#### Constraints on inflation revisited: an analysis including the latest local measurement of the Hubble constant

Eur. Phys. J. C
Constraints on inflation revisited: an analysis including the latest local measurement of the Hubble constant
Rui-Yun Guo 1
Xin Zhang 0 1
0 Center for High Energy Physics, Peking University , Beijing 100080 , China
1 Department of Physics, College of Sciences, Northeastern University , Shenyang 110004 , China
We revisit the constraints on inflation models by using the current cosmological observations involving the latest local measurement of the Hubble constant (H0 = 73.00 ± 1.75 km s −1 Mpc−1). We constrain the primordial power spectra of both scalar and tensor perturbations with the observational data including the Planck 2015 CMB full data, the BICEP2 and Keck Array CMB B-mode data, the BAO data, and the direct measurement of H0. In order to relieve the tension between the local determination of the Hubble constant and the other astrophysical observations, we consider the additional parameter Neff in the cosmological model. We find that, for the CDM+r +Neff model, the scale invariance is only excluded at the 3.3σ level, and Neff > 0 is favored at the 1.6σ level. Comparing the obtained 1σ and 2σ contours of (ns , r ) with the theoretical predictions of selected inflation models, we find that both the convex and the concave potentials are favored at 2σ level, the natural inflation model is excluded at more than 2σ level, the Starobinsky R2 inflation model is only favored at around 2σ level, and the spontaneously broken SUSY inflation model is now the most favored model.
1 Introduction
Inflation is the leading paradigm to explain the origin of the
primordial density perturbations and the primordial
gravitational waves, which is a period of accelerated expansion
of the early universe. It can resolve a number of puzzles
of the standard cosmology, such as the horizon, flatness, and
monopole problems [
1–4
], and offer the initial conditions for
the standard cosmology. During the epoch, inflation can
generate the primordial density perturbations, which seeded the
cosmic microwave background (CMB) anisotropies and the
large-scale structure (LSS) formation in our universe. Thus,
current cosmological observations can be used to explore
the nature of inflation. For example, the measurements of
CMB anisotropies have confirmed that inflation can provide
a nearly scale-invariant primordial power spectrum [
5–8
].
Although inflation took place at energy scale as high as
1016 GeV, where particle physics remains elusive, hundreds
of different theoretical scenarios have been proposed. Thus
selecting an actual version of inflation has become a major
issue in the current study. As mentioned above, the
primordial perturbations can lead to the CMB anisotropies and LSS
formation, so comparing the predictions of these inflation
models with cosmological data can provide the possibility to
identify the suitable inflation models.
The astronomical observations measuring the CMB
anisotropies have provided an excellent opportunity to
explore the physics in the early universe. The Planck
collaboration [
9
] has measured the primordial power
spectrum of density perturbations with an unprecedented
accuracy. Namely, the spectral index is measured to be ns =
0.968 ± 0.006 (1σ ), ruling out the scale invariance at more
than 5σ , and the running of the spectral index is measured to
be dns/d ln k = −0.003 ± 0.007 (1σ ), from the Planck
temperature data combined with the Planck lensing likelihood.
The constraint on the tensor-to-scalar ratio is r0.002 < 0.11
at the 2σ level, also derived by using the Planck
temperature data combined with the Planck lensing likelihood. In
addition, the Keck Array and BICEP2 collaborations [
10
]
released a highly significant detection of B-mode
polarization with inclusion of the first Keck Array B-mode
polarization at 95 GHz. These data were taken by the BICEP2
and Keck Array CMB polarization experiments up to and
including the 2014 observing season to improve the
current constraints on primordial power spectra. The constraint
on the tensor-to-scalar ratio is r0.05 < 0.09 at the 2σ level
from the B-mode only data of BICEP2 and Keck Array. The
tighter constraint is r0.05 < 0.07 at the 2σ level when the
BICEP2/Keck Array B-mode data are combined with the
Planck CMB data plus other astrophysical observations.
The baryon acoustic oscillation (BAO) data can
effectively break the degeneracies between cosmological
parameters and further improve the constraints on inflation
models (see, e.g., Refs. [
11–15
]). In this paper, we employ the
latest BAO measurements including the Date Release 12
of the SDSS-III Baryon Oscillation Spectroscopic Survey
(BOSS DR12) [
16
], the 6dF Galaxy Survey (6dFGS)
measurement [
17
], and the Main Galaxy Sample of Data Release
7 of Sloan Digital Sky Survey (SDSS-MGS) [
18
].
Recently, Riess et al. [
19
] reported their new result of a
direct measurement of the Hubble constant, H0 = 73.00 ±
1.75 km s−1 Mpc−1, which is 3.3σ higher than the fitting
result, H0 = 66.93 ± 0.62 km s−1 Mpc−1, derived by the
Planck collaboration [
20
] based on the CDM model
assuming mν = 0.06 eV using the Planck TT, TE, EE+lowP
data. The strong tension between the new measurement of
H0 and the Planck data may be from some systematic
uncertainties in the measurements or some new physics effects.
In order to reconcile the new measurement of H0 and the
Planck data, one can consider the new physics by adding
some extra parameters, such as the parameters describing a
dynamical dark energy [
21,22
], extra relativistic degrees of
freedom [
19,23–26
] and light sterile neutrinos [
23,24,27–
31
].
Although there are strong tensions between the new
measurement of H0 and other cosmological observations, the
result of H0 = 73.00 ± 1.75 km s−1 Mpc−1 can play an
important role in current cosmology due to its reduced
uncertainty from 3.3 to 2.4%. In this paper, we combine the new
measurement of H0 with the Planck data, the BICEP2/Keck
Array data and the BAO data to constrain inflation
models. The aim of this work is to investigate whether the local
determination H0 = 73.00 ± 1.75 km s−1 Mpc−1 will have
a remarkable influence on constraining the primordial power
spectra of scalar and tensor perturbations. In order to relieve
the tension between the local determination of the Hubble
constant and other astrophysical observations, we decide to
consider dark radiation, parametrized by Neff (defined by
Neff −3.046), in the cosmological model in our analysis. The
constraint results of (ns , r ) will be compared with the
theoretical predictions of some typical inflation models to make
a model selection analysis.
The structure of the paper is organized as follows. In
Sect. 2, we briefly introduce the single-field slow-roll
inflationary scenario. In Sect. 3, we report the results of the
constraints on the primordial power spectra with the
combination of the Planck data, the BICEP2/Keck Array data, the
BAO data and the latest measurement of H0. In Sect. 4, we
compare the constraint results of (ns , r ) with the theoretical
predictions of some typical inflationary models and show the
H 2
V (φ)
≈ 3Mp2l
,
3H φ˙ ≈ −V (φ).
Mp2l V (φ) 2
= 2
η = Mp2l
ξ 2 =
Mp4lV (φ)V (φ)
V 2(φ)
,
,
impacts of the latest measurement of H0 on the selection of
the inflation model. A conclusion is given in Sect. 5.
2 Slow-roll inflationary scenario
In this paper, we only consider the simplest inflationary
scenario within the slow-roll paradigm, for which the
accelerated expansion of early universe is driven by a homogeneous,
slowly rolling scalar field φ. According to the energy density
of the inflaton ρφ = φ˙ 2/2 + V (φ), the Friedmann equation
becomes
H 2
1
= 3Mp2l
1 2
2 φ˙ + V (φ) ,
where H = a˙ /a (with a the scale factor of the universe) is
the Hubble parameter, Mpl = 1/√8π G is the reduced Planck
mass, V (φ) is the inflaton potential, and the dot denotes the
derivative with respect to the cosmic time t .
The equation of motion for the inflaton satisfies
φ¨ + 3H φ˙ + V (φ) = 0,
where the prime is the derivative with respect to the inflaton
φ. Due to the slow-roll approximation, φ˙ 2 0 and φ¨ 0,
Eqs. (1) and (2) can be reduced to
Usually, the inflationary universe can be characterized with
the slow-roll parameters, which can be defined as
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
and so on. The inflaton slowly rolls down its potential V (φ)
as long as 1 and |η| 1.
The tensor-to-scalar ratio, which is defined to be the ratio
of the tensor spectrum Pt(k) to the scalar spectrum Ps(k),
can be given by the slow-roll approximation as
Pt(k)
r = Ps(k) = 16 .
ns = 1 − 6 + 2η,
Similarly, according to the slow-roll approximation, we
can obtain the spectral index,
(10)
and the running spectral index,
dns/d ln k = 16 η − 24 2 − 2ξ 2.
By constraining these parameters using cosmological
observations, we can effectively distinguish between different
inflation models.
3 Constraints on primordial power spectra
In this section, we make a comprehensive analysis of
constraining the primordial power spectra of scalar and tensor
perturbations by combining the new measurement of the
Hubble constant, H0 = 73.00 ± 1.75 km s −1 Mpc−1 [
19
],
with the Planck data, the BICEP2/Keck Array data and the
BAO data, to investigate how the new measurement of H0
affects the constraint results of inflation models. We employ
the Planck CMB 2015 data set including the temperature
power spectrum (TT), the polarization power spectrum (EE),
the cross-correlation power spectrum of temperature and
polarization (TE), and the Planck low- ( ≤ 30) likelihood
(lowP), as well as the lensing reconstruction, which is
abbreviated as “Planck”. We employ all the BICEP2 and Keck
Array B-mode data with inclusion of 95 GHz band,
abbreviated as “BK”. The BAO data include the CMASS and LOWZ
samples from the BOSS DR12 at zeff = 0.57 and zeff = 0.32
[
16
], the 6dFGS measurement at zeff = 0.106 [
17
], and the
SDSS-MGS measurement at zeff = 0.15 [
18
], abbreviated
as “BAO”.
The primordial power spectra of scalar and tensor
perturbations can be expressed as
(11)
Ps(k) = As
k
k∗
ns−1+ 21 ddlnnsk ln kk
∗ ,
nt+ 12 ddlnntk ln kk
∗ ,
k
Pt(k) = At k∗ (12)
where As and At correspond to the scalar and tensor
amplitudes at the pivot scale k∗, respectively. For the canonical
single-field slow-roll inflation model without the inclusion
of the running of the spectral index, we have the
consistency relation nt = −r/8. When the running spectral index
is considered, we then have nt = −r (2 − r/8 − ns)/8 and
dnt/d ln k = r (r/8 + ns − 1)/8. We uniformly set the pivot
scale as k∗ = 0.002 Mpc−1 in this work.
There are seven independent free parameters in the base
CDM+r model:
P = { bh2, ch2, 100θMC, τ, ln(1010 As), ns, r },
where bh2 and ch2 denote the present-day densities of
baryon and cold dark matter; θMC denotes the ratio of the
sound horizon rs to the angular diameter distance DA at the
last-scattering epoch; τ denotes the optical depth to
reionization; As and ns denote the amplitude and the spectral index of
the primordial power spectra of scalar perturbations,
respectively; r denotes the tensor-to-scalar ratio. When the
running is considered, the parameter dns/d ln k is added to the
cosmological model. In this work, we derive the posterior
parameter probabilities by using the Markov Chain Monte
Carlo (MCMC) sampler CosmoMC [
32
].
In Fig. 1, we give one-dimensional marginalized
distributions and two-dimensional contours (1σ and 2σ ) for the
parameters ns, r0.002 and H0 in the CDM+r model using
the Planck+BK+BAO+H0 data. The constraint results of
the CDM+r model are summarized in the second column
of Table 1. Here we quote ±1σ limits for every
parameter in the CDM+r model, except for r , which is quoted
with the 2σ upper limit. We obtain the constraints on r and
ns:
The result of ns for the primordial power spectrum of scalar
perturbations excludes the Harrison–Zel’dovich (HZ)
scaleinvariant spectrum with ns = 1 at the 7.5σ level.
In addition, the constraint on the Hubble constant is H0 =
68.23+−00..4476 km s−1 Mpc−1, which is 2.6σ less than the local
determination H0 = 73.00 ± 1.75 km s−1 Mpc−1. Namely,
the direct measurement of H0 = 73.00±1.75 km s−1 Mpc−1
is in tension with the fit result derived by the Planck+BK
+BAO+H0 data based on the CDM+r model. As shown in
Fig. 2, the green line denotes the one-dimensional posterior
distribution for the parameter H0 in the CDM+r model
using the Planck+BK+BAO+H0 data, and the light red band
denotes the new local measurement of H0. Obviously, there
is a strong tension between the two results.
Next, we consider the extra relativistic degrees of
freedom (i.e., the additional parameter Neff ) in the cosmological
model to relieve the tension between the latest measurement
of H0 and other observational data. The total radiation energy
density in the universe is given by
Table 1 The fitting results of the cosmological parameters in the
CDM+r +dns/d ln k+Neff models using the Planck+BK+BAO+H0 data
CDM+r
4/3
ργ ,
where ργ is the energy density of the photons. If there are
only three-species active neutrinos in the universe, we have
the standard value of Neff = 3.046. Any additional value
of Neff = Neff − 3.046 > 0 indicates the existence of
some dark radiation in the universe. Now, we follow Planck
collaboration [
9
] to constrain Neff as a free parameter,
varying within its prior range of [
0, 6
]. Values of Neff < 3.046
are less well motivated, because such values would require
that standard neutrinos are incompletely thermalized or
additional photons are produced after the neutrino decoupling, but
we still include this range for completeness.
The third column of Table 1 gives the constraint results of
the cosmological parameters in the CDM+r +Neff model
using the Planck+BK+BAO+H0 data. We obtain the
constraints on r and ns:
(13)
CDM + r + Neff .
The value of ns becomes larger than that without
considering Neff . The fit result of Neff = 3.30 ± 0.16 indicates that
Neff > 0 is favored at the 1.6σ level. Due to a positive
correlation between ns and Neff , as shown in Fig. 3, Neff > 0
will lead to a larger ns.
On the other hand, a larger Hubble constant, H0 = 69.63±
0.99 km s−1 Mpc−1, is obtained when the parameter Neff is
considered, which is only 1.7σ less than the local
determination H0 = 73.00 ± 1.75 km s−1 Mpc−1. Namely, the tension
between H0 = 73.00 ± 1.75 km s−1 Mpc−1 and other
observational data is greatly alleviated by introducing the
parameCDM+r ,
CDM+r +Neff ,
CDM+r +dns/d ln k, and
CDM+r +dns/d ln k
CDM+r +dns/d ln k+Neff
ter Neff in the cosmological model. As showed in Fig. 2, the
constraint on H0 derived using the Planck+BK+BAO+H0
data in the CDM+r +Neff model is much closer to the
local measurement of H0. In addition, when the free
parameter Neff is included in the cosmological model, χ 2 decreases
from 13616.988 to 13612.184. The big χ 2 difference, χ 2 =
−4.804, implies that the CDM+r +Neff model, compared
to the CDM+r model, is more favored by the current
Planck+BK+BAO+H0 data. Here we note that in this paper
2
we compare models through only a χmin comparison, because
we constrain these models using the same data
combination. In this situation, if one additional parameter can lead
2
to χmin decreasing by more than 2, then we say that adding
this parameter is reasonable statistically. Thus, we do not
employ Bayesian information criterion or Bayesian evidence
2
in this paper, since a χmin comparison is sufficient for our
task.
Furthermore, we consider the inclusion of the
running of the spectral index, dns/d ln k, in the fit to the
Planck+BK+BAO+H0 data. Figure 4 gives one-dimensional
marginalized distributions and two-dimensional contours
(1σ and 2σ ) for parameters ns, dns/d ln k, r0.002, and H0
in the CDM+r +dns/d ln k model using the Planck+BK
+BAO+H0 data. We obtain the constraints on r , ns and
dns/d ln k (see also the fourth column in Table 1):
We find that dns/d ln k = 0 is well consistent with the
Planck+BK+BAO+H0 data in this case, and the fit result
H0 = 68.37+−00..4570 km s−1 Mpc−1 is still in tension with the
direct H0 measurement. The comparison with the CDM+r
model gives χ 2 = −1.664, implying that adding the
parameter dns/d ln k does not effectively improve the fit. The
comparison with the CDM+r +Neff model gives χ 2 =
3.14, explicitly showing that Neff is much more worthy to be
added than dns/d ln k in the sense of improving the fit.
In Fig. 5, we give one-dimensional marginalized
distributions and two-dimensional contours (1σ and 2σ ) for
the parameters Neff , ns, dns/d ln k, r0.002, and H0 in the
CDM+r +dns/d ln k+Neff model using the Planck+BK
+BAO+H0 data. We obtain the constraints on r , ns and
dns/d ln k (see also the last column in Table 1):
r0.002 < 0.074 (2σ ) ⎫
⎪⎪⎪
ns = 0.9781 ± 0.0080(1σ ) ⎬⎪
dns/d ln k=0.0010−+00..00007734 (1σ ) ⎪⎪⎪⎭⎪
We find that the fitting results are almost unchanged
comparing to the case of the CDM+r +Neff model (although the
parameter space is slightly amplified), as shown in the third
CDM+r +dns/d ln k+Neff .
and fifth columns of Table 1. The results explicitly show that
dns/d ln k = 0 is in good agreement with the current
observations. A χ 2 comparison shows that, when the additional
2
parameter dns/d ln k is included, the χmin value decreases
only by 1.062 (i.e., χ 2 = −1.062), which implies that
the running of the spectral index dns/d ln k is not deserved
to be considered in the cosmological model in the sense of
statistical significance.
4 Inflation model selection
In this section, we consider a few simple and representative
inflation models and compare them with the constraint results
given in the former section. See also Ref. [
33
] for a
preliminary research. In what follows, we give the predictions of
these inflation models for r and ns. For these inflation
models, we uniformly take the number of e-folds N ∈ [50, 60].
In principle, adding the parameter Neff modifies the radiation
density and thereby changes the post-inflationary expansion
history, so that the e-folding number N becomes dependent
on the value of Neff . However, practically it is hard to link
N to the actual observations. Thus, the usual treatment of
considering N ∈ [50, 60] is of course applicable for our
analysis.
The simplest class of inflation models has a monomial
potential V (φ) ∝ φn [
34
], which is the prototype of the
chaotic inflation model. They lead to the predictions
The Starobinsky R2 inflation model is described by the
action S = M2p2l d4x √−g(R + R2/6M 2) (where M denotes
an energy scale) [
1
], with the predictions
12
N 2 ,
r
In Fig. 6, we plot two-dimensional contours (1σ and
2σ ) for ns and r0.002 using the Planck+BK+BAO and
Planck+BK+BAO+H0 data, compared to the theoretical
predictions of selected inflation models. The orange
contours denote the constraints on the CDM+r model with the
Planck+BK+BAO data, the green contours denote the
constraints on the CDM+r model with the Planck+BK+BAO
Fig. 6 Two-dimensional contours (1σ and 2σ ) for ns and r0.002 using +H0 data, the gray contours denote the constraints on
tthhee tPhleaonrcekti+caBlKpr+edBiActOionasndofPslealnecckte+d BinKfl+atBioAnOm+odHe0lsd.(aIt)aa,ncdo m(IpI)acreodrreto- the CDM+r +Neff model with the Planck+BK+BAO
spond to the constraints on the CDM+r and CDM+r +Neff models, data, and the blue contours denote the constraints on the
respectively CDM+r +Neff with the Planck+BK+BAO+H0 data.
Comparing the orange and green contours, we find that
r = 4Nn , (14) cwohmenbitnhaetiodnir,etchtemceoanssutrraeimntenotnothfeH0 CisDiMnc+lurdemdoidnelthies odnaltya
changed a little, i.e., a little right shift of ns is yielded, which
ns = 1 − n2+N2 , (15) (dsoeeesanlsoot gRreefa.t[l1y1c]hfaonrgtheethceasreesouflotroafnigneflcatoinotnomuros)d.eAl scecloercdtiinogn
where n is any positive number. We take n = 2/3, 1, and 2 to the cases of both the orange and the green contours, the
as typical examples in this work. See also Refs. [
35–38
] for inflation model with a convex potential is not favored; both
relevant studies of this class of models. the inflation model with a monomial potential (φ and φ2/3
The natural inflation model has the effective one-dimensional cases) and the natural inflation model are marginally favored
potential V (φ) = 4(1 + cos(φ/ f )) [
39,40
], with the pre- at around the 2σ level; the SBS inflation model is located at
dictions: out of the 2σ region; the Starobinsky R2 inflation model is
the most favored model in this case.
8 1 + cos θN , (16) When the parameter Neff is considered in the
analyr = ( f /Mpl)2 1 − cos θN sis, and if the H0 measurement is not used (i.e., using the
ns = 1 − ( f / M1pl)2 31 +− ccooss θθNN , (17) iPslagnrcekat+lyBaKm+pBliAfieOd d(mataa)in,lwyefofirnnds )th.aCtotmhepaprairnagmtehteerorsapnagcee
and gray contours, we find that without using the H0
meawhere θN is given by surement the addition of Neff can only amplify the range of
ns but cannot lead to an obvious right shift of ns .
cos θ2N = exp − 2( f /NMpl)2 . (18) graWyhanend bthlueeHc0onmtoeuarssu,rewmeesneteisthaaltsotheusaeddd, ictioomn poafritnhge tHhe0
prior in the combination of data sets for constraining the
CDM+r +Neff model leads to a considerable right shift of
ns (and also a slight shrink of width for the range of ns ).
In Fig. 3, we explicitly show that H0 is positively correlated
with Neff and Neff is positively correlated with ns , which
well explains why the H0 prior (with a larger value of H0)
will lead to a larger value of ns in a cosmological model with
Note that different values of ns and r result from the different
decay constant f when the number of e-folds N is set to be
a certain value.
The spontaneously broken SUSY (SBS) inflation model
has the potential V (φ) = V0(1 + c ln(φ/Q)) (where V0 is
dominant and the parameter c 1) [
41–45
], with the
predictions:
(19)
(20)
Neff .
Next, we compare the green and blue contours, which is
for the comparison of the CDM+r and CDM+r +Neff
models with the Planck+BK+BAO+H0 data, and we see
that using the same data sets including the H0 measurement,
r
0,
1
ns = 1 − N .
the consideration of Neff yields a tremendous right shift of ns
(see also Ref. [
12
]), which largely changes the result of the
inflation model selection. As discussed in the last section, the
CDM+r +Neff model is much better than the CDM+r
model for the fit to the current Planck+BK+BAO+H0 data,
since the inclusion of Neff makes the tension between H0
measurement and other observations be greatly relieved and
2
also leads to a much better fit (i.e., the χmin value is largely
reduced).
We now compare the predictions of the above typical
inflation models with the fit results of (ns , r ) corresponding to the
blue contours. We see that, in this case, neither the concave
potential nor the convex potential is excluded by the current
data. But it seems that, when comparing the two, the inflation
model with the concave potential is more favored by the data.
The natural inflation model is now excluded by the data at
more than the 2σ level. For the inflation models with a
monomial potential, we find that the φ2 model is entirely excluded,
the φ model is only marginally favored (at the edge of the 2σ
region), and the φ2/3 model is still well consistent with the
current data (located in the 1σ region). Now, the Starobinsky
R2 inflation model is not well favored, because it is located at
the edge of the 2σ region and actually the N = 50 point even
lies out of the 2σ region. We find that in this case the most
favored model is the SBS inflation model, which locates near
the center of the contours.
Actually, the brane inflation model is also well consistent
with the current data in this case (for previous analyses of
brane inflation, see, e.g., Refs. [
46,47
]). We leave a
comprehensive analysis for the brane inflation model to a future
work.
From the analysis in this paper, we have found that the
inclusion of the latest local measurement of the Hubble
constant can exert significant influence on the model selection of
inflationary models, but one must be aware of that the result
is dependent on the assumption of dark radiation in the
cosmological model. Without the addition of the parameter Neff ,
the H0 measurement is in tension with the Planck
observation, and the H0 prior actually does not greatly influence the
fit result of the primordial power spectra (see the comparison
of the orange and green contours in Fig. 6). The H0 tension
can be largely relieved provided that the parameter Neff is
considered in the model (the tension is reduced from 2.6σ
to 1.7σ ). The inclusion of the H0 measurement in the
combination of data sets, together with the consideration of Neff
in the cosmological model, leads to a tremendous right shift
of ns (see the comparison of the green and blue contours in
Fig. 6), which greatly changes the situation of the inflation
model selection. Future experiments on accurately
measuring the Hubble constant and searching for light relics (dark
radiation) would further test the robustness of our result in
this paper.
5 Conclusion
In this paper, we investigate how the constraints on the
inflation models are affected by considering the latest local
measurement of the Hubble constant in the
cosmological global fit. We constrain the primordial power spectra
of both scalar and tensor perturbations by using the
current cosmological observations including the Planck 2015
CMB full data, the BICEP2 and Keck Array CMB B-mode
data, the BAO data, and the direct measurement of H0.
In order to relieve the tension between the local
determination of the Hubble constant and the other
astrophysical observations, we consider the additional parameter Neff
in the cosmological model. We make comparison for the
CDM+r , CDM+r +Neff , CDM+r +dns/d ln k, and
CDM+r +dns/d ln k+Neff models.
We find that the inclusion of Neff indeed effectively
relieves the tension. Comparing the CDM+r and
CDM+r +Neff models, the tension is reduced from 2.6σ
to 1.7σ . The comparison also shows that the addition of
one parameter, Neff , leads to the decrease of χ 2 by 4.804.
When the running of the spectral index dns/d ln k is
considered, we find that the fit results are basically not changed
and dns/d ln k = 0 is well consistent with the current data.
Therefore, it is meaningful to consider the CDM+r +Neff
model when the latest measurement of the Hubble constant
is included in the analysis.
We constrain the CDM+r +Neff model using the
current Planck+BK+BAO+H0 data. We find that, in this case,
the scale invariance is only excluded at the 3.3σ level and
Neff > 0 is favored at the 1.6σ level. We then compare
the obtained 1σ and 2σ contours of (ns , r ) with the
theoretical predictions of some selected typical inflation models.
We find that, in this case, both the convex and the concave
potentials are favored at the 2σ level, although the concave
potential is more favored. The natural inflation model is now
excluded at more than 2σ level, the Starobinsky R2 inflation
model becomes only favored at around 2σ level, and the most
favored model becomes the SBS inflation model.
Acknowledgements This work was supported by the National Natural
Science Foundation of China (Grants nos. 11522540 and 11690021),
the National Program for Support of Top-notch Young Professionals,
and the Provincial Department of Education of Liaoning (Grant no.
L2012087).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. A.A. Starobinsky , A new type of isotropic cosmological models without singularity . Phys. Lett. 91B , 99 ( 1980 )
2. A.H. Guth , The inflationary universe: a possible solution to the horizon and flatness problems . Phys. Rev. D 23 ( 2 ), 347 ( 1980 )
3. A.D. Linde , A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems . Phys. Lett. 108B , 389 ( 1982 )
4. A. Albrecht , P.J. Steinhardt , Cosmology for grand unified theories with radiatively induced symmetry breaking . Phys. Rev. Lett . 48 , 1220 ( 1982 )
5. G. Hinshaw et al. [ WMAP Collaboration], Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological parameter results . Astrophys. J. Suppl . 208 , 19 ( 2013 ). https:// doi.org/10.1088/ 0067 -0049/208/2/19. arXiv: 1212 . 5226 [astroph .CO]
6. J.L. Sievers et al. [ Atacama Cosmology Telescope Collaboration], The Atacama cosmology telescope: cosmological parameters from three seasons of data . JCAP 1310 , 060 ( 2013 ). https://doi.org/10. 1088/ 1475 - 7516 / 2013 /10/060. arXiv: 1301 .0824 [ astro-ph .CO]
7. Z. Hou et al., Constraints on cosmology from the cosmic microwave background power pectrum of the 2500 deg2 SPT-SZ survey . Astrophys. J . 782 , 74 ( 2014 ). https://doi.org/10.1088/ 0004 -637X/782/ 2/74. arXiv: 1212 .6267 [ astro-ph .CO]
8. P.A.R. Ade et al. [Planck Collaboration], Planck 2013 results. XXIV. Constraints on primordial non-Gaussianity . Astron. Astrophys . 571 , A24 ( 2014 ). https://doi.org/10.1051/ 0004 -6361/ 201321554. arXiv: 1303 .5084 [ astro-ph .CO]
9. P.A.R. Ade et al. [Planck Collaboration], Planck 2015 results. XIII. Cosmological parameters . Astron. Astrophys . 594 , A13 ( 2016 ). https://doi.org/10.1051/ 0004 -6361/201525830. arXiv: 1502 .01589 [ astro-ph .CO]
10. P.A.R. Ade et al. [ BICEP2 and Keck Array Collaborations], Improved constraints on cosmology and foregrounds from BICEP2 and Keck array cosmic microwave background data with inclusion of 95 GHz band . Phys. Rev. Lett . 116 , 031302 ( 2016 ). https://doi. org/10.1103/PhysRevLett.116.031302
11. Q.G. Huang , K. Wang , S. Wang , Inflation model constraints from data released in 2015 . Phys. Rev. D 93 ( 10 ), 103516 ( 2016 ). https:// doi.org/10.1103/PhysRevD.93.103516. arXiv: 1512 . 07769 [astroph .CO]
12. T. Tram, R. Vallance , V. Vennin , Inflation model selection meets dark radiation . JCAP 1701 ( 01 ), 046 ( 2017 ). https://doi.org/10. 1088/ 1475 - 7516 / 2017 /01/046. arXiv: 1606 .09199 [ astro-ph .CO]
13. M. Gerbino , K. Freese , S. Vagnozzi , M. Lattanzi , O. Mena , E. Giusarma , S. Ho , Impact of neutrino properties on the estimation of inflationary parameters from current and future observations . Phys. Rev. D 95 ( 4 ), 043512 ( 2017 ). https://doi.org/10.1103/PhysRevD. 95.043512. arXiv: 1610 .08830 [ astro-ph .CO]
14. E. Di Valentino , L. Mersini-Houghton , Testing predictions of the quantum landscape multiverse 2: the exponential inflationary potential . JCAP 1703 ( 03 ), 020 ( 2017 ). https://doi.org/10.1088/ 1475 - 7516 / 2017 /03/020. arXiv: 1612 .08334 [ astro-ph .CO]
15. E. Di Valentino , L. Mersini-Houghton , Testing predictions of the quantum landscape multiverse 1: the starobinsky inflationary potential . JCAP 1703 ( 03 ), 002 ( 2017 ). https://doi.org/10.1088/ 1475 - 7516 / 2017 /03/002. arXiv: 1612 .09588 [ astro-ph .CO]
16. H. Gil-Marn et al., The clustering of galaxies in the SDSS-III baryon oscillation spectroscopic survey: BAO measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies . Mon. Not. R. Astron. Soc . 460 ( 4 ), 4210 ( 2016 ). https://doi.org/10.1093/ mnras/stw1264. arXiv: 1509 .06373 [ astro-ph .CO]
17. F. Beutler et al., The 6dF galaxy survey: baryon acoustic oscillations and the local Hubble constant . Mon. Not. R. Astron. Soc . 416 , 3017 ( 2011 ). arXiv:1106.3366 [astro-ph.CO]
18. A.J. Ross , L. Samushia , C. Howlett , W.J. Percival , A. Burden , M. Manera , The clustering of the SDSS DR7 main galaxy sample I: a 4 percent distance measure at z = 0.15 . Mon . Not. R. Astron. Soc. 449 ( 1 ), 835 ( 2015 ). arXiv:1409.3242 [astro-ph.CO]
19. A.G. Riess et al., A 2 . 4% determination of the local value of the Hubble constant . Astrophys. J . 826 ( 1 ), 56 ( 2016 ). https://doi.org/ 10.3847/ 0004 -637X/826/1/56. arXiv: 1604 .01424 [ astro-ph .CO]
20. N. Aghanim et al. [Planck Collaboration], Planck intermediate results . XLVI. Reduction of large-scale systematic effects in HFI polarization maps and estimation of the reionization optical depth . Astron. Astrophys . 596 , A107 ( 2016 ). https://doi.org/10.1051/ 0004 -6361/201628890. arXiv: 1605 .02985 [ astro-ph .CO]
21. M. Li , X.D. Li , Y.Z. Ma , X. Zhang , Z. Zhang, Planck constraints on holographic dark energy . JCAP 1309 , 021 ( 2013 ). https://doi. org/10.1088/ 1475 - 7516 / 2013 /09/021. arXiv: 1305 . 5302 [astroph .CO]
22. Q.G. Huang , K. Wang , How the dark energy can reconcile Planck with local determination of the Hubble constant . Eur. Phys. J. C 76 ( 9 ), 506 ( 2016 ). https://doi.org/10.1140/epjc/ s10052-016-4352-x. arXiv:1606.05965 [astro-ph.CO]
23. J.F. Zhang , Y.H. Li , X. Zhang, Sterile neutrinos help reconcile the observational results of primordial gravitational waves from Planck and BICEP2 . Phys. Lett. B 740 , 359 ( 2015 ). https://doi.org/10. 1016/j.physletb. 2014 . 12 .012. arXiv: 1403 .7028 [ astro-ph .CO]
24. J.F. Zhang , J.J. Geng , X. Zhang, Neutrinos and dark energy after Planck and BICEP2: data consistency tests and cosmological parameter constraints . JCAP 1410 ( 10 ), 044 ( 2014 ). https://doi. org/10.1088/ 1475 - 7516 / 2014 /10/044. arXiv: 1408 . 0481 [astroph .CO]
25. E. Di Valentino , F.R. Bouchet , A comment on power-law inflation with a dark radiation component . JCAP 1610 ( 10 ), 011 ( 2016 ). https://doi.org/10.1088/ 1475 - 7516 / 2016 /10/011. arXiv: 1609 .00328 [ astro-ph .CO]
26. M. Benetti , L.L. Graef , J.S. Alcaniz , Do joint CMB and HST data support a scale invariant spectrum ? JCAP 1704 ( 04 ), 003 ( 2017 ). https://doi.org/10.1088/ 1475 - 7516 / 2017 /04/003. arXiv: 1702 .06509 [ astro-ph .CO]
27. J.F. Zhang , Y.H. Li , X. Zhang, Cosmological constraints on neutrinos after BICEP2 . Eur. Phys. J. C 74 , 2014 ( 2954 ). https://doi. org/10.1140/epjc/s10052-014-2954-8. arXiv: 1404 . 3598 [astroph .CO]
28. L. Feng , J.F. Zhang , X. Zhang, A search for sterile neutrinos with the latest cosmological observations . Eur. Phys. J. C 77 ( 6 ), 418 ( 2017 ). https://doi.org/10.1140/epjc/ s10052-017-4986-3. arXiv: 1703 .04884 [ astro-ph .CO]
29. M.M. Zhao , D.Z. He , J.F. Zhang , X. Zhang, Search for sterile neutrinos in holographic dark energy cosmology: reconciling Planck observation with the local measurement of the Hubble constant . Phys. Rev. D 96 ( 4 ), 043520 ( 2017 ). https://doi.org/10.1103/ PhysRevD.96.043520. arXiv: 1703 .08456 [ astro-ph .CO]
30. L. Feng , J.F. Zhang , X. Zhang, Searching for sterile neutrinos in dynamical dark energy cosmologies . arXiv:1706 . 06913 [astroph .CO]
31. M.M. Zhao , J.F. Zhang , X. Zhang, Measuring growth index in a universe with massive neutrinos: a revisit of the general relativity test with the latest observations . arXiv:1710.02391 [astro-ph.CO]
32. A. Lewis , S. Bridle , Cosmological parameters from CMB and other data: a Monte Carlo approach . Phys. Rev. D 66 , 103511 ( 2002 ). arXiv:astro-ph/0205436
33. X. Zhang , Impact of the latest measurement of Hubble constant on constraining inflation models . Sci. China Phys. Mech. Astron . 60 ( 6 ), 060421 ( 2017 ). https://doi.org/10.1007/ s11433-017-9017-7. arXiv: 1702 .05010 [ astro-ph .CO]
34. A.D. Linde , Chaotic inflation . Phys. Lett. 129B , 177 ( 1983 )
35. E. Silverstein , A. Westphal , Monodromy in the CMB: gravity waves and string inflation . Phys. Rev. D 78 , 106003 ( 2008 ). https:// doi.org/10.1103/PhysRevD.78.106003. arXiv: 0803 .3085 [hep-th]
36. L. McAllister , E. Silverstein , A. Westphal , Gravity waves and linear inflation from axion monodromy . Phys. Rev. D 82 , 046003 ( 2010 ). https://doi.org/10.1103/PhysRevD.82.046003. arXiv: 0808 .0706 [hep-th]
37. F. Marchesano , G. Shiu , A.M. Uranga , F-term axion monodromy inflation . JHEP 1409 , 184 ( 2014 ). https://doi.org/10.1007/ JHEP09( 2014 ) 184 . arXiv: 1404 .3040 [hep-th]
38. L. McAllister , E. Silverstein , A. Westphal , T. Wrase, The powers of monodromy . JHEP 1409 , 123 ( 2014 ). https://doi.org/10.1007/ JHEP09( 2014 ) 123 . arXiv: 1405 .3652 [hep-th]
39. F.C. Adams , J.R. Bond , K. Freese , J.A. Friedmann , A.V. Olinto , Natural inflation: particle physics models, power law spectra for large scale structure, and constraints from COBE . Phys. Rev. D 47 , 426 ( 1993 ). https://doi.org/10.1103/PhysRevD.47.426. arXiv: hep-ph/9207245
40. K. Freese , Natural inflation . arXiv:astro-ph/9310012
41. E.J. Copeland , A.R. Liddle , D.H. Lyth , E.D. Stewart , D. Wands , False vacuum inflation with Einstein gravity . Phys. Rev. D 49 , 6410 ( 1994 ). https://doi.org/10.1103/PhysRevD.49.6410. arXiv: astro-ph/9401011
42. G.R. Dvali , Q. Shafi , R.K. Schaefer , Large scale structure and supersymmetric inflation without fine tuning . Phys. Rev. Lett. 73 , 1994 ( 1886 ). https://doi.org/10.1103/PhysRevLett.73. 1886 . arXiv:hep-ph/9406319
43. E.D. Stewart , Inflation, supergravity and superstrings . Phys. Rev. D 51 , 6847 ( 1995 ). https://doi.org/10.1103/PhysRevD.51.6847. arXiv: hep-ph/9405389
44. P. Binetruy, G.R. Dvali , D term inflation . Phys. Lett. B 388 , 241 ( 1996 ). https://doi.org/10.1016/S0370- 2693 ( 96 ) 01083 - 0 . arXiv: hep-ph/9606342
45. D.H. Lyth , A. Riotto , Particle physics models of inflation and the cosmological density perturbation . Phys. Rep . 314 , 1 ( 1999 ). https://doi.org/10.1016/S0370- 1573 ( 98 ) 00128 - 8 . arXiv: hep-ph/9807278
46. Y.Z. Ma , X. Zhang, Brane inflation revisited after WMAP five year results . JCAP 0903 , 006 ( 2009 ). https://doi.org/10.1088/ 1475 - 7516 / 2009 /03/006. arXiv: 0812 .3421 [astro-ph]
47. Y.Z. Ma , Q.G. Huang , X. Zhang, Confronting brane inflation with Planck and pre-Planck data . Phys. Rev. D 87 ( 10 ), 103516 ( 2013 ). https://doi.org/10.1103/PhysRevD.87.103516. arXiv: 1303 .6244 [ astro-ph .CO]