1. 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 University 2012-09-10 1

2. 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? 2

3. 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? 3

4. 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 4

5. Lagrangian: can be transformed to Einstein frame of through the transformation: 5 where so that

6. 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 frameEinstein frame Equation of motion for curvature perturbation The variables defined as: Equation of motion for tensor perturbation The variables defined as:

7. Assume the action of the Einstein frame of our model with the form: have inflationary solution as 7 where

8. By assuming Lagrangian: we can have: 8 Main result (I)

9. The numerical result: 9 Conclusions: 1) the universe expands when or while contracts when 2) some critical points: The value of f_I The value of w_J The physical meaning slow expansion/ contraction trivial inflation division of accelerated/ decelerated expansion

10. Lagrangian: whereandare constants. Examples: 1) 2) working as inflation working as slow- expansion 10 Assume After some manipulations, we get: Main result (II)

11. Assume the action in the Einstein frame of our model with the form: have the matter-like contractive solution as 11

12. Lagrangian: Following the same procedure, we have: 12 with Main result (I)

13. The numerical results: 13 The value of f_M The value of w_J The physical meaning slow expansion/ contraction trivial inflation division of accelerated/ decelerated expansion Conclusions: 1) the universe expands when or while contracts when 2) some critical points:

14. Lagrangian: 14 whereandare 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)

15. 15 Reconstructed from inflation:Reconstructed from matter-like contraction: in both cases: either contraction with w>-1/3 ( ) or expansion 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 matter-like contraction) (proof omitted) Avoiding horizon problem!!!

16. 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). 16

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