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Reconstruction of a Nonminimal Coupling Theory with Scale-invariant Power Spectrum. Taotao Qiu LeCosPA Center, National Taiwan University 2012-09-10. Based on T. Qiu, “ Reconstruction of a Nonminimal Coupling Theory with Scale-invariant Power Spectrum”, JCAP 1206 (2012) 041 - PowerPoint PPT Presentation
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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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