Cosmology of the string tachyon

Transcription

1 Cosmology of the string tachyon Progress in cubic string field theory Gianluca Calcagni March 28th, 2007

2 Outline 1 Motivations

3 Outline 1 Motivations 2 The DBI tachyon

4 Outline 1 Motivations 2 The DBI tachyon 3 The CSFT tachyon

5 Outline 1 Motivations 2 The DBI tachyon 3 The CSFT tachyon 4 Cosmology

6 Outline 1 Motivations 2 The DBI tachyon 3 The CSFT tachyon 4 Cosmology 5 Nonlocal solutions

7 The many faces of the string tachyon

8 The many faces of the string tachyon Conformal field theory

9 The many faces of the string tachyon Conformal field theory [Sen 2002]

10 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action

11 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action [Garousi 2000; Bergshoeff et al. 2000; Klusoň 2000; Gibbons, Hori, and Yi 2001]

12 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action [Garousi 2000; Bergshoeff et al. 2000; Klusoň 2000; Gibbons, Hori, and Yi 2001] Boundary string field theory (BSFT)

13 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action [Garousi 2000; Bergshoeff et al. 2000; Klusoň 2000; Gibbons, Hori, and Yi 2001] Boundary string field theory (BSFT) [Witten 1992,1993; Shatashvili 1993a,b]

14 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action [Garousi 2000; Bergshoeff et al. 2000; Klusoň 2000; Gibbons, Hori, and Yi 2001] Boundary string field theory (BSFT) [Witten 1992,1993; Shatashvili 1993a,b] Cubic string field theory (CSFT)

15 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action [Garousi 2000; Bergshoeff et al. 2000; Klusoň 2000; Gibbons, Hori, and Yi 2001] Boundary string field theory (BSFT) [Witten 1992,1993; Shatashvili 1993a,b] Cubic string field theory (CSFT) [Witten 1986a,b; Preitschopf et al. 1990; Aref eva et al. 2002; Aref eva et al. 1990a,b; Berkovits 1996,1999]

16 The many faces of the string tachyon Conformal field theory [Sen 2002] Dirac Born Infeld (DBI) effective action [Garousi 2000; Bergshoeff et al. 2000; Klusoň 2000; Gibbons, Hori, and Yi 2001] Boundary string field theory (BSFT) [Witten 1992,1993; Shatashvili 1993a,b] Cubic string field theory (CSFT) [Witten 1986a,b; Preitschopf et al. 1990; Aref eva et al. 2002; Aref eva et al. 1990a,b; Berkovits 1996,1999] For a review see Sen (hep-th/ ).

17 Tachyon condensation (in Minkowski) All methods agree: as the tachyon rolls down towards the asymptotic minimum of the potential, the brane it lives on decays into a lower-dimensional brane or the closed string vacuum

18 Tachyon condensation (in Minkowski) All methods agree: as the tachyon rolls down towards the asymptotic minimum of the potential, the brane it lives on decays into a lower-dimensional brane or the closed string vacuum ( Sen s conjecture...

19 Tachyon condensation (in Minkowski) All methods agree: as the tachyon rolls down towards the asymptotic minimum of the potential, the brane it lives on decays into a lower-dimensional brane or the closed string vacuum ( Sen s conjecture... proven by Schnabl 2005)

20 Two actions for one tachyon (in Minkowski)

21 Two actions for one tachyon (in Minkowski) DBI tachyon (negligible higher-than-first-order derivatives): S T = d D xv(t) 1 + µ T µ T

22 Two actions for one tachyon (in Minkowski) DBI tachyon (negligible higher-than-first-order derivatives): S T = d D xv(t) 1 + µ T µ T Cubic string field theory: S = 1 ( 1 g 2 o 2α Φ Q BΦ + 1 ) 3 Φ Φ Φ

23 Two actions for one tachyon (in Minkowski) DBI tachyon (negligible higher-than-first-order derivatives): S T = d D xv(t) 1 + µ T µ T Cubic string field theory: S = 1 ( 1 g 2 o 2α Φ Q BΦ + 1 ) 3 Φ Φ Φ At level (0, 0): Φ = Φ = φ(x) and (metric )

24 Two actions for one tachyon (in Minkowski) DBI tachyon (negligible higher-than-first-order derivatives): S T = d D xv(t) 1 + µ T µ T Cubic string field theory: S = 1 ( 1 g 2 o 2α Φ Q BΦ + 1 ) 3 Φ Φ Φ At level (0, 0): Φ = Φ = φ(x) and (metric ) S φ = 1 [ 1 g 2 d D x o 2α φ(α µ µ + 1)φ λ ( ) ] 3 λ α µ µ /3 φ Λ 3 where λ = 3 9/2 /

25 Two actions for one tachyon (in Minkowski) DBI tachyon (negligible higher-than-first-order derivatives): S T = d D xv(t) 1 + µ T µ T Cubic string field theory: S = 1 ( 1 g 2 o 2α Φ Q BΦ + 1 ) 3 Φ Φ Φ At level (0, 0): Φ = Φ = φ(x) and (metric ) S φ = 1 [ 1 g 2 d D x o 2α φ(α µ µ + 1)φ λ ( ) ] 3 λ α µ µ /3 φ Λ 3 where λ = 3 9/2 / See Ohmori (hep-th/ ).

26 Do they give the same predictions?

27 Do they give the same predictions? DBI tachyon: extensively studied both as inflaton and dark energy field

28 Do they give the same predictions? DBI tachyon: extensively studied both as inflaton and dark energy field (problematic or ineffective in both cases)

29 Do they give the same predictions? DBI tachyon: extensively studied both as inflaton and dark energy field (problematic or ineffective in both cases) SFT tachyon: poor control both on Minkowski and FRW backgrounds...?

30 Do they give the same predictions? DBI tachyon: extensively studied both as inflaton and dark energy field (problematic or ineffective in both cases) SFT tachyon: poor control both on Minkowski and FRW backgrounds...? A comparison would open up interesting possibilities!

31 Friedmann Robertson Walker background Flat FRW metric: ds 2 = dt 2 + a 2 (t) dx i dx i. The Hubble parameter is defined as H ȧ/a = d t a/a.

32 Dynamics DBI action: S T = d D x gv(t) 1 + µ T µ T

33 Dynamics DBI action: S T = d D x gv(t) 1 + µ T µ T Perfect fluid (FRW): p T = V(T) 1 Ṫ 2 ρ T = V(T)/ 1 Ṫ 2

34 Dynamics DBI action: S T = d D x gv(t) 1 + µ T µ T Perfect fluid (FRW): p T = V(T) 1 Ṫ 2 ρ T = V(T)/ 1 Ṫ 2 w p T /ρ T = Ṫ 2 1

35 Dynamics DBI action: S T = d D x gv(t) 1 + µ T µ T Perfect fluid (FRW): p T = V(T) 1 Ṫ 2 ρ T = V(T)/ 1 Ṫ 2 w p T /ρ T = Ṫ 2 1 < 0

36 Dynamics DBI action: S T = d D x gv(t) 1 + µ T µ T Perfect fluid (FRW): p T = V(T) 1 Ṫ 2 ρ T = V(T)/ 1 Ṫ 2 w p T /ρ T = Ṫ 2 1 < 0 Equations of motion (EH action): H 2 = ρ T + ρ m + ρ r T 1 Ṫ 2 + 3HṪ + V,T V = 0

37 Tachyon condensation Condensation into the closed string vacuum (Carrollian limit).

38 Tachyon condensation Condensation into the closed string vacuum (Carrollian limit). Near the minimum g s = O(1) and the perturbative description may fail down (for simple compactification).

39 Tachyon condensation Condensation into the closed string vacuum (Carrollian limit). Near the minimum g s = O(1) and the perturbative description may fail down (for simple compactification). This is a toy model valid on cosmological scales.

40 Tachyon condensation Condensation into the closed string vacuum (Carrollian limit). Near the minimum g s = O(1) and the perturbative description may fail down (for simple compactification). This is a toy model valid on cosmological scales. More problematic to justify at late times.

41 DBI tachyon inflation

42 DBI tachyon inflation No reheating with runaway string effective potentials.

43 DBI tachyon inflation No reheating with runaway string effective potentials. Large density perturbations with such potentials.

44 DBI tachyon inflation No reheating with runaway string effective potentials. Large density perturbations with such potentials. Reheating can be achieved with a negative KKLT-like Λ. Anisotropies adjusted with small warp factor [Garousi, Sami, Tsujikawa 2004]

45 DBI tachyon inflation No reheating with runaway string effective potentials. Large density perturbations with such potentials. Reheating can be achieved with a negative KKLT-like Λ. Anisotropies adjusted with small warp factor [Garousi, Sami, Tsujikawa 2004] With phenomenological potentials, good inflation and non-gaussianity but non-characteristic predictions.

46 DBI tachyon as dark energy with Andrew R. Liddle PRD 74, (2006), astro-ph/

47 DBI tachyon as dark energy with Andrew R. Liddle PRD 74, (2006), astro-ph/ Different predictions between the DBI tachyon and canonical quintessence?

48 DBI tachyon as dark energy with Andrew R. Liddle PRD 74, (2006), astro-ph/ Different predictions between the DBI tachyon and canonical quintessence? For a wide choice of potentials (ad-hoc or motivated), fine tuning on either the i.c. or the parameters of the potential.

49 DBI tachyon as dark energy with Andrew R. Liddle PRD 74, (2006), astro-ph/ Different predictions between the DBI tachyon and canonical quintessence? For a wide choice of potentials (ad-hoc or motivated), fine tuning on either the i.c. or the parameters of the potential. High-precision observational cosmology allows to constrain the theory in a remarkable way.

50 DBI tachyon as dark energy with Andrew R. Liddle PRD 74, (2006), astro-ph/ Different predictions between the DBI tachyon and canonical quintessence? For a wide choice of potentials (ad-hoc or motivated), fine tuning on either the i.c. or the parameters of the potential. High-precision observational cosmology allows to constrain the theory in a remarkable way. The tachyon is poorly effective as dark energy (the cosmological constant problem is NOT solved).

51 DBI tachyon as dark energy with Andrew R. Liddle PRD 74, (2006), astro-ph/ Different predictions between the DBI tachyon and canonical quintessence? For a wide choice of potentials (ad-hoc or motivated), fine tuning on either the i.c. or the parameters of the potential. High-precision observational cosmology allows to constrain the theory in a remarkable way. The tachyon is poorly effective as dark energy (the cosmological constant problem is NOT solved). The tachyon cannot decay faster than dust matter, ρ T a 3(1+w T) > ρ m a 3 cannot be used as quintessential inflaton.

52 Cubic SFT action S = S g + S φ,

53 Cubic SFT action S = S g + S φ, S g = 1 2κ 2 D d D x g R

54 Cubic SFT action S = S g + S φ, S g = 1 2κ 2 d D x g R D S φ = d D x [ ] 1 g 2 φ( m2 )φ U( φ) Λ

55 Cubic SFT action S = S g + S φ, S g = 1 2κ 2 d D x g R D S φ = d D x [ ] 1 g 2 φ( m2 )φ U( φ) Λ 1 µ ( g µ ) g

56 Cubic SFT action S = S g + S φ, S g = 1 2κ 2 d D x g R D S φ = d D x [ ] 1 g 2 φ( m2 )φ U( φ) Λ 1 µ ( g µ ) g φ λ /3 φ e r φ, r ln λ

57 Cubic SFT action S = S g + S φ, S g = 1 2κ 2 d D x g R D S φ = d D x [ ] 1 g 2 φ( m2 )φ U( φ) Λ 1 µ ( g µ ) g φ λ /3 φ e r φ, r ln λ e r r l + = l! l c l l l=0 l=0

58 Some other definitions Ṽ( φ) 1 2 m2 φ 2 + U( φ) + Λ

59 Some other definitions Ṽ( φ) 1 2 m2 φ 2 + U( φ) + Λ U δu δφ = er Ũ

60 Some other definitions Ṽ( φ) 1 2 m2 φ 2 + U( φ) + Λ U δu δφ = er Ũ Ũ U φ

61 Choice for the potential Pure monomial potential:

62 Choice for the potential Pure monomial potential: U( φ) = σ n φ n U r = σe φn 1

63 Choice for the potential Pure monomial potential: U( φ) = σ n φ n U r = σe φn 1 Bosonic CSFT when m 2 = 1, σ = λ, n = 3.

64 Choice for the potential Pure monomial potential: U( φ) = σ n φ n U r = σe φn 1 Bosonic CSFT when m 2 = 1, σ = λ, n = 3. Susy CSFT when m 2 = 1/2, σ = σ(λ), n = 4.

65 Equations of motion: scalar field ( m 2 )φ + U = 0

66 Equations of motion: scalar field ( m 2 )φ + U = 0 In terms of φ: ( m 2 )e 2r φ + Ũ = 0

67 Energy density and pressure ρ = T 0 0 = φ 2 2 (1 O 2) + Ṽ O 1 p = T i i = φ 2 2 (1 O 2) Ṽ + O 1

68 Energy density and pressure ρ = T 0 0 = φ 2 2 (1 O 2) + Ṽ O 1 p = T i i = φ 2 O 1 = r 0 2 (1 O 2) Ṽ + O 1 ds (e s Ũ )( e s φ),

69 Energy density and pressure ρ = T 0 0 = φ 2 2 (1 O 2) + Ṽ O 1 p = T i i = φ 2 O 1 = r 0 O 2 = 2 φ2 2 (1 O 2) Ṽ + O 1 ds (e s Ũ )( e s φ), r 0 ds (e s Ũ ). (e s φ)..

70 Equation of state

71 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1

72 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1

73 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1 A new condition for acceleration: φ 2 (1 O 2 )/2 + Ṽ O 1

74 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1 A new condition for acceleration: φ 2 (1 O 2 )/2 + Ṽ O 1 w φ 1

75 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1 A new condition for acceleration: φ 2 (1 O 2 )/2 + Ṽ O 1 w φ 1 Pseudo kinetic regime (beyond the DBI barrier): φ 2 Ṽ and φ2 O 2 O 1

76 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1 A new condition for acceleration: φ 2 (1 O 2 )/2 + Ṽ O 1 w φ 1 Pseudo kinetic regime (beyond the DBI barrier): φ 2 Ṽ and φ2 O 2 O 1 w φ 1

77 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1 A new condition for acceleration: φ 2 (1 O 2 )/2 + Ṽ O 1 w φ 1 Pseudo kinetic regime (beyond the DBI barrier): φ 2 Ṽ and φ2 O 2 O 1 w φ 1 Pseudo phantom regime: φ 2 Ṽ O 1 and O 2 > 1

78 Equation of state Pseudo slow-roll condition: φ 2 Ṽ and φ2 O 2 O 1 w φ 1 A new condition for acceleration: φ 2 (1 O 2 )/2 + Ṽ O 1 w φ 1 Pseudo kinetic regime (beyond the DBI barrier): φ 2 Ṽ and φ2 O 2 O 1 w φ 1 Pseudo phantom regime: φ 2 Ṽ O 1 and O 2 > 1 w φ 1

79 Non-locality The cosmological equations of motion are nonlinear and involve all the derivatives φ (n) and H (n), that is, all the infinite SR tower.

80 Non-locality The cosmological equations of motion are nonlinear and involve all the derivatives φ (n) and H (n), that is, all the infinite SR tower. Non-standard Cauchy problem: infinite initial conditions, to know them means to find the solution! [Moeller and Zwiebach 2002]

81 Non-locality The cosmological equations of motion are nonlinear and involve all the derivatives φ (n) and H (n), that is, all the infinite SR tower. Non-standard Cauchy problem: infinite initial conditions, to know them means to find the solution! [Moeller and Zwiebach 2002] Nonlocal theories are at odds with the inflationary paradigm: while the latter tends to erase any memory of the initial conditions, the formers do preserve this memory.

82 Unviable cosmologies?

83 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q

84 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q Constant SR parameters when q = 1.

85 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q Constant SR parameters when q = 1. l 1 l φ = ( 1) l t p 2l (p 2n)(p 2n 1 + 3H 0 ). n=0

86 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q Constant SR parameters when q = 1. l 1 l φ = ( 1) l t p 2l (p 2n)(p 2n 1 + 3H 0 ). n=0 If p R \ N +, φ is ill-defined

87 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q Constant SR parameters when q = 1. l 1 l φ = ( 1) l t p 2l (p 2n)(p 2n 1 + 3H 0 ). n=0 If p R \ N +, φ is ill-defined lim c l+1 l+1 φ l c l l φ = lim c 1(p 2l)(p 2l 1+3H 0 ) t 2 l l + 1 = +,

88 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q Constant SR parameters when q = 1. l 1 l φ = ( 1) l t p 2l (p 2n)(p 2n 1 + 3H 0 ). n=0 If p R \ N +, φ is ill-defined lim c l+1 l+1 φ l c l l φ = lim c 1(p 2l)(p 2l 1+3H 0 ) t 2 l l + 1 = +, Possibility: p is a positive even number, so that the series ends at l p/2 and φ t p.

89 Unviable cosmologies? Very important class of dynamics: φ = t p, H = H 0 t q Constant SR parameters when q = 1. l 1 l φ = ( 1) l t p 2l (p 2n)(p 2n 1 + 3H 0 ). n=0 If p R \ N +, φ is ill-defined lim c l+1 l+1 φ l c l l φ = lim c 1(p 2l)(p 2l 1+3H 0 ) t 2 l l + 1 = +, Possibility: p is a positive even number, so that the series ends at l p/2 and φ t p. Another case is p 1 + 3H 0 = 2n.

90 Unviable cosmologies? Can we conclude that power-law cosmology cannot be used as a base for nonlocal solutions?

91 Unviable cosmologies? Can we conclude that power-law cosmology cannot be used as a base for nonlocal solutions? NO! What is not defined is the nonlocal solution expressed as an infinite series of powers of the d Alembertian.

92 Localization GC, G. Nardelli, and M. Montobbio, to appear; GC et al., to appear

93 Localization GC, G. Nardelli, and M. Montobbio, to appear; GC et al., to appear Interpret r as a fixed value of an auxiliary evolution variable r.

94 Localization GC, G. Nardelli, and M. Montobbio, to appear; GC et al., to appear Interpret r as a fixed value of an auxiliary evolution variable r. The scalar field φ(r, t) is thought to live in dimensions.

95 Localization GC, G. Nardelli, and M. Montobbio, to appear; GC et al., to appear Interpret r as a fixed value of an auxiliary evolution variable r. The scalar field φ(r, t) is thought to live in dimensions. The solution φ loc (t) = φ(r = 0, t) of the local system (r = 0) is the initial condition.

96 Localization GC, G. Nardelli, and M. Montobbio, to appear; GC et al., to appear Interpret r as a fixed value of an auxiliary evolution variable r. The scalar field φ(r, t) is thought to live in dimensions. The solution φ loc (t) = φ(r = 0, t) of the local system (r = 0) is the initial condition. Define φ(r, t) e r(β+ /α) φ loc (t)

97 Localization Property 1: it satisfies the diffusion equation: α r φ(r, t) = αβ φ(r, t) + φ(r, t)

98 Localization Property 1: it satisfies the diffusion equation: α r φ(r, t) = αβ φ(r, t) + φ(r, t) Property 2: e q is simply a shift of the auxiliary variable r. e q φ(r, t) = e αq r φ(r, t) = φ(r + αq, t)

99 Localization Property 1: it satisfies the diffusion equation: α r φ(r, t) = αβ φ(r, t) + φ(r, t) Property 2: e q is simply a shift of the auxiliary variable r. e q φ(r, t) = e αq r φ(r, t) = φ(r + αq, t) The system becomes local in t!

100 Localization Property 1: it satisfies the diffusion equation: α r φ(r, t) = αβ φ(r, t) + φ(r, t) Property 2: e q is simply a shift of the auxiliary variable r. e q φ(r, t) = e αq r φ(r, t) = φ(r + αq, t) The system becomes local in t! ( m 2 )φ(r, t) = e rβ Ũ [e rβ φ((1 + 2α)r, t)]

101 Steps towards localized solutions

102 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator:

103 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t).

104 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t). 2 Write the local solution as an expansion in the basis of eigenstates of the (Mellin Barnes transform):

105 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t). 2 Write the local solution as an expansion in the basis of eigenstates of the (Mellin Barnes transform): φ(0, t) = dµ [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ).

106 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t). 2 Write the local solution as an expansion in the basis of eigenstates of the (Mellin Barnes transform): φ(0, t) = dµ [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ). 3 Write the nonlocal solution (Gabor transform):

107 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t). 2 Write the local solution as an expansion in the basis of eigenstates of the (Mellin Barnes transform): φ(0, t) = dµ [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ). 3 Write the nonlocal solution (Gabor transform): φ(r, t) = dµ e r(β+µ2 /α) [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ).

108 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t). 2 Write the local solution as an expansion in the basis of eigenstates of the (Mellin Barnes transform): φ(0, t) = dµ [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ). 3 Write the nonlocal solution (Gabor transform): φ(r, t) = dµ e r(β+µ2 /α) [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ). It satisfies the heat equation by definition!

109 Steps towards localized solutions 1 Find the eigenstates of the d Alembertian operator: G(µ, t) = G(µ, t) 3HĠ(µ, t) = µ 2 G(µ, t). 2 Write the local solution as an expansion in the basis of eigenstates of the (Mellin Barnes transform): φ(0, t) = dµ [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ). 3 Write the nonlocal solution (Gabor transform): φ(r, t) = dµ e r(β+µ2 /α) [C 1 G 1 (µ, t) + C 2 G 2 (µ, t)] f (µ). It satisfies the heat equation by definition! 4 Check that φ(r, t) is an approximated solution of the nonlocal e.o.m.s for some α, β, C i.

110 Present status and advantages

111 Present status and advantages 1 SUSY Minkowski solution successfully found (to appear)

112 Present status and advantages 1 SUSY Minkowski solution successfully found (to appear) 2 Only known method allowing to keep the whole series e

113 Present status and advantages 1 SUSY Minkowski solution successfully found (to appear) 2 Only known method allowing to keep the whole series e 3 Quantization is well-defined, as in the closely related Hamiltonian formalism [Llosa and Vives 1994; Gomis et al. 2001,2004; Cheng et al. 2002]

114 Unviable cosmologies revisited a = t p, H = H 0t 1

115 Unviable cosmologies revisited a = t p, H = H 0t 1 ψ(r, t) ( ) 4r p/2 ) p2 αt2 Ψ ( ; 1 ν; α 4r

116 Unviable cosmologies revisited a = t p, H = H 0t 1 ψ(r, t) ( ) 4r p/2 ) p2 αt2 Ψ ( ; 1 ν; α 4r ν = (1 3H 0 )/2

117 Unviable cosmologies revisited a = t p, H = H 0t 1 ψ(r, t) ( ) 4r p/2 ) p2 αt2 Ψ ( ; 1 ν; α 4r ν = (1 3H 0 )/2 Ψ(α; β; z) = Φ(α; β; x) = Γ(β) Γ(α) [ π Φ(α; β; z) sin πβ Γ(1 + α β)γ(β) ] 1 β Φ(1 + α β; 2 β; z) z, Γ(α)Γ(2 β) + k=0 Γ(α + k) x k Γ(β + k) k!

118 Unviable cosmologies revisited a = t p, H = H 0t 1 ψ(r, t) ( ) 4r p/2 ) p2 αt2 Ψ ( ; 1 ν; α 4r ν = (1 3H 0 )/2 Ψ(α; β; z) = Φ(α; β; x) = Γ(β) Γ(α) [ π Φ(α; β; z) sin πβ Γ(1 + α β)γ(β) ] 1 β Φ(1 + α β; 2 β; z) z, Γ(α)Γ(2 β) + k=0 Γ(α + k) x k Γ(β + k) k!

119 Unviable cosmologies revisited a = t p, H = H 0t 1 ψ(r, t) ( ) 4r p/2 ) p2 αt2 Ψ ( ; 1 ν; α 4r ν = (1 3H 0 )/2 Ψ(α; β; z) = Φ(α; β; x) = Γ(β) Γ(α) [ π Φ(α; β; z) sin πβ Γ(1 + α β)γ(β) ] 1 β Φ(1 + α β; 2 β; z) z, Γ(α)Γ(2 β) + k=0 Γ(α + k) x k Γ(β + k) k!

120 Unviable cosmologies revisited Figure: p = 1/2, ν = 3/2, r = 1

121 Unviable cosmologies revisited Figure: p = 1/2, ν = 3/2, r = 1

122 Conclusions Non-locality generates new dynamics (at classical and quantum level)

123 Conclusions Non-locality generates new dynamics (at classical and quantum level) The CSFT tachyon is cosmologically inequivalent to (and maybe more viable than) the DBI one

124 Conclusions Non-locality generates new dynamics (at classical and quantum level) The CSFT tachyon is cosmologically inequivalent to (and maybe more viable than) the DBI one Open issues

125 Conclusions Non-locality generates new dynamics (at classical and quantum level) The CSFT tachyon is cosmologically inequivalent to (and maybe more viable than) the DBI one Open issues Search for analytic cosmological solutions in progress (to appear)

126 Conclusions Non-locality generates new dynamics (at classical and quantum level) The CSFT tachyon is cosmologically inequivalent to (and maybe more viable than) the DBI one Open issues Search for analytic cosmological solutions in progress (to appear) Evidence for a nontrivial relation between SBFT and CSFT! (to appear)