preferred direction is selected at a certain Lorentz frame, which implies that the equivalence-principle is locally broken for that particular frame. Observations of Lorentz violation can emerge if particles or elds interact with the Bumblebee eld [10]. It is worthy to repeat that within this local frame of reference, local particle Lorentz transformations (referred to in chapter3) can be performed without changing the local Bumblebee eld, as under these transformations Bµ behaves as a set of four scalars. Rotations and boosts that change the local Lorentz frame (observer Lorentz transformations) allow us to choose arbitrarily the local Lorentz frame being observed, as under these transformations the Bumblebee eld behaves covariantly as a four-vector. This allows us to maintain local Lorentz invariance despite having local particle Lorentz violation. As referred in Ref. [34], a smooth quadratic potential of the form,
V = A(B Bµ b2)2 (4.54) µ ± with A a dimensionless constant, is chosen. This potential allows both Nambu-Goldstone excitations (massless bosons) besides the massive excitations for the cases of V = 0 and V = 0 respectively [10]. 6 The other case mirrors one of the presented potentials in the aether model section above, i.e. a linear Lagrange-multiplier potential which takes the form
V = λ(B Bµ b2). (4.55) µ ± Both cases where studied from the particle physics point of view and, besides the spontaneous lorentz breaking, these potentials present also the breaking of the U(1) gauge invariance and other implications to the behavior of the matter sector, the photon and the graviton. A good review of experimental proposals to test the result of Bumblebee models can also be found in [63].
Notice that the potential V is assumed to depend on BµB b2 with b = 0 the non-vanishing vev µ ± 6 signalling the spontaneous Lorentz symmetry breaking. Variation of Eq. (4.52) with respect to the metric yield the modi ed equations of motion [10],
1 R Rg = 8πG(T M + T B ), (4.56) µν − 2 µν µν µν
M B where T µν is the matter stress-energy tensor and T µν the Bumblebee stress-energy tensor, de ned as
1 ξ 1 T B B Bα B Bαβg V g + 2V 0B B + BαBβR g B BαR µν ≡ − µα ν − 4 αβ µν − µν µ ν 8πG 2 αβ µν − µ αν (4.57) 1 1 1 1 + (BαB ) + (BαB ) 2(B B ) g D (BαBβ) . 2∇α∇µ ν 2∇α∇ν µ − 2∇ µ ν − 2 µν α∇β The equations for the Bumblebee eld are
ξ Bµν = 2V 0Bν B Rµν , (4.58) ∇µ − 8πG µ where the prime represents derivative in respect to the argument.
35 4. Vector Theories
No separate conservation laws are assumed for matter and the Bumblebee vector eld in the work that will be presented in the following chapter. The covariant (non)conservation law (which is not used) can be obtained directly from the Bianchi identities µG applied to both sides of the modi ed ∇ µν eld equations (5.7): this leads to µT M = µT B = 0, which may be interpreted as an energy ∇ µν −∇ µν 6 transfer between the Bumblebee and matter. Studies using these models have recently emerged in the literature. The vacuum solutions for the Bumblebee eld for purely radial, temporal/radial and temporal/axial Lorentz symmetry breaking where obtained in [64]. For the rst case, a new black-hole solution was found where its Schwarzschild radius presents itself with a slight perturbation. The second case was analyzed through the PPN formalism where a set of PPN parameters was obtained. The nal case, due to a breaking of isotropy, was not possible to analyze directly through the PPN formalism, although an estimation of the PPN parameter γ was obtained. Other work which directly deals with these models was done by Bluhm in Ref. [34], where the possibility of a Higgs mechanism was analyzed. Studies referring to the electrodynamics of these elds was done in Ref. [65], where the Bumblebee eld was interpreted as a photon eld and its propagation velocity was studied, along with its implications on acelerator physics and cosmic ray observations. The following chapter will feature original work which closely follows the work done in Ref. [1]. In it, these Bumblebee models are used in order to constrain the possibility of Lorentz violating elds existing in astrophysical bodies such as the Sun.
36 5 Astrophysical constraints on the Bumblebee
Contents 5.1 Introduction...... 38 5.2 Static, spherically symmetric scenario...... 38 5.3 Perturbative Eect of the Bumblebee Field...... 39 5.4 Numerical analysis...... 41
37 5. Astrophysical constraints on the Bumblebee
5.1 Introduction
This chapter presents the results of a perturbation induced by the Bumblebee eld on a system like the Sun. It follows the original work presented in Ref. [1].
5.2 Static, spherically symmetric scenario
Given that the relevant quantities such as the density, pressure and scalar curvature inside a spherical symmetric body such as the Sun present a very strong radial variation in comparison with the temporal component, the Bumblebee eld is chosen to be
Bµ = (0,B(r), 0, 0). (5.1)
Accordingly, a static Birkho metric is also selected,
2Gm(r) −1 g = diag e2ν(r), 1 , r2, r2 sin2 θ . (5.2) µν − − r " # where m(r) is the mass pro le in function of the radial coordinate, and it is assumed that the potential takes a quadratic form, for simplicity,
V = A(B Bµ b2)2, (5.3) µ − with the adopted sign re ecting the spacelike nature of the Bumblebee eld.
For the radial case µ = r, the Ricci tensor is given by,
G(m0r m)(2 + rν0) R = − (ν0)2 ν00. (5.4) rr r2(r 2Gm) − − − The only non-vanishing component of Eq. (4.58) is for the component µ = r, 16πGV 0g ξR = 0 rr − rr which yields,
ξ 2A(g B2 b2) = Gr2[m0ν0 + 2m(ν00 + ν02)] + Gr(2m0 mν0) 2mG r3(ν00 + ν0) . (5.5) rr − 16πGr3 − − − Through some algebraic manipulation we can the calculate the bumblebee eld strength,
2Gm ξ B2 = 1 b2 + (Gr2[m0ν0 + 2m(ν00 + ν02)] + Gr(2m0 mν0) 2mG r3[ν00 + ν0]) . − r 32πGAr3 − − − (5.6)
In order to obtain the pressure and density equations, the trace-reversed eld equations are con- sidered, giving,
1 E R 8πG T M + T B g (T M + T B) = 0, (5.7) µν ≡ µν − µν µν − 2 µν with T M and T B the traces of the stress energy tensors for normal matter and the Bumblebee eld, respectively. The stress-energy tensor for normal matter in the perfect uid hypothesis is given by,
M (5.8) Tµν = (ρ + p)uµuν + pgµν ,
38 5.3 Perturbative Eect of the Bumblebee Field
where u is the four velocity; in the static scenario and given that u uµ = 1, we have u = (e2ν(r),~0), µ µ − µ so that
p T M = diag e2ν ρ, , r2p, r2p sin2(θ) , (5.9) µν 1 2Gm − r ! with trace T = 3p ρ. − If we then consider the following combinations of the trace-reversed eld equations,
tt rr g Ett g Err = 0, − (5.10) θθ g Eθθ = 0. we can derive the equations that will allow us to obtain p(r), ρ(r) and ν(r). Without the Bumblebee
eld, these quantities (denoted with the subscript 0) are simply r(r 2Gm )ν0 Gm p (r) = − 0 0 − 0 , (5.11) 0 4πGr3 m0 ρ (r) = 0 , (5.12) 0 4πr2 m + 4πp r3 ν0 (r) = G 0 0 , (5.13) 0 r(r 2Gm ) − 0 which, along with a state equation that relates p0 and ρ0, yields a closed set of four dierential equations with four unknowns. In the presence of the Bumblebee eld, the related eld equation (4.58) also has to be included; solving Eq. (5.7), the pressure and density are then given by 1 p(r) = rξB(r 2Gm)2B0(2 + rν0) + r(8πGV r3 2Gm + 2r(r 2Gm)ν0) 8πGr4 − − − − B2(r 2Gm)( 2ξGm( 1 + r(ν0( 2 + rν0) + rν00)) + r[16πGV 0r2 (5.14) − − − − ξ + rξ(ν0( 2 + rν0) + rν00)])] , − − and 1 ρ(r) = [ r2(8πGV r2 + ξ(r 2Gm)2B02 2Gm0) + rξB(r 2Gm)(B0( 4r + 3Gm 8πGr4 − − − − − +5Grm0) r(r 2Gm)B00) + ξB2(3G2m2. (5.15) − − 2G2rm(3m0 + rm00) + r2[ 1 + G(m0(4 Gm0) + rm00)])]. − − − Although we have a complete set of equations that describe the behaviour of our system, the solution of that set of equations implies very intensive numerical computations. This is expected as we have up to second order derivatives on the ν(r) function inside the expression for the Bumblebee eld, which itself appears inside both the pressure and density equations as a second derivative. The combined expansion of all the terms in these expressions, combined with the fact that we would have a fourth order system in our hands, brings a great deal of di culty to the problem. The linearisation of the system of equations (not shown) does not simplify the problem signi cantly, as the number of terms in the equation would increase immensely.
5.3 Perturbative Eect of the Bumblebee Field
Since the stellar structure of the Sun is known to be well described by General Relativity, we consider the perturbation to be of zero order, i.e. we replace the quantities on the r.h.s. of the
39 5. Astrophysical constraints on the Bumblebee
equations mentioned above by the unperturbed expressions for m0(r) and ν0(r) and the Bumblebee eld. Regarding the latter, it is more straightforward to resort instead to Eq. (5.5), since at zeroth order one has
1 R = 8πG T M g T M = 4πG(ρ p )g , (5.16) rr rr − 2 rr 0 − 0 rr which leads to the following expression for the Bumblebee eld equation
2Gm ρ p B2(r) = 1 b2 + 0 − 0 g . (5.17) − r 8A rr
Since the unperturbed solutions ρ0(χ) and p0(χ) vanish at the boundary of the spherical body, the above shows that the Bumblebee eld collapses onto its VEV as it crosses to its outer solution (where
M ), µ 2. This is consistent with the approach followed in Ref. [64], where the latter Tµν = 0 BµB = b condition was also assumed. Following the changes above, the expressions for the pressure and density can now be obtained, giving
ξ2(ρ0 p0) 2Gm 3 ξ 2Gm 2 p(r) = p + [ξ(p ρ)]2 + − 1 (2 + ν0r) + 1 [8Ab2 0 − 2AGπr − r Aπr3 − r r 2Gm +ξ(ρ p)] (2 + ν0r)(m m0r) + 1 1 + r(4Gπpr + 2ν0 (5.18) − × − G − r 2Gm r[4Gπρ + (ν0)2 + ν00]) [1 + r(ν0[2 ν0r] rν00)] , − − r − − ξ 2 ξ2 2Gm 2 m ρ(r) = ρ (ρ p) + 1 4 + G 9m0 (p0 ρ0) 0 − 8 − 128AGπr − r r − − h i 2Gm 3 ξ 2Gm + 1 (p00 ρ00)r + 1 [8Ab2 (5.19) − r − 64AGπr2 − r Gm 2 2Gm +ξ(ρ p)] 2 + 6Gm0(1 Gm0) + 2Gm00r 1 (1 + 2Gm00r) , − r − − − r The advantage of considering the simplistic model provided by the polytropic EOS Eq. (2.25) lies in the possibility of rewriting the rather convoluted expressions above only in terms of the LE solution
θ(χ). For this, we now introduce the dimensionless parameters
ξ2 ξ3b2 R α , β , γ s , (5.20) ≡ R2G ≡ R2G ≡ R where R 2GM is the Schwarzschild radius of the star, together with the form factor s ≡ 3M (5.21) φ 3 , ≡ 4πρcR and the EOS parameter ω p /ρ . c ≡ c c Using the relations (2.28) and the expression for 0 from Eq. (5.11), we can obtain ν0(r)
1+n 0 0 3γ(χωcθ θ ) (5.22) ν0(χ) = 2 − 0 , 2φχf + 6γχθ and the form factor becomes φ = 3θ0(χ )/χ displaying the homology invariance of the LE Eq. − f f (2.31).
40 5.4 Numerical analysis
It is now possible to rewrite the above expressions for the pressure and density in terms of the
LE solution θ(χ) and its derivatives. This allows us to obtain a more manageable form: separating the contributions to the pressure and density arising from the non-vanishing vev b and the potential strength A as
p(χ) = p0(χ) + pb(χ) + pV (χ) + δ(χ), (5.23)
ρ(χ) = ρ (χ) + ρ (χ) + ρ (χ) δ(χ), (5.24) 0 b V − we have
β 2 0 2 2 n (5.25) pb(χ) = 3 2 4 (φχf + 3γχθ ) [3γχ θ ( 1 + ωcθ) 16πφ ξ χf χ × − − 2( 1 + χ[ν0( 2 + χν0) + χν00])(φχ2 + 3γχθ0) + 3γχ(2 + χν0)(θ0 + χθ00)], − − f β 2 0 2 4 02 (5.26) ρb(χ) = 3 2 4 (φχf + 3γχθ ) [2φ χf + 3γχ(45γχθ + 16πφ ξ χf χ × 2 00 2 002 2 000 0 2 2 00 000 χ[14φχf θ + 9γχ θ + 2φχf χθ ] + 2θ [7φχf + 3γχ (10θ + χθ )])], 3γα2θn p (χ) = (φχ2 + 3γχθ0)2 (ω 1)(3γχ2θn[1 ω θ] + (5.27) V 1024π2Aφ4ξ2χ4 χ2 f × c − − c f χ 2[ 1 + χ(ν0[χν0 2] + χν0)](φχ2 + 3γχθ0)) (2 + χν0)(θ0[θ([1 + n]φχ2 ω + 3γ[ω θ 1]) + − − f − θ f c c − 3γχ([1 + n]ω θ n)θ0 nφχ2 ] + 3γχθ[ω θ 1]θ00) , c − − f c − 3γα2θn χ ρ (χ) = (φχ2 + 3γχθ0) (φχ2 + 3γχθ0)(2[1 n]nχθ02[φχ2 + 3γχθ0] + (5.28) V 2048π2Aφ4ξ2χ4 χ2 f × θ2 f − f f nθ[θ0(χθ0[ 51γ + 2(1 + n)φχ2 ω + 6γ(1 + n)χω θ0] 8φχ2 ) − f c c − f − χ(2φχ2 + 33γχθ0)θ00] + [1 + n]ω θ2[θ0(8φχ2 + 51γχθ0) + χ(2φχ2 + 33γχθ0)θ00]) f c f f − 2 4 02 2 00 2 002 2 000 2(2φ χf + 3γχ[45γχθ + χ(14φχf θ + 9γχ θ + 2φχf χθ ) +
0 2 2 00 000 2 4 02 2θ (7φχf + 3γχ [10θ + χθ ])]) + 2ωcθ(2φ χf + 3γ + χ[45γχθ +
2 00 2 002 2 000 0 2 2 00 000 χ(14φχf θ + 9γχ θ + 2φχf χθ ) + 2θ (7φχf + 3γχ [10θ + χθ ])]) , and
9γ2α2θ2n(ω θ 1)2 δ(χ) = c − . (5.29) 4096π2φ2ξ2 The latter appears in both the pressure and density perturbations and, as shall be shown in the following section, has a negligible impact when compared with the remaining contributions.
5.4 Numerical analysis
In Fig. 2, the pro le of the contributions to Eq. (5.23) was shown for values between 0.1 and 10 solar masses. The radii varied between 0.1 and 1000 times those of the Sun. The motivation behind having a much bigger variation in the radii in comparison to the masses stems from the fact that this is what is observed in the set of chosen stars. Stars that have a big variance in radius show relatively low variance in their mass. This can be seen by the chosen set of known stars in Table 5.1:
41 5. Astrophysical constraints on the Bumblebee
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
Figure 5.1: Pro le 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, showed by the horizontal line in each plot.
42 5.4 Numerical analysis
Star Name Mass Radius
Wolf 359 0.09M 0.16R Betelgeuse 7.7 20M 950 1200R − − Antares 12.4M 883R VY Canis Majoris 17M 1420R
Table 5.1: Selected stars that used as models for the numerical analysis of the Bumblebee perturbation.
The abrupt variations in the perturbations for χ 1 and χ 2.7 are the result of the pro le ≈ ≈ changing between positive and negative values around those points and the adopted logarithmic scale.
The variation of the polytropic index n with the size of the star is also shown. The values of the parameters (ξ, b, A) are chosen so that the maximum of the relative perturbations is of the order of 1%, the order of magnitude of the current accuracy of the central temperature of the Sun [6668]. For reference, the values of these parameters for ξ = 10−11, b = 10−8 are shown in table 5.2 for all the model stars considered in the numerical analysis.
5 − pb ρb 10 −
) 15
b − GeV
log( 20 −
25 −
30 − 20 10 0 10 20 − − log(ξ[GeV 2])
Figure 5.2: Allowed region (in grey) for a relative perturbation of less than 1% for pb and ρb.
A small variation in the polytropic index does not cause signi cant changes on the obtained bounds for the system: in particular, n does not impact signi cantly the value of ρb, as can be seen directly in the equation for ρb. If we increase the radius (thus lowering γ and α) the impact of the non-vanishing vev increases, while leading to a lower contribution from the potential term. If we increase the mass, we end up having smaller eects on all quantities except ρV , which is rather insensitive to variations of M.
If we x M = M and R = R , we can then nd values for the parameters of the model (ξ, b, A) that lead to relative perturbations of less than 1%. The allowed parameter space can be obtained, as depicted in Fig. 3. Notice that the allowed values for A are bounded from below, since this quantity 2 appears in the denominator of pV and ρV ; conversely, the region allowed for ξb is bounded from above. Having said this, we can now obtain the bounds for the parameters of our model,
2 −23 ξ 2 −3 ξ 34 ξb . 10 , GeV . 10 . 10 . (5.30) √A → √AG
43 5. Astrophysical constraints on the Bumblebee
0
10 − ) A log( 20 −
pV ρV 30 − 20 18 16 14 12 10 8 6 − − − − − − − − log(ξ[GeV 2])
Figure 5.3: Allowed region (in grey) for a relative perturbation of less than 1% for pV and ρV .
It is also worth noting that for the considered sets of parameters, the term δ(χ) is negligible in comparison with the other terms in both the pressure and density equations, as mentioned after Eq.
(5.29): indeed, one can calculate numerically that −34 for the considered masses and radii. δ . 10 pV
−33 −49 M,R α 10 β 10 γ ωc × × −7 −7 0.1 M , R 1.18 1.18 4.07 10 6.946 10 × −6 × −6 M , R 1.18 1.18 4.07 10 6.946 10 × −5 × −5 10 M , R 1.18 1.18 4.07 10 6.946 10 × −5 × −5 M , 0.1 R 1.18 1.18 4.07 10 6.946 10 × −9 × −9 M , 1000 R 1.18 1.18 4.07 10 6.946 10 × × 11 Table 5.2: Adimensional parameters used in the perturbative expressions for the pressure with ξ = 10− , 8 b = 10− and polytropic index n = 3.
And so we reach the end of this work. The equations, although quite complex in form, present themselves in a more obvious way through this numerical analysis. Through the employment of the LE equations and some dimensional analysis, we were able to separate the contributions of the Bumblebee potential from the eld itself. This in turn allowed us to study their individual contribution on a set of astrophysical bodies modeled through a polytropic equation of state. The obtained constraints are, as will be discussed in the next section, a lot more stringent than the ones previously obtained in the literature, thus giving relevance to the results.
44 6 Conclusions
45 6. Conclusions
After a brief review of the relevant literature that brings context to the Bumblebee models, we treated it as a zeroth order perturbation on a set of stars. We employed a set of stars which mimic, in terms of order of magnitude, the radii and masses of some known stars. These stars where then described by the Lane-Emden solution for a spherically symmetric body. Assuming that it follows the underlying symmetry of the problem, we choose a Bumblebee eld with a radial component only. Because the impact of the eld is considered as a perturbation, an attempt was made in order to constrain the parameters of the model in such a way as to only cause a variation on the system of roughly 1%, following the accuracy of our present modelling of the Sun. 2 −23 The obtained constraint for the value of the potential driving the Bumblebee eld, ξb . 10 , is 2 −9 many orders of magnitude more stringent than the previously available bound ξb . 10 , obtained by resorting to tests of Kepler's law using the orbit of Venus [64]; by assuming that, in the presence of matter, the Bumblebee eld is not relaxed at its vev, this study has also yielded a constraint on the strength of the corresponding potential, ξ < 1034√AG, which is a new result for these models. Although only a quadratic potential was considered in this study, the change of the power n showed very little eect on the pro les of the perturbation, as it s eect comes mostly as multiplicative factor on the potential itself. Future re nements of this method could clearly include the use of a more accurate model for stellar structure, as well as following a more thorough numerical analysis procedure, eectively solving the (dierential) modi ed eld equations to rst order in the model's parameters. However, we must still take into account that this should only re ne the obtained bounds, having no eect on the order of magnitude of the eect. The application of this same methodology to the study of galaxies is also possible, in order to gain further knowledge on the constraints to the parameters of our model, as well as the possibility of describing galactic dark matter as a manifestation of the Bumblebee dynamics following analogue eorts in both scalar eld [66] and vectorial Aether models [61, 69, 70]. In doing so, the addition of a non-vanishing temporal component for a time evolving Bumblebee eld could also be considered, in order to provide a results relevant at cosmological scales.
46 1
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