1. Inflation with a Gauss-Bonnet coupling
Zong-Kuan Guo
ACGRG9
Nov 27, 2017

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

3. Outline Introduction Inflation with a Gauss-Bonnet coupling
slow-roll inflation
predictions and Planck data
Conclusions

4. Introduction Inflationary scenario
Some cosmological puzzles, such as the horizon problem, flatness problem and relic density problem, can be explained in the inflationary 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.
𝑃 𝑠 𝑘 = 𝐴 𝑠 𝑘 𝑘 𝑛 𝑠 −1
𝑃 𝑡 𝑘 = 𝐴 𝑡 𝑘 𝑘 𝑛 𝑡

5. CMB constraints on inflationary models
constraints on primordial non-Gaussianity
constraints on isocurvature modes
constraints on 𝑛 𝑠 and 𝑟
(R^2 and Rφ^2 are favored by CMB data)
Planck Collaboration, arXiv:

6. Higher order terms in the curvature
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.
In the weak coupling limit, 𝜉(𝜙)∝ exp (−𝜙)
Supersymmetry breaking or nonperturbative effects may give a non-vanishing potential, 𝑉(𝜙).
𝑅 𝐺𝐵 2 ≡ 𝑅 𝜇𝜈𝜌𝜎 𝑅 𝜇𝜈𝜌𝜎 −4 𝑅 𝜇𝜈 𝑅 𝜇𝜈 + 𝑅 2

7. Questions Does the GB coupling drive acceleration of the Universe?
Yes. (arXiv: )
If so, is it possible to generate nearly scale-invariant curvature and tensor perturbations?
No. (arXiv: )
If not, when the GB coupling is sub-dominated, what is the influence on the power spectra?
Then how strong CMB data constrain the GB coupling?

8. Inflation coupled to a GB term
Our model:
𝑆= 𝑆 𝐸𝐻 + 𝑆 𝜙 + 𝑆 𝐺𝐵
𝑆 𝐸𝐻 = 𝑑 4 𝑥 −𝑔 𝑅
𝑆 𝜙 =− 𝑑 4 𝑥 −𝑔 𝛻𝜙 2 +𝑉(𝜙)
𝑆 𝐺𝐵 =− 𝑑 4 𝑥 −𝑔 𝜉 𝜙 𝑅 𝐺𝐵 2

9. Background equations in a spatially-flat FRW Universe:
6 𝐻 2 = 𝜙 2 +2𝑉+24 𝜉 𝐻 3 ,
2 𝐻 =− 𝜙 𝜉 𝐻 2 +4 𝜉 𝐻 2 𝐻 − 𝐻 2 ,
𝜙 +3𝐻 𝜙 + 𝑉 ,𝜙 +12 𝜉 ,𝜙 𝐻 2 𝐻 + 𝐻 2 =0.

10. Scalar perturbation equations in Fourier space:
𝑢 𝑠 ′′ + 𝑐 𝑠 2 𝑘 2 − 𝑧 𝑠 ′′ 𝑧 𝑠 𝑢 𝑠 =0,
𝑐 𝑠 2 =1+ 8Δ 𝜉 𝐻 𝐻 +2 Δ 2 𝐻 2 𝜉 − 𝜉 𝐻 𝜙 2 +6Δ 𝜉 𝐻 3 ,
𝑧 𝑠 2 = 𝑎 2 𝜙 2 +6Δ 𝜉 𝐻 −Δ/2 2 𝐻 2 ,
Δ= 4 𝜉 𝐻 1−4 𝜉 𝐻 .
The power spectrum of scalar perturbations reads
𝑃 𝑠 = 𝑘 3 2 𝜋 𝑢 𝑠 𝑧 𝑠 2 ,
𝑛 𝑠 −1≡ 𝑑 ln 𝑃 𝑠 𝑑 ln 𝑘

11. For long wavelength perturbations
Assuming 𝑧 𝑠 = 𝑄 𝑠 −𝜏 1/2− 𝜈 𝑠 , the general solution is a linear combination of Hankel functions
𝑢 𝑠 = 𝑒 𝑖 1+2 𝜈 𝑠 𝜋/4 −𝜋𝜏 2 𝑐 1 𝐻 𝜈 𝑠 − 𝑐 𝑠 𝑘𝜏 + 𝑐 2 𝐻 𝜈 𝑠 − 𝑐 𝑠 𝑘𝜏 .
Choose 𝑐 1 =1 and 𝑐 2 =0, so that as
For long wavelength perturbations as
The power spectrum reads
𝑃 𝑠 = 𝑐 𝑠 −2 𝜈 𝑠 4 𝜋 2 𝑄 𝑠 Γ 𝜈 𝑠 Γ 3/ 𝑘 2 3−2 𝜈 𝑠 ,
𝑛 𝑠 −1=3−2 𝜈 𝑠

12. Tensor perturbation equations in Fourier space:
𝑢 𝑡 ′′ + 𝑐 𝑡 2 𝑘 2 − 𝑧 𝑡 ′′ 𝑧 𝑡 𝑢 𝑡 =0,
𝑐 𝑡 2 =1− 4 𝜉 − 𝜉 𝐻 1−4 𝜉 𝐻 ,
𝑧 𝑡 2 = 𝑎 2 1−4 𝜉 𝐻 .
The power spectrum of tensor perturbations reads
The tensor-to-scalar ratio is defined as

13. Slow-roll inflation Introducing Hubble and GB flow parameters:
the slow-roll approximation: 𝜀 𝑖 ≪1 and 𝛿 𝑖 ≪1.
The background equations are approximately given as
𝐻 2 ≈ 1 3 𝑉(𝜙),
𝐻 ≈− 𝜙 2 −2 𝜉 𝐻 3 ,
𝜙 ≈− 1 3𝐻 ( 𝑉 ,𝜙 +12 𝜉 ,𝜙 𝐻 4 ).

14. 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.
To first order in the slow-roll approximation
𝑛 𝑠 −1≈−2 𝜀 1 − 2 𝜀 1 𝜀 2 − 𝛿 1 𝛿 2 2 𝜀 1 − 𝛿 1 ,
𝑟≈8 2 𝜀 1 − 𝛿 1 ,
𝑛 𝑡 ≈−2 𝜀 1 ≠− 𝑟 8 .
The scalar spectral index contains not only the Hubble but also GB flow parameters.
The degeneracy of standard consistency relation is broken.
the horizon-crossing time Δ𝑁~ ln ( 𝑐 𝑡 / 𝑐 𝑠 ) ~ 𝛿 1 /2.
The scalar spectral index contains not only the Hubble but also GB flow parameters.
The degeneracy of standard consistency relation is broken.

15. Predictions and Planck data
Power-law inflation with an exponential coupling
𝑉 𝜙 = 𝑉 0 𝑒 −𝜆𝜙 , 𝜉 𝜙 = 𝜉 0 𝑒 𝜆𝜙
Defining 𝛼≡4 𝑉 0 𝜉 0 /3 , the spectral index and the tensor-to-scalar ratio can be written as:
𝑛 𝑠 −1=− 𝜆 2 (1−𝛼),
𝑟=8 𝜆 2 (1−𝛼) 2

16. Chaotic inflation with an inverse power-law coupling
𝑉 𝜙 = 𝑉 0 𝜙 𝑛 , 𝜉 𝜙 = 𝜉 0 𝜙 −𝑛
The spectral index and the tensor-to-scalar ratio can be written in terms of the function of N:
𝑛 𝑠 −1= 2 𝑛+2 4𝑁+𝑛 ,
𝑟= 16𝑛 1−𝛼 4𝑁+𝑛 .

17. Chaotic inflation with a dilaton-like coupling
𝑉 𝜙 = 𝑉 0 𝜙 𝑛 , 𝜉 𝜙 = 𝜉 0 𝑒 −𝜆𝜙 .
The spectral index and the tensor-to-scalar ratio are
𝑛 𝑠 −1= −𝑛 𝑛+2 +𝛼𝜆 𝑒 −𝜆𝜙 𝜙 𝑛+1 2𝜆𝜙−𝑛 𝜙 2
𝑟= 8 (𝑛−𝛼𝜆 𝑒 −𝜆𝜙 𝜙 𝑛+1 ) 2 𝜙 2
𝑛=2, 𝑁=60
There exist parameter regions in which the predictions are consistent with the Planck data.

18. Conclusions
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.
The standard consistency relation does not hold because of the GB coupling.
If the tensor spectral index is allowed to vary freely, the Planck constraints on the tensor-to-scalar ratio are slightly improved.
The quadratic potential is consistent with Planck data with the help of the GB coupling.

19. Recent development Some models Non-Gaussianity Reheating
arXiv: , arXiv: , arXiv: , arXiv:
Non-Gaussianity
arXiv: , arXiv:
Reheating
arXiv:
Reconstruction of the potential and coupling
arXiv:

20. Thanks for your attention!