Constraining Newtonian stellar configurations in f(R) theories of gravity
aa r X i v : . [ a s t r o - ph ] O c t Constraining Newtonian stellar configurations in f ( R ) theories of gravity T. Multam¨aki ∗ and I. Vilja † Department of Physics, University of Turku, FIN-20014 Turku, FINLAND (Dated:)We consider general metric f ( R ) theories of gravity by solving the field equations in the presence ofa spherical static mass distribution by analytical perturbative means. Expanding the field equationssystematically in O ( G ), we solve the resulting set of equations and show that f ( R ) theories whichattempt to solve the dark energy problem very generally lead to γ PPN = 1 / γ PPN and show that it cannot have a significant effect.
I. INTRODUCTION
The dark energy problem remains central in modernday cosmology. Since the matter only, homogeneous uni-verse within the framework of general relativity is in con-flict with cosmological observations, the assumptions be-hind this model have been questioned. The most popularmodification is to consider a universe filled with other,more exotic forms of matter, the cosmological constantbeing the leading natural candidate. Other ways to tacklethe dark energy problem are then to relax the assumptionof homogeneity or modify the theory of gravity.In recent years, a particular modification of gravity, the f ( R ) gravity models that replace the Einstein-Hilbert ac-tion of general relativity (GR) with an arbitrary functionof the curvature scalar (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9]and references therein) have been extensively studied.Naive modification of the gravitational action is not with-out challenges, however, and obstacles including cosmo-logical constraints (see e.g. [38, 39, 40] and referencestherein), instabilities [11, 12, 13], solar system constraints(see e.g. [14, 15, 16, 21] and references therein) and evo-lution large scale perturbations [17, 18, 19] need to beovercome. In addition, a number of consistency require-ments need to be satisfied (see e.g. [20, 22] and referencestherein).One of the most direct and strictest constraints on anymodification of gravity comes from observations of outnearby space-time i.e. the solar system. This is oftendone by conformally transforming the theory to a scalar-tensor theory and then considering the ParameterizedPost-Newtonian (PPN) limit [23, 24] (see also [25, 26] fora discussion). The question of validity of the solar systemconstraints f ( R ) theories has been extensively discussedin the literature and not completely without controversy.The opinions on the viability of f ( R ) theories have beendivided from more or less skeptical [27, 28, 29, 30] toapproving [31, 32] depending on the point of view of theauthor.The essence of the discussion has been the question ∗ Electronic address: tuomul@utu.fi † Electronic address: vilja@utu.fi of validity of the Schwarzschild-de Sitter (SdS) metric asthe correct metric in the solar system. The SdS metricis a vacuum solution to a large class of f ( R ) theories ofgravity but due to the higher-derivative nature of metric f ( R ) theories, it is not unique. Other solutions can alsobe constructed in empty space, in the presence of matterand in a cosmological setting (see eg. [10, 33, 34]).In light of recent literature [27, 28, 29], the validity ofthe solar system constraints has become clear and it isnow understood that the equivalent scalar-tensor theoryresults are valid in a particular limit that corresponds tothe limit of light effective scalar in the In terms of the f ( R ) theory, this is equivalent to requiring that one canapproximate the trace of the field equations by Laplace’sequation [28]. As a result, the often considered R − µ /R theory [1] (the CDTT model) is not consistent with theSolar System constraints in this limit, if the 1 /R term isto drive late time cosmological acceleration.In [27] the CDTT model was considered by lineariz-ing around a static de Sitter spacetime and solving thetrace equation in terms of R ( r ), resulting in a spacetimeoutside the star where γ = 1 /
2. This result was thengeneralized for a general f ( R ) theory in [28] by study-ing the space-time outside a spherical mass distributionand expanding f ( R ) in terms of a perturbation in R .Again solving the trace equation leads to an outside so-lution with γ = 1 / F ( r ) ≡ df /dR in the perturbative expansion. Solving thetrace equation then leads to γ = 1 / F along with the metric as independent functions.By expanding all quantities in G and solving the result-ing equations inside and outside the star for a general f ( R ) theory, we find that generally, γ P P N = 1 / O ( G )outside the star and the scalar curvature is O ( G ) ev-erywhere. We also idenfity the first order correction to γ P P N and show that it cannot have a significant effect.Only if initial conditions inside the star are fine-tunedsuch that the scalar curvature follows the matter densitylike in GR [21, 35] can these bounds be evaded.
II. f ( R ) GRAVITY FORMALISM
The action for f ( R ) gravity is ( c = 1) S = Z d x √− g (cid:16) πG f ( R ) + L m (cid:17) . (1)The field equations resulting in the so-called metric ap-proach are reached by variating with respect to g µν : F ( R ) R µν − f ( R ) g µν −∇ µ ∇ ν F ( R )+ g µν (cid:3) F ( R ) = 8 πGT mµν , (2)where T mµν is the standard minimally coupled stress-energy tensor and F ( R ) ≡ df /dR . Contracting the field equations and assuming that wecan describe the stress-energy tensor with a perfect fluid,we get F ( R ) R − f ( R ) + 3 (cid:3) F ( R ) = 8 πG ( ρ − p ) . (3)In this letter we consider spherically symmetric staticfluid configurations and adopt a metric, which reads inspherically symmetric coordinates as ds = B ( r ) dt − A ( r ) dr − r dθ − r sin θdϕ . (4)By taking suitable linear combinations of the field equa-tions they can be written in the following form: F A ′ r A + F B ′ r B + A ′ F ′ A + B ′ F ′ B − F ′′ = 8 G π A ( ρ + p ) (5) − Fr + A Fr + F A ′ r A + F B ′ r B − F A ′ B ′ A B − F B ′ B − F ′ r + B ′ F ′ B + F B ′′ B = 8 G π A ( ρ + p ) (6) A (2 f ( R ) − R F ( R )) + 6 F ′ r − A ′ F ′ A + 3 B ′ F ′ B + 3 F ′′ = − G π A ( ρ − p ) , (7)where prime indicates a derivation with respect to r , ′ ≡ d/dr and we have written f and F as functions of the ra-dial coordinate r expect in combination 2 f ( R ) − R F ( R ),which we will expand in terms of curvature R .The corresponding equation of continuity is p ′ ( r ) ρ ( r ) + p ( r ) = − B ′ ( r ) B ( r ) . (8)When pressure is negligible, it is easy to see that B mustbe a constant. This is, however, not acceptable and there-fore an adequate perturbation expansion is needed. III. PERTURBATIVE EXPANSION AND ITSSOLUTIONS
We expand the metric as well as F with G as an ex-pansion parameter: A ( r ) = 1 + G A ( r ) + O ( G ) ,B ( r ) = B + G B ( r ) + O ( G ) ,F ( r ) = F + G F ( r ) + O ( G ) ,p ( r ) = p + G p ( r ) + O ( G ) . Note, that we consider the density profile ρ ( r ) to be afixed function and also that B and F are constants.From the expansion of A and B , one can also read outan expansion for R : R = R + GR + O ( G ) . (9) From the equation of continuity we see that at O ( G )pressure is constant and exactly zero, p = 0, simplybecause it vanishes in empty space. Therefore pressureeffects are always O ( G ) and do not contribute to the O ( G ) expansion.The 2 f − F R term in the third field equations is cru-cial in determining the behaviour of the solution. In gen-eral, for a f ( R ) dark energy model, this term is neg-ligible and can be omitted, at least in the first orderapproximation. This is demonstrated explicitly for theCDTT in model and discussed more generally in [35],where it is argued that the non-linear term is completelynegligible, barring fine tuning. This argument is easilyunderstandable in a general model since in the vacuum2 f − F R ∼ Gρ DE ≪ Gρ for any stellar matter configu-ration. Note that this will in general be true also outsidea stellar configuration as the dark matter will completelydominate over the cosmological term. Hence, in the traceequation, the non-linear terms can be dropped, unlessthe initial conditions are fine-tuned. We will return tothe fine-tuned condition, or the Palatini limit [21, 35],later.More formally, the same conclusion can be confirmedby using an expansion in G for the non linear-terms aswell: 2 f ( R ) − F ( R ) R = 2 f ( R ) − F ( R ) R + ( F ( R ) − F ′ ( R ) R ) R + O ( G ) Evidently, the expansion point R has to be such, that it corresponds to the correct back-ground of the theory, i.e. f ( R ) − F ( R ) R = 0. Thenexpanding up to first order in G , the field equations are F A ′ r + F B ′ B r − F ′′ = 8 π ρ,F A r − F A ′ r + F B ′ B r − F ′ r + F B ′′ B = 8 π ρ, (10) I R + 6 F ′ r + 3 F ′′ = − π ρ, where I = F ( R ) − F ′ ( R ) R is a constant and F = F ( R ).The set of equations (10) can be straightforwardlysolved leading to O ( G ) functions: F ( r ) = F − G Z r m ( r ) r dr (11) A ( r ) = 1 + 4 G F m ( r ) r (12) B ( r ) = B (1 + 8 G F Z r m ( r ) r ) dr, (13)where m ( r ) ≡ Z r πr ρ dr. (14)Inserting this solution back to expression of the curvaturescalar, we find that R = 0, i.e. , R is O ( G ). It is crucialthat in deriving the solution (11), we have assumed that A, B, F are regular at the origin.The PPN-parameter is now straightforwardly calcula-ble: γ P P N = 12 (1 − r m ′ m )+ 2 G F (cid:0) R r mr dr + m ′ (cid:1) ( m − r m ′ ) m . (15)It is easy to see that, at the boundary of the star γ P P N → / O ( G ). This behaviour was also observedin numerical studies [35, 36]. From the first order cor-rection one can furthermore conclude that if one wishescorrections to be effective at zeroth order, F needs to beof order G . However, looking at the continuity equation,Eq. (8), we find that − r p ′ = 43 GF ρ m ( r ) + O ( G ) . (16)Comparing this with the Newtonian result, − r p ′ =4 Gρm ( r ), we see that if F ∼ O ( G ), the effective New-ton’s constant is orders of magnitude larger than the onein Newton’s theory (or GR), resulting in stars with acompletely different mass to radius relationship than theone observed. Furthermore, from the continuity equa-tion, we can read that unless F ≈ / f − F R ∼ R , i.e. when f ( R ) = R + c R . Because in the approximation described above, R ∼ O ( G ), it is easy to see that that this term will playno role in the trace equation. Similarly for higher orderterms in R . One can avoid the constraint only if F hasno G order correction. In this case, the (cid:3) F , term is neg-ligible in trace equation and we recover the GR results,or the Palatini limit [21, 35]. Alternatively, if one relaxesthe regularity constraint of the metric at the origin, onecan also avoid the constraint as demonstrated in [36] forthe CDTT model. A. Recovering the general relativity
In the Palatini limit, where the trace-equation is sim-ilar than in the Palatini formalism, the theory is fine-tuned so that 2 f − F R ≈ R ≈ − πGρ throughout (see[35] for a numerical example). This is the mechanismthat allows one to construct solutions that are consistentwith solar system observations [21]. In the Palatini limit,the field equations read as F ( r ) ≃ A ( r ) ≃ Gm ( r ) r (18) B ( r ) ≃ B (1 + 2 G Z r m ( r ) r ) dr. (19)The γ P P N parameter is easily calculable: γ P P N ≃ − r m ′ m + O ( G ) . (20)Therefore, in this limit γ P P N → IV. DISCUSSION AND CONCLUSIONS
In this letter we have considered a general metric f ( R ) theory in the presence of matter by analyzing thefield equations by perturbative means in linear order inthe Newton’s constant G . We have shown explicitlythat for a typical star, any modification of gravity fromGR will naturally lead to physically unacceptable value γ P P N = 1 /
2. This places a very strong constraint on any f ( R ) theory, in particular when acting as a dark energycandidate. Furthermore, even if the gravity theory is notmotivated by cosmology, but by other arguments, suchas quantum gravity, the presence of non-linear terms canstill lead to a space-time inconsistent with observations.In this order of perturbation theory we can recoverthe observationally acceptable space-time, only when F = df /dR has no order G correction. Such a constraintindicates fine-tuning in the initial values of the solutionso that one remains in the high curvature limit, R ∼ Gρ throughout. However, the stability of such a fine-tunedsolution may be problematic [35], although possible toobtain [21, 37, 38].Since our analysis is of order O ( G ), further study onthe system, in particular second order perturbations in G , may affect the conclusions. Indeed, our analysis showsthat the first order perturbation theory is essentially in-dependent on the details of the underlying f ( R ) theory.The only piece of information used was the knowledgethat there are higher order derivatives in the equationsof motion, i.e. that the theory is not GR. New effectsmay appear in higher order perturbation theory, wherefinally the dependence on the functional form of f ( R ) should become evident. However, our results suggest thatunless the solution is fine-tuned so that R ∼ Gρ through-out the mass distribution, a naive modification where asmall correction is added to the Einstein-Hilbert actionto solve the dark energy problem is not likely to pass thesolar system constraints. Acknowledgments
TM is supported by the Academy of Finland. [1] S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner,Phys. Rev. D , 043528 (2004).[2] S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson,M. Trodden and M. S. Turner, Phys. Rev. D , 063513(2005).[3] G. Allemandi, A. Borowiec and M. Francaviglia, Phys.Rev. D , 103503 (2004).[4] X. Meng and P. Wang, Class. Quant. Grav. , 951(2004).[5] S. Nojiri and S. D. Odintsov, Phys. Rev. D , 123512(2003).[6] S. Capozziello, Int. J. Mod. Phys. D , 483 (2002).[7] S. Nojiri and S. D. Odintsov, Phys. Lett. B , 5 (2003).[8] R. P. Woodard, arXiv:astro-ph/0601672.[9] For dicussion of other generalized theories, see e.g. :S. Nojiri and S. D. Odintsov, arXiv:hep-th/0601213.[10] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsovand S. Zerbini, JCAP , 010 (2005)[arXiv:hep-th/0501096].[11] A. D. Dolgov and M. Kawasaki, Phys. Lett. B , 1(2003).[12] M. E. Soussa and R. P. Woodard, Gen. Rel. Grav. ,855 (2004) [arXiv:astro-ph/0308114].[13] V. Faraoni, Phys. Rev. D , 124005 (2005)[arXiv:gr-qc/0511094].[14] T. Chiba, Phys. Lett. B , 1 (2003).[15] E. E. Flanagan, Class. Quant. Grav. , 417 (2003);Class. Quant. Grav. , 3817 (2004); G. Magnano andL. M. Sokolowski, Phys. Rev. D , 5039 (1994).[16] T. Clifton and J. D. Barrow, Phys. Rev. D , 103005(2005) [arXiv:gr-qc/0509059].[17] R. Bean, D. Bernat, L. Pogosian, A. Silvestri andM. Trodden, arXiv:astro-ph/0611321.[18] Y. S. Song, W. Hu and I. Sawicki,arXiv:astro-ph/0610532.[19] Y. S. Song, H. Peiris and W. Hu, arXiv:0706.2399 [astro-ph].[20] I. Sawicki and W. Hu, Phys. Rev. D , 127502 (2007)[arXiv:astro-ph/0702278]. [21] W. Hu and I. Sawicki, arXiv:0705.1158 [astro-ph].[22] S. A. Appleby and R. A. Battye, arXiv:0705.3199 [astro-ph].[23] T. Damour and G. Esposito-Farese, Class. Quant. Grav. , 2093 (1992).[24] G. Magnano and L. M. Sokolowski, Phys. Rev. D ,5039 (1994) [arXiv:gr-qc/9312008].[25] G. J. Olmo, Phys. Rev. Lett. , 261102 (2005)[arXiv:gr-qc/0505101].[26] S. Capozziello, A. Stabile and A. Troisi,arXiv:gr-qc/0603071; G. Allemandi, M. Francav-iglia, M. L. Ruggiero and A. Tartaglia, Gen. Rel. Grav. , 1891 (2005) [arXiv:gr-qc/0506123]; T. P. Sotiriou,arXiv:gr-qc/0507027.[27] A. L. Erickcek, T. L. Smith and M. Kamionkowski,arXiv:astro-ph/0610483.[28] T. Chiba, T. L. Smith and A. L. Erickcek,arXiv:astro-ph/0611867.[29] X. H. Jin, D. J. Liu and X. Z. Li, arXiv:astro-ph/0610854.[30] T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao,arXiv:astro-ph/0612569.[31] V. Faraoni, Phys. Rev. D , 023529 (2006)[arXiv:gr-qc/0607016].[32] G. J. Olmo, Phys. Rev. D , 023511 (2007)[arXiv:gr-qc/0612047].[33] T. Multamaki and I. Vilja, Phys. Rev. D , 024018(2006) [arXiv:astro-ph/0506692].[34] T. Multamaki and I. Vilja, Phys. Rev. D , 064022(2006) [arXiv:astro-ph/0606373].[35] K. Kainulainen, J. Piilonen, V. Reijonen and D. Sunhede,Phys. Rev. D76