Cosmology of Chameleons with Power-Law Couplings
aa r X i v : . [ a s t r o - ph . C O ] M a r Draft version November 7, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
COSMOLOGY OF CHAMELEONS WITH POWER-LAW COUPLINGS
David F. Mota & Hans A. Winther
Institute of Theoretical Astrophysics, University of Oslo, Box 1029, 0315 Oslo, Norway
Draft version November 7, 2018
ABSTRACTIn chameleon field theories a scalar field can couple to matter with gravitational strength and stillevade local gravity constraints due to a combination of self-interactions and the couplings to matter.Originally, these theories were proposed with a constant coupling to matter, however, the chameleonmechanism also extends to the case where the coupling becomes field-dependent. We study thecosmology of chameleon models with power-law couplings and power-law potentials. It is found thatthese generalized chameleons, when viable, have a background expansion very close to
ΛCDM , butcan in some special cases enhance the growth of the linear perturbations at low redshifts. For themodels we consider it is found that this region of the parameter space is ruled out by local gravityconstraints. Imposing a coupling to dark matter only, the local constraints are avoided, and it ispossible to have observable signatures on the linear matter perturbations.
Subject headings: (cosmology:) cosmic background radiation; cosmology: miscellaneous; cosmology:observations; cosmology: theory; cosmology: large-scale structure of universe INTRODUCTION
The origin of dark energy (DE) responsible for the cos-mic acceleration remains a mystery. A host of indepen-dent observations have supported the existence of darkenergy over the past decade, however, no strong evidencewas found yet implying that dynamical DE models arebetter than a cosmological constant (see e.g. Davis et al.(2007) for results on observation of dark energy and(Copeland et al. 2006; Durrer & Maartens 2008) for the-oretical overviews). The first step towards understandingthe origin of DE would be to detect some clear deviationfrom the
ΛCDM model observationally and experimen-tally.Scalar fields (quintessence) are natural DE candi-dates, with an equation of state different from − .In quintessence models (Wang et al. 2000; Zlatev et al.1999) the scalar field is slowly rolling down its potential,its energy density is dominated by the potential energyand almost remaining constant provided that the poten-tial is flat enough. However, this means that the massof the scalar field is in general very light and as a resultthe scalar field almost does not cluster so that its effectsin cosmology are mainly on the (modified) backgroundexpansion rate.However, a certain class of theories have been pro-posed, in which the scalar field(s) properties dependon the environment: these are the class of chameleonfield theories, proposed in Khoury & Weltman (2004),that employed a combination of self-interactions of thescalar-field and couplings to matter to avoid the mostrestrictive of the current bounds. In the models thatthey proposed, a scalar field couples to matter withgravitational strength, in harmony with general expec-tations from string theory, whilst, at the same time, re-maining relatively light on cosmological scales. Origi-nally, the chameleon was proposed with a constant cou-pling to matter, however, it was shown in Brax et al.(2010b) that the chameleon mechanism extends to thecase where the coupling becomes field-dependent. The chameleon mechanism can also be present in other typesof models like shown in Wei & Cai (2005) for the caseof k-essence. Because chameleons are allowed to cou-ple strongly to matter and at the same time exhibit along ranged fifth-force in space, they could leave a strongsignature on the growth of the matter perturbations.Chameleon models also have signatures in local gravityexperiments which can be within reach of near-future ex-periments (Khoury & Weltman 2004; Brax et al. 2010a;Steffen & Gammev Collaboration 2008).The modified evolution of the matter density pertur-bations can provide an important tool to distinguishgenerally modified gravity DE models, from DE modelsinside GR like the ΛCDM model (Carroll et al. 2006;Faulkner et al. 2007; Song et al. 2007a; Bean et al.2007; Song et al. 2007b; Pogosian & Silvestri 2008;Tatekawa & Tsujikawa 2008; Mota & Barrow 2004;Oyaizu et al. 2008; Koyama et al. 2009; Mota et al.2007; Koivisto & Mota 2008; Bourliot et al. 2007; Mota2008). In fact the effective gravitational constant G eff which appears in the source term driving theevolution of matter perturbations can change signif-icantly relative to the gravitational constant G . Auseful way to describe the perturbations is to writethe growth function f = d log δ m log a as f = Ω m ( z ) γ where Ω m is the density parameter of non-relativistic matter(baryonic and dark matter) (Peebles 1984; Lahav et al.1991). One has γ ≈ . at low redshifts in the ΛCDM -model (Wang & Steinhardt 1998; Linder 2005;Huterer & Linder 2007). While γ is quasi-constant instandard (non-interacting) DE models inside GR, thisneeds not be the case in modified gravity models. Anadditional important point is whether γ can exhibitscale dependence (dispersion). When this happens theresulting matter power spectrum acquires an additionalscale dependence which is not found in ΛCDM .In Gannouji et al. (2010) a chameleon model, with aconstant coupling, was constructed where the presentvalue of the growth index γ can be as small as γ = 0 . together with a significant redshift dependence. This al-lows to clearly discriminate this model from ΛCDM .In this paper, we investigate the cosmological proper-ties of chameleon models where the matter-coupling isa power-law function of the scalar-field. This article isdivided in 5 main sections. In section 2 we give a shortreview of chameleon models and define the models we willbe looking more closely at. In section 3 we discuss thebackground evolution. In section 4 we discuss local andcosmological constraints on our models, and in section 5we consider the evolution of the matter perturbations inour models. CHAMELEON THEORIES
In this section we review the basic properties ofchameleon models. We start by considering the scalar-tensor theory described by the action S = Z dx √− g " RM −
12 ( ∂φ ) − V ( φ ) + S m (˜ g ( i ) µν , ψ i ) (1)where M pl = √ πG is the reduced Planck scale, g isthe determinant of the metric g µν , R is the Ricci-scalarwith respect to g µν and ψ i are the different matter-fields(including radiation). The matter field i couples to theJordan-frame metric ˜ g ( i ) µν which is related to the Einstein-frame metric g µν via a conformal rescaling on the form ˜ g ( i ) µν = e β i ( φ ) g µν (2)In the following we focus, for simplicity, on the case whereall the matter-fields couple to φ with the same β i ( φ ) ≡ β ( φ ) . Thus, a general model of this kind is then uniquelyspecified by stating the two functions V ( φ ) and β ( φ ) .A variation of the action Eq. (1) with respect to φ yields the field-equation (cid:3) φ = V ,φ − β, φ ( φ ) e β ( φ ) ˜ g ( i ) µν ˜ T µν ( i ) (3)where ˜ T µν = √− g ∂ L m ∂g µν is the energy-momentum ten-sor of the matter fields. The energy-momentum ten-sor for radiation vanishes meaning that chameleons donot couple directly to photons. However, a coupling tophotons, which have some interesting observable signa-tures (Davis et al. 2009; Schelpe 2010; Davis et al. 2010;Brax & Zioutas 2010), can be introduced by modifyingthe electromagnetic field strength F µν → e β γ ( φ ) F µν .In the perfect fluid approximation, we have ˜ g µν ˜ T µν = − ˜ ρ m where ˜ ρ m is the Jordan-frame energy density ofnon-relativistic matter. The density ˜ ρ m is conservedwith respect to the Jordan-frame metric ˜ g µν . In theEinstein-frame the density ρ m ≡ ˜ ρ m e β ( φ ) is both con-served and φ -independent. The field equation, in theEinstein-frame, can be written (cid:3) φ = V eff ,φ V eff ( φ ) = V ( φ ) + ρ m e β ( φ ) (4)where V eff is the effective potential. In the following, thequantity β, φ will be referred to as the coupling and β ( φ ) as the coupling-function. We will also use the notation A, φ ≡ dAdφ and A, φ a ≡ dAdφ (cid:12)(cid:12)(cid:12) φ = φ a . Chameleon thin-shell mechanism
We will in this section give a short review of thechameleon thin-shell mechanism in which these theoriescan have a strong matter coupling and still be viable.The coupling Eq. (2) of φ to matter leads to the in-troduction of a fifth-force in nature. In linear theo-ries of massive scalar fields the superposition principleholds, meaning that the larger a massive body is thestronger the fifth-force becomes. Chameleon theories be-have quite opposite: In situations where massive bodiesare involved, the chameleon field is trapped inside thebodies and its influence on other bodies is only due to athin shell close to the surface of the bodies. This leadsto a shielded fifth-force which becomes hard to detect.In order to have a chameleon mechanism in a theorythe following conditions must be satisfied • For a given density ρ the effective potential V eff = V ( φ ) + ρ e β ( φ ) must exhibit a minimum φ min . • The mass, m min ≡ p V eff ,φφ ( φ min ) , of small oscilla-tions about this minimum must be a real increasingfunction of ρ .and puts the following constraints on the potential V ( φ ) and coupling-function β ( φ ) : i) V, φ β, φ < , ii) V, φφφ V, φ > , iii) β, φφφ β, φ > (5)Note that the second condition above is required if wewant the chameleon to exhibit a long-ranged force in thevacuum of space. The final requirement, which is themost important, is that there exists a chameleon thin-shell mechanism in which the fifth-force, for sufficientlylarge bodies, becomes suppressed relative to the gravita-tional force.To see how this mechanism works in practice, we con-sider a spherical object with constant density ρ c , radii R c and mass M c embedded in a background of homogeneousdensity ρ b ≪ ρ c . In such a static spherical symmetricspace time, the field equation reads d φdr + 2 r dφdr = V eff ,φ = (cid:26) V, φ + ρ c β, φ e β ( φ ) r < R c V, φ + ρ b β, φ e β ( φ ) r > R c (6)The effective potential have two minimums at φ = φ c and φ = φ b satisfying V, φ ( φ c ) + β, φ ( φ c ) ρ c e β ( φ c ) = 0 (7) V, φ ( φ b ) + β, φ ( φ b ) ρ b e β ( φ b ) = 0 (8)The φ -mediated fifth-force per unit mass is given by ~F φ = − β, φ ~ ∇ φ for r > R c (9)In the following we will consider the simplest case β ( φ ) = QφM pl , in which the coupling β, φ = Q/M pl is constant,together with an arbitrary runaway potential like e.g. V ( φ ) = M n +4 φ n .For this case, the field profile outside the body havebeen derived in Khoury & Weltman (2004) and reads φ = φ b − Q eff πM pl M c r e − m b r (10)where m b = p V eff ,φφ ( φ b ) is the mass of the field in thebackground and Q eff is the effective coupling. The valuefor Q eff depends on the value of the thin-shell factor ǫ th ≡ φ b − φ c QM pl Φ c (11)where Φ c is the Newtonian gravitational potential. Whenthe body is small in the sense that ǫ th > the field actslike a linear scalar field and Q eff = Q .For large bodies, ǫ th ≪ , the field will be stuck at theminimum φ c inside the body, and the only changes in φ takes place in a thin-shell close to the surface. We saythat the body has developed a thin-shell.In this case we find Q eff = 3 Qǫ th . Thus, the cou-pling strength is suppressed relative to Q . More accu-rate formulas for Q eff when ǫ th . can be found inTamaki & Tsujikawa (2008).The amplitude of the fifth-force on a test-particle ofunit mass outside the body is F φ = 2 QQ eff GM c r for r < m − b (12)For a sufficiently large body we have QQ eff ≪ , andthe resulting fifth-force is suppressed relative to the grav-itational force. When test-masses used in local gravityexperiments have thin-shells, the experimental boundsare easily evaded.For more general coupling-functions β ( φ ) the analysisabove can be much more complicated due to the cou-pling β, φ in Eq. (9) being field-dependent, see Brax et al.(2010b)). We present a derivation of the thin-shell solu-tion for the power-law coupling β ( φ ) = (cid:16) λφM pl (cid:17) m with m > in section 4. Our models
We will look more closely on the following models, inwhich the cosmological constant is part of the potential:
Model A : V ( φ ) = M h (cid:16) Mφ (cid:17) n i β ( φ ) = (cid:16) λφM pl (cid:17) m (13)with n > and m ≥ . This model is a generalizationof the original chameleon model ( m = 1 ) introduced inKhoury & Weltman (2004). Model B : V ( φ ) = M h (cid:16) φM (cid:17) n i β ( φ ) = (cid:16) λH φ (cid:17) m (14)with n > , m > and H being the current Hubbleparameter. This model was introduced, and a wide rangeof local gravity constraints was calculated, in Brax et al.(2010b).In order for the chameleon to act as dark energy weneed to choose M ∼ Λ ( M ∼ − eV). Note that thisvalue is also required in order to satisfy local gravityconstraints. Thus, the fine-tuning problem of the cosmo-logical constant is also present in these models. BACKGROUND EVOLUTION
In this section we discuss the background cosmologicalevolution of chameleon models. We consider a flat Friedmann-Lemaitare-Robertson-Walker (FLRW) background metric ds = − dt + a ( t ) ( dx + dy + dz ) (15)The corresponding background equations are given by H M = ρ m e β ( φ ) + ρ r + 12 ˙ φ + V ( φ ) (16) ˙ ρ m + 3 Hρ m = 0 (17) ˙ ρ r + 4 Hρ r = 0 (18)The field equation for φ becomes ¨ φ + 3 H ˙ φ + V eff ,φ = 0 V eff = V ( φ ) + ρ m e β ( φ ) (19)We also introduce the density parameters Ω r = ρ r H M Ω m = ρ m e β ( φ H M Ω φ = V + ˙ φ H M (20)The background evolution of the original chameleonmodel introduced in Khoury & Weltman (2004), whichcorresponds to Model A Eq. (13) with m = 1 , was thor-oughly discussed in Brax et al. (2004b). We will in thenext section show that the chameleon behaves very sim-ilar in the general setting. There are however some im-portant differences. Attractor solution
We show the existence of an attractor solution wherethe chameleon follow the minimum of its effective poten-tial φ = φ min ( t ) as long as the condition m φ ≫ H issatisfied.Suppose the field is at the minimum at some time t i .Then a time later due to the red shifting of the mat-ter density the minimum φ min has moved to a slightlylarger value (or smaller, depending on the form of thecoupling). The characteristic timescale for this evolutionis the Hubble time /H . Meanwhile the characteristictimescale of the evolution of φ is given by /m φ . When m φ ≪ H the response-time of the chameleon is muchlarger than /H , the chameleon cannot follow the min-imum and starts to lag behind. But if m φ ≫ H thenthe response-time of the chameleon is much smaller than /H , the chameleon will adjust itself and adiabaticallystart to oscillate about the minimum. This can also beseen from the analogy of Eq. (19) with a driven harmonicoscillator ¨ x + 2 ζω ˙ x + ω x = 0 (21)This equation will have a solution which oscillates witha decreasing amplitude as long as ζ < . When the fieldis close to the minimum we can approximate ¨ φ + 3 H ˙ φ + m φ ( φ − φ min ) = 0 (22)The condition ζ < reduces to m φ H > , and we willrequire that m φ ≫ H is satisfied from the early universeand until the present era. Reaching the attractor
We discuss how the chameleon acts in the early uni-verse, and how the field converges to the attractor. Forsimplicity we will only focus on the case where ˙ φ min < (Model B Eq. (14)), the case ˙ φ min > (Model A Eq. 13))is analogous.If we release the field at some time t i , at some initialvalue φ i , we can have two cases: Undershooting: φ i ≫ φ min ( t i ) In this case the field equation can be approximated by ¨ φ + 3 H ˙ φ ≈ − V, φ (23)which is the same equation as in quintessence. Thedriving term will dominate over the friction term when V ,φφ ≫ H and will start driving the field down towardsthe minimum. If this condition is not satisfied, the fieldwill be fixed at φ i until the Hubble factor has had time tobe sufficiently redshifted allowing the field to start rollingdown the potential.When the field starts to roll it drops to, and go past, φ = φ min . Here the approximation Eq. (23) cannot beused anymore, but the field will usually have to muchkinetic energy to settle at the minimum and will bedriven past it. The further below the minimum the fieldis driven, the larger the factor β, φ ρ m becomes. Eventhough ρ m is very small in the radiation era it will even-tually kick in and drive the field up again. More impor-tantly, we will also have a contribution from the decou-pling of relativistic matter which will be discussed in thenext section. This will make the field oscillate aroundthe minimum, and as long as m φ /H ≫ the ampli-tude of the oscillations will be damped, making sure thatthe field quickly converges to the attractor. This canbe showed explicitly as done in Brax et al. (2004b), thederivation found there is general and applies to our caseas well. Overshooting: φ i ≪ φ min ( t i ) In this case the potential term V, φ can be ignored andthe φ -equation becomes ¨ φ + 3 H ˙ φ ≈ β, φ T µµ (24)where we have restored the trace of the energy-momentum (EM) tensor. In the radiation-dominatedera this trace is very small since radiation does not con-tribute to the trace, and the field will be frozen at itsinitial value. As the universe expands and cools the dif-ferent matter-species decouple from the radiation heatbath when the temperature is of the same order as themass of the matter-particles. This gives rise to a trace-anomaly where the trace of the EM-tensor gets non-zerofor about one e-fold of expansion leading to a ’kick’ in thechameleon pushing it to larger field-values. This tracecan be written for a single matter-species, see Brax et al.(2004b), as T µ ( i ) µ = − π H M g i g ∗ ( T ) τ ( m i /T ) (25)where g ∗ is the effective number of relativistic degreesof freedom, g i , T i and m i is the degrees of freedom the temperature and the mass of species i respectively. The τ -function is given by τ ( x ) = x Z ∞ x p y − x dye y ± (26)and ± refers to bosons and fermions respectively.See Fig. (1) for a plot of T µµ / (3 H M ) in the radiationera. The plot shows that each kick contributes to the field Fig. 1.—
The trace of the EM-tensor, Ω m eff = − T µµ / (3 H M ) ,in the radiation dominated era for all the different matter speciesdecoupling from the radiation heat bath. equation approximately as an effective matter-density Ω m eff ∼ O (0 . . By using a delta-function source asthe kick we can show that the result is to push the fieldup a distance | ∆ φ | = O (cid:18) | β, φ i | M g i g ∗ ( m i ) (cid:19) (27)where φ i is the field-value before the kick sets in. When β, φ = constant , the field will be kicked almost the sameamount each time a new species freezes out. But when β, φ = constant the smaller the initial value φ i the moreeffective the kick is in bringing it back up. When abovethe minimum the resulting kicks will be balanced by theterm V, φ which drives the field down again making thefield oscillate above the minimum before eventually set-tling down.Due to this mechanism, the chameleon can have initialvalues far below the minimum and still be able to getclose to the attractor relative quickly. The initial valuewill of course depend on the how the chameleon behavesunder inflation. If the chameleon couples to the infla-ton and sits at the minimum at the onset of inflation,then after the inflaton decays to reheat the universe thedensity of matter-species coupled to the chameleon willdecrease rapidly since most of the energy will go to ra-diation. This will lead to a release of the field at a valuewell above the minimum, where the undershoot solutionapplies. As long as m φ ≫ H , the field will typically settleat the minimum before the onset of the Big Bang Nucle-osynthesis (BBN). See Fig. (2) for a typical evolution of φ in the early universe with and without the inclusion ofthe kicks. Dynamics of φ along the attractor Fig. 2.—
The evolution of φ in the early universe as a function ofthe redshift z for two types of initial conditions: φ i ≫ φ min (above)and φ i ≪ φ min (below). In this example V ( φ ) = M + φ , β ( φ ) = H φ . The dashed (full) line corresponds to the solution where weneglect (include) the kicks. The kicks-solution does not reach theminimum until z ≈ , but due to the large mass of the field itstarts to follow it right away. When the field follows the attractor φ ≈ φ min , we have V eff ,φ ≈ . Taking the time-derivative yields ˙ φ ≈ − H V, φ m φ (28)for β ( φ ) ≪ . We further find ˙ φ V = 9 H m φ (29)where Γ =
V m φ V ,φ (30)As long as Γ > , the field will be slow-rolling wheneverthe condition m φ ≫ H is satisfied.The equation of state for a minimal coupled scalarfield (quintessence) is given by ω φ = ˙ φ − V ˙ φ +2 V . Sincethe chameleon is not minimal coupled, the time evo-lution of ρ φ must be computed directly from ˙ ρ φ /ρ φ = − H (1 + ω eff ) . Using Eq. (28), we find ω eff = − (31)From this equation we see that the chameleon acts as a dark energy fluid as long as < Γ , and its only for modelswhere Γ ∼ O (1) where we can have a significant deviationfrom ω = − . See Fig. (3) for a typical evolution of ω eff . Fig. 3.—
The effective equation of state for the chameleon when V ( φ ) = M + φ , β ( φ ) = H φ . We have ω = − m + nn = − . during the period before BBN. After the transition to φ < M ,the chameleon behaves like a cosmological constant with ω = − .The oscillations in ω eff comes from the field oscillating around theminimum before eventually settling down. Model A Eq. (13) yields
Γ = 1 + mn + ( n + m ) n (cid:18) φ min M (cid:19) n (32)The transition from φ min < M to φ min > M takes placefor a redshift (1 + z ) = (cid:20) n Ω φ m Ω m (cid:18) M pl M λ (cid:19) m (cid:21) / ≈ m λ − m/ (33)which for typical values of ( m, λ ) is before the time ofBBN ( z ≈ ) . This means that in the backgroundtoday we have φ ≫ M and therefore V ( φ ) ≈ M and Γ ≫ .For Model B Eq. (14) we find Γ = 1 + mn + ( n + m ) n (cid:18) Mφ min (cid:19) n (34)The transition from φ min > M to φ min < M takes placefor a redshift (1 + z ) = (cid:20) m Ω φ n Ω m (cid:18) MH λ (cid:19)(cid:21) m ≈ m λ − m (35)where we have used MH ∼ M pl M ∼ . For typical valuesof ( m, λ ) we find a redshift which also is before the timeof BBN. Today this translates into φ ≪ M giving V ( φ ) ≈ M and Γ ≫ .Thus the models we consider here, will have a back-ground evolution very close to that of ΛCDM . Statefinder parameters
The statefinder diagnostics, introduced in Sahni et al.(2003), can be a useful tool for distinguishing differentDE models. The statefinder parameters are defined by r = ¨ aaH , s = r − q − / (36)where q = − ¨ aaH is the deceleration parameter. Upondefining h ≡ H , it follows q = − h ′ h , r = 1 + h ′′ h − h ′ h (37)where a prime denoted a derivative relative to x = − log( a ) . ΛCDM , neglecting the contribution from ra-diation, corresponds to the fixed point ( r, s ) = (1 , .For a general chameleon (in units of M pl ≡ ) we find r = 1 + 32 ( φ ′ ) −
32 Ω m β, φ φ ′ + V, φ h φ ′ (38) s = (cid:18) V, φ V φ ′ + 13 ρ m β, φ V − ( φ ′ ) hV (cid:19) − ( φ ′ ) h V (39)When the chameleon is slow rolling along the minimumof the effective potential, that is φ ′ ≈ V, φ m φ and m φ ≫ H ,we can simplify the above equations to r − ≃
272 1Γ Ω φ (40) s ≃ − (41)When Γ ≫ we find r ≈ and s ≈ . It is onlyfor Γ ∼ O (1) that we can have an (observable) devi-ation from ΛCDM . For the models considered here, Γ ≫ and the statefinders will be the same as in ΛCDM .See Gannouji et al. (2010) for a viable chameleon modelwhere the statefinder parameters deviates significantlyfrom
ΛCDM . CONSTRAINTS
Local gravity constraints
Experimental tests of general relativity in the solarsystem, see e.g. Will (1993), and searches for a fifth-force in nature (Kapner et al. 2007; Hoyle et al. 2001)gives strong constraints on any new interactions. Forchameleon models, these constraints are usually avoideddue to the chameleon mechanism as long as typical test-masses used in the experiments have thin-shells.Local gravity constraints for Model B Eq. (14) wasfound in Brax et al. (2010b), see Fig. (4) for the con-straints when m = 1 and n = 6 , .In order for the chameleon to affect large scale struc-ture formation we need the coupling, evaluated in thecosmological background Q φ ≡ | β, φ today M pl | , to be ofgravitational strength: O (1) . Q φ .For Model B, this corresponds to the r.h.s of the dashedline in Fig. (4). From this figure we see that this requires M . − eV as claimed below Eq. (14).Local gravity constraints for the original chameleonmodel (Model A with m = 1 ) have been calcu-lated in several papers, see e.g. Khoury & Weltman(2004); Mota & Shaw (2007); Brax et al. (2009);Steffen & Gammev Collaboration (2008); Brax et al.(2007); Brax et al. (2007); Gies et al. (2008);Mota & Shaw (2006).Below, we derive thin-shell solutions for Model AEq. (13), and use these solutions to find the the localconstraints coming from tests of the equivalence princi-ple in the solar-system. Thin-shell solutions for the power-law coupling
We show the existence of thin-shell solutions for thepower-law coupling-function β ( φ ) = (cid:16) λφM pl (cid:17) m togetherwith an arbitrary run-away potential like e.g. V ( φ ) = M n +4 φ n (Model A). This model was given a treatment inBrax et al. (2004a) and it was found that there do notexist thin-shell solutions for m > when λ = O (1) . Forvalues λ ≫ , which is equivalent to a coupling scale M ∗ = M pl λ ≪ M pl , there can indeed exist thin-shells.Another conclusion the authors of Brax et al. (2004a)reached was the existence of a possible singularity in thefield-profile. We have calculated the field-profile numer-ically and no such singularity was found. The solutioncorresponding to the claimed singularity seems to comefrom a mathematical correct, but non-physical, solutionto the field equation.The field equation in a static spherical symmetric met-ric with weak gravity reads d φdr + 2 r dφdr = V, φ + ρβ, φ (42)where we have assumed β ( φ ) ≪ . We consider a bodywith constant density ρ c , radii R c and mass M c embed-ded in a background of homogeneous density ρ b and im-pose the boundary conditions dφdr (cid:12)(cid:12)(cid:12) r =0 = 0 dφdr (cid:12)(cid:12)(cid:12) r = ∞ = 0 φ ( r → ∞ ) = φ b (43)Lets start by considering a test body. The field insideand outside the body is then just a small perturbation inthe background φ b . Solving the linearized field equationswe find φ = φ b − Q b πM pl M c r e − m b r r > R c (44) Q b = β, φ b M pl (45)The amplitude of the fifth-force between two test bodies,located in the vacuum of space, is F ( r ) = 2 Q b F gravity ( r ) for m b r < (46)Now lets see what happens for macroscopic bodies. Thethin-shell solution for m = 1 (see Khoury & Weltman(2004)) is characterized by the field being stuck at theminimum, φ c , inside the body. We therefore look for so-lutions where φ (0) ≡ φ i ≈ φ c . The linear approximation V eff ,φ = m c ( φ − φ c ) is now valid close to r = 0 with thesolution φ = φ c + φ c sinh( m c r ) m c r δ (47) δ ≡ φ i − φ c φ c (48)We assume that this solution is valid all the way to r = R c . For this to be true, the linear term in the Taylorexpansion of V eff ,φ must dominate over the higher orderterms inside the body. Since φ is increasing in < r < R c the largest value of | φ − φ c | occurs at r = R c , and leadsto the condition | φ ( R c ) − φ c | φ c ≪ | n − m − | (49)Outside the body the linear approximation V eff ,φ = m b ( φ − φ b ) is valid with the solution φ = φ b − AR c r e − m b r (50)where we have assumed m b R c < as would be the casein most interesting cases. Matching the two solutions at r = R c gives us A = ( φ b − φ c ) (cid:18) m c R c ) m c R c (cid:19) (51) δ = φ b − φ c φ c cosh( m c R c ) (52)The condition Eq. (49) becomes φ b − φ c φ c (cid:18) tanh( m c R c ) m c R c (cid:19) ≪ | n − m − | (53)For x ≫ we have tanh( x ) ≃ , and by using φ b ≫ φ c we find that the condition above is satisfied for all m c R c ≫ φ b φ c (54)The far-away field can now be written φ = φ b − Q eff πM pl M r e − m b r (55)where Q eff = 3 β, φ b M pl ǫ th (56) ǫ th = φ b − φ c β, φ b M Φ c (57)and Φ c is the gravitational potential for the body. Thethin-shell factor ǫ th is on the same form as found inKhoury & Weltman (2004), but it does not have the ge-ometrical interpretation as an explicit thin-shell. It ishowever this factor which determines the suppression ofthe fifth-force.Comparing the effective coupling Eq. (56) (for ’large’bodies: ǫ th ≪ ) with the corresponding expressionEq. (45) (for ’small’ bodies: ǫ th ≫ ) we see that Q eff Q b = 3 ǫ th ≪ (58)This shows that the chameleon force, relative to gravity,between two bodies is suppressed as long as one (or both)of the bodies have a thin-shell, and demonstrates that thechameleon mechanism is present in this model. Lunar Laser Ranging
We will restrict our attention to tests of the equiva-lence principle using Lunar Laser Ranging (LLR), seee.g. Will (1993). LLR measures the free-fall accelerationof the moon and the earth relative to the sun. The ac-celeration induced by a fifth force with the field profile φ ( r ) and effective coupling Q eff is a fifth = | Q eff ∇ φ | /M pl .In most interesting cases, m − b > , the chameleon isa free field in the solar-system. This leads to the follow-ing constraint (see Khoury & Weltman (2004) for a moredetailed derivation) | a moon − a ⊕ || a moon + a ⊕ | ≈ Q ⊙ eff | Q meff − Q ⊕ eff | . − (59)where Q ⊙ eff , Q m eff and Q ⊕ eff is the effective coupling forthe sun, moon and earth respectively. The backgrounddensity in the solar-system is ρ b ≈ − g/cm corre-sponding to the average matter (dark and cold) densityin our galaxy.The resulting bounds for m = 2 and m = 3 are shownin Fig. (5).The strongest constraints on chameleon models (withnatural parameters) typically comes from the Eot-Washexperiment (Kapner et al. 2007). As we can see inFig. (5), the interesting part of the parameter space is n ≪ . In this regime the test-masses used in the Eot-Wash experiment will typically not have thin-shells andthe experiment cannot, by design, detect the chameleon.Nevertheless, the LLR constraints are good enough toconstraint this part of the parameter space where inter-esting cosmological signatures take place. Cosmological constraints
Due to the conformal coupling, Eq. (2), of φ to matter,a constant mass scale m in the Jordan-frame is relatedto a φ -dependent mass scale m ( φ ) in Einstein-frame by m ( φ ) = m e β ( φ ) . A variation in φ leads to a variation inthe various masses (cid:12)(cid:12)(cid:12)(cid:12) ∆ mm (cid:12)(cid:12)(cid:12)(cid:12) ≈ ∆ β ( φ ) (60)BBN constrains the variation in m ( φ ) from the time ofnucleosynthesis until today to be less than around .If the field has settled at the minimum before BBN, theresulting bound turns into β ( φ today ) . . since β ( φ ) isan increasing function of time. When the field is not atthe minimum at BBN we must also require β ( φ BBN ) . . .For our models this constraints only the parameterswhere the coupling satisfies Q φ = | β, φ today M pl | ≫ , inwhich both the background and the matter perturbationsare completely similar to ΛCDM . The bounds for ModelB Eq. (14) are shown in Fig. (4).Another important restriction on chameleon theoriescomes out from considering the isotropy of the Cosmo-logical Microwave Background (CMB) (Hinshaw et al.2003). A difference in the value of φ today and the valueit had during the epoch of recombination would meanthat the electron mass at that epoch differed from itspresent value by ∆ m e m e ≈ ∆ β ( φ ) . Such a change in m e would, in turn, alter the redshift at which recombinationoccurred, z rec : (cid:12)(cid:12)(cid:12)(cid:12) ∆ z rec z rec (cid:12)(cid:12)(cid:12)(cid:12) ≈ ∆ β ( φ ) (61)WMAP bounds z rec to be within (at σ , 23% at σ )of the value that has been calculated using the present Fig. 4.—
Local gravity constraints for the Model B Eq. (14)when n = 6 (top) and n = 10 (bottom). The horizontal lineshows M = M DE = 10 − eV. The vertical line shows λ = 1 .The dashed line shows Q φ = | β, φ today | M pl = 1 . The labels’Irvine’ and ’Eot-Wash’ refers to test of the Newtonian gravita-tional law, ’Casimir’ refers to Casimir experiments, ’PPN’ refers totest of post-Newtonian gravity in the solar-system and ’BBN’ toconstraints from Big Bang Nucleosynthesis. day value of m e (Nagata et al. 2004). Denoting φ , φ rec and φ BBN with the field value today, at recombinationand BBN respectively. Then β ( φ ) > β ( φ rec ) > β ( φ BBN ) and the CMB bound is weaker then the bound comingfrom BBN. PERTURBATIONS
In this section we will study the growth of perturba-tions in chameleon models. We start by consider thegeneral scalar-tensor model given by the action Eq. (1)with a universal matter coupling-function β ( φ ) and po-tential V ( φ ) . In deriving the perturbation we will workin units of M pl = √ πG ≡ . For simplicity, we willconsider the Jordan-frame matter-density satisfying ˙ ρ m + (cid:16) H − β, φ ˙ φ (cid:17) ρ m = 0 (62)since this choice will simplify the field equation. In termsof the Einstein-frame density ρ EFm this choice correspondsto ρ m = e β ( φ ) ρ EFm . This is just a matter of conveniencesince e β ( φ ) ≈ in the late universe whenever the the-ory satisfies the BBN bounds. With this choice the fieldequation reads ¨ φ + 3 H ˙ φ + V, φ + β, φ ρ m = 0 (63)The most general metric in a FLRW space time withscalar perturbations is given by ds = − (1 + 2 α ) dt − aB ,i dtdx i (64) + a ((1 + 2 ψ ) δ ij + 2 γ ,i ; j ) dx i dx j where the covariant derivative is given in terms of thethree-space metric which in the case of a flat backgroundreduces to δ ij . We decompose the field φ into the back-ground and perturbations parts: φ ( x , t ) = φ ( t )+ δφ ( x , t ) .The EM-tensors of non-relativistic matter can be decom-posed as T = − ρ m (1 + δ m ) , T i = − ρ m v, i (65)where v is the peculiar velocity of non-relativistic matterand δ m is the matter-density perturbations defined by δ m ≡ δρ m ρ m − ˙ ρ m ρ m v ≡ δρ m ρ m in the co-moving gauge (66)The equation determining the evolution of the per-turbations follows from the Einstein-equations. Inthe gauge-ready formulation (Hwang 1991), the scalarperturbations equations are ¨ δφ + 3 H ˙ δφ + ( V ,φφ − ∆ a ) δφ + β, φφ ρ m δφ + 2 αV ,φ + β, φ (2 αρ m + δρ m ) − ˙ φ ( ˙ α − Hα + κ ) = 0 (67) ˙ δρ m + 3 Hδρ m − ρ m (cid:18) κ − Hα + ∆ a v (cid:19) − β, φ ( ρ m ˙ δφ + δρ m ˙ φ ) − β, φφ ρ m ˙ φδφ = 0 (68) ˙ κ + 2 Hκ + 3 α ˙ H + ∆ a α − (cid:16) δρ m − α ˙ φ + 4 ˙ φ ˙ δφ − V ,φ δφ (cid:17) = 0 (69) Hκ + ∆ a ψ − (cid:16) − δρ m + α ˙ φ − ˙ φ ˙ δφ − V ,φ δφ (cid:17) = 0 (70) κ + ∆ a χ −
32 ( ρ m v + ˙ φδφ ) = 0 (71) ˙ v − α + β, φ ( ˙ φv − δφ ) = 0 (72) ˙ χ + Hχ − α − ψ = 0 (73)with χ = a ( B + a ˙ γ ) (74) κ = 3( − ˙ ψ + Hα ) − ∆ a χ (75)and ∆ being the co-moving covariant three-space Lapla-cian. In the list of equations above Eq. (67) is the scalarfield equation of motion, Eq. (68) the continuity equa-tion, Eq. (69) the Raychauhuri equation, Eq. (70) theADM energy constraint, Eq. (71) the momentum con-servation constraint and Eq. (73) the ADM propagation equation. In these equations we have not yet fixed thegauge-degrees of freedom. The choice of a gauge willsimplify the system and we will work in the so-called co-moving gauge ( v = 0) . This gauge leaves no residualgauge freedom and we can solve the system for the twovariables ( δφ, δ m ) directly.From Eq. (72) we have α = − β, φ δφ . Solving Eq. (69)for κ and inserting this into Eq. (67,69) we find, af-ter transforming to Fourier space, the following equations ¨ δ m +2 H ˙ δ m − ρ m δ m + δφ (cid:18) V ,φ − β, φ [6 H + 6 ˙ H − k a + 2 ˙ φ ] (cid:19) − δφ (cid:16) β, φφ [2 H ˙ φ − V eff ,φ ] + β, φφφ ˙ φ (cid:17) − β, φ ¨ δφ − ˙ δφ (cid:16) β, φ H + 2 ˙ φ + 2 β, φφ ˙ φ (cid:17) = 0 (76) ¨ δφ + (3 H + 2 β, φ ˙ φ ) ˙ δφ + β, φ ρ m δ m − ˙ φ ˙ δ m + (cid:18) m φ + k a − β, φ V eff ,φ + 2 β, φφ ˙ φ (cid:19) δφ = 0 (77)where k is a co-moving wavenumber.When the field is slow rolling along the minimum wecan neglect all terms proportional to ˙ φ and the oscil-lating term V eff ,φ . The perturbations in φ will evolvemore slowly than the perturbations in δ m for scales deepinside the Hubble radius, thus, the term ρ m β, φ δ m and ( m φ + k a ) δφ will dominate over the δφ time derivativesin Eq. (77). Using these approximations, we can simplifyEq. (76) and Eq. (77) to ¨ δ m + 2 H ˙ δ m = Ω m H β, φ a m φk ! δ m δφ = − β, φ Ω m (cid:16) H m φ (cid:17) k a m φ (78)Note that the perturbations in φ satisfies δφ ≪ δ m (in M pl = 1 units) as long as m φ ≫ H and β, φ . O (1) justifying dropping the δφ -derivatives in Eq. (77). Restoring M − ≡ πG and defining Q φ = | β, φ M pl | we can write this first equation on the same form as in ΛCDM ¨ δ m + 2 H ˙ δ m = 4 πG eff ρ m δ m (79)where G eff = G Q φ a m φ k (80)The quantity G eff is seen to encode the modification ofgravity due to the chameleon in the weak-field regime.The chameleon will also exhibit an oscillating term,but this term is time-decreasing and hence negligible forsmall redshifts. In some f ( R ) -models however, this oscil-lating term can grow to infinity because the mass of thescalaron is not bounded above. The divergence of thismass can be removed by adding a UV-term as shown in0Thongkool et al. (2009).In studying perturbations, it is convenient to introducethe growth-factor f = d log( δ m ) d log( a ) . In Λ CDM f → at highredshifts and f → in an Einstein-de Sitter universe. Itis important to find a characteristics in the perturbationsthat can discriminate between different DE models andthe Λ CDM. Writing the growth factor as f = Ω γm (81)can be a parametrization that is useful for this purpose,see e.g. Gannouji et al. (2009a,b). In Λ CDM we have toa good accuracy γ ≈ . for low redshifts. Of course insome models γ will vary to much for it to be considereda constant, and there can also be a scale dependence, sowe should write γ = γ ( z, k ) .We will be most interested in scales k relevant to thegalaxy power spectrum (Tsujikawa et al. 2009) . hM pc − . k . . hM pc − (82)where h = 0 . ± . corresponds to the uncertaintyin the Hubble factor today. These scales are also in thelinear regime of perturbations.It is also convenient to introduce the length scale ofthe perturbations λ p = πak and the length scale of thechameleon λ φ = πm φ . In Eq. (80) we have two asymptoticregimes: G eff = (cid:26) G λ p ≫ λ φ G (1 + 2 Q φ ) λ p ≪ λ φ (83)In the GR regime, λ p ≫ λ φ , the perturbations show nodeviation from ΛCDM .In the scalar regime, λ p ≪ λ φ , however, the matter-perturbations will feel a stronger gravitational constantthan in GR. The coupling Q φ is in general a dynam-ical quantity, which will increase with time when thechameleon follows the minimum. Thus, when we reach atime where Q φ > the perturbations will start to growwith increasing amplitude and will quickly enter the non-linear regime.See Baldi (2010); Li et al. (2010a,b); Li & Barrow(2010); Li & Zhao (2009); Zhao et al. (2011); Baldi et al.(2010) for a numerical N-body simulation analysiswith this type of models, and Nunes & Mota (2006);Mota et al. (2008a,b); Brax et al. (2010) for a study ofthe spherical collapse in cosmological models with a timedependent coupling between dark energy, dark matterand other matter fields. The critical length scale λ φ In order to study the perturbations more closely, welook at the value of the critical length scale today λ φ, for our models.The critical length scale for Model A Eq. (13) satisfies λ φ, ∼ − n +1 | Q φ | − ( n +2)2( n +1) pc (84)where Q φ = | β, φ today M pl | . In order for this length scaleto affect the matter perturbations: λ φ, = O (1Mpc) and Q φ ∼ , we need to impose n < . . See Fig. 5 for aplot of the growth factor γ ( z = 0) . The plot shows thetwo regimes: • (i): Phase space where γ < . for all relevantscales. The perturbations are in the scalar regime. • (ii): Phase space where γ ≈ . for all relevantscales. This is the GR regime.For our Model B Eq. (14), the critical length scale isgiven by λ φ, = r mn ( n + m ) (cid:18) M pl M (cid:19) ( n − n − Q − ( n − n − φ πM (85)which gives λ φ, ∼ − − n − Q − ( n − n − φ pc (86)In order to have λ φ, = O (1Mpc) together with a cou-pling Q φ of the order of unity we need n . . . However,for n < the model is no longer a chameleon accordingto our definition in section 1: the range of the field isshorter in the low density cosmological background thanin a high density environment. This also means that localgravity bounds will most certainly be violated.The only way to increase λ φ, up to a mega-parsecvalue is by decreasing the coupling strength. For n > we need Q φ . − (87)which is to small to significantly affect the growth of theperturbations. Thus the perturbations in Model B arealways in the GR regime.It is only in Model A Eq. (13) that we can have in-teresting signatures on the matter perturbations. Butafter imposing local gravity constraints we find that theperturbations are confined to be in the GR regime withno signature on the matter perturbations or on the back-ground evolution relative to ΛCDM . This agrees with theresult found in Gannouji et al. (2010) (for m = 1 ). Theonly way to have observable signatures in these models isto restrict the coupling to dark matter only, and therebyavoiding the local constraints.The different regimes shown in Fig. (5) have been de-rived by considering an universal coupling. Since darkmatter is dominating over baryonic matter at large scalesthe regimes in this figure is expected to be similar if werestrict the coupling to dark matter only.In Gannouji et al. (2010), the chameleon model β ( φ ) = λφM pl (88) V ( φ ) = M (1 − µ (1 − e − φM pl ) n ) (89)where < µ < and < n < , was found tohave observable signatures on the growth of the mat-ter perturbations even when local gravity constraintswas taken into account. We note that this potentialdo not directly generalize to a more general coupling-function β ( φ ) = (cid:16) λφM pl (cid:17) m . This is because the require-ment < n < , which is required in order to have apositive definite mass of the field, leads to violation oflocal gravity constraints for parameters where signaturesare present.1 Fig. 5.—
The two regimes for the growth factor γ in Model AEq. (13): (i) γ < . , and (ii) γ ≈ . for all relevant scales. In re-gion ( i ) we have a significant dispersion between scales. Above, thequadratic coupling m = 2 , and below the cubic coupling m = 3 .The label ’Local constraints’ shows the allowed region from ex-perimental tests of the equivalence principle (Lunar Laser Rang-ing). Due to numerical difficulties the perturbations have beenintegrated using the approximation Eq. (78) rather than using thefull equations Eq. (76)-(77) SUMMARY AND CONCLUSIONS
We have discussed the cosmological evolution ofchameleon models with power-law couplings and power-law potentials. The chameleon follows the attractor so-lution φ = φ min as long as m φ ≫ H , and the attractorcan be reached for a large span of initial conditions. Infact, non-constant couplings can allow for a larger off-set from the attractor in the early universe and still bein agreement with BBN bounds. Along the attractorthe chameleon is slow rolling and can account for thelate time acceleration of the universe. The background evolution is however found to be very close to that of ΛCDM , as found in many other similar models (see e.g.Faulkner et al. (2007); Brax et al. (2008)). The mass-scale M in the potential is fine-tuned in the same manneras a cosmological constant, thus not providing a solutionto the fine-tuning problem. This is however motivatedand required by local gravity experiments.Even though the background expansion is very close ΛCDM , the growth of the linear perturbations can bequite different. The reason is the fifth-force acting onboth dark matter and baryons, which leads to a differentgrowth rate of matter perturbations on cosmic scales.For our Model B Eq. (14), the linear perturbationsare not affected by the chameleon since the field-rangeis in general too small compared to cosmic scales, or, ifthe field-range is large enough then the coupling is toosmall to produce observable effects. Otherwise the modelwould be ruled out by local gravity constraints.In Model A Eq. (13), the range of the chameleon can belarge enough as to affect the matter perturbations, whichleads to a growth-rate different from
ΛCDM . Since thecoupling, in general, varies with time, we will also have adispersion for scales within the linear regime. However,for this to be the case, it must be emphasized that lo-cal gravity constraints force us to have a gravitationalcoupling of the chameleon field to dark matter only. Byneglecting the coupling to baryonic matter, the purposeof the chameleon mechanism are lost. But with this con-sideration, the growth of matter perturbations can inprinciple allow us to discriminate between Model A and
ΛCDM .Naively, one would expect that introducing a more gen-eral coupling would lead to a more richer phenomenology.The reason why this is not the case in the models consid-ered here is because local constraints are so stringent itforces the chameleon to stay at the local minimum almosteverywhere in space. This again means that the kineticterm of the chameleon is suppressed and the dynamics ofthe theory is determined solely by the interplay betweenthe potential and the coupling. In the power-law mod-els considered here this interplay is very similar to thestandard chameleon model Brax et al. (2010b).If future galaxy surveys manages to pin down the mat-ter power spectrum to a greater accuracy, and detect aclear deviation from
ΛCDM (especially if a scale depen-dence in the growth index γ is detected), then it wouldbe interesting to see how good these chameleon modelscan fit the data.Even though these models, for most parameterschoices, do not leave an imprint on the linear matterperturbations, it may have an impact on the small scalestructure formation. It would be interesting to investi-gate the effect of these models on the non-linear regime ofstructure formation by means of high resolution N-bodysimulations. We leave this for future work. ACKNOWLEDGMENTS