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MNRAS000,1–8(2016) Preprint2ndMarch2016 CompiledusingMNRASLATEXstylefilev3.0 Conformal Gravity Rotation Curves with a Conformal Higgs Halo Keith Horne1(cid:63) 1SUPA Physics and Astronomy, University of St. Andrews, KY16 9SS, Scotland, UK Accepted2016February28.Received2016February25;inoriginalform2016January27 6 1 0 ABSTRACT 2 WediscusstheeffectofaconformallycoupledHiggsfieldonConformalGravity(CG) r predictions for the rotation curves of galaxies. The Mannheim-Kazanas (MK) metric a isavalidvacuumsolutionofCG’s4-thorderPoissonequationifandonlyiftheHiggs M field has a particular radial profile, S(r) = S a/(r+a), decreasing from S at r = 0 0 0 1 with radial scale length a. Since particle rest masses scale with S(r)/S0, their world lines do not follow time-like geodesics of the MK metric g , as previously assumed, µν butratherthoseoftheHiggs-frameMKmetricg˜ =Ω2g ,withtheconformalfactor ] µν µν A Ω(r)=S(r)/S .WeshowthattherequiredstretchingoftheMKmetricexactlycancels 0 G the linear potential that has been invoked to fit galaxy rotation curves without dark matter. We also formulate, for spherical structures with a Higgs halo S(r), the CG . h equations that must be solved for viable astrophysical tests of CG using galaxy and p cluster dynamics and lensing. - o Key words: gravitation – galaxies: kinematics and dynamics – cosmology: theory, r dark matter, dark energy t s a [ 2 1 INTRODUCTION 2nd-order field equations, v 7 TheneedfordarkmatteranddarkenergytoreconcileEin- G +Λg =−8πGT , (1) µν µν µν 3 stein’s General Relativity (GR) with observations, together 5 whereG =R −(R/2)g istheEinsteintensor,G isNew- with the lack of other tangible evidence for their existence, µν µν µν 7 ton’s constant, and Λ is the cosmological constant. These motivatesthestudyofalternativegravitytheoriesaimingto 0 terms are excluded in CG because Λ and G build in fun- achieve similar success without resort to the dark sector. 1. Conformal Gravity (CG), like GR, employs a metric damental scales, and R violates conformal symmetry. In- 0 stead the CG action allows only conformally-invariant scal- to describe gravity as curved space-time. But the CG field 6 ars linked by dimensionless coupling constants. R can ap- equations, which dictate how matter and energy generate 1 pear if it is coupled to a scalar field S in the particular v: scpipalcee,-ctoimnfeorcmuarvlastyumrem,eatrryis,ewhfriocmhhaoldloscfaolrtshyemsmtreotnrgy,wpreiank- conformally-invariant combination S;µS;µ − R6 S2 . Particle i and electro-magnetic interactions, but is violated by GR. rest masses cannot be fundamental, but may instead arise X through Yukawa couplings to the conformal Higgs field S. Conformal symmetry means that stretching the metric by ar a factor Ω2(x), and scaling all other fields by appropriate Likewise, the Higgs mass cannot be fundamental, but may powersofΩ,hasnophysicaleffects.Inparticular,localcon- arisethroughdynamicalsymmetrybreaking.Conformalin- variance replaces Einstein’s 2nd-order field equations with formal transformations preserve all angles and the causal the 4th-order CG field equations Mannheim & Kazanas relations among space-time events, but physical distances, (1989); Mannheim (2006), time intervals, and masses change, so that only local ratios of these quantities have physical significance. 4α W =T , (2) g µν µν UnlikeGR,andmanyrelatedalternativegravitytheor- where α is a dimensionless coupling constant. The Bach ies, terms allowed in the CG action are highly restricted by g tensor W and stress-energy tensor T are both traceless, therequiredconformalsymmetry.GR’sEinstein-Hilbertac- µν µν and scale as Ω−4. tion adopts the Ricci scalar R, leading to Einstein’s famous Despite their complexity, the 4th-order CG field equations admit analytic solutions for systems with sufficient symmetryMannheim(2006).Forhomogeneousandisotropic space-times Mannheim (2001), the Robertson-Walker met- (cid:63) E-mail:[email protected] ric is a solution with W = 0, providing a dynamical cos- µν (cid:13)c 2016TheAuthors 2 Horne mological model identical to that of GR, except that the then require α <0 so that a localised source with ρ−p>0 g Friedmann equation has a negative effective gravitiational generates attractive gravity. 2 constant Geff = −3/4πS02, where S0 is the vacuum expect- The remaining constraint can then be the 3rd-order ation value of a conformally-coupled scalar field with va- equation cuumenergydensityλS4.ThisCGcosmologygivesanopen universe, but it can fit0luminosity distances from super- Wr = 1−B2 + 2BB(cid:48) − BB(cid:48)(cid:48)+(B(cid:48))2 nwoitvhae0M<aΩnnΛh<eim1, n(2e0a0tl3y)saonlvdinfgeatthuerecsoscmosomloicgicaaclcecloenrasttaionnt r + B(cid:48)3Br(cid:48)4(cid:48)3−rBB3(cid:48)(cid:48)r(cid:48)3+ 2B(cid:48)B(cid:48)(cid:48)1(cid:48)32−r2(B(cid:48)(cid:48))2 = 41αg Trr , (5) problem without dark energy Mannheim (2001, 2011), see which can be imposed as a boundary condition Brihaye & also Nesbet (2011). Growth of structure in CG cosmology Verbin (2009). isstartingtobeinvestigatedMannheim(2012),buthasnot yet produced predictions for the CMB. The plan of this paper is as follows: In Sec. 2 we re- 2.1 Vacuum Solution : The MK metric viewthestaticsphericalsolutionsthathavebeenusedwith somesuccessMannheim(1993,1997);Mannheim&O’Brien A source-free vacuum solution requires T00 = Trr, so that (2012)tofittherotationcurvesofspiralgalaxies,largeand f(r)=0. Since small.InSec.3wediscusstheneedforaconformallycoupled (cid:16)rn+1(cid:17)(cid:48)(cid:48)(cid:48)(cid:48)=(n+1)n(n−1)(n−2)rn−3 (6) Higgs field S(r), and show that it makes a non-zero contri- butiontothesource f(r)inCG’s4th-orderPoissonequation vanishes for n=−1, 0, 1, and 2, the homogeneous 4th-order unlessithasaparticularradialprofileS(r)=S a/(r+a).In Poisson equation then integrates 4 times to give 0 Sec. 4 we stretch the MK metric with the conformal factor 2β Ω (r) = S(r)/S , and show that the linear potential used in B(r)=w− +γr−κr2 , (7) S 0 r previousfitstogalaxyrotationcurvesiseffectivelyremoved. InSec.5wesummarisethecoupledsystemofequationsthat with 4 integration constants w, β, γ and κ. The 3rd-order must be solved in order to make astrophysical tests of CG constraint 4αgWrr =Trr then gives predictions for static spherically symmetric structures. We 3r4 summarise and conclude in Sec. 6. w2=1−6βγ+ Tr . (8) 4α r g For Tr = 0, or Tr ∝ r−4, this gives 1 constraint on the 4 r r coefficients, leaving the metric with 3 parameters: β, γ, κ. ThisMannheim-Kazanas(MK)metricmatchesthesuc- 2 STATIC SPHERICAL SOLUTIONS cesses of GR’s Schwarzschild metric in the classic solar sys- For static and spherically symmetric spacetime geometries, tem tests, if we identify β = M and require |βγ| (cid:28) 1. The analytic solutions to CG include the Mannheim-Kazanas quadratic potential, −κr2, embeds the spherical structure metric Mannheim & Kazanas (1989) (MK), an extension into a curved space at large r. of GR’s Schwarzschild metric. Co-movingcoordinatesrenderWµ andTµ diagonal,giv- ν ν ing in principle 4 CG field equations. But only 2 are inde- 2.2 Rotation Curves pendent, given that spherical symmetry requires Wθ = Wφ, θ φ The MK metric’s linear potential, γr, enjoys some suc- and the Bianchi identities require a traceless Bach tensor, cesssinfittinggalaxyrotationcurvesMannheim&O’Brien Wµ=0, and hence Tµ=0. µ µ (2012). For circular orbits, the rotation curve for the MK MK show that for any static spherically symmetric metric is spacetime,aparticularconformaltransformationbringsthe β γ metric into a standard form 1, v2≡ r2θ˙2 = dln(|g00|) = rB(cid:48) = r + 2r−κr2 , (9) ds2=−B(r)dt2+ dr2 +r2dθ2+r2 sin2θdφ2 . (3) B dln(|gθθ|) 2B w− 2β +γr−κr2 B(r) r We refer to this standard form, in which −g = g = B(r), where dotted quantities denote time derivatives, e.g. θ˙ = 00 rr dθ/dt. Fig. 1 shows the metric potential B(r) and the cor- as the“MK frame”. responding rotation curve v(r) for a compact point mass With this metric ansatz, MK show that the CG field M = β = 1011M . In the weak field limit relevant to as- equationsboildowntoanexact4th-orderPoissonequation (cid:12) trophysics, B≈1, so that the three terms in the numerator for B(r): of Eqn. (9) determine the shape of the velocity curve. The 3 (cid:16)W0−Wr(cid:17)= 1 (rB)(cid:48)(cid:48)(cid:48)(cid:48)= 3 (cid:16)T0−Tr(cid:17)≡ f(r) , (4) Newtonianpotential−2β/rgivesaKeplerianrotationcurve B 0 r r 4α B 0 r v2=β/r.Thelinearpotentialγrgivesarisingrotationcurve g v2=γr/2.Aflatrotationcurveresemblingthoseobservedon where (cid:48) denotes d/dr, and f(r) is the CG source. Note that T0 =−ρ and Tr = p for a perfect fluid (representing matter 0 r and radiation) with energy density ρ and pressure p. We 2 WhileFlanagan(2006)arguethatCGisrepulsiveintheNewto- nianlimit,Mannheim(2007)showthatthismistakenconclusion arisesfromasubtletyintakingtheNewtonianlimitinisotropic coordinates,andthatαg<0giveslocallyattractivegravityinthe 1 Hereandhenceforthweadoptnaturalunits,(cid:126)=c=G=1. Newtonianlimit. MNRAS000,1–8(2016) Conformal Gravity Rotation Curves 3 5.42×10−41 cm−1, and M ≡ γ M /γ = 5.6×1010M . This 0 0 (cid:12) (cid:63) (cid:12) metric fits the rotation curves of a wide variety of spiral galaxies,replacingtheirindividualdarkmatterhalosbyjust 2 free parameters Mannheim (1993, 1997). To justify this particular form, it is argued Mannheim (1993)thatγ isgeneratedbymatterexternaltorwhileγ is 0 (cid:63) generatedbymatterinternaltor.Thisargumentisplausible butinourviewnotreallyconvincinguntilitbecomesclearer howtocalculateγ giventheexternalmatterdistributionin 0 the expanding universe. The most recent fits Mannheim & O’Brien (2012) to a sampleof111galaxiesrequireinvokingthequadraticpoten- tial−κr2tocountertherisingγrpotentialontheoutskirtsof particularly large galaxies like Malin 1. The required scale, −κ=9.54×10−54 cm−2∼−(100Mpc)−2,isplausiblyidentified with the observed size of typical structures in the cosmic web. For the 1011M point mass illustrated in Fig. 1, the (cid:12) transitionfromrisinglineartofallingquadraticpotentialhas theeffectofextendingtherelativelyflatpartoftherotation curveouttoaround100kpc,andeliminatingboundcircular orbitsoutsidethe“watershedradius”r=|γ/2κ|1/2=144kpc, where the potential B(r) reaches a maximum. 2.3 Concerns about using the MK metric GiventhenotablesuccessoftheMKmetricinfittingawide variety galaxy rotation curves with just 3 parameters, it is temptingtoconcludethatCGprovidesasimplerdescription ofgalaxydynamicsthananalternativemodelwithhundreds of individual dark matter halos. However, the vacuum has a non-zero Higgs field, with radial profile S(r). This raises two potential problems with using the MK metric’s linear potential γr to fit galaxy rotation curves. First, with a non-constant S(r), test particles Figure 1. Top panel: The MK metric potential B(r) = w− find their rest masses changing with position, causing them 2β/r+γr−κr2 (blue curve) and the associated Higgs halo S(r) to deviate from geodesics of the MK metric Mannheim (red dashed) for a point mass β = M = 1011M(cid:12), with the MK (1993b); Wood & Moreau (2001). Second, if S(r) fails to parameters (β,γ,κ) used by (Mannheim & O’Brien 2012) to fit satisfy (1/S)(cid:48)(cid:48) = 0, then f(r) is non-zero, causing the MK therotationcurvesofspiralgalaxies.Bottompanel:Circularor- bit velocity curve v2 = rB(cid:48)/2B for the MK potential B(r) (black coefficients w, β, γ, κ to be functions of r, and altering their curve), v2 =β/rB for the Newtonian potential (red dashed) and radialdependencewithinand outside extendedmass distri- v2=γr/2Bforthelinearpotential(bluedot-dash).Threefiducial butions such as galaxies and galaxy clusters. radiiattransitionsbetweentheNewtonian,linear,andquadratic TheroleofS(r)insourcing B(r)hasbeenpreviouslyin- potentials are also marked. The rotation curve is relatively flat vestigated Mannheim (2007); Brihaye & Verbin (2009), but from 10 to 100 kpc as a result of cancelling contributions from thebestCGanalysisofgalacticrotationcurvestodateMan- the three potentials. The potential B(r) has a maximum at the nheim & O’Brien (2012) omits this effect, arguing that it is watershedradiusr=|γ/2κ|≈144kpc,outsidewhichthereareno negligible. In our view conclusions about the success of CG stablecircularorbits. in fitting galaxy rotation curves are unsafe unless it can be justified to neglect radial gradients in S(r). We argue be- theoutskirtsoflargespiralgalaxiesRubin,Ford,Thonnard lowthateventhoughS(r)isverynearlyconstant,itsradial (1978) then corresponds to the transition between these re- gradientislargeenoughtosignificantlyalterpredictionsfor gimes, at r2 ∼ 2β/γ, or r ∼ 19 kpc for the case in Fig. 1. galaxyrotationcurves,andmoreovertheeffectontherota- The velocity at the transition radius, v ≈ (2βγ)1/4, can ap- tion curve is to cancel that of the linear potential. proximate the observed Tully-Fisher relation v4 ∝ M Tully Nullgeodesics(photontrajectories)areindependentof & Fisher (1977), provided γ has the right magnitude and is conformal transformations, and those of the MK metric are independent of β. wellstudiedEdery&Paranjape(1998);Pireaux(2004);Sul- To fit the rising rotation curves observed in smaller tana & Kazanas (2012); Villanueva & Olivares (2013). A dwarf galaxies, however, it is necessary to assume a rela- majorchallengetoCGisthatalinearpotentialwithγr>0 tionship between γ and M: is needed to fit galaxy rotation curves, and this produces (cid:32) (cid:33) (cid:32) (cid:33) light bending in the wrong direction, away from the central M M γ(M)=γ +γ =γ 1+ , (10) massratherthantowardit,makingitdifficulttoaccountfor 0 (cid:63) M 0 M (cid:12) 0 observedgravitationallensingeffects.However,sinceS(r)af- where M is the galaxy mass, γ = 3.06×10−30 cm−1, γ = fectsB(r),analysisoflensingbyextendedsourceslikegalax- 0 (cid:63) MNRAS000,1–8(2016) 4 Horne ies and clusters must also include the Higgs halo. We show where below that the Higgs halo S(r) outside a point mass effect- Tµ(S) = 2S;µS − 1SS;µ− 1S2Rµ ivelyeliminatestheγrpotential,sothatrotationcurvesmay ν 3 ;ν 3 ;ν 6 ν no longer constrain the sign of γ. We may then reconsider (19) using γ<0 when analysing gravitational lensing effects. −δµν (cid:16)16S;αS;α− 13SS;;αα− 112RS2+ σλ S4(cid:17) . The trace (cid:32) (cid:33) 1 3 CONFORMALLY COUPLED HIGGS FIELD Tα=ψ¯D(cid:54) ψ+σ SS;α+ RS2 −4λS4 (20) α ;α 6 VacuumsolutionsofGR,suchastheSchwarzschildandKerr metrics, assume Tµ =0, and the MK metric of CG assumes vanishes by virtue of the Dirac and Higgs equations, (15) ν f ∝ T0 −Tr = 0. However, the vacuum now has a Higgs and (16) respectively. field S0, for wr hich Tµ and/or f may well not vanish. A fam- For static spherically-symmetric fermion fields, the ν stress-energy tensor takes the form ily of analytic solutions of GR with a conformally-coupled scalar field Wehus & Ravndal (2007) includes the extreme Tµ(ψ)=diag(−ρ,p,p ,p ) , (21) ν r ⊥ ⊥ Reissner-Nordström black hole metric, with withenergydensityρ,radialpressurep,andazimuthalpres- (cid:18) M(cid:19)2 r B(r)= 1− , (11) sure p⊥. The Dirac equation (15) then gives r Tα(ψ)= p +2p −ρ=ψ¯D(cid:54) ψ=µSψ¯ψ . (22) sourced by the scalar field profile α r ⊥ (cid:32) 3 (cid:33)1/2 M WecanconsideraHiggsfieldS(r,t),allowingforapos- S(r)= . (12) sibletimedependence,withtheunderstandingthatthecon- 4π r−M sequent stress-energy tensor and gravitational source f(r) Below we discuss a similar solution for CG. mustbetimeindependentforthestaticsphericalstructures For the CG matter action ofprimaryinteresthere.AnexampleisacomplexHiggsfield (cid:90) √ with S ∝e−iωt, for which (S˙)2=−ω2S2 and S¨ =−ω2S. IM = d4x −gLM , (13) SpecialisingtotheMKmetric,theHiggsequation(16) evaluates as the Lagrangian density Mannheim (2007) is L =−σ(cid:32)S;αS − RS2(cid:33)−λS4−ψ¯ (D(cid:54) −µS)ψ . (14) SB¨ = r12 (cid:16)r2BS(cid:48)(cid:17)(cid:48)+ R6 S − 4σλS3+ pr+σ2pS⊥−ρ , (23) M 2 ;α 6 with the Ricci scalar This features a Dirac 4-spinor field ψ, with Dirac operator (cid:16) (cid:17)(cid:48)(cid:48) D(cid:54) andYukawacouplingtotheconformalHiggsfieldS with R= r2B −2 = 2(w−1) + 6γ −12κ . (24) dimensionless coupling constant µ. Note that a conformal r2 r2 r √ factor Ω stretches the volume element −gd4x by Ω4, so thatconformalsymmetryrequiresL ∝Ω−4.WithS ∝Ω−1, M conformalsymmetryholdsforS;αS −RS2/6,forthequartic ;α 4 THE HIGGS FRAME self-couplingpotentialλS4,andaswellforthefermionterms with ψ∝Ω−3/2. By rescaling S the dimensionless parameter A conformal transformation Ω maps the Higgs field S to σcanbesetto+1fora“right-sign”or−1fora“wrong-sign” S →S˜ =Ω−1S . (25) scalar field kinetic energy. Varying IM with respect to ψ gives the Dirac equation Transforming to the“Higgs frame”, where S˜ = S0, requires D(cid:54) ψ=µSψ , (15) the specific conformal factor ΩS(x) = S(x)/S0. In the static spherical geometry, given any CG solution B(r) and S(r) in with the fermion mass mass m = µS induced by Yukawa the MK frame, where −g = 1/g = B(r), we can“stretch” 00 rr coupling. Varying I with respect to S gives the 2nd-order the metric into the Higgs frame: M Higgs equation (cid:32)S (cid:33)2 S;;αα= √1−g (cid:16)√−ggαβS,β(cid:17),α=−R6 S + 4σλS3− σµ ψ¯ψ . (16) gµν →g˜µν =Ω2gµν = S0 gµν . (26) IntheHiggsframe,testparticleshavespace-timeindepend- This is the Klein-Gordon equation in curved spacetime, for ent rest masses, m˜ =µS , and thus they follow geodesics of a massless scalar field S with a fermion source µψ¯ψ/σ and 0 thisstretchedmetricg˜ ,ratherthanthoseoftheMKmetric µν a space-time dependent“Mexican hat”potential g . µν R λ V(S)=− S2+ S4 . (17) 12 σ 4.1 Vacuum Stability and Spontaneous Symmetry Varying I with respect to the metric gives the con- Breaking M formal stress-energy tensor, with mixed components Using a tilde to denote the Higgs-frame counterparts of the 2 δI MK-frameHiggsandfermionfields,wehaveS˜ =Ω−1S =S , Tµ≡ √ gµα M =Tµ(ψ)+σTµ(S) , (18) 0 ν −g δgαν ν ν and ψ˜ = Ω−3/2ψ = (S/S0)−3/2 ψ. The fermion stress-energy MNRAS000,1–8(2016) Conformal Gravity Rotation Curves 5 components are then (ρ˜,p˜ ,p˜ ) = (ρ,p,p )(S/S )−4. The where h˜ ≡ ha/(a+h), r˜ ≡ ra/(a+r), and K ≡ −2λS2. This r ⊥ r ⊥ 0 0 MK-frame Higgs equation is then metric has a Schwarzschild-like horzon, with B(r) ∝ (r−h) S¨ = 1 (cid:16)r2BS(cid:48)(cid:17)(cid:48)+ RS −4λ¯S3 , (27) vanisEhxinpganadtinrg=Ehq.n.(33)inpowersofr,wecanreadoffthe B r2 6 3independentMKparameters(β,γ,κ)intermsof the3BV where we define parameters (h,a,K): λ ρ˜−p˜ −2p˜ λ¯ ≡ σ + 4σr S4 ⊥ . (28) 2β=h˜ (cid:16)1−Kh˜2(cid:17) , (34) 0 Note that the fermions effectively strengthen the quartic 1(cid:32) h˜ (cid:16) (cid:17)(cid:33) Higgs self-coupling constant. The corresponding Higgs po- γ= 2−3 1−Kh˜2 , (35) tential is a a V(S)=−R S2+λ¯S4=λ¯ (cid:32)S2− R (cid:33)2− R2 . (29) 1 (cid:32) h˜ (cid:16) (cid:17)(cid:33) 12 24λ¯ 576λ¯ κ=K− 1− 1−Kh˜2 . (36) a2 a A stable vacuum in the MK-frame requires λ¯ > 0, so that From these one can verify that the constant term, V(S)isboundedfrombelow.Spontaneoussymmetrybreak- ing to induce non-zero fermion masses can then occur for h˜ (cid:16) (cid:17) 6β w=1−3 1−Kh˜2 =γa−1=1− , (37) positive curvature R > 0. The minimum of V(S) occurs a a at S2 = R/24λ¯. This gives the vacuum energy density satisfies the Wr = 0 constraint w2 = 1−6βγ. The inverse V(S)=−R2/576λ¯ =−λ¯S4. r relations are: Note that λ¯ > 0 and R > 0 are not required, however, 1+w 6β since a time-dependent S(r,t) in the MK frame corresponds a= = , (38) to a constant S in the Higgs frame. γ 1−w 0 (cid:18) γ (cid:19)2 (cid:18) γ (cid:19)3 4.2 Source-Free Solution : The BV Metric K=κ+ 1+w −2β 1+w , (39) Thesource f(r)inCG’s4-thorderPoissonequationincludes andfinally,the horizon radius h, where B(r) vanishes, is the both fermion and Higgs contributions Mannheim (2007); smallest positive real root of the cubic Brihaye & Verbin (2009): 0=−2β+wr+γr2−κr3 . (40) 3 (cid:32)(cid:32)1(cid:33)(cid:48)(cid:48) 1 (cid:32)1(cid:33)··(cid:33) 4αg f(r)= B (pr−ρ)+σS3 S + B2 S . (30) The MK and BV metrics are thus equivalent, repres- entingthesame3-parametersource-freesolutiontotheCG TheHiggsfieldS(r,t)makesnoexplicitcontributionto f(r) field equations. However, as BV show, the Higgs field has if and only if it takes the specific form a radial profile S(r). Massive test particles therefore do not S t a follow the time-like geodesics of the MK metric. S(r,t)= 0 0 , (31) (t+t )(r+a) Fortunately,sinceweknowS(r),weknowtheconformal 0 declining from S at time t = 0 and radius r = 0 with a transformationbetweentheMKframeandtheHiggsframe: 0 timescale t0 and radial length scale a. This holds for either S = S0a →S˜ =Ω−1S =S , (41) sign σ. r+a 0 The time dependence included here may have applica- tions, for example when embedding static spherical struc- (cid:32)S (cid:33)2 (cid:18) a (cid:19)2 turesinanexpandinguniverse,witht0∼1/H0,orforacom- gµν →g˜µν =Ω2gµν = S gµν = r+a gµν . (42) plex Higgs field varying as S(r,t)=S(r)e−iωt. We set t =∞ 0 0 The stretched metric’s circumferential radius to focus on static solutions. NoteinEqn.(30)thatastatic“Higgshalo”S(r)makesa r˜≡ (cid:112)|g˜ |= rS = ra (43) contributionto f(r)thatdoesnotdependonB(r).Thuswhen θθ S r+a 0 S(r)isknownitisstraightforwardtointegratethe4th-order maps 0<r<∞ to 0<r˜<a. The stretched metric has Poisson equation (4) to determine the corresponding B(r). One cannot specify an arbitrary S(r), however, since B(r) |g˜ |=(cid:18) a (cid:19)2 B(r)=1− 2M −Kr˜2 , (44) appearsintheHiggsequation(23)forS(r).Remarkably,the 00 r+a r˜ MKpotential B(r)andsource-freeS(r)doadmitananalytic featuring a Newtonian potential with mass solution Brihaye & Verbin (2009), as we see below. h(cid:16) (cid:17) Brihaye & Verbin (2009) (BV) use numerical methods M= 1+Kh2 (45) 2 to investigate static spherical solutions of CG with various assumptions about f(r). Among these BV identify one 3- embedded in an external space with curvature K. The con- parameter analytic solution with a source-free scalar field, formal transformation does not move the horizon at r = h, which remains at r˜=h˜. S a S(r)= 0 , (32) Note, however, that while the original MK metric g r+a µν has a linear potential γr, the corresponding Higgs-frame for which the MK potential is metric g˜ has no term in g˜ linear in r˜. Thus even though (cid:18)a+r(cid:19)2(cid:32) h˜ (cid:33) (cid:32) h˜3 (cid:33) the Higgµsν field S(r) decline0s0 only slightly from its central B(r)= 1− −Kr2 1− , (33) a r˜(r) r˜3(r) value S0, this has a significant effect on the shape of the MNRAS000,1–8(2016) 6 Horne potential and the resulting rotation curve. With B rising as B≈1+γr,S fallsasS/S ≈1−γr/2,sothatS2Blacksalinear 0 potential.Whilethisresultisdemonstratedhereforapoint mass,ratherthanforamorerealisticextendedsourcestruc- ture,itindicatesthepotentialdangerwhenusingthelinear potential in the MK metric to fit galaxy rotation curves. Fig. 2 further illustrates this point by showing the Higgs-frame potential (S/S )2B, and the corresponding ro- 0 tation curve, for the same 1011M point mass as in Fig. 1. (cid:12) TheHiggsfieldisconstant,bydefinition,intheHiggsframe. Because a ≈ 2/γ = 7.6×1010 pc is by far the longest scale in the problem, r and r˜ = ra/(r+a) are nearly identical, and the Higgs-frame mass M and curvature K are essen- tially unchanged from their MK-frame counterparts β and κ. The Higgs-frame potential (S/S )2B retains the Newto- 0 nian potential −2β/r˜ and the quadratic potential −κr˜2, but lacks a linear potential term. The rotation curve thus follows a Keplerian profile out to ∼ 20 kpc, bending down as the quadratic potential takes hold, and the watershed radius, where −g˜ = (S/S )2B has a maximum, is now at 00 0 r˜=|β/κ|1/3=37.5 kpc. 5 ASTROPHYSICAL TESTS ToreallytestCGwithastrophysicalobservationsisconsid- erably harder than simply using geodesics of the MK metric with B(r) sourced by matter. The source f(r) in CG’s 4th-orderPoissonequationmustincludecontributionsfrom the Higgs halo S(r), in addition to those from the matter (+radiation) energy density ρ(r) and pressure p(r). These are specified in the Higgs frame, ρ˜(r˜) and p˜(r˜), and moved to the MK frame using r˜ = rS(r)/S , ρ(r) = (S/S )4ρ˜(r˜) 0 0 and p(r) = (S/S )4p˜(r˜). For example, to model spherical 0 structuressimilartothematterdistributioningalaxiesand Figure2. SameasinFig.1butafteraconformaltransformation clusters,itmaybeappropriatetoadoptaHearnquistprofile Ω(r)=S(r)/S0 stretchesthegeometryfromtheMKframe,where Hearnquist (1990) B(r) = −g00 = 1/grr, to the Higgs frame, where the Higgs field is ρ constant.TheHiggs-framepotentialB˜≡(S/S0)2Bhasamaximum ρ˜(r˜)= x(x+01)3 , (46) awthtichhetwhaetreersahreednroadsituasblr˜e≡ci(rSc/uSl0a)rro=rb|βit/sκ.|1/N2o≈te37t.h5aktpBc(,r)ouhtassidae with x = r˜/r in units of the scale radius r , and with ρ = rising linear potential γr, but this is effectively canceled by the 0 0 0 M/(2πr3) for total mass M. The enclosed mass profile is decline in S2, so that the rotation curve is Keplerian out to ∼ 0 20kpc. (cid:18) x (cid:19)2 M(r˜)=M . (47) x+1 r2 This Hearnquist profile ρ˜(r˜) is specified in the Higgs frame, γ(cid:48)=− f(r) , γ(0)=γ , (51) 2 0 then scaled by (S/S )4 for use in the MK frame where the 0 4th-order Poisson equation and 2nd-order Higgs equation r are more easily solved. κ(cid:48)=−6 f(r) , κ(∞)=κ∞ , (52) In the MK frame, B(r) and S(r) satisfy their equations of motion. The 4th-order Poisson equation for B(r), r3 3r4 4αg (rB)(cid:48)(cid:48)(cid:48)(cid:48) =σS3(cid:18)(cid:16)1(cid:17)(cid:48)(cid:48)+ 1 (cid:16)1(cid:17)··(cid:19) w(cid:48)= 2 f(r) , w2(r)=1−6β(r)γ(r)+ 4αg Trr(r) . (53) r − 3 (cid:18)S (cid:19)S4 (ρ˜+Bp˜)2≡S4α f(r) , (48) The MK parameters, β(r), γ(r), κ(r), w(r), are then internal B S g and/or external moments of f(r). For non-singular struc- 0 tures, appropriate boundary conditions at the origin are is convenient because solutions of the form β(0) = 0 and γ(0) = 0, though non-zero values may also be 2β(r) B(r)=w(r)− +γ(r)r−κ(r)r2 (49) chosenforanunresolvedcentralsourcesuchasanucleon,a r star, or a black hole. The curvature of the external 3-space can be found for extended sources f(r) by integrating 1st- issetbyκ(∞).The3rd-orderconstraintonwcanbesetany order equations, with appropriate boundary conditions: radius where Tr is known. r r4 Note that even for an extended source f(r), the first 3 β(cid:48)= 12 f(r) , β(0)=β0 , (50) derivatives of B(r) evaluate as if the MK parameters were MNRAS000,1–8(2016) Conformal Gravity Rotation Curves 7 r-independent. For example: to modelling galaxies and clusters can fit the light bending B(cid:48) = 2r2β +γ−2κr− 2rβ(cid:48) +w(cid:48)+γ(cid:48)r−κ(cid:48)r2 ahnogpleestofraodmdrleesnssitnhgisaisnwfuetlulraeswtohrek.flat rotation curves. We (54) = 2β +γ−2κr+(cid:16)−1 + 1 − 1 + 1(cid:17)r3 f . r2 6 2 2 6 6 CONCLUSIONS As a consequence, the Ricci scalar remains The 4th-order field equations of Conformal Gravity have 2(w(r)−1) 6γ(r) R= + −12κ(r) , (55) vacuum solutions that augment the Schwarzschild metric r2 r with linear and quadratic potentials Mannheim & Kazanas and the 3rd-order constraint remains (1989). This MK metric has been used to fit the rotation w(r)2+6β(r)γ(r)−1 1 curves of a wide variety of galaxies with only 3 free para- Wr = = Tr . (56) r 3r4 4αg r meters Mannheim & O’Brien (2012). We highlight two potential problems with using Here Tr includes radial pressure from both matter (+radi- r geodesicsoftheMKmetrictostudyrotationcurvesofgalax- ation) and from the Higgs halo, ies. First, the MK metric is a source-free solution to CG’s (cid:32)S (cid:33)4 4th-order Poisson equation, but the conformally-coupled Trr = p˜r S +σTrr(S) , (57) Higgs field makes an extended halo S(r) that also contrib- 0 utestothegravitationalsourceunlessithasaspecificradial with the Higgs halo contribution being, profile,S(r)=S a/(r+a).Second,sinceparticlemassesscale 0 Tr(S) = (cid:16)S˙(cid:17)2−2SS¨ + B(S(cid:48))2+ SS(cid:48) (cid:16)B(cid:48)+ 4B(cid:17) woffithgetohdeeHsicigsgosffitehlde,MthKeHmiegtgrsich.aloS(r)pushestestparticles r + S62 (cid:18)B6r(cid:48)B+ Br−21(cid:19)2− σλ S4 . 6 r (58) Ω(r)T=oSa(dr)d/rSe0sssttrheetsceheisssuthees,mweetrnioctteothaaftoramcotnhfaotrmmaalkfeasctthoer Higgs field constant. Test particles then follow geodesics of Because f(r) depends on B(r) and S(r), the moment in- this stretched“Higgs-frame”metric. tegralsfor B(r)mustbeiteratedalongwithsolvingthe2nd- For the analytic solution to the source-free CG equa- order MK-frame Higgs equation for S(r): tions Brihaye & Verbin (2009), which is equivalent to the (cid:32) (cid:33) SB¨ = r12 (cid:16)r2BS(cid:48)(cid:17)(cid:48)+ R6 S −4 σλ + ρ˜−4p˜σr−S42p˜⊥ S3 , (59) MtoKthmeHetirgigc,swfreamfinedistthoaetlitmheineaffteecttheoflisnteraertcphointegntthiaelmtheattriics 0 with boundary conditions S(0)=S and S(cid:48)(0)=S . usedtofitgalaxyrotationcurves.Thustheremarkableres- 0 1 ults of Mannheim & O’Brien (2012), using geodesics of the Having solved for B(r) and S(r) in the MK frame, we MK metric to fit a large variety of galaxy rotation curves movebacktotheHiggsframe,andusegeodesicsoftheres- ulting Higgs-frame metric g˜ = (S(r)/S )2g to test CG in with just 3 parameters, may be testing an empirical model µν 0 µν rather than the actual CG predictions. 3 ways: We collect the equations and outline the procedure for 1. galaxy rotation curves. astrophysicaltestsofCGinstaticsphericalgeometries.Spe- 2. galaxy cluster potentials probed by X-ray gas. cifically, the sources for CG’s 4th-order Poisson equation 3. lensing by galaxies and galaxy clusters. include not only the energy density and pressure of dis- For example, the circular orbit rotation curve is: tributed matter (stars+gas), and radiation if relevant, but (cid:16) (cid:17) dln(|g˜ |) dln S B1/2 v2 +v2 also the associated Higgs halo S(r). The resulting MK met- v2= 00 = = B S , (60) dln(|g˜θθ|) dln(Sr) 1+v2S rg˜ic g=µν(mS/uSst)2thgen. bTeh“estHreigtgchs-efdra”mtoe tgheeodHeisgicgss tfrhaemnepmroevtirdiec µν 0 µν where predictions for testing CG against observations. β γ v2 = rB(cid:48) = r + 2r−κr2 (61) B 2B 2β Acknowledgements w− +γr−κr2 r KH would like to thank Amy Deacon, Aidan Farrell and is the rotation curve arising from the MK potential B(r), as Indar Ramnarine for hospitality at the University of the used by Mannheim & O’Brien (2012), and v2S ≡ rS(cid:48)/S im- West Indies in Trinidad, Alistair Hodson, Alasdair Leggat, plements the corrections arising from the Higgs halo profile and Carl Roberts for helpful comments on the present- S(r). ation, an anonymous referee for thought provoking criti- One of the objections to CG is that for the MK metric cism, and the UK Science and Technology Facilities Coun- withγ>0thelinearpotentialcauseslightraystobendaway cil(STFC)forfinancialsupportthroughconsolidatedgrant from the point mass, rather than toward it. But our results ST/M001296/1. show that a rising MK-frame potential B(r) is compensated by a corresponding decline in the Higgs halo S(r). In light of this, the galaxy rotation curves may not in fact require References γ>0,andwemaynowreconsideradjustingthestrengthand sign of the linear potential when testing CG predictions for Brihaye,Y.,Verbin,Y.,2009,Phys.Rev.D,80,124048 gravitational bending of light rays. It remains to be shown Edery,A.,Paranjape,M.B.,1998,Phys.Rev.D,58,24011 whether the effect of an extended source f(r) appropriate Flanagan,E.,2006,Phys.Rev.D,74,023002 MNRAS000,1–8(2016) 8 Horne Hearnquist,L.,1990,ApJ,356,359 Mannheim,P.D.,1993,ApJ,419,150 Mannheim,P.D.,1993,Gen.Rel.Grav.,25,697 Mannheim,P.D.,1997,ApJ,479,659 Mannheim,P.D.,2001,ApJ,561,1 Mannheim,P.D.,2003,IJMPD,12,893 Mannheim,P.D.,2006,Prog.Part.Nucl.Phys.,56,340 Mannheim,P.D.,2007,Phys.Rev.D,75,124006 Mannheim,P.D.,2011,Gen.Rel.Grav.,43,703 Mannheim,P.D.,2012,Phys.Rev.D,85,124008 Mannheim,P.D.,Kazanas,D.,1989,ApJ,342,635 Mannheim,P.D.,Kazanas,D.,1991,Phys.Rev.D,44,417 Mannheim,P.D.,O’Brien,J.G.,2012,Phys.Rev.D,85,124020 Nesbet,R.K.,2011,MPLA,26,893 Pireaux,S.,2004,Class.Quant.Grav.,21,1897 Rubin,V.C.,Ford,W.K.,Thonnard,N.,1978,ApJL,225,L107 Sultana,J.,Kazanas,D.,2010,Phys.Rev.D,81,7502 Tully,R.B.,Fisher,J.R.,1977,A&A,54,661 Villanueva,J.R.&Olivares,M.,2013,JCAP,06,40 Wehus,I.K.,Ravndal,F.,2007,J.Phys.Conf.Ser,66,012044 Wood,J.,Moreau,W.,2001,arXiv:gr-qc/0102056 ThispaperhasbeentypesetfromaTEX/LATEXfilepreparedby theauthor. 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