Note on the Evolution of the Gravitational Potential in Rastall Scalar Field Theories
aa r X i v : . [ a s t r o - ph . C O ] A p r Note on the Evolution of the Gravitational Potential inRastall Scalar Field Theories
J. C. Fabris, M. Hamani Daouda, O. F. Piattella
Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, avenida Ferrari 514,29075-910 Vit´oria, Esp´ırito Santo, Brazil
Abstract
We investigate the evolution of the gravitational potential in Rastall scalarfield theories. In a single component model a consistent perturbation theory,formulated in the Newtonian gauge, is possible only for γ = 1, which isthe General Relativity limit. On the other hand, the addition of anothercanonical fluid component allows to consider the case γ = 1. Keywords:
Gravitational potential, Rastall’s theory
Email address: [email protected], [email protected],[email protected] (J. C. Fabris, M. Hamani Daouda, O. F. Piattella)
Preprint submitted to Physics Letters B May 29, 2018 ote on the Evolution of the Gravitational Potential inRastall Scalar Field Theories
J. C. Fabris, M. Hamani Daouda, O. F. Piattella
Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, avenida Ferrari 514,29075-910 Vit´oria, Esp´ırito Santo, Brazil
1. Introduction
The nature of dark matter and dark energy is one of the most impor-tant issues today in physics. There are strong observational evidences inastrophysics and cosmology for the existence of these two exotic componentsof the cosmic energy budget, indicating that about 95% of the Universe iscomposed of dark matter (about 25%) and dark energy (about 70%), but nodirect detection has been reported until now. The usual candidates to darkmatter (neutralinos and axions, for example) and dark energy (cosmologicalconstant, quintessence, etc.) lead to very robust scenarios, but at same timethey must face theoretical and observational issues. For recent reviews onthe subject, see for example [1, 2, 3, 4, 5].An interesting proposal concerning the nature of dark matter and darkenergy are the unification models. According to the latter, the whole darksector is a manifestation of a single entity. The paradigm is a perfect fluidmodel called Chaplygin gas [6], but recently it has also been shown thatviscous models may lead to unification scenarios [7]. In spite of their greatappealing, however, unification models suffer from severe problems whenconfronted with observations, since the parameter estimations from differenttests lead to contradictory values [8]. One way to surmount this conflict is toencode the unification model in a scalar-tensor framework, the exotic fluidbeing described by a self-interacting scalar field. It is not easy to implementthis idea, since a canonical self-interacting scalar field has a sound speed
Email address: [email protected], [email protected],[email protected] (J. C. Fabris, M. Hamani Daouda, O. F. Piattella)
Preprint submitted to Physics Letters B May 29, 2018 qual to the speed of light, and it cannot represent dark matter in the pastevolution of the universe, as required by the unification program [9].A scalar model that is able to represent a realization of dark matter anddark energy can be obtained with non-canonical self-interacting scalar fields.One example is a scalar field obeying the structure of Rastall’s theory, forwhich the usual conservation law for the matter content is modified [10].The price to pay is the loss of a Lagrangian formulation, at least in thecontext of Riemannian geometry. In Rastall’s theory a new dimensionlessparameter γ is introduced, which measures the departure from the usualequations of General Relativity. When γ = 1, the General Relativity theoryis recovered. As in the case of other unification scenarios, the theory is ableto satisfactorily reproduce the kinematic background observational tests (e.g.based on type Ia supernovae surveys), but essentially reduces to GeneralRelativity if a hydrodynamical approach is used for the fluid obeying thenew conservation laws and the matter power spectrum data are used [11].However, the agreement improves if a non-canonical scalar field is employedinstead of a fluid. Moreover, if γ = 2 this non-canonical scalar field maybehave as dark matter, and may lead to a unification scenario [12].Can Rastall’s theory pass another important test, represented by the Inte-grated Sachs-Wolfe effect? In order to answer this question, the gravitationalpotential Φ must be analysed. Using the newtonian gauge, we find an aston-ishing property of Rastall’s theory: in a scenario with just one component,given by the non-canonical Rastall scalar field, only homogeneous solutionsfor Φ are admitted, unless γ = 1. But homogeneous solutions for Φ are notreal perturbations, since they can be re-absorbed in the background metricthrough a suitable reformulation of the background functions. Thus, thecase γ = 1 seems to be forced. This fundamental drawback can be cured if atwo-fluid model is formulated: one scalar field obeying the modified conser-vation laws, and a fluid obeying the canonical conservation law. The mainconclusion is: Rastall non-canonical scalar field requires a matter compo-nent (baryons, for example) in order for the theory to make sense at theperturbative level.In the next section, we introduce the Rastall non-canonical scalar field.In section 3, we discuss the sound speed issue in this theory, and in section4 we investigate the evolution of the gravitational potential and find theconstraint γ = 1. In section 5, we show how to soothe such constraint, byadding a canonical fluid component, and we study the gravitational potentialin a special case. In section 6, we present our conclusions.3 . Scalar field in Rastall’s theory In Rastall’s theory [10] a scalar field φ is characterised by the followingstress-energy tensor: T µν = φ ,µ φ ,ν − − γ g µν φ ,α φ ,α + g µν (3 − γ ) V ( φ ) , (1)where γ is a parameter. When γ = 1, we recover the corresponding theoryin General Relativity.We consider a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker met-ric ds = dt − a ( t ) δ ij dx i dx j , (2)and its perturbation written in the newtonian gauge ds = [1 + 2Φ( t, x i )] dt − a ( t )[1 − t, x i )] δ ij dx i dx j , (3)where Φ( t, x i ) is the gravitational potential.Consider a scalar field perturbation of the form φ ( t, x i ) = φ ( t ) + δφ ( t, x i )and derive from Eq. (1) together with Eqs. (2) and (3) the background andfirst-order perturbed mixed-component stress-energy tensors. The former is (0) T = ρ = γ φ + (3 − γ ) V ( φ ) , (4) (0) T i = 0 , (5) (0) T ij = − pδ ij = − (cid:20) − γ φ − (3 − γ ) V ( φ ) (cid:21) δ ij , (6)where the dot denotes derivative with respect to t and ρ and p are the scalarfield energy density and pressure, respectively. The perturbative quantitieshave the form δT = δρ = γ ˙ φ ˙ δφ − γ Φ ˙ φ + (3 − γ ) V ,φ δφ , (7) δT i = ˙ φ δφ ,i , (8) δT ij = − δpδ ij = h ( γ −
2) ˙ φ ˙ δφ + (2 − γ )Φ ˙ φ + (3 − γ ) V ,φ δφ i δ ij , (9)where , i denotes the spatial derivative with respect to the coordinate x i and V ,φ := dV ( φ ) /dφ . The modified Klein-Gordon equation has the covariantform (cid:3) φ + (3 − γ ) V ,φ = (1 − γ ) φ ,ρ φ ,σ φ ; ρ ; σ φ ,α φ ,α , (10)4here (cid:3) φ := φ ; α ; α . From Eq. (10), it appears clearer that γ = 1 restores theGeneral Relativity case. Insert again φ = φ + δφ into Eq. (10) and employmetric (2) and (3), in order to find γ ¨ φ + 3 H ˙ φ + (3 − γ ) V ,φ = 0 , (11)where H := ˙ a/a is the Hubble parameter. Employing the conformal time dη = dt/a ( t ) we write γφ ′′ + (3 − γ ) H φ ′ + (3 − γ ) a V ,φ = 0 , (12)where the prime denotes derivation with respect to the conformal time and H := a ′ /a . The perturbed modified Klein-Gordon equation has the followingform: γδφ ′′ + (3 − γ ) H δφ ′ − ∇ δφ − (3 + γ ) φ ′ Φ ′ + 2(3 − γ ) a V ,φ Φ + (3 − γ ) a V ,φφ δφ = 0 , (13)where V ,φφ := d V ( φ ) /dφ .Differently from the γ = 1 case, in Rastall’s theory we have more degreesof freedom. Indeed, considering T µν ; µ = 0 : φ ,ν [ (cid:3) φ + (3 − γ ) V ,φ ] = (1 − γ ) φ ,µ φ ; µ ; ν , (14)one notices that the contraction with φ ,ν gives the Klein-Gordon equation,Eq. (10). However, Eq. (14) are actually four independent equations. Onlyin the γ = 1 case they reduce to only one, namely the usual Klein-Gordonequation. Using Eq. (10) into Eq. (14) one obtains φ ,ν φ ; α ; β φ ,α φ ,β − φ ,α φ ,α φ ; ν ; β φ ,β = 0 . (15)At the background level, being φ dependent only on the time, Eq. (15) isidentically satisfied. For small perturbations, Eq. (15) gives the perturbedKlein-Gordon equation for ν = 0, whereas for ν = i one has(1 − γ ) ¨ φ δφ ,i = (1 − γ ) ˙ φ δ ˙ φ ,i − (1 − γ ) ˙ φ Φ ,i . (16)For γ = 1 one can cast the above equation as follows: (cid:18) δφ ,i ˙ φ (cid:19) · = Φ ,i , (17)which appears to be an additional constraint that we have to take into ac-count together with the Einstein’s equations.5 . The scalar field speed of sound in Rastall’s theory In [12] the authors investigate the case corresponding to γ = 2, which isable to reproduce the ΛCDM scenario both at the background and at theperturbative level. A possible explanation for the success of the case γ = 2may reside in the fact that in such instance the speed of sound vanishes, aswe show now. The speed of sound is defined as the ratio c s := δp/δρ , whichis gauge-dependent. Therefore, it makes sense to consider its value in thereference frame where the substance whose collapse is being investigated isat rest; we denote such quantity as ˆ c s .Following [13], we employ the formula δp = ˆ c s δρ + 3 aHρ (1 + w ) (cid:0) ˆ c s − c a (cid:1) θk , (18)which links the pressure perturbations to the energy density ones, both in ageneric gauge, via ˆ c s . In this formula, c a is the adiabatic speed of sound,defined as c a := ˙ p/ ˙ ρ and which, for the Rastall scalar field that we areinvestigating, has the form c a = 3 H (2 − γ ) ˙ φ + 2(3 − γ ) V ,φ Hγ ˙ φ , (19)where we have employed Eqs. (4), (6) and the equation of motion (11). More-over, in formula (18), w := p/ρ , k is the wavenumber coming from a normalmode decomposition and θ is defined via a ( ρ + p ) θ := ∂ i δT i = ˙ φ ∂ i δφ ,i . (20)Substituting in Eq. (18) the expressions for δρ and δp , that we have foundin Eqs. (7) and (9), and Eqs. (19) and (20) we obtain γ ˆ c s h γ ˙ φ ˙ δφ − γ Φ ˙ φ + (3 − γ ) V ,φ δφ + 3 H ˙ φ δφ i == (2 − γ ) h γ ˙ φ ˙ δφ − γ Φ ˙ φ + (3 − γ ) V ,φ δφ + 3 H ˙ φ δφ i , (21)which clearly gives that ˆ c s = (2 − γ ) /γ and therefore the case γ = 2 impliesˆ c s = 0, which is a favouring case for the collapse.6 . Evolution of the gravitational potential Following [14], we calculate the Einstein tensor from metric (3) and com-bine it with the perturbed stress-energy tensor in Eqs. (7)–(9). In particu-lar, as we implicitly anticipated in writing the perturbed metric (3), since δT ij ∝ δ ij we have only one gravitational potential. See [14] for more detail.We obtain ∇ Φ − H ( H Φ + Φ ′ ) + γ (cid:0) H − H ′ (cid:1) Φ =4 πG (cid:2) γφ ′ δφ ′ + (3 − γ ) a V ,φ δφ (cid:3) , (22) H Φ ,i + Φ ′ ,i = 4 πGφ ′ δφ ,i , (23)Φ ′′ + 3 H Φ ′ + (cid:0) H ′ + H (cid:1) Φ + (2 − γ ) (cid:0) H − H ′ (cid:1) Φ =4 πG (cid:2) (2 − γ ) φ ′ δφ ′ − (3 − γ ) a V ,φ δφ (cid:3) , (24)where we have also used the background relation 4 πGφ ′ = H − H ′ .Now we reduce the above system in two different ways which, however,will give different results unless we choose γ = 1. Let us employ a normalmode decomposition. The second equation can thus be written as H Φ + Φ ′ =4 πGφ ′ δφ and ∇ = − k .Now, if we sum or subtract the first equation with the third, use H Φ+Φ ′ =4 πGφ ′ δφ in order to eliminate δφ and the equation of motion (12) in orderto eliminate φ ′′ we obtain: Φ ′′ + 3 H Φ ′ + (cid:0) H ′ + H (cid:1) Φ =2 − γγ (cid:2) − k Φ − H ( H Φ + Φ ′ ) (cid:3) − V ,φ a γφ ′ (3 − γ ) ( H Φ + Φ ′ ) , (25)Φ ′′ + 3 H Φ ′ + (cid:0) H ′ + H (cid:1) Φ = − k Φ − H − γγ ( H Φ + Φ ′ ) − V ,φ a γφ ′ (3 − γ ) ( H Φ + Φ ′ ) . (26)These equations can be identical only if γ = 1, which is the General Relativitylimit of Rastall’s theory.Actually, these equations could be consistent also if k = 0, but what doesthis mean? Going back to the Einstein equations, before the normal modes7ecomposition, k = 0 would mean ∇ Φ = 0, which implies that Φ should bea homogeneous field, i.e. Φ = Φ( η ). Note that any spatial linear dependencefor Φ would be unacceptable since the field would diverge at infinity.But if Φ is homogeneous, then we are not facing real perturbations. Infact, it is sufficient in metric (3) to redefine the time and the scale factor asfollows: d ¯ t = [1 + 2Φ( t )] dt , ¯ a ( t ) = a ( t )[1 − t )] , (27)and we obtain again the FLRW metric in the usual reference frame, with ¯ t the cosmic time.Therefore we conclude that a consistent perturbation theory, formulatedin the newtonian gauge, is possible only for γ = 1.Is such result compatible with the new constraint (17) that we haveshowed to exist in Rastall’s theory, for γ = 1? The answer is yes, sincecombining Eqs. (22) and (23) it is possible to obtain Eq. (17). Therefore,the latter is not an actual constraint: it is already embedded in Einstein’sequations.
5. The role of a fluid component
The result found in the previous section appears to be different if weintroduce another component together with the Rastall scalar field. Indeed,let us consider a perfect fluid with equation of state p = wρ and w constant.We write its total (i.e. background plus perturbed) stress-energy tensor asfollows: T = ρ (1 + δ ) , (28) T i = − ρ (1 + w ) v i , (29) T ij = − ( p + δp ) δ ij , (30)where δ := δρ/ρ is the usual density contrast, δρ is the density perturba-tion, δp is the pressure one and v i is the velocity. We assume adiabaticperturbations, i.e. δp = c δρ , where c = w .Employing again the normal mode decomposition, we rewrite the system8f linearised Einstein equations as follows: − k Φ − H ( H Φ + Φ ′ ) + γ (cid:0) H − H ′ (cid:1) Φ = 4 πGa δρ φ + 4 πGa δρ , (31) k ( H Φ + Φ ′ ) = 4 πGkφ ′ δφ + 4 πGρ (1 + w ) v , (32)Φ ′′ + 3 H Φ ′ + 3 H Φ − γ (cid:0) H − H ′ (cid:1) Φ = 4 πGa δp φ + 4 πGa δp , (33)where we have defined: δρ φ := 1 a γφ ′ δφ ′ + (3 − γ ) V ,φ δφ , (34) δp φ := 1 a (2 − γ ) φ ′ δφ ′ − (3 − γ ) V ,φ δφ , (35)and v is the velocity potential defined by v i = − v ,i /k .With the fluid variables and the relation δp = c δρ we have a total of fourunknowns (Φ , δφ, δρ, v ) but only 3 equations. Therefore, it is impossible toobtain again a constraint as strong as γ = 1.Let us investigate in some detail the coupled system fluid plus Rastallscalar field. Multiplying Eq. (31) by 2 − γ , Eq. (33) by γ and subtractingthe two we obtain: γ Φ ′′ + 6 H Φ ′ + 6 H Φ − γ (cid:0) H − H ′ (cid:1) Φ + (2 − γ ) k Φ = − πGa (3 − γ ) V ,φ δφ + 4 πGa (cid:0) γc + γ − (cid:1) δρ . (36)From this equation we eliminate δφ with the help of Eq. (32), obtaining γ Φ ′′ + 6 H Φ ′ + 6 H Φ − γ (cid:0) H − H ′ (cid:1) Φ + (2 − γ ) k Φ = − a (3 − γ ) V ′ (cid:20) H Φ + Φ ′ φ ′ − πGa ρ (1 + w ) vkφ ′ (cid:21) +4 πGa ρ (cid:0) γc + γ − (cid:1) δ , (37)where we have used V ,φ = V ′ /φ ′ . We now assume the fluid to satisfy its ownenergy-momentum conservation, separately from the scalar field, in order togain one more equation necessary to solve the system. From T µν ; µ = 0, forthe fluid component only, we get: δ ′ = − (1 + w )( kv − ′ ) , (38) v ′ = −H (1 − w ) v + kc w δ + k Φ . (39)9ow we have to specify the background evolution, i.e. the function H . Itsgeneral form is H a = 8 πG (cid:20) ρ + γφ ′ a + (3 − γ ) V (cid:21) . (40)In order to simplify considerably Eq. (37), we assume that the potential is aconstant, i.e. V ′ = 0. Therefore, it is going to play the role of an effectivecosmological constant. If the potential is a constant, it turns out from theKlein-Gordon equation (11) that φ ′ = u a − (3 − γ ) /γ , (41)and the Friedmann equation takes on the following form: H H a = Ω a − w ) + Ω V + Ω u a − /γ , (42)with the definitionsΩ V := 8 πG (3 − γ ) V H , Ω u := 8 πGγu H . (43)Finally, trading the conformal time for the scale factor, we write the coupledsystem of Einstein equations plus the fluid equations as γ Φ aa + " γ ˙ HH + 6 + γa Φ a + " γ ˙ HH a + 6 − γa Φ + (2 − γ ) k H a Φ =3 H H Ω a − w ) (cid:0) γc + γ − (cid:1) δ , (44) δ a = − (1 + w ) (cid:18) kv H a − a (cid:19) , (45) v a = − a (1 − w ) v + kc (1 + w ) H a δ + k H a Φ , (46)where the subscript a denotes derivation with respect to the scale factor.We start the evolution from a i = 10 − and choose as initial conditionsΦ ′ i = v i = 0, Φ i = − δ i = − w = 0 and γ = 2, Eqs. (42) and (44) reproduce the samedynamics of the ΛCDM model. It is curious that this seems to happen only10 .0 0.2 0.4 0.6 0.8 1.00.60.70.80.91.0 a F ∆ Figure 1: Evolution of Φ and δ for the choice w = 0 and Ω = 0 .
04. The black solid linesrepresent the case γ = 2, i.e. the ΛCDM one. The red dashed lines correspond to γ = 1 . γ = 2 .
05. We have chosen a scale k = 10 − h Mpc − . F ∆ Figure 2: Evolution of Φ and δ for the choice w = 0 and Ω = 0 .
04. The black solid linesrepresent the case γ = 2, i.e. the ΛCDM one. The red dashed lines correspond to γ = 1 . γ = 1. We have chosen a scale k = 10 − h Mpc − . δ for the choice w = 0 and Ω = 0 .
04. Thatis, we are assuming that the perfect fluid under consideration is a baryoncomponent. We have also chosen a scale k = 10 − h Mpc − . It seems that,for γ >
2, the growth of the density contrast of the fluid component isenhanced. This is probably due to the fact that ˆ c becomes negative andtherefore the collapse of the scalar field is unimpeded. For γ < δ is sensibly hampered. It is curious in Figure 2 how the gravitationalpotential suffers a larger decrease for the case γ = 1 . γ = 1one. We would have expect the contrary, since for γ = 1 the speed of soundis ˆ c = 1, whereas for γ = 3 / c = 1 /
3, i.e. smaller.Note that such discrepancy is not present in the plots for δ , i.e. the growthfor γ = 1 . γ = 1. Therefore, such effect is probablydue to the different background evolution.For completeness, we display here also the evolution of δ φ := δρ φ /ρ φ ,which can be easily calculated from Eq. (31), once we know Φ and δ . Theresults are plotted in Figure 3. ∆ Φ Figure 3: Evolution of δ φ for the choice w = 0 and Ω = 0 .
04. From top to bottom: γ = 2 . , , . , . ,
1. We have chosen a scale k = 10 − h Mpc − . Some comments about Eq. (17) are in order, since the latter establishes a12trong connection between perturbations in the scalar field and in the gravita-tional potential. For a single scalar field component, Rastall’s theory reducesto General Relativity, thus everything runs as in the standard lore. On theother hand, the coupling with matter brings new features. Here also comesthe complexity given by the new conservation law of Rastall’s theory, whichadmits many consistent alternatives. If the matter component conserves sep-arately (as we have investigated in this paper), Eq. (17) remains untouched.However, there are examples in which it may change, e.g. the case in whichmatter exchanges energy with the scalar field or the one investigated in [15],where two fluid components were considered, one of them still conservingseparately, whereas the conservation of other depending on the curvature. Inthis case, it is not difficult to show that relation (17) gives place to (cid:18) δφ ,i ˙ φ (cid:19) · = Φ ,i −
12 ( δρ ,i − δp ,i ) , (47)rendering the situation somewhat different. In [15] other possible couplingsbetween the two components are also evoked, which may lead to other vari-ants for Eq. (17).Even if our interest in this work is the late-times universe, we may ask forpotential consequences of Eq. (17) and its possible variants for the primordialspectrum. One point is that Rastall’s scalar model requires another compo-nent in order to make sense. But, an adiabatic primordial spectrum can benaturally implemented mainly if the matter component is subdominant withrespect to the scalar component. The isocurvature component can also beimplemented, in principle, since it requires a zero total (scalar plus matter)density fluctuation δρ tot = 0, and this even if the relation (17) still holds, asin the case where the other component conserves separately, without directinteraction with the scalar field. When other types of interactions betweenboth components are considered, as in the equation above, the isocurvatureperturbation can still exist. But, in general, the detailed predictions for thespectrum must differ from the standard cases, mainly in the isocurvaturecase. This may open a new path of investigation concerning the specificpredictions of Rastall’s theory for the primordial spectrum of perturbations.Equation (17) also reminds the relation between the spatial curvature andthe inflaton field, in the standard inflationary scenario [16]. However, suchconnection still has to be investigated in detail, in order to understand howinflation could be implemented into Rastall’s theory.13 . Conclusions The scalar formulation of Rastall’s theory of gravity may allow a consis-tent unification of dark matter and dark energy for the background evolutionof the universe. We have shown in this paper that, on the other hand, itssingle component version (with the non-canonical Rastall scalar field as theonly matter content) is perturbatively inconsistent: the compatibility of theperturbed equations requires a homogeneous gravitational potential, whichis equivalent to a redefinition of the background functions and not to realperturbations. The Rastall scalar theory may admit consistent perturbativescenario if another (canonical) fluid component is added.For a two-fluid model, we numerically evaluate the behaviour of the grav-itational potential and that of the density contrast for the scalar field com-ponent. For some cases, as γ ∼ .
5, a behaviour very similar to that ob-tained in the General Relativity case with a quintessence field is obtained.Although this does not represent an exhaustive study of the Rastall two-component model, the results here reported indicate that consistent scenar-ios may emerge from a Rastall unification model for dark energy. We hope topresent in future a more exhaustive study with a detailed comparison withcosmological observational data.
Acknowledgements
We thank CNPq (Brazil) for partial financial support. We also acknowl-edge the anonymous referee for his useful remarks and suggestions. We areindebted to W. Zimdahl for many enlightening discussions.