1. Cosmology in Eddington- inspired Born-Infeld gravity Hyeong-Chan Kim Korea National University of Transportation 19 Feb 2013 The Ocean Suites Jeju, Asia Pacific School/Workshop on Gravitation and Cosmology

2. Hubble’s Law(1927): big bang! Flatness, horizon, monopole problems Inflation…. Fine-tuning problem, Entropy problem.. These are the question for the initial state of the Universe. Oldest question Where are we from? Where all in the Universe come from? However, in Einstein’s General Relativity, it is impossible to answer this question because GR predicts singularity at t=0.

3. Eq. of state Non-singular origin? Eternal inflation with no origin at the beginning. Quantum cosmological origin (no-boundary proposal) Bouncing or Circulating universes Etc. Proposals: Quantum gravitational effect may modify the gravity interaction at high energy so that the singularity can be controllable.  Modified gravity. For example, in the case of f(R) gravity, one can get power-law inflating initial state. (need to fix the form of action) Too many ways of generalizations. The Universe still contains initial singularity. However, inflation requires extremely special initial condition. Carroll,Chen;05

4. EiBI gravity ( Banados, Ferreira; 2010 ) Eddington-inspired Born-Infeld gravity One additional parameter denotes the determinant of the metric is a dimensionless constant related with the cosmological const is dependent only on the connection. The connection is treated as independent field. The matter field couples only with the metric For vacuum, it is the same as GR. Exist non-singular initial state for radiation filled Universe for positive ; bouncing universe for negative.

5. Progress Singularity free solutions for stars composed of dust, polytropic fluids ( Pani,Cardoso, Delsate, 2011 ) Cosmological and astrophysical constraints are satisfied( Felice, Gumjudpai,Jhingan;Avelino, 2012 ) Tensor perturbations ( Escamilla-Rivera,Banados,Ferreira, 2012 ) 5d brane model ( Liu, Yang, Guo, Zhong, 2012 ) Effective stress tensor ( Delsate, Steinhoff, 2012 ) | |<3 x 10 5 m 5 s -2 /kg ( Casanellas,Pani,Lopes,Cardoso,2011) Non singular initial state for perfect fluid with positive equation of state(EoS; w>0 ). (Cho,K,Moon 2012) de Sitter state for w=0. Anisotropic universe with perfect fluid Surface singularity for compact star (Pani, Sotiriou, 2012)

6. Why non-singular? From the equation for the Hubble parameter, the reason is quite obvious: –There exist a maximum value of the energy density! This term gives and the maximum energy density, The scale factor around there behaves as Minimal scale factor (Cho,K,Moon;2012)

7. Fundamental Question We have found that the universe filled with perfect fluid with positive equation of state starts from a regular initial state. Is this right even for the case of a realistic field?

8. Cosmology with a scalar field, Consider a scalar field in a Robertson-Walker spacetime with metric Matter action: Equation of motions: Hubble parameter: This part remains Even if this part vanishes, (Cho,K,Moon, to appear soon)

9. Maximal pressure condition The maximal pressure condition for a given potential: The scalar evolution equation determine the Hubble parameter: The Hubble parameter equation is trivially satisfied. Consider upper sign only. (time reversal)  The scalar field is non-decreasing. Expanding universe exists for

10. Exact solution: For positive For negative Bounce back to contract. Monotonically increases. Positive definite always.Presents an expanding sol if time reversed. Initially de Sitter spacetime

11. Solution for positive.

12. Perturbations Introduce linear perturbations: It reproduces the unperturbed solution when The scalar evolution eqThe Hubble parameter The linear perturbations grows exponentially: In other word, it goes to zero as we go past.

13. Non-linear stability I Evolve backward and forward in time numerically by introducing arbitrary initial data at later times when the scalar field goes to zero: Initially, approach to the exact sol. Later, it goes to the oscillatory phase (inflaton decay) Any small deviation from the exact solution leads to this phase.

14. Non-linear stability II Evolution of (H, ) Fine tuning problem? NO. Initial state: Inflaton decay

15. Fine tuning? If the MPS is not satisfied, for large. Then, the Hubble parameter is much bigger than that of the exact solution. Q: Is it possible the initial state of the Universe has other initial state? Scalar evol. Eq: Solution: Observing backward in time, continually increases. Therefore, soon, the value of reaches its maximum: Around this point, the Hubble parameter suddenly drops to its MPS value, and the Universe starts from the MPS. The yellow curve denotes this behavior.

16. Summary and Discussions We have found an exact solution of the Universe filled with a free scalar field in EiBI gravity. The solution is shown to describe a regular initial state of the Universe. The initial state of the Universe is nothing but the de Sitter state. Further questions: What happens for other fields and potentials? Density perturbations? Anisotropic universe? Quantum corrections?

17. Thank you for Listening!