Fading Gravity and Self-InflationFading Gravity and Self-Inflation

Justin Khoury Justin Khoury (Perimeter Institute)(Perimeter Institute)

hep-th/0610???hep-th/0610???

Motivation: Motivation: Widely studied modifications of gravity, e.g. largeWidely studied modifications of gravity, e.g. largeextra dimensions, typically lead to extra dimensions, typically lead to stronger gravitystronger gravity at short distances. at short distances.

1. Deviation from inverse-square law:1. Deviation from inverse-square law:

€

V (r) =GN M

r→

GN RD−4 M

rD−3

2 ways to see this:2 ways to see this:

2. Modified Friedmann eqn in brane-world models:2. Modified Friedmann eqn in brane-world models:

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3H 2 = 8πGN ρ 1+ρ

2σ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

σ ≡brane tension

Need Need faster expansionfaster expansion at fixed density in order for universe to at fixed density in order for universe to expand forever (since expand forever (since k=0k=0). ).

• Can you still have inflation? Can you still have inflation?

• Do you need inflation? Do you need inflation?

• Does it offer alternatives to the inflationary paradigm?Does it offer alternatives to the inflationary paradigm?

BUT: What if gravity gets weaker (and shuts off) at short BUT: What if gravity gets weaker (and shuts off) at short distances? distances?

Genuinely different than scalar-driven inflation Genuinely different than scalar-driven inflation

Our idea: Our idea: Shut off the graviton propagator at large momentumShut off the graviton propagator at large momentum

€

1

∇ 2→

F(∇ 2L2)

∇ 2

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F(∇ 2L2) →1 for ∇ 2L2 → 0

→ 0 for ∇ 2L2 → ∞

• Find Find novel inflationary solutionsnovel inflationary solutions without scalar field or other stress energy without scalar field or other stress energy • Not equivalentNot equivalent to a scalar-tensor theory to a scalar-tensor theory

wherewhere

Non-perturbative effects in string theory (Tseytlin ‘95)Non-perturbative effects in string theory (Tseytlin ‘95)Tachyon action in open string field theoryTachyon action in open string field theoryp-adic string theory (Brekke, Freund, Olson, Witten ‘88)p-adic string theory (Brekke, Freund, Olson, Witten ‘88)Fat Graviton scenario (Sundrum ‘97)Fat Graviton scenario (Sundrum ‘97)

Form factor must be Form factor must be analyticanalytic, so no new degrees of freedom, so no new degrees of freedom

e.g., e.g.,

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F(∇ 2L2) = exp(−∇4L4 )

Modified (horrific) Einstein’s equations:Modified (horrific) Einstein’s equations:

1. Pure de Sitter doesn’t care1. Pure de Sitter doesn’t care

2 key observations:2 key observations:

necessary to maintain Bianchi identitynecessary to maintain Bianchi identity

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F−1(∇ 2L2)Gαβ + O(R2) = Tαβ

Effective Newton’s constantEffective Newton’s constant

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RαβdS = λgαβ

⇒

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∇2RαβdS = 0 (since )(since )

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∇2gαβ ≡ 0

⇒

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F−1(∇ 2L2)RαβdS = F−1(0)Rαβ

dS = RαβdS

Thus pure dS is Thus pure dS is obliviousoblivious to form factor to form factor

2. But small deviations from dS care a lot:2. But small deviations from dS care a lot:

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Rαβ = 3H 2gαβ + rαβ

⇒

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F−1(∇ 2L2)rαβ ≈ F−1(H 2L2)rαβ >> rαβ

of order dH/dtof order dH/dt

Modified cosmological eqn roughly of the form:Modified cosmological eqn roughly of the form:

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3H 2 + 8F−1(8H 2L2) ˙ H ≈ 0

i.e., i.e.,

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˙ H

H 2≈ −

8

3F(8H 2L2) = −

8

3exp(−64H 4L4 )

Inflating solution if Inflating solution if HL > 1HL > 1

for HL > 1for HL > 1

More explicit sketch:More explicit sketch:

• Relativistic and covariant action is:• Relativistic and covariant action is:

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S = d4 x −g R + Gαβ F−1(∇ 2L2) −1

∇ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Rαβ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

∫

• Quick check: in weak-field limit, have etc.• Quick check: in weak-field limit, have etc.

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Rαβ = −1

2∇ 2hαβ +K

⇒

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S = d4 x −g1

4hαβ −

1

2η αβ h

⎛

⎝ ⎜

⎞

⎠ ⎟∫ F−1 ∇ 2L2

( )∇2hαβ +K

which is action for massless spin-2 with modified propagatorwhich is action for massless spin-2 with modified propagator

Note: as F theory reduces to GR Note: as F theory reduces to GR

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F−1(∇ 2L2)Gαβ = Iαβ(1) + ds Iαβ

(2) + Iαβ(3)

( ) −gαβ

2Rγδ

0

1

∫ F−1(∇ 2L2) −1

∇ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Rγδ

+ Rαβ −gαβ

4R

⎛

⎝ ⎜

⎞

⎠ ⎟F−1(∇ 2L2) −1

∇ 2

⎛

⎝ ⎜

⎞

⎠ ⎟R + 2δ(α

γδβ )δ − gαβ gγδ

( )∇ρ∇γ

F−1(∇ 2L2) −1

∇ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Gρδ

A fun weekend in Waterloo….A fun weekend in Waterloo….

wherewhere

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Iαβ(i) ≡ −∇α Bγδ∇ β Aγδ +

gαβ

2∇κ Bγδ∇κ Aγδ − Bα

γ∇ 2Aβγ +gαβ

2Bγδ∇ 2Aγδ − Bγδ∇γ∇α Aβδ

+ Aγδ∇2Bβ

γ −∇γ Bγδ∇α Aβδ + Aδα∇γ∇ β Bγ

δ +∇α Bγδ∇γ Aβδ + Bαγ∇δ∇ β Aγδ

+∇γ Bαδ∇ β Aγδ − Aγδ∇

γ∇α Bβδ −∇α Bβ

γ∇δ Aγδ

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Aαβ(1) ≡

F−1 −1

∇ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Rαβ ; Bαβ

(1) ≡1

∇ 2Gαβ

Aαβ(2) ≡ L4∇ 2F s−1Rαβ ; Bαβ

(2) ≡F−s

∇ 2Gαβ

Aαβ(3) ≡ L2F sRαβ ; Bαβ

(3) ≡ F−sGαβ

withwith

Another fun weekend in Waterloo…Another fun weekend in Waterloo…

• Substitute FRW ansatz keeping terms at most of order dH/dt (i.e. ignoring d2H/dt2, (dH/dt)2, …). • Substitute FRW ansatz keeping terms at most of order dH/dt (i.e. ignoring d2H/dt2, (dH/dt)2, …).

• Equations simplify tremendously and find:• Equations simplify tremendously and find:

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˙ H

H 2≈ −

8

3F(8H 2L2)

Exiting self-inflating phase:Exiting self-inflating phase:

• Since dH/dt < 0, eventually reach F and inflation ends.• Since dH/dt < 0, eventually reach F and inflation ends.

• Meanwhile, when dH/dt ~ H2 particle production is efficient. This is the dominant reheating mechanism here.

• Meanwhile, when dH/dt ~ H2 particle production is efficient. This is the dominant reheating mechanism here.

• Moreover, recall that as F theory reduces to GR• Moreover, recall that as F theory reduces to GR

Ford (‘87)Ford (‘87)Grishchuk & Sidorov (‘90) Grishchuk & Sidorov (‘90) Spokoiny (‘93)Spokoiny (‘93)

Conjecture: graceful exit into normal radiation-dominated phase at T~L-1

Conjecture: graceful exit into normal radiation-dominated phase at T~L-1

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Treheat ~ Hend ~ L−1

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˙ H

H 2≈ −

8

3F(8H 2L2)

Perturbations: Perturbations: Can study evolution of fluctuations about self-Can study evolution of fluctuations about self-inflating solution.inflating solution.

• Density (scalar) perturbations are nearly scale-inv, with amplitude• Density (scalar) perturbations are nearly scale-inv, with amplitude

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δρρ

~H

εand spectral tiltand spectral tilt

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ε ≡−˙ H /H 2wherewhere

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n s≈1+ 8ε ln3ε

8

⎛

⎝ ⎜

⎞

⎠ ⎟≈ 0.96

• BUT tensor spectrum is very blue• BUT tensor spectrum is very blue

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nT ≈10 /3

Amplitude of gravity waves too small to be observed, like ekpyrotic.Amplitude of gravity waves too small to be observed, like ekpyrotic.

Robustness of effective actionRobustness of effective action

• Slow-evolution approximation gets better as HL • Slow-evolution approximation gets better as HL

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˙ H

H 2≈ −

8

3F(8H 2L2) → 0

• Generic corrections to effective action are less and less relevant as HL • Generic corrections to effective action are less and less relevant as HL

e.g.e.g.

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Rαβ∇α∇ β R ~ H 4 ˙ H

negligible compared to from form factor

negligible compared to from form factor

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exp(64H 4L4 ) ˙ H ⇒e.g.e.g.

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R2L2 ~ H 4L2

Effectively “renormalizes” form factor Effectively “renormalizes” form factor ⇒

Summary and future avenuesSummary and future avenues

• Presented a theory where gravity shuts off at short distances• Presented a theory where gravity shuts off at short distances

• To my knowledge, theory can’t be rewritten as some scalar-tensor theory• To my knowledge, theory can’t be rewritten as some scalar-tensor theory

• Find novel inflationary solutions to the vacuum eqns.• Find novel inflationary solutions to the vacuum eqns.

• Density perturbations basically degenerate with scalar-driven inf. • Density perturbations basically degenerate with scalar-driven inf.

• BUT gravity waves have blue tilt distinguishing feature from scalar infn.

• BUT gravity waves have blue tilt distinguishing feature from scalar infn.

⇒