Based on T. Qiu, “Reconstruction of a Nonminimal Coupling Theory with Scale-invariant Power Spectrum”,

JCAP 1206 (2012) 041 T. Qiu, “Reconstruction of f(R) Theory with Scale-invariant Power Spectrum”, arXiv: 1208.4759

Taotao Qiu LeCosPA Center, National Taiwan

University2012-09-10

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In order to form structures of our universe that can be observed today.

Power spectrum: With spectral index:

Observationally, nearly scale-invariant power spectrum ( ) is favored by data!D. Larson et al. [WMAP collaboration], arXiv:1001.4635 [astro-ph.CO].

Variables for testing perturbations:

Others: bispectrum, trispectrum, gravitational waves, etc.

Why perturbations?

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In GR+single scalar field, there are two ways to get scale-invariant power spectrum:• De Sitter expansion with w=-1 (applied in inflation scenarios)• Matter-like contraction with w=0 (applied in bouncing scenarios)

Proof: see my paper JCAP 1206 (2012) 041 (1204.0189)

However, there are large possibility that GR might be modified!e.g. F(R), F(G), scalar-tensor theory, massive gravity,…

Question: How can these theories generate scale-invariant power spectrum?

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Focus: scalar tensor theory with lagrangian:

Note: First nonminimal coupling model

Brans-Dicke model

Two approaches:Direct calculation from the original action: difficulty & complicated due to the coupling to gravity

Making use of the conformal equivalence

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Lagrangian:

can be transformed to Einstein frame of

through the transformation:

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where so that

The perturbations in two frames obey the same equations, so the nonminimal coupling theory can generate scale-invariant power spectrum as long as its Einstein frame form can generate power spectrum (which is inflation or matter-like contraction). 6

Perturbations:Jordan frame Einstein frame

Equation of motion for curvature perturbationThe variables defined as:

Equation of motion for tensor perturbation

The variables defined as:

Assume the action of the Einstein frame of our model with the form:

have inflationary solution as

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where

By assuming

Lagrangian:

we can have:

8Main result (I)

The numerical result:

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Conclusions: 1) the universe expands when or while

contracts when 2) some critical points:The value of f_IThe value of w_J

The physical meaning

slow expansion/ contraction

trivial inflationdivision of accelerated/ decelerated expansion

Lagrangian:

where and are constants.

Examples: 1)

2)

working as inflation

working as slow-expansion

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Assume

After some manipulations, we get:

Main result (II)

Assume the action in the Einstein frame of our model with the form:

have the matter-like contractive solution as

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Lagrangian:

Following the same procedure, we have:

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with

Main result (I)

The numerical results:

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The value of f_MThe value of w_J

The physical meaning

slow expansion/ contraction

trivial inflationdivision of accelerated/ decelerated expansion

Conclusions: 1) the universe expands when or while contracts when

2) some critical points:

Lagrangian:

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where and are constants.

Assume

Examples: 1)

2)

working as inflation

working as slow-expansion/contraction depending on sign of

After some manipulations, we get:

with

Main result (II)

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Reconstructed from inflation: Reconstructed from matter-like contraction:

in both cases: either contraction with w>-1/3 ( ) orexpansion with w<-1/3 ( )

A condition for avoidance of conceptual problems such as horizon, etc is to have the universe expand with w<-1/3 (including inflation) or contract with w>-1/3 (including matter-like contraction) (proof omitted)

Avoiding horizon problem!!!

Observations suggest scale-invariant power spectrum.• In GR case: (generally) inflation or matter-like contraction.• In Modified Gravity case: possibility could be enlarged.For general nonminimal coupling theory, we can

construct models with scale-invariant power spectrum making use of conformal equivalence.

PROPERTIES:PROPERTIES:• The behavior of the universe is more free• Models reconstructed from both inflation and matter-like

contraction allow contracting and expanding phases, respectively.

• One can have more fruitful forms of field theory models.Models are constrainted to be free of theoretical

problems (due to the conformal equivalence).

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