Scale-invariance from spontaneously broken conformal invariance - - PowerPoint PPT Presentation

Scale-invariance from spontaneously broken conformal invariance

Austin Joyce

Center for Particle Cosmology University of Pennsylvania

Hinterbichler, Khoury arXiv:1106.1428 Hinterbichler, AJ, Khoury arXiv:1202.6056

March 17, 2012

Austin Joyce (UPenn) March 17, 2012 1 / 20

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 φ4 — Rubakov 0906.3693; Hinterbichler, Khoury

1106.1428

◮ Galilean Genesis — Creminelli, Nicolis & Trincherini 1007.0027 Austin Joyce (UPenn) March 17, 2012 2 / 20

Simple example — negative quartic potential

Consider coupling a weight-0 field χ in the CFT Sχ =

• d4x
• −1

2φ2(∂χ)2 − m2

χ

2 λφ4χ2 + κ 2φφχ2

• .

Austin Joyce (UPenn) March 17, 2012 3 / 20

Φ VΦ

Sφ =

• d4x
• −1

2(∂φ)2 + λ 4 φ4

• symmetry algebra so(4, 2)

δPµφ = −∂µφ , δJµνφ = (xµ∂ν − xν∂µ)φ , δDφ = −(∆ + xµ∂µ)φ , δKµφ =

• −2∆xµ − 2xµxν∂ν + x2∂µ
• φ .

Simple example — negative quartic potential

Consider coupling a weight-0 field χ in the CFT Sχ =

• d4x
• −1

2φ2(∂χ)2 − m2

χ

2 λφ4χ2 + κ 2φφχ2

• .

This field couples to the effective metric geff

µν = φ2ηµν

Austin Joyce (UPenn) March 17, 2012 3 / 20

Φ VΦ

Sφ =

• d4x
• −1

2(∂φ)2 + λ 4 φ4

• symmetry algebra so(4, 2)

δPµφ = −∂µφ , δJµνφ = (xµ∂ν − xν∂µ)φ , δDφ = −(∆ + xµ∂µ)φ , δKµφ =

• −2∆xµ − 2xµxν∂ν + x2∂µ
• φ .

Negative quartic model cont.

Equation of motion: ¨ φ − λφ3 = 0 Zero energy solution: ¯ φ(t) =

• 2

λ 1 (−t)

with −∞ < t < 0 attractor! This breaks some of the symmetries—the generators

• δPi, δD, δJij, δKi
• still annihilate the background; they can be repackaged as

δJij; δJ56 = δD; δJ5i = 1 2 (δPi + δKi) ; δJ6i = 1 2 (δPi − δKi) . which have the commutation relations of the so(4, 1) algebra, [δJab, δJcd] = ηacδJbd − ηbcδJad + ηbdδJac − ηadδJbc , where ηab = diag (δij, 1, −1).

Austin Joyce (UPenn) March 17, 2012 4 / 20

Negative quartic model cont.

Equation of motion: ¨ φ − λφ3 = 0 Zero energy solution: ¯ φ(t) =

• 2

λ 1 (−t)

with −∞ < t < 0 attractor! This breaks some of the symmetries—the generators

• δPi, δD, δJij, δKi
• still annihilate the background; they can be repackaged as

δJij; δJ56 = δD; δJ5i = 1 2 (δPi + δKi) ; δJ6i = 1 2 (δPi − δKi) . which have the commutation relations of the so(4, 1) algebra, [δJab, δJcd] = ηacδJbd − ηbcδJad + ηbdδJac − ηadδJbc , where ηab = diag (δij, 1, −1). symmetry breaking: so(4, 2) → so(4, 1). In the broken phase, χ couples to geff

µν = ¯

φ2ηµν ∼ 1

t2 ηµν.

Austin Joyce (UPenn) March 17, 2012 4 / 20

Perturbations — φ

Writing φ = ¯ φ + ϕ, the Fourier modes of the perturbations ϕ satisfy ¨ ϕk +

• k2 − 6

t2

• ϕk = 0

As k → 0, this is solved by ϕk ∼ 1

t2 and ϕk ∼ (−t)3.

The growing mode is just a constant time shift of the background solution ¯ φ(t + ε) = ¯ φ(t) + ε ˙ ¯ φ(t) = ¯ φ(t) + ε

• 2

λ 1 t2 , hence the φ ∼ 1/t solution is an attractor.

Austin Joyce (UPenn) March 17, 2012 5 / 20

Perturbations — φ

Writing φ = ¯ φ + ϕ, the Fourier modes of the perturbations ϕ satisfy ¨ ϕk +

• k2 − 6

t2

• ϕk = 0

As k → 0, this is solved by ϕk ∼ 1

t2 and ϕk ∼ (−t)3.

The growing mode is just a constant time shift of the background solution ¯ φ(t + ε) = ¯ φ(t) + ε ˙ ¯ φ(t) = ¯ φ(t) + ε

• 2

λ 1 t2 , hence the φ ∼ 1/t solution is an attractor. Quantum fluctuations — |ϕk|2 ∼ 1/k5t4 Red spectrum blue spectrum for ζ

Austin Joyce (UPenn) March 17, 2012 5 / 20

Perturbations — χ

Expanding around χ = 0, the quadratic χ lagrangian is Lχ = − 1 λt2 (∂χ)2 − 2m2

χ + 2κ

λt4 χ2 Defining ˆ χ =

• 2

λ 1 (−t) · χ, its mode functions satisfy

¨ ˆ χk +

• k2 − 2(1 − m2

χ − κ)

t2

• ˆ

χk = 0

Austin Joyce (UPenn) March 17, 2012 6 / 20

Perturbations — χ

Expanding around χ = 0, the quadratic χ lagrangian is Lχ = − 1 λt2 (∂χ)2 − 2m2

χ + 2κ

λt4 χ2 Defining ˆ χ =

• 2

λ 1 (−t) · χ, its mode functions satisfy

¨ ˆ χk +

• k2 − 2(1 − m2

χ − κ)

t2

• ˆ

χk = 0 If |m2

χ|, |κ| ≪ 1, then this is solved by (assuming adiabatic vacuum initial

conditions) ˆ χk = e−ikt √ 2k

• 1 − i

kt

• In the long-wavelength limit, χ is scale-invariant

Pχ = 1 2π2 k3|χk|2 ≃ λ 2(2π)2

Austin Joyce (UPenn) March 17, 2012 6 / 20

Coupling to gravity — cosmology

Einstein Frame We consider minimal coupling to gravity S =

• d4x√−g

M2

Pl

2 R + LCFT [gµν]

• .

Conformal symmetry is broken at the

1 MPl level.

Austin Joyce (UPenn) March 17, 2012 7 / 20

Coupling to gravity — cosmology

Einstein Frame We consider minimal coupling to gravity S =

• d4x√−g

M2

Pl

2 R + LCFT [gµν]

• .

Conformal symmetry is broken at the

1 MPl level.

At early times, gravity is negligible, ¯ φ ∼

1 (−t) is an approximate

solution

Austin Joyce (UPenn) March 17, 2012 7 / 20

Coupling to gravity — cosmology

Einstein Frame We consider minimal coupling to gravity S =

• d4x√−g

M2

Pl

2 R + LCFT [gµν]

• .

Conformal symmetry is broken at the

1 MPl level.

At early times, gravity is negligible, ¯ φ ∼

1 (−t) is an approximate

solution Dilatation symmetry then implies ρCFT ∼ PCFT ∼ 1

t4 — however, we

know ρ = const. to lowest order in

1 MPl so

ρCFT ≃ 0 , PCFT ≃ β t4 . Quartic model, β > 0, Galilean genesis β < 0.

Austin Joyce (UPenn) March 17, 2012 7 / 20

Cosmology cont.

We integrate M2

Pl ˙

H = − 1

2(ρCFT + PCFT) to find

H(t) ≃ − β 6(−t)3M2

Pl

, a(t) ≃ 1 − β 12t2M2

Pl

. The universe is therefore contracting (expanding) for β > 0 (β < 0)

Austin Joyce (UPenn) March 17, 2012 8 / 20

Cosmology cont.

We integrate M2

Pl ˙

H = − 1

2(ρCFT + PCFT) to find

H(t) ≃ − β 6(−t)3M2

Pl

, a(t) ≃ 1 − β 12t2M2

Pl

. The universe is therefore contracting (expanding) for β > 0 (β < 0) The universe is nearly static until tend = −

√β MPl (φ ∼ MPl in φ4 model)

Austin Joyce (UPenn) March 17, 2012 8 / 20

Cosmology cont.

We integrate M2

Pl ˙

H = − 1

2(ρCFT + PCFT) to find

H(t) ≃ − β 6(−t)3M2

Pl

, a(t) ≃ 1 − β 12t2M2

Pl

. The universe is therefore contracting (expanding) for β > 0 (β < 0) The universe is nearly static until tend = −

√β MPl (φ ∼ MPl in φ4 model)

The CFT equation of state wCFT ≃ PCFT ρCFT = 12 β t2M2

Pl .

decreases from +∞ to O(1).

Austin Joyce (UPenn) March 17, 2012 8 / 20

Cosmology cont.

We integrate M2

Pl ˙

H = − 1

2(ρCFT + PCFT) to find

H(t) ≃ − β 6(−t)3M2

Pl

, a(t) ≃ 1 − β 12t2M2

Pl

. The universe is therefore contracting (expanding) for β > 0 (β < 0) The universe is nearly static until tend = −

√β MPl (φ ∼ MPl in φ4 model)

The CFT equation of state wCFT ≃ PCFT ρCFT = 12 β t2M2

Pl .

decreases from +∞ to O(1). |w| ≫ 1 drives the background to be flat, homogeneous and isotropic

Gratton, Khoury, Steinhardt, Turok astro-ph/0301395

3H2M2

Pl = k

a2 + Cmatter a3 + Cradiation a4 + Canisotropy a6 + . . . + C a3(1+w)

Austin Joyce (UPenn) March 17, 2012 8 / 20

Other examples

Quartic U(1) model — Rubakov 0906.3693

LU(1) = −1 2∂ ¯ ψ∂ψ + λ 4 |ψ|4 In polar coordinates, ψ = φeiχ, this is a special case of the quartic model LU(1) = −1 2(∂φ) + λ 4 φ4 − 1 2φ2(∂χ)2

Austin Joyce (UPenn) March 17, 2012 9 / 20

Other examples

Quartic U(1) model — Rubakov 0906.3693

LU(1) = −1 2∂ ¯ ψ∂ψ + λ 4 |ψ|4 In polar coordinates, ψ = φeiχ, this is a special case of the quartic model LU(1) = −1 2(∂φ) + λ 4 φ4 − 1 2φ2(∂χ)2

Galilean genesis — Creminelli, Nicolis & Trincherini 1007.0027

LGal = 1 2e2φ(∂φ)2 + 1 3H2 φ(∂φ)2 + 1 6H2 (∂φ)4 This has a solution eφ =

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, 2012 9 / 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, 2012 10 / 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

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 g = ex·Peξ·Z From the Maurer–Cartan form ω = g−1dg = ωP · P + ωz · Z + ωV · V we can build covariant derivatives for the Goldstones.

Austin Joyce (UPenn) March 17, 2012 10 / 20

Coset construction for so(4, 2) − → so(4, 1) — Hinterbichler, AJ,

Khoury 1202.6056

Parameterize the conformal algebra by Jµν, Kµ, D and ˆ Pµ ≡ Pµ + 1

4H2Kµ.

Benefit is that (ˆ Pµ, Jµν) generates an so(4,1) algebra

Austin Joyce (UPenn) March 17, 2012 11 / 20

Coset construction for so(4, 2) − → so(4, 1) — Hinterbichler, AJ,

Khoury 1202.6056

Parameterize the conformal algebra by Jµν, Kµ, D and ˆ Pµ ≡ Pµ + 1

4H2Kµ.

Benefit is that (ˆ Pµ, Jµν) generates an so(4,1) algebra The broken generators are D and Kµ, have Goldstones π and ξµ 5 Broken generators, but only 1 independent Goldstone Inverse Higgs constraint = ⇒ ξµ ∼ e−π∂µπ

Austin Joyce (UPenn) March 17, 2012 11 / 20

Coset construction for so(4, 2) − → so(4, 1) — Hinterbichler, AJ,

Khoury 1202.6056

Parameterize the conformal algebra by Jµν, Kµ, D and ˆ Pµ ≡ Pµ + 1

4H2Kµ.

Benefit is that (ˆ Pµ, Jµν) generates an so(4,1) algebra The broken generators are D and Kµ, have Goldstones π and ξµ 5 Broken generators, but only 1 independent Goldstone Inverse Higgs constraint = ⇒ ξµ ∼ e−π∂µπ End result of coset construction (ωP)

a µ (ωP) b ν

ηab = e2π¯ gdS

µν

Dµξν = 1 2∂µπ∂νπ − 1 2 ¯ ∇µ ¯ ∇νπ − 1 4 ¯ gαβ∂απ∂βπ¯ gµν − H2 4 e2π¯ gµν + H2 4 ¯ gµν . The field π transforms as δπ = −ξµ∂µπ − 1

4 ¯

∇µξµ, where ξµ are conformal Killing vectors

Austin Joyce (UPenn) March 17, 2012 11 / 20

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 geff

µν = e2π¯

geff

µν

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, 2012 12 / 20

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 geff

µν = e2π¯

geff

µν

Curvature invariants of this metric give the action for the field π and we can couple spectator fields using ∇[g] Ingredients:

• geff

µν , Rµν, ∇µ

• equivalent to
• geff

µν , Dµξν, ∇µ

• Austin Joyce (UPenn)

March 17, 2012 12 / 20

Goldstone action

The simplest invariant term is just the measure L0 ∼ d4x√−geff ∼ d4x√−¯ geffe4π

Austin Joyce (UPenn) March 17, 2012 13 / 20

Goldstone action

The simplest invariant term is just the measure L0 ∼ d4x√−geff ∼ d4x√−¯ geffe4π The kinetic term is the Ricci scalar L1 ∼ d4x√−geffR ∼ d4x√−¯ geff 1 2e2π(∂π)2 + 1 2e2π ¯ π − H2e2π

• Austin Joyce (UPenn)

March 17, 2012 13 / 20

Goldstone action

The simplest invariant term is just the measure L0 ∼ d4x√−geff ∼ d4x√−¯ geffe4π The kinetic term is the Ricci scalar L1 ∼ d4x√−geffR ∼ d4x√−¯ geff 1 2e2π(∂π)2 + 1 2e2π ¯ π − H2e2π

• At four derivative order, R2 and R2

µν terms are degenerate

L2 ∼ d4x√−geffR2 ∼ d4x√−¯ geff

• (¯

π)2+2¯ π(∂π)2+(∂π)4−4H2(∂π)2 However an orthogonal term can be constructed as a Wess–Zumino term — Goon, Hinterbichler, AJ, Trodden 1203.3191 Lwz ∼ d4x√−¯ geff

• (∂π)4 + 2¯

π(∂π)2 + 6H2(∂π)2 .

Austin Joyce (UPenn) March 17, 2012 13 / 20

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(χ3) and O(∂2)

Austin Joyce (UPenn) March 17, 2012 14 / 20

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(χ3) and O(∂2) S =

• d4x√−¯

geff

• M2
• − 1

2e2π(∂π)2 − H2e2π + H2 2 e4π

• +

− M2

χ

2 e2π(∂χ)2 + m2

χ

2 e4πχ2 + λχe4πχ3+ + ¯ M2 1 2e2π(∂π)2 + 1 2e2π ¯ π − H2e2π + H2 2 e4π

• (χ2 + αχ3)
• .

Austin Joyce (UPenn) March 17, 2012 14 / 20

Analysis of the general effective action — π

The quadratic action for π that derives from this action is Sπ = M2

• d4x√−¯

geff

• −1

2(∂π)2 + 2H2π2

• .

It is convenient to work with the flat slicing ds2

eff = 1 H2t2

• −dt2 + d

x2 . The π action takes the form Sπ = M2

• d4x
• 1

2H2t2 ˙ π2 − 1 2H2t2 ( ∇π)2 + 2 H2t4 π2

• .

Define the canonically-normalized variable, v =

M0 (−Ht)π, which satisfies

¨ vk +

• k2 − 6

t2

• vk = 0 .

Austin Joyce (UPenn) March 17, 2012 15 / 20

Analysis of the general effective action — π

The quadratic action for π that derives from this action is Sπ = M2

• d4x√−¯

geff

• −1

2(∂π)2 + 2H2π2

• .

It is convenient to work with the flat slicing ds2

eff = 1 H2t2

• −dt2 + d

x2 . The π action takes the form Sπ = M2

• d4x
• 1

2H2t2 ˙ π2 − 1 2H2t2 ( ∇π)2 + 2 H2t4 π2

• .

Define the canonically-normalized variable, v =

M0 (−Ht)π, which satisfies

¨ vk +

• k2 − 6

t2

• vk = 0 .

Therefore, πk = H(−t)3/2 4π √ 2M0 H(1)

5/2(−kt) =

⇒ Pπ ∼ 9H2 32π5M2 1 (−kt)2 red

Austin Joyce (UPenn) March 17, 2012 15 / 20

Analysis of the general effective action — χ

At quadratic order in χ, the action gives Sχ =

• d4x√−¯

geff

• −M2

χ

2 (∂χ)2 − m2

χ + ¯

M2

0H2

2 χ2

• ,

Action for a scalar on dS. If |m2

χ/(M2 χH2)|, | ¯

M2

0/M2 χ| ≪ 1 the field χ will have a scale-invariant

spectrum of perturbations

Austin Joyce (UPenn) March 17, 2012 16 / 20

Analysis of the general effective action — χ

At quadratic order in χ, the action gives Sχ =

• d4x√−¯

geff

• −M2

χ

2 (∂χ)2 − m2

χ + ¯

M2

0H2

2 χ2

• ,

Action for a scalar on dS. If |m2

χ/(M2 χH2)|, | ¯

M2

0/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 ˆ χ =

Mχ (−Ht)χ is

ˆ χk = 1 √ 2k

• 1 − i

kt

• e−ikt ,

This implies that the long-wavelength power spectrum for χk is scale invariant Pχ = 1 2π2 k3|χk|2 ∼ H2 (2π)2M2

χ

.

Austin Joyce (UPenn) March 17, 2012 16 / 20

3-point function for χ

The cubic action for χ is (working in the exact scale-invariant limit)

• d4x√−¯

geff

• −M2

χ

2 (∂χ)2 − M2

χπ(∂χ)2 + λχχ3

• .

From this, we can compute the χχχ correlator χk1χk2χk3 = λχH2 2M6

χ

(2π)3δ(3)( k1 + k2 + k3) 1

• i k3

i

×  k1k2k3 −

• i=j

k2

i kj −

• i

k3

i (1 − γ − log ktt∗)

  . 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 1104.2846;

Creminelli 1108.0874

Austin Joyce (UPenn) March 17, 2012 17 / 20

Constraints from non-linearly realized conformal symmetry

Linearly-realized de Sitter symmetry acts as the conformal group on R3. For example lim

t→0 ϕ1(

x1, t)ϕ2( x2, t)ϕ3( x3, t) = C123 x∆1+∆2−∆3

12

x∆2+∆3−∆1

23

x∆1+∆3−∆2

13

.

Austin Joyce (UPenn) March 17, 2012 18 / 20

Constraints from non-linearly realized conformal symmetry

Linearly-realized de Sitter symmetry acts as the conformal group on R3. For example lim

t→0 ϕ1(

x1, t)ϕ2( x2, t)ϕ3( x3, t) = C123 x∆1+∆2−∆3

12

x∆2+∆3−∆1

23

x∆1+∆3−∆2

13

. However, these theories are additionally constrained—consider Sχ =

• d4x√−¯

geff

• −1

2e2π(∂χ)2

• =
• d4x√−¯

geff

• −1

2(∂χ)2 − π(∂χ)2 − 2π2(∂χ)2 + . . .

• Conformal symmetry fixes the relative coefficients between the

terms—should lead to particular relations between n-point functions

Austin Joyce (UPenn) March 17, 2012 18 / 20

DBI — non-linearly realized so(4, 2) from the start —

Hinterbichler, Khoury, Miller, to appear

Imagine a flat brane probing an AdS5 bulk — lowest order world-volume action LDBI = −φ4

• 1 − (∂φ)2

φ4 +

• 1 + λ

4

• φ4

Austin Joyce (UPenn) March 17, 2012 19 / 20

DBI — non-linearly realized so(4, 2) from the start —

Hinterbichler, Khoury, Miller, to appear

Imagine a flat brane probing an AdS5 bulk — lowest order world-volume action LDBI = −φ4

• 1 − (∂φ)2

φ4 +

• 1 + λ

4

• φ4

The so(4, 2) isometries act non-linearly δDφ = −(1 + xµ∂µ)φ; δKµ =

• −2xµ − 2xµxν∂ν + x2∂µ −

1 2φ2 ∂µ

• φ

Austin Joyce (UPenn) March 17, 2012 19 / 20

DBI — non-linearly realized so(4, 2) from the start —

Hinterbichler, Khoury, Miller, to appear

Imagine a flat brane probing an AdS5 bulk — lowest order world-volume action LDBI = −φ4

• 1 − (∂φ)2

φ4 +

• 1 + λ

4

• φ4

The so(4, 2) isometries act non-linearly δDφ = −(1 + xµ∂µ)φ; δKµ =

• −2xµ − 2xµxν∂ν + x2∂µ −

1 2φ2 ∂µ

• φ

Equations of motion are more intricate, but still allow φ ∼ 1/t: ¯ φ(t) = 1 + λ/4

• 1 + λ/8
• 2

λ 1 (−t)

Austin Joyce (UPenn) March 17, 2012 19 / 20

DBI — non-linearly realized so(4, 2) from the start —

Hinterbichler, Khoury, Miller, to appear

Imagine a flat brane probing an AdS5 bulk — lowest order world-volume action LDBI = −φ4

• 1 − (∂φ)2

φ4 +

• 1 + λ

4

• φ4

The so(4, 2) isometries act non-linearly δDφ = −(1 + xµ∂µ)φ; δKµ =

• −2xµ − 2xµxν∂ν + x2∂µ −

1 2φ2 ∂µ

• φ

Equations of motion are more intricate, but still allow φ ∼ 1/t: ¯ φ(t) = 1 + λ/4

• 1 + λ/8
• 2

λ 1 (−t) Possible to couple massless spectator field by instead considering a brane probing AdS5 × S1 — Possible to extend to warped-throat type compactifications?

Austin Joyce (UPenn) March 17, 2012 19 / 20

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, 2012 20 / 20

Cosmology — Jordan frame

Consider the effective metric geff

µν = φ2ηµν, in Jordan frame, the action

takes the form S =

• d4x√−geff

M2

Pl

2φ2 Reff + 3M2

Pl

φ4 gµν

eff ∂µφ∂νφ + 1

φ4 LCFT

• φ−2geff

µν

• .

With the cosmological ansatz ds2

J = −dt2 J + a2 J(tJ)d

x2, the EOM are 3H2

J ≃ 6HJ

˙ φ φ2 − 3 ˙ φ2 φ4 , ¨ φ φ3 + 3HJ ˙ φ φ2 − 3 ˙ φ2 φ4 − Reff 6 = − β 4φ2M2

Plt4 ,

At early times

β 4φ2M2

Plt4 ∼ 0 and the equations admit a solution

φ ∼ 1 t , HJ = constant However, this is not inflation in any normal sense — Meff

Pl ∼ 1/φ varies by

O(1) in a Hubble time.

Austin Joyce (UPenn) March 17, 2012 20 / 20