Inflation coupled to the GB correction

Zong-Kuan Guo

Hangzhou workshop on gravitation and cosmology

Sep 4, 2014

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].

outline

• motivations• inflation coupled to the Gauss-Bonnet term– power-law inflation– slow-roll inflation– predictions and the Planck data

• conclusions

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.

• 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

inflation coupled to the GB term

matGBEH SSSSS

RgxdSEH 4

21

24 )(21

GBGB RgxdS

)()(

224

VgxdS

our action:

22 4 RRRRRRGB

here the GB term is defined as

background equations in a spatially-flat FRW Universe:

.0)(12)3(

),2(442

,2426

22,,

222

322

HHHVH

HHHHH

HVH

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,

312 HV

|| > 1: a potential-dominated model,|| < 1: a GB-dominated model.

scalar perturbation equations in Fourier space:

,022

s

s

sss u

zzkcu

,6)(281

,)1()6(

32

222

2221

3222

HHHHHc

HHaz

s

s

with ).41/(4 HH

the power spectrum of scalar perturbations

2

2

3

2 s

ss z

ukP

assuming sss Qz 2/1)(

The general solution is a linear combination of Hankel functions

.)()(2)2(

2)1(

14/)21(

kcHckcHceu ss

is ss

s

Choose and , so that kcekc

u skic

ss

s

21

for long wavelength perturbations

12)2/3(

)(4

)( 2/14/)12(

kckceu s

ssis

ssss

.23lnln1

,2)2/3(

)(4

232

22

2

ss

s

s

s

ss

kdPdn

kQ

cPss

the power spectrum

as

as

tensor perturbation equations in Fourier space:

,022

t

t

ttt u

zzkcu

.41

)(41

),41(

2

22

HHc

Haz

t

t

the power spectrum of tensor perturbations

,2)2/3(

)(482

22 232

22

22

2

3 tt kQ

czukP t

t

t

t

tt

the tensor-to-scalar ratio

.23lnln

tt

t kdPdn

.2)(

)(8222

2

2

2

2 ts

t

s kcc

QQ

PPr

s

t

t

s

t

s

s

t

0,0,4/1 22 ts cctts nrcc 8,1,2 22

.)6(12)4)(22(16

,133

,4)1(41

,1331

,)4](12)4)(22[()5(161

2

2

2

22

2

22

222

s

t

s

t

t

s

s

ccr

n

c

n

c

power-law inflation

.,,)( 2/1

tVttta

power-law inflation:

power spectra

Two limiting cases:

If =0,

If =0,

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: 44 10410

.22/2

,exp)(,exp2

)(

2

V

slow-roll inflation

introducing Hubble and GB flow parameters:

.1,lnln

,4,lnln

, 11121 iad

dH

add

HH i

ii

i

the slow-roll approximation: and

The kinetic, potential and coupling can be written in terms of the flow parameters

).(

,)5(26

,)1(2

21141

221112

1

22111

2

HV

H

Is the slow-roll solution an attractor under the slow-roll condition?

Suppose and is the slow-roll solution where .

Considering a linear homogeneous perturbation and , one has

𝛿𝐻 ,𝜙=− 3𝐻Π [1+

𝛿1𝜀1

2 𝜀1−𝛿1+𝒪(𝛿1𝜀1 , 𝛿1 𝛿2)]𝛿𝐻

𝛿Π ,𝜙=− 3𝐻Π [1+

2 𝜀1𝜀2−8𝜀1𝛿1− 𝛿1 𝛿2−8 𝛿12

6(2𝜀1−𝛿1)+𝒪(𝛿1𝜀1 ,𝛿1𝛿2)]𝛿Π

𝐻2=𝑉3 ,Π=− 1

3𝜔 𝐻 (𝑉 ,𝜙+12 𝜉 ,𝜙Π 4)

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

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

.82

,28

,2221

1

11

11

21211

rn

r

n

t

s

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.

2/~)/ln(~ 1st ccN

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:

.4

)1(16

,4

)2(21

nNnr

nNnns

13/4 00 V

𝑉 (𝜙 )=𝑉 0𝜙𝑛 ,𝜉 (𝜙 )=𝜉0 𝜙−𝑛

chaotic inflation with a dilaton-like coupling

𝑉 (𝜙 )=𝑉 0𝜙𝑛 ,𝜉 (𝜙 )=𝜉0𝑒−𝜆𝜙 .the spectral index and the tensor-to-scalar ratio

𝑛𝑠−1=−𝑛 (𝑛+2 )+𝛼𝜆𝑒−𝜆𝜙 𝜙𝑛+1 (2 𝜆𝜙−𝑛)

𝜙2

𝑟=8(𝑛−𝛼𝜆𝑒−𝜆𝜙𝜙𝑛+1)2

𝜙2𝑛=2 ,𝑁=60

There exist parameter regions in which the predictions are consistent with the Planck data.

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.

Thanks for your attention!