1. Gauss-Bonnet inflation 郭宗寛 (Zong-Kuan Guo) ITP, CAS 3rd Joint Retreat on Cosmology and LHC Physics November 2, 2012

2. Based on collaboration with D.J. Schwarz Phys. Rev. D 80 (2009) 063523 [arXiv:0907.0427], Phys. Rev. D 81 (2010) 123520 [arXiv:1001.1897].

3. outline motivations GB inflation and cosmological perturbations – power-law inflation – slow-roll inflation 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 perturbations? – If not, when the GB term is sub-dominated, what is the influence on the power spectra? – How strong WMAP data constrain the GB coupling? two cases – the simplest case: power-law inflation – to generalize it to slow-roll inflation

6. GB inflation and cosmological perturbation 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 Choose c 1 = 1 and c 2 = 0, so that 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 spectra Two limiting cases:   =0    =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 constraints on the GB coupling

13. slow-roll inflation introducing Hubble and GB flow parameters: the slow-roll approximation: |  i |  1 and |  i |  1 The kinetic, potential and coupling can be written in terms of the flow parameters Is the slow-roll solution an attractor under the slow-roll condition? Yes.

14. 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.

15. Consider a specific inflation model: Defining in the case, the spectral index and the tensor-to-scalar ratio can be written in terms of the function of N: n = 2 n = 4 The GB term may revive the quartic potential ruled out by recent cosmological data.

16. 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 quartic potential can be revived by a positive GB coupling. ④ The standard consistency relation does not hold because of the GB coupling.

17. Thanks for your attention!