1. Inflation coupled to the GB correction Zong-Kuan Guo Hangzhou workshop on gravitation and cosmology Sep 4, 2014

2. Based on collaboration with J.W. Hu, P.X. Jiang, N. Ohta, D.J. Schwarz and S. Tsujikawa Phys. Rev. D 75 (2007) 023520 [hep-th/0610336], Phys. Rev. D 80 (2009) 063523 [arXiv:0907.0427], Phys. Rev. D 81 (2010) 123520 [arXiv:1001.1897], Phys. Rev. D 88 (2013) 123508 [arXiv:1310.5579].

3. outline motivations inflation coupled to the Gauss-Bonnet term – power-law inflation – slow-roll inflation – predictions and the Planck data conclusions

4. motivations inflation scenario – Some cosmological puzzles, such as the horizon problem, flatness problem and relic density problem, can be explained in the inflation scenario. – The most important property of inflation is that it can generate irregularities in the Universe, which may lead to the formation of structure and CMB anisotropies. – So far the nature of inflation has been an open question. higher order corrections – It is known that there are correction terms of higher orders in the curvature to the lowest effective supergravity action coming from superstrings. The simplest correction is the Gauss-Bonnet (GB) term.

5. questions – Does the GB term drive acceleration of the Universe? – If so, is it possible to generate nearly scale-invariant curvature and tensor perturbations? – If not, when the GB term is sub-dominated, what is the influence on the power spectra? – How strong CMB data constrain the GB coupling? three steps – the simplest case: power-law inflation – to generalize it to slow-roll inflation – to confront the specific models with observational data

6. inflation coupled to the GB term our action: here the GB term is defined as

7. background equations in a spatially-flat FRW Universe: To compare the contributions from the potential and the GB term, we use the ratio of the second to the third term on the right-hand side of the Friedmann equation, | | > 1: a potential-dominated model, | | < 1: a GB-dominated model.

8. scalar perturbation equations in Fourier space: with the power spectrum of scalar perturbations

9. assuming The general solution is a linear combination of Hankel functions for long wavelength perturbations the power spectrum as

10. tensor perturbation equations in Fourier space: the power spectrum of tensor perturbations the tensor-to-scalar ratio

11. power-law inflation power-law inflation: power spectra Two limiting cases:  If  =0,  If  =0,

12. ① an exponential potential and an exponential GB coupling ② In the GB-dominated case, ultra-violet instabilities of either scalar or tensor perturbations show up on small scales. ③ In the potential-dominated case, the GB correction with a positive (or negative) coupling may lead to a reduction (or enhancement) of the tensor-to-scalar ratio. ④ WMAP5 constraints on the GB coupling:

13. slow-roll inflation introducing Hubble and GB flow parameters:

14. All linear perturbations die away exponentially fast as the number of e-folds increases.

15. to first order in the slow-roll approximation a)The scalar spectral index contains not only the Hubble but also GB flow parameters. b)The degeneracy of standard consistency relation is broken. c)the horizon-crossing time Assuming that time derivatives of the flow parameters can be neglected during slow-roll inflation, we get the power spectra of scalar and tensor perturbations.

16. predictions and the Planck data  chaotic inflation with an inverse power-law coupling Defining in the case, the spectral index and the tensor-to-scalar ratio can be written in terms of the function of N:

17.  chaotic inflation with a dilaton-like coupling the spectral index and the tensor-to-scalar ratio There exist parameter regions in which the predictions are consistent with the Planck data.

18. conclusions ① For GB-dominated inflation ultra-violet instabilities of either scalar or tensor perturbations show up on small scales. ② The GB term with a positive (or negative) coupling may lead to a reduction (or enhancement) of the tensor-to- scalar ratio in the potential-dominated case. ③ The standard consistency relation does not hold because of the GB coupling. ④ If the tensor spectral index is allowed to vary freely, the Planck constraints on the tensor-to-scalar ratio are slightly improved. ⑤ The quadratic potential is consistent with Planck data.

19. Thanks for your attention!