Mon.Not.R.Astron.Soc.000,1–15(2004) Printed6February2008 (MNLATEXstylefilev2.2) Inhomogeneous Gravity Timothy Clifton1⋆, David F. Mota2† and John D. Barrow1‡ 1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 2Astrophysics, Department of Physics, Universityof Oxford, Keble Road, Oxford, OX1 3RH, UK 5 0 6February2008 0 2 n ABSTRACT a We study the inhomogeneous cosmological evolution of the Newtonian gravitational J ’constant’G inthe frameworkofscalar-tensortheories.We investigatethe differences 9 that arise between the evolution of G in the background universes and in local inho- 1 mogeneitiesthathaveseparatedoutfromthe globalexpansion.Exactinhomogeneous solutions are found which describe the effects of masses embedded in an expanding 2 FRW Brans-Dicke universe. These are used to discuss possible spatial variations of v 1 G in different regions. We develop the technique of matching different scalar-tensor 0 cosmologies of different spatial curvature at a boundary. This provides a model for 0 the linear and non-linear evolution of spherical overdensities and inhomogeneities in 6 G.This allowsus tocomparethe evolutionofGandG˙ thatoccursinsidea collapsing 0 overdenseclusterwiththatinthebackgrounduniverse.Wedevelopasimplevirialisa- 4 tioncriterionandapply the method to arealistic lambda-CDMcosmologycontaining 0 sphericaloverdensities.Typically,farslowerevolutionofG˙ willbefoundinthebound / c virialisedclusterthaninthecosmologicalbackground.Weconsiderthebehaviourthat q occurs in Brans-Dicke theory and in some other representative scalar-tensor theories. - r g Key words: Cosmology:theory : v i X r 1 INTRODUCTION carries variations in G, (Bergmann 1968),(Wagoner 1970; a Nordtvedt1970). Many of the most interesting extensions of the general the- Extensivestudieshavebeenmadeofcosmological solu- oryofrelativityarescalar-tensortheoriesofgravity.Theyin- tionsofscalar-tensor gravitytheories (Fuji& Maeda2003), corporatescalarfieldswhichcancarryspace-timevariations although they are limited in two respects. First, they focus inscalarquantitiesthataretraditionallyassumedtobecon- on the simplest case of isotropic expansion with zero spa- stant in general relativity. In this respect they provide nat- tial curvature, where simple exact solutions exist. Second, uralarenas in which toexploretheconsequencesof varying they are exclusively concerned with spatially homogeneous ’constants’ of Nature, like Newton’s ’constant’ G, or Som- cosmologies. The latter restriction means that the value of merfeld’s finestructure’constant’, α.Inthispaperweshall G and its rate of change in time, G˙, are required to be the focus uponthefirst of theseapplications, pioneered byJor- same everywhere in the universe. This assumption has run dan(Jordan1949,1955,1959)andthenrefinedintothemost throughtheentireliteratureonvaryingGandsoitisgener- familiargeneralisation ofEinstein’stheoryofgravitationby ally assumed, for example, that local observational bounds Brans and Dicke in 1961, (Brans & Dicke 1961). This the- onvaryingGderivedfrom geophysics,solarsystemdynam- oryfeaturesbothindirectexplorations ofthepossible vari- ics, stellar evolution, or white dwarf cooling can be applied ation of G, in dimensional reduction of higher-dimensional directly to constrain cosmological variations in G on extra- theories, and in string theories (Green, Schwartz& Witten galactic scales or in the very early universe. There is no 1987), where is appears containing a dilaton with non- justification for this simplifying assumption, as pointed out minimal coupling. The original theory of Brans-Dicke is by Barrow and O’Toole (Barrow & O’Toole 2001). The lo- recognised as the simplest case of a family of scalar-tensor cal bounds on varying G are all derived from observations gravity theories in which the original Brans-Dicke coupling of gravitationally bound ’lumps’ which are in gravitational parameter, ω, becomes a function of the scalar field that equilibriumanddonottakepartintheexpansionoftheuni- verse. Beforetheycanbeextrapolatedtoconstrainpossible variations of G in a background Friedmann universe (as is ⋆ E-mail:[email protected] † E-mail:[email protected] habitually done in the literature, without justification) we ‡ E-mail:[email protected] need to understand how G and G˙ are expected to vary in 2 T. Clifton, D. F. Mota and J. D. Barrow space in a realistic inhomogeneous universe. Since, even on Dicketheory,ω isconstant.Theactionaboveisvariedwith the scale of a typical galaxy, the amplitude of visible den- respect to g to givethe field equations µν sityinhomogeneitiesareoforder106,weneedtogobeyond linear perturbation theory for such an analysis. 1 1 In this paper we begin to confront this deficiency by Rµν gµνR+ (gµρgνσ gµνgρσ)φ − 2 φ − ;ρσ tracingtheevolutionofG,firstinexactinhomogeneoussolu- tionsandthenasimple,butnotunrealistic,inhomogeneous + ω(φ)(gµρgνσ 1gµνgρσ)φ φ = 8πTµν, (2) universe in which a zero-curvature Brans-Dicke-Friedmann φ2 − 2 ,ρ ,σ − φ background universe is populated by spherical overdensi- and with respect to φ to obtain thepropagation equation ties which are modelled by positive curvature Brans-Dicke- Friedmann universes in the dust-dominated era of the uni- vluetrisoen’sfhoilslotowreyd.TbhyisGw(til)liennatbhleebuasctkogtrroaucnkdthuenidvieffresreenatndevion- (cid:3)φ= 2ω(φ1)+3(8πT −ω′(φ)gabφ;aφ;b), (3) theoverdenseregions,whicheventuallyseparateofffromthe whereprimedenotesdifferentiationwithrespecttoφ.Inthis background universe and start to contract to high-density paper we will often consider Friedmann-Robertson-Walker likeseparatecloseduniverses.Thisprocesscanproducesig- (FRW) universes with themetric nificantdifferencesbetweenGandG˙ inthebackgroundand in the overdensities. Eventually, the collapse of the spheri- cal overdensities will be stopped by pressure and a compli- ds2=dt2 a2(t) dr2 +r2(dθ2+sin2θdφ2) , cated sequence of dissipative and relaxation processes will − 1 kr2 (cid:18) − (cid:19) lead to virialisation and some final state of gravitational wherea(t)is thescale factor andk is thecurvatureparam- equilibrium. This state will provide the gravitational envi- eter. Using theFRW metric in eq. (2) gives ronment out of which which stars and planetary systems like our own will form, directly reflecting the local value of G(t)inheritedfromtheirvirialisedprotogalaxyoritsparent ωφ˙2 φ˙ φ¨ 8π k protocluster. The simple model we use for inhomogeneities 2H˙ +3H2+ 2 φ2 +2Hφ + φ =− φ p− a2, (4) in density and in G has many obvious limitations, notably in its neglect of pressure, deviations from spherical sym- metry,accretion,andinteractionsbetweeninhomogeneities. φ¨ 8π (ρ 3p) φ˙ ω˙ φ˙ = − 3H , (5) Nonetheless, we expect that it will be indicative of the im- φ φ (2ω+3) − φ − (2ω+3)φ portanceoftakingspatialinhomogeneityintoaccountinany and attemptstouseobservationaldatatoconstraincosmological modelswhichpermitvaryingG.Itprovidesthefirststepin 8π φ˙ ωφ˙2 k a clear path towards improved realism in the modelling of 3φρ=H2+Hφ − 6 φ2 + a2, (6) inhomogeneitiesthatmirrorstheroutefollowed instandard cosmological studies of galaxy formation with constant G. where H a˙/a is the Hubble rate, over-dot denotes differ- ≡ In section 2 we give the field equations and the field entiation with respect to comoving proper time, t, ρ is the equationsforscalar-tensor gravitytheories. Tofixideas, we matter density, and p is the pressure. Each non-interacting use some exact Brans–Dicke–Friedmann cosmological solu- fluidsourcep(ρ)separatelysatisfiesaconservationequation: tions in section 3 to model the time variation of G in some new exact inhomogeneous solutions of the field equations ρ˙+3H(ρ+p)=0. (7) which describe a spherical inhomogeneity in an expanding universe.Insection4weuseexactsolutionsofflatandclosed Substitutingeqs. (5) and (6) intoeq. (4) gives vacuum Brans–Dicke–Friedmann cosmologies to illustrate the use of the spherical collapse model for non-linear over- densities. Wethen apply thesame techniquestodust–filled φ˙ ωφ˙2 H˙ +H2 H + Brans–Dicke–FriedmannuniversesinSection5.Thenumer- − φ 3 φ2 icalsolutionoftheequationsarepresentedanddiscussedin 8π(3pω+3ρ+ρω) 1 ω˙ φ˙ this section for Brans-Dicke and some other scalar–tensor = + . (8) −3φ (2ω+3) 2(2ω+3)φ theories. A summary of our principal results is given in 6. A general feature of the scalar-tensor field equations is that any solution of general relativity (hence ω and φ both constant) for which the energy-momentum tensor of mat- 2 SCALAR-TENSOR COSMOLOGIES ter has vanishing trace (eg vacuum, black-body radiation, The action for a scalar-tensor theory of gravity is given by Yang-Millsfield,ormagneticfield)isaparticular(φ=con- stant) exact solution of the scalar-tensor gravity theory. A specificationofω(φ)isrequiredtodeterminethetheoryand 1 ω(φ) S = d4x√ g(φR+ gµν∂ φ∂ φ+16π ). (1) close the system of equations. In general, we do not know 16π − φ µ ν Lm Z theformofω(φ)butifthetheoryistoapproachgeneralrel- Hereφ is a scalar field,R is theRicci scalar, and L is the ativityinanappropriatelimitthenwerequirebothω m →∞ Lagrangian for matter fields in the space-time and the free andω′(φ)ω−3 0to holdsimultaneously in theweak–field → function ω(φ) specifies the scalar-tensor theory. In Brans- limit. Inhomogeneous Gravity 3 3 BRANS-DICKE COSMOLOGIES in a vacuum under the transformation (12). From (12) we immediately obtain (Fuji & Maeda 2003) Consider the simplest case of Brans-Dicke (BD) theory (Brans & Dicke1961)tofixideasaboutvaryingG.Inthese gµν =φg¯µν, √ g=φ−2√ g¯ − − theories ω(φ) ω is a constant. The three essential field ≡ and equations for the evolution of the BD scalar field φ(t) and the expansion scale factor a(t) in a BD universe are (5), R=φ(R¯+6(cid:3)Γ+6g¯µνΓ,µΓ,ν) (13) (6), and (7). Now, ω is a constant parameter and the the- where here R is the Ricci scalar, Γ = lnφ−1/2 and (cid:3)Γ = ory reduces to general relativity in the limit ω where φ = G−1 → constant. The form of the gener→al ∞solutions w√i1−thg¯∂aµ(m√in−igm¯g¯aµlνly∂νcΓo)u.pTleoddsecraivlaertfiheeldE,inwseteciannfieexldtreeqmuiasetiothnes to the Friedmann metric in BD theories are fully under- Lagrangian density stood(Barrow1992),(Gurevich, Finkelstein & Ruban1973; Holden & Wands 1998). The vacuum solution is the t 0 attractorfortheperfect-fluidsolutions.Thegeneralperf→ect- =√ g¯ 1 R¯+ 1g¯µνψ, ψ, (14) fluid solutions with equation of state L − 16π 2 µ ν (cid:18) (cid:19) to get Einstein’s field equations, G = 8πT , where µν µν − T = ψ, ψ, 1g¯ g¯αβψ, ψ, . We now set φ = p=(γ 1)ρ (9) µν µ ν−2 µν α β − exp[ψ 8π ] so that under the conformal transformation andk=0canallbefound.Atearlytimestheyapproachthe ω+23 vacuumsolutions butat late time theyapproach particular prescrqibed by(12) we find that (14) becomes power-law exact solutions (Nariai 1968): 1 a(t)=t[2+2ω(2−γ)]/[4+3ωγ(2−γ)] (10) L=φ2√−g(16π(φ−1R−6(cid:3)Γ−6φ−1gµνΓ,µΓ,ν) 1 3 φ, φ, + (ω+ )φ−1gµν µ ν). (15) 16π 2 φ2 φ(t)=φ0t[2(4−3γ)/[4+3ωγ(2−γ)] (11) We see that (cid:3)Γ disappears on integration by parts and so we can discard it. We also note that gµνΓ, Γ, = µ ν Theseparticularexactpower-lawsolutionsfora(t)and 41gµνφ−2φ,µφ,ν, so that (15) simplifies to φ(t) are ’Machian’ in the sense that the cosmological evo- lution is driven by the matter content rather than by the 1 ω kineticenergyofthefreeφfield.Thesignofφ˙ isdetermined L=√−g16π(φR+ φgµνφ,µφ,ν). (16) by thesign of 4 3γ. These soluti−ons are spatially homogeneous and so can- ThisistheLagrangiandensitythatgivestheBDaction(1), not tell us about the effects of any spatial inhomogeneity with matter =0. L in φ and ρ on observational tests of time-varying G=φ−1. Starting with a solution of Einstein’s field equations Next, we consider some simple exact inhomogeneous solu- with a scalar field we can apply this conformal transfor- tions of BD theory in order to gain some intuition about mationtoarriveatasolution oftheBDfieldequationsina the likely effects of spatial inhomogeneity in G . We will vacuum.Aspherically symmetricexactsolution forthecol- findthat theseexact simple homogeneous solutions playan lapse of a minimally-coupled scalar field, ψ,in general rela- important role in determining the time dependence of G in tivityis known and is given by(Husain, Martinez & Nunez inhomogeneous solutions. 1994) 3.1 An Inhomogeneous vacuum Brans-Dicke ds2=(qt+b)(f2(r)dt2 f−2(r)dr2) − Solution R2(r,t)(dθ2+sin2θdφ2), (17) − It is well known that BD theory is related to general rel- where f2(r) = (1− 2rc)α, R2(r,t) = (qt+b)r2(1− 2rc)1−α ativity through a conformal transformation of the form andα= √3.Theevolutionoftheminimallycoupledscalar (Fuji & Maeda 2003) ± 2 field,ψ, in theEinstein frame, is given by 1 α gµν = φg¯µν (12) ψ(r,t)= 1 ln d 1 2c √3 (qt+b)√3 . (18) ±4√π " (cid:18) − r (cid:19) # where φ is the BD scalar field. Symbols with bars refer to quantities in the Einstein (general relativistic) conformal Here, q, b, c and d are constants. Now under the transfor- frame and symbols without bars refer to quantities in the mation (12),where φ=exp[ψ 8π ], we obtain ω+3 Jordan (BD) conformal frame. This conformal equivalence 2 q tailolonwsswuisthtoaexspcalolaitrkfineolwdntosofiluntdionsosluoftiEoninsstteoint’hsefieBldDeqfiueald- ds¯2= B(t)1−√3/β A(r)αdt2 A(r)−αdr2 (19) d1/βA(r)α/√3β − − equations in a vacuum. (cid:2) (cid:3) equivWaleenpcreocoefegdenbeyrafilrrsetlasthivoiwtyinwgitehxpalicscitallyarthfieeldcoannfodrmBaDl A(r)1−α1+√√3β3βB(t)1−√3/βr2(dθ2+sin2θdφ2) d1/β 4 T. Clifton, D. F. Mota and J. D. Barrow and φ(r,t)= dA(r)α/√3B(t)√3 1/β (20) h i whereA(r)=1 2c,B(t)=qt+b andβ = √2ω+3.We 1.1 − r ± now assume that q = 0 (i.e. the metric is not static) and 11 definethenew time6coordinate t¯=(qt+b)23−√2β3. 0000....8899 €G€€€ G€H€€ €2H€€€.r€€€1,€€€,€€t€€€L5€€L€€€€ In the limit that c 0 the r-dependence of the met- 20 ric isremoved and thesp→acebecomes homogeneous. Inthis 1155 40 case we expect (20) to reduce to the FRW Brans-Dicke rr 1100 60 t metric given in the last section. We see from the form of (20) that the metric should reduce to that of a flat FRW 80 55 Brans-Dicke universe. Insisting on this limit requires us to set β = √2ω+3, q = 2β and d = 1. This leaves the 100 3β √3 metric: − ds¯2=A(r)α(1−√13β)dt¯2 Figure1.Thisgraphillustratesthepossiblespaceandtimevari- ations that can arise in G in inhomogeneous solutions to the A(r)−α(1+√13β)t¯23(ββ−√√33) Brans–Dicke field equations, normalised at r=2.1 and t=5. − − × dr2+A(r)r2(dθ2+sin2θdφ2) . (21) Rewriting (20) with(cid:2)these coordinates and constant(cid:3)s gives eµ= 1+ c 4 1− 2kcr 2(k−1)(k+2)/k, (25) φ(r,t)= 1 2c ±21β t¯2/(√3β−1). (22) (cid:16) 2kr(cid:17) (cid:18)1+ 2kcr(cid:19) − r (cid:18) (cid:19) 2ω(2 γ)+2 t 3ωγ(2−γ)+4 A comparison of (21) with (32) shows that (21) does a(t)=a − , (26) 0 t indeed reduce to a flat vacuum FRW metric in the limit (cid:18) 0(cid:19) c 0 (an inhomogeneous universe requires c = 0). The and → 6 metric (21) is asymptotically flat and has singularities at t¯= 0 and r = 2c; the coordinates r and t¯therefore cover therTanhgeeesq0ua6tit¯on<s∞(47a)ndan2d6(2rc2)<ca∞n.now be used to con- φ(r,t)=φ0(cid:18)tt0(cid:19)3ω2γ((42−−3γγ))+4 (cid:18)11−+ 22kkccrr(cid:19)−2(k2−1)/k (27) struct a plot of G(r,t); this is done in Figure 1 which was for thematter distribution constructed by choosing 1 in (22). From the form of −2β G(r,t) we see that this choice corresponds to an overden- sityin themassdistribution (identifiedbycomparison with ρ(r,t)=ρ a0 3γ 1− 2kcr −2k (28) theinhomogeneousBrans–Dickesolutionwithmatter,found 0 a(t) 1+ c (cid:18) (cid:19) (cid:18) 2kr(cid:19) below). In this figure, ω was set equal to 100, and c set equal to 1. This plot shows how G can vary in space and wherek= 4+2ω.Thisseparablesolutiondisplaysthesame 3+2ω timeinaninhomogeneousuniversewhichconsistsofastatic time depenqdence as the power–law FRW Brans-Dicke uni- Schwarzschild-like mass sitting at r = 0 in an expanding verses,(10)–(11),butwith an additional inhomogeneous r– universe. As r 0 the solution approaches the behaviour dependence created by the matter source at r = 0. Such a → ofthestaticsphericalvacuumBDsolution butasr it distributionofmatterinspaceisillustratedbyfigure2.Here →∞ approaches thebehaviour of a BD Friedmann universe. we have chosen, for illustrative purposes, γ = 1, ω = 100 andabackgroundvaluesetbythechoiceρ=ρ (ρ FRW FRW beingthematterdensitythatwouldbeexpectedinthecor- 3.2 An Inhomogeneous Brans-Dicke Solution respondinghomogeneousuniverse).Thetemporalevolution With Matter of ρ is exactly thesame as theFRW case. We now seek a solution of the Brans-Dicke field equations WeseefromFigure2thatthematterdensityisisotropic (2) with theform andasymptotically constant asr with asharppower- →∞ law peak near theorigin. Now (27) gives us ds2 =eνdt2 eµa2(dr2+r2dθ2+r2sin2θdφ2) (23) Awhwereesehνow=theνa(tr−)a, esoµlu=tioenµ(orf) tahnedfiaeld=eaq(uta)t.ioInnsA(p2p)efnodrixa G(r,t)=G0 11−+ 2kccr 2(k2−1)/kt−3ω2γ((42−−3γγ))+4. (29) (cid:18) 2kr(cid:19) metric of theform (23) is given by Equation (29) is used, with the values ω = 100, γ = 1 andc=0.5 tocreate Figure3,which showsthespace–time 1 c 2k evolution of G(r,t). eν = − 2kr , (24) 1+ c These results show how G(r,t) can vary in space and (cid:18) 2kr(cid:19) Inhomogeneous Gravity 5 €Ρ€€€ €H€€€r€€€L€€€€ moreslowlythanthebackground,beforereachinganexpan- Ρ FRW sion maximum and collapsing back to high density, whilst 2.75 thebackgroundcontinuestoexpand.Inthissectionwecon- 2.5 sidervacuumuniversesonly,sothereexistssphericallysym- metricinhomogeneityintheexpansionrateandinφ G−1, 2.25 ∼ but ρ=p=0. 2 For flat vacuum FRW universeseq. (6) gives 1.75 1.5 a˙ 2 a˙ φ˙ ω φ˙ 2 + b = b (30) 1.25 (cid:18)a(cid:19) aφb 6 (cid:18)φb(cid:19) 5 10 15 20 r where˙= ddt andφb=φb(t)istheBDscalarfieldandaisthe scale factor in the flat background. For a positively curved Figure 2. Distribution of ρ as a function of r, with ω = 100, (k =+1) region the scale factor is taken to be S(τ), which from eq. (28)and c=0.5. satisfies the Friedmann equation for the closed vacuum BD universe: S′ 2+ S′φ′p = ω φ′p 2 k , (31) (cid:18)S (cid:19) S φp 6 (cid:18)φp(cid:19) − S2 where ′ = ddτ, φp =φp(τ) and τ and k are the proper time 1 and curvatureof this perturbed region. €G€€€ €H€€€r€€€,€€€€€t€€€L€€€ 0.99 In matching these two regions at t = t0 = τ0 we must G H1,5L satisfy the boundary conditions 0.98 20 1155 40 dS da rr 1100 60 t S(τ0)=a(t0), dτ = dt , (cid:18) (cid:19)0 (cid:18) (cid:19)0 55 80 φ (τ )=φ (t ) and dφp = dφb . p 0 b 0 dτ dt (cid:18) (cid:19)0 (cid:18) (cid:19)0 100 4.1 The Background Universe Figure 3.Evolution of G(r,t) in space and time inan inhomo- geneousmatterdominatedUniverse,withω=100,fromeq.(29) Assuming solutions of the form φb tx and a ty and ∝ ∝ and c=0.5. settinga(0)=0gives,onsubstitutioninto(30)and(5),the k=0 BD vacuum solutions (O’Hanlon & Tupper1972) timeinanasymptotically–flat universewith apeakof mat- ter at the origin. Observers located near the mass concen- a(t)=t13(1+2(1−√3(3+2ω))−1) (32) tration will determinedifferentvaluesof Glocally although theywillfindthesamevaluesofG˙/Geverywherebecauseof and theseparablenatureoftheG(r,t)evolutionineq.(28).This was also the case for solution (22) given in subsection 3.2. t −2(1−√3(3+2ω))−1 φ (t)=φ . In thenextsection weshall consider a more realistic model b b0 t inwhich bothGandG˙/Garedifferentfrom placetoplace. (cid:18) 0(cid:19) PlotslikeFigure3canbegeneratedforuniversesdominated byothertypesofcosmological fluidandwithdifferentrates 4.2 A Collapsing Universe of density fall off with r. Fortheclosedregionwenowfollowthemethodgiveninrefs. (Barrow 1992; Barrow & Parsons 1997) to find expressions 4 MATCHING TWO VACUUM FRW forS(τ)andφp(τ).Westartbyintroducingconformaltime, BRANS-DICKE UNIVERSES η, definedby Sdη=dτ;then eq. (5) becomes We will now consider a simple model of a spherically sym- 2 metric cosmological inhomogeneity produced by matching φ , + S, φ , =0. together flat and positively curved vacuum FRW–BD uni- p ηη S η p η verses. This is a well studied technique, first introduced by This integrates directly to yield Lemaˆıtre, for studying the non-linear evolution of overden- sities in general relativistic FRW universes. The overdense region ismodelledasaclosed universethatatfirstexpands φp,ηS2 =√3A(2ω+3)−1/2 (33) 6 T. Clifton, D. F. Mota and J. D. Barrow where A is a constant. We now introduce the variable y = RHtL, SHtL φ S2 to write (6) as p 1.5 y,2= 4ky2+ 1φ ,2S4(2ω+3). (34) 1.25 η − 3 p η 1 Now, eqs. (33) and (34) give 0.75 φ , p η =√3Ay−1(2ω+3)−1/2 0.5 φ p 0.25 and y,2η=−4ky2+A2. (35) 1 2 3 4 5 t The solutions of eqs.(35), when k>0, are given by Figure 4.Evolution of the scale factor S in the perturbed over- dense region (dashed line) and in the background a(solid line) A with respect tothe comoving proper time inthe flat background. y(η)= sin(2√k(η+B), (36) 2√k and GHtL(cid:144)GH5L 7 φp(η)=Ctan (2ω3+3)(√k(η+B)) (37) 6 q 5 where B and C are arbitrary constants. We now fix the conformal time origin by setting B = 0, so that y = φpS2 4 gives 3 sin1/2(2√kη) 2 S(η) . (38) ∝ tan 4(2ω3+3)(√kη) 1 q t 1 2 3 4 5 4.3 From η to t Figure5.EvolutionofG(t)intheoverdenseperturbedoverdense The function τ(η) is now obtained by integrating Sdη=dτ regionofpositivecurvature(dashedline)andinthespatiallyflat and τ(η) can then be used to obtain S(τ). We now re- background universe (solid line). quire a relation between t and τ, for this we proceed as in ref. (Barrow & Kunze 1997) and use the equation of relativistic hydrostatic equilibrium (Harrison 1970, 1973), S(t) from (38). This is done numerically. Now fixing the (Landau & Lifshitz 1975) constants of proportionality together with k in eqs. (38), (37) and (32), in order to satisfy the boundary conditions, we find equations for the evolution of the scale factors and ∂Φ ∂p/∂r = (39) scalar fields in the flat background and perturbed region. ∂r − p+ρ These are matched at a boundary,at time t =τ =η . 0 0 0 where Φ is the Newtonian gravitational potential and r is radial distance. Eqn. (39) is derived underthe assumptions thattheconfiguration is staticandthegravitational fieldis 4.4 Results weak, so Φ completely determines the metric; then (39) is Figure 4 shows the evolution of a(t) and S(t) when the re- givenbytheconservationofenergy-momentumforaperfect fluid. We now use dτ =eΦdt, and for a scalar field we have gion described by S(t) becomes positively curved at initial aneffectivestatewithp=ρandρ= ωφ˙2.Combiningthese time t0 =1. In Figure 4 we choose ω =100, for illustrative φ purposes, and a = S =1 so that the boundary condition results gives 0 0 for the matching of the first derivatives of the scale factors is given by dS = dS = da . ddτt = φφ˙′pb φφ11p//22. (40) (cid:0)dτ(cid:1)0 (cid:16)dη(cid:17)0 (cid:0)dt(cid:1)0 b Nowφ˙b∝a−3 andφ′p ∝S−3,andsowith(37)and(32) ferenWt ceucravnatunroewuseixnpgreesqsuGatio=nsG((4t7)),in(3t7h)eanredgi(o3n2s),oaflodnifg- this gives withtheappropriatecoordinatetransformations.Thisgives Figure5,below. Itisclearly seen thattheevolutionof G(t) dτ sin1/2(2√kη) S2a√1+23ω−4 is quite different in the two regions, as expected. The col- = (41) dt sin1/2(2√kη0)S02a√0 1+23ω−4 lpaopsssiensgseosvaersdmenalslietryveavlouleveosfGfasbteurttahlaanrgtehrevbalaucekgorfouG˙n/dGanadt | | Eq. (41) and the relation Sdη = dτ allow us to obtain all times after theevolution commences. Inhomogeneous Gravity 7 5 A MORE REFINED SPHERICAL COLLAPSE while (5) reduces to MODEL We now generalise the spherical collapse model described φ¨ +3S˙φ˙ = 8π (ρ +4ρ ) ω˙cφ˙c (45) in the last section to the more astronomically realistic case c S c (2ωc+3) cdm Λ − (2ωc+3) ofaflatuniversecontainingmatterandacosmological con- where S = S(t) is the scale factor and φ = φ (t) is the c c stant(see.e.g.refs.(Padmanabhan1995),(Peacock1999)or BDscalarfieldinthecollapsingregionofpositivecurvature (Lahav et al.1991)).Asbefore,wematchaflatBrans-Dicke where ρcdm S(t)−3 and ρΛ = constant. These equations FRWbackgroundtoaspherically symmetricoverdensityat ∝ give theevolution of φ (t) and S(t). c anappropriateboundaryandallowthetworegionstoevolve We have assumed that the equation of motion of the separately. field inside the cluster overdensity is described by the local space-time geometry. This means that the field follows the dark-mattercollapsefromthebeginningofthecluster’sfor- 5.1 The background universe mation. We do not consider this to be fully realistic since thereisexpectedtobeanoutflowofenergyassociated with Again,weconsideraflat(k=0),homogeneousandisotropic φ from the overdensity to the background universe, as first backgrounduniverse.Sinceweareinterestedin thematter- noticed by Mota and van de Bruck (Mota & van de Bruck dominated epoch, when structure formation starts, we can (2004)). The details of this outflow of energy and its effect assumethatouruniversecontainsonlymatterandavacuum on the collapse can only be determined by a fully relativis- energycontributionsothatρ=ρ +ρ andp=p = ρ m Λ Λ − Λ tichydrodynamicalcalculation,whichisbeyondthescopeof give thetotal densityand pressure, respectively. So, for the thisstudy.Nevertheless,atlatetimesduringthecollapse of flat background, eq.(6) gives for a general ω(φ)theory, thedarkmatter(andespeciallywhenthedensitycontrastin thedarkmatterisverylarge)thefieldshouldnolongerfeel a˙ 2 a˙ φ˙ ωφ˙2 8π the effects of the expanding background and will decouple a + aφ = 6 φ2 + 3φ(ρm+ρΛ) (42) fromit.Wearealsoneglectingtheeffectsofdeviationsfrom (cid:18) (cid:19) spherical symmetry, which grow during the collapse in the and eq. (5) gives absence of pressure, along with rotation, gravitational tidal interactionsbetweendifferentoverdensities,andallformsof non-linear hydrodynamicalcomplexity. a˙ 8π ω˙φ˙ φ¨+3 φ˙ = (ρ +4ρ ) . (43) a (2ω+3) m Λ − (2ω+3) 5.3 Evolution of the overdensity Here,ρm a(t)−3 andρΛ =constant.Theseequations govern the evo∝lution of φ(t) and a(t) in the flat expanding Considerthespherical perturbationin thecdm fluidwith a cosmological background. The contribution of the vacuum spatially constant internal density. Initially, this perturba- energy stress (p = ρ) to the Friedmann equation (42) in tion is assumed to have a density amplitude δi > 0 where BDcosmologydiffer−sfromthatingeneralrelativitybecause δi 1. The initial cdm density inside the overdensity is | | ≪ of the presence of the variable φ field: it is not the same as therefore ρcdm =ρm(1+δi). theaddition of a cosmological constant term totheRHS of Fourcharacteristicphasesoftheoverdensity’sevolution (42). However, with this proviso, we shall continue to refer can beidentified: to Λcdm models in Brans-Dicke theories in the following Expansion: we employ the initial boundary condition sections. • φ = φ and assume that at early times the overdensity ex- c pandsalong with thebackground. Turnaround: forasufficientlylarge δ ,gravityprevents i • 5.2 The overdensity the overdensity from expanding forever; the spherical over- densitybreaksawayfromthegeneralexpansionandreaches Again, we consider a spherical overdense region of radius S amaximumradius.Turnaroundisdefinedasthetimewhen and model the interior space-time as a closed FRW Brans- S =S , S˙ =0 and S¨<0. max Dicke theory universe, ignoring any anisotropic effects of Collapse: the overdensity subsequently collapses (S˙ < gravitational instability or collapse. As usual, we assume • 0). If pressure and dissipative physics are ignored the over- thereisnoshell-crossing; thisimplies massconservation in- density would collapse to a singularity where the density side the overdensity and independence of the radial coordi- of matter would tend to infinity. In reality this singular- nate(Padmanabhan1995).Theevolutionequationscannow ity does not occur; instead, the kineticenergy of collapse is be written in a form that ignores thespatial dependenceof transformed into random motions. the fields. Put ρ = ρ +ρ and p = p = ρ , where cdm Λ Λ − Λ Virialisation:dynamicalequilibriumisreachedandthe ’cdm’ corresponds to cold dark matter, so that (8) gives • systembecomesstationarywithafixedradiusandconstant energy density. φ˙ ω φ˙ 2 1 ω˙ φ˙ Werequireoursphericaloverdensitytoevolvefromthe S¨ S˙ c = S c c c c − φc − 3 φ2c − 2(2ωc+3)φc linear perturbation regime at high redshift untilit becomes non-linear,collapses,andvirialises.Thereafter,theoverden- 8π (ρ (3+ω )+ρ (3 2ω )) + cdm c Λ − c (44) sitywillbecomegravitationallystableandfurtherlocalevo- 3φc (2ωc+3) (cid:19) lution of the scale factor and of the scalar field will cease. 8 T. Clifton, D. F. Mota and J. D. Barrow However, the background scale factor will continue to ex- where pand,andsothebackgrounddensityandbackgroundvalue ofGwillcontinuetodecrease. Asaresult,asignificant dis- parity between the evolution of G inside and outside the ∂U = 16π2 ∂Gcρ ρ S5+ cluster can result. ∂S − 15 ∂S tot x (cid:20) ∂ρ ∂ρ G totρ S5+G ρ xS5+G ρ ρ 5S4 (49) c ∂S x c tot ∂S c tot x 5.4 Virialisation (cid:21) and we have used U = U +U +U , ρ = ρ + tot cdm Λ φc tot cdm In scalar-tensor theories we expect that the gravitational ρ +ρ , ρ =ω φ˙2/φ and potential will not be of the standard local r−1 form. This Λ φc φc c requires reconsideration of the virial condition. According to the virial theorem, equilibrium will be reached when ∂Gc = G˙c = G φ˙ 3+2ωc G + 2ωc′(φc) . (Goldstein 1980) ∂S S˙ − c c4+2ωc c (3+2ωc)2! The other components of eq. (49) are obtained from eqs. 1 ∂U T = S (46) (44) and (45). 2 ∂S Using eq. (46), together with energy conservation at whereT istheaveragetotalkineticenergy,U istheaverage turnaroundandvirialisation, weobtainanequilibriumcon- total potential energy and S here denotes the radius of the dition in terms of potential energies only spherical overdensity. The potential energy for a given component x can be calculated from its general form in a spherical region 1 ∂U S +U (z )=U (z ), (50) (Landau & Lifshitz 1975) 2 vir ∂S tot v tot ta (cid:18) (cid:19)zv wherez istheredshiftatvirialisationandz istheredshift v ta S U =2π ρ Φ r2dr, of theover-densityat itsturnaroundradius. Thebehaviour x tot x of G during theevolution of an overdensity can now be ob- Z0 tainedbynumericallyevolvingthebackgroundeq.(42)-(43) whereρ isthetotalenergydensityandΦ isthegravita- tot x and the overdensity eqs. (44) and (45) until the virial con- tional potential dueto thedensity component ρ . x dition (50) holds. The gravitational potential Φ can be obtained from x Wepointouthereaninconsistencywhenonemakesuse theweak-fieldlimitofthefieldequations(2).Thisresultsin ofequation(50)togetherwiththeassumptionthatenergyis aPoisson equationwherethetermsassociated tothescalar notconserved.Thisinconsistencyisremovedbyassuminga field can be absorbed into the definition of the Newtonian negligibleoutflowofφfromtheoverdensity,inwhichcasewe constant as (Will 1993) regain energy conservation within the system and so retain self-consistency. 4+2ω (φ ) 1 G = c c . (47) c 3+2ω (φ )φ c c c ThisresultsintheusualformfortheNewtonianpoten- 5.5 Overdensities vs Background tial 5.5.1 Brans–Dicke theory r2 TheBrans–Dickecouplingparameterωisconstantandcon- Φx(s)=−2πGcρx(3γx−2) S2− 3 strained by a variety of local gravitational tests (see (Uzan (cid:18) (cid:19) 2003) for a review). The strongest constraint to date is de- whereGcisgivenbyequation(47)andγx 1ispx/ρxforthe rived from observations of the Shapiro time delay of sig- − fluid component with density ρx and pressure px (appear- nals from the Cassini space craft as it passes behind the ing due to the relativistic correction to Poisson’s equation: Sun. These considerations led Bertotti, Iess and Tortora ∆Φ=4πG(ρ+3p)). (Bertotti, Iess & Tortora 2003), after a complicated data In Λcdm models of structure formation it is entirely analysis process, to claim that ω must have a value greater plausibletosetγx =1astheenergydensityofthecosmolog- than 40000 (to 2σ). This limit on ω must besatisfied at all icalconstantisnegligibleonthevirialisedscaleswearecon- timesinallpartsoftheuniverse,andleadstotheconclusion sidering (Wang & Steinhardt 1998; Mota & van de Bruck that Brans-Dicke theory must be phenomenologically very 2004). The potential energies associated with a given com- similar to general relativity throughout most of the history ponent (x) inside theoverdensity are now given by of the universe. However, we do still expect a cosmologi- cal evolution of the Brans-Dicke field φ which determines 16π2 the value of Newton’s G; and we expect this evolution to Ux =− 15 GcρtotρxS5. (48) be different in regions that collapse to form the structure probedbyCassinicomparedtothatintheidealisedexpand- Therefore, the virial theorem will be satisfied when ingcosmological background,asdescribed above.Hencewe expect the measurable value of G to be different in these 1 ∂U two distinct regions with different histories. It is quite pos- T = S , vir 2 vir ∂S sible that theregion of gravitational equilibrium probed by (cid:18) (cid:19)vir Inhomogeneous Gravity 9 x 10−8 x 10−8 6.88 6.675 6.86 6.6745 −810) 66..8824 G , G cB66.6.677345 ×G , G ( c666...6777.4688 6.673 6.72 6.7 6.6725 6.68 10−1 100 100 log ( 1+z ) log ( 1+z ) (a)ω=40000 (b)ω=500 Figure 6. Plots of G against ln(1+z) in the background (dashed-line) and in an overdensity (solid-line), for different values of ω. Initial conditions are chosen in both cases so as to give G =G0, the present value of the Newton constant, at virialisation. We note that increasing ω decreases the difference inG betweenthe overdensity and the background. Cassini and other local observations would find no percep- choice of coupling function tible change in G locally despite the presence of change on cosmological scales outside of boundinhomogeneities. 2ω(φ)+3=2A 1 φ −p, In this section we quantify this difference in G by nu- (cid:12) − φ∞(cid:12) (cid:12) (cid:12) merically evolving a, S, φ and φc until virialisation occurs. where A, φ , and p are positiv(cid:12)e definit(cid:12)e constants. We re- At virialisation, the evolution of S and φc is expected to fer to this a∞s Theory 1. Such a(cid:12)choice o(cid:12)f coupling was con- end,givingavalueofGthatislocallyconstantintimeeven sidered by Barrow and Parsons and was solved exactly for though thecosmological evolution of a and φ continues. the case of a flat FRW universe containing a perfect fluid The plots in figure 6 were constructed using the repre- (Barrow & Parsons 1997). sentative values ω =40000 and ω =500 and the boundary Settingtheconstantsas2A=(φ /β)2 andp=2gives conditionφc0 ≃G−01,sothatthevalueofGmeasuredinside us the scalar–tensor theory considere∞d by Damour and Pi- theoverdensityat present isequaltothevalueof Newton’s chon (Damour & Pichon 1999) and by Santiago, Kalligas G,asmeasuredlocally.Theevolutionofthebackgroundwas and Wagoner (Santiago, Kalligas & Wagoner 1997). This determinedbymatchingφc toφatthetimewhentheover- choice of ω(φ) corresponds to setting lnA(φ) = lnA(φ0)+ density decouples from the background, ti. We see a clear 1β(φ φ)2,whereA2(φ)istheconformalfactor1/φfrom difference in the evolution of G in the two regions, as ex- e2q.(1∞2)−.DamourandNordvedt(Damour & Nordvedt1993) pected. This example shows that we expect different values consider this function as a potential and therefore justify of G and G˙/G inside and outside virialised overdensities. its choice in relation to the generic parabolic form near a The present value of G and G˙/G dependson thehistory of potential minimum. Expecting the function to be close to the region where it was sampled, as well as on the Brans– zero (i.e. GR), Santiago, Kalligas and Wagoner justify its Dickecoupling parameter, ω. expression as a perturbativeexpansion.This choice of ω(φ) Itcanbeseenfrom theplotsinfigure6thatincreasing with p>1/2 corresponds to a general two–parameter class ω has the effect of decreasing the difference in G between of scalar–tensor theories that are close to GR and will be the background universe and the overdensity. The size of drawn ever closer to it with ω and ω′/ω3 0 as this inhomogeneity is found to be of order 1/ω and, corre- the universe expands and φ φ→.∞We therefore c→onsider spondingly,reducestozeroasω .Thisisanimportant it as a representative example→of a∞wide family of plausible →∞ consistencycheckforthemethodsusedasweexpectBrans– varying-Gtheories that generalise Brans-Dicke. Dicketheorytoreducetogeneralrelativity,withaconstant Theevolution ofthisformofω(φ)isshowngraphically G, in thislimit. infigure7fordifferentvaluesofAandp.Clearly theevolu- tionofω(φ)issensitivetobothAandpandsothechoiceof theseparametersisimportantfortheformoftheunderlying theory. For illustrative purposes we choose here the values p=1.5,2 and 5 and A=1,2 and 5. p 5.5.2 Scalar-tensor theory with 2ω+3=2A 1 φ − In a similar way to the BD case we now create an (cid:12) − φ∞(cid:12) evolution of φc that virialises at z = 0 to give the value Next, we consider a scalar–tensor theory wit(cid:12)(cid:12)h a var(cid:12)(cid:12)iable Gc0 =6.673×10−11,asobservedexperimentally.Thecorre- ω(φ). We investigate the class of theories defined by the sponding evolution for φ is calculated as before by match- b 10 T. Clifton, D. F. Mota and J. D. Barrow 2ΩHΦL+3 2ΩHΦL+3 1000 600 800 500 400 600 300 400 200 200 100 0.2 0.4 0.6 0.8 Φ(cid:144)Φ¥ 0.2 0.4 0.6 0.8 Φ(cid:144)Φ¥ (a) 2ω(φ)+3 for p=2 and A=1, 2 and 5 (solid, dashed (b)2ω(φ)+3forA=1andp=1.5,2and2.5(solid,dashed anddottedlines,respectively). anddottedlines,respectively). Figure 7. In these gravity theories there is fastapproach to general relativity at late times when φ→φ , but significantly different ∞ behaviouratearlytimes. ing it to the value of φ at the time the overdensity decou- c plesfromthebackgroundandbeginstocollapse.Increating ∆G G(t) G δG G (t) G (t) these plots we have used the conservative parameter values (t) − 0, (t) c − b p=2,ωc0 =1.2 105 andA=6 10−7 whichareconsistent G ≡ G0 G ≡ Gb(t) × × with observation and allow structure formation to occur in and a similar way to general relativity. The results of this are plotted in figure 8. ρ (z ) ∆ cdm v , Again,wenotethedifferentevolutionofG(t)inthetwo c ≡ ρm(zv) regions,andthedifferenceintheasymptoticvaluesofG.We notethatexperimentalmeasurementsofGonEarthhavea whereGc and Gb correspond toGas measured in theover- significantuncertaintywiththe1998CODATAvaluecarry- density and in thebackground universerespectively. inganuncertainty12timesgreater thanthestandardvalue The results of our numerical calculations, for a cluster adoptedin1987.The1998valueisgivenas(Mohr & Taylor which virialises at zv = 0, are presented in Figures 9, 10 2000;Scherrer 1999) and 11, respectively. These plots display the evolution of G˙/G,∆G/G, and δG/G with redshift for Brans-Dicke, the theory of subsection 5.5.2, and some other choices of ω(φ) G1998 =6.673 0.010 10−8cm3gm−1s−2, that are specified in the captions. The parameters used in ± × generating these plots are B = 0.4, C = 10−16, D = 80, while the 2002 CODATA pre-publication announcement A = 6 10−7, p = 2 and ω = 4 106. These values were reverts to the earlier higher accuracy consensus with × × chosen so as to agree with observation and so that struc- (National Instituteof Standardsand Technology 2002) tureformation isnot significantly differentfrom thatwhich occurs in general relativity. G2002 =6.6742 0.0010 10−8cm3gm−1s−2 Itisclearfromtheplotsthatdifferentscalar-tensorthe- ± × ories lead to different variations of G. The predictions of Wecouldre–runtheaboveanalysiswithdifferentvalues these models can be quite diverse. While some models pro- of A and p, but expect that the results would look qualita- duce higher values of G inside the overdensity, others pro- tively similar. From Figure 7 we see that increasing (de- duce a lower one. A feature common to all models is that creasing) the values of A and p will increase (decrease) the the value of G and G˙/G inside an overdensity is different valueofω(φ)foragivenφ,therebymakingthetheorymore from G and G˙/G in the background universe. The reason (less)likeGR.Wethereforeexpectananalysiswithahigher for these differences is that in the non–linear regime, when (lower)valueofAand/orptolookverysimilartotheanal- theoverdensitydecouplesfromthebackgroundexpansionat ysis presented above with a less (more) rapid evolution of turnaround,thefieldφ thatdrivesvariationsin theNewto- G(t).Forthesakeofbrevityweomitsuchananalysishere. nian gravitational “constant” stops feeling the background expansion.Afterturnaround,thefieldinsidetheoverdensity, φ , deviates from the field, φ, in the background universe, c leadingtospatialvariationsinG.Inreality,suchspatialin- 5.6 Space and Time variations of G homogeneities in the value of G are small: δG/G 10−6, ≈ WenowcalculatehowtimeandspacevariationsofGevolve Figure 11. The time variations of G are even smaller than with redshift and depend on the cdm density contrast, ∆ . the spatial inhomogeneities but with a marked difference c In order to dothis, we make thedefinitions between the inside and outside rates of change. We find