Supersymmetric chameleons and ultra-local models Philippe Brax, Luca Alberto Rizzo, Patrick Valageas
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Philippe Brax, Luca Alberto Rizzo, Patrick Valageas. Supersymmetric chameleons and ultra- local models. Physical Review D, American Physical Society, 2016, 94 (023512 ), 10.1103/Phys- RevD.94.023512. cea-01461805
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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:1605.05946v1 [astro-ph.CO] 19 May 2016 once 4.Ti stecs o oesa ies as diverse as models for intimately case are the and is scales f energy This large realized dark on [4]. been often gravity connected very has of that it modification years Moreover a ten last the ago. over years more fifteen appeared existence its than of observations first the when rKmuae[6.Frti esn n eas fits of because promising a and be reason, might this [6] supersymmetry For stability, cases Galileons radiative some for [16]. in K-mouflage alleviated e.g. or only theorems, La- one, non-renormalisation bare thorny by the low- a to of as corrections is issue radiative grangian the appear the particular, of such In structure as the theories. and field effective terms energy kinetic or potential mechanism. screening the specific precluding a [15] force for situations fifth need no (quasi)-static couplings, in disformal present out- For models is related, chameleon mass. be large the can a to case with wavelength, ab- We this Compton theory. the that the the paper from side of this locality arises in the find [14], and will terms paper kinetic companion of sence a Another intro- in models [13]. ultra-local duced Vainshtein the with and associated Damour-Polyakov 12] mechanism, [9], feasi- [11, chameleon be K-mouflage ways: to few [10], appears a This only locally. in screened ble force be fifth to to scalar-induced needs resulting con- most the For or the couplings, conformal consequences. In formal either physical be different matter. can with coupling to coupling disformal the this freedom [8], on of case been degree general depend have scalar dynamics order the second Their of of are [7]. mere motion dynam- classified a the of beyond where equations going theories ical of Such way simplest constant. the cosmological as field scalar ( R akeeg 13 ssila ytrosnwa twas it as now mysterious as still is [1–3] energy Dark l hs hoisivlennlnaiis ihri the in either non-linearities, involve theories these All hois[]o aien 6.Teemdl tls a utilise models These [6]. Galileons or [5] theories ) ASnmes 98.80.-k ca numbers: PACS induces it halos that matter interaction dark force interaction. of Newton’s fifth number scales, the the larger and where On structure galaxies of field. growth super-chameleon the gra the on of in of energy modification energy presence any dark dark of the the the su existence from the by of preclude decouples set mechanisms screening sector mass chameleon matter matt gravitino baryonic of inverse dark the version the and extreme than field an shorter bo energy comprising dark i.e. sector, the short, dark between d the range In be to gravity. interaction shown modified are of sectors, models matter and breaking supersymmetry ue-hmlo oesweealtpso atrbln ot to belong matter of types all where models Super-chameleon .INTRODUCTION I. uesmerccaeen n lr-oa models ultra-local and chameleons Supersymmetric hlpeBa,Lc let iz,adPtikValageas Patrick and Rizzo, Alberto Luca Brax, Philippe -19 i-u-vte ´dx France C´edex, Gif-sur-Yvette, F-91191 nvri´ ai-alyCA CNRS, CEA, Paris-Saclay Universit´e ntttd hsqeTh´eorique, Physique de Institut Dtd a 0 2016) 20, May (Dated: aino rvt nteSlrSystem. 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Damour-Polyakov a nmclyeuvln oultra-local to equivalent ynamically rescue etr,ie h dark, the i.e. sectors, secluded hree tcnas ffc h yaisof dynamics the affect also can It . ri osrie ob extremely be to constrained is er neato.O h te hand, other the On interaction. | ln A | . 10 − 6 to 2 is quite different from the usual screening mechanisms no direct interaction between the super-chameleon Φ and encountered in other modified-gravity scenarios as it di- matter, we take for the total K¨ahler potential which gov- rectly follows from the locality of the fifth-force inter- erns the kinetic terms of the model action. The intermediate region between the very large † and very small scales is not amenable to our analysis and K = K(ΦΦ )+ K✟SG✟ + KM (1) would require numerical simulations which go beyond our analysis, although we present a thermodynamic approach and similarly for the superpotential which is responsible to investigate the fifth-force non-linear regime. We find for the interactions between the fields that the number of intermediate dark matter halos is af- W = W (Φ) + W✟✟ + W . (2) fected by the presence of the super-chameleon. This is all SG M the more true for galactic size and mass halos where the The kinetic terms for the complex scalar fields φi of the fifth force is of the same magnitude as Newton’s force. A model obtained as the scalar components of the super- more complete analysis would require numerical simula- fields Φi are given by tions which are left for future work. This paper is organized as follows. In section II we i µ ¯¯j kin = Ki¯j ∂µφ ∂ φ (3) describe the supersymmetric chameleon models and the L − dark and baryonic sectors. Next, in section III we show where we have defined that these models can be identified with ultra-local mod- 2 els introduced in a companion paper, over the scales that ∂ K ¯ Ki¯j = ∂i∂¯j K (4) are relevant for cosmological purposes. We describe the ∂Φi∂Φ¯ ¯j ≡ background dynamics and the growth of large-scale struc- i¯j i tures in section IV, considering both linear perturbation and its matrix inverse such that K Kk¯j = δk. The scalar theory and the spherical collapse dynamics. In section V potential obtained from the F -terms of the superfields is we estimate the magnitude of the fifth force within spher- given by ical halos and on cluster and galaxy scales. In section VI i¯j ¯ ¯ we use a thermodynamic approach to investigate the non- VF = K ∂iW ∂¯j W, (5) linear fifth-force regime for the cosmological structures where W¯ is the complex conjugate of W . This is the that turn non-linear at high redshift and for the cores of only term in the scalar potential when the fields are not dark matter halos. We briefly investigate the dependence charged under gauge groups. on the parameter α of our results in section VII and we We will also need to add a D-term potential to the conclude in section VIII. scalar potential when some extra fields in the dark sec- tor are charged under a gauge symmetry. We will also consider the corrections due to supergravity induced by II. SUPERSYMMETRIC CHAMELEONS the presence of the supersymmetry breaking sector. This will be dealt with in the corresponding sections. A. Super-chameleons
The nature of the dark part of the Universe, i.e. dark B. The supersymmetric model matter and dark energy, is still unknown. It is not ruled out that both types of dark elements belong to a secluded We consider supersymmetric models where the scalar sector of the ultimate theory of physics describing all the potential and the coupling to Cold Dark Matter (CDM) interactions of the Universe. In this paper, we will use arise from a particular choice of the K¨ahler potential a supersymmetric setting at low energy and assume that for the dark energy superfield Φ which is non-canonical the theory comprises three sectors with only gravitational whilst the dark matter superfields Φ± have a canonical interaction between each other. We will assume that the normalisation standard model of particles to which baryons belong is 2 † γ one of them. We will also add a supersymmetry break- Λ Φ Φ † † K(ΦΦ†)= 1 +Φ Φ +Φ Φ . (6) ✟✟ 2 + + − − ing sector SG and a dark sector comprising both the dark 2 Λ1 energy field, which will turn out to be a supersymmet- ric version of a chameleon dark energy model, and dark The self-interacting part of the superpotential is matter represented by fermions in separate superfields γ Φω 1 Φγ from the super-chameleon one. For details about super- W = + , 0 < ω < γ, (7) symmetry and its relation to cosmology, see for instance √2ω Λω−3 √2 γ−3 0 Λ2 ! [22]. Baryons are introduced in a secluded sector defined by where Φ contains a complex scalar φ whose modulus acts the K¨ahler potential KM and the superpotential WM . as super-chameleon and Φ± are chiral superfields con- This is the matter sector which complements the dark taining dark matter fermions ψ±. Defining the super- sector and the supersymmetry breaking one. Assuming chameleon field as φ(x)= φ eiθ and identifying φ φ , | | ≡ | | 3 one can minimise the potential over the angular field θ second part of the new scalar potential is far more com- and after introducing the new scales plicated with the addition of these new fields but when π− = 0 it simplifies and the sum of both terms yields (γ−1)/2 (ω−3)/(γ−ω) h i Λ1 Λ2 Λ=Λ2 , φmin =Λ2 , 1 2 Λ Λ V (π )= qπ2 ξ2 + g′2φ2π2 ; π =0, (15) 2 0 + + + − (8) 2 − h i the scalar potential becomes where we have put π = π . It can be shown [19] + | +| n 2 that π− = 0 minimises the whole potential so we only 2 2 h i ΦΦ† dW 4 φmin consider the effects of the new term V (π+). In particular, VF(φ)= K =Λ 1 , (9) dΦ − φ the mass of the charged scalar π+ is " #
m2 =2g′2φ2 2qξ2. (16) with π+ −
n = 2(γ ω) for n 2, γ ω +1. (10) At early times the super-chameleon is small (φ φmin) − ≥ ≥ and this mass is negative. The U(1) symmetry≪ is there- When φ φ equation (9) reduces to the Ratra- fore broken ( π+ = 0). However, as the cosmological min h i 6 Peebles potential≪ [23] field evolves towards its minimum this mass increases until it reaches zero, restoring the symmetry so that n 4 φmin π+ = 0. Minimising (15) with respect to π+ one finds φ φmin : VF(φ) Λ , (11) h i ≪ ≈ φ 4 4 √q mπ ξ φ< ξ : V = + + , (17) which has been well studied in the context of dark en- g′ min − 8q2 2 ergy and used to define chameleons. This is the reason √q ξ4 why this model is called super-chameleon. At larger field φ> ξ : V = . (18) g′ min 2 values the potential has a minimum at φ = φmin where VF(φmin) = 0 and dW/dφ = 0. Supersymmetry is there- Therefore, at late times we recover the present-day dark fore broken whenever φ = φmin and restored at the min- energy density by taking imum where the supersymmetric6 minimum always has a 4 vanishing energy (this follows from the supersymmetry ξ = 2¯ρde0, (19) algebra). Then, a new mechanism must be introduced in order to have a non-vanishing cosmological constant at which gives ξ 10−3eV. This mechanism requires that ′∼ the minimum of the potential. φmin > √qξ/g , which imposes restrictions on the param- eter space,
− − C. The Fayet-Iliopoulos mechanism Λ (ω 3)/(γ ω) √q Λ 2 > (2¯ρ )1/4 . (20) 2 Λ g′ de0 0 An effective cosmological constant can be implemented by introducing two new scalars Π± = π± + ... with charges q under a local U(1) gauge symmetry in the D. The coupling to Cold Dark Matter dark sector.± These have the canonical K¨ahler potential Dark energy in the form of Φ is coupled to dark matter. † 2qX † −2qX K(Π±) = Π+e Π+ + Π−e Π−, q> 0, (12) The coupling function between the two dark sides of the model is found by considering the interaction of Φ and where X is the U(1) vector multiplet containing the U(1) Φ± gauge field Aµ. They are chosen to couple to the super- chameleon via the superpotential g Φσ Wint = m 1+ σ−1 Φ+Φ−, σ> 0, (21) ′ m Λ3 Wπ = g ΦΠ+Π− (13) which gives a super-chameleon dependent mass to the ′ where g = (1) is a coupling constant. This construc- dark matter fermions tion gives riseO to new terms in the scalar potential. The 2 first contribution is the D-term potential coming from ∂ Wint ψ+ψ−. (22) the fact that the Π± fields are charged L⊃ ∂Φ+∂Φ−
1 2 2 2 2 When the dark matter condenses to a finite density, VD = qπ+ qπ− ξ , (14) 2 − − ρ = m ψ+ψ− , this term gives a density-dependent con- h i tribution to the scalar potential where we have included a Fayet-Illiopoulos term ξ2 which will later play the role of the cosmological constant. The A(φ)ρ, (23) L⊃ 4 from which one can read off the coupling function where we have defined the energy density
g φσ n 4 3 ρ∞ = Λ =ρ ¯0(1 + z∞) , and 0 < ϕ ϕmin , (32) A(φ)=1+ σ−1 . (24) ασ ≤ m Λ3 where z∞ is the redshift below which the field becomes This function reappears in the form of the conformal cou- close to its supersymmetric minimum ϕmin[34]. pling between dark matter and dark energy considered as As in scalar tensor theories, such as dilaton models or a scalar-tensor theory f(R) theories, it is convenient to introduce the coupling β(ϕ) defined by
E. The normalised dark-energy scalar field ϕ d ln A β(ϕ)= M (33) Pl dϕ −1 Because Kφφ¯ = 1 the field φ is not canonically nor- σ/γ σ/γ−1 6 ασ MPl ϕ ϕ malised, since the kinetic term in the Lagrangian reads = 1+ α (34) γ ϕ ϕ ϕ min " min # min γ2 φ 2(γ−1) µ ¯ µ ¯ 2 2 2 kin = Kφφ¯ ∂µφ∂ φ = | | ∂µφ∂ φ. and the effective mass m = ∂ Veff /∂ϕ at the minimum L − − 2 Λ1 eff (25) of the effective potential, The normalised field is then easily defined by σ/γ (n+σ)/γ 2 ασ ρ∞ ϕ n ϕmin γ meff (ϕ)= 2 φ γ ϕ ϕmin " γ ϕ ϕ =Λ1 , (26) Λ (n+2σ)/2γ 1 n ϕ σ ρ min + , (35) −2γ ϕ γ ρ and the coupling function (24) becomes ∞ #
ϕ σ/γ gφσ where we used Eq.(31). The quasi-static approximation min 2 2 A(ϕ)=1+ α with α σ−1 , (27) (32) applies if meff H . This holds for redshifts z ϕmin ≡ mΛ3 ≫ ≤ z∞ provided and 2 αρ∞ 2 ϕmin γ γ γ(ω−3)/(γ−ω) 2 H∞, whence α, (36) φmin Λ2 Λ2 ϕmin ≫ MPl ≪ ϕmin =Λ1 =Λ1 , Λ1 Λ1 Λ0 where in the second inequality we assumed z∞ zeq. At (28) higher redshifts, m (z) grows at least as fast as≤H(z) in while the effective potential V (ϕ)+ ρ(A(ϕ) 1) is eff F − both the matter and radiation eras if we have 2 ϕ n/2γ ϕ σ/γ matter era: σ 2γ, radiation era: σ γ + ω/2. (37) V (ϕ)=Λ4 min 1 + αρ . ≤ ≤ eff ϕ − ϕ " # min (29) F. Supersymmetry breaking Notice that the effective potential in this model coincides with the one obtained in a scalar tensor theory with the Supersymmetry is broken by values much larger than potential VF(ϕ) and the coupling function A(ϕ). We will exploit this fact below. Since we require the cosmology the energy density of CDM. This is achieved in a dedi- to remain close to the Λ-CDM scenario, i.e. the fifth cated sector of the theory which we do not need to specify force must not be much greater than Newtonian grav- here. Gravitational interactions lead to a correction to ity, within this framework we can infer that the coupling the scalar potential coming from supersymmetry break- function A(ϕ) must remain close to unity. This provides ing [20] the constraint m2 K 2 m2 φ2γ 3/2 | Φ| 3/2 ∆V✟SG✟ = 2γ−2 , (38) α 1 (30) KΦΦ† ∼ Λ ≪ 1 on the parameter combination α of Eq.(27). where m3/2 is the gravitino mass. This competes with The dynamics of the model can be determined by mini- the density dependent term in the effective potential (29). mizing the effective potential. This leads to the minimum This correction does not upset the dynamics of the model ϕ of the theory in the presence of matter (CDM) as long as
(n+σ)/γ (n+2σ)/2γ 2 ϕmin ϕmin ρ ϕmin αρ∞ = , (31) 2 2 . (39) ϕ − ϕ ρ MPl ≪ MPlm3/2 ∞ 5
(i) This is typically much more stringent than the quasi- where the dark matter fields ψm follow the Jordan-frame static condition (36). Using Eq.(35) this can also be metric gµν , with determinant g, which is related to the shown to correspond to a condition on the mass of Einstein-frame metricg ˜µν by the scalar field ϕ at the supersymmetric minimum, for 2 z z∞, gµν = A (ϕ)˜gµν . (45) ≤ 2 αρ∞ 2 meff (ϕmin) 2 m3/2. (40) We explicitly take no coupling between baryons and the ∼ ϕmin ≫ scalar field to make possible the equivalence with the As the gravitino mass is always greater than 10−5 eV in supersymmetric chameleon models. In this paper we re- realistic models of supersymmetry breaking [24], we de- strict ourselves to large cosmological scales, which are duce that the range of the scalar interaction mediated by dominated by the dark matter, and we neglect the im- ϕ is very small, at most at the cm level. Because the pact of baryons. Ultra-local models are defined by the scalar interaction has such a short range, we call these property that their scalar-field kinetic term is negligible, models ultra-local. In fact, we shall see below that they ˜ (ϕ)= V (ϕ). (46) can be related to the so-called“ultra-local models” intro- Lϕ − duced in the companion paper [14]. Introducing the characteristic energy scale 4 of the po- tential and the dimensionless fieldχ ˜ as M G. Coupling to baryons V (ϕ) χ˜ , and A(˜χ) A(ϕ), (47) ≡− 4 ≡ We consider that matter fermions ψ belong to a super- M these models are fully specified by a single function, A(˜χ), field ΦM . The mass of the canonically normalised matter fermions becomes which is defined from the initial potential V (ϕ) and cou- † 2 pling function A(ϕ) through Eq.(47). In other words, K(Φ,Φ )/2MPl (0) mψ = e mψ , (41) because the kinetic term is negligible there appears a de- generacy between the potential V (ϕ) and the coupling (0) ∂2W where m is the bare mass of the baryons 2 . The ψ ∂ΦM function A(ϕ). The change of variable (47) absorbs this exponential prefactor is at the origin of the coupling func- degeneracy and we are left with a single free function tion between the matter fields and the super-chameleon A(˜χ). in the Einstein frame. This leads to the identification of coupling function in the matter sector
2 2 B. Cosmological background of ultra-local models ϕ /2MPl AM (ϕ)= e (42) for the canonically normalised super-chameleon, and the Because the matter fields follow the geodesics set by coupling to baryons the Jordan frame and satisfy the usual conservation equa- tions in this frame, we mostly work in the Jordan frame. ϕ βM (ϕ)= , (43) We introduce the time dependent coupling MPl ¯ which is the coupling of a dilaton to matter. As long as d ln A ǫ2(t) , (48) ϕ M , which is already required to suppress the ≡ d ln a min ≪ Pl supergravity corrections to the scalar potential, the cou- such that, as shown in the companion paper, the Fried- pling to baryons is negligible. Hence this model describes mann equation reads as a scenario where dark energy essentially couples to dark matter and decouples from ordinary matter. 3M 2 2 = (1 ǫ )−2a2(¯ρ +¯ρ +ρ ¯ ), (49) PlH − 2 rad χ˜ where τ is the conformal time, the conformal Hubble III. THE SUPERSYMMETRIC CHAMELEON expansion rate, and the Jordan-frameH Planck mass is AS AN ULTRA-LOCAL MODEL 2 ¯−2 ˜ 2 MPl(t)= A (t) MPl, (50) A. Definition of ultra-local models whileρ ¯,ρ ¯rad andρ ¯χ˜ are the matter, radiation and scalar field energy densities. In particular, the background mat- We define ultra-local scalar field models by the action ter and radiation densities evolve as usual as [14] ρ¯0 ρ¯rad0 ˜ 2 ρ¯ = , ρ¯rad = , (51) 4 MPl a3 a4 S = d x g˜ R˜ + ˜ϕ(ϕ) − " 2 L # Z p while the scalar field energy density is given by 4 (i) + d x √ g m(ψm ,gµν ), (44) ¯−4 4 ¯ − L ρ¯χ˜ = A χ,˜ (52) Z − M 6 and the equation of motion of the background scalar field (the only place where deviations of A from unity are im- is portant is for the computation of the fifth force through the gradient ln A). ¯ ¯ 4 4 d ln A dχ˜ 4 ρ¯ In general∇ configurations including perturbations, the = A¯ ρ¯ hence = A¯ ǫ2 . (53) M dχ˜¯ dτ 4 H equation of motion of the scalar field reads as M It is convenient to write the Friedmann equation (49) in d ln A 4 = M . (62) a more standard form by introducing the effective dark dχ˜ ρ energy densityρ ¯de defined by The dark matter component obeys the continuity and 3M 2 2 = a2(¯ρ +ρ ¯ +ρ ¯ ), (54) Euler equations PlH rad de ∂ρ which gives + (v )ρ + (3 + v)ρ =0, (63) ∂τ · ∇ H ∇ · 2ǫ ǫ2 2 2 and ρ¯de =ρ ¯χ˜ + − 2 (¯ρ +ρ ¯rad +ρ ¯χ˜). (55) (1 ǫ2) − ∂v + (v )v + v = Φ. (64) ∂τ · ∇ H −∇ C. Cosmological perturbations of ultra-local models From Eq.(60) we have Φ = ΨN + ln A, and then the scalar field equation∇ (62) gives∇ ∇ We write the Newtonian gauge metric as 4 ln A = M χ.˜ (65) ∇ ρ ∇ ds2 = a2[ (1 + 2Φ)dτ 2 + (1 2Ψ)dx2], (56) − − Thus in terms of matter dynamics, the scalar field ap- so that the Einstein- and Jordan-frame metric potentials pears via the modification of the Poisson equation (58), are related by because of the additional source associated to the scalar field and the time dependent Planck mass, and via the A2 A2 1+2Φ= (1+2Φ)˜ , 1 2Ψ = (1 2Ψ)˜ , (57) appearance of the “new” term (65) in the Euler equation A¯2 − A¯2 − (64), which is due to the spatial variation of ln A. while the Jordan-frame Newtonian potential is defined On large scales we may linearize the equations of mo- by tion. Expanding the coupling function A(˜χ) as ∞ 2 βn(t) δρ + δρχ˜ ln A(˜χ) = ln A¯ + (δχ˜)n, (66) ∇ ΨN 2 . (58) n! a2 ≡ 2M n=1 Pl X Because we wish the deviations of Φ and Ψ from the New- the scalar field equation (62) gives at the background and tonian potential ΨN to remain modest, and we typically linear orders −5 have ΨN . 10 for cosmological and astrophysical 4 β | | −5 1 structures, we require δ ln A . 10 and δρ . δρ . β1 = M , δχ˜ = δ. (67) χ˜ ρ¯ −β This first constraint is fulfilled| | by choosing coupling| | func-| | 2 tions A(˜χ) that are bounded and deviate from unity by Defining less than 10−5, which reads as β 4 β2 ǫ (t) 1 M = 1 , (68) ln A(˜χ) . 10−5, (59) 1 ≡ β ρ¯ β | | 2 2 while the second constraint will follow naturally because we have for the linear matter density contrast δ the characteristic scalar field energy density is the dark ∂δ2 ∂δ ρa¯ 2 energy density today. Then, we can linearize Eq.(57) in 2 2 2 + + ǫ1c δ = 2 (1 + ǫ1)δ, (69) δ ln A. This leads to ∂τ H ∂τ ∇ 2MPl which also reads in Fourier space as Φ=ΨN + δ ln A, Ψ=ΨN δ ln A, (60) − 2 ∂ δ ∂δ 3 2 while the dark energy density fluctuations read as + Ωm(τ) [1 + ǫ(k, τ)] δ =0, (70) ∂τ 2 H ∂τ − 2 H 4 δρχ˜ = δχ.˜ (61) where ǫ(k, τ), which corresponds to the deviation from −M the Λ-CDM cosmology, is given by In Eq.(61) and in the following we use the characteristic property (59) of ultra-local models to write A 1 wher- 2 c2k2 −5≃ ǫ(k, τ)= ǫ1(τ) 1+ . (71) ever this approximation is valid within a 10 accuracy 3Ω a2H2 m 7
4 The k-dependent term dominates when ck/aH > 1, i.e. We have seen in Eq.(19) that ξ = 2¯ρde0 to recover on sub-horizon scales. Moreover, we have (ck/aH)2 the cosmological constant associated with the current ex- 7 −1 ∼ 4 10 today at scales of about 1 h Mpc. Therefore, we pansion of the Universe. We can also take =ρ ¯de0 must have without loss of generality, as this only sets theM choice of normalization ofχ ˜. To simplify the model we also take −7 4 ǫ1 . 10 (72) Λ =ρ ¯de0, which avoids introducing another scale. This | | gives to ensure that the growth of large-scale structures is not too significantly modified. This small value does not re- n/2γ 2 4 4 ϕmin quire introducing additional small parameters as it will =Λ =ρ ¯de0 :χ ˜ = 1 1 (77) M − − ϕ − follow from the constraint (59), which already leads to " # the introduction of a small parameter α . 10−5 that and gives the amplitude of the coupling function ln A. The quantity ǫ2 introduced in Eq.(48) is related to the −2σ/n A(˜χ)=1+α 1+ 1 χ˜ withχ ˜ 1, (78) quantity ǫ1 defined in Eq.(68) by − − ≤− −7 p ǫ2 =3ǫ1, hence ǫ2 . 10 . (73) which is the expression of the coupling function in terms | | of the ultra local scalar field. The comparison with the This implies that at the background level the ultra-local supersymmetric model can be completed by verifying model behaves like the Λ-CDM cosmology, see Eqs.(52)- that the cosmological perturbations also obey the same (55), as the scalar field and dark energy densities coincide dynamics. and are almost constant at low z, within an accuracy of The coupling of dark energy to dark matter implies 10−6. that the growth of the density contrast of CDM is mod- ified [25–27] and the linear density contrast δ = δρ/ρ of the super-chameleon model in the conformal Newtonian D. Super-chameleon identification Gauge evolves on sub-horizon scales according to
∂δ ∂δ 3 2β2(ϕ) Super-chameleon models are such that the mass of the 2 2 + Ωm(τ) 1+ 2 2 δ =0. (79) ∂τ H ∂τ − 2 H meff a scalar field is so large that the kinetic terms are negli- 1+ k2 ! gible. They behave like ultra-local models on distances −1 Physically, the last term in (79) corresponds to a scale r & meff . It is only on very short distances, which are negligible on astrophysical and cosmological scales, that dependent enhancement of Newton’s constant. As the the kinetic terms play a role. The identification with an mass of the scalar field is always very large compared to ultra-local model is therefore valid on scales astrophysical wave numbers, we can simplify (79) to find
2 2 k k ∂δ ∂δ 3 2 2k β (ϕ) . meff ; this includes the range . m3/2 meff , + Ω (τ) 1+ δ = 0 (80) a a ≪ ∂τ 2 H ∂τ − 2 m H m2 a2 (74) eff where we used Eq.(40). Even as early as a 10−10, BBN ∼ for k/a meff . This equation is the same as the equation the model is equivalent to an ultra-local model on comov- (70) obtained≪ for the ultra-local models, on sub-horizon ing scales larger than 10 km, well below the distances of scales where we can neglect the unit factor in Eq.(71). interest in the growth of cosmological structures. As a Indeed, the chameleon coupling β(ϕ) defined in Eq.(33), result, for all practical purposes super-chameleon mod- β = MPld ln A/dϕ, and the ultra-local coupling β1(˜χ) els can be identified with ultra-local models. Thus, the defined in Eq.(66), β1 = d ln A/dχ˜, are related by coupling function A(ϕ) and the potential V (ϕ) defined in Eqs.(45)-(46) for the ultra-local model can be read dχ˜ β = β M . (81) from the effective potential (29) of the super-chameleon 1 Pl dϕ model, to which we must add the cosmological constant contribution (18). Using the mapping (47) in terms of From the identification (76) we can write the effective the dimensionless fieldχ ˜ this yields chameleon potential of Eq.(29) as
4 ϕ σ/γ Veff (ϕ)= χ˜ + ρ(A 1) ρ¯de0, (82) A(˜χ)=1+ α (75) −M − − ϕ min where we explicitly subtract the cosmological constant. Then, the quasi-static equation (31) for ϕ, which corre- and sponds to the minimum of the potential ∂Veff /∂ϕ = 0, 4 2 yields β1 = /ρ, where we used Eq.(81) and A 1, ϕ n/2γ ξ4 M ≃ 4χ˜ = V =Λ4 min 1 + . (76) and we recover the ultra-local equation of motion (62)- −M ϕ − 2 " # (67). Next, from the definition of the chameleon effective 8
2 2 2 mass, meff = ∂ Veff /∂ϕ , we obtain using Eq.(82) and while the self-interacting part of the superpotential is 4 the result β1 = /ρ, M 3Ω √ 2 de0 2 2 W = 2Λ0Φ+ H0Φ , (86) 2 ρβ2β r 2 meff (ϕ)= 2 2 , (83) MPlβ1 which contains a linear term and a mass term, with Λ2 = √3Ωde0H0. Both Λ0 and H0 are protected by su- 2 2 where the ultra-local factor β2 = d ln A/dχ˜ = dβ1/dχ˜ persymmetry under renormalisation. 2 2 was introduced in Eq.(66). This gives 2β /meff = The supersymmetric minimum φmin of Eq.(8) becomes 2 2 2M β /ρβ2 and we find that Eq.(80) coincides with Pl 1 2 Eq.(70) over the range H k/a meff , using the sec- Λ0 ≪ ≪ φmin = . (87) ond expression (68) for ǫ1(t). Λ2 This identification of the super-chameleon model with ′ Requiring that φmin > √qξ/g to recover the late cos- the ultra-local model shows that on cosmological scales, mological constant behavior (18) and using Eq.(19) we H k/a meff , the dynamics is set by the single obtain the lower bound on Λ function≪ A(˜≪χ) obtained in Eq.(78). This implies that 0 structure formation is only sensitive to two combina- H 3/2 Λ2 & M 2 0 . (88) tions of the parameters introduced in the supersymmetric 0 Pl M chameleon setting, namely the exponent ratio σ/n and Pl 4 the ratio Λ /ρ¯de0 (which we set to unity in this paper), The normalized chameleon field ϕ of Eq.(26) reads as in addition to the cosmological constant ξ4/2=ρ ¯ . de0 2 4 Conversely, there is a wide model degeneracy and the ϕ φ ϕmin Λ0 = 2 , = 2 2 , (89) same coupling function (78) corresponds to many differ- MPl MPl MPl 3Ωde0MPlH0 ent chameleon models. while the characteristic density ρ of Eq.(32) is We can note here that in the context of usual ∞ chameleon models such as f(R) theories, where β 1, 2 ρ¯de0 2 −1∼ ρ∞ = ρ¯de0 . (90) having a very large effective mass meff , with meff ασ ∼ α 10−4mm, would lead to negligible departure from the Λ-≪ CDM cosmology for the formation of large scale struc- We must also satisfy the constraint (39), which yields the tures, as seen from Eq.(80). This is not the case for the upper bound on Λ0 super-chameleon models studied in this paper because 3/2 1/2 2 2 H0 MPl the coupling β is also very large and much greater than Λ0 MPl . (91) ≪ MPl m3/2 unity. Indeed, from Eq.(34) we have β αMPl/ϕmin 2 ∼ 2 ≫ 1, whereas from Eq.(35) we have m αρ∞/ϕ . This eff ∼ min As we always have m3/2 MPl, the comparison of yields Eq.(91) with Eq.(88) shows≪ that the range of values for Λ0 is fairly large. β2 α2M 2 Pl The scales m and Λ3 of the dark matter interaction 2 4 , (84) meff ∼ Λ Wint in Eq.(21) are only constrained through their combi- nation with φmin in the coupling parameter α of Eq.(27), 2 2 2 2 and β k /meff a can be of order unity on kpc to Mpc which must be small as noticed in Eq.(30). In fact, the scales, even with α 1, as we typically have Λ4 identification with the ultra-local model and the study 2 2 ≪ ∼ MPlH0 . presented in the companion paper shows that we must require α . 10−6 to keep the formation of large cosmo- logical structures close to the Λ-CDM behavior. From E. Example of models Eq.(37) the exponent σ should satisfy σ 5/2 if we wish to ensure that the quasi-static approximation≤ re- mains valid up to arbitrarily high redshifts, which gives It is interesting to consider templates for ultra-local 0 < σ/n 5/4. More generally, combining Eqs.(10) and models coming from super-chameleons. (37) we have≤ A good set of models can be obtained for instance by taking the cut-off of the theory Λ1 = MPl in the K¨ahler σ γ + ω/2 σ 3γ 1 4 0 < hence 0 < − . (92) potential (6). To obtain Λ =ρ ¯de0 as in Eq.(77) this n ≤ 2(γ ω) n ≤ 4 requires the non-renormalised scale in the superpotential − 4 1/(6−2γ) It is interesting to obtain the characteristic scales of W of Eq.(7) to be Λ2 = MPl(¯ρde0/MPl) . A simple choice for the exponents ω and γ is ω = 1 and γ = 2, the coupling β and effective mass meff of these super- which gives n = 2 and the K¨ahler potential becomes chameleon models. Using the bounds (88) and (91) we obtain 2 † 2 2 2 † MPl Φ Φ † † αMPlH0 αm3/2 αMPl K(ΦΦ )= +Φ Φ+ +Φ Φ− (85) β hence β . , (93) 2 M 2 + − ∼ Λ4 H ≪ H Pl 0 0 0 9 and At the background level, we switch from the high-density 4 5 regime (98) to the low-density regime (99) at the redshift 2 MPlH0 2 2 2 zα, with meff 8 hence m3/2 meff . MPl. (94) ∼ Λ0 ≪ 1/3 −1/3 aα = α . 0.01, zα = α & 100, ρ¯(zα)= ρα. We can check that both β and meff are large in these (100) super-chameleon models. Thus, together with the density scale ρα these ultra-local As noticed above from Eq.(78), eventually we will models also select a particular redshift z & 100. This is study the super-chameleon models of this type where the α −6 the redshift where the fifth force effects are the strongest, only parameters are α, which will be chosen to be 10 in terms of the formation of cosmological structures, even or lower, and ζ = σ/n, of order unity. though at the background level the scalar field energy density only becomes dominant at low z as a dark en- ergy contribution. Up to factors of order unity, the den- IV. ULTRA-LOCAL DYNAMICS sity ρα and redshift zα also correspond to the density ρ∞ and redshift z∞ introduced in Eq.(32), where the super- A. Chameleon and ultra-local potentials and chameleon field ϕ reaches the supersymmetric minimum coupling functions 4 ϕmin (we chose Λ =ρ ¯de0). Thus, within this supersym- metric setting the density and redshift (ρα,zα) obtain an −6 As the total variation of A(˜χ) is bounded by α . 10 , additional physical meaning. we can approximate Eq.(78) as From Eqs.(98) and (99) we also obtain the behavior
−2ζ of the coupling function ln A(ρ) in terms of the matter ln A(˜χ)= α 1+ 1 χ˜ , ζ> 0, (95) density, − − p −ζ/(1+ζ) where we defined ζ = σ/n. Equation (95) fully de- ζρ ρ ρα : ln A(ρ) α , (101) fines the ultra-local model that corresponds to the super- ≫ ∼ ρα chameleon models considered in this paper. For the numerical applications below we take α = 10−6 and ζ ρ among 1/2, 1, 3/2 . The first two choices can be ob- ρ ρ : ln A(ρ) α 1 2ζ2 . (102) { } ≪ α ≃ − ρ tained with σ = 1 and σ = 2 for the explicit super- α chameleon model described in section III E with γ = n = 2. The choice ζ = 3/2 requires a model with γ 7/3 As shown in the companion paper, the derived function or corresponds to a model with γ < 7/3 where the≥ field ln A(ρ) is particularly important when applied to static ϕ has not yet reached the quasi-static equilibrium (31) configurations and can be used to probe the existence of at very high redshift (which is not very important as the a screening mechanism for this theory as we will show in dark energy and the fifth force do not play a significant sec.V A. role at high redshifts far in the radiation era). We show in Fig. 1 the characteristic functions that Using Eq.(95), the equation for the evolution of the define the super-chameleon models and the associated scalar field (62) becomes ultra-local models, for the choice of chameleon expo- nents γ = 2,ω = 1,n = 2 for the K¨ahler potential K ρ 1 2ζ+1 and the superpotential W , and σ = 1, 2, 3 for the in- = 1 χ˜ 1+ 1 χ˜ , (96) teraction potential W . This gives ζ = 1/2, 1, 3/2 for ρα ζ − − − − int p p the ultra-local coupling function ln A(˜χ). The left panel where we introduced shows the normalized chameleon potential V/ 4, which M 4 is also equal to the opposite of the ultra-local fieldχ ˜ ρ¯de0 ρα = M = . (97) from Eq.(76). It is identical for the three models that we α α consider in the numerical computations presented in this This explicitly shows that, because of the small param- paper. The middle panel shows the chameleon coupling eter α, such models introduce a second density scale function ln A(ϕ) for the three choices for the exponent 6 ρα & 10 ρ¯de0 in addition to the current dark energy den- σ. The right panel shows the ultra-local coupling func- sityρ ¯de0. tion ln A(˜χ) for the corresponding three choices of the Eq.(96) can be used to expressχ ˜ as a function of the exponent ζ. In terms of the ultra-local model, or for the density in the high- and low-density limits, dynamics of cosmological perturbation in the chameleon model over scales H k/a meff , this function ln A(˜χ) ζρ 1/(1+ζ) fully defines the system.≪ ≪ ρ ρα :χ ˜(ρ) , (98) ≫ ∼− ρα In the right panel of Fig. 1 we show the coupling func- tion ln A as a function of the normalized scalar field ϕ for different values of the parameter ζ. For all the models 2 ζρ we have A¯ 1 . 10−6 1 which means that we re- ρ ρα :χ ˜(ρ) 1 . (99) | − | ≪ ≪ ≃− − ρ cover the Λ-CDM cosmology at the background level to α 10
10 1 ζ=1/2 ζ=1/2 1 ζ=1 ζ=1 8 0.8 ζ=3/2 ζ=3/2 0.8 4
6 α 0.6 α 0.6 = V / M 4 ln(A)/ 0.4 ln(A)/ ~ χ 0.4 -
2 0.2 0.2
0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -3 -2.5 -2 -1.5 -1 ϕ ϕ ϕ ϕ ~ / min / min χ
FIG. 1: Left panel: ultra-local scalar field or chameleon potential, −χ˜ = V/M4, as a function of the chameleon scalar field ϕ/ϕmin, as in Eq.(77) for γ = 2, n = 2. Middle panel: coupling function ln A(ϕ) as a function of the chameleon scalar field from Eq.(27), with γ = 2, σ = 1, 2, 3, which corresponds to ζ = 1/2, 1, 3/2 with n = 2. Right panel: coupling function ln A(˜χ) as a function of the ultra-local scalar fieldχ ˜ from Eq.(95), for ζ = 1/2, 1, 3/2.
10-6 which for the models presented in the previous sections is equal to 10-7 −2ζ √ 1 χ˜ 1+ √ 1 χ˜ 10-8 ǫ =2αζ − − − − , (103) 1 1+2(ζ + 1)√ 1 χ˜ − − 10-9
1 where we used the definition (68). From Eq.(98) and ε -10 10 (99) we have the following simplified expressions for ǫ1 as function of the density 10-11 ζ − =1/2 αζ ζρ ζ/(1+ζ) 10-12 ζ=1 ρ ρ : ǫ (ρ) , (104) ≫ α 1 ∼ 1+ ζ ρ ζ=3/2 α 10-13 0.001 0.01 0.1 1 2 ρ a ρ ρα : ǫ1(ρ) 2αζ . (105) ≪ ∼ ρα
This explicitly shows that ǫ1 decreases both at high and FIG. 2: Time evolution of the factor ǫ1(a) as a function of low densities and peaks around ρα. This also gives the the scale factor for ζ = 1/2, 1, 3/2. evolution of ǫ1(t) as a function of the scale factor a(t) −3 usingρ ¯ =ρ ¯0a ,
−6 3ζ/(1+ζ) a 10 accuracy: in particular as we increase ζ the cou- 1/3 a pling function becomes steeper making the effect of the a aα = α : ǫ1(a) α , (106) ≪ ∼ aα presence of the scalar field on the growth of structure more relevant, as we will demonstrate in section IV B. a −3 a a = α1/3 : ǫ (a) α , (107) ≫ α 1 ∼ a α B. Cosmological background and perturbations which peaks at the scale factor aα that corresponds to ρ¯ = ρα. In Fig. 2 we show the evolution of ǫ1, for For all the models we have A¯ 1 . 10−6 1, which ζ = 1/2, 1, 3/2, as a function of the scale factor. It is means that we recover the Λ-CDM| − | cosmology≪ at the always positive for these models leading to an amplifica- −6 background level to a 10 accuracy. Therefore, to dis- tion of the Newtonian gravity. We can check that ǫ1 has 1/3 tinguish such models from the Λ-CDM scenario we must a maximum around aα = α , which for this paper cor- consider the dynamics of cosmological perturbations. As responds to a value of aα =0.01. At low redshifts we re- −3 we can see from Eq.(70), the linear growth D+(k,t) of the cover the same decrease as ǫ1 a of Eq.(107), whereas dark matter density contrast is modified with respect to at high redshift the decrease∝ is stronger for higher ex- the Λ-CDM case only by the presence of the factor ǫ(k,t), ponent ζ, in agreement with Eq.(106). At its peak at 11
101 101 101
ζ=1/2 ζ=1 ζ=3/2 100 100 100
-1 -1 -1
(k,a) 10 (k,a) 10 (k,a) 10 + + + D D D
k=1 k=1 k=1 10-2 10-2 10-2 k=5 k=5 k=5 k=10 k=10 k=10 ΛCDM ΛCDM ΛCDM 10-3 10-3 10-3 0.001 0.01 0.1 1 0.001 0.01 0.1 1 0.001 0.01 0.1 1 a a a
FIG. 3: Linear growing mode D+(k,a) for the models defined by Eq.(95), as a function of the scale factor for k = 1, 5 and 10 hMpc−1, and for the Λ-CDM cosmology. We consider the cases ζ = 1/2, 1 and 3/2 (respectively left, center and right panel).
102 102 102 ζ=1/2 ζ=1 ζ=3/2 101 101 101 0 0 0 10 0 10 0 10 0 10-1 10-1 10-1
(k) -2 (k) -2 (k) -2
2 L 10 10 2 L 10 10 2 L 10 10 ∆ ∆ ∆ 10-3 10-3 10-3 10-4 100 10-4 100 10-4 100 10-5 z=500 10-5 z=500 10-5 z=500 10-6 10-6 10-6 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 k [h Mpc-1] k [h Mpc-1] k [h Mpc-1]
2 FIG. 4: Logarithmic linear power spectra ∆L(k,z) at redshifts z = 0, 10, 100 and 500 (from top to bottom) at fixed ζ = 1/2, 1 and 3/2 (respectively left, center and right panel).
−6 min aα, we have ǫ1 α = 10 , whereas today we have Therefore, low wave numbers k < kα are never sensitive ǫ α2 = 10−12.∼ to the fifth force whereas high wave numbers k > kmin 1 ∼ α As shown in the companion paper, the growth of struc- are sensitive to the fifth force around aα. The range of ture is vastly enhanced by the presence of the scalar field scale factors [a−(k),a+(k)] where a wave number k feels when ǫ(k,a) 1 in Eq.(70). Because ǫ(k,a) grows as k2 the fifth force broadens at higher k. From Eq.(108) we ≫ at high k, there exists a time dependent scale kα(a) such obtain that for any scale smaller than the latter D (k,a) devi- + −(2ζ+2)/(4ζ+1) ates significantly from the Λ-CDM one. This threshold k k > kmin : a−(k) aα , (110) kα(a) can be computed from the condition ǫ[kα(a),a]=1 ∼ kmin in Eq.(71), to obtain
aH H0 k kα(a)= , (108) a+(k) aα . (111) c√ǫ1 ∼ c√ǫ1a ∼ kmin
2 −3 In Fig. 3 we show the evolution of the linear growing where we used H a in the matter era. Because ǫ1 decreases at both high∝ and low redshifts, with a peak at mode D+(k,a) obtained numerically solving Eq.(70) at three different scales, for the models considered in this aα, the threshold kα(a) is minimum at the scale factor paper. In agreement with the discussion of Eq.(109) aα, min above, low wave numbers k < kα are never sensitive H to the fifth force and follow the Λ-CDM growth. Higher kmin = k (a ) 0 3hMpc−1, (109) α α α ∼ c α2/3 ∼ wave numbers depart from the Λ-CDM behavior around 12 a 0.01 and show a faster growth over a limited time of the matter density contrast, σ2 = δ (x)2 its vari- α ∼ x h L< i range [a−,a+], resuming the Λ-CDM growth at later ance within radius x, which defines a sphere of volume V ; times. This transient speed-up increases with k. This and W˜ (kx) = 3[sin(kx) kx cos(kx)]/(kx)3 the Fourier effect becomes stronger at higher ζ because of the higher transform of the 3D top hat− of radius x. The profile (116) amplitude of ǫ1 found in Fig. 2. is the typical profile around a density fluctuation at scale The presence of the scalar field leads to a very steep x in the initial Gaussian field and provides a convenient −1 increase of D+(k,a) at k 1 h Mpc and so these ansatz (here we use the initial linear power spectrum or scales enter the nonlinear regime≫ much earlier than in its Λ-CDM amplified value at the redshift of interest). the Λ-CDM cosmology, at z z . This can be seen in We show in Fig. 5 the time evolution of the nonlinear ∼ α Fig. 4 where we plot the logarithmic linear power spec- density contrast δ<(r) within a shell of mass M, given by 2 3 trum ∆L(k,z)=4πk PL(k,z). the spherical dynamics (115), for different values of the mass M, fixing the initial linear density contrast so that Λ−CDM δL =1.6 today (the initial condition is set at high C. Spherical collapse redshift before the onset of the fifth force and it is com- mon to all models and the Λ-CDM cosmology; as usual On large scales where the baryonic pressure is negligi- it is convenient to describe this initial condition by its ble, the particle trajectories r(t) follow the equation of value today using the Λ-CDM linear growth factor). In motion agreement with what we found by studying the evolution of linear perturbations, we can see that at large masses, 2 2 12 −1 d r 1 d a M & 10 h M⊙, the evolution of δ<(r) closely follows r = r (Ψ +Ψ ) , (112) dt2 − a dt2 −∇ N A the Λ-CDM one, whereas the collapse of small masses is strongly accelerated around a . This faster growth r x α where = a is the physical coordinate, ΨN the Newto- occurs earlier for smaller mass, as a−(k) decreases on 2 nian potential and ΨA = c ln A the fifth-force potential. smaller scales. To study the spherical collapse before shell crossing, it We show in Fig. 6 the spherical dynamics for a fixed 8 −1 is convenient to label each shell by its Lagrangian radius value of the mass M = 10 h M⊙ and several initial den- q or enclosed mass M, and to introduce its normalized sity contrasts. The acceleration of the growth of struc- radius y(t) by ture due to the presence of the scalar field makes halos collapse before a = 1, even starting from δΛ−CDM 0.1. 1/3 L r(t) 3M In agreement with previous figures, the acceleration≃ of y(t)= with q = , y(t =0)=1. a(t)q 4πρ¯ the collapse occurs around aα. For sufficiently high ini- 0 (113) tial conditions this leads to a collapse at high redshift In particular, the matter density contrast within radius around zα. For lower initial conditions the dynamics is r(t) reads as still in the linear regime after the fifth force has van- ished, at low redshift, but with a higher amplitude than −3 1+ δ<(r)= y(t) . (114) in the Λ-CDM cosmology and a higher final collapse red- shift. Again, we can see that the effect of the fifth force The equation of motion becomes increases with ζ. We show in the upper panel of Fig. 7 the linear den- 2 Λ−CDM d y 1 dH dy Ωm + 2+ + y(y−3 1) = sity contrast threshold, measured by δL (i.e., the d(ln a)2 H2 dt d ln a 2 − extrapolation up to z = 0 of the linear initial density c 2 d ln A r ∂δ contrast by the Λ-CDM growth rate), required to reach a y . (115) − Hr d ln ρ 1+ δ ∂r nonlinear density contrast δ< = 200 today. In agreement with Figs. 5 and 6, at large mass we recover the Λ-CDM Λ−CDM The fifth force introduces a coupling as it depends on the linear density threshold, δ 1.6, whereas at small L ≃ density profile, through the local density ρ(r)=ρ ¯(1 + mass we obtain a much smaller linear density threshold, Λ−CDM δ(r)). δ 1, because of the acceleration of the collapse L ≪ In the following, we use the density profile defined by by the fifth force. Again, at small masses the threshold δL becomes smaller for larger exponent ζ as the effect of δ (x) dx′′ the fifth force increases. δ(x′)= < ξ (x′, x′′) σ2 V L x ZV δ (x) +∞ dk sin(kx′) = < ∆2 (k)W˜ (kx) . (116) D. Halo mass function σ2 k L kx′ x Z0 Here x(t)= a(t)r(t) is the comoving radius of the spher- As for the Λ-CDM cosmology, we write the comoving ical shell of mass M that we are interested in while x′ halo mass function as 2 is any radius along the profile; ξL and ∆L are the lin- dM ρ¯ dν n(M) = 0 f(ν) , (117) ear correlation function and logarithmic power spectrum M M ν 13
102 ζ=1/2 102 ζ=1 102 ζ=3/2
101 101 101
0 0 0
< 10 < 10 < 10 δ δ δ
6 6 6 10-1 M=10 10-1 M=10 10-1 M=10 M=108 M=108 M=108 M=1010 M=1010 M=1010 10-2 10-2 10-2 M=1012 M=1012 M=1012 M=1014 M=1014 M=1014 10-3 10-3 10-3 0.001 0.01 0.1 1 0.001 0.01 0.1 1 0.001 0.01 0.1 1 a a a
FIG. 5: Time evolution of the nonlinear density contrast δ(< r) given by the spherical dynamics, as a function of the scale −1 factor, for several masses (in units of h M⊙) at fixed ζ = 1/2, 1, 3/2 (respectively left, center and right panel). The initial Λ−CDM condition corresponds to the same linear density contrast δL = 1.6 today, using the Λ-CDM growth factor.
1.6 1.6 1.6 102 102 102 0.1 0.1 0.1 101 0.01 101 0.01 101 0.01 0.001 0.001 0.001 100 100 100 10-1 10-1 10-1 < < < δ 10-2 δ 10-2 δ 10-2 10-3 10-3 10-3 -4 -4 -4 10 ζ=1/2 10 ζ=1 10 ζ=3/2 -5 -5 -5 10 8 h-1 10 8 h-1 10 8 h-1 M= 10 MO· M= 10 MO· M= 10 MO· 10-6 10-6 10-6 0.001 0.01 0.1 1 0.001 0.01 0.1 1 0.001 0.01 0.1 1 a a a
FIG. 6: Time evolution of the nonlinear density contrast δ(< r) given by the spherical dynamics, as a function of the scale 8 −1 factor, for several values of the initial density contrast and for a mass of M = 10 h M⊙ at fixed ζ = 1/2, 1, 3/2(respectively Λ−CDM left, center and right panel). We show our results for different initial conditions, which correspond to δL = 1.6, 0.1, 0.01 and 0.001 from top to bottom. where the scaling variable ν(M) is defined as sity power spectrum as for the Λ-CDM reference at high redshift. δΛ−CDM(M) ν(M)= L , (118) The threshold δΛ−CDM(M) was shown in the upper σΛ−CDM(M) L panel of Fig. 7. We show the mass function in the Λ−CDM lower panel of Fig. 7. Once again, we can notice that and δL (M) is again the initial linear density con- trast (extrapolated up to z = 0 by the Λ-CDM linear at large mass all the mass functions are close to the Λ-CDM prediction whereas at smaller masses, M growth factor) that is required to build a collapsed halo 8 10 −1 ∼ 10 10 h M⊙, they are higher. This is because the (which we define here by a nonlinear density contrast of − 200 with respect to the mean density of the Universe) and fifth force has no effect on very large scales and accel- Λ−CDM erates the formation of structures on small scales. At σ its variance. The variable ν measures whether 7 −1 such an initial condition corresponds to a rare and very lower mass, M . 10 h M⊙, the mass function becomes high overdensity in the initial Gaussian field (ν 1) or smaller than in the Λ-CDM cosmology, because both to a typical fluctuation (ν . 1). In the Press-Schechter≫ mass functions are normalized to unity (the sum over 2 all halos cannot give more matter than the mean matter approach, we have f(ν) = 2/πνe−ν /2. Here we use density). the same function as in [28]. Then, the impact of the p 12 −1 modified gravity only arises through the linear threshold At large masses, M > 10 h M⊙, where the forma- ΛCDM δL (M), as we assume the same initial matter den- tion of large-scale structures remains close to the Λ-CDM 14
V. ASTROPHYSICAL EFFECTS 100 A. Screening within spherical halos
10-1 1. Radial profiles
CDM -2
Λ L 10 z=0
δ We first consider here how the ratio of the fifth force ΛCDM to Newtonian gravity behaves within spherical halos with 10-3 ζ=1/2 a mean density profile such as the Navarro-Frenk-White ζ=1 (NFW) [29] density profile. In particular, we wish to find ζ=3/2 the conditions for the fifth force not to diverge at the -4 10 center of the halos and to remain modest at all radii, to 106 107 108 109 1010 1011 1012 1013 1014 be consistent with observations of X-ray clusters. Within M [h-1 M ] O· spherical halos, the Newtonian force reads as
NM(< r) Ωm 2 FN = G 2 = ∆(< r)rH , (119) 10-1 − r − 2 where ∆(< r) is the mean overdensity within radius r. The fifth force reads 2
) n(M) 2 d ln A c d ln A d ln ρ 0 FA = c = . (120) ± ρ 10-2 − dr − r d ln ρ d ln r ΛCDM
(M / ζ=1/2 We can also use FN and FA to define characteristic ve- ζ=1 locity scales, z=0 ζ=3/2 2 2 -3 vN(r) cs(r) 10 FN = , FA = , (121) 108 109 1010 1011 1012 1013 1014 1015 − r − r h-1 M [ MO· ] with M(< r) d ln A v2 = GN , c2 = c2 , (122) N r s d ln r FIG. 7: Upper panel: Initial linear density contrast, as mea- Λ−CDM where v is the Newtonian circular velocity. Therefore, sured by δ , that gives rise to a nonlinear density con- N L the ratio of the fifth force to the Newtonian force is trast δ< = 200 at z = 0, as a function of the halo mass M for fixed ζ = 1/2, 1 and 3/2. Lower panel: Halo mass function 2 2 FA cs 2 c d ln A d ln ρ at z = 0 for fixed ζ = 1/2, 1 and 3/2, and for the ΛCDM η = 2 = . (123) ≡ FN vN Ωm∆(< r) rH d ln ρ d ln r cosmology. From Eq.(102), we have at moderate densities, ρ ρ¯(z), ∼ α2ζ2 c 2 ρ ρ : η . (124) ≪ α | |∼ a3 rH Thus, in the late Universe the ratio η is suppressed by a case, with only a modest acceleration, and the mass func- 2 −1 2 factor α so that η only reaches unity at r 3h kpc, i.e. tion is dominated by the Gaussian tail e−ν /2, we at galaxy scales (see also Sec. V A 3 below).∼ At higher ∼ can expect that the results obtained are robust, since in densities, we obtain from Eq.(101) this regime the shape of the halo mass function is dom- 2 (1+2ζ)/(1+1ζ) inated by the exponential tail e−ν /2. At low masses, α2ζ a3 c 2 12 −1 ρ ρα : η . M < 10 h M⊙, where the history of gravitational clus- ≫ | |∼ a3 αζ∆ rH tering is significantly different from the Λ-CDM scenario, (125) as a large range of masses have collapsed together before We plot the ratio η for several halo masses, with an a redshift of 100, and the halo mass function is no longer NFW density profile in Fig. 8. In agreement with the dominated by its universal Gaussian tail, these results are results obtained in previous sections, we can see that the unlikely to be accurate. Nevertheless, we can still expect fifth force is more important for smaller halos, which also the halo mass function to be significantly higher than in correspond to smaller scales. For a power-law density 8 11 −1 the Λ-CDM case for masses M 10 10 h M⊙, al- profile, of exponent γp > 0 and critical radius rα, though it is difficult to predict∼ the maximum− deviation −γ and the transition to a negative deviation at very low r p ρ(r)= ρ , (126) masses. α r α 15
102 ratory, because within the supersymmetric setting con- sidered in this paper baryons do not couple to the fifth 1 force. Therefore, astrophysical systems which are domi- 10 11 h-1 10 MO· nated by baryons do not feel the effect of the fifth force 13 0 10 and we automatically recover the General Relativity or N 10 Newtonian dynamics in these systems.
/ F 15 We have seen in Eq.(123) that η = c2/v2 , whence η A -1 10 s N F 10 (c/v )2 d ln A/d ln ρ if we take d ln ρ/d ln r 1. From∼ N | | ∼ Eq.(102), we also have at moderate densities below ρα -2 ζ=1/2 6 2 ∼ 10 10 ρ¯0, d ln A/d ln ρ α ∆ at redshift z = 0. This gives ζ=1 ∼− ζ=3/2 2 10-3 αc 10-4 10-3 10-2 10-1 100 z = 0 : η ∆. (128) ∼ vN r/R200c 3 For clusters of galaxies, with ∆ 10 and vN 500 km/s, this yields ∼ ∼
FIG. 8: Ratio η = FA/FN as a function of the radius r, 4 2 within spherical halos with an NFW profile. We display the clusters: η (10 α) 1. (129) 11 13 15 −1 ∼ ≪ cases of halo masses M = 10 , 10 and 10 h M⊙ from top to bottom, for the ultra-local model exponent ζ = 1/2, 1 Therefore, the fifth force is negligible on cluster scales. and 3/2. However, as seen in Fig. 8, this is no longer the case far inside the cluster, where the characteristic scales are smaller and the density greater, which gives rise to a we have greater fifth force.
γp(1+2ζ)/(1+ζ)−2 r < rα, η r . (127) ∼ 3. Galaxies If we consider halos with a mean NFW density profile, −1/(1+ζ) which has γp = 1, we find that η r and the We now consider a typical galaxy, such as the Milky ∼ relative importance of the fifth force does not vanish at Way, with ∆ 106, which is at the upper limit of the ∼ the center for the models, whatever the value of the expo- regime ρ . ρα, and vN 200 km/s. This gives nent ζ, in agreement with Fig. 8. This suggests that these ∼ models would lead to significant modifications in the clus- galaxies: η (106 α)2 1. (130) ter dynamics with respect to the Λ-CDM model and so ∼ ∼ would be ruled out by the observations, which show a Thus, the fifth force is of the same order as the Newtonian good agreement with the Λ-CDM cosmology. However, gravity on galaxy scales. This suggests that interesting as we can see from Fig. 8, for typical cluster masses η phenomena could occur in this regime and that galaxies only becomes of the order of unity far within the virial could provide a useful probe of such models, as we can 13 −1 11 −1 radius, r . 0.01R200c for M & 10 h M⊙. Because see from Fig. 8 for low-mass halos M . 10 h M⊙. at these scales clusters have significant substructures the approximation of a smooth profile is not any more cor- rect. Then, deeper analyses are needed to unravel the B. Fifth-force dominated regime dynamics of clusters of galaxies considering the ultra- local behaviour of the theory. We leave these analysis for It is useful to reformulate the analysis presented above future studies when we may need to use numerical sim- for clusters and galaxies and to determine the domain of ulations and to estimate the observational accuracy of length, density and mass scales where the fifth force is the measured halo profiles. On the other hand, we will dominant. Taking d ln ρ/d ln r 1, we write for struc- perform a thermodynamic analysis of the system in VI tures of typical radius R, density∼ ρ and mass M = where we find that for large enough clusters, the mean 4πρR3/3, density approximation is valid. 2 ρ¯ c 2 d ln A η 0 . (131) | |∼ Ω ρ RH d ln ρ 2. Clusters of galaxies m0 0
Then, the fifth force is greater than Newtonian gravity if We now estimate the fifth force to Newtonian grav- we have ity ratio η on a global scale, for clusters and for galax- 2 ies. In contrast with the companion paper, we do not c 2 ρ¯0 d ln A η 1 : R2 R2 . (132) need to study the Solar System, the Earth or the labo- | |≥ ≤ η ≡ H Ω ρ d ln ρ 0 m0
16
10-1 100 ζ=1/2 groups ζ=1 -2 -1 10 ζ=3/2 10
10-3 10-2 galaxies Mpc ] Mpc ] -1 -1
h -4 h -3 10 large 10 large
R [ 5th force R [ 5th force ζ 10-5 10-4 =1/2 ζ=1 ζ=3/2 10-6 10-5 10-2 100 102 104 106 108 1010 1012 106 107 108 109 1010 1011 1012 1013 1014 ρ / ρ± M [ h-1 M ] 0 O·
FIG. 9: Domain in the density-radius plane where the fifth FIG. 10: Domain in the mass-radius plane where the fifth force is greater than Newtonian gravity (bottom left area be- force is greater than Newtonian. The horizontal axis is the low the curves), for the ultra-local exponents ζ = 1/2, 1 and typical mass M of the structure and the vertical axis its typi- 3/2. cal radius R. The rectangles show the typical scales of galaxies and groups of galaxies.
At low densities, using Eq.(102) we obtain M Rη Rα for M coll 10 redshift, when the fifth force is dominant, regions that c ΛCDM collapse and turn non-linear because of the fifth-force in- ζ=1/2 teraction relax towards the thermodynamic equilibrium. ζ=1 Then, if this equilibrium is strongly inhomogeneous the ζ=3/2 mean field approach used in the previous sections breaks 1 down, whereas if this equilibrium is homogeneous we can 0.01 0.1 1 conclude that the system does not develop strong small- a scale inhomogeneities and the previous analysis is correct. FIG. 11: Upper panel: collapse radius xcoll(z) (in comoving A. Cosmological non-linear transition coordinates) as a function of the scale factor, for the ultra- local models and the Λ-CDM cosmology. Lower panel: col- lapse velocity scale ccoll(z). We first study in this section the evolution with red- shift of the comoving cosmological scales xcoll(z) that en- ter the non-linear regime, which we define by eras. For a 10 10 10 ζ=1/2 ζ=1 ζ=3/2 5 5 5 0 0 0 θ θ θ -5 -5 -5 -10 -10 -10 -15 -15 -15 1 10 100 1 10 100 1 10 100 β^ β^ β^ FIG. 12: Thermodynamic phase diagram for the ultra-local models with ζ = 1/2, 1 and 3/2. The shaded area is the region of initial inverse temperature βˆ and density θ where the thermodynamic equilibrium is inhomogeneous. The solid line is the cosmological trajectory (βˆcoll(z),θcoll(z)). This trajectory moves downward to lower densities with sis and structure formation proceeds as in the standard cosmic time, followingρ ¯(z). In agreement with the lower Λ-CDM case. panel of Fig. 11, the inverse temperature βˆcoll first de- creases until aα, as the velocity ccoll(z) grows. Next, C. Halo centers βˆcoll increases while ccoll(z) decreases along with the fifth force, until we recover the Λ CDM behavior at late − times and βˆcoll decreases again thereafter. We are inter- It is interesting to apply the thermodynamic analysis ested in the first era, a 4 ζ=1/2 4 ζ=1 4 ζ=3/2 2 2 2 M =1015 h-1 M M =1015 h-1 M M =1015 h-1 M 200c O· 200c O· 200c O· 0 0 0 -2 -2 -2 θ θ θ -4 -4 -4 -6 -6 -6 -8 1013 -8 1013 -8 1013 1011 1011 1011 -10 -10 -10 0.1 1 β^ 10 100 0.1 1 β^ 10 100 0.1 1 β^ 10 100 15 13 FIG. 13: Radial trajectory (βˆr,θr) over the thermodynamic phase diagram inside NFW halos of mass M = 10 , 10 and 11 −1 10 h M⊙. at z = 0. 2 2 cs + vN that we used in Eq.(139) for the cosmological Fig. 8 at z = 0. As we move inside the halo, towards analysis for illustrative convenience. Indeed, the discon- smaller radii r, the density θr grows and the trajectory tinuous changes of slope in Fig. 13 show at once the lo- moves upward in the figure. The turn-around of βˆr at cation of the transitions η = 1 between the fifth-force θ 4 corresponds to the NFW radius r where the | | r s and Newtonian gravity regimes. local≃ slope − of the density goes through γ = 2 and the cir- When the density grows at small radii as a power law, cular velocity is maximum. At smaller radii, r rs, the −γp −1 ≪ ρ r , we have seen in Eq.(127) that the fifth-force to NFW profile goes to ρ r , hence γp = 1. In agreement ∝ − gravity ratio η behaves as η rγp(1+2ζ)/(1+ζ) 2 with with the asymptotic analysis∝ above, this implies that we ∼ move farther into the fifth-force regime and we follow v2 r2−γp , c2 rγpζ/(1+ζ), (149) N ∼ s ∼ the upper boundary θ+ of the inhomogeneous phase do- main, so that the dimensional analysis of section V A 1 is at high density ρ ρ , where we used Eq.(101). This α valid. This also leads to an increasingly dominant fifth gives in the Newtonian≫ gravity and fifth-force regimes force at small radii and characteristic velocities that are γp higher than the Newtonian circular velocity. This may η < 1 : θr ln βˆr, (150) | | ∼ 2 γp rule out these ultra-local scenarios. However, on small − scales the baryonic component is non-negligible and it ac- tually dominates on kpc scales inside galaxies. Since the 1+ ζ η > 1 : θr ln βˆr. (151) baryons do not feel the fifth force this could keep these | | ∼ ζ models consistent with observations. On the other hand, 11 −1 2 for low-mass halos, M . 10 h M⊙ at z = 0, we find For γp > 2 we are in the Newtonian regime vN , that a significant part of the halo is within the inhomo- ˆ → ∞ βr 0, so that we are in the homogeneous phase of geneous thermodynamic phase. This may leave some sig- → ˆ ˆ the thermodynamic phase diagram as βr < βc. For nature as a possible fragmentation of the system on these (2+2ζ)/(1+2ζ) <γp < 2 Newtonian gravity still dom- intermediate scales into higher-density structures. This inates at small radii and we have the asymptote (150) process would next lead to a screening of the fifth force, with γp/(2 γp) > (1+ζ)/ζ, so that the radial trajectory because of the ultra-local character of the fifth force. In- − (βˆr,θr) moves farther above from the upper bound θ+ of deed, because it is set by the local density gradients, the Eq.(145) of the inhomogeneous phase and small radii are fragmentation of the system leads to a disappearance of within the homogeneous phase. For γp < (2+2ζ)/(1+2ζ) large-scale collective effects and the fifth force behaves we are in the fifth-force regime and we obtain θ θ , like a surface tension at the boundaries of different do- r ∼ + so that the radial trajectory (βˆr,θr) follows the upper mains. Such a process may also happen in the case of boundary of the inhomogeneous phase domain. This massive halos at earlier stages of their formation, which means that the dimensional analysis of section V A 1 is could effectively screen the fifth force whereas the simple valid as the fifth force does not push towards a fragmen- static analysis leads to a dominant fifth force at small tation of the system down to very small scales. radii. However, a more precise analysis to follow such These asymptotic results apply to the small-radius evolutionary tracks and check the final outcomes of the limit r 0. In Fig. 13 we show the full radial trajecto- systems requires numerical studies that are beyond the → ries (βˆr,θr) over the thermodynamic phase diagram, from scope of this paper. R200c inward, for the NFW halos that were displayed in 21 VII. DEPENDENCE ON THE α PARAMETER simplification as the latter involve a single free function, ln A(˜χ). As in most dark energy and modified gravity In this section we investigate how the results obtained models, we also need to introduce a cosmological con- in the previous sections change when we vary the param- stant and the associated energy scale. In addition, we −6 eter α. As a matter of example, we consider the model need a small parameter α . 10 , which however ap- with ζ = 1 and we show our results in Fig. 14, where we pears as a ratio of several energy scales. This provides compare the case α = 10−6 considered in the previous a natural setting to explain why this quantity can be sections with the two cases α = 10−7 and α = 10−8. significantly different from unity. In agreement with the discussion in section IV B, as Next, we have used the ultra-local models identifica- α decreases the maximum amplitude of ǫ1 decreases as tion to study the cosmological properties of these scenar- ǫ1(aα) α while the associated scale factor decreases ios. We have considered both the background dynamics ∼ 1/3 as aα α . This implies that the effect of the fifth and the evolution of linear perturbations. Whereas the force is∼ shifted to higher redshift with a lower ampli- background remains very close to the Λ-CDM evolution, tude, whence a smaller impact of the scalar field on the within an accuracy of 10−6, the growth of cosmologi- matter power spectrum, P (k,z), and on the halo mass cal structures is significantly amplified on scales below function, as we can check in the upper right and lower 1h−1Mpc. This fifth-force effect shows a fast increase at left panels in Fig. 14. The area in the (M, R) plane high k as it corresponds to a pressure-like term in the where the fifth force is greater than Newtonian gravity linearized equations of motion. Another property that is also shrinks as α decreases, as we can see in the lower peculiar to these models, as opposed to most dark energy right panel. This is because Rα α, which moves or modified gravity models, is that the fifth force is the ∝ −1/3 the upper branch down towards small radii, whereas greatest at a high redshift zα α 100 and for ∼ ∼ the lower branch slowly moves upward because at fixed galaxies (among cosmological structures). mass we have R(M) α−1/(4ζ+1). Therefore, galaxies We have also considered the modifications to the ∼ are no longer sensitive to the modification of gravity if spherical collapse of cosmological structures. The faster α . 5 10−7. growth of structures at z z leads to an accelera- × α tion of the collapse at these∼ early times and to a lower Λ−CDM linear density threshold δL required to reach a VIII. CONCLUSION non-linear density contrast of 200 today, especially on smaller scales where the fifth force is greater. This leads We have considered in this paper supersymmetric to a higher halo mass function at intermediate masses, 8 14 −1 chameleon models with a very large mass, 1/meff 10 . M < 10 h M⊙, as compared with the Λ-CDM 10−4mm, and coupling β 1. This makes the range≪ cosmology. Next, we have considered the behavior of of the fifth force very small≫ and leads to an equivalence the fifth force inside spherical halos. We find that the between these supersymmetric chameleon models and the fifth force increasingly dominates at smaller radii in ha- ultra-local models studied in a companion paper, for cos- los with a shallow density profile, γp . 1, as for NFW mological scales with H k/a meff . The background profiles. On the other hand, the fifth force is negligible on remains very close to the≪ Λ-CDM≪ cosmology in both sets cluster scales and of the same order as Newtonian grav- of models. However, in contrast with the more general ity on galaxy scales. This suggests that galaxies could be ultra-local models, in this supersymmetric context only the best probes of such models. the dark matter is sensitive to the fifth force. Therefore, To investigate the non-linear fifth force regime, and although the ultra-local character of the models gives rise to check that the previous cosmological analysis is not to an automatic screening mechanism that ensures that violated by small-scale non-linear effects, we have used we satisfy Solar System tests of gravity in that more gen- the thermodynamic analysis developed in the compan- eral framework, in the context studied in this paper this ion paper. Again, we find that for these supersymmetric mechanism is not so critical as baryons, which dominate chameleon models the cosmological scales that turn non- on small scales and in the Solar System, never feel the linear at high redshift because of the fifth force are at fifth force (except through its effects on the dark matter the boundary of the inhomogeneous domain in the ther- Newtonian potential) and follow General Relativity. modynamic phase diagram. This suggests that they do We have first described how to build such chameleon not develop strong small-scale inhomogeneities and that models in this supersymmetric context. This involves the standard mean field cosmological analysis is valid. several characteristic functions that enter the K¨ahler po- The same behavior is found at small radii in spheri- tential K, which governs the kinetic terms of the model, cal halos, which again suggests that the spherically av- the superpotential W responsible for the interactions be- eraged analysis applies. However, for low-mass halos, 11 −1 tween the fields, and the coupling between the dark mat- M . 10 h M⊙ at z = 0, intermediate radii fall within ter and the dark energy. This also introduces several the inhomogeneous phase. This could lead to some frag- energy scales that may be different. We have shown mentation of the system with the formation of interme- in details how these models are equivalent to ultra-local diate mass clumps. On the other hand, this same pro- models for cosmological purposes. This leads to a great cess leads to a self-screening of the fifth force as isolated 22 10-6 102 -7 ζ=1, z=0 10 ζ=1 10-8 101 10-9 (k) 1 2 L ε ∆ 10-10 ΛCDM -11 10 0 α=10-6 -6 10 α=10 α -7 -12 =10 α -7 -8 10 =10 α=10 α=10-8 10-13 0.1 1 10 100 0.001 0.01 0.1 1 -1 k [h Mpc ] groups a 10-1 10-1 10-2 galaxies ) n(M) Mpc ] -3 0 10 -1 ± ρ -2 10 h ΛCDM large 5th force (M / α -6 R [ =10 -4 -7 10 -6 α=10 α=10 -7 ζ=1, z=0 α=10-8 α=10 α=10-8 10-3 -5 8 9 10 11 12 13 14 15 10 10 10 10 10 10 10 10 10 105 106 107 108 109 1010 1011 1012 h-1 -1 M [ MO· ] h M [ MO· ] FIG. 14: Dependence on the parameter α of the deviations from the Λ-CDM predictions. We plot models with ζ = 1 and −6 −7 −8 α = 10 , 10 and 10 . Upper left panel: ǫ1(a) as a function of the scale factor, as in Fig. 2. Upper right panel: logarithmic 2 linear power spectrum ∆L(k,z) at redshift z = 0, as in Fig. 4. Lower left panel: halo mass function as in the lower panel of Fig. 7. Lower right panel: domain in the mass-radius plane where the fifth force is greater than Newtonian gravity, as in Fig. 10. clumps no longer interact through the fifth force because ysis presented in this paper may not be sufficient as the of its ultra-local character. Finally, we have considered systems may not reach this equilibrium because of incom- the dependence of our results on the value of the pa- plete relaxation. To go beyond the analytic approaches rameter α. We find that for α 10−7 the deviations used in this paper and to make an accurate compari- from the Λ-CDM cosmology are≪ likely to be negligible son with data on galaxy scales requires numerical simu- (contrary to the models studied in the companion paper) lations, which we leave to future work. because they have a lower amplitude and are pushed to lower scales where baryons are dominant. Thus, we find that although such models follow the Λ-CDM behavior at the background level they display a Acknowledgments non-standard behavior for the dark matter perturbations on small scales, below 1h−1Mpc. At the level of the pre- This work is supported in part by the French Agence liminary analysis presented in this paper they appear to Nationale de la Recherche under Grant ANR-12-BS05- remain globally consistent with observational constraints. 0002. This project has received funding from the Euro- However, the effects of the fifth force deep inside halos, pean Unions Horizon 2020 research and innovation pro- on kpc scales, may provide strong constraints and rule gramme under the Marie Skodowska-Curie grant agree- out this models. In particular, the thermodynamic anal- ment No 690575. 23 [1] A. G. Riess et al. (Supernova Search Team), Astron.J. 210 (2013), 1212.4392. 116, 1009 (1998), astro-ph/9805201. [21] S. M. Carroll, I. Sawicki, A. Silvestri, and M. Trodden, [2] S. Perlmutter et al. (Supernova Cosmology Project), New J. Phys. 8, 323 (2006), astro-ph/0607458. Bull.Am.Astron.Soc. 29, 1351 (1997), astro-ph/9812473. [22] P. Binetruy, Supersymmetry: Theory, experiment and [3] E. J. Copeland, M. Sami, and S. Tsujikawa, cosmology (2006). Int.J.Mod.Phys. D15, 1753 (2006), hep-th/0603057. [23] B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 [4] A. Joyce, B. Jain, J. Khoury, and M. Trodden, Phys. (1988). Rept. 568, 1 (2015), 1407.0059. [24] P. Fayet, Phys. Lett. B175, 471 (1986). [5] W. Hu and I. Sawicki, Phys. Rev. D76, 064004 (2007), [25] P. Brax, C. van de Bruck, A.-C. Davis, J. Khoury, and 0705.1158. A. Weltman, Phys. Rev. D70, 123518 (2004), astro- [6] A. Nicolis, R. Rattazzi, and E. Trincherini, Phys.Rev. ph/0408415. D79, 064036 (2009), 0811.2197. [26] P. Brax, C. van de Bruck, A.-C. Davis, and A. M. Green, [7] G. W. Horndeski, Int.J.Theor.Phys. 10, 363 (1974). Phys.Lett. B633, 441 (2006), astro-ph/0509878. [8] J. D. Bekenstein, Phys. Rev. D48, 3641 (1993), gr- [27] P. Brax, A.-C. Davis, B. Li, and H. A. Winther (2012), qc/9211017. 1203.4812. [9] J. Khoury and A. Weltman, Phys. Rev. D69, 044026 [28] W. H. Press and P. Schechter, Astrophys. J. 187, 425 (2004), astro-ph/0309411. (1974). [10] T. Damour and A. M. Polyakov, Nucl. Phys. B423, 532 [29] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro- (1994), hep-th/9401069. phys. J. 462, 563 (1996), astro-ph/9508025. [11] E. Babichev, C. Deffayet, and R. Ziour, Int.J.Mod.Phys. [30] P. H. Chavanis, J. Vatteville, and F. Bouchet, Euro- D18, 2147 (2009), 0905.2943. pean Physical Journal B 46, 61 (2005), arXiv:cond- [12] P. Brax and P. Valageas, Phys. Rev. D 90, 023507 mat/0408117. (2014), 1403.5420. [31] R. Balian, From Microphysics to Macrophysics, The- [13] A. Vainshtein, Phys.Lett. B39, 393 (1972). oretical and Mathematical Physics (Springer-Verlag, [14] P. Brax, L. A. Rizzo, and P. Valageas (2016), 1605.02938. Berlin, Germany, 2007), ISBN 978-3-540-45469-4, URL [15] P. Brax, Phys.Lett. B712, 155 (2012), 1202.0740. http://www.springer.com/fr/book/9783540454694. [16] C. de Rham and R. H. Ribeiro, JCAP 1411, 016 (2014), [32] D. Lynden-Bell, Mon. Not. R. Astr. Soc. 136, 101 1405.5213. (1967). [17] P. Brax and J. Martin, Phys. Lett. B468, 40 (1999), [33] T. Padmanabhan, Phys. Rep. 188, 285 (1990). astro-ph/9905040. [34] Although the vacuum energy due to VD lifts the vanishing [18] P. Brax, C. van de Bruck, J. Martin, and A.-C. Davis, energy density of a true supersymmetric minimum and JCAP 0909, 032 (2009), 0904.3471. therefore supersymmetry is broken by ξ, we will still refer [19] P. Brax, A.-C. Davis, and J. Sakstein, JCAP 1310, 007 to ϕmin as the supersymmetric minimum as it minimises (2013), 1302.3080. VF . [20] P. Brax, A.-C. Davis, and J. Sakstein, Phys. Lett. B719,