The detection of B-mode power spectrum by the BICEP2 collaboration constrains
the tensor-to-scalar ratio $r=0.20^{+0.07}_{-0.05}$ for the lensed-$\Lambda$CDM
model. The consistency of this big value with the {\em Planck} results requires
a large running of the spectral index. The large values of the tensor-to-scalar
ratio and the running of the spectral index put a challenge to single field
inflation. For the chaotic inflation, the larger the value of the
tensor-to-scalar ratio is, the smaller the value of the running of the spectral
index is. For the natural inflation, the absolute value of the running of the
spectral index has an upper limit.Comment: 4 figures, 3 pages, some references added, PLB in pres

Gao, Qing

Gong, Yungui

Physics Letters B

English

arXiv

AbstractThe detection of B-mode power spectrum by the BICEP2 collaboration constrains the tensor-to-scalar ratio r=0.20−0.05+0.07 for the lensed-ΛCDM model. The consistency of this big value with the Planck results requires a large running of the spectral index. The large values of the tensor-to-scalar ratio and the running of the spectral index put a challenge to single field inflation. For the chaotic inflation, the larger the value of the tensor-to-scalar ratio is, the smaller the value of the running of the spectral index is. For the natural inflation, the absolute value of the running of the spectral index has an upper limit

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Physics Letters B 734 (2014) 41–43Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The challenge for single ﬁeld inﬂation with BICEP2 result
Qing Gao a, Yungui Gong a,b,∗
a MOE Key Laboratory of Fundamental Quantities Measurement, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, PR China
b Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, PR China
a r t i c l e i n f o a b s t r a c t
Article history:
Received 23 March 2014
Received in revised form 24 April 2014
Accepted 12 May 2014
Available online 15 May 2014
Editor: J. Hisano
The detection of B-mode power spectrum by the BICEP2 collaboration constrains the tensor-to-scalar
ratio r = 0.20+0.07−0.05 for the lensed-CDM model. The consistency of this big value with the Planck
results requires a large running of the spectral index. The large values of the tensor-to-scalar ratio
and the running of the spectral index put a challenge to single ﬁeld inﬂation. For the chaotic inﬂation,
the larger the value of the tensor-to-scalar ratio is, the smaller the value of the running of the spectral
index is. For the natural inﬂation, the absolute value of the running of the spectral index has an upper
limit.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.1. Introduction
The detection of the primordial B-mode power spectrum by the
BICEP2 collaboration conﬁrms the existence of primordial gravi-
tational wave, and the observed B-mode power spectrum gives
the constraint on the tensor-to-scalar ratio with r = 0.20+0.07−0.05 at
1σ level for the lensed-CDM model [1]. Furthermore, r = 0 is
disfavored at 7.0σ level. The new constraints on r and the spec-
tral index ns exclude a wide class of inﬂationary models. For the
inﬂation model with non-minimal coupling with gravity [2], a uni-
versal attractor at strong coupling was found with ns = 1 − 2/N
and r = 12/N2. This model is inconsistent with the BICEP2 re-
sult r  0.1 at 2σ level because the BICEP2 constraint on r re-
quires the number of e-folds N = √12/r  √120 ≈ 11 which is
not enough to solve the horizon problem. If we require N = 50,
then r = 0.0048, so the model is excluded by the BICEP2 result. For
the small-ﬁeld inﬂation like the hilltop inﬂation with the potential
V (φ) = V0[1 − (φ/μ)p] [3,4], r ∼ 0, so the model is excluded by
the BICEP2 result.
Without the running of the spectral index, the combination of
Planck + WP + highL data gives ns = 0.9600 ± 0.0072 and r0.002 <
0.0457 at the 68% conﬁdence level for the CDM model [5,6]
which is in tension with the BICEP2 result. When the running of
the spectral index is included in the data ﬁtting, the same combi-
nation gives ns = 0.957±0.015, n′s = dns/d lnk = −0.022+0.020−0.021 and
r0.002 < 0.263 at the 95% conﬁdence level [5,6]. To give a consis-
* Corresponding author.
E-mail addresses: gaoqing01good@163.com (Q. Gao), yggong@mail.hust.edu.cn
(Y. Gong).http://dx.doi.org/10.1016/j.physletb.2014.05.018
0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article
SCOAP3.tent constraint on r for the combination of Planck + WP + highL
data and the BICEP2 data, we require a running of the spectral in-
dex n′s < −0.002 at the 95% conﬁdence level. For the single ﬁeld
inﬂation, the spectral index ns for the scalar perturbation deviates
from the Harrison–Zel’dovich value of 1 in the order of 10−2, so
n′s is in the order of 10−3. The explanation of large r and n′s is a
challenge to single ﬁeld inﬂation. In light of the BICEP2 data, sev-
eral attempts were proposed to explain the large value of r [7–30].
In this Letter, we use the chaotic and natural inﬂation models to
explain the challenge.
2. Slow-roll inﬂation
The slow-roll parameters are deﬁned as
 = M
2
plV
2
φ
2V 2
, (1)
η = M
2
plVφφ
V
, (2)
ξ = M
4
plVφVφφφ
V 2
, (3)
where M2pl = (8πG)−1, Vφ = dV (φ)/dφ, Vφφ = d2V (φ)/dφ2 and
Vφφφ = d3V (φ)/dφ3. For the single ﬁeld inﬂation, the spectral in-
dices, the tensor-to-scalar ratio and the running are given by
ns − 1 ≈ 2η − 6, (4)
r ≈ 16 ≈ −8nt, (5)
n′s = dns/d lnk ≈ 16η − 242 − 2ξ. (6)under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
42 Q. Gao, Y. Gong / Physics Letters B 734 (2014) 41–43Fig. 1. The ns–r diagrams for the chaotic inﬂation with p = 1, p = 2, p = 3 and
p = 4. The 68% and 95% conﬁdence contours from the Planck + WP + highL data
[5,6] and the Planck + WP + highL + BICEP2 data [1] for the CDM model are also
shown.
Fig. 2. The ns–n′s diagrams for the chaotic inﬂation with p = 1, p = 2, p = 3 and
p = 4. The 95% conﬁdence contour for the CDM model from the Planck + WP +
highL data [5,6] is also shown.
The number of e-folds before the end of inﬂation is given by
N(t) =
te∫
t
Hdt ≈ 1
M2pl
φ∫
φe
V (φ)
Vφ(φ)
dφ, (7)
where the value φe of the inﬂaton ﬁeld at the end of inﬂation is
deﬁned by (φe) = 1. The scalar power spectrum is
PR = As
(
k
k∗
)ns−1+n′s ln(k/k∗)/2
, (8)
where the subscript “∗” means the value at the horizon crossing,
the scalar amplitude
As ≈ 1
24π2M4pl
Λ4

. (9)
With the BICEP2 result r = 0.2, the energy scale of inﬂation is Λ ∼
2.2× 1016 GeV.
For the chaotic inﬂation with the power-law potential V (φ) =
Λ4(φ/Mpl)p [31], the slow-roll parameters are  = p/(4N∗), η =
(p − 1)/(2N∗) and ξ = (p − 1)(p − 2)/(4N2∗). The spectral in-
dex ns = 1 − (p + 2)/(2N∗), the running of the spectral index
n′s = −(2+ p)/(2N2∗) = −2(1− ns)2/(p + 2) < 0 and the tensor-to-
scalar ratio r = 4p/N∗ = 8p(1−ns)/(p+2). We plot the ns − r and
ns −n′s relations in Figs. 1 and 2 for p = 1, p = 2, p = 3 and p = 4.
In Fig. 1, we also show the points with N∗ = 50 and N∗ = 60. FromFig. 3. The ns–r diagrams for the natural inﬂation with f = 5Mpl , f = 7Mpl , f =
10Mpl and f = 20Mpl . The 68% and 95% conﬁdence contours from the Planck +
WP+ highL data [5,6] and the Planck+WP+ highL+ BICEP2 data [1] for the CDM
model are also shown.
Fig. 4. The ns–n′s diagrams for the natural inﬂation with f = 5Mpl , f = 7Mpl , f =
10Mpl and f = 20Mpl . The 95% conﬁdence contour for the CDM model from the
Planck + WP + highL data [5,6] is also shown.
Figs. 1 and 2, we see that r increases with the power p, but |n′s|
decreases with the power p. Therefore, it is not easy to satisfy
both the requirements r  0.1 and n′s < −0.002. The chaotic inﬂa-
tion with 2 < p < 3 is marginally consistent with the observation
at the 95% conﬁdence level.
For the natural inﬂation with the potential V (φ) = Λ4[1 +
cos(φ/ f )] [32], the slow-roll parameters are
 = M
2
pl
2 f 2
[
sin(φ/ f )
1+ cos(φ/ f )
]2
, (10)
η = −M
2
pl
f 2
cos(φ/ f )
1+ cos(φ/ f ) , (11)
ξ = −M
4
pl
f 4
[
sin(φ/ f )
1+ cos(φ/ f )
]2
= −2M
2
pl
f 2
. (12)
Inﬂation ends when  ∼ 1, so
φe
f
= arccos
[
1− 2( f /Mpl)2
1+ 2( f /Mpl)2
]
, (13)
and the number of e-folds before the end of inﬂation is
N∗ = 2 f
2
M2pl
ln
[
sin(φe/2 f )
sin(φ∗/2 f )
]
. (14)
Combining Eqs. (4)–(6) with (10)–(12), we plot the ns–r and ns–n′s
relations for the natural inﬂation with f = 5Mpl , f = 7Mpl , f =
10Mpl and f = 20Mpl in Figs. 3 and 4. In Fig. 3, we also show the
points with N∗ = 50 and N∗ = 60. The results show that both r and
Q. Gao, Y. Gong / Physics Letters B 734 (2014) 41–43 43|n′s| increase with the global symmetry breaking scale f . However,
there is an upper limit on |n′s| which is only marginally consistent
with the observation at the 95% conﬁdence level. When f /Mmp 
1, the potential can be approximated by V (φ) = Λ4(φ/ f − π)2/2
which is the power-law potential with p = 2, this is the reason for
the upper limit on n′s .
3. Conclusions
For a single inﬂaton ﬁeld with slow-roll, the tensor-to-scalar ra-
tio r ≈ 16 which is linear with the slow-roll parameter  , but
the running of the spectral index n′s depends on the second or-
der slow-roll parameters, so n′s is at most in the order of 10−3.
The BICEP2 and the Planck data constrain n′s = −0.0221+0.011−0.0099 and
r = 0.20+0.07−0.05 at the 1σ conﬁdence level. Both the chaotic and nat-
ural inﬂation are inconsistent with the observation at the 1σ level.
The chaotic inﬂation with 2 < p < 3 and the natural inﬂation with
f  10Mpl are marginally consistent with the observation at the
95% conﬁdence level. In conclusion, it is a challenge to simulta-
neously explain r as large as 0.2 and n′s as large as −0.01 for
single ﬁeld inﬂation. Unless the Planck and the BICEP2 data can
be reconciled without large n′s , the challenge to single ﬁeld inﬂa-
tion remains.
Acknowledgements
This work was partially supported by the National Basic Science
Program (Project 973) of China under grant No. 2010CB833004,
the NNSF of China under grant No. 11175270, the Program for
New Century Excellent Talents in University under grant No. NCET-
12-0205 and the Fundamental Research Funds for the Central Uni-
versities under grant No. 2013YQ055.
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The challenge for single field inflation with BICEP2 result 

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