Accelerating cosmology in Rastall's theory
Monica Capone, Vincenzo Fabrizio Cardone, Matteo Luca Ruggiero
aa r X i v : . [ a s t r o - ph . C O ] O c t ACCELERATING COSMOLOGY IN RASTALL’S THEORY
Monica Capone ∗ Dipartimento di Matematica, Universit`a di Torino, Via Carlo Alberto 10, 10125 - Torino, ItalyINFN, Sezione di Torino, Via Pietro Giuria 1, 10125 - Torino, Italy
Vincenzo Fabrizio Cardone † Dipartimento di Fisica Generale “Amedeo Avogadro”,Universit`a di Torino, Via Pietro Giuria 1, 10125 - Torino, ItalyINFN, Sezione di Torino, Via Pietro Giuria 1, 10125 - Torino, Italy
Matteo Luca Ruggiero ‡ UTIU, Universit`a Telematica Internazionale Uninettuno,Corso Vittorio Emanuele II 39, 00186 - Roma, ItalyDipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 23, 10129 - Torino, ItalyINFN, Sezione di Torino, Via Pietro Giuria 1, 10125 - Torino, Italy (Dated: October 30, 2018)In an attempt to look for a viable mechanism leading to a present-day accelerated expansion, weinvestigate the possibility that the observed cosmic speed up may be recovered in the frameworkof the Rastall’s theory, relying on the non-conservativity of the stress-energy tensor, i.e. T µν ; µ = 0.We derive the modified Friedmann equations and show that they correspond to Cardassian-likeequations. We also show that, under suitable assumptions on the equation of state of the matter termsourcing the gravitational field, it is indeed possible to get an accelerated expansion, in agreementwith the Hubble diagram of both Type Ia Supernovae (SNeIa) and Gamma Ray Bursts (GRBs).Unfortunately, to achieve such a result one has to postulate a matter density parameter much largerthan the typical Ω M ≃ . I. INTRODUCTION
The observed cosmic speed up [1–3] questions the validity of General Relativity (GR) on large scales. In fact,if on one hand the model of gravitational interaction as described by Einstein’s theory is in agreement with manyobservational tests on relatively small scales, as Solar System and binary pulsars observations show [4], it is wellknown that in order to make GR agree with the observed acceleration of the Universe the existence of dark energy ,a cosmic fluid having exotic properties, has been postulated. Actually, many candidates for explaining the nature ofdark energy have been proposed (see e.g. [5], [6] and references therein), some of them relying on the modificationof the geometrical structure of the theory, some others on the introduction of physically (up to day) unknown fluidsinto the equations governing the behaviour of our universe. Moreover, it is interesting to point out that the problemof explaining the acceleration of the Universe has been addressed also in the framework of GR (see [7] and referencestherein).In this context, we want to consider here a generalization of Einstein’s theory, the so-called Rastall’s model [8],based on the requirement that the stress-energy tensor for the matter/energy content is not conserved, i.e. T µν ; µ = 0.Rastall’s model has been initially motivated by the need for a theory able to allow a non-conservativity of thesource stress-energy tensor without violating the Bianchi identities. As such, the original theory was based on purelyphenomenological motivations and directly started with the field equations without any attempt to derive them from avariational principle (even if there have been subsequent attempts to deduce Rastall’s field equations from a variationalprinciple, but none of them have succeeded [9, 10]).As for the confrontation with the data, it is interesting to point out that Rastall’s field equations, in vacuum, areequivalent to GR ones: as a consequence, all classical tests of GR are correctly reproduced. On the other hand, it couldbe useful to test the cosmological predictions of the theory, by considering the solutions within the cosmological fluid.Our work is motivated by the fact that Rastall’s theory was introduced more than 30 years ago, so it is interestingto test it against the recent cosmological data. In particular, we focus on the possibility of describing the accelerated ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] expansion of the Universe in Rastall’s framework, by investigating the conditions that the parameters of the theoryhave to fulfill in order to reproduce the data. Furthemore, prompted by a recent paper [11], we check whether theCardassian model [12, 13] can be derived by Rastall’s tensorial equations, because, despite the fact that this modelpassed almost all observational tests, it is purely phenomenological.The plan of the paper is as follows: we will firstly give an introduction to Rastall’s model in Sect. II, while thecorresponding cosmological scenario and the analogy with the Cardassians expansion model is worked out in Sect.III. In order to check the possible viability of the Rastall’s proposal, we test the model with respect to the SNeIa andGRBs Hubble diagram, as detailed in Sect. IV. Conclusions are finally presented in Sect. V.
II. RASTALL’S MODEL
In 1972 P. Rastall [8] explored a model in which the stress-energy tensor of the source of the gravitational field, T µν , was not conserved, i.e. the condition T µν ; µ = 0 is imposed a priori.Indeed, Einstein equations read G µν . = R µν − g µν R = κ GR T µν , (1)where the Ricci tensor is obtained from a metric connection, so that R µν = R µν ( g ) and the scalar curvature R has tobe intended as R ≡ R ( g ) = g αβ R αβ ( g ); furthermore we have set κ GR = πGc .These equations naturally imply the stress-energy tensor conservation as a consequence of the contracted Bianchiidentities, G µν ; µ = 0 . (2)It is therefore worth wondering whether it is possible to fulfill the requirement T µν ; µ = 0 without violating Eqs.(2). Apossible way out could be introducing further geometrical terms on the right hand side of Einstein equations, even ifone should ask whether this makes sense. Actually, if we insist in deriving these relations from a metric variationalapproach, the sudden answer would be of course negative: in this case the stress-energy tensor would be surelyconserved by construction, so no way to escape the conditions T µν ; µ = 0.Another remark against the non-conservativity focuses on the equivalence principle: as a matter of fact, the con-servation of the stress-energy tensor is tested with high accuracy in the realm of Special Relativity (SR). Then, onejumps to the realm of GR just invoking the principle of minimal coupling. However, one has to go easy with such anapproach, as this principle could be misleading [14]. To give an example, when passing from GR to SR, we completelymiss the information provided by terms explicitly depending on the curvature tensor, R αµβν , as it becomes identicallyzero when the spacetime becomes flat. This means that the two sets of equations ∇ α j α = 0 (3)and ∇ σ j σ + R αµβν ∇ α j µ ∇ β j ν = 0 , (4)give exactly the same equations, i.e. ∂ α j α = 0, in SR. So, the straightforward application of the equivalence principlein writing conservation laws should be carefully considered.The question is now how to pick up a proper geometrical term such that the Bianchi identities are still valid, butnevertheless the conservation of the stress-energy tensor of the gravity source is violated. To resume, we ask for afour-vector, say a ν , such that (i) T µν ; µ = a ν ; (ii) a ν = 0 on curved spacetime, but a µ = 0 on flat spacetime in ordernot to conflict with the validity of SR. Both these properties hold for the Rastall’s proposal, that is T µν ; µ = λR ,ν , (5) Throughout the paper, spacetime is assumed to have the signature (+ , − , − , − ), and Greek indices run from 0 to 3. λ being a suitable non-null dimensional constant.Because of the assumption (5), the field equations are obviously modified and now read R µν −
12 (1 − κ r λ ) Rg µν = κ r T µν , (6)where κ r is a dimensional constant to be determined in order to give the right Poisson equation in the static weak-fieldlimit. It is manifest that in vacuum, where T µν = 0, Rastall’s field equations (6) are equivalent to GR ones.As a matter of fact, the same set of equations can be obtained as the result of guesswork, that is assuming theleft hand side of the sought after equations to be a symmetric tensor only consisting of terms that are linear in thesecond derivative and/or quadratic in the first derivatives of the metric [15]. Moreover, the time-time component ofsuch equations must give the Poisson equations back for a stationary weak-field. Accordingly, the only requirementwe drop with respect to the derivation of the Einstein equations is the one concerning the conservation of T µν . Hence,starting from G µν = C R µν + C Rg µν , with C and C appropriate constants, we end up with Eqs.(6) again, providedthat we set: C = C ( C − − C ) , (7) κ r = 8 πGC c ≡ κ GR C , (8) κ r λ = C (1 − C )2 (3 − C ) , (9)where we have chosen to rewrite all the other constants in terms of the C . Note, in particular, that the couplingconstant between matter and geometry, κ r , is not the same as in GR, unless C = 1, that is λ = 0 (i.e., we consistentlygo back to GR).Taking the trace of Eqs.(6) gives us the structural or master equation [16] :(4 κ r λ − R = κ r T . (10)For a traceless stress-energy tensor, T = 0 (as for the electromagnetic tensor) and two possibilities arise. The first isthat R = 0 so that we get no differences with standard GR. On the other hand, one could also solve Eq.(10) setting κ r λ = 1 /
4, whatever the value of R is. However, inserting this condition in Eq.(9), we get a complex value for C which is clearly meaningless. Therefore, we hereafter assume that κ r λ = 1 / T µν ; µ = 0 are nothingbut the equations of motion of the fluid we are dealing with. The problem is then, what sort of curves are describedin a curved spacetime by a fluid whose stress-energy tensor is not conserved. Following the calculations made byRastall, we find that in his model geodesics are those curves characterized by the fact that the scalar curvature R isconstant along them. Moreover, it is still possible to speak of conservation of energy for an ideal fluid [8], but againprovided that R is constant along the time-like four-velocity vector of the fluid, u µ . The question remains whetherparticles creation takes place in the regions where this condition does not hold.It is worth mentioning that the Rastall’s equations (6) can be recast into the same form as the usual Einstein ones.Indeed, one can immediately write G µν = κ r S µν , (11)where S µν = T µν − κ r λ κ r λ − g µν T . (12)By construction, this new stress-energy tensor is conserved, S µν ; µ = 0. On introducing S µν , we can recover all theknown solutions of Einstein GR by simply taking care of the difference between S µν and T µν . Furthermore, if weassume T µν = ( ρ + p ) u µ u ν − pg µν , i.e. the source is a perfect fluid with energy density ρ and pressure p , we canexplicitely work out an expression for S µν . This turns out to be still a perfect fluid, provided we redefine its energydensity and pressure as ρ S = (3 κ r λ − ρ + 3 κ r λp κ r λ − , (13) p S = κ r λρ + ( κ r λ − p κ r λ − . (14)In order to obtain the value of the coupling constant κ r , we remember that the time-time component of the modifiedequations should recover the Poisson equation in the static weak-field limit. One thus gets: κ r κ r λ − (cid:18) κ r λ − (cid:19) = κ GR , (15)whence it is immediate to derive exactly the same coupling as in Einstein gravity only when λ = 0, that is when theconservation of T µν is granted. III. A CARDASSIAN ANALOG AND THE COSMIC SPEED UP
It has been recently claimed [11] that a Cardassian-like [12, 13] modification of the Friedman equation in the form H = 8 πG c [ ρ + B ( t )( ρ − p ) n ] , (16)can be obtained from Rastall-like equations, where B ( t ) is a function of the cosmic time t . We would now like toshow that, although it is indeed possible to recast the Rastall’s theory equations in such a way that a Cardassian-likemodel is recovered, the parameter B in (16) must be a constant.To this aim, we derive the cosmological equations for the Rastall’s theory. We first remember that, when theisotropic and homogenous Robertson-Walker (RW) metric is adopted, in GR one gets the usual Friedmann equations H = κ GR c ρ , (17)˙ H + H = − κ GR c ρ + 3 p ) , (18)where H = ˙ a/a is the Hubble parameter, a the scale factor and a dot denotes the derivative with respect to cosmictime t . To get the corresponding equations for the Rastall’s theory, one has to insert the RW metric into Eqs.(6) andconsider the only independent equations that can be obtained, that is3 ˙ a − κ r λ (cid:0) ˙ a + a ¨ a (cid:1) = κ r a ρ , (19)˙ a + 2 a ¨ a + 6 κ r λ (cid:0) ˙ a + a ¨ a (cid:1) = − κ r a p , (20)respectively. The master equation thus becomes6 (cid:0) ˙ a + a ¨ a (cid:1) = − κ r a ( ρ − p ) , (21)so that multiplying Eq.(20) by − aa = κ r ρ + 3 p − κ r λ ( ρ + p )4 κ r λ − , (22)which makes it possible to directly infer the sign of the acceleration. To obtain the second modified Friedmann’sequation, it is easier to proceed in a slightly different way. Let us first take the Rastall’s equations in the form R µν = κ r (cid:18) T µν −
12 2 κ r λ − κ r λ − T g µν (cid:19) . (23)By inserting the RW metric and adding up Eq.(19) with three times Eq.(20), we eventually obtain H = κ r (cid:20) ( ρ + 3 p ) + 2 κ r λ − κ r λ − ρ − p ) (cid:21) . (24)It is then only a matter of algebra to rearrange Eqs.(22) and (24) to write them as H = κ r (cid:20) ρ − κ r λ κ r λ − ρ − p ) (cid:21) , (25)˙ H = κ r κ r λ −
1) [ ρ + p − κ r λ ( ρ − p )] , (26)which reduce to the standard Friedmann equations (17) and (18) when the parameter λ is switched off. Note also thatEq.(25) has indeed the same expression as the Cardassian-like Eq.(16) provided we set n = 1 and accordingly redefinethe parameter B . However, it is straightforward to show that B must be a constant. Indeed, from the condition S µν ; µ = 0 , (27)it is immediate to demonstrate that κ r λ must be a constant by simply inserting the master equation (21) into Eq.(27)and using the Rastall’s requirement T µν ; µ = λR ,ν . So, an equation like (16), cannot be self-consistently obtained inRastall’s model.Moreover, since T µν is a perfect fluid and remembering the definition of S µν , Eq.(27), we get˙ ρ + 3 H ( ρ + p ) = κ r λ κ r λ − ρ − p ) , (28)which generalizes the continuity equation for the Rastall’s theory.It is worth noticing that, even without integrating the equations, one can immediately predict whether the universeexpansion is accelerating or not by simply studying the sign of the right hand side of Eq.(22).Assuming for simplicity that the equation of state of the perfect fluid is a constant, i.e. setting p = wρ , the condition¨ a > w > κ r λ − − κ r λ ) , (29)provided κ r λ > /
4. When λ = 0, however, the above relation reduces to w > − / w < κ r λ − − κ r λ ) , (30)with κ r λ < /
4. The right hand side of (30) may be positive or negative depending on the value of κ r λ . Moreprecisely, if 1 / < κ r λ < /
4, then the right hand side is positive, while it is negative for 0 < κ r λ < /
6. It isworth stressing that, however, the model always gives a monotonic behaviour: always decelerated or, as in the aboveanalyzed case, always accelerated.By the way, in the spirit of Cardassians, the only sources of gravity are radiation and matter. In particular, therecent epoch is driven by the matter content, described as a perfect fluid with equation of state w = 0. With thisconstraint, it is easy to show that an accelerated behaviour is obtained for < κ r λ < , whereas we have a deceleratedexpansion choosing the following values κ r λ < or κ r λ > . z µ FIG. 1: Comparison among predicted and observed SNeIa and GRBs Hubble diagram.
IV. RASTALL’S MODEL CONFRONTED WITH THE DATA
Neglecting the radiation component, the only fluid sourcing the gravitational field is the standard matter, whichcan be modeled as dust, i.e. p = 0. In such a case, the continuity equation (28) is straightforwardly integrated giving : ρ ∝ ρ (1 + z ) w eff , (31)with a 0 subscript denoting present day quantities, z − a the redshift (having set a = 1 for our flat-spaceuniverse), and w eff = 1 − κ r λ κ r λ − , (32)an effective equation of state (EoS) for the dust matter, from which a relation between w eff and κ r λ is easily deduced.Note that, for λ = 0, one recovers the usual matter scaling ρ ∝ (1 + z ) , while deviation from the standard behaviouroccurs in λ = 0 Rastall’s theory. Such a different scaling is not surprising at all being an expected consequence of thenon-conservativity of the stress-energy tensor. Inserting back Eq.(31) into Eq.(25), we get E . = H /H = (1 + z ) w eff , (33)which is all what we need to compute the luminosity distance D L ( z, w eff , h ) = d H (1 + z ) Z z E ( z ′ ) dz ′ , (34)with the Hubble radius d H = c/H ≃ h − Gpc and h the Hubble constant in units of 100 km / sMpc. We have nowall the main ingredients to test the viability of the Rastall’s model by fitting the predicted luminosity distance tothe data on the combined Hubble diagram of SNeIa and GRBs. To this aim, we maximize the following likelihoodfunction: L ( w eff , h ) ∝ exp (cid:18) − χ SNeIa + χ GRB (cid:19) × exp " − (cid:18) h HST − hσ HST (cid:19) , (35)with χ SNeIa = N SNeIa X i =1 (cid:20) µ obs ( z i ) − µ th ( z i , w eff , h ) σ i (cid:21) , (36) χ GRB = N GRB X i =1 (cid:20) µ obs ( z i ) − µ th ( z i , w eff , h ) σ i (cid:21) . (37)The χ terms in (35) take care of the Hubble diagram of SNeIa and GRBs, respectively, and rely on the distancemodulus defined as µ th ( z, w eff , h ) = 25 + 5 log D L ( z, w eff , h ) . (38)We use the Union [17] dataset for SNeIa and the GRBs sample assembled in Cardone et al. [18] to set the observedquantities ( µ obs , σ i ) for the SNeIa and GRBs, respectively. Since the Hubble constant h is degenerate with the(unconstrained) absolute magnitude of a SN, we have added a Gaussian prior on h using the results from the HSTKey Project [19] thus setting ( h HST , σ
HST ) = (0 . , . w eff , h ) = (0 . , . , giving χ SNeIa /d.o.f. = 1 . , χ GRB /d.o.f. = 2 . , where d.o.f. = N SNeIa + N GRB − N p is the number of degree of freedom of the model, with N SNeIa = 307 thenumber of SNeIa in the Union sample, N GRB = 69 the number of GRBs, and N p = 2 the number of parameters ofthe Rastall’s model. While for SNeIa we get a very good reduced χ , this is not the case for GRBs so that one couldbe tempted to deem as unsuccessfull the fit. Actually, Fig. 1 shows that the model is indeed fitting quite well boththe SNeIa and GRB data so that the large value of χ GRB /d.o.f. should be imputed to the large scatter of the highredshift data around the best fit line, not taken into account by the statistical error on the GRBs distance modulus.In order to further test the model, one can consider the constraints on the matter density parameter. Since our modelonly contains matter, one could naively think that Ω M = 1. Actually, one must also take into account that Ω M isdefined using the GR coupling constant κ GR which is related to the Rastall’s coupuling κ r through Eq.(15). It is thena matter of algebra to show that Ω M = w eff so that, after marginalizing over h , we get the following constraints:Ω M = 0 . +0 .
02 +0 . − . − . , where we have used the notation x + x + x − y − y to mean that x is the median value of the parameter and ( x + x , x − y ) , ( x + x , x − y ) are the 68% and 95% confidence ranges respectively. Note that the value thus obtained is instrong disagreement with the typical Ω M ≃ . M and leading to the final disagreement.Inverting the relation between Ω M and κ r λ , we get κ r λ = (1 − Ω M ) / (3 − M ) so that, for Ω M ≃ .
3, we get κ r λ ≃ . > /
4. Indeed, our best fit value for w eff gives back a value for κ r λ that falls outside the suitable rangeto reproduce an accelerated behaviour. V. CONCLUSIONS
In this paper we have focused on Rastall’s theory of gravity, which has been initially motivated by the need for atheory able to allow a non conservativity of the source stress-energy tensor without violating the Bianchi identities.In particular, we have reexamined this model of gravity to investigate the possibility that it could reproduce theobserved cosmic speed up. First, we have explicitly worked out the modified Friedmann equations and we haveshown that Cardassian-like modifications of Friedmann equations are obtained in Rastall’s model but, contrary torecent claims, they cannot contain time-dependent parameters. Then, we have confronted the model predictionswith the available data concerning type Ia Supernovae (SNeIa) and Gamma Ray Bursts (GRBs): what we haveshowed is that it is possible to get an accelerated expansion that is in agreement with the Hubble diagram of bothSNeIa and GRBs, even if there is unfortunately no possibility to reproduce an accelerated-decelerated-acceleratedexpansion for our universe as it seems to be requested. These results have also a major drawback: indeed,to get them it is necessary to postulate a matter density parameter much larger than the typical Ω M ≃ . Note added.
After the publication of a preprint of this paper, the problem of structure formation in Rastall’stheory has been studied in [20], where the authors point out the difficulties of finding an agreement between thismodified gravity model and the observational data.
Acknowledgments
The authors warmly thank the attendants of the Journal Club on Extended and Alternative Theories of Gravityfor useful discussions. MC and VFC are supported by University of Torino and Regione Piemonte. Partial supportfrom INFN projects PD51 and NA12 is acknowledged too. [1] Riess, A.G., et al.,
Astron. J. , 1009 (1998)[2] Perlmutter, S., et al.,
Astrophys. J. , 565 (1999)[3] Bennet, C.L., et al.,
Astrophys. J. Suppl. , 1 (2003)[4] Will, C.M.,
Living Rev. Relativity Rev. Mod. Phys. , 559 (2003)[6] Kamionkowski, M., arXiv:0706.2986 [astro-ph] (2007)[7] Kolb, E.W., Matarrese, S., Riotto, A. New J. Phys. , 322 (2006)[8] Rastall, P., Phys. Rev. D , 3357 (1972)[9] Smalley, L. L., Class. Quantum Grav. , 1179 (1993)[10] Lindblom, L., Hiscock, W.A., J. Phys. A , 1827 (1982)[11] Al - Rawaf, A.S., Mod. Phys. Lett. A , 2691 (2008)[12] Freese, K., Lewis, M., Phys. Lett. B , 1 (2002)[13] Fay, S., Amarzguioui, M.,
Astronomy and Astrophysics , 37 (2006)[14] Trautman, A., in
Lectures on General Relativity , edited by Deser, S., and Ford, K.W., Prentice-Hall, Englewood Cliffs,New Jersey (1965)[15] Weinberg, S.,
Gravitation and Cosmology , J. Wiley and Sons, New York (1972)[16] Ferraris, M., Francaviglia, M., Volovich, I.,
Class. Quant. Grav. , 1505 (1994)[17] Kowalski M., et al., Astrophys. J. , 749 (2008)[18] Cardone, V.F., Capozziello, S., Dainotti, M.G., arXiv:0901.3194 [astro-ph.CO] (2009)[19] Freedman, W.L. et al.,
Astrophys. J.553