Mimicking general relativity in the solar system
aa r X i v : . [ g r- q c ] O c t Mimi king general relativity in the solar systemL. Amendola,1 C. Charmousis,2 and S. C. Davis31INAF/Osservatorio Astronomi o di Roma, Viale Fras ati 33, 00040 Monteporzio Catone (Roma), Italy2LPT, Universitï¾½ Paris(cid:21)Sud, Bï¾½timent 210, 91405 Orsay CEDEX, Fran e3Servi e de Physique Thï¾½orique, Orme des Merisiers,CEA/Sa lay, 91191 Gif-sur-Yvette Cedex, Fran eIn order for a modi(cid:28)ed gravity model to be a andidate for osmologi al dark energy it has topass stringent lo al gravity experiments. We (cid:28)nd that a Brans-Di ke (BD) theory with well-de(cid:28)nedse ond order orre tions that in lude the Gauss-Bonnet term possess this feature. We onstru t thegeneri se ond order theory that gives, to linear order, a BD metri solution for a point-like masssour e. We (cid:28)nd that the Eddington parameter γ , heavily onstrained by time delay experiments, an be arbitrarily lose to the GR value of 1, with an arbitrary BD parameter ω BD . We (cid:28)nd theregion where the solution is stable to small timelike perturbations.Brans-Di ke (BD) theory is a simple modi(cid:28) ation ofgeneral relativity (author?) [1℄ (see also (author?) [2℄for generalisations) as it is a single massless s alar-tensortheory whose only parameter is the kineti oupling term ω BD , S BD = Z √− g h Φ R − ω BD Φ ( ∇ Φ) i − π L matter . (1)Its GR limit is obtained for ω BD → ∞ . BD gravitybreaks the strong equivalen e prin iple and yields at lo als ales di(cid:27)ering Eddington parameters β and γ to thoseof GR, whi h are stri tly equal to 1. In parti ular theparameter γ , whi h measures how mu h spatial urva-ture is produ ed by unit rest mass (see (author?) [3℄),is given by γ = (1 + ω BD ) / (2 + ω BD ) . It is strongly on-strained by time delay experiments, su h as the one on-du ted with the Cassini spa e raft, whi h re ently gave | γ − | . − (author?) [4℄ (for a re ent review and al-ternative methods to measure γ see (author?) [3℄). Thisimplies ω BD > , therefore the s alar se tor is veryweakly oupled.On the other hand modi(cid:28) ation of general relativity ispossible or even needed in order to explain e(cid:27)e ts on osmologi al and gala ti s ales, or at s ales just be-yond the solar system (su h as the Pioneer anomaly (au-thor?) [5℄). At osmologi al s ales, some times big-ger than solar system s ales, supernovae data (author?)[7℄ entertain the possibility that GR may be modi(cid:28)edat large distan es (author?) [6℄. Modi(cid:28) ation of GR isalso being envisaged at gala ti s ales in order to explaindeviations from standard Newtonian gravity in gala -ti rotation urves, as in MOND or Bekenstein-Sanderstheor (author?) [8℄. S alars have been quite naturallyintrodu ed in order to mediate gravity modi(cid:28) ation oreven as sour es of osmologi al dark energy (author?)[9℄. These modi(cid:28) ations are well into the lassi al, in-frared se tor of gravity at very low energies, very far fromthe the Plan k s ale UV se tor, where quantum gravitybe omes important. Even from the point of view of UVmodi(cid:28) ations, string theory predi ts a zoo of s alars that,if massless, would give BD-type phenomenology in the so- lar system. It is therefore quite fair to say that there isin reasing tension between gravitational onstraints im-posed experimentally for weak gravity in the solar systemand laboratory tests, as well as strong gravity from bi-nary pulsars (see for example (author?) [10℄), and onthe other hand theories of modi(cid:28)ed gravity or dark en-ergy that aim to explain unexpe ted out omes of novelexperimental data. The aim of this letter is to showthat well motivated se ond order orre tions to simpleBD theory (with no potential) an mimi GR at the so-lar system s ale, in the sense of giving γ = 1 indepen-dently of the parameter ω BD . This does not mean thatthe toy s alar-tensor theory in question and GR wouldnot be distinguishable, quite the ontrary, on osmologi- al s ales their phenomenology would be totally di(cid:27)erentand even at the solar system level one ould dete t somee(cid:27)e t, most probably by arrying out an experiment tomeasure β , as we will dis uss in the on luding remarks.Our starting point is the general (modulo (cid:28)eld rede(cid:28)ni-tions) s alar-tensor Lagrangian of se ond order in powersof the urvature tensor whi h has the unique property ofgiving se ond order (cid:28)eld equations, L = √− g [ f R − f ( ∇ φ ) + ξ L GB + ξ G µν ∇ µ φ ∇ ν φ + ξ ( ∇ φ ) ∇ φ + ξ ( ∇ φ ) − V ] − πG L mattter . (2)The theory (2) is parametrised by the potential V and ouplings f i , ξ i whi h are all fun tions of the s alar (cid:28)eld φ . The Gauss-Bonnet term is L GB = R µνσρ R µνσρ − R µν R µν + R and is a topologi al invariant in 4 dimen-sions for GR but not for s alar-tensor (2). Theories ofthe above form (2) have been proposed as a solution tothe dark energy problem (author?) [11℄. For the aseof f = 1 , it has been shown that solar system data animpose severe restri tions on the ouplings ξ i (author?)[12℄, whi h allows the range of possible gravity modi(cid:28) a-tion to be narrowed down. Throughout this arti le wework in units with c = 1 . We will assume ξ = ξ = 0 inthe following as a ompromise between introdu ing newfree oupling fun tions in the theory and generality ofthe setup. We hoose to keep the non-minimal intera -tion terms between the graviton and the s alar ξ and ξ rather than higher order s alar orre tions. When sear h-ing for solutions we will assume also a vanishing potential V (as in BD) sin e invoking a large mass is an obviousbut totally ad ho way to evade solar system onstraintsand would suppress observable e(cid:27)e ts at all s ales. Ourwork also ontrasts with ` hameleon' models (author?)[13℄, in whi h the s alar's gravity is suppressed at lo al(but not osmologi al) s ales by a suitably hosen poten-tial that yields a ba kground dependent mass in eq. (2).The higher order orre tions are hosen so that the grav-itational propagator does not pi k up extra degrees offreedom. Therefore the resulting (cid:28)eld equations will bea-priori ghost-free around the Minkowski va uum, se ondorder in the derivatives, with well-de(cid:28)ned Dira distribu-tional terms.In order to derive the post-Newtonian equations, wenow assume a stati , spheri ally symmetri metri inisotropi oordinates, ds = − (1 − U ) dt + (1 + 2Υ) δ ij dx i dx j + O ( ǫ / ) , (3)where U is the Newtonian potential and Υ is the lead-ing post-Newtonian spatial ontribution. They are bothfun tions of the radial o-ordinate r only and are assumedto be of the order of the smallness dimensionless param-eter ǫ = Gm ⊙ /r where m ⊙ is the solar mass and r is a hara teristi length s ale of the problem. For the solarsystem ǫ . − for r greater than the sun's radius. Thepost-Newtonian parameter (PPN) we will be al ulatingis Eddington's parameter de(cid:28)ned as γ = Υ /U . Matterenergy density ρ m is given by the mass of the sun and isas usual assumed to be a distributional sour e at r = 0 : ρ m = m ⊙ δ (3) ( x ) . This is an ex ellent assumption giventhat the S hwarzs hild radius of the sun is of the orderof 3 km ompared to s ales of the order of astronomi alunits. Note however, that higher order terms in ǫ have tobe in luded in (3) in order to al ulate β (for the advan eof Mer ury's perihelion for example). For the relativisti experiment we will onsider here, namely time delay, ourexpansion is ne essary and su(cid:30) ient. Let us now de(cid:28)nethe operators, ∆ F = X i F ,ii , D ( X, Y ) = X i,j X ,ij Y ,ij − ∆ X ∆ Y , (4)whi h will be the te hni al tool essential for ouranalysis. For fun tions with only r -dependen e theyredu e to ∆ F = r − ∂ r ( r ∂ r F ) and D ( X, Y ) = − r − ∂ r ( r∂ r X∂ r Y ) and in parti ular, D ( r − n , r − m ) = 2 nmn + m + 2 ∆ r − ( n + m +2) , ∆ r − n = n ( n − r n +2 − πnδ (3) ( x ) r n − , (5)and thus we an easily evaluate the relevant distribu-tional parts asso iated with D . We do not make any assumptions about the relative sizes of f i , ξ i , V , or theirderivatives (sin e we expe t the higher order terms toplay a signi(cid:28) ant role), and instead in lude the leadingorder in ǫ ontribution from ea h term in the (cid:28)eld equa-tions f ∆ U = − πG ρ m + V + ∆ f − D ( U + Υ , ξ )+ O ( ǫ , f , ǫV, ǫξ , ǫ ξ ) (6) f ∆Υ = − πG ρ m − V − ∆ f − D (Υ , ξ ) + D ( φ, ζ )4+ O ( ǫ , ǫV, f ǫξ , ǫ ξ ) (7)where we have de(cid:28)ned ξ = ∂ φ ζ and f = ∂ φ h . We willalso expand f to (cid:28)rst order in ǫ , f = Φ + O ( ǫ ) . Thes alar (cid:28)eld equation on the other hand is globally of oneorder higher and gives to leading order ∂ φ h ∆ φ + ∆ h = 2 ∂ φ V + 2 ∂ φ f ∆(2Υ − U ) − ∂ φ ξ D ( U, Υ) + D ( U − Υ , ζ )+ ∂ φ ζ D ( U − Υ , φ ) + O ( ǫ , ǫV, f ǫ, ǫ ξ , ǫ ξ ) . (8)For omparison, BD theory ( V, ξ i ≡ ) has f = Φ ≡ Φ + φ , f ≡ ω BD (Φ)Φ ≈ ω BD Φ + O ( φ ) (9)Sin e we are expanding the equations to the lowest non-trivial order, we annot estimate the se ond PPN param-eter β , whi h requires higher order terms in the metri oe(cid:30) ients. This loss in generality in the metri oe(cid:30)- ients is ompensated by the great generality of our so-lution.It is useful to de(cid:28)ne η ≡ − γ whereupon the various onstants in the model are related by ω BD = − /η and G = 2 G / [(2 − η )Φ ] . We see that η = 0 givesexa tly GR. We are interested in gravitational theoriesemanating from (2), whi h while not identi al to generalrelativity, give almost identi al predi tions for the weak(cid:28)eld of the solar system. One approa h to this problemwould be to solve the (cid:28)eld equations of the previous se -tion for a range of oupling fun tions ξ i , f i , and then ompare the resulting potentials U and Υ , with those ofEinstein gravity. We will not take this approa h sin e wehave no interest in the solutions of the (cid:28)eld equations, ex- ept for the spe ial ases where they give (to this order in ǫ ) pre isely the Newtonian result U ≈ Υ ≈ Gm ⊙ /r . Thisin parti ular gives us agreement with tests of Newton'slaw from planetary orbits. Therefore, instead of tryingto (cid:28)nd the metri (3) whi h solves the (cid:28)eld equations forgiven ξ i , f i , we will inversely start by assuming the de-sired Newtonian form of U and Υ , and then view (6)(cid:21)(8)as equations for the oupling fun tions f i , ξ i parametris-ing the theory (2). As dis ussed above, we will now set V = 0 , just as in the standard BD model and we allowthe PPN parameter γ = 1 − η to take any value. Hen ewe take U = Gm ⊙ r , Υ = (1 − η ) Gm ⊙ r . (10)Note also that the e(cid:27)e tive gravitational oupling G neednot be equal to the fundamental parameter for the grav-itational oupling of matter G . The general solution of(6)(cid:21)(8) is then f = Φ + Gm ⊙ (cid:18) Φ η + λr + 2 − η Z ∂ r Sr dr (cid:19) ,r ( ∂ r φ ) f = ( Gm ⊙ ) [Φ η + λ − + λ ) η − − η + 2 η ) S − η (3 − η ) r∂ r S ] ,ξ = − ηλr Z r ∂ r S dr , ( ∂ r φ ) ξ = Gm ⊙ [2 λr (1 − η ) − − η ∂ r S ] , (11)where λ is an arbitrary, dimensionful onstant, obtainedby the distributional part appearing in the equations ofmotion (6)(cid:21)(8) as the boundary ondition at r = 0 . Onthe other hand, S ( r ) is an arbitrary fun tion with the reg-ularity ondition rS ′ = 6 S as r → , i.e. S = r∂ r S = 0 at r = 0 . Viewing the above expressions as the solution ofordinary inhomogeneous di(cid:27)erential equations for f i , ξ i ,the fun tion S ( r ) parametrises the general homogeneoussolution of (6)(cid:21)(8) whereas λ parametrises the parti ularsolution. The integrals range from ∞ to r . The gravita-tional oupling satis(cid:28)es G G = Φ h − η i − λ (cid:2) − η + 2 η (cid:3) . (12)The above equations fully spe ify the ouplings neededto reprodu e a PPN parameter γ and an exa tly New-tonian /r gravitational potential. In fa t λ and S nowparametrise the theory (2). Setting S = λ = 0 gives uspure BD (9) with φ = Gm ⊙ Φ η/r . Setting on top ofthat η = 0 gives GR. The key point however is that ifwe set η = 0 keeping S and λ non-zero we have the samepost-Newtonian limit as standard GR, i.e. γ = 1 in (3)without the theory a tually being GR. Indeed note thatthe orresponding kineti oupling f an take arbitraryvalues parametrised by λ and S . Indeed when the higher urvature terms are in luded, the Newtonian potential isstill proportional to /r . For non-trivial S , this is be- ause all the orre tions to standard gravity an el out,making the gravity modi(cid:28) ations `invisible' to this order.On the other hand, if λ is non-zero, the orre tions do not an el, but instead `mimi ' Newtonian gravity. This anbe seen from the fa t that the e(cid:27)e tive gravitational ou-pling G re eives a λ dependent orre tion (12). A similare(cid:27)e t was found for f ( L GB ) gravity in (author?) [14℄,although the resulting γ was too large.This fa t is made learer when we note that the abovesolution (11) does not give a spe i(cid:28) form for φ . This isnatural, sin e by a hange of variables, φ an be made to take any desired form. Sin e we wish to express thefun tions in terms of φ , let us take φ = φ r g r . (13)The onstant φ simply orresponds to a re-s aling of φ .De(cid:28)ning r g = Gm ⊙ , the expressions (11) then give f = Φ + (Φ η + λ ) φφ + 2 − η φ Z φ∂ φ S dφ ,f = 1 φ [Φ η + λ − + λ ) η − − η + 2 η ) S + η (3 − η ) φ∂ φ S ] ,ξ = r g φ (cid:20) − ηλφ + Z ∂ φ S φ dφ (cid:21) ,ξ = r g φ (cid:20) λφ (1 − η ) + 3 − η φ ∂ φ S (cid:21) , (14)with the regularity onditions at r = 0 now implying S and φ∂ φ S tending to zero as φ → ∞ . S an thenbe expanded as S = P n ≥ c n φ − n , obtaining the gen-eral asymptoti solution to all orders in φ . Note alsothat the higher order ouplings are r g dependent whi hfollows from the fa t that they are of dimension lengthsquared. In parti ular this means that if we introdu e theGauss-Bonnet oupling onstant α then it is related viaa multipli ative number to the only length s ale of theproblem r g , namely, α = − φ ηλr g / . In fa t the multi-pli ative onstant is the hierar hy generated between the lassi al s ale r g and √ α .To illustrate our result we onsider the simplest aseof S = 0 , and take | η | < − to agree with solarsystem onstraints. Without loss of generality we set φ = λ + Φ η . We see that even with η = 0 we have a BD Lagrangian (2) with the additional term ξ G µν ∇ µ φ ∇ ν φ whi h an reprodu e general relativity upto the (cid:28)rst post-Newtonian parameter γ . Therefore (cid:28)nd-ing γ = 1 does not guarantee the absen e of a s alarintera tion even in this simplest of ases sin e the s alar oupling ω BD is still freely given by Φ /λ . Note also thatthe strength of the s alar intera tion ω BD is inversely pro-portional to the strength of the higher order orre tions,as parametrised by λ .We will now examine the stability of the solution (14)with respe t to time-dependent perturbations. We take U → U + δU , et . and keep the leading order time deriva-tives of δU (up to O ( δU ) , for linear gravity terms, and O ( ǫδU ) for quadrati terms). For simpli ity we will re-stri t ourselves to the extreme ase of solution (14) with η = 0 and S ≡ . The orresponding BD-like param-eter for the higher order theory is ω BD = Φ /λ . Theperturbation equations then redu e to ω BD ¨ δ Υ −
32 ¨ δφφ = − ω BD ∆ δU + 12 ∆ δφφ (15) − ω BD ∆ δ Υ + 12 ∆ δφφ = − r ∂ r (cid:18) ∂ r ( r δφ ) rφ (cid:19) (16) − δ Υ − ¨ δφφ = ∆ δU − ∆ δφφ + 2 r∂ r (cid:18) ∂ r ( δU − δ Υ) r (cid:19) (17)where we have assumed that the perturbations are moreregular than the leading order solution /r as r → .If (14) is to be a viable gravity model, there needs tobe a reasonable range of parameters for whi h δU , et .os illate, rather than growing over time. Substituting δU ( t, −→ r ) = δU ( t ) exp i −→ k −→· r and analogously for δφ, δ Υ ,we (cid:28)nd ¨ δφ = − ω BD + 32 ω BD + 9 − k r ( kr − k r +4 kr − k δφ → ω BD + 39 − ω BD k δφ (18)(the limit is for r → ∞ ). In the same limit, the time-dependen e of δU, δ Υ is the same as for δφ .If ω BD > / or ω BD < − / , the perturbations willos illate for large r , rather than grow exponentially, in-di ating that our gravitational solution mimi king GR is lassi ally stable for a reasonable range of parameters,at least at (cid:28)rst order. Although it might be that in- luding higher order terms in the perturbation expan-sion the os illations turn out to be of a growing na-ture, one has to be reminded that the os illations willbe naturally damped by the emission of gravitationalwaves and by the ba kground osmologi al expansion.A further onstraint omes from requiring positive grav-itational oupling. For the above ase, (12) redu es to G = ( G / Φ )2 ω BD / (2 ω BD − . Hen e G > implies ω BD < or ω BD > / , whi h are already overed by theabove ranges. We expe t qualitatively similar results formore general (14) with η or S ( φ ) non-zero, although aproof of this is beyond the s ope of this paper. Let usremember that S ( r ) is anyway an arbitrary fun tion andone an always hoose it a posteriori in su h a way tomaintain stability.In this paper we exhibited a sensible se ond order (inpowers of the urvature tensor not derivative) s alar-tensor theory whi h shares some hara teristi s of ordi-nary BD or GR, in parti ular, well-de(cid:28)ned se ond order(cid:28)eld equations, distributional boundary onditions andwell de(cid:28)ned stable va ua. We found that su h a theory,given the right oupling fun tions, an mimi a GR Ed-dington parameter γ exa tly equal to with virtually no onstraint on the kineti oupling ω BD (in ordinary BDtheory a tual measurements of γ give ω BD > ). Inthis sense we saw that the in lusion of higher order op-erators in the a tion an mimi GR with a s alar-tensortheory. We do not view the solutions we have found (14)or even the model in question (2), as some fundamen-tal s alar-tensor theory; our aim was rather to see howrobust were the solar system predi tions to higher or-der orre tions. Our on lusion is that ertain solar sys-tem onstraints known to rule out theories su h as BD are not as robust in their GR predi tion as one mightthink. In fa t similar results have been shown for er-tain ve tor-tensor theories (author?) [15℄ although oneexpe ts loser agreement with the Eddington parametersin the ase of ve tors rather than s alars.This does not mean that one annot distinguish be-tween su h higher order s alar tensor theories and GR.For a start the se ond PPN parameter β may not beunity for su h theories, although if we allow for the re-maining higher order operators ξ , ξ in (2), in prin iplewe have the mathemati al (cid:29)exibility in the equations toagain (cid:28)x β = 1 by solving for the oupling fun tions. Themain di(cid:27)eren e between GR, BD and these higher ordertheories is that the oupling fun tions are dimensionful.Thus the relevant solutions su h as (14) will depend onthe length s ale of the solution, namely r g , times some di-mensionless number whose magnitude will determine the(cid:16)(cid:28)ne-tuning(cid:17) one has to impose between the length s alesof the theory and the solar system. In other words wewould view the experimental error bars as hierar hies be-tween the higher order ouplings and lo al s ales wherethe experiment is arried out. This relation may alsobe relaxed by allowing for the general se ond order the-ory at the expense of introdu ing further free parametersin the theory (2). We further note that we have on-stru ted gravitational theories whi h exa tly reprodu ethe Newtonian potential for the sun: U = r g /r . In fa tit is perfe tly a eptable to have U − r g /r non-zero, butsmaller than the experimental bounds from planetary or-bits. The above issues as well as a al ulation of ββ