Conformal Gravity Rotation Curves with a Conformal Higgs Halo

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provided by St Andrews Research Repository MNRAS 458, 4122–4128 (2016) doi:10.1093/mnras/stw506 Advance Access publication 2016 March 7

Conformal Gravity rotation curves with a conformal Higgs halo

Keith Horne‹ SUPA Physics and Astronomy, University of St Andrews, KY16 9SS Scotland, UK

Accepted 2016 February 28. Received 2016 February 25; in original form 2016 January 27

ABSTRACT We discuss the effect of a conformally coupled Higgs ﬁeld on conformal gravity (CG) predictions for the rotation curves of galaxies. The Mannheim–Kazanas (MK) metric is a valid

vacuum solution of CG’s fourth-order Poisson equation if and only if the Higgs ﬁeld has a Downloaded from particular radial proﬁle, S(r) = S0 a/(r + a), decreasing from S0 at r = 0 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 gμν , as previously assumed, but rather those of the Higgs-frame 2 MK metric g˜μν = gμν , with the conformal factor (r) = S(r)/S0. We show that the required

stretching of the MK metric exactly cancels the linear potential that has been invoked to fit http://mnras.oxfordjournals.org/ galaxy rotation curves without dark matter. We also formulate, for spherical structures with a Higgs halo S(r), the CG equations that must be solved for viable astrophysical tests of CG using galaxy and cluster dynamics and lensing. Key words: gravitation – galaxies: kinematics and dynamics – cosmology: theory – dark energy – dark matter. at University of St Andrews on April 11, 2016 are excluded in CG because and G build in fundamental scales, 1 INTRODUCTION and R violates conformal symmetry. Instead the CG action allows The need for dark matter and dark energy to reconcile Einstein’s only conformally invariant scalars linked by dimensionless cou- general relativity (GR) with observations, together with the lack pling constants. R can appear if it is coupled to a scalar field S in of other tangible evidence for their existence, motivates the study ;μ − R 2 the particular conformally invariant combination S S;μ 6 S . of alternative gravity theories aiming to achieve similar success Particle rest masses cannot be fundamental, but may instead arise without resort to the dark sector. through Yukawa couplings to the conformal Higgs field S.Like- Conformal gravity (CG), like GR, employs a metric to describe wise, the Higgs mass cannot be fundamental, but may arise through gravity as curved space–time. But the CG field equations, which dynamical symmetry breaking. Conformal invariance replaces Ein- dictate how matter and energy generate space–time curvature, arise stein’s second-order field equations with the fourth-order CG field from a local symmetry principle, conformal symmetry, which holds equations (Mannheim & Kazanas 1989; Mannheim 2006), for the strong, weak and electro-magnetic interactions, but is vio- = lated by GR. Conformal symmetry means that stretching the met- 4 αg Wμν Tμν , (2) ric by a factor 2(x), and scaling all other fields by appropriate where αg is a dimensionless coupling constant. The Bach tensor powers of , has no physical effects. In particular, local confor- Wμν and stress–energy tensor Tμν are both traceless, and scale as mal transformations preserve all angles and the causal relations −4. among space–time events, but physical distances, time intervals, Despite their complexity, the fourth-order CG field equations and masses change, so that only local ratios of these quantities have admit analytic solutions for systems with sufficient symmetry physical significance. (Mannheim 2006). For homogeneous and isotropic space–times Unlike GR, and many related alternative gravity theories, terms (Mannheim 2001), the Robertson–Walker metric is a solution with allowed in the CG action are highly restricted by the required confor- Wμν = 0, providing a dynamical cosmological model identical mal symmetry. GR’s Einstein–Hilbert action adopts the Ricci scalar to that of GR, except that the Friedmann equation has a nega- R, leading to Einstein’s famous second-order field equations, =− π 2 tive effective gravitational constant Geff 3/4 S0 ,whereS0 is the vacuum expectation value of a conformally coupled scalar Gμν + gμν =−8 π GTμν , (1) field with vacuum energy density λS4. This CG cosmology gives = − R 0 where Gμν Rμν ( /2) gμν is the Einstein tensor, G is New- an open universe, but it can fit luminosity distances from super- ton’s constant, and is the cosmological constant. These terms novae (Mannheim 2003) and features cosmic acceleration with 0 < < 1, neatly solving the cosmological constant problem without dark energy (Mannheim 2001, 2011; see also Nesbet 2011). E-mail: [email protected] Growth of structure in CG cosmology is starting to be investigated

C 2016 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society Conformal gravity rotation curves 4123

(Mannheim 2012), but has not yet produced predictions for the 2.1 Vacuum solution: the MK metric cosmic microwave background. A source-free vacuum solution requires T 0 = T r ,sothatf(r) = 0. The plan of this paper is as follows: in Section 2, we review the 0 r Since static spherical solutions that have been used with some success (Mannheim 1993b, 1997; Mannheim & O’Brien 2012)toﬁtthe rn+1 = (n + 1) n (n − 1) (n − 2) rn−3 (6) rotation curves of spiral galaxies, large and small. In Section 3, we =− discuss the need for a conformally coupled Higgs ﬁeld S(r), and vanishes for n 1, 0, 1, and 2, the homogeneous fourth-order show that it makes a non-zero contribution to the source f(r)in Poisson equation then integrates four times to give

CG’s fourth-order Poisson equation unless it has a particular radial 2 β 2 = + B(r) = w − + γr− κr , (7) profile S(r) S0 a/(r a). In Section 4, we stretch the Mannheim– r Kazanas (MK) metric with the conformal factor (r) = S(r)/S , S 0 with four integration constants w, β, γ ,andκ. The third-order and show that the linear potential used in previous fits to galaxy constraint 4 α W r = T r then gives rotation curves is effectively removed. In Section 5, we summarize g r r the coupled system of equations that must be solved in order to 3 r4 2 = − + r make astrophysical tests of CG predictions for static spherically w 1 6 βγ Tr . (8) 4 αg symmetric structures. We summarize and conclude in Section 6. − r = r ∝ 4 Downloaded from For Tr 0, or Tr r , this gives one constraint on the four coefficients, leaving the metric with three parameters: β, γ , κ. 2 STATIC SPHERICAL SOLUTIONS This MK metric matches the successes of GR’s Schwarzschild = For static and spherically symmetric space–time geometries, ana- metric in the classic Solar system tests, if we identify β M | | − 2 lytic solutions to CG include the MK metric (Mannheim & Kazanas and require βγ 1. The quadratic potential, κ r , embeds the spherical structure into a curved space at large r. 1989), an extension of GR’s Schwarzschild metric. Analytic so- http://mnras.oxfordjournals.org/ lutions are also available for charged and rotating black holes (Mannheim & Kazanas 1991). 2.2 Rotation curves μ μ Comoving coordinates render Wν and Tν diagonal, giving in principle four CG field equations. But only two are independent, The MK metric’s linear potential, γ r, enjoys some success in fitting θ = φ galaxy rotation curves (Mannheim & O’Brien 2012). For circular given that spherical symmetry requires Wθ Wφ , and the Bianchi μ = orbits, the rotation curve for the MK metric is identities require a traceless Bach tensor, Wμ 0, and hence μ = Tμ 0. β γ + − 2 MK show that for any static spherically symmetric space–time, r2 θ˙ 2 dln(|g |) rB r κr v2 ≡ = 00 = = r 2 , a particular conformal transformation brings the metric into a stan- | | 2 β B dln( gθθ ) 2 B 2 at University of St Andrews on April 11, 2016 1 w − + γr− κr dard form, r dr2 (9) ds2 =−B(r)dt 2 + + r2 dθ 2 + r2 sin2 θ dφ2 . (3) B(r) where dotted quantities denote time derivatives, e.g. θ˙ = dθ/dt. Fig. 1 shows the metric potential B(r) and the corresponding rotation We refer to this standard form, in which −g00 = 1/grr = B(r), as the 11 ‘MK frame’. curve v(r) for a compact point mass M = β = 10 M. In the weak With this metric ansatz, MK show that the CG field equations field limit relevant to astrophysics, B ≈ 1, so that the three terms in boil down to an exact fourth-order Poisson equation for B(r): the numerator of equation (9) determine the shape of the velocity curve. The Newtonian potential −2 β/r gives a Keplerian rotation 3 1 3 0 − r = = 0 − r ≡ curve v2 = β/r. The linear potential γ r gives a rising rotation W0 Wr (rB) T0 Tr f (r) , (4) B r 4 αg B curve v2 = γ r/2. A flat rotation curve resembling those observed 0 =− on the outskirts of large spiral galaxies (Rubin, Ford & Thonnard where denotes d/dr,andf(r) is the CG source. Note that T0 ρ and T r = p for a perfect fluid (representing matter and radiation) 1978) then corresponds to the transition between these regimes, at r 2 ∼ ∼ with energy density ρ and pressure p. We then require α < 0so r 2 β/γ,orr 19 kpc for the case in Fig. 1. The velocity at g ≈ 1/4 that a localized source with ρ − p > 0 generates attractive gravity.2 the transition radius, v (2 βγ) , can approximate the observed 4 ∝ The remaining constraint can then be the third-order equation Tully–Fisher relation v M (Tully & Fisher 1977), provided γ has the right magnitude and is independent of β. 2 1 − B2 2 BB BB + B To fit the rising rotation curves observed in smaller dwarf galax- W r = + − r 3 r4 3 r3 3r2 ies, however, it is necessary to assume a relationship between γ and M: 2 B B − BB 2 B B − B 1 + + = r M M Tr , (5) = + = + 3 r 12 4 αg γ (M) γ0 γ γ0 1 , (10) M M0 which can be imposed as a boundary condition (Brihaye & Verbin −30 −1 where M is the galaxy mass, γ 0 = 3.06 × 10 cm , 2009). −41 −1 10 γ = 5.42 × 10 cm ,andM0 ≡ γ 0 M/γ = 5.6 × 10 M. This metric fits the rotation curves of a wide variety of spiral galax- 1 Here and henceforth, we adopt natural units, = c = G = 1. ies, replacing their individual dark matter haloes by just two free 2 While Flanagan (2006) argues that CG is repulsive in the Newtonian limit, parameters (Mannheim 1993b, 1997). Mannheim (2007) shows that this mistaken conclusion arises from a subtlety To justify this particular form, it is argued (Mannheim 1993b) in taking the Newtonian limit in isotropic coordinates, and that αg < 0gives that γ 0 is generated by matter external to r while γ is generated locally attractive gravity in the Newtonian limit. by matter internal to r. This argument is plausible but in our view

MNRAS 458, 4122–4128 (2016) 4124 K. Horne

2.3 Concerns about using the MK metric Given the notable success of the MK metric in fitting a wide variety galaxy rotation curves with just three parameters, it is tempting to conclude that CG provides a simpler description of galaxy dynamics than an alternative model with hundreds of individual dark matter haloes. 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 γ r to fit galaxy rotation curves. First, with a non-constant S(r), test particles find their rest masses changing with position, causing them to deviate from geodesics of the MK metric (Mannheim 1993a; Wood & Moreau 2001). Secondly, if S(r) fails to satisfy (1/S) = 0, then f(r) is non-zero, causing the MK coefficients w, β, γ , κ to be functions of r, and altering their radial dependence within and outside extended mass distributions such as

galaxies and galaxy clusters. Downloaded from TheroleofS(r) in sourcing B(r) has been previously investigated (Mannheim 2007; Brihaye & Verbin 2009), but the best CG analysis of galactic rotation curves to date (Mannheim & O’Brien 2012) omits this effect, arguing that it is negligible. In our view conclusions about the success of CG in fitting galaxy rotation curves are unsafe unless it can be justified to neglect radial gradients in S(r). We http://mnras.oxfordjournals.org/ argue below that even though S(r) is very nearly constant, its radial gradient is large enough to significantly alter predictions for galaxy rotation curves, and moreover the effect on the rotation curve is to cancel that of the linear potential. Null geodesics (photon trajectories) are independent of conformal transformations, and those of the MK metric are well studied (Edery & Paranjape 1998; Pireaux 2004; Sultana & Kazanas 2012; Villanueva & Olivares 2013). A major challenge to CG is that a

linear potential with γ r > 0 is needed to fit galaxy rotation curves, at University of St Andrews on April 11, 2016 and this produces light bending in the wrong direction, away from the central mass rather than towards it, making it difficult to account Figure 1. Top panel: the MK metric potential B(r) = w − 2 β/r + γ r − κ r2 for observed gravitational lensing effects. However, since S(r)af- (blue curve) and the associated Higgs halo S(r) (red dashed) for a point mass fects B(r), analysis of lensing by extended sources like galaxies 11 β = M = 10 M, with the MK parameters (β,γ ,κ) used by Mannheim & and clusters must also include the Higgs halo. We show below that O’Brien (2012) to fit the rotation curves of spiral galaxies. Bottom panel: the Higgs halo S(r) outside a point mass effectively eliminates the circular orbit velocity curve v2 = rB/2 B for the MK potential B(r) (black γ r potential, so that rotation curves may no longer constrain the curve), v2 = β/rBfor the Newtonian potential (red dashed) and v2 = γ r/2 B for the linear potential (blue dot–dashed). Three fiducial radii at transitions sign of γ . We may then reconsider using γ<0 when analysing between the Newtonian, linear, and quadratic potentials are also marked. The gravitational lensing effects. rotation curve is relatively flat from 10 to 100 kpc as a result of cancelling contributions from the three potentials. The potential B(r) has a maximum 3 CONFORMALLY COUPLED HIGGS FIELD at the watershed radius r =|γ/2 κ|≈144 kpc, outside which there are no stable circular orbits. Vacuum solutions of GR, such as the Schwarzschild and Kerr μ = metrics, assume Tν 0, and the MK metric of CG assumes ∝ 0 − r = f T0 Tr 0. However, the vacuum now has a Higgs field μ S, for which Tν and/or f may well not vanish. A family of analytic solutions of GR with a conformally coupled scalar field (Wehus not really convincing until it becomes clearer how to calculate γ 0 given the external matter distribution in the expanding universe. & Ravndal 2007) includes the extreme Reissner–Nordstrom¨ black hole metric, with The most recent fits (Mannheim & O’Brien 2012)toasam- ple of 111 galaxies require invoking the quadratic potential M 2 − 2 B(r) = 1 − , (11) κr to counter the rising γ r potential on the outskirts of r particularly large galaxies like Malin 1. The required scale, −κ = 9.54 × 10−54 cm−2 ∼ (100 Mpc)−2, is plausibly identi- sourced by the scalar field profile fied with the observed size of typical structures in the cosmic web. 3 1/2 M For the 1011M point mass illustrated in Fig. 1, the transition from S(r) = . (12) 4 π r − M rising linear to falling quadratic potential has the effect of extending the relatively flat part of the rotation curve out to around 100 kpc, Below we discuss a similar solution for CG. For the CG matter action and eliminating bound circular orbits outside the ‘watershed radius’ r =|γ/2 κ|1/2 = 144 kpc, where the potential B(r) reaches a √ = 4 − L maximum. IM d x g M , (13)

MNRAS 458, 4122–4128 (2016) Conformal gravity rotation curves 4125 the Lagrangian density Mannheim (2007)is with the Ricci scalar R 2 − σ ;α 2 4 r B 2 2 (w − 1) 6 γ LM =− S S;α − S − λS − ψ¯ (D − μS) ψ. (14) R = = + − 12 κ. (24) 2 6 r2 r2 r This features a Dirac four-spinor field ψ, with Dirac operator D and Yukawa coupling to the conformal Higgs field S with dimension- 4 THE HIGGS FRAME less coupling constant√μ. Note that a conformal factor stretches the volume element −g d4x by 4, so that conformal symmetry A conformal transformation maps the Higgs field S to L ∝ −4 ∝ −1 requires M .WithS , conformal symmetry holds for → ˜ = −1 ;α 2 4 S S S. (25) S S;α − R S /6, for the quartic self-coupling potential λ S ,and ∝ −3/2 as well for the fermion terms with ψ . By rescaling S the Transforming to the ‘Higgs frame’, where S˜ = S0, requires the + − dimensionless parameter σ can be set to 1 for a ‘right-sign’ or 1 specific conformal factor S(x) = S(x)/S0. In the static spherical for a ‘wrong-sign’ scalar field kinetic energy. geometry, given any CG solution B(r)andS(r) in the MK frame, Varying IM with respect to ψ gives the Dirac equation where −g00 = 1/grr = B(r), we can ‘stretch’ the metric into the D = Higgs frame: ψ μSψ, (15) 2 with the fermion mass m = μ S induced by Yukawa coupling. Vary- 2 S Downloaded from gμν → g˜μν = gμν = gμν . (26) ing IM with respect to S gives the second-order Higgs equation S0 √ R In the Higgs frame, test particles have space–time independent rest ;α 1 αβ 4 λ 3 μ S = √ −gg S,β =− S + S − ψψ.¯ = ;α −g ,α 6 σ σ masses, m˜ μS0, and thus they follow geodesics of this stretched (16) metric g˜μν , rather than those of the MK metric gμν . http://mnras.oxfordjournals.org/ This is the Klein–Gordon equation in curved space–time, for a massless scalar field S with a fermion source μ ψψ/σ¯ and a space– 4.1 Vacuum stability and spontaneous symmetry breaking time dependent ‘Mexican hat’ potential Using a tilde to denote the Higgs-frame counterparts of the MK- ˜ = −1 = ˜ = R λ frame Higgs and fermion fields, we have S S S0,andψ V (S) =− S2 + S4 . (17) −3/2 = −3/2 12 σ ψ (S/S0) ψ. The fermion stress–energy components −4 are then (ρ,˜ p˜ , p˜ ⊥) = (ρ,p ,p⊥) (S/S ) . The MK-frame Higgs Varying I with respect to the metric gives the conformal stress– r r 0 M equation is then energy tensor, with mixed components ¨ R S 1 2 3 μ 2 μα δIM μ μ = + − ¯ T ≡ √ g = T (ψ) + σT (S) , (18) 2 r BS S 4 λS , (27) ν −g δgαν ν ν B r 6 at University of St Andrews on April 11, 2016 where where we define − − 2 1 1 λ ρ˜ p˜ r 2p˜ ⊥ μ = ;μ − ;μ − 2 Rμ λ¯ ≡ + . (28) Tν (S) S S;ν SS;ν S ν σ 4 3 3 6 4σS0 1 1 1 λ Note that the fermions effectively strengthen the quartic Higgs self- −δμ S;α S − SS;α − R S2 + S4 . (19) ν 6 ;α 3 ;α 12 σ coupling constant. The corresponding Higgs potential is 2 2 The trace R R R V (S) =− S2 + λS¯ 4 = λ¯ S2 − − . (29) 1 12 24 λ¯ 576 λ¯ α = ¯ D + ;α + R 2 − 4 Tα ψ ψ σ SS;α S 4 λS (20) 6 A stable vacuum in the MK-frame requires λ>¯ 0, so that V(S) vanishes by virtue of the Dirac and Higgs equations (15) and (16), is bounded from below. Spontaneous symmetry breaking to in- respectively. duce non-zero fermion masses can then occur for positive curvature 2 For static spherically symmetric fermion fields, the stress–energy R > 0. The minimum of V(S) occurs at S = R/24 λ¯.Thisgives 2 4 tensor takes the form the vacuum energy density V (S) =−R /576 λ¯ =−λS¯ . Note that λ>¯ 0andR > 0 are not required, however, since a μ = − Tν (ψ) diag ( ρ,pr ,p⊥,p⊥) , (21) time-dependent S(r, t) in the MK frame corresponds to a constant with energy density ρ, radial pressure pr, and azimuthal pressure S0 in the Higgs frame. p⊥. The Dirac equation (15) then gives α = + − = ¯ D = ¯ 4.2 Source-free solution: the BV metric Tα (ψ) pr 2 p⊥ ρ ψ ψ μSψψ. (22) We can consider a Higgs field S(r, t), allowing for a possible time The source f(r) in CG’s fourth-order Poisson equation includes dependence, with the understanding that the consequent stress– both fermion and Higgs contributions (Mannheim 2007; Brihaye & energy tensor and gravitational source f(r) must be time indepen- Verbin 2009): ·· dent for the static spherical structures of primary interest here. 3 1 1 1 −i ω t 3 An example is a complex Higgs field with S ∝ e , for which 4 αg f (r) =− (ρ + pr ) − σS + . (30) B S B2 S (S˙ )2 =−ω2 S2 and S¨ =−ω2 S. Specializing to the MK metric, the Higgs equation (16) evaluates The Higgs field S(r, t) makes no explicit contribution to f(r), if and as only if it takes the specific form

S¨ 1 R 4 λ p + 2 p⊥ − ρ S0 t0 a = 2 + − 3 + r S(r,t) = , (31) 2 r BS S S , (23) + + B r 6 σ σS (t t0)(r a)

MNRAS 458, 4122–4128 (2016) 4126 K. Horne declining from S0 at time t = 0 and radius r = 0 with a time-scale 2 2 2 S a t and radial length-scale a. This holds for either sign σ . gμν → g˜μν = gμν = gμν = gμν . (42) 0 S r + a The time dependence included here may have applications, for 0 example when embedding static spherical structures in an expanding The stretched metric’s circumferential radius universe, with t ∼ 1/H , or for a complex Higgs ﬁeld varying as 0 0 ≡ | | = rS = ra S(r, t) = S(r) e−i ω t.Wesett =∞to focus on static solutions. r˜ g˜θθ (43) 0 S0 r + a Note in equation (30) that a static ‘Higgs halo’ S(r) makes a < r < ∞ < r

transformation does not move the horizon at r h, which remains Downloaded from about f(r). Among these BV identify one three-parameter analytic at r˜ = h˜. solution with a source-free scalar field, Note, however, that while the original MK metric gμν has a linear S a S r = 0 , potential γ r, the corresponding Higgs-frame metric g˜μν has no term ( ) + (32) r a in g˜00 linear in r˜. Thus even though the Higgs field S(r) declines for which the MK potential is only slightly from its central value S0, this has a significant effect on the shape of the potential and the resulting rotation curve. With B http://mnras.oxfordjournals.org/ + 2 ˜ ˜ 3 a r h 2 h 2 = − − − rising as B ≈ 1 + γ r, S falls as S/S0 ≈ 1 − γ r/2, so that S B lacks B(r) 1 Kr 1 3 , (33) a r˜(r) r˜ (r) a linear potential. While this result is demonstrated here for a point ˜ ≡ + ≡ + ≡− 2 mass, rather than for a more realistic extended source structure, it where h ha/(a h), r˜ ra/(a r), and K 2 λS0 .This metric has a Schwarzschild-like horizon, with B(r) ∝ (r − h)van- indicates the potential danger when using the linear potential in the ishing at r = h. MK metric to fit galaxy rotation curves. Expanding equation (33) in powers of r, we can read off the three Fig. 2 further illustrates this point by showing the Higgs-frame 2 independent MK parameters (β, γ , κ) in terms of the three BV potential (S/S0) B, and the corresponding rotation curve, for the 11 parameters (h, a, K): same 10 M point mass as in Fig. 1. The Higgs field is constant, 10 by definition, in the Higgs frame. Because a ≈ 2/γ = 7.6 × 10 pc at University of St Andrews on April 11, 2016 = ˜ − ˜ 2 2 β h 1 K h , (34) 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 1 h˜ γ = 2 − 3 1 − K h˜ 2 , (35) K are essentially unchanged from their MK-frame counterparts β a a 2 and κ. The Higgs-frame potential (S/S0) B retains the Newtonian 1 h˜ potential −2 β/r˜ and the quadratic potential −κ r˜2, but lacks a κ = K − 1 − 1 − K h˜ 2 . (36) a2 a linear potential term. The rotation curve thus follows a Keplerian profile out to ∼20 kpc, bending down as the quadratic potential From these one can verify that the constant term, 2 takes hold, and the watershed radius, where −g˜00 = (S/S0) B has h˜ 6 β a maximum, is now at r˜ = |β/κ|1/3 = 37.5 kpc. w = 1 − 3 1 − K h˜ 2 = γa− 1 = 1 − , (37) a a r = 2 = − satisfies the Wr 0 constraint w 1 6 βγ. The inverse relations 5 ASTROPHYSICAL TESTS are 1 + w 6 β To really test CG with astrophysical observations is considerably a = = , (38) harder than simply using geodesics of the MK metric with B(r) γ 1 − w sourced by matter. The source f(r) in CG’s fourth-order Poisson γ 2 γ 3 K = κ + − 2 β , (39) equation must include contributions from the Higgs halo S(r), 1 + w 1 + w in addition to those from the matter (+radiation) energy density and finally, the horizon radius h,whereB(r) vanishes, is the smallest ρ(r) and pressure p(r). These are specified in the Higgs frame, = positive real root of the cubic ρ˜ (r˜)andp˜ (r˜), and moved to the MK frame using r˜ rS(r)/S0, 4 4 ρ(r) = (S/S0) ρ˜ (r˜), and p(r) = (S/S0) p˜ (r˜). For example, to =− + + 2 − 3 0 2 β wr γr κr . (40) model spherical structures similar to the matter distribution in galax- The MK and BV metrics are thus equivalent, representing the ies and clusters, it may be appropriate to adopt a Hearnquist profile same three-parameter source-free solution to the CG field equations. (Hearnquist 1990) However, as BV show, the Higgs field has a radial profile S(r). ρ ρ˜ (r˜) = 0 , (46) Massive test particles therefore do not follow the time-like geodesics x (x + 1)3 of the MK metric. with x = r/r˜ in units of the scale radius r ,andwithρ = Fortunately, since we know S(r), we know the conformal trans- 0 0 0 M/(2 π r3) for total mass M. The enclosed mass profile is formation between the MK frame and the Higgs frame: 0 2 S a x S = 0 → S˜ = −1 S = S , (41) M(r˜) = M . (47) r + a 0 x + 1

MNRAS 458, 4122–4128 (2016) Conformal gravity rotation curves 4127

r2 γ =− f (r) ,γ(0) = γ , (51) 2 0 r κ =− f (r) ,κ(∞) = κ∞ , (52) 6 r3 3 r4 = 2 = − + r w f (r) ,w(r) 1 6 β(r) γ (r) Tr (r) . (53) 2 4 αg The MK parameters, β(r), γ (r), κ(r), w(r), are then internal and/or external moments of f(r). For non-singular structures, appropriate boundary conditions at the origin are β(0) = 0andγ (0) = 0, though non-zero values may also be chosen for an unresolved central source such as a nucleon, a star, or a black hole. The curvature of the external three-space is set by κ(∞). The third-order constraint on r w can be set any radius where Tr is known. Note that even for an extended source f(r), the ﬁrst three deriva-

tives of B(r) evaluate as if the MK parameters were r-independent. Downloaded from For example: 2 β 2 β B = + γ − 2 κr− + w + γ r − κ r2 r2 r 2 β 1 1 1 1 3 = + γ − 2 κr+ − + − + r f. (54) http://mnras.oxfordjournals.org/ r2 6 2 2 6 As a consequence, the Ricci scalar remains 2 (w(r) − 1) 6 γ (r) R = + − 12 κ(r) , (55) r2 r and the third-order constraint remains w(r)2 + 6 β(r) γ (r) − 1 1 r = = r Wr 4 Tr . (56) 3 r 4 αg

r at University of St Andrews on April 11, 2016 Here, Tr includes radial pressure from both matter (+radiation) and from the Higgs halo, Figure 2. Same as in Fig. 1 but after a conformal transformation S 4 r = + r (r) = S(r)/S0 stretches the geometry from the MK frame, where Tr p˜ r σTr (S) , (57) S0 B(r) =−g00 = 1/grr, to the Higgs frame, where the Higgs field is con- 2 stant. The Higgs-frame potential B˜ ≡ (S/S0) B has a maximum at the with the Higgs halo contribution being, 1/2 watershed radius r˜ ≡ (S/S0) r = |β/κ| ≈ 37.5 kpc, outside which there ˙ 2 ¨ are no stable circular orbits. Note that B(r) has a rising linear potential γ r, S − 2 S S B 2 SS 4 B T r (S) = + S + B + but this is effectively cancelled by the decline in S2, so that the rotation curve r 6 B 2 6 r is Keplerian out to ∼20 kpc. S2 B B − 1 λ + + − S4. (58) 2 This Hearnquist profileρ ˜ (r˜) is specified in the Higgs frame, then 6 r r σ 4 scaled by (S/S0) for use in the MK frame where the fourth-order Because f(r) depends on B(r)andS(r), the moment integrals for Poisson equation and second-order Higgs equation are more easily B(r) must be iterated along with solving the second-order MK-frame solved. Higgs equation for S(r): In the MK frame, B(r)andS(r) satisfy their equations of motion. S¨ 1 R λ ρ˜ − p˜ − 2 p˜ ⊥ The fourth-order Poisson equation for B(r), = r2 BS + S − 4 + r S3, B r2 6 σ 4σS4 ·· 0 4 α 1 1 1 g (rB) =−σS3 + (59) r S B2 S = = with boundary conditions S(0) S0 and S (0) S1. 3 S 4 Having solved for B(r)andS(r) in the MK frame, we move back − (ρ˜ + p˜ ) ≡ 4 αg f (r), (48) B S0 to the Higgs frame, and use geodesics of the resulting Higgs-frame 2 metric g˜μν = (S(r)/S0) gμν to test CG in three ways: is convenient because solutions of the form 2 β(r) (1) galaxy rotation curves; B(r) = w(r) − + γ (r) r − κ(r) r2 (49) (2) galaxy cluster potentials probed by X-ray gas; r (3) lensing by galaxies and galaxy clusters. can be found for extended sources f(r) by integrating first-order equations, with appropriate boundary conditions: For example, the circular orbit rotation curve is 1/2 2 2 4 dln(|g˜00|) dln SB v + v = r = v2 = = = B S , (60) β f (r) ,β(0) β0 , (50) | | + 2 12 dln( g˜θθ ) dln(Sr) 1 vS

MNRAS 458, 4122–4128 (2016) 4128 K. Horne where We collect the equations and outline the procedure for astro- β γ physical tests of CG in static spherical geometries. Speciﬁcally, the + − 2 r κr sources for CG’s fourth-order Poisson equation include not only 2 rB r 2 v = = (61) + B 2 B 2 β the energy density and pressure of distributed matter (stars gas), w − + γr− κr2 r and radiation if relevant, but also the associated Higgs halo S(r). The resulting MK metric g must then be ‘stretched’ to the Higgs is the rotation curve arising from the MK potential B(r), as used μν frame metric g˜ = (S/S )2 g . The Higgs-frame geodesics then by Mannheim & O’Brien (2012), and v2 ≡ rS/S implements the μν 0 μν S provide predictions for testing CG against observations. corrections arising from the Higgs halo proﬁle S(r). One of the objections to CG is that for the MK metric with γ>0 the linear potential causes light rays to bend away from ACKNOWLEDGEMENTS the point mass, rather than towards it. But our results show that a KH would like to thank Amy Deacon, Aidan Farrell and Indar rising MK-frame potential B(r) is compensated by a corresponding Ramnarine for hospitality at the University of the West Indies in decline in the Higgs halo S(r). In light of this, the galaxy rotation Trinidad, Alistair Hodson, Alasdair Leggat, and Carl Roberts for curves may not in fact require γ>0, and we may now reconsider helpful comments on the presentation, an anonymous referee for adjusting the strength and sign of the linear potential when testing thought provoking criticism, and the UK Science and Technology

CG predictions for gravitational bending of light rays. It remains to Downloaded from Facilities Council (STFC) for financial support through consoli- be shown whether the effect of an extended source f(r) appropriate dated grant ST/M001296/1. to modelling galaxies and clusters can fit the light bending angles from lensing as well as the flat rotation curves. We hope to address this in future work. REFERENCES

Brihaye Y., Verbin Y., 2009, Phys. Rev. D, 80, 124048 http://mnras.oxfordjournals.org/ 6 CONCLUSIONS Edery A., Paranjape M. B., 1998, Phys. Rev. D, 58, 24011 Flanagan E., 2006, Phys. Rev. D, 74, 023002 The fourth-order field equations of CG have vacuum solutions that Hearnquist L., 1990, ApJ, 356, 359 augment the Schwarzschild metric with linear and quadratic poten- Mannheim P. D., 1993a, Gen. Relativ. Gravit., 25, 697 tials (Mannheim & Kazanas 1989). This MK metric has been used Mannheim P. D., 1993b, ApJ, 419, 150 to fit the rotation curves of a wide variety of galaxies with only three Mannheim P. D., 1997, ApJ, 479, 659 free parameters (Mannheim & O’Brien 2012). Mannheim P. D., 2001, ApJ, 561, 1 We highlight two potential problems with using geodesics of the Mannheim P. D., 2003, Int. J. Mod. Phys. D, 12, 893 MK metric to study rotation curves of galaxies. First, the MK metric Mannheim P. D., 2006, Prog. Part. Nucl. Phys., 56, 340 Mannheim P. D., 2007, Phys. Rev. D, 75, 124006 is a source-free solution to CG’s fourth-order Poisson equation, at University of St Andrews on April 11, 2016 Mannheim P. D., 2011, Gen. Relativ. Gravit., 43, 703 but the conformally coupled Higgs field makes an extended halo Mannheim P. D., 2012, Phys. Rev. D, 85, 124008 S(r) that also contributes to the gravitational source unless it has a Mannheim P. D., Kazanas D., 1989, ApJ, 342, 635 = + specific radial profile, S(r) S0 a/(r a). Secondly, since particle Mannheim P. D., Kazanas D., 1991, Phys. Rev. D, 44, 417 masses scale with the Higgs field, the Higgs halo S(r) pushes test Mannheim P. D., O’Brien J. G., 2012, Phys. Rev. D, 85, 124020 particles off geodesics of the MK metric. Nesbet R. K., 2011, Mod. Phys. Lett. A, 26, 893 To address these issues, we note that a conformal factor Pireaux S., 2004, Class. Quantum Gravity, 21, 1897 (r) = S(r)/S0 stretches the metric to a form that makes the Higgs Rubin V. C., Ford W. K., Thonnard N., 1978, ApJ, 225, L107 field constant. Test particles then follow geodesics of this stretched Sultana J., Kazanas D., 2010, Phys. Rev. D, 81, 7502 ‘Higgs-frame’ metric. Tully R. B., Fisher J. R., 1977, A&A, 54, 661 For the analytic solution to the source-free CG equations (Brihaye Villanueva J. R., Olivares M., 2013, J. Cosmol. Astropart. Phys., 06, 40 Wehus I. K., Ravndal F., 2007, J. Phys. Conf. Ser., 66, 012044 &Verbin2009), which is equivalent to the MK metric, we find that Wood J., Moreau W., 2001, preprint (arXiv:gr-qc/0102056) the effect of stretching the metric to the Higgs frame is to eliminate the linear potential that is used to fit galaxy rotation curves. Thus, the remarkable results of Mannheim & O’Brien (2012), using geodesics of the MK metric to fit a large variety of galaxy rotation curves with just three parameters, may be testing an empirical model rather than the actual CG predictions. This paper has been typeset from a TEX/LATEX file prepared by the author.

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