Abelian-Higgs strings in Rastall gravity
Eugenio R. Bezerra de Mello, Julio C. Fabris, Betti Hartmann
aa r X i v : . [ g r- q c ] A p r Abelian-Higgs strings in Rastall gravity
Eugˆenio R. Bezerra de Mello ( a ) , ∗ J´ulio C. Fabris ( b ) , † and Betti Hartmann ( b ) , ( c ) ‡ ( a ) Departamento de F´ısica, Universidade Federal da Para´ıba,58.059-970, Caixa Postal 5.008, Jo˜ao Pessoa, PB, Brazil ( b ) Universidade Federal do Esp´ırito Santo, Departamento de F´ısica, CEP 29075-910, Vit´oria (ES), Brazil ( c ) School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany (Dated: October 9, 2018)In this paper we analyze Abelian-Higgs strings in a phenomenological model that takes quantumeffects in curved space-time into account. This model, first introduced by Rastall, cannot be derivedfrom an action principle. We formulate phenomenological equations of motion under the guidingprinciple of minimal possible deformation of the standard equations. We construct string solutionsthat asymptote to a flat space-time with a deficit angle by solving the set of coupled non-linearordinary differential equations numerically. Decreasing the Rastall parameter from its Einsteingravity value we find that the deficit angle of the space-time increases and becomes equal to 2 π atsome critical value of this parameter that depends on the remaining couplings in the model. Forsmaller values the resulting solutions are supermassive string solutions possessing a singularity at afinite distance from the string core. Assuming the Higgs boson mass to be on the order of the gaugeboson mass we find that also in Rastall gravity this happens only when the symmetry breaking scaleis on the order of the Planck mass. We also observe that for specific values of the parameters in themodel the energy per unit length becomes proportional to the winding number, i.e. the degree ofthe map S → S . Unlike in the BPS limit in Einstein gravity, this is, however, not connect to anunderlying mathematical structure, but rather constitutes a would-be-BPS bound . PACS numbers: 98.80.Cq, 11.27.+d ∗ emello@fisica.ufpb.br † [email protected] ‡ [email protected] I. INTRODUCTION
One of the well-known ingredients of Einstein’s theory of General Relativity is the covariant conservation of theenergy-momentum tensor which leads, via Noether’s Theorem, to the conservation of globally defined quantities. Thesequantities appear as integrals of the components of the energy-momentum tensor over suitable space-like surfaces thattypically have one of the Killing vectors of the space-time as their normal. As such, the total rest energy/mass of asystem is conserved in General Relativity. Now the question is whether this is a suitable assumption as there is (up todate) no clear experimental evidence for this. Hence, models have been developed that relax the condition of covariantenergy-momentum conservation. In this paper we are interested in a modification of General Relativity suggested byRastall [1]. In this model, the gravitational fields are sourced by the energy and momentum, as in General Relativity,but also by the metric of the external space. Since in empty space, i.e. for a vanishing energy-momentum tensorRastall gravity agrees with General Relativity, this can be seen as a direct implementation of Mach’s principle statingthat the inertia of a mass distribution should be dependent on the mass and energy of the external space-time [2].This model has been studied extensively in the context of cosmology [3–5].At first sight Rastall’s theory has major drawbacks: its phenomenological formulation and in addition the absenceof a variational principle. However, it contains a rich structure that may be easily connected with many fundamentalaspects of a gravity theory. First of all, the usual energy-momentum conservation law of Special Relativity may begeneralized to curved space-time in many different ways, including geometric terms. General Relativity is one possibleextension and constitutes a minimal implementation of such a generalization by replacing the standard derivative witha covariant derivative. This by itself is not completely free of unclear aspects. On the other hand, if quantum effectsare taken into account in curved space-time the classical expression for the energy-momentum tensor must be modifiedintroducing quantities related to the curvature of the space-time [6]. Moreover, the propagation of quantum fieldsin space-times with horizons may lead to a violation of the classical conservation law (due to the chirality of thequantum modes) leading to a so-called gravitational anomaly [7]. In this sense, Rastall’s theory is a phenomenologicalprocedure to consider effects of quantum fields in curved space-time and to investigate in a completely covariant waysuch effects. Even if there is in principle no action leading to the Rastall equation it is possible to find such anaction if an external field is introduced in the Einstein-Hilbert action through a Lagrange multiplier. This is somehowa reminiscence of the quantum effects described by the Rastall equation. Other geometrical frameworks, like Weylgeometry, may lead to equations similar to the ones in Rastall gravity [8, 9].In this paper we are interested in static, cylindrically symmetric solutions to the Rastall gravity model coupled tothe U(1) Abelian-Higgs model. These solutions constitute field theoretical realizations of a specific type of topologicaldefect called Abelian-Higgs string [10] which could e.g. describe infinite straight cosmic strings whose properties havebeen analyzed in [11, 12]. Topological defects are believed to have formed in the numerous phase transitions in theearly universe due to the Kibble mechanism [13]. While magnetic monopoles and domain walls, which result from thespontaneous symmetry breaking of a spherical and parity symmetry, respectively, are catastrophic for the universesince they would overclose it, cosmic strings are an acceptable remnant from the early universe. These objects formwhenever an axial symmetry gets spontaneously broken and, due to topological arguments, are either infinitely long orexist in the form of cosmic string loops. Numerical simulations of the evolution of cosmic string networks have shownthat these networks reach a scaling solution, i.e. their contribution to the total energy density of the universe becomesconstant at some stage. The main mechanism that allows cosmic string networks to reach this scaling solution is theformation of cosmic string loops due to self-intersection and the consequent decay of these loops under the emissionof gravitational radiation.For some time, cosmic strings were believed to be responsible for the structure formation in the universe. NewCosmic Microwave Background (CMB) data clearly shows that the theoretical power spectrum associated to cosmicstrings is in stark contrast to the observed power spectrum. However, other effects might be caused by moving cosmicstrings that can potentially be observed in the CMB data (see e.g. [14] for a recent discussion). Moreover, there hasbeen a recent revival of cosmic strings since it is now believed that cosmic strings might be linked to the fundamentalstrings of String Theory [15]. While perturbative fundamental strings were excluded to be observable on cosmic scalesfor many reasons [16], there are now new theories containing extra dimensions, so-called brane world model, thatallow to lower the fundamental Planck scale down to the TeV scale. Moreover, cosmic strings are interesting due tothe recent BICEP2 data [17] that showed evidence for a B-mode polarization at small ℓ in the Cosmic Microwavebackground (CMB). While topological defects cannot be accounted for the B-mode polarization alone [18] it has beensuggested that this polarization results from gravitational waves, i.e. tensor modes, originating from an inflationaryepoch in the early universe. If that turns out to be correct the question remains what the origin of the field drivinginflation is and how it can be embedded into suitable Unified Theories, which are certainly necessary to describe thephysics at the energy scales relevant for the inflationary epoch. Now, it is interesting that cosmic strings genericallyform at the end of inflation in inflationary models resulting from String Theory [19] and Supersymmetric GrandUnified Theories [20]. Hence, it is conceivable that cosmic strings show up in the CMB and, indeed, CMB data(power and polarization spectra) are well compatible with a substantial amount of the total energy density of theuniverse coming from cosmic strings [21, 22].While field theoretical cosmic strings would always form loops when self-intersecting (and hence providing a “short-cut” for the magnetic flux), this can well be different for cosmic superstrings. The question then is how these networksof cosmic superstrings loose energy to reach a scaling solution and hence be not dangerous for the universe today.One of the suggestions is that they can form bound states. Since it is difficult to study cosmic superstrings withrespect to the formation of bound states, the interaction of cosmic strings has been investigated in the context of fieldtheoretical models describing bound systems of D- and F-strings, so-called p-q-strings [23, 24].In order to understand the lensing properties of cosmic strings (and hence their impact on the CMB spectrum) aswell as the evolution of cosmic string networks in different gravity models it is important to study their properties indetail. This is the aim of this paper which considers cosmic string in the Rastall gravity model.Our paper is organized as follows: In Section II, we briefly introduce the gravity model proposed by Rastall. InSection III we present the U(1) Abelian-Higgs model that we study coupled to Rastall gravity. We present the set ofdifferential equations associated with this system and give the asymptotic behavior for the matter and for the metricfunctions. In Section IV we present our numerical results. In section V we give our conclusions. The Appendixcontains an outline on the procedure we used to derive the equations of motion for the matter fields. II. THE MODEL
Rastall’s generalization of General Relativity [1] uses the idea of covariant non-conservation of the energy-momentumtensor and has been cast into the following form (where µ, ν = 0 , , , D µ T µν = κD ν R , (1)or D µ T µν = ¯ κD ν T , (2)where D .. denotes the covariant derivative, T µν the energy-momentum tensor with trace T ≡ T µµ and R the Ricciscalar. κ and ¯ κ are some coupling constants such that in the limit κ → R µν − g µν R = 8 πG (cid:18) T µν − γ − g µν T (cid:19) (3)and D µ T µν = γ − D ν T . (4)Of course the constants κ , ¯ κ and γ can be easily related and γ = 1 corresponds to the Einstein gravity limit.The equations (3) and (4) can be written in the form R µν = 8 πG (cid:18) T µν − g µν T + 12 g µν ( γ − T (cid:19) (5)and 1 √− g ∂ µ (cid:0) √− gT µν (cid:1) + Γ νµλ T µλ = γ − ∂ ν T , (6)where Γ νµλ are the Christoffel symbols.
III. ABELIAN-HIGGS STRINGS
In this section we would like to study Abelian-Higgs strings in Rastall gravity. The matter Lagrangian density, L m ,is given by L m = D µ φ ( D µ φ ) ∗ − F µν F µν − λ φφ ∗ − η ) , (7)with the covariant derivative D µ φ = ∇ µ φ - ieA µ φ of the complex scalar field φ . The field strength tensor is F µν = ∇ µ A ν − ∇ ν A µ = ∂ µ A ν − ∂ ν A µ , of the U(1) gauge potential A µ with coupling constant e . ∇ µ denotes the gravitationalcovariant derivative. Finally, λ is the self-coupling of the scalar field, while η denotes the vacuum expectation value.We define the energy momentum tensor to be given in the standard way by the variation of the matter Lagrangian L m with respect to the metric T µν = − δ L m δg µν + g µν L m . (8)Note, however, that this is not linked to an action principle in which the variation of the total action (matter plusmetric) with respect to the metric gives the gravity equations. The model we are studying here is a phenomenologicalmodel in which we insert “by hand” the non-conservation of the energy-momentum tensor.The symmetry breaking pattern is such that U (1) → M H = √ λη , while the gauge boson mass is M W = √ eη .By using the standard cylindrical coordinates ( r, ϕ, z ), the most general static cylindrically symmetric line elementinvariant under boosts along the z -direction is: ds = N dt − dr − L dϕ − N dz , (9)where N and L are functions of r only.The non-vanishing components of the Ricci tensor R νµ then read [25]: R tt = − ( LN N ′ ) ′ N L , R rr = − N ′′ N − L ′′ L , R ϕϕ = − ( N L ′ ) ′ N L , R zz = R tt (10)where the prime now and in the following denotes the derivative with respect to r .For the matter and gauge fields, we have [10] φ ( r, ϕ ) = ηf ( r ) e inϕ , A µ dx µ = 1 e ( n − P ( r )) dϕ , (11)where n is an integer indexing the vorticity of the Higgs field around the z − axis, i.e. corresponds to the degree of themap S → S . In the following we will study only the case n = 1.Now let us define the following dimensionless quantities r → reη , L → Leη , (12)such that r measures the radial distance in units of M W / √
2. Then, the Lagrangian density, L m → L m / ( η e ),depends only on the following dimensionless coupling constants α = 8 πGη = 8 π η M , β = λe = M H M W , (13)where M pl is the Planck mass.The non-vanishing components of the energy-momentum tensor read [25] − T tt = ( f ′ ) + ( P ′ ) L + P f L + β (cid:0) f − (cid:1) , T zz = T tt , (14) − T rr = − ( f ′ ) − ( P ′ ) L + P f L + β (cid:0) f − (cid:1) , (15) − T ϕϕ = ( f ′ ) − ( P ′ ) L − P f L + β (cid:0) f − (cid:1) (16)and the trace T ≡ T µµ is given by T = − (cid:20) ( f ′ ) + P f L + β (cid:0) f − (cid:1) (cid:21) . (17)Now, since the trace of the energy-momentum tensor is independent of the t and the z coordinate we have D µ T µt = 0 , D µ T µz = 0 . (18)Note that this is not a consequence of the gravity model used, as it would be in General Relativity, but rather aconsequence of the particular choice of the matter content. Then, (18) allows us to define globally conserved charges,namely the energy per unit length, µ , as well as the tension along the string axis, τ , in the usual way (with (2) g denoting the determinant of the induced metric on spatial sections perpendicular to z ) µ = Z π Z ∞ p (2) g T tt drdθ = 2 π Z ∞ L T tt dr and τ = Z π Z ∞ p (2) g T zz drdθ = 2 π Z ∞ L T zz dr , (19)where obviously from the fact that T tt = T zz we find that µ = τ . Note that we “measure” µ in units of η . A quantityoften cited when discussing the observational effects of cosmic strings is G ˜ µ , where ˜ µ is the dimensionful energy perunit length of the string. This quantity enters in both the expression for the deficit angle as well as the temperatureanisotropies ∆ T /T in the CMB. In our dimensionless units the quantity G ˜ µ is equal to αµ/ (8 π ). A. Equations of motion
We use the ( tt )- and ( ϕϕ )-components of the Rastall equation (5):( LN N ′ ) ′ N L = α (cid:20) ( γ − f ′ ) + ( P ′ ) L + ( γ − P f L + (cid:18) γ − (cid:19) β ( f − (cid:21) (20)and ( N L ′ ) ′ N L = α (cid:20) ( γ − f ′ ) − ( P ′ ) L + ( γ − P f L + (cid:18) γ − (cid:19) β ( f − (cid:21) . (21)The ( rr ) component reads2 N ′′ N + L ′′ L = α (cid:20) ( γ − f ′ ) − ( P ′ ) L + ( γ − P f L + (cid:18) γ − (cid:19) β ( f − (cid:21) , (22)which can be combined with (20) and (21) to obtain a constraint that is first order in derivatives2 N ′ L ′ N L + ( N ′ ) N = α (cid:20) γ ( f ′ ) + ( P ′ ) L + ( γ − P f L + (cid:18) γ − (cid:19) β ( f − (cid:21) (23)Finally, the modified conservation law (6) reads4 N ′ N (cid:18) ( f ′ ) + ( P ′ ) L (cid:19) + 2 L ′ L (cid:18) ( f ′ ) − P f L (cid:19) + γ (cid:0) ( f ′ ) (cid:1) ′ + (cid:18) ( P ′ ) L (cid:19) ′ +( γ − (cid:18) P f L (cid:19) ′ + (cid:18) γ − (cid:19) β (cid:16)(cid:0) f − (cid:1) (cid:17) ′ = 0 . (24)For γ = 1 we can derive the Euler-Lagrange equations by the variation of the action with respect to the matter fields.This is not possible here, hence we “read-off” the equations from the conservation law (24). This is motivated by thefact that in standard Einstein gravity the conservation law holds on shell , i.e. for solutions of the Euler-Lagrangeequations. Moreover, we require that the equations become equal to the standard equations in the γ = 1 limit andconstitute a minimal extension of the given model (see also the Appendix).It hence makes sense to consider the following equations of motion for the matter fields: γf ′′ + (cid:18) N ′ N + L ′ L (cid:19) f ′ + ( γ − P fL + (2 γ − βf ( f −
1) = 0 (25)and LN (cid:18) N P ′ L (cid:19) ′ + 2(1 − γ ) (cid:18) P P ′ L ′ L f (cid:19) = 2(2 − γ ) f P . (26)Note that these, indeed, reduce to the standard Euler-Lagrange equations in the limit γ = 1.Hence we have to solve a system of four coupled, non-linear ordinary differential equations numerically subject toappropriate boundary conditions. These read L (0) = 0 , L ′ (0) = 1 , N (0) = 1 , N ′ (0) = 0 , f (0) = 0 , P (0) = 1 , (27)to ensure regularity on the z -axis. Furthermore, we want the matter fields to reach their vacuum expectation valuesasymptotically. So, we require f ( r → ∞ ) → , P ( r → ∞ ) → . (28) B. Behaviour close to the string axis
The behaviour at r = 0 is determined by the requirement of imposing globally regular solutions. From the boundaryconditions it follows for the metric functions N ( r ) ∼ n r and L ( r ) ∼ r , (29)where n is a constant.Inserting this into (25) we find the following behaviour of f ( r ) at small rf ( r ) ∼ f r d with d = γ − γ ± s(cid:18) γ − γ (cid:19) − γ . (30)For γ = 1 this reduces to f ( r ) ∼ f r , but here the behaviour is much more complicated and depends strongly on γ .In particular we note that we have to require d > P ( r ) ∼ P r . (31)Consequently, the behaviour of the gauge field at the origin does not depend on γ . C. Asymptotic behaviour
In the absence of matter sources, Rastall gravity reduces to standard Einstein gravity. Since the energy-momentumtensor associated with the matter fields in our model falls off exponentially fast and since we want the Abelian-Higgs string to be well-localized, we can assume that far away from the string core the space-time corresponds to acylindrical vacuum space-time. Now, it is well known that cylindrical solutions of the vacuum Einstein equations areof the Kasner type. Hence, we would expect our solutions to asymptote to these such that the fall-off of the metricfunctions is N ( r ) ∼ N r a and L ( r ) ∼ L r b , (32)where the coefficients a and b have to fulfill the Kasner conditions2 a + b = 2 a + b = 1 , (33)with N and L being constants. The two possible solutions to (33) are( a, b ) = (0 ,
1) and ( a, b ) = (cid:18) , − (cid:19) . (34)The first set of parameters corresponds to string-like solutions, in which case L determines the deficit angle of thespace-time. The second set of possible values are the so-called Melvin solutions, which are not of physical interest incosmological settings, however are mathematical solutions to the equations of motion. The string-like solution thenpossesses a deficit angle which can be expressed as follows:∆ = 2 π (1 − L ′ ( ∞ )) = 2 π (1 − L ) . (35)The matter field functions have the following behaviour at r → ∞ f ( r ) ∼ − f √ r exp( − m f,γ r ) with m f,γ = s β (cid:18) γ − (cid:19) (36)and P ( r ) ∼ P √ r exp ( − m P,γ r ) with m P,γ = p − γ , (37)where f and P are constants and m f,γ and m P,γ correspond to the effective Higgs and gauge boson mass, respectively.For γ = 1 the Higgs field reaches its vacuum value quicker than the gauge field if β > β <
1. Thevalue β = 1 corresponds to the BPS limit. In this limit the masses of the gauge and Higgs bosons are equal. Nowfor γ < β equal = (2 γ − γ ) / (3 − γ ), which isequal unity for γ = 1 and decreases monotonically with decreasing γ . IV. NUMERICAL RESULTS
To the best of our knowledge there are no explicit solutions to the set of coupled differential equations presentedabove. We have hence solved the equations numerically using the ODE solver COLSYS [27]. Relative errors of thesolutions are typically on the order of 10 − to 10 − (and sometimes even better).The limit γ = 1 corresponds to standard Einstein gravity. From (25) it is apparent that γ = 0 is excluded.Furthermore, we see from (30) that in order for the solutions at γ = 1 ± δ , where δ is small, to behave like thesolutions in the γ = 1 limit we must choose the positive sign. This means that for γ > < d < L ( r ) ∼ r at r ∼
0, otherwise the space-time would not be regular, this leads toinfinities on the right hand side of the gravity equations (20) and (21). So, we conclude that we have to choose γ ∈ (0 , γ = 1 it is well known that the coupled system of equations admits gravitating Abelian-Higgs string solutions.In this paper we are interested to investigate the influence of the Rastall parameter, γ , on the behavior of the matterfields and on the metric. Since Rastall gravity reduces to Einstein gravity in the absence of sources our asymptoticspace-time (in which the matter fields reach their vacuum values) should be a solution to Einstein gravity. We canhence use the standard definition of a deficit angle given in (35).Let us first recall what is know about the Einstein gravity case γ = 1. In this case, it has been observed thatthe value of ∆ depends on both β and α and increases with increasing α . At some maximal value of α = α max the deficit angle becomes equal to 2 π . For α > α max no globally regular string solutions exist, but only solutionswith singularities (so-called “supermassive” or “inverted” string solutions). These supermassive solutions possess asingularity at some finite value of the radial coordinate r = r max , at which L ( r max , ) = 0, while N ( r max , ) stays finite[25, 26].In Fig.1 we show the behavior of a typical Abelian-Higgs string solution for β = 1 and α = 0 . γ . For β = 1 and γ = 1 these solutions fulfill a Bogomolnyi-Prasad-Sommerfield (BPS) bound. In this limit thecomponents of the energy-momentum tensor in the direction perpendicular to the string axis vanish implying via (20)that N ( r ) ≡
1. This means that there is no gravitational force acting perpendicular to the string axis. In this limit,the remaining equations can be recast into the form f ′ = PL f , P ′ = L ( f −
1) and L ′′ = − αL (cid:18) ( f − + 2 P f L (cid:19) , for γ = β = 1 . (38)A solution of this type is shown in Fig.1. Decreasing the Rastall parameter γ we observe that both the gauge fieldfunction P ( r ) as well as the Higgs field function f ( r ) are stronger localized around the string axis implying that thewidth of the string decreases. At the same time, the metric function N ( r ) starts to deviate from its constant value ofunity stronger and stronger when decreasing γ . Furthermore, the metric function L ( r ) possesses a decreasing slope atlarge r implying an increase in the deficit angle ∆ with decreasing γ . This seems natural since the energy-momentumcontent is localized inside a smaller region of space-time.Let us now discuss the value of the deficit angle, ∆, in more detail since this is an important quantity whenpredicting observational consequences of strings. The deficit angle leads to gravitational lensing as well as to red-andblue-shift of photons towards which and away from which, respectively, the string is moving (the Kaiser-Stebbinseffect). Strings (if they existed) would hence have an important impact on the temperature anisotropies of the CMB.It is often stated that the deficit angle ∆ = 8 πG ˜ µ = αµ . This is strictly speaking only true in the BPS limit β = 1.In this case, it is easy to see from (38) and the definition of µ that this relation indeed holds. Since furthermore in P (r) r α =0.5, β =1 γ =1.0 γ =0.8 γ =0.6 γ =0.4 γ =0.2 γ =0.11 (a) gauge field function P ( r ) f (r) r α =0.5, β =1 γ =1.0 γ =0.8 γ =0.6 γ =0.4 γ =0.2 γ =0.11 (b) Higgs field function f ( r ) N (r) r α =0.5, β =1 γ =1.0 γ =0.8 γ =0.6 γ =0.4 γ =0.2 γ =0.11 (c) metric function N ( r ) L (r) r α =0.5, β =1 γ =1.0 γ =0.8 γ =0.6 γ =0.4 γ =0.2 γ =0.11 (d) metric function L ( r )FIG. 1: We show the profiles of the matter functions P ( r ) and f ( r ) (top) and the metric functions N ( r ) and L ( r ) (bottom) for Abelian-Higgsstrings in Rastall gravity with α = 0 . β = 1 .
0. The γ = 1 curves correspond to the Einstein gravity limit, while the γ = 0 .
11 casegives the profiles of the solution which has ∆ ≈ π . this limit, the energy per unit length in the BPS limit is µ = 2 π we find that the deficit angle in this specific case is∆ = 2 πα .We have studied the case α = 0 . β = 0 . β = 1 and β = 2, respectively. Our results are shown in Fig.2, wherewe give the deficit angle ∆ as function of γ . The function N ( r ) stays finite all along and varies only little. This iswhy we do not present any detailed results about it here.For γ = 1 the deficit angle has the known value (see e.g.[26]). E.g. in the BPS limit, β = 1, we have ∆ = π .Decreasing γ we find that the deficit increases until it reaches ∆ = 2 π at some value of γ = γ cr . For γ < γ cr thesolutions have a singularity at a finite value of the radial coordinate r = r max with L ( r = r max ) = 0.The critical value for γ depends on the value of β . Considering α = 0 .
5, we observe that for β = 2 the critical valueis γ cr ≈ . β = 1 we have γ cr ≈ . β = 0 . π for all values of γ ∈ (0 , α = 0 . β . . γ ∈ (0 , β the value of γ cr increases, but reaches γ = 1 onlyexponentially slow. Hence, we find that for all reasonable values of the Higgs to gauge boson mass ratio as well asthe ratio between the symmetry breaking scale and the Planck mass regular Abelian-Higgs strings in Rastall gravitycan be constructed.As stated above, for γ = 1 and/or β = 1 there is no linear relation between the deficit angle ∆ and the energy perunit length µ . We have hence also studied the energy per unit length in dependence on the parameters of the model.Our results are shown in Fig.4.As is clearly seen from this figure, we have µ = 2 πα for γ = β = 1. For the other cases, the energy per unit lengthstill depends (nearly) linearly on α and behaves as µ = 2 π (1 + ǫ ( β, γ )) α , (39) ∆ / ( π ) γα =0.5 β =0.5 β =1.0 β =2.0 FIG. 2: We show the value of the deficit angle ∆ = 2 π (cid:0) − L ′ ( ∞ ) (cid:1) in dependence on γ for α = 0 . β . γ β α =0.7 α =0.5 α =0.3 FIG. 3: We show the value of γ cr in dependence on β for three different values of α . String solutions with deficit angle smaller than 2 π and hencewithout singularity exist only for values of γ above the curve. where ǫ ( β, γ ) is a function that depends on β and γ and fulfills the condition ǫ (1 ,
1) = 0. Now, we find that thisfunction is monotonically increasing with β and becomes nearly constant for very large values of β . For β < β > γ = 1. This function also increases with decreasing γ , i.e. moving away more and more from the Einstein gravity limit. The increase is stronger for larger values of β .Now, we can make an interesting observation. If we choose β <
1, we can find values of γ for which µ = 2 π , i.e.corresponding to the energy per unit length that the solution has in the BPS limit in standard Einstein gravity. Inthe following we will refer to this as a would-be-BPS limit . For γ = β = 1 this is true for all values of α . For a fixedvalue of β < α we can then decrease γ such that at some specific value of γ = ˜ γ , the energy per unit length0 µ / ( π ) αβ =0.5, γ =1.0 β =1.0, γ =1.0 β =2.0, γ =1.0 β =0.5, γ =0.5 β =1.0, γ =0.5 β =2.0, γ =0.5 FIG. 4: We show the value of the energy per unit length µ/ (2 π ) in dependence on α for three different values of β and two different values of γ ,respectively. becomes again equal to unity (in units of 2 π ). We find that the lower β the lower we have to choose γ to achieve thiscondition. E.g. for α = 0 . γ ≈ .
73 for β = 0 .
75, while ˜ γ ≈ .
56 for β = 0 .
5. We also find a (albeitsmaller) dependence on α . E.g. for β = 0 . γ ≈ .
45 for α = 0 .
5. Hence, ˜ γ decreases with increasing α . V. CONCLUSIONS
In this paper we have studied Abelian-Higgs strings in the context of Rastall gravity. Rastall theory touches oneof the cornerstones of General Relativity, namely the conservation of the energy-momentum tensor. The violationof the conservation law is parametrized in terms of a parameter γ with γ = 1 constituting the General Relativitylimit. In spite of its phenomenological character Rastall gravity may be related to an effective (and hence classical)implementation of a gravitational anomaly that might appear due to quantum effects. Our main purpose here was toinvestigate the impact of these effects on a field theoretical realization of line-like topological defects, so-called cosmicstrings. Our main results can be summarized as follows: • we find that singularity-free space-times are possible only when 0 < γ ≤ • depending on the other parameters in the model (the two ratios between the fundamental mass scales) the soliddeficit angle becomes equal to 2 π at a value of γ cr > • a BPS limit in which the energy per unit length saturates a bound (and hence becomes equal to 2 πn ) does notseem to exist here unlike for the Einstein gravity limit, where it exists for equal gauge and Higgs boson mass, • a would-be-BPS bound exists at which the energy per unit length becomes equal to 2 πn , i.e. fulfills the abovementioned bound. However, this is not related to an underlying mathematical structure.Our results are interesting because of the recently presented BICEP2 data. If the measurements of the B-modepolarization are confirmed (preferably additionally through other measurements like e.g. the PLANCK collaboration)we do have a window to the very early universe and the phase of inflation. Now inflation seems to be driven by scalarfields and the question remains where these originate from. Most unifying models that are able to model inflationpredict the production of cosmic strings at the end of inflation. Hence, it might turn out that if inflation took place,cosmic strings become a prediction rather than a speculation. Since it is certain that the energy conditions at theepoch of inflation are extreme, we would expect that quantum effects play a rˆole (even if one treats the gravity sideclassically this can certainly not be said for the matter side). Rastall gravity is one possibility to model these quantum1effects effectively. Our results presented above suggest that taking such corrections into account could have strongereffects on the CMB due to an increased deficit angle.While in this paper we have studied string-like objects without additional structure one could also think of in-vestigating strings with additional degrees of freedom inside their core. These could be in the form of fermionic orbosonic currents and the corresponding objects have been coined superconducting strings [28]. Since the stability ofthese objects is of huge importance in the context of the formation of loops of cosmic string, so-called vortons (seee.g. [29] for more details) a macroscopic stability criterion has been developed [30, 31] and used in a detailed analysisfor superconducting string solutions of the U(1) × U(1) model in flat space-time [32] as well as in curved space-time[33]. It would be very interesting to study possible quantum effects on the stability of these objects and the Rastallgravity model would implement these quantum effects naturally.2
Acknowledgment
B.H. would like to acknowledge the CNPq for financial support. B.H. would also like toacknowledge the Deutsche Forschungsgemeinschaft (DFG) for support within the framework of the DFG ResearchGroup 1620
Models of Gravity . E.R.B.M. and J.C.F. would also like to acknowledge CNPq and FAPES for partialfinancial support. [1] P. Rastall, Phys. Rev. D Rastall’s gravity equations and Mach’s principle , gr-qc/0610070.[3] C. E. M. Batista, J. C. Fabris, O. F. Piattella and A. M. Velasquez-Toribio, Eur. Phys. J. C Quantum fields in curved space , Cambridge University Press, Cambridge (1982).[7] R. Bertlmann,
Anomalies in Quantum Field Theory , Oxford University Press, Oxford (2000).[8] L. L. Smalley, Nuovo Cim. B
42 (1984).[9] T. S. Almeida, M. L. Pucheu, C. Romero and J. B. Formiga, Phys. Rev. D , 45 (1973).[11] D. Garfinkle, Phys. Rev. D Cosmic strings and other topological defects , Cambridge University Press (2000); M. Hind-marsh and T. Kibble,
Cosmic strings , Rept. Prog. Phys. , 477 (1995).[14] Planck Collaboration
Collaboration, P. Ade et al., Planck 2013 result XXV. Searches for cosmic strings and othertopological defects, [arXiv:1303.5085].[15] See for instance, J. Polchinski, Proceedings of the NATO Advanced Study Institute and EC Summer School Cargese 2004,p.229-253, Dordrecht, Netherlands: Springer (2006); A. C. Davis and T. W. B. Kibble, Contemp. Phys.
313 (2005).[16] E. Witten, Phys. Lett. B , 243 (1985).[17] P. A. R. Ade et al. [BICEP2 Collaboration], Phys. Rev. Lett. (2002) 056; S. Sarangi and S. H. H. Tye, Phys. Lett. B , 185 (2002);N. T. Jones, H. Stoica and S. H. H. Tye, Phys. Lett. B , 103514 (2003).[21] N. Bevis et al , Phys. Rev. D75 , 065015 (2007); N. Bevis et al , Phys. Rev. Lett. et al , Phys.Rev.
D76 , 043005 (2007); N. Bevis, M. Hindmarsh, M. Kunz and J. Urrestilla, Phys. Rev. Lett. , 021301 (2008); Phys.Rev. D , 065015 (2007); Phys. Rev. D , 065004 (2010); JCAP , 021 (2011).[22] J. Urrestilla, P. Mukherjee, A. R. Liddle, N. Bevis, M. Hindmarsh and M. Kunz, Phys. Rev. D , 123005 (2008); Phys.Rev. D , 043003 (2011).[23] P. Saffin, JHEP , 011 (2005).[24] B. Hartmann and J. Urrestilla, JHEP , 006 (2008).[25] M. Christensen, A.L. Larsen and Y. Verbin, Phys. Rev. D , 125012 (1999).[26] Y. Brihaye and M. Lubo, Phys. Rev. D (1979), 659; ACM Trans. Math. Softw. (1981), 209.[28] E. Witten, Nucl. Phys. B , 557 (1985).[29] P. Peter and J.-P. Uzan, Primordial Cosmology , Oxford University Press, 2009.[30] B. Carter, Phys. Lett.B
466 (1989).[31] B. Carter, in
Formation and Evolution of Cosmic strings , edited by G. W. Gibbons, S. W. Hawking and T. Vachaspati,Cambridge University Press, 1990.[32] P. Peter, Phys. Rev. D , 1091 (1992).[33] B. Hartmann and F. Michel, Phys. Rev. D VI. APPENDIX: THE CONSERVATION LAW