Cosmological Tests of Coupled Galileons
Philippe Brax, Clare Burrage, Anne-Christine Davis, Giulia Gubitosi
aa r X i v : . [ a s t r o - ph . C O ] N ov Preprint typeset in JHEP style - HYPER VERSION
Cosmological Tests of Coupled Galileons
Philippe Brax
Institut de Physique Th´eorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/YvetteCedex, FranceE-mail: [email protected]
Clare Burrage
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD,United KingdomE-mail:
Anne-Christine Davis
DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA, UKE-mail:
Giulia Gubitosi
Dipartimento di Fisica, Universit`a di Roma “La Sapienza” and INFN sez. Roma1, P.leAldo Moro 2, Roma, ItalyTheoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, U.K.E-mail: [email protected]
Abstract:
We investigate the cosmological properties of Galileon models with positivekinetic terms. We include both conformal and disformal couplings to matter and focus onconstraints on the theory that arise because of these couplings. The disformal coupling tobaryonic matter is extremely constrained by astrophysical and particle physics effects. Thedisformal coupling to photons induces a cosmological variation of the speed of light andtherefore distorsions of the Cosmic Microwave Background spectrum which are known to bevery small. The conformal coupling to baryons leads to a variation of particle masses sinceBig Bang Nucleosynthesis which is also tightly constrained. We consider the backgroundcosmology of Galileon models coupled to Cold Dark Matter (CDM), photons and baryonsand impose that the speed of light and particle masses respect the observational boundson cosmological time scales. We find that requiring that the equation of state for theGalileon models must be close to -1 now restricts severely their parameter space and canonly be achieved with a combination of the conformal and disformal couplings. This leadsto large variations of particle masses and the speed of light which are not compatiblewith observations. As a result, we find that cosmological Galileon models are viable darkenergy theories coupled to dark matter but their couplings, both disformal and conformal,to baryons and photons must be heavily suppressed making them only sensitive to CDM. ontents
1. Introduction 12. Disformally Coupled Galileons 3
3. Cosmological Galileons 8
4. Exploring the Parameter Space 11 c and G N . 124.2 Background Cosmology 124.3 Growth of Structure 134.4 Variation of c and G N
5. On the attractor 156. Conclusions 18
1. Introduction
Galileon models have received significant attention since their first proposal [1] and theirgeneralization to curved backgrounds [2, 3] as a promising scenario for infrared modifica-tions of gravity. A first reason of interest is the fact that the symmetry properties of thescalar degree of freedom guarantee that only second-order derivatives enter the equationsof motion. Moreover the peculiar field self-interactions ensure validity of General Relativityat small scales and near massive objects thanks to the Vainshtein mechanism [4], so thatsolar system constraints are satisfied.Galileons have been widely studied both on a purely theoretical ground, with re-sults showing that this kind of models arise also in the context of massive gravity [5] andbraneworld models [6], and from the observational point of view, unveiling a very richphenomenology [7–21].While originally Galileon models assumed no interaction between matter and the scalarfield, it was suggested in [22] that both conformal and derivative (disformal) couplings to– 1 –atter should be considered. And indeed these kind of interactions emerge naturally withinthe context of massive gravity and braneworld cosmology [23]. In this work we investigatethe phenomenological viability of Galileon models taking into account the possibility ofhaving both conformal and disformal couplings.Recent works including [14, 24, 25] have concentrated on the Galileon models withnegative-sign kinetic terms for which Minkowski space is an unstable vacuum solution,however it is difficult to explain how such terms could arise from a healthy massive gravityor brane world scenario [5, 6]. Although Galileon models with negative-sign kinetic termscan be ghost-free in a Friedman-Robertson-Walker (FRW) Universe corresponding to ourpresent knowledge of the cosmos, we restrict our study to Galileon models which are ghostfree in both Minkowski space and the FRW solution describing our cosmological Universe.Besides requiring that the model is stable, we look for regions of the parameter space wherethe background cosmology and the growth of structure is compatible with observations. Itturns out that the presence of disformal and conformal couplings is fundamental in orderto match these constraints. However some tension arises when comparing the backgroundcosmology with the structure growth. This motivates us to investigate further whethercouplings to matter are in agreement with other kinds of observations.As far as standard matter is concerned, a conformal coupling can only affect baryonsand induces a variation of the particle masses which can be viewed as a time variationof the effective Newton constant. If Vainshtein screening were absent this coupling wouldbe severely constrained by searches for fifth forces [26], however in the Galileon modelonly very weak constraints can be derived from galaxy clusters [1, 27]. Disformal coupling,on the other hand, must be strongly suppressed for baryons, due to astrophysical andparticle physics constraints, and so can only affect photons, inducing a variation of thespeed of light. When taking into account the observational bounds on the variation ofNewton’s constant and the variation of the speed of light, coming respectively from BigBang Nucleosynthesis on the one hand and distance duality relations and CMB spectraldistortion on the other hand, we find that the values of the coupling constants that arepreferred by standard cosmological constraints, such as having an equation of state nowclose to -1, are ruled out. Only a coupling to Dark Matter is allowed.The paper is organized as follows. In Section 2 we briefly discuss conformal and dis-formal couplings in a general setting, we review the Galileon model and the Vainshteinmechanism. We also show that astrophysical and particle physics constraints rule outdisformal coupling with baryons and we derive the effects on duality relation and spectraldistortion due to speed of light variation. In Section 3 we write the equations for cosmolog-ical evolution in the Galileon framework and the no-ghost and Laplace stability conditions.We derive the modified equations for structure growth and we give the explicit relationbetween variations of speed of light and Newton’s constant and the disformal and confor-mal coupling coefficients of the model. In Section 4 we explore the parameter space of themodel. We first take into account the background cosmology constraints, showing thatthey require non-zero coupling parameters. We then observe that the growth of structureconstraints already generate some tension with the background cosmology. The situationis only worsened by introducing the constraints coming from the variation of Newton’s con-– 2 –tant and of the speed of light. In particular the preferred value of the couplings selected bybackground cosmology and structure growth produce variations of both Newton’s constantand speed of light that are far too big compared to current constraints.
2. Disformally Coupled Galileons
Matter can couple to scalars via a metric ˜ g µν which can differ from the Einstein metric g µν describing the behaviour of gravity. Bekenstein has shown [28] that the most generalmetric that can be constructed from g µν and a scalar field that respects causality and theweak equivalence principle is˜ g µν = A ( φ, X ) g µν + B ( φ, X ) ∂ µ φ∂ ν φ , (2.1)where the first term gives rise to conformal couplings between the scalar field and mat-ter, and the second term leads to the disformal coupling. Here X = (1 / g µν ∂ µ φ∂ ν φ isthe kinetic term of standard scalar field theories. The conformal coupling gives rise toLagrangian interaction terms of the form L ⊃ A ( φ, X ) T µJµ . (2.2)and the disformal interactions give rise to Lagrangian interaction terms of the form L ⊃ B ( φ, X )2 ∂ µ φ∂ ν φT µνJ . (2.3)where T µνJ is the energy momentum tensor of matter fields in the Jordan frame, definedby the metric g Jµν = A ( φ, X ) g µν . The conformal coupling gives rise to Yukawa type longrange forces between matter fields. In the following we shall use˜ g µν = A ( φ ) g µν + 2 M ∂ µ φ∂ ν φ . (2.4)This is not the most general scalar metric as given by Bekenstein in Equation (2.1), howeverit describes all the leading order effects of both the conformal and disformal couplings, andis much simpler to work with. The coupling scale M is constant and should be fixed byobservations. The disformal coupling has no influence on static configurations of matteras no disformal interaction between static non-relativistic objects is generated. As we willrecall, photons are particularly sensitive to the disformal coupling whereas they see noinfluence of the conformal coupling. We embed the coupled scalar field that we have just defined into a wider setting definedby the Galileon models [1]. These are scalar field theories which have equations of motionthat are at most second order in derivatives, despite the presence of non-trivial derivativeself-interactions. Moreover they are interesting dark energy candidates where an explicit– 3 –osmological constant is not compulsory. Their Lagrangian reads in the Einstein framedefined by the metric g µν L = − c ∂φ ) − c Λ (cid:3) φ ( ∂φ ) − c Λ L − c Λ L + X i c i φm Pl T i − X i c iG Λ ∂ µ φ∂ ν φT µνi , (2.5)where we have introduced different conformal c i and disformal c iG couplings to each matterspecies with an energy momentum tensor T µνi in the Einstein frame. The common scaleΛ = H m Pl (2.6)is chosen to be of cosmological interest as we focus on cosmological Galileon models whichcan lead to dark energy in the late time Universe. We also require that c > M i = − Λ m Pl c Gi . (2.7)for the coupling scale of the i -th species to the metric˜ g iµν = A i ( φ ) g µν + 2 M i ∂ µ φ∂ ν φ . (2.8)where the conformal coupling for a given species is is A i ( φ ) = 1 + c i φm Pl . (2.9)The complete Galileon Lagrangian depends on the higher order terms which are given by L =( ∂φ ) (cid:20) (cid:3) φ ) − D µ D ν φD ν D µ φ − R ( ∂φ ) (cid:21) L =( ∂φ ) (cid:2) ( (cid:3) φ ) − (cid:3) φ ) D µ D ν φD ν D µ φ + 2 D µ D ν φD ν D ρ φD ρ D µ φ (2.10) − D µ φD µ D ν φD ρ φG νρ ] . and these terms play an important role cosmologically.When baryons are not conformally coupled, the disformal coupling does not induce anyadditional forces in the static case [29]. In cosmology, as we will see in section 3.2, the higherorder terms of the Galileon models involving c , c and c all lead to a modification of gravityon cosmological scales even when c = 0. On the other hand in a Minkowski background,when c b = 0, the Galileons evade the solar system tests thanks to the Vainshtein mechanism[1, 27, 30, 31]. Around a spherically symmetric source of mass m the scalar field profile is,for the cubic Galileon with c = c = 0, dφdr = − Λ r − s (cid:18) R ∗ r (cid:19) , (2.11)– 4 –nd the non-linearities dominate the evolution of the scalar within the Vainshtein radius R ∗ = 1Λ (cid:18) c b m πc m Pl (cid:19) / . (2.12)Within this radius the non-linearities act to suppress the scalar force, F φ , compared to thatof Newtonian gravity, F N , so that F φ F N = ( c b ) (cid:16) rR ∗ (cid:17) / . Outside the Vainshtein radius, thenon-linearities in the kinetic terms become irrelevant and the dominant kinetic term reducesto − ( ∂φ ) /
2. Inside the Vainshtein radius, any perturbations around the background ofequation (2.11) inherit a wave function renormalisation such that the kinetic terms of theperturbations read Z ( ∂δφ ) where we have | Z | ∼ φ ′ r Λ ∼ (cid:18) R ⋆ r (cid:19) / (2.13)and a prime denotes a derivative with respect to radius. Therefore inside the Vainshteinradius Z can be large and after canonically normalising the field the effective coupling tomatter c b → c bZ = c b √ Z (2.14)becomes small enough to evade gravitational tests for massless scalar fields in the solarsystem. For the earth embedded in the Milky Way halo, the Z factor becomes Z ⊕ ∼ (cid:18) c b ρ G πc Λ m Pl (cid:19) / (2.15)As the density of the Milky Way halo is ρ G ∼ ρ c where ρ c is the critical density of theUniverse, we find that Z ⊕ ∼ . The disformal couplings are rescaled too c iG → c iZG = c iG Z . (2.16)Similar phenomena occur for quartic and quintic Galileons. It has been suggested thatwhen the Galileon is considered as an effective field theory its cut-off is rescaled by thelarge Z factor in a similar way to the rescaling of the Lagrangian parameters above [32],which means that the Galileon remains a valid effective field theory beyond its naive cut-offon appropriate non trivial backgrounds, however it has been shown that this is not possiblefor all UV completions of the Galileon [33]. The disformal coupling to electrons, protons and neutrons leads to a faster burning rateof stars and supernovae. The detailed processes have been studied in [29] where the moststringent bound has been found to be M SNb &
92 GeV (2.17)– 5 –rom the explosion of the SN1987a supernovae. A more severe constraint can even beobtained by particle colliders such as the LHC where two quarks would lead to two invisiblescalars and the corresponding missing energy. The result from ATLAS implies that M LHCb &
490 GeV . (2.18)This can be translated into the bound | c b,LHCG | ≤ − (2.19)in the terrestrial environment where Z ⊕ ∼ , the Z -factor cannot render the bare couplingof order one as c bG = Z ⊕ c b,LHCG . − in order to be compatible with experimentalbounds. In dense matter such as supernovae cores with a density of ρ SN ∼ · g . cm − ,the normalisation factor is of order Z SN ∼ and this implies that c bG . − . All inall we find that c bG must be minuscule, implying that baryons are effectively disformallydecoupled. We next consider the effect of the disformal coupling to photons defined by the followingaction to leading order: S = Z d x √− g (cid:18) R κ −
12 ( ∂φ ) − F + 1 M γ ∂ µ φ∂ ν φT µν ( γ ) (cid:19) , (2.20)where T µν ( γ ) = F µα F να − g µν F is the Einstein frame energy-momentum tensor of the photon.The equation of motion resulting from the Lagrangian in Equation (2.20) gives the gener-alised form of Maxwell’s equation. In the cosmological setting where we use the conformalLorentz gauge ∂ α A α = 0 and A = 0, we obtain ∂ a i + ( c p k − C − ¨ C ) a i = 0 , (2.21)where ∆ = ∂ i ∂ i , the index i runs only over spatial directions, and C ( ˙ φ ) = 1 + M γ a ˙ φ , D ( ˙ φ ) = 1 − M γ a ˙ φ , where ˙ = ∂ is the derivative in conformal time η with ds = a ( − dη + dx ). The new canonically normalised vector field is A i = C − a i , and theeffective speed of light is c p = D ( ˙ φ ) /C ( ˙ φ ). If C and D are close to one, we find that c p = 1 − M γ a ˙ φ . (2.22)This is expected as the metric ˜ g µν is the one directing the photon trajectories. Hence thephotons experience a time-varying speed of light in the course of the cosmological evolution.When the functions C ( ˙ φ ) and D ( ˙ φ ) vary over cosmological times, we expect that¨ C/C ∝ H , and similarly for D . Then in the sub-horizon limit k/a ≫ H , we can neglectthe effect of C ′′ in equation (2.21) and write the time dependent dispersion relation as ω = c p ( η ) k . The solution to Maxwell’s equation can be written as A i = e i A cos (cid:18) k (cid:20)Z c p dη (cid:21) − kx + ϕ (cid:19) , (2.23)– 6 –here A is the amplitude of the photon, whose time variation we assume to be negligiblecompared to the variation of the phase, which we write as ϕ = k (cid:0)R c p dη (cid:1) − kx + ϕ . The polarisation vector e i , satisfies e i k i = 0 and e = 1. Using this solution of Maxwell’sequation, the energy momentum tensor in the disformal frame defined by ˜ g µν can be writ-ten in the form of the energy-momentum tensor typically used in geometrical optics anddescribing light rays [34] ˜ T ( γ ) µν = A a k µ k ν , (2.24)where k µ = ( − c p k sin( ϕ ) , k i sin( ϕ )), and k µ = ∂ µ g , where g = cos( ϕ ). This confirms that c p is the speed for the propagation of light rays.The variation of the speed of light with time implies that the duality relation betweenthe luminosity and the angular distances is modified [34]. The angular diameter distance d A of an object is obtained by considering a bundle of geodesics converging at the observerunder a solid angle d Ω obs and coming from a surface area dS emit : d A = dS emit d Ω obs . Theluminosity distance is given in terms of the emitter luminosity L emit and the radiation fluxreceived by the observer F obs by d L = L emit πF obs . For a unit sphere, L emit = R F emit d Ω emit =4 πF emit . where F emit is the emitted flux. The duality relation is then modified d L = (cid:18) c obs c emit (cid:19) (1 + z ) d A , (2.25)corresponding to a variation of the speed of light from emission to observation.The intensity of the photon radiation in the conformal gauge is given I = a ρ γ whichreads I = [( ∂ A i ) + B i B i ], where indices are raised with η µν and B i = ǫ ijk ∂ j A k . Ourunderstanding of the primordial Universe leads us to expect that the CMB will display analmost perfect black body spectrum. We consider that the CMB spectrum is initially ablack body spectrum, so that I ( k, η i ) = k e k/T − , and we assume that the only distortionsappear through the influence of the scalar field as the light propagates towards us fromthe time of last scattering. The measured spectrum will be related to the intensity I ( k, η )by a geometrical factor which depends on the way the reciprocity relation is modified bythe variation of the speed of light [35] I obs ( k, η ) = (cid:16) d A r S (cid:17) I ( k, η ), where r S is the sourcearea distance, which we define by considering a bundle of null geodesics diverging from thesource and subtending a solid angle d Ω S at the source. The combined effect is then I obs ( k, η ) = (cid:18) c emit c obs (cid:19) I ( k, η i ) , (2.26)a result which is valid on subhorizon scales and when the speed of light varies only oncosmological timescales. We can rewrite the intensity as I obs ( k, η ) = k e k/T µ − , where thedimensionless chemical potential is given by µ = 2( e − k/T − δc p c p , (2.27)where δ ( . ) denotes the difference in the quantities between their current and their initialvalues [34, 36]. The tight constraints on µ will be discussed below. We have implicitly averaged over ϕ to define the flux and the luminosity. – 7 – . Cosmological Galileons The cosmological Galileons can play the role of dark energy despite the absence of a cos-mological constant. Here we focus on the cosmological models where the baryons aredisformally decoupled while the disformal coupling to photons and CDM is universal M γ = M CDM = M , similarly all the conformal couplings are equal to c . In a Friedmann-Robertson-Walker background, the equations of motion can be simplified using x = φ ′ /m Pl where a prime denotes ′ = d/d ln a = − d/d ln(1 + z ) where a is the scale factor and z theredshift. Defining ¯ y = φm Pl x , ¯ x = x/x and ¯ H = H/H where H is the Hubble rate, andthe rescaled couplings ¯ c i = c i x i , i = 2 . . . , ¯ c = c x , ¯ c G = c G x where x is the valueof x now, the cosmological evolution satisfies [22]¯ x ′ = − ¯ x + αλ − σγσβ − αω ¯ y ′ = ¯ x ¯ H ′ = − λσ + ωσ ( σγ − αλσβ − αω )where we have introduced α = − c ¯ H ¯ x + 15¯ c ¯ H ¯ x + ¯ c ¯ H + ¯ c ¯ H ¯ x −
352 ¯ c ¯ H ¯ x − c G ¯ H ¯ x (3.1) β = − c ¯ H ¯ x + ¯ c ¯ H c ¯ H ¯ x − c ¯ H ¯ x − ¯ c G ¯ H (3.2) γ =2¯ c ¯ H − ¯ c ¯ H ¯ x + ¯ c ¯ H ¯ x c ¯ H ¯ x − c G ¯ H ¯ x (3.3) σ =2(1 − c ¯ y ) ¯ H − c ¯ H ¯ x + 2¯ c ¯ H ¯ x − c ¯ H ¯ x + 21¯ c ¯ H ¯ x + 6¯ c G ¯ H ¯ x (3.4) λ =3(1 − c ¯ y ) ¯ H − c ¯ H ¯ x − c ¯ H ¯ x + ¯ c ¯ H ¯ x r a + 152 ¯ c ¯ H ¯ x (3.5) − c ¯ H ¯ x − ¯ c G ¯ H ¯ x (3.6) ω = − c ¯ H + 2¯ c ¯ H ¯ x − c ¯ H ¯ x + 15¯ c ¯ H ¯ x + 4¯ c G ¯ H ¯ x. (3.7)It is important to notice that the intrinsic values of the c i coefficients cannot be probedcosmologically. Only the various combinations of the c i ’s and x are relevant. On the otherhand, we will use a naturality criterion and impose that x cannot be arbitrarily large.The Friedmann equation which governs the evolution of the Hubble rate can be writtenin a similar way(1 − c ¯ y ) ¯ H = Ω m a + Ω r a + 2¯ c ¯ H ¯ x + ¯ c ¯ H ¯ x − c ¯ H ¯ x + 152 ¯ c ¯ H ¯ x − c ¯ H ¯ x − c G ¯ H ¯ x (3.8)– 8 –here the final six terms on the right hand side of Equation (3.8) correspond to the scalarenergy density ρ φ H m = 6¯ c ¯ H ¯ x + ¯ c ¯ H ¯ x − c ¯ H ¯ x + 452 ¯ c ¯ H ¯ x − c ¯ H ¯ x − c G ¯ H ¯ x (3.9)and the scalar pressure is p φ H m = − ¯ c [4 ¯ H ¯ x + 2 ¯ H ( ¯ H ¯ x ) ′ ] + ¯ c H ¯ x + 2 c ¯ H ¯ x ( ¯ H ¯ x ) ′ − ¯ c [ 92 ¯ H ¯ x + 12 ¯ H ¯ x ¯ x ′ + 15 ¯ H ¯ x ¯ H ′ ] + 3¯ c ¯ H ¯ x (5 ¯ H ¯ x ′ + 7 ¯ H ′ ¯ x + 2 ¯ H ¯ x ] (3.10)+ c G [6 ¯ H ¯ x ¯ H ′ + 4 ¯ H ¯ x ¯ x ′ + 3 ¯ H ¯ x ]from which we define the equation of state ω φ = p φ ρ φ which must be close to -1 now.Normalising y = 0, the Friedmann equation gives the constraint on the parameters1 = Ω m + Ω r + 2¯ c + ¯ c − c + 152 ¯ c − c − c G (3.11)which reduces the dimension of the parameter space by one unit. In the following, wechoose ¯ c = 1 without any loss of generality implying that the parameter space comprises(¯ c , ¯ c , ¯ c , ¯ c G ) and ¯ c is determined using (3.11).It is important to stress that the validity of the Galileon scenario can only be guaran-teed in the absence of ghosts and when the speed of sound squared for the scalar pertur-bations is positive. The no-ghost condition reads¯ c − c ¯ H ¯ x − c G ¯ H + 27¯ c ¯ H ¯ x − c ¯ H ¯ x > (3¯ c ¯ H ¯ x + 6¯ c G ¯ H ¯ x − c ¯ H ¯ x + ¯ c ¯ H ¯ x − c ) − − c ¯ y ) − c G ¯ H ¯ x + 9¯ c ¯ H ¯ x − c ¯ H ¯ x ! . (3.12)The speed of sound is given by c s = 4 κ κ κ − κ κ − κ κ κ (2 κ κ + 3 κ ) (3.13)where the various κ i ’s are defined by κ = − c ¯ H ¯ x (( ¯ H ¯ x ) ′ + ¯ H ¯ x c G ( ¯ H ( ¯ H ¯ x ) ′ + ¯ H ¯ x ) − c + ¯ c ¯ H ¯ x (12 ¯ H ¯ x ′ + 15 ¯ H ′ ¯ x + 3 ¯ H ¯ x ) κ = − ¯ c c ¯ H ¯ x + 3¯ c G ¯ H − c ¯ H ¯ x + 30¯ c ¯ H ¯ x κ = − (1 − c ¯ y ) − ¯ c ¯ H ¯ x c G ¯ H ¯ x − c ¯ H ¯ x ( ¯ H ¯ x ) ′ κ = − − c ¯ y ) + 3¯ c ¯ H ¯ x − c G ¯ H ¯ x − c ¯ H ¯ x κ =2¯ c ¯ H ¯ x − c ¯ H ¯ x + 4¯ c G ¯ H ¯ x − c + 15¯ c ¯ H ¯ x κ = ¯ c − c ( ¯ H ( ¯ H ¯ x ) ′ + 2 ¯ H ¯ x ) + ¯ c (12 ¯ H ¯ x ¯ x ′ + 18 ¯ H ¯ x ¯ H ′ + 13 ¯ H ¯ x ) − ¯ c G (2 ¯ H ¯ H ′ + 3 ¯ H ) − ¯ c (18 ¯ H ¯ x ¯ x ′ + 30 ¯ H ¯ x ¯ H ′ + 12 ¯ H ¯ x )– 9 –e must impose that c s > c = 0, the Galileon equations of motion in a FRW background reduce to apair of equations as the dynamics of ¯ y decouple and only ¯ x ′ and ¯ H ′ matter. In this case,the Galileon models admit a long time attractor where both ¯ x ′ and ¯ H ′ vanish (see forinstance [37] for a discussion of the more general attractor where ˙ φH = constant). Theequation of state on the attractor is a constant which depends on the parameters of themodel.Numerically we have integrated the equations of motion running time backwards usingln(1+ z ) as the time variable and starting with the initial conditions ¯ H = 1 , ¯ x = 1 , ¯ y = 0.We have verified that for a large portion of the parameter space, the dynamics are driventowards the attractor.On the other hand, when ¯ c = 0, the three differential equations defining the dynamicshave no attractor as ¯ x = 0 implying that ¯ y ′ = 0. Nevertheless, we have found that close tothe origin in z , where ¯ y ∼
0, the dynamics mimic the one when c = 0 and admit a quasiattractor as long as ¯ y ∼
0. In the future of the Universe, this condition breaks down andthe Hubble rate starts departing from a constant as can be seen in Figure 1. In general, wefind that our present Universe has not quite reached this (quasi) attractor. We will discussthe numerical results in detail in Section 4.
The Galileon models modify gravity and in particular the growth of structure is altered.Defining by δ the density contrast of Cold Dark Matter (CDM), the growth equationbecomes δ ′′ + (cid:18) H ′ ¯ H (cid:19) δ ′ − g eff δ = 0 (3.14)where we have introduce the effective Newton constant in the FRW background g eff ≡ G eff G N = 4( κ κ − κ ) κ ( κ κ − κ κ ) − κ ( κ κ − κ κ ) . (3.15)It is convenient to introduce the growth factor f = δ ′ which measures the growth ofstructure and its deviation from the pure Einstein-de Sitter case where f ≡
1. The growthfactor satisfies f ′ + f + f (cid:18) H ′ ¯ H (cid:19) − g eff = 0 . (3.16)In the following, we shall use Λ-CDM as a template for cosmology. Although reasonabledeviations from Λ-CDM are presently allowed by cosmological data, see [14] for instance,we will treat the deviation from Λ-CDM as small and expand f = f Λ CDM (1 + g ) where, tofirst order, we have g ′ + (cid:18) f CDM − f Λ CDM (cid:19) g = 32 f Λ CDM ( g eff − − ∆ (cid:18) ¯ H ′ H (cid:19) (3.17)where ∆ (cid:16) ¯ H ′ H (cid:17) = H ′ H − H ′ H | Λ CDM . The deviation of the growth factor is given by g (ln a ) = u − (ln a ) Z ln a ln a i u ( v ) (cid:20) f Λ CDM ( g eff − − ∆ (cid:18) ¯ H ′ H (cid:19)(cid:21) dv (3.18)– 10 –here u (ln a ) = exp (cid:16)R ln a ln a i dv ( f CDM − f Λ CDM )( v ) (cid:17) . We can clearly see that the deviations fromΛ-CDM have two origins: the modification of the background as defined by ∆( ¯ H ′ H ) andthe change in Newton constant in g eff −
1. It should be also noted that this is only anindication on the behaviour of structures in Galileon models as non-linear effects have beenshown to be important even on quasi-linear scales [37–40]. Nevertheless, this will be enoughfor an order of magnitude estimate of the deviations from Λ-CDM implied by the Galileonon the growth of structures.
In the Galileon models, the speed of light in Equation (2.22) can be expressed as c p = 1 + 2¯ c G ¯ H ¯ x (3.19)which is completely determined by the dynamics at the background level. The variation ofthe speed of light is tightly constrained by both the duality relation at low z and the CMBdistortions at large z .Particle masses in the Einstein frame, such as the electron mass, will vary in this theoryaccording to the universal rescaling m ψ = A ( φ ) m ψ where m ψ is the Jordan frame masswhich is identified with their masses now. Because the rescaling is universal, this can alsobe reformulated as the variation of Newton’s constant in the Jordan frame of the conformalcoupling. In this frame, the interaction of particles are the ones of the standard model ofparticle physics. As such processes involved in Big Bang Nucleosynthesis (BBN) are notaltered. On the other hand, the Hubble rate at BBN is modified due to the change ofNewton’s constant. In the vicinity of the earth where the Vainshtein mechanism applies,Newton’s constant is simply G N . On cosmological scales, it evolves according to G N = G N (1 − c ¯ y ) , (3.20)where G N is the experimental value on earth now, as can be read off from Friedmann’sequation. The variation between the value of Newton’s constant in the Jordan frame nowand at a given cosmological time in the past of the Universe is then given by δG N G N = − c ¯ y (3.21)We find that the cosmological evolution of y in the distant past and in particular its valueat BBN imply a change in the formation of the elements as it modifies the Hubble rate inthe Jordan frame at BBN. This will give a bound on ¯ c .
4. Exploring the Parameter Space
The aim of this section is to explore the Galileon parameter space, to determine whetherit is possible to find Galileons with c > .1 Experimental bounds on the equation of state and variations of c and G N . We begin by listing the constraints that we will impose. It is well know that the accelerationof the expansion of the universe can be explained if the dominant component of the Universehas an equation of state that is close to minus one. The best current bounds come fromthe analysis of the Plank survey combined with the polarizations of the CMB from theWMAP satellite and observations of baryon acoustic oscillations [41] w = − . +0 . − . (4.1)at 95% confidence.The best current constraint from the duality relation is provided in reference [42]. bycomparison between galaxy cluster mass fraction estimations obtained from X-ray mea-surements (which probe d L /d A ) and observations of the Sunyaev-Zeldovic effect (whichprobe d A ). The clusters considered are all in the redshift range z ∈ (0 . , . (cid:12)(cid:12)(cid:12)(cid:12) δc p c p (cid:12)(cid:12)(cid:12)(cid:12) < . , (4.2)at 68% c.l. assuming a gaussian distribution of errors.The present limits on the amount of µ distortion in the CMB spectrum come fromCOBE/FIRAS observations. At 95% c.l. they are | µ | < . × − at wavelengths of cmand dm [43]. The ARCADE2 balloon also provided constraints on µ spectral distortion, | µ | < · − at 95 % c.l. between 3 and 90 GHz [44]. We assume that the constrainingpower of observations of the black body spectrum of the CMB comes from observations atfrequencies corresponding to T ∼ . K . Therefore we find | µ | < . × − ⇒ | δc p /c p | < . × − . (4.3)This is an extremely tight constraint on the variation of c p since last scattering which willtranslate in a strong bound on ¯ c G .The variation of Newton’s constant between local measurements in the earth’s envi-ronment and at the time of BBN for z ∼ must be [45] δG N G N = − c ¯ y BBN = − . ± .
05 (4.4)at the 1 σ level by combining the deuterium and He abundances with the baryon to photonratio extracted from the Cosmic Microwave Background (CMB) and Large Scale Structure(LSS) data. Stronger bounds on the local variation of G N and of particle masses exist.As they require a detailed analysis of the Vainshtein mechanism in a galactic background,we leave their analysis for further study. The background cosmology of Galileon models, given the values of Ω m and Ω r , is deter-mined by four independent parameters (¯ c , ¯ c , ¯ c , ¯ c G ). The quartic Galileon model corre-sponds to ¯ c = ¯ c G = ¯ c = 0 and depends only on the value of ¯ c . The value of ¯ c is derived– 12 – igure 1: The reduced Hubble rate ¯ H = H/H as a function of the redshift z for the quarticGalileon with ¯ c = 1 .
2, ¯ c G = − .
02 and ¯ c = 0 .
32. The background cosmology now is close to aquasi attractor for z ∼
0. The effect of the coupling ¯ c is to destabilise the attractor in the verydistant future. from the Friedmann equation at z = 0. In this section, we will explore the parameterspace of Galileon models by first considering the quartic model and varying ¯ c to obtainan equation of state today w φ ( z = 0) ≡ w close to −
1. We will also impose that thereis no ghost and that the speed of sound squared is positive. Recall that we have fixed¯ c = 1 to guarantee that the models are ghost-free in a Minkowski setting. Starting from¯ c = 0 corresponding to the pure quartic model, we find that the equation of state cannotapproach w = − c . By increasing ¯ c to value greater than ¯ c ∼ . z close to zero.As a result, the equation of state w cannot reach values which are close enough to -1 tocomply with data, for instance for ¯ c = 1 .
2, which we will choose as our template value,we find that w = − .
58. The value of w can be lowered by taking negative values of¯ c G . Decreasing ¯ c G to values lower than ¯ c G ∼ − .
05, the square speed of sound becomesnegative again for small z . This implies that one can only reach value of w = − . c G = − .
02. Again one can lower w by taking positive values of ¯ c andreach w = − c = 0 .
32 (see Figures 1 and 2) for which the square speed of sound isalways positive and varies significantly (Figure 3). Finally, changing ¯ c by more than 0 . . c = 0 in what follows. The modification of gravity induced by the Galileon field has direct consequences on thegrowth of structure. We have seen that the difference between the Λ-CDM growth factorand the Galileon one depends on the difference ∆ (cid:16) ¯ H ′ ¯ H (cid:17) between the Hubble rate and– 13 – igure 2: The equation of state as a function of the redshift z for the quartic Galileon with¯ c = 1 .
2, ¯ c G = − .
02 and ¯ c = 0 .
32. The equation of state now is w = − Figure 3:
The speed of sound squared as a function of the redshift z for the quartic Galileon with¯ c = 1 .
2, ¯ c G = − .
02 and ¯ c = 0 . ( g eff −
1) which measures the cosmological deviation of the effective Newtonian constantfrom the GR case. The background cosmology differs from the Λ-CDM case but thisdifference is small as can be seen in Figure 4. However this deviation is not negliglible andis large enough to slow down the growth of structures at moderate redshifts z & . g eff at small redshift impliesthat the growth of structure is enhanced in this case (see Figures 5 and 8). The resultingeffect on the growth factor is large (see Figure 8) although the increase of g eff by a factor– 14 – igure 4: The deviation from Λ-CDM as measured by the difference ∆( ¯ H ′ H ) as a function of z forthe quartic Galileon with ¯ c = 1 .
2, ¯ c G = − .
02 and ¯ c = 0 .
32. The variation is relatively small. around 4 at small redshift only results in an increase of f by 30 %. Imposing that thesmall deviation of f from f Λ CDM remains small reveals a tension between requiring theGalileon to be ghost-free in Minkowski space with ¯ c >
0, an equation of state close to − c and G N When both ¯ c and ¯ c G are non-vanishing, as required to have w = −
1, both Newton’sconstant and the speed of light vary cosmologically. The variation of Newton’s constant isshown in Figure 6 where we see that it exceeds the allowed bound. Similarly the variationof the speed of light is far too large to be compatible with the spectral distortions of theCMB, see Figure 7. The latter implies that effectively ¯ c G . − (see figure 6). For suchlow values of ¯ c G and ¯ c = 0 . G N is around 15%, the equation ofstate becomes w = − .
72 and the cosmological values of g eff for small z is very largeimplying that growth of structure is a crucially discriminating test of these models [38].
5. On the attractor
We now recall that the Galileon models admit a long time attractor where x ′ = H ′ = 0for ¯ c = 0 which acts as a quasi attractor in the vicinity of z = 0 when ¯ c = 0. We haveseen that the numerical results (see Figure 1) tend to confirm that our Universe must beclose to this quasi-attractor today. In the following we shall impose that our Universe ison the quasi attractor and study the parameter values that allow this. Setting y = 0 and¯ x = ¯ H = 1, we obtain a restricted parameter space which can be taken to depend on only– 15 – igure 5: The reduced Newton constant g eff as a function of the redshift z for the quartic Galileonwith ¯ c = 1 . c G = − . , c = 0 . Figure 6:
The variation G N ( z ) /G N as a function of the redshift z for the quartic Galileon with¯ c = 1 .
2, ¯ c G = − .
02 and ¯ c = 0 .
32. The variation between BBN and now where locally Newton’sconstant is equal to G N is larger than the allowed bound around 15 %. two parameters. When on this attractor the equations of motion, (3.1)-(3.1), become β = − γ (5.1) λ = − ω (5.2)where β , γ , λ and ω are given in terms of the c -parameters in Equations (3.2), (3.3),(3.6) and (3.7) respectively. When combined with the Friedmann equation (3.11), the– 16 – igure 7: The variation of the speed of light as a function of the redshift z for the quartic Galileonwith ¯ c = 1 .
2, ¯ c G = − .
02 and ¯ c = 0 .
32. The variation between last scattering and now is muchtoo large.
Figure 8:
The deviation from Λ-CDM of the growth factor as a function of the redshift z for thequartic Galileon with ¯ c = 1 . c G = − . , c = 0 . parameters are constrained to be¯ c = 3Ω m + Ω r c = −
139 + 8¯ c − c G m r
27 (5.4)¯ c = −
53 + 2¯ c − c G m + 20Ω r c is completely determined by the requirement that the Galileon is on theattractor today.The equation of state of the Galileon fluid can also be determined on the attractor: w = 3 + Ω r − m + Ω r ) (5.6)This is also independent of the remaining Galileon parameters ¯ c and ¯ c G . If we takeΩ m = 0 .
279 and Ω r = 5 . × − as representative values of the observed cosmologytoday we find that the Galileon equation of state is w = − .
39 to three significant figures.On the attractor the speed of light is given by c p = 1 + ¯ c G (5.7)The constraint on allowed variation of the speed of light from observations of the CMB inEquation (4.3) requires ¯ c G < . × − if the Universe is on the attractor today.The ratio of the effective to true Newton’s constants is g eff = (cid:0) − . . c + 3 . c G + 0 . c − . c ¯ c G + 10 . c G (cid:1) / (cid:0) . − . c + 31 . c G + 5 . c + 12 . c ¯ c G − . c G + ¯ c − . c ¯ c G + 33 . c ¯ c G − . c G (cid:1) (5.8)where we have assumed our fiducial cosmology, Ω m = 0 .
279 and Ω r = 5 . × − ,and numbers have been quoted to three significant figures. Therefore, on the attractor,measurements of the speed of light directly determine ¯ c G and then measurements of theeffective Newton constant can be used to place constraints on ¯ c . In Figure 9 we show theconstraints imposed on the ¯ c and ¯ c G parameter space that result from requiring | g eff − | < . c G coming from variation in the speed of light. Therefore on the attractor it is not possiblefor the Galileon to mimic Λ-CDM.Whilst it is not necessary for the Galileon to be exactly on the attractor, we expectthat the system will be evolving towards the attractor, and that the Universe is currentlyapproaching this solution. The tension between the different observational bounds on theGalileon parameter space discussed above gives an indication of why it is so difficult tofind Galileon parameters that fit all current observables.
6. Conclusions
We have shown that Galileons that are universally coupled to all particle species and remainghost free around both Minkowski and FRW backgrounds cannot drive the backgroundevolution of the Universe today whilst remaining in agreement with all other observationalconstraints. Even if one were to consider Galileon models with the negative-sign kineticterms, the large value of the disformal coupling found in [21] is not compatible with CMBdistorsions. In fact, the constraints can only be met when different species couple differentlyto the Galileon. – 18 – c c G Figure 9:
The shaded region shows the portion of parameter space (¯ c G , ¯ c ) in which | g eff − | < . Matching the background cosmology requires a large value of ¯ c , the parameter thatcontrols the conformal coupling. This is not compatible with the observed absence ofvariation of Newton’s constant and observations of BBN. One way of alleviating this tensionis to decouple baryons completely. In this case, the baryonic masses do not vary at all fromBBN onwards. Similarly, the tight bound on ¯ c G , the parameter controlling the disformalcoupling, from the CMB distortion is complemented by the tighter bound on M γ from thePrimakoff effect in helium burning stars [29] M γ ≥
346 MeV (6.1)which implies that photons must be effectively disformally decoupled. Therefore we con-clude that both baryons and photons must be conformally and disformally decoupled fromthe Galileon field. Hence cosmological Galileons with positive kinetic energy terms at thelowest order in their effective Lagrangian are nothing but a new type of coupled quintessencemodels, where CDM is the only species which can have significant interactions with thedark energy scalar field. However such a theory requires a mechanism to explain why thecoupling between the Galileon and dark matter do not get communicated to the visiblesector. This appears to require additional fine tunings, although this is very dependent onthe assumptions made about the theory of dark matter.Alternatively we could assume that the Galileon field never dominates the expansionhistory of the universe. Then it would be possible to make the field sufficiently weaklycoupled that it is in agreement with all current observations at the cost of the theorybecoming less cosmologically interesting. Such a Galileon field could be a remnant of amechanism at high energies that solves the cosmological constant problem, or to arise froma brane world scenario that has no connection to explanations of the current expansion of– 19 –he universe. Less is currently know about the constraints on such theories, and exploringthem remains an interesting topic for research.
Acknowledgments
We would like to thank A. Barreira and B. Li for stimulating comments and exchanges.CB is supported by a Royal Society University Research Fellowship. GG was supported inpart by a grant from the John Templeton Foundation. P.B. acknowledges partial supportfrom the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN- GA-2011-289442) and from the Agence Nationale de la Recherche under contract ANR 2010 BLANC0413 01. ACD acknowledges partial support from STFC under grants ST/L000385/1 andST/L000636/1.
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