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

The European Physical Journal C, Dec 2017

We revisit the constraints on inflation models by using the current cosmological observations involving the latest local measurement of the Hubble constant (\(H_{0} = 73.00\pm 1.75\) 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 \(H_0\). In order to relieve the tension between the local determination of the Hubble constant and the other astrophysical observations, we consider the additional parameter \(N_\mathrm{eff}\) in the cosmological model. We find that, for the \(\Lambda \)CDM\(+\) \(r\) \(+\) \(N_\mathrm{eff}\) model, the scale invariance is only excluded at the 3.3\(\sigma \) level, and \(\Delta N_\mathrm{eff}>0\) is favored at the 1.6\(\sigma \) level. Comparing the obtained 1\(\sigma \) and 2\(\sigma \) contours of \((n_s,r)\) with the theoretical predictions of selected inflation models, we find that both the convex and the concave potentials are favored at 2\(\sigma \) level, the natural inflation model is excluded at more than 2\(\sigma \) level, the Starobinsky \(R^2\) inflation model is only favored at around 2\(\sigma \) level, and the spontaneously broken SUSY inflation model is now the most favored model.

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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). 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Rui-Yun Guo, Xin Zhang. Constraints on inflation revisited: an analysis including the latest local measurement of the Hubble constant, The European Physical Journal C, 2017, 882, DOI: 10.1140/epjc/s10052-017-5454-9