Scale-invariance from spontaneously broken conformal invariance

Transcription

1 Scale-invariance from spontaneously broken conformal invariance Austin Joyce Center for Particle Cosmology University of Pennsylvania Hinterbichler, Khoury arxiv: Hinterbichler, AJ, Khoury arxiv: March 17, 2012 Austin Joyce (UPenn) March 17, / 20

2 Introduction At early times, the universe is described by a (nearly) CFT on (nearly) flat space Universe is driven to be homogeneous and flat Explain scale invariance of primordial perturbations in terms of the symmetry breaking pattern so(4, 2) so(4, 1) from some scalar operator in the CFT getting a VEV 1 t Under general conditions, scale-invariance for spectator fields follows solely from symmetry breaking Specific examples: Quartic U(1), Negative φ 4 Rubakov ; Hinterbichler, Khoury Galilean Genesis Creminelli, Nicolis & Trincherini Austin Joyce (UPenn) March 17, / 20

3 Simple example negative quartic potential Φ S φ = d 4 x ( 12 ( φ)2 + λ4 ) φ4 symmetry algebra so(4, 2) δ Pµ φ = µ φ, δ Jµν φ = (x µ ν x ν µ )φ, δ D φ = ( + x µ µ )φ, δ Kµ φ = ( 2 x µ 2x µ x ν ν + x 2 µ ) φ. Consider coupling a weight-0 field χ in the CFT ( ) S χ = d 4 x 1 2 φ2 ( χ) 2 m2 χ 2 λφ4 χ 2 + κ 2 φ φχ2. Austin Joyce (UPenn) March 17, / 20

4 Simple example negative quartic potential Consider coupling a weight-0 field χ in the CFT ( ) S χ = d 4 x 1 2 φ2 ( χ) 2 m2 χ 2 λφ4 χ 2 + κ 2 φ φχ2. This field couples to the effective metric g eff µν = φ 2 η µν Φ S φ = d 4 x ( 12 ( φ)2 + λ4 ) φ4 symmetry algebra so(4, 2) δ Pµ φ = µ φ, δ Jµν φ = (x µ ν x ν µ )φ, δ D φ = ( + x µ µ )φ, δ Kµ φ = ( 2 x µ 2x µ x ν ν + x 2 µ ) φ. Austin Joyce (UPenn) March 17, / 20

5 Negative quartic model cont. Equation of motion: φ λφ 3 = λ ( t) Zero energy solution: φ(t) = with < t < 0 attractor! This breaks some of the symmetries the generators { } δ Pi, δ D, δ Jij, δ Ki still annihilate the background; they can be repackaged as δ Jij ; δ J56 = δ D ; δ J5i = 1 2 (δ P i + δ Ki ) ; δ J6i = 1 2 (δ P i δ Ki ). which have the commutation relations of the so(4, 1) algebra, [δ Jab, δ Jcd ] = η ac δ Jbd η bc δ Jad + η bd δ Jac η ad δ Jbc, where η ab = diag (δ ij, 1, 1). Austin Joyce (UPenn) March 17, / 20

6 Negative quartic model cont. Equation of motion: φ λφ 3 = λ ( t) Zero energy solution: φ(t) = with < t < 0 attractor! This breaks some of the symmetries the generators { } δ Pi, δ D, δ Jij, δ Ki still annihilate the background; they can be repackaged as δ Jij ; δ J56 = δ D ; δ J5i = 1 2 (δ P i + δ Ki ) ; δ J6i = 1 2 (δ P i δ Ki ). which have the commutation relations of the so(4, 1) algebra, [δ Jab, δ Jcd ] = η ac δ Jbd η bc δ Jad + η bd δ Jac η ad δ Jbc, where η ab = diag (δ ij, 1, 1). symmetry breaking: so(4, 2) so(4, 1). In the broken phase, χ couples to g eff µν = φ 2 η µν 1 t 2 η µν. Austin Joyce (UPenn) March 17, / 20

7 Perturbations φ Writing φ = φ + ϕ, the Fourier modes of the perturbations ϕ satisfy ( ϕ k + k 2 6 ) t 2 ϕ k = 0 As k 0, this is solved by ϕ k 1 t 2 and ϕ k ( t) 3. The growing mode is just a constant time shift of the background solution φ(t + ε) = φ(t) + ε φ(t) 2 1 = φ(t) + ε λ t 2, hence the φ 1/t solution is an attractor. Austin Joyce (UPenn) March 17, / 20

8 Perturbations φ Writing φ = φ + ϕ, the Fourier modes of the perturbations ϕ satisfy ( ϕ k + k 2 6 ) t 2 ϕ k = 0 As k 0, this is solved by ϕ k 1 t 2 and ϕ k ( t) 3. The growing mode is just a constant time shift of the background solution φ(t + ε) = φ(t) + ε φ(t) 2 1 = φ(t) + ε λ t 2, hence the φ 1/t solution is an attractor. Quantum fluctuations ϕ k 2 1/k 5 t 4 Red spectrum blue spectrum for ζ Austin Joyce (UPenn) March 17, / 20

9 Perturbations χ Expanding around χ = 0, the quadratic χ lagrangian is Defining ˆχ = 2 λ 1 ( t) L χ = 1 λt 2 ( χ)2 2m2 χ + 2κ λt 4 χ 2 ˆχ k + χ, its mode functions satisfy ( k 2 2(1 ) m2 χ κ) t 2 ˆχ k = 0 Austin Joyce (UPenn) March 17, / 20

10 Perturbations χ Expanding around χ = 0, the quadratic χ lagrangian is Defining ˆχ = 2 λ 1 ( t) L χ = 1 λt 2 ( χ)2 2m2 χ + 2κ λt 4 χ 2 ˆχ k + χ, its mode functions satisfy ( k 2 2(1 ) m2 χ κ) t 2 ˆχ k = 0 If mχ, 2 κ 1, then this is solved by (assuming adiabatic vacuum initial conditions) ( ˆχ k = e ikt 1 i ) 2k kt In the long-wavelength limit, χ is scale-invariant P χ = 1 2π 2 k3 χ k 2 λ 2(2π) 2 Austin Joyce (UPenn) March 17, / 20

11 Coupling to gravity cosmology Einstein Frame We consider minimal coupling to gravity S = d 4 x ( M 2 ) g Pl 2 R + L CFT [g µν ]. Conformal symmetry is broken at the 1 M Pl level. Austin Joyce (UPenn) March 17, / 20

12 Coupling to gravity cosmology Einstein Frame We consider minimal coupling to gravity S = d 4 x ( M 2 ) g Pl 2 R + L CFT [g µν ]. 1 Conformal symmetry is broken at the M Pl level. At early times, gravity is negligible, φ 1 ( t) is an approximate solution Austin Joyce (UPenn) March 17, / 20

13 Coupling to gravity cosmology Einstein Frame We consider minimal coupling to gravity S = d 4 x ( M 2 ) g Pl 2 R + L CFT [g µν ]. 1 Conformal symmetry is broken at the M Pl level. At early times, gravity is negligible, φ 1 ( t) is an approximate solution Dilatation symmetry then implies ρ CFT P CFT 1 however, we t 4 know ρ = const. to lowest order in 1 M Pl so ρ CFT 0, P CFT β t 4. Quartic model, β > 0, Galilean genesis β < 0. Austin Joyce (UPenn) March 17, / 20

14 Cosmology cont. We integrate M 2 PlḢ = 1 2 (ρ CFT + P CFT ) to find β β H(t) 6( t) 3 MPl 2, a(t) 1 12t 2 MPl 2. The universe is therefore contracting (expanding) for β > 0 (β < 0) Austin Joyce (UPenn) March 17, / 20

15 Cosmology cont. We integrate M 2 PlḢ = 1 2 (ρ CFT + P CFT ) to find β β H(t) 6( t) 3 MPl 2, a(t) 1 12t 2 MPl 2. The universe is therefore contracting (expanding) for β > 0 (β < 0) The universe is nearly static until t end = β M Pl (φ M Pl in φ 4 model) Austin Joyce (UPenn) March 17, / 20

16 Cosmology cont. We integrate M 2 PlḢ = 1 2 (ρ CFT + P CFT ) to find β β H(t) 6( t) 3 MPl 2, a(t) 1 12t 2 MPl 2. The universe is therefore contracting (expanding) for β > 0 (β < 0) The universe is nearly static until t end = The CFT equation of state decreases from + to O(1). β M Pl w CFT P CFT ρ CFT = 12 β t2 M 2 Pl. (φ M Pl in φ 4 model) Austin Joyce (UPenn) March 17, / 20

17 Cosmology cont. We integrate M 2 PlḢ = 1 2 (ρ CFT + P CFT ) to find β β H(t) 6( t) 3 MPl 2, a(t) 1 12t 2 MPl 2. The universe is therefore contracting (expanding) for β > 0 (β < 0) The universe is nearly static until t end = The CFT equation of state β M Pl w CFT P CFT ρ CFT = 12 β t2 M 2 Pl. (φ M Pl in φ 4 model) decreases from + to O(1). w 1 drives the background to be flat, homogeneous and isotropic Gratton, Khoury, Steinhardt, Turok astro-ph/ H 2 M 2 Pl = k a 2 + C matter a 3 + C radiation a 4 + C anisotropy a C a 3(1+w) Austin Joyce (UPenn) March 17, / 20

18 Other examples Quartic U(1) model Rubakov L U(1) = 1 2 ψ ψ + λ 4 ψ 4 In polar coordinates, ψ = φe iχ, this is a special case of the quartic model L U(1) = 1 2 ( φ) + λ 4 φ4 1 2 φ2 ( χ) 2 Austin Joyce (UPenn) March 17, / 20

19 Other examples Quartic U(1) model Rubakov L U(1) = 1 2 ψ ψ + λ 4 ψ 4 In polar coordinates, ψ = φe iχ, this is a special case of the quartic model L U(1) = 1 2 ( φ) + λ 4 φ4 1 2 φ2 ( χ) 2 Galilean genesis Creminelli, Nicolis & Trincherini L Gal = 1 2 e2φ ( φ) H 2 φ( φ) H 2 ( φ)4 This has a solution e φ = 1 H( t), which also breaks so(4, 2) so(4, 1). This solution violates the NEC Perturbations can propagate superluminally Austin Joyce (UPenn) March 17, / 20

20 Phenomenological lagrangians Weight-0 fields acquiring a scale invariant spectrum is a generic feature of the symmetry breaking pattern so(4, 2) so(4, 1) We are therefore motivated to construct the most general phenomenological lagrangian for this symmetry breaking Austin Joyce (UPenn) March 17, / 20

21 Phenomenological lagrangians Weight-0 fields acquiring a scale invariant spectrum is a generic feature of the symmetry breaking pattern so(4, 2) so(4, 1) We are therefore motivated to construct the most general phenomenological lagrangian for this symmetry breaking Coset construction Callan, Coleman, Wess & Zumino; Volkov Given a Lie group G and a Lie subgroup H, technique for constructing the most general H-invariant lagrangian that non-linearly realizes full G. The Goldstone fields parameterize the coset G/H by From the Maurer Cartan form g = e x P e ξ Z ω = g 1 dg = ω P P + ω z Z + ω V V we can build covariant derivatives for the Goldstones. Austin Joyce (UPenn) March 17, / 20

22 Coset construction for so(4, 2) so(4, 1) Hinterbichler, AJ, Khoury Parameterize the conformal algebra by J µν, K µ, D and ˆP µ P µ H2 K µ. Benefit is that (ˆP µ, J µν ) generates an so(4,1) algebra Austin Joyce (UPenn) March 17, / 20

23 Coset construction for so(4, 2) so(4, 1) Hinterbichler, AJ, Khoury Parameterize the conformal algebra by J µν, K µ, D and ˆP µ P µ H2 K µ. Benefit is that (ˆP µ, J µν ) generates an so(4,1) algebra The broken generators are D and K µ, have Goldstones π and ξ µ 5 Broken generators, but only 1 independent Goldstone Inverse Higgs constraint = ξ µ e π µ π Austin Joyce (UPenn) March 17, / 20

24 Coset construction for so(4, 2) so(4, 1) Hinterbichler, AJ, Khoury Parameterize the conformal algebra by J µν, K µ, D and ˆP µ P µ H2 K µ. Benefit is that (ˆP µ, J µν ) generates an so(4,1) algebra The broken generators are D and K µ, have Goldstones π and ξ µ 5 Broken generators, but only 1 independent Goldstone Inverse Higgs constraint = End result of coset construction ξ µ e π µ π (ω P ) a µ (ω P) b ν η ab = e 2π ḡµν ds D µ ξ ν = 1 2 µπ ν π 1 2 µ ν π 1 4ḡαβ α π β πḡ µν H2 4 e2π ḡ µν + H2 4 ḡµν. The field π transforms as δπ = ξ µ µ π 1 4 µ ξ µ, where ξ µ are conformal Killing vectors Austin Joyce (UPenn) March 17, / 20

25 Geometric construction In the case of interest, the coset construction turns out to be equivalent to a simple geometric construction Linearly realize de Sitter group construct theories on a fictitious ds. To non-linearly realize the conformal group, we merely add the conformal mode gµν eff = e 2π ḡµν eff Curvature invariants of this metric give the action for the field π and we can couple spectator fields using [g] Austin Joyce (UPenn) March 17, / 20

26 Geometric construction In the case of interest, the coset construction turns out to be equivalent to a simple geometric construction Linearly realize de Sitter group construct theories on a fictitious ds. To non-linearly realize the conformal group, we merely add the conformal mode gµν eff = e 2π ḡµν eff Curvature invariants of this metric give the action for the field π and we can couple spectator fields using [g] Ingredients: { gµν eff } { }, R µν, µ equivalent to g eff µν, D µ ξ ν, µ Austin Joyce (UPenn) March 17, / 20

27 Goldstone action The simplest invariant term is just the measure L 0 d 4 x g eff d 4 x ḡ eff e 4π Austin Joyce (UPenn) March 17, / 20

28 Goldstone action The simplest invariant term is just the measure L 0 d 4 x g eff d 4 x ḡ eff e 4π The kinetic term is the Ricci scalar L 1 d 4 x g eff R d 4 x ḡ eff ( 1 2 e2π ( π) e2π π H 2 e 2π ) Austin Joyce (UPenn) March 17, / 20

29 Goldstone action The simplest invariant term is just the measure L 0 d 4 x g eff d 4 x ḡ eff e 4π The kinetic term is the Ricci scalar L 1 d 4 x g eff R d 4 x ḡ eff ( 1 2 e2π ( π) e2π π H 2 e 2π ) At four derivative order, R 2 and R 2 µν terms are degenerate L 2 d 4 x g eff R 2 d 4 x ḡ eff [ ( π) 2 +2 π( π) 2 +( π) 4 4H 2 ( π) 2] However an orthogonal term can be constructed as a Wess Zumino term Goon, Hinterbichler, AJ, Trodden L wz d 4 x ḡ eff [( π) π( π) 2 + 6H 2 ( π) 2]. Austin Joyce (UPenn) March 17, / 20

30 Coupling weight-0 matter fields Matter fields couple via the covariant derivative of the conformal metric, µ. Additionally, we are free to promote any of the mass scales in the Goldstone lagrangian to an arbitrary function of χ For concreteness, work to O(χ 3 ) and O( 2 ) Austin Joyce (UPenn) March 17, / 20

31 Coupling weight-0 matter fields Matter fields couple via the covariant derivative of the conformal metric, µ. Additionally, we are free to promote any of the mass scales in the Goldstone lagrangian to an arbitrary function of χ For concreteness, work to O(χ 3 ) and O( 2 ) S = d 4 x ( ḡ eff [M ) e2π ( π) 2 H 2 e 2π + H2 2 e4π + M2 χ 2 e2π ( χ) 2 + m2 χ 2 e4π χ 2 + λ χ e 4π χ 3 + ( 1 + M e2π ( π) ) ] 2 e2π π H 2 e 2π + H2 2 e4π (χ 2 + αχ 3 ). Austin Joyce (UPenn) March 17, / 20

32 Analysis of the general effective action π The quadratic action for π that derives from this action is S π = M0 2 d 4 x [ ḡ eff 1 ] 2 ( π)2 + 2H 2 π 2. It is convenient to work with the flat slicing dseff 2 = 1 ( H 2 t dt 2 + d x 2). 2 The π action takes the form S π = M 2 0 [ d 4 x 1 2H 2 t 2 π2 1 2H 2 t 2 ( π) H 2 t 4 π2 Define the canonically-normalized variable, v = M 0 ( v k + k 2 6 ) t 2 v k = 0. ( Ht) ] π, which satisfies. Austin Joyce (UPenn) March 17, / 20

33 Analysis of the general effective action π The quadratic action for π that derives from this action is S π = M0 2 d 4 x [ ḡ eff 1 ] 2 ( π)2 + 2H 2 π 2. It is convenient to work with the flat slicing dseff 2 = 1 ( H 2 t dt 2 + d x 2). 2 The π action takes the form S π = M 2 0 [ d 4 x 1 2H 2 t 2 π2 1 2H 2 t 2 ( π) H 2 t 4 π2 Define the canonically-normalized variable, v = M 0 Therefore, ( v k + k 2 6 ) t 2 v k = 0. ( Ht) ] π, which satisfies π k = H( t)3/2 4π H (1) 5/2 2M ( kt) = P π 9H π 5 M0 2 ( kt) 2. red Austin Joyce (UPenn) March 17, / 20

34 Analysis of the general effective action χ At quadratic order in χ, the action gives S χ = d 4 x ḡ eff [ M2 χ 2 ( χ)2 m2 χ + M ] 0 2H2 χ 2 2, Action for a scalar on ds. If m 2 χ/(m 2 χh 2 ), M 2 0 /M2 χ 1 the field χ will have a scale-invariant spectrum of perturbations Austin Joyce (UPenn) March 17, / 20

35 Analysis of the general effective action χ At quadratic order in χ, the action gives S χ = d 4 x ḡ eff [ M2 χ 2 ( χ)2 m2 χ + M ] 0 2H2 χ 2 2, Action for a scalar on ds. If mχ/(m 2 χh 2 2 ), M 0 2/M2 χ 1 the field χ will have a scale-invariant spectrum of perturbations Indeed, in this case the solution for the canonically normalized variable ˆχ = Mχ ( Ht) χ is ˆχ k = 1 ( 1 i ) e ikt, 2k kt This implies that the long-wavelength power spectrum for χ k is scale invariant P χ = 1 2π 2 k3 χ k 2 H 2 (2π) 2 Mχ 2. Austin Joyce (UPenn) March 17, / 20

36 3-point function for χ The cubic action for χ is (working in the exact scale-invariant limit) d 4 x [ ] ḡ eff M2 χ 2 ( χ)2 Mχπ( χ) λ χ χ 3. From this, we can compute the χχχ correlator χ k1 χ k2 χ k3 = λ χh 2 (2π) 3 δ (3) ( k 1 + k 2 + k 3 ) 2M 6 χ 1 i k3 i k 1 k 2 k 3 ki 2 k j i j i ki 3 (1 γ log k t t ). Unsurprisingly, this is the same as the pure 3-point function for a massless specator field in inflation Linearly realized SO(4,1) symmetry Maldacena, Pimentel ; Creminelli Austin Joyce (UPenn) March 17, / 20

37 Constraints from non-linearly realized conformal symmetry Linearly-realized de Sitter symmetry acts as the conformal group on R 3. For example lim ϕ 1( x 1, t)ϕ 2 ( x 2, t)ϕ 3 ( x 3, t) = t 0 C 123 x x x Austin Joyce (UPenn) March 17, / 20

38 Constraints from non-linearly realized conformal symmetry Linearly-realized de Sitter symmetry acts as the conformal group on R 3. For example lim ϕ 1( x 1, t)ϕ 2 ( x 2, t)ϕ 3 ( x 3, t) = t 0 C 123 x x x However, these theories are additionally constrained consider S χ = d 4 x [ ḡ eff 1 ] 2 e2π ( χ) 2 = d 4 x ḡ eff [ 1 ] 2 ( χ)2 π( χ) 2 2π 2 ( χ) Conformal symmetry fixes the relative coefficients between the terms should lead to particular relations between n-point functions. Austin Joyce (UPenn) March 17, / 20

39 DBI non-linearly realized so(4, 2) from the start Hinterbichler, Khoury, Miller, to appear Imagine a flat brane probing an AdS 5 bulk lowest order world-volume action L DBI = φ 4 ( 1 ( φ)2 φ λ 4 ) φ 4 Austin Joyce (UPenn) March 17, / 20

40 DBI non-linearly realized so(4, 2) from the start Hinterbichler, Khoury, Miller, to appear Imagine a flat brane probing an AdS 5 bulk lowest order world-volume action L DBI = φ 4 ( 1 ( φ)2 φ λ 4 ) φ 4 The so(4, 2) isometries act non-linearly ( δ D φ = (1 + x µ µ )φ; δ Kµ = 2x µ 2x µ x ν ν + x 2 µ 1 ) 2φ 2 µ φ Austin Joyce (UPenn) March 17, / 20

41 DBI non-linearly realized so(4, 2) from the start Hinterbichler, Khoury, Miller, to appear Imagine a flat brane probing an AdS 5 bulk lowest order world-volume action L DBI = φ 4 ( 1 ( φ)2 φ λ 4 ) φ 4 The so(4, 2) isometries act non-linearly ( δ D φ = (1 + x µ µ )φ; δ Kµ = 2x µ 2x µ x ν ν + x 2 µ 1 ) 2φ 2 µ φ Equations of motion are more intricate, but still allow φ 1/t: φ(t) = 1 + λ/ λ/8 λ ( t) Austin Joyce (UPenn) March 17, / 20

42 DBI non-linearly realized so(4, 2) from the start Hinterbichler, Khoury, Miller, to appear Imagine a flat brane probing an AdS 5 bulk lowest order world-volume action L DBI = φ 4 ( 1 ( φ)2 φ λ 4 ) φ 4 The so(4, 2) isometries act non-linearly ( δ D φ = (1 + x µ µ )φ; δ Kµ = 2x µ 2x µ x ν ν + x 2 µ 1 ) 2φ 2 µ φ Equations of motion are more intricate, but still allow φ 1/t: φ(t) = 1 + λ/ λ/8 λ ( t) Possible to couple massless spectator field by instead considering a brane probing AdS 5 S 1 Possible to extend to warped-throat type compactifications? Austin Joyce (UPenn) March 17, / 20

43 Summary Alternative to inflation based upon breaking so(4, 2) so(4, 1) At early times, gravity is negligible universe is is driven to be flat and homogeneous by symmetry. Low energy effective action fixed by symmetry = χ acquires a scale-invariant spectrum Non-linearly realized conformal symmetry should constrain correlators in beyond SO(4,1) symmetry of spectators in inflation Austin Joyce (UPenn) March 17, / 20

44 Cosmology Jordan frame Consider the effective metric gµν eff = φ 2 η µν, in Jordan frame, the action takes the form S = d 4 x g eff ( M 2 Pl 2φ 2 R eff + 3M2 Pl φ 4 g µν eff µφ ν φ + 1 φ 4 L CFT [ φ 2 g eff µν ] ). With the cosmological ansatz ds 2 J = dt2 J + a2 J (t J)d x 2, the EOM are At early times φ 3HJ 2 6H J φ 2 3 φ 2 φ 4, φ φ 3 + 3H φ J φ 2 3 φ 2 φ 4 R eff 6 = β 4φ 2 MPl 2, t4 β 4φ 2 M 2 Pl t4 0 and the equations admit a solution φ 1 t, H J = constant However, this is not inflation in any normal sense M eff Pl 1/φ varies by O(1) in a Hubble time. Austin Joyce (UPenn) March 17, / 20