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
T. Qiu, “Reconstruction of f(R) Theory with Scale-invariant Power Spectrum”, arXiv: 1208.4759
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Why perturbations?
In order to form structures of our
universe that can be observed today.
Variables for testing perturbations:
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].
Others: bispectrum, trispectrum, gravitational waves, etc.
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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 scaleinvariant 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:
where
so that
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Perturbations:
Jordan frame
Einstein frame
Equation of motion
for curvature
perturbation
The variables
defined as:
Equation of motion
for tensor
perturbation
The variables
defined as:
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).
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Assume the action of the Einstein frame of our model
with the form:
have inflationary solution as
where
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Lagrangian:
By assuming
we can have:
Main result (I)
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The numerical result:
Conclusions: 1) the universe expands when
contracts when
2) some critical points:
or
The value of f_I
The value of w_J
The physical slow expansion/ division of accelerated/
meaning
contraction
decelerated expansion
while
trivial
inflation
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Lagrangian:
Assume
where
and
are constants.
After some manipulations, we get:
Main result (II)
Examples: 1)
2)
working as inflation
working as slowexpansion
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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:
with
Following the same procedure, we have:
Main result (I)
12
The numerical results:
Conclusions: 1) the universe expands when
contracts when
2) some critical points:
The value of f_M
The value of w_J
The physical slow expansion/ division of accelerated/
meaning
contraction
decelerated expansion
or
while
trivial
inflation
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Lagrangian:
Assume
where
and
are constants.
After some manipulations, we get:
Main result (II)
Examples: 1)
2)
working as inflation
with
working as slow-expansion/contraction depending on sign of
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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)
Reconstructed from inflation:
in both cases:
either
contraction with w>-1/3 (
or
expansion with w<-1/3 (
Reconstructed from matter-like
contraction:
)
)
Avoiding
horizon
problem!!!
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 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:
• 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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