Reconstruction of a Nonminimal Coupling Theory with Scale-invariant Power Spectrum

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

2. Observations from the early universe: Power Spectrum of primordial perturbations 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.

3. Theoretical Model-building • 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? 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)

4. How can these theories generate scale-invariant power spectrum? 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 Focus: scalar tensor theory with lagrangian:

5. Conformal transformation between Modified Gravity and GR (I) Lagrangian: can be transformed to Einstein frame of through the transformation: so that where

6. Conformal transformation between Modified Gravity and GR (II) 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).

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

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

9. Reconstruction from inflation 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 division of accelerated/ decelerated expansion trivial inflation The numerical result:

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

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

12. Reconstruction from matter-like contraction Lagrangian: with Following the same procedure, we have: Main result (I)

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

14. Reconstruction from matter-like contraction 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

15. Recompare the numerical results 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: Reconstructed from matter-like contraction: in both cases: either contraction with w>-1/3 ( ) or expansion with w<-1/3 ( ) Avoiding horizon problem!!!

16. Summary • 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).

17. Thank you very much!