Dark Energy. RESCEU APcosPA Summer School on Cosmology and Particle Astrophysics Matsumoto city, Nagano. July 31 - August

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1 RESCEU APcosPA Summer School on Cosmology and Particle Astrophysics Matsumoto city, Nagano LCC, Université Montpellier 2 July 31 - August More :

2 Friedmann-Lemaître-Robertson-Walker (FLRW) universes: k = 0, 1, -1 ds 2 = dt 2 a 2 (t) dl 2 dl 2 = dr 2 1 k r 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) flat, closed, open spatial sections More : If gravity is described by General Relativity (GR): R µν 1 2 g µν R = 8πG T µν

3 Comoving perfect fluids: T µν = (ρ + p) U µ U ν p g µν U 0 = U 0 = 1, spatial components vanish. dρ dt = 3H(ρ + p) = 3H(1 + w) ρ w p ρ w = ρ a 3(1+w) w m = 0 ρ m a 3 w r = 1 3 ρ m a 4 More :

4 Friedmann equations: (ȧ ) 2 + k a a 2 = 8πG 3 Ω i = ρ i ρ cr ä a = 4πG 3 i ρ i (ρ i + 3p i ) i H 2 8πG 3 ρ cr Ω k = k a 2 H 2 Ω i + Ω k = 1 i q ä ah 2 = 1 Ω i (1 + 3w i ) 2 w m = 0, w r = 1 3 > 0 ä < 0 ȧ a H H 0 = 67 km s 1 Mpc 1 i More :

5 replacements 0, He 2 H More He Y 10 6 : Li Ω b h 2

6 More :

7 paradigm comes from observations: SNIa Luminosity-distances F = L 4πd L 2 m M = 5 log d L + 25 ( c d L (z) = a 0 (1 + z) r = a 0 (1 + z) S Μ a 0 z z ΛCDM Ω m,0 = 0.3 Ω k,0 = 0 EdS 0 dz ) H(z ) More :

8 Expansion no longer as in standard cosmology ä < 0 ä > z 0.5 puzzle: What is the origin of this accelerated expansion? We are not really unhappy... Ω m,0 0.3, Ω DE,0 0.7, Ω k,0 0 A new vision has emerged supported by many observations: SNIa, CMB, BAO,... More :

9 Introduce (DE) For k = 0, z z eq : h 2 (z) = ( H(z) 3H 2 = 8πG (ρ m + ρ DE ) 2Ḣ = 8πG (ρ m + ρ DE + p DE ) ) 2 = Ωm,0 (1 + z) 3 + Ω DE,0 f (z) [ z f (z) = exp 3 dz 1 + w DE (z ] ) z H 0 w DE (z) = 1+z 3 d ln H 2 dz 1 1 Ω m,0 H 2 0 H 2 (1 + z) 3 d L (z) H(z) w DE (z) Large errors! More :

10 Old simple solution: cosmological Λ...My greatest blunder... A. Einstein ä a = 4πG 3 ρ m + Λ 3 More Conceptual problem : Λ l 2 Pl...But consistent with observations w Λ = 1, and ρ Λ is exactly Other have generically: w DE (z)!! :

11 : Is ρ DE / Is w DE = 1? w DE (z)? Is Ω DE 1 at z 1? Coupling in the dark sector? Is DE connected to dark matter? Is DE a perfect fluid? Is gravity described by GR? Is our universe homogeneous? More :

12 Early DE (k=0) Ω DE (x) = Ω DE,0 Ω e DE (1 x 3w 0) Ω DE,0 + Ω m,0 x 3w +Ω e DE (1 x 3w 0 ) 0 x a a 0 w 0 < 0 Ω e DE = const w 0 < 0 Ω DE (x) Ω DE,0 for x 1 Ω DE (x) Ω e DE for x 0 w DE,0 = w 0 Ω DE (x) = Ω Λ (x) for Ω e DE = 0, w 0 = 1 More :

13 (Generalized) Chaplygin gas p = A ρ α A > 0, α > 0 ( ρ(a) = A + B ) 1 1+α a 3(1+α) B is an arbitrary integration More w 0 for a 0 w 1 for a ( ) B 1 w(a) = 1 + A a 3(1+α) : Genuine barotropic equation of state p(ρ) c 2 s = α w dw d ln a = 3(1 + w)(c2 s w)

14 : (minimally coupled) scalar field φ(t) S φ = d 4 x ( ) 1 g 2 gµν φφ V (φ) ρ φ = 1 2 φ 2 + V (φ) p φ = 1 2 φ 2 V (φ) w φ = p φ ρ φ = φ 2 2V φ 2 + 2V 1 w φ 1 ρ φ + p φ 0 No phantom! Dynamical properties: attractors, thawing and freezing Many generalizations: DBI, tachyons, k-essence,... More :

15 Ratra-Peebles model V (φ) = M4+α φ α α > 0 field φ is subdominant in the early universe: ρ φ ρ B a(t) t q 2 q = 3(1 + w B ) Equation of motion (Klein-Gordon equation) has solution: φ + 3H φ + dv dφ = 0 φ t p, p = α More :

16 field is catching up: ρ φ t 2p 2 a α p q ρ φ ρ B t 2p ρ B t 2 Can also be understood from the behaviour of EoS parameter w φ w φ = α w B 2 α + 2 < w B = w B 2+α α More :

17 Reconstruction of the quintessence potential V (φ) Friedmann equations: H 2 = 8π G ( ρ m + 1 ) 3 2 φ 2 + V (φ) Ḣ = 4π G (ρ m + φ 2) 8π G 3 V = H2 Ω m,0 H0 2 (1 + z)3 4π G φ 2 3 = H Ω m,0h0 2 (1 + z)3 + Ḣ 3 (1) More :

18 8π G V = 3H Ω m,0h 2 0 (1 + z)3 ( ) dφ 2 4πG(1+z) 2 H 2 (1 + z) = dz 2 (1 + z) 2 dh 2 dz dh 2 dz 3 2 Ω m,0h 2 0 (1+z)3 Reconstruction from the background expansion: D L (z) 1 H(z) = ( ) DL (z) +Ω m,0 V (z) V (φ φ 0 ) 1 + z Reconstruction also possible using perturbations More :

19 Another scalar field model (Tachyon): S T = = d 4 x g L T d 4 x [ g V (T ) ] 1 g µν µ T ν T More T µ,ν = V (T ) µ T ν T 1 g µν µ T ν T g µ,ν L T FLRW : p = V 1 Ṫ 2 ρ = Equation of motion: T 1 Ṫ 2 + 3HṪ + V V = 0 V 1 Ṫ 2 :

20 In general: S φ = d 4 x g L(X, φ) X 1 2 gµν µ φ ν φ T µν = L X µφ ν φ g µν L (p + ρ) U µ U ν g µν p p = L ρ = 2 X L X L U µ = µφ 2 X More :

21 Dynamical system analysis Introduce dimensionless variables x 2 = 1 2 φ 2 ρ cr y 2 = V ρ cr ρ cr = 3 H2 8π G 3 M2 p H 2 More dx dn = F V 1(x, y; w B, λ) λ M p V dy dn = F 2(x, y; w B, λ) N ln a λ = corresponds to exponential potential V = V 0 exp( λ φ M p ) :

22 Critical points: F 1 (x c, y c ) = 0 F 2 (x c, y c ) = 0 We linearize around the critical points: F i (x, y) = F i x (x c, y c ) (x x c ) + F i y (x c, y c ) (y y c ) M i1 δx + M i2 δy dδx dn = M 11 δx + M 12 δy dδy dn = M 21 δx + M 22 δy More :

23 Eignevalues ω 1, ω 2 real and nondegenerate: 1) Same sign node Both negative stable, otherwise unstable 2) Opposite sign saddle Eigenvalues a ± ib are complex conjugate: Stable spiral if a < 0 Eigenvalues are real, degenerate:... Ḣ = 3 2 H2 (1 + w eff ) w eff = i w i Ω i More :

24 Eignevalues ω 1, ω 2 real and nondegenerate: 1) Same sign node Both negative stable, otherwise unstable 2) Opposite sign saddle Eigenvalues a ± ib are complex conjugate: Stable spiral if a < 0 Eigenvalues are real, degenerate:... Ḣ = 3 2 H2 (1 + w eff ) w eff = i w i Ω i More :

25 Eignevalues ω 1, ω 2 real and nondegenerate: 1) Same sign node Both negative stable, otherwise unstable 2) Opposite sign saddle Eigenvalues a ± ib are complex conjugate: Stable spiral if a < 0 Eigenvalues are real, degenerate:... Ḣ = 3 2 H2 (1 + w eff ) w eff = i w i Ω i More :

26 Critical points: 1) (x c, y c ) = (0, 0) Ω φ = 0 w B = w eff, w φ not determined ω 1 = 3(1 w B) 2, ω 2 = 3(1+w B) 2 It is a saddle ( 2) (x c, y c ) = ) λ 6, 1 λ2 6 Ω φ = 1, w eff = w φ = 1 + λ2 3 ω 1 = λ2 6 2, ω 2 = λ 2 3(1 + w B ) Stable for λ 2 < 3(1 + w B ) Accelerated expansion for λ 2 < 2 3) (x c, y c ) = ( w B λ, w 2 B λ Ω φ = 3(1+w B) w λ 2 eff = w φ = w B Scaling solution, stable for λ 2 > 3(1 + w B ) 4) (x, y ) = (±1, 0) Ω = 1 w = w = 1 ) More :

27 Generically λ is not! dw φ dn = (w φ 1) ( 3(1 + w φ ) λ 3(1 + w φ ) Ω φ ) d ln Ω φ dn = 3(w φ w B ) (1 Ω φ ) dλ dn = 3(1 + w φ ) Ω φ (Γ 1) λ 2 Γ V V V 2 A tracker solution attracts evolutions with various initial conditions: Ω φ = 3(1+w φ) λ 2 Γ > 1 w φ = d ln Ω φ dn = 2 dλ dn λ w φ = w B 2(Γ 1) 2Γ 1 for Ω φ 1 More :

28 Ratra-Peebles model: Γ = α > 1 w φ = α w B 2 α+2 From observations w φ < or α < at 2 σ More :

29 DE Maybe gravity differs from GR on large scales? Can this explain the accelerated expansion without introducing a new DE component? We keep the RW metric ds 2 = dt 2 a 2 (t) dl 2 We get modified Friedmann equations They can be often recast in an Einsteinian way The growth of perturbations get modified as well! Crucial probe of modified gravity More :

30 ) L = 1 16πG (F(Φ) R Z µ Φ µ Φ 2U(Φ) + L m (g µν ) Brans-Dicke parametrization F(Φ) = Φ Another choice Z (Φ) = ω BD(Φ) Φ F(Φ) = arbitrary Z = 1 ω BD > 0 ω BD = F (df/dφ) 2 > 3 2 V = G eff M 1 M 2 r G eff = G N (1 + G eff,0 G N,0 G ) 1 2ω BD + 3 ω BD,0 > massless Φ field G N = G F More :

31 ) L = 1 16πG (F(Φ) R Z µ Φ µ Φ 2U(Φ) + L m (g µν ) Brans-Dicke parametrization F(Φ) = Φ Another choice Z (Φ) = ω BD(Φ) Φ F(Φ) = arbitrary Z = 1 ω BD > 0 ω BD = F (df/dφ) 2 > 3 2 V = G eff M 1 M 2 r G eff = G N (1 + G eff,0 G N,0 G ) 1 2ω BD + 3 ω BD,0 > massless Φ field G N = G F More :

32 ) L = 1 16πG (F(Φ) R Z µ Φ µ Φ 2U(Φ) + L m (g µν ) Brans-Dicke parametrization F(Φ) = Φ Another choice Z (Φ) = ω BD(Φ) Φ F(Φ) = arbitrary Z = 1 ω BD > 0 ω BD = F (df/dφ) 2 > 3 2 V = G eff M 1 M 2 r G eff = G N (1 + G eff,0 G N,0 G ) 1 2ω BD + 3 ω BD,0 > massless Φ field G N = G F More :

33 ) L = 1 16πG (F(Φ) R Z µ Φ µ Φ 2U(Φ) + L m (g µν ) Brans-Dicke parametrization F(Φ) = Φ Another choice Z (Φ) = ω BD(Φ) Φ F(Φ) = arbitrary Z = 1 ω BD > 0 ω BD = F (df/dφ) 2 > 3 2 V = G eff M 1 M 2 r G eff = G N (1 + G eff,0 G N,0 G ) 1 2ω BD + 3 ω BD,0 > massless Φ field G N = G F More :

34 Modified background equations 3FH 2 = 8πG ρ m + Φ U 3HḞ 2FḢ = 8πG ρ m + Φ 2 + F HḞ Define ρ DE and p DE : 3H 2 = 8πG N,0 (ρ m + ρ DE ) 2Ḣ = 8πG N,0 (ρ m + ρ DE + p DE ) dh2 dz < 3 Ω m,0 (1 + z) Ω k,0 (1 + z) phantom Possible in scalar-tensor 8π G (ρ DE + p DE ) = Φ 2 + F HḞ + 2(F F 0) Ḣ More :

35 Modified background equations 3FH 2 = 8πG ρ m + Φ U 3HḞ 2FḢ = 8πG ρ m + Φ 2 + F HḞ Define ρ DE and p DE : 3H 2 = 8πG N,0 (ρ m + ρ DE ) 2Ḣ = 8πG N,0 (ρ m + ρ DE + p DE ) dh2 dz < 3 Ω m,0 (1 + z) Ω k,0 (1 + z) phantom Possible in scalar-tensor 8π G (ρ DE + p DE ) = Φ 2 + F HḞ + 2(F F 0) Ḣ More :

36 F z Figure: The function F F 0 is shown for several with vanishing potential U = 0. We have from left to right the following equation of state parameter w DE : 2, 1.5, polynomial expression, 1, 0.5. More :

37 Growth of matter perturbations is modified: δ m + 2H δ m 4πG eff ρ m δ m = 0 ( (h h 2 δ m 2 ) ) + 2 h2 δ m = z 2 (1 + z)g eff G Growth of perturbations must be consistent with background expansion! h 2 = Ω m,0 δ m [ ] (1 + z z)2 δ 2 δ Ω G eff δ δ m,0 0 G 1 + z dz More :

38 Technical details ds 2 = (1 + 2φ)dt 2 a 2 (1 2ψ)d~x V = φ δ m = k 2 a 2 V + 3 (HV ) + 3 ψ φ = ψ δf F In quintessence: φ = ψ! V : peculiar velocity potential δ m = δρm ρ m + 3HV φ, ψ, δ m are gauge-invariant quantities In quasi static limit: δ m + 2H δ m + k 2 a 2 φ = 0 More :

39 Perturbed dilaton equation of motion: δφ = (φ 2ψ) df dφ = φ F ωbd 2 + ω BD Some small clustering in contrast to quintessence ( k 2 a 2 δϕ φ) More : Combination of the perturbed Einstein equations: k 2 a 2 φ = 4πG eff ρ m δ m δ m + 2H δ m 4πG eff ρ m δ m = 0

40 Perturbed dilaton equation of motion: δφ = (φ 2ψ) df dφ = φ F ωbd 2 + ω BD Some small clustering in contrast to quintessence ( k 2 a 2 δϕ φ) More : Combination of the perturbed Einstein equations: k 2 a 2 φ = 4πG eff ρ m δ m δ m + 2H δ m 4πG eff ρ m δ m = 0

41 L = R 16πG + 1 [ 2 gµν µ φ ν φ V (φ) + L m Ψm ; A 2 ] (φ) g µν A 2 = e 2βφ/M PL V = M 4 e ( M φ )n M φ M PL V is like Λ! Growth of perturbations is modified: M G eff (z, k) V (r) = G 1 M ( 2 r β 2 e m φr ) In this model, m φ is too large, no influence on cosmological scales! More :

42 Interacting dark sector ρ φ + 3H (ρ φ + p φ ) = α φ ρ m ρ m + 3H ρ m = α φ ρ m α d ln A dφ More φ + 3H φ = V α ρ m = V eff V eff V + ρ m ρ m A mechanism Coupling to dark matter only? A b = 0 A dm 0 :

43 f (R) modified gravity DE : R f (R) Basic idea: No new component, just change gravity! Many (e.g. R + µ2 R ) lead to unviable cosmic expansion with a t 2 3 a t 1 2 Some interesting viable f (R) still remain: f (R) = R λr c f 1 (x) x R/R c ( ) ) n e.g. R λr c (1 1 + R2, n, λ > 0(n 2) Rc 2 F df (R) dr > 0, df(r) dr > 0 More :

44 G eff = G F δ m + 2H δ m 4πG eff ρ m δ m = 0 ( ) k 2 a 2 m k 2 a 2 m 2 G eff = G eff (z, k) V (r) = G F M 1 M 2 r Deep in matter-domination and R R c : F F 3 m 2 ( e mr) G eff = 4 3 G δ 33 1 m a 4 k am = G δ m a k am More :

45 galileon model modified Friedmann eqs with effective ρ DE (φ, φ) w DE,0 = 1, w DE can be < 1 G eff (z) signature in the perturbation growth Laboratory and solar system constraints: Vainshtein mechanism More :

46 Matter perturbations can be characterized by the growth function f = d ln δ d ln a d ln δ dx df dx + f (1 3 w eff) f = 3 G eff 2 G Ω m A convenient parameterization f = Ω γ m. Actually δ m (z, k) γ = γ(z, k) In ΛCDM: γ 0.55 It can be very different in modified gravity! More :

47 Matter perturbations can be characterized by the growth function f = d ln δ d ln a d ln δ dx df dx + f (1 3 w eff) f = 3 G eff 2 G Ω m A convenient parameterization f = Ω γ m. Actually δ m (z, k) γ = γ(z, k) In ΛCDM: γ 0.55 It can be very different in modified gravity! More :

48 Matter perturbations can be characterized by the growth function f = d ln δ d ln a d ln δ dx df dx + f (1 3 w eff) f = 3 G eff 2 G Ω m A convenient parameterization f = Ω γ m. Actually δ m (z, k) γ = γ(z, k) In ΛCDM: γ 0.55 It can be very different in modified gravity! More :

49 Γ Γ 0 Ω m,0 = 0.3 and various values of w DE,0 We have from top to bottom: w DE,0 = 1.4, 1.3, 1.2, 1, 0.8. The black line gives the true value of γ 0 realized: same non vanishing γ More :

50 Exact results if γ is Requiring w DE 0 deep in the matter era γ = 0.5 is singular 0.5 < γ 0.6 γ = 0.6: DE behaves like dust γ = 6 11 = if DE = Λ 0.5 < γ < 6 11 : phantom regime Deep in the DE dominated era: phantom regime Cannot be realized by quintessence! More :

51 f (R) = R λr c x 2n x 2n + 1 CDM k 0.01 h Mpc 1 k h Mpc 1 k 0.1 h Mpc 1 k 0.2 h Mpc 1 x R R c More Γ z n = 1, λ = 1.55 :

52 f (R) = R λr c Γ x 2n (x 2n + 1) CDM k 0.01 h Mpc 1 k h Mpc 1 k 0.1 h Mpc 1 k 0.2 h Mpc 1 x R R c More : n λ = 1.55

53 We need complementary probes: Supernovae Clusters Weak lensing Baryon Acoustic Oscillations Cosmic Microwave Background Gamma Ray Bursts? Gravitational waves? Redshift drift?...theoretical tools: parametrizations, Fisher matrices, Bayesian approach,.. More :

54 We need complementary probes: Supernovae Clusters Weak lensing Baryon Acoustic Oscillations Cosmic Microwave Background Gamma Ray Bursts? Gravitational waves? Redshift drift?...theoretical tools: parametrizations, Fisher matrices, Bayesian approach,.. More :

55 parametrizations: assume for some physical quantity a functional dependence depending on a restricted number of parameters e.g. w DE (z) = ( 1 + α) + β (1 x) w 0 + w a (1 x) w DE (z) = ( 1 + α p ) + β (x p x) f (z) = (1 + z) 3(α+β) e 3β z 1+z Construct a quantity which assumes a special value, typically for Λ e.g. Om(z) = h2 (z) 1 (1 + z) 3 1 Om(z) = Ω m,0 = for ΛCDM Om(z) evolves for all other DE! More :

56 Union2.1 set, 580 SNIa, z wcdm : w = (1σ C.L.) Suzuki N.et al., ApJ 746,85 (2012), arxiv: More 7-year WMAP Shift parameter: zrec R = Ω m,0 0 dz = ± h(z) E. Komatsu et al.,apj Suppl. 192, 18 (2011) :

57 Baryon Acoustic Oscillations: D V (z) = H 1 0 [ z h(z) ( z for z = 0.106, 0.2, 0.35, 0.44, 0.6, dz ) ] 2 1/3 h(z ) From the following galaxy surveys: SDSS DR7,W. Percival et al., MNRAS 401,2148 (2010) WiggleZ, C. Blake et al., MNRAS 415, 2892 (2011); MNRAS 418, 1707 (2011) 6dF GS, F. Beutler et al., MNRAS 416, 3017 (2011). SDSS-3BOSS: L. Anderson et al., arxiv: D A (0.57) = 1408 ± 45 Mpc and H(0.57) = 92.9 ± 7.8 km/s/mpc More :

58 Planck results, arxiv: Planck+WP+BAO w = (2σ C.L.) Planck+WP+Union2.1 w = 1.09 ± 0.17 (2σ C.L.) Planck+WP+SNLS w = (2σ C.L.) Planck+WP+H 0 w = (2σ C.L.) Planck+WP+BAO w 0 = w a < 1.32 (2σ C.L.) w(a) = w 0 + w a (1 a) More :

59 BAO Acoustic scale at various z D A (z) and H(z) BAO forecasts for a Full-Sky survey z min z max Vol % Err D A (z) % Err H(z) Ω Λ σ w More :

60 The art of inducing accelerated expansion seems to close on ΛCDM... Smooth component in GR with w 1 Λ? w(z)? w < 1? Clustering? Modifying gravity? Modified growth of perturbations Challenging cosmological framework? Future of our universe? will single out viable A new vision with challenging! More :