The dilaton and modified gravity

Transcription

1 The dilaton and modified gravity Carsten van de Bruck University of Sheffield Work in collaboration with P. Brax, A. Davis and D. Shaw arxiv:

2 Scalar Fields and Modified Gravity Scalar fields are common in particle physics (although not a single one has been observed). Usually those fields couple to matter. Properties are often difficult to reconcile with cosmology/gravity experiments. Examples: a) Dilaton in string theory, which couples to all matter forms and is rather light. b) Moduli fields in supersymmetry/gravity and/or string theory. Need mechanism to hide those type of fields from current experiments, otherwise ruled out!

3 The typical action: S = d 4 x M 2 g Pl 2 R h(φ) 2 gµν µ φ ν φ V (φ) +S matter (χ,a 2 (φ)g µν ) A(φ) = coupling function h(φ) = arbitrary function V (φ) = potential χ(φ) = matter field(s)

4 The typical action: S = d 4 x M 2 g Pl 2 R h(φ) 2 gµν µ φ ν φ V (φ) +S matter (χ,a 2 (φ)g µν ) A(φ) = coupling function h(φ) = arbitrary function V (φ) = potential χ(φ) = matter field(s) Potential assumed to have standard form.

5 Here we consider two mechanisms to hide the scalar field(s): Damour-Polyakov (DP) mechanism (Damour&Polyakov, Nucl.Phys.B 423, 532 (1994)) The Chameleon Mechanism (Khoury&Weltman, Phys.Rev.D 69, (2004))

6 Chameleon mechanism (for h(φ)=1): Chameleon effect is based on interactions of field with ambient matter and nonlinear self-interactions. Essential for chameleon mechanism is the thin-shell solutions to the field equation! 60!!! [M] r [M -1 ]

7 Chameleon mechanism: The force (given by the gradient of the scalar field) is suppressed outside the body. Not all bodies have this thin shell! Field can behave as a standard massive scalar field and force can be long-ranged. Field follows an effective potential, which consists of the bare potential and a part which depends on the ambient matter density: V eff (φ) =V (φ)+ρ matter A(φ)

8 The coupling, usually given by β = ln A(φ) φ is in the case of a thin shell given by β 2 eff = β 2 φ φ c βφ N (R) This results in constraints on the theory!

9 Not all scalar-tensor theories are chameleon theories!! Thin-shell solution essential!! Whether or not a thin-shell solution exists, depends on the potential V and the coupling function A! What about the string-theory dilaton? We require ßeff to be small. Consider a theory for which A(φ) =e βφ V (φ) =V 0 e φ

10 This results in φ φ c = 1 1 3β ln ρ ρ 0. Now, with densities of order 1 g/cm 3 and g/cm 3, and the effective coupling is not small! Φ N 10 9 So, if ß is constant and the potential is an exponential, the chameleon mechanism is not at play, i.e. there is no thin-shell solution!

11 The Damour-Polyakov mechanism This mechanism is operating in theories in which the coupling function A(ϕ) has a minimum. The coupling β = ln A(φ) φ is zero at this point. Cosmologically, the scalar field is driven to the minimum of A(Φ) if the bare potential vanishes! Damour and Polyakov (1994) applied this mechanism to the dilaton, ignoring the influence of V(Φ).

12 In our work we took the influence of the potential into account and promoted the dilaton to be a dark energy scalar field. Damour, Piazza & Veneziano (Phys.Rev.D 66, (2002)) considered the dilaton as a dark energy field too, but in their case the minimum of A(Φ) was at Φ=! Here, we consider the minimum of A(Φ) at finite value: very different constraints on the theory and also cosmological behaviour is very different!

13 S = gd 4 x Einstein frame action: (Damour, Piazza, Veneziano PRD 66, (2002)) R(g) 2κ 2 k2 (φ) κ 2 ( φ) 2 V (φ) +S m χi,a 2 (φ)g µν ; φ, and in the strong coupling limit (λ O(1) or larger) k(φ) = 1 λ 1+3λ2 β 2 (φ) V (φ) =V 0 e φ + O(e 2φ ) Near the minimum: A(φ) 1+ A 2 2 (φ φ 0) 2

14 Field equation for scalar field: dϕ = k(φ)dφ ϕ = κ2 2k(φ) [ V (φ) β(φ)(a(φ)t m 4V (φ))]. The field evolves under the influence of an effective minimum! At the minimum: β(φ min )= V (φ min ) 4V (φ min )+A(φ min )ρ m This implies β(φ min ) 1 4 (today 0.23)

15 Due to the non-canonical nature of the scalar field (k(φ) is not 1!), the coupling to matter is not β, but α = β(φ)2 k(φ) 2 = β(φ) 2 λ 2 +3β(φ) 2 Cosmologically, if field is near the minimum: α = depending on λ. The field is long-ranged (cosmological scales)!

16 The mass is given by m 2 H 2 3A 2 2 (Ω m +4Ω Λ ) λ Ωm +4 Ω Λ so A 2 sets the mass-scale with a small dependence on λ. Locally, the coupling is not given by the cosmological value of α, but a local one!

17 Note that β(φ min )= V (φ min ) 4V (φ min )+A(φ min )ρ m therefore β 0 for ρ The coupling α therefore tends to zero in large density regions. We need it to be smaller than 10-5 in the solar system (the strongest local constraint, from Cassini experiment). One can indeed show that β is driven to small values, provided A 2 is large.

18 A technical note: Demanding that β < β 3(β 2 β 2 )+λ 2 ln β 2 leads to < 2A 2 δφ N (r). β 2 with 2 δφ N =4πG N δρ m We demand α<10-5 and assume NFW for density profile.

19 10 2 Allowed Parameter Space for Environmentally Dependent Dilaton Allowed Region Value of A Lower bound on A from necessary condition Unshaded region violates Cassini bound Value of s

20 This translates into the allowed cosmological interaction range 2.5 Allowed Values of the Cosmological Fifth Force Range: cos 2 Ruled out by local tests Cosmological Range of Fifth Force: (in Mpc) cos Allowed Region Value of s

21 Interaction range important for structure formation (on scales Mpc). No modification of gravity well above that scale. Background evolution very similar to ΛCDM. Deviations from w=-1 very small (not checked but expect situation to be similar as in chameleon cosmology). Coupling rather small ( ), but could be significant, even if additional force is below gravity!

22 Demand that inside our galaxy scalar field is essentially decoupled. Not necessarily true for dark matter halos. So, growth for baryons and dark matter different (situation as in talk by Philippe Brax, see Brax; Brax, vdb, Davis, Shaw, JCAP 1004, 032 (2010) for details): δ g + δ m + α = 2 3Ω m 2 2 3Ω m 2 3+λ 2 4+ δ g = 3 2 Ω mδ m δ m = 3 2 [1 + α eff] Ω m δ m Ω m0 a 3 (1 Ω m0 2 1, α eff = Fifth force inside galaxies suppressed, assume here that this is true for all type of galaxies, so coupling α = 0 for galaxies ( baryons ). More realistically, the coupling will be time-(and scale)-dependent, but need more sophistical calculations to determine the full evolution of the density contrast for both CDM and baryons. α 1+ am k 2

23 Observables are f gal =dln δ g /dln a, and the functions Σ κm and Σ κi and the indicator E G: k 2 (Φ + Ψ) = 8πGa 2 ρσ κm D GR δ i H 1 k 2 ( Φ + Ψ) = 8πGa 2 ρσ κi (f GR 1)D GR δ i E G = k2 (Ψ + Φ) 3H 2 0 a 1 θ and θ = dδ g dlna covered in talk by Brax; Brax, vdb, Davis, Shaw, JCAP 1004, 032 (2010) and for E G see Zhang, Ligouri, Bean & Dodelson, Phys.Rev.Lett. 99, (2007).

24 Relative Modified Gravity Parameter: E G /E G GR z=0 z=1 z=2 Modified Gravity Parameter, E G, for s = Spatial Scale: (A 2 1/2 H0 ) 1 k

25 Relative Modified Gravity Parameter: E G /E G GR x z=0 z=1 z=2 Modified Gravity Parameter, E G, for s = Spatial Scale: (A 2 1/2 H0 ) 1 k

26 1.2 ISW Slip Parameter, I m, for s = 10 1 ISW Slip Parameter, I m z=0 z=1 z= Spatial Scale: (A 2 1/2 H0 ) 1 k

27 1.005 ISW Slip Parameter, I m, for s = 1 1 ISW Slip Parameter, I m z=0 z=1 z= Spatial Scale: (A 2 1/2 H0 ) 1 k

28 Weak Lensing Slip Parameter, m Weak Lensing Slip Parameter, m, for s = 10 z=0 z=1 z= Spatial Scale: (A 2 1/2 H 0 ) 1 k

29 Weak Lensing Slip Parameter, m z=0 z=1 z=2 Weak Lensing Slip Parameter, m, for s = Spatial Scale: (A 2 1/2 H 0 ) 1 k

30 Conclusions Considered dilaton action in strong coupling regime, assuming minimum in coupling function A(Φ) (motivated by work of Damour & Polyakov) in order to hide the field. Promoted field to be responsible for dark energy: field displaced from minimum of effective coupling function. Coupling therefore not zero, but can be small locally (depending on parameter). As a side remark: if A(Φ) would not have minimum, chameleon mechanism cannot come to rescue. Constraints on parameter (λ, A 2 ) given from local constraints. Cosmological interaction range below 2 Mpc, constraints are currently weak, but hopefully soon much better constraints available. Leads to more stringent constraints on strong coupling limit of string theory. For this, a more detailed calculation of the non-linear regime is necessary.