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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
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!