Constraints on inflation revisited: An analysis including the latest local measurement of the Hubble constant
CConstraints on inflation revisited: An analysis including the latest local measurementof the Hubble constant
Rui-Yun Guo and Xin Zhang ∗
1, 2, † Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China Center for High Energy Physics, Peking University, Beijing 100080, China
We revisit the constraints on inflation models by using the current cosmological observations in-volving the latest local measurement of the Hubble constant ( H = 73 . ± .
75 km s − Mpc − ).We constrain the primordial power spectra of both scalar and tensor perturbations with the obser-vational data including the Planck 2015 CMB full data, the BICEP2 and Keck Array CMB B-modedata, the BAO data, and the direct measurement of H . In order to relieve the tension between thelocal determination of the Hubble constant and the other astrophysical observations, we consider theadditional parameter N eff in the cosmological model. We find that, for the ΛCDM+ r + N eff model,the scale invariance is only excluded at the 3.3 σ level, and ∆ N eff > σ level.Comparing the obtained 1 σ and 2 σ contours of ( n s , r ) with the theoretical predictions of selectedinflation models, we find that both the convex and concave potentials are favored at 2 σ level, thenatural inflation model is excluded at more than 2 σ level, the Starobinsky R inflation model is onlyfavored at around 2 σ level, and the spontaneously broken SUSY inflation model is now the mostfavored model. I. INTRODUCTION
Inflation is the leading paradigm to explain the originof the primordial density perturbations and the primor-dial gravitational waves, which is a period of acceleratedexpansion of the early universe. It can resolve a num-ber of puzzles of the standard cosmology, such as thehorizon, flatness, and monopole problems [1–4], and offerthe initial conditions for the standard cosmology. Duringthe epoch, inflation can generate the primordial densityperturbations, which seeded the cosmic microwave back-ground (CMB) anisotropies and the large-scale structure(LSS) formation in our universe. Thus, current cosmo-logical observations can be used to explore the natureof inflation. For example, the measurements of CMBanisotropies have confirmed that inflation can provide anearly scale-invariant primordial power spectrum [5–8].Although inflation took place at energy scale as highas 10 GeV, where particle physics remains elusive, hun-dreds of different theoretical scenarios have been pro-posed. Thus selecting an actual version of inflation hasbecome a major issue in the current study. As mentionedabove, the primordial perturbations can lead to the CMBanisotropies and LSS formation, so comparing the pre-dictions of these inflation models with cosmological datacan provide the possibility to identify the suitable infla-tion models.The astronomical observations measuring the CMBanisotropies have provided an excellent opportunity toexplore the physics in the early universe. The Planck col-laboration [9] has measured the primordial power spec-trum of density perturbations with an unprecedented ac-curacy. Namely, the spectral index is measured to be ∗ Corresponding author † Electronic address: [email protected] n s = 0 . ± .
006 (1 σ ), ruling out the scale invarianceat more than 5 σ , and the running of the spectral indexis measured to be dn s /d ln k = − . ± .
007 (1 σ ), fromthe Planck temperature data combined with the Plancklensing likelihood. The constraint on the tensor-to-scalarratio is r . < .
11 at the 2 σ level, also derived byusing the Planck temperature data combined with thePlanck lensing likelihood. In addition, the Keck Arrayand BICEP2 collaborations [10] released a highly signif-icant detection of B-mode polarization with inclusion ofthe first Keck Array B-mode polarization at 95 GHz.These data were taken by the BICEP2 and Keck Ar-ray CMB polarization experiments up to and includingthe 2014 observing season to improve the current con-straints on primordial power spectra. The constraint onthe tensor-to-scalar ratio is r . < .
09 at the 2 σ levelfrom the B-mode only data of BICEP2 and Keck Array.The tighter constraint is r . < .
07 at the 2 σ level whenthe BICEP2/Keck Array B-mode data are combined withthe Planck CMB data plus other astrophysical observa-tions.The baryon acoustic oscillation (BAO) data can effec-tively break the degeneracies between cosmological pa-rameters and further improve the constraints on inflationmodels (see, e.g., Refs. [11–15]). In this paper, we employthe latest BAO measurements including the Date Release12 of the SDSS-III Baryon Oscillation Spectroscopic Sur-vey (BOSS DR12) [16], the 6dF Galaxy Survey (6dFGS)measurement [17], and the Main Galaxy Sample of DataRelease 7 of Sloan Digital Sky Survey (SDSS-MGS) [18].Recently, Riess et al. [19] reported their new resultof direct measurement of the Hubble constant, H =73 . ± .
75 km s − Mpc − , which is 3 . σ higher than thefitting result, H = 66 . ± .
62 km s − Mpc − , derivedby the Planck collaboration [20] based on the ΛCDMmodel assuming (cid:80) m ν = 0 .
06 eV using the Planck TT,TE, EE+lowP data. The strong tension between thenew measurement of H and the Planck data may be a r X i v : . [ a s t r o - ph . C O ] O c t from some systematic uncertainties in the measurementsor some new physics effects. In order to reconcile the newmeasurement of H and the Planck data, one can con-sider the new physics by adding some extra parameters,such as the parameters describing a dynamical dark en-ergy [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 newmeasurement of H and other cosmological observations,the result of H = 73 . ± .
75 km s − Mpc − can playan important role in current cosmology due to its reduceduncertainty from 3.3% to 2.4%. In this paper, we com-bine the new measurement of H with the Planck data,the BICEP2/Keck Array data and the BAO data to con-strain inflation models. The aim of this work is to inves-tigate whether the local determination H = 73 . ± . − Mpc − will have a remarkable influence on con-straining the primordial power spectra of scalar and ten-sor perturbations. In order to relieve the tension betweenthe local determination of the Hubble constant and otherastrophysical observations, we decide to consider dark ra-diation, parametrized by ∆ N eff (defined by N eff − . n s , r ) will be compared with the theoreticalpredictions of some typical inflation models to make amodel selection analysis.The structure of the paper is organized as follows. InSec. II, we briefly introduce the single-field slow-roll in-flationary scenario. In Sec. III, we report the results ofthe constraints on the primordial power spectra with thecombination of the Planck data, the BICEP2/Keck Arraydata, the BAO data and the latest measurement of H .In Sec. IV, we compare the constraint results of ( n s , r )with the theoretical predictions of some typical inflation-ary models and show the impacts of the latest measure-ment of H on the inflation model selection. Conclusionis given in Sec. V. II. SLOW-ROLL INFLATIONARY SCENARIO
In this paper, we only consider the simplest inflation-ary scenario within the slow-roll paradigm, for which theaccelerated expansion of early universe is driven by a ho-mogeneous, slowly rolling scalar field φ . According tothe energy density of the inflaton ρ φ = ˙ φ / V ( φ ), theFriedmann equation becomes H = 13 M (cid:20)
12 ˙ φ + V ( φ ) (cid:21) , (1)where H = ˙ a/a (with a the scale factor of the universe)is the Hubble parameter, M pl = 1 / √ πG is the reducedPlanck mass, V ( φ ) is the inflaton potential, and the dotdenotes the derivative with respect to the cosmic time t .The equation of motion for the inflaton satisfies¨ φ + 3 H ˙ φ + V (cid:48) ( φ ) = 0 , (2) where the prime is the derivative with respect to the in-flaton φ . Due to the slow-roll approximation, ˙ φ (cid:28) φ (cid:28)
0, Eqs. (1) and (2) can be reduced to H ≈ V ( φ )3 M , (3)3 H ˙ φ ≈ − V (cid:48) ( φ ) . (4)Usually, the inflationary universe can be characterizedwith the slow-roll parameters, which can be defined as (cid:15) = M (cid:20) V (cid:48) ( φ ) V ( φ ) (cid:21) , (5) η = M (cid:20) V (cid:48)(cid:48) ( φ ) V ( φ ) (cid:21) , (6) ξ = M V (cid:48) ( φ ) V (cid:48)(cid:48)(cid:48) ( φ ) V ( φ ) , (7)and so on. The inflaton slowly rolls down its potential V ( φ ) as long as (cid:15) (cid:28) | η | (cid:28) P t ( k ) to the scalar spectrum P s ( k ), can be given by the slow-roll approximation as r = P t ( k ) P s ( k ) = 16 (cid:15). (8)Similarly, according to the slow-roll approximation, wecan obtain the spectral index n s = 1 − (cid:15) + 2 η, (9)and the running spectral index dn s /d ln k = 16 (cid:15)η − (cid:15) − ξ . (10)By constraining these parameters using cosmological ob-servations, we can effectively distinguish between differ-ent inflation models. III. CONSTRAINTS ON PRIMORDIAL POWERSPECTRA
In this section, we make a comprehensive analysis ofconstraining the primordial power spectra of scalar andtensor perturbations by combining the new measurementof the Hubble constant, H = 73 . ± .
75 km s − Mpc − [19], with the Planck data, the BICEP2/KeckArray data and the BAO data, to investigate how thenew measurement of H affects the constraint results ofinflation models. We employ the Planck CMB 2015 dataset including the temperature power spectrum (TT), thepolarization power spectrum (EE), the cross-correlation TABLE I: The fitting results of the cosmological parameters in the ΛCDM+ r , ΛCDM+ r + N eff , ΛCDM+ r + dn s /d ln k , andΛCDM+ r + dn s /d ln k + N eff models using the Planck+BK+BAO+ H data.Parameter ΛCDM+ r ΛCDM+ r + N eff ΛCDM+ r + dn s /d ln k ΛCDM+ r + dn s /d ln k + N eff Ω b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . θ MC . ± . . +0 . − . . +0 . − . . ± . τ . ± .
012 0 . ± .
012 0 . ± .
012 0 . ± . A s ) 3 . ± .
023 3 . ± .
023 3 . ± .
023 3 . ± . n s . +0 . − . . +0 . − . . +0 . − . . ± . dn s /d ln k ... ... − . +0 . − . . +0 . − . r . (2 σ ) < . < . < . < . N eff ... 3 . ± .
16 ... 3 . ± . m . ± . . ± . . +0 . − . . +0 . − . H . +0 . − . . ± .
99 68 . +0 . − . . +1 . − . σ . +0 . − . . ± . . +0 . − . . ± . χ .
988 13612 .
184 13615 .
324 13611 . power spectrum of temperature and polarization (TE),and the Planck low- (cid:96) ( (cid:96) ≤
30) likelihood (lowP), as wellas the lensing reconstruction, which is abbreviated as“Planck”. We employ all the BICEP2 and Keck ArrayB-mode data with inclusion of 95 GHz band, abbrevi-ated as “BK”. The BAO data include the CMASS andLOWZ samples from the BOSS DR12 at z eff = 0 .
57 and z eff = 0 .
32 [16], the 6dFGS measurement at z eff = 0 . z eff = 0 .
15 [18],abbreviated as “BAO”.The primordial power spectra of scalar and tensor per-turbations can be expressed as P s ( k ) = A s (cid:18) kk ∗ (cid:19) n s − dn s d ln k ln ( kk ∗ ) , (11) P t ( k ) = A t (cid:18) kk ∗ (cid:19) n t + dn t d ln k ln ( kk ∗ ) , (12)where A s and A t correspond to the scalar and ten-sor amplitudes at the pivot scale k ∗ , respectively. Forthe canonical single-field slow-roll inflation model with-out the inclusion of the running of the spectral index,we have the consistency relation n t = − r/
8. Whenthe running spectral index is considered, we then have n t = − r (2 − r/ − n s ) / dn t /d ln k = r ( r/ n s − / k ∗ = 0 .
002 Mpc − inthis work.There are seven independent free parameters in thebase ΛCDM+ r model: P = { Ω b h , Ω c h , θ MC , τ, ln(10 A s ) , n s , r } , where Ω b h and Ω c h denote the present-day densities ofbaryon and cold dark matter; θ MC denotes the ratio ofthe sound horizon r s to the angular diameter distance D A H r . n s H r . Λ CDM+r
FIG. 1: One-dimensional marginalized distributions andtwo-dimensional contours (1 σ and 2 σ ) for parameters n s , r . and H in the ΛCDM+ r model using thePlanck+BK+BAO+ H data. at the last-scattering epoch; τ denotes the optical depthto reionization; A s and n s denote the amplitude and thespectral index of the primordial power spectra of scalarperturbations, respectively; r denotes the the tensor-to-scalar ratio. When the running is considered, the pa-rameter dn s /d ln k is added to the cosmological model.In this work, we derive the posterior parameter probabil-ities by using the Markov Chain Monte Carlo (MCMC)sampler CosmoMC [32]. H P / P m a x Λ CDM+r Λ CDM+r+ N eff H by Riess et al (2016). FIG. 2: The one-dimensional posterior distributions for theparameter H in the ΛCDM+ r and ΛCDM+ r + N eff modelsusing the Planck+BK+BAO+ H data. The light red banddenotes the new local measurement of H [19]. In Fig. 1, we give one-dimensional marginalized dis-tributions and two-dimensional contours (1 σ and 2 σ ) forthe parameters n s , r . and H in the ΛCDM+ r modelusing the Planck+BK+BAO+ H data. The constraintresults of the ΛCDM+ r model are summarized in thesecond column of Table I. Here we quote ± σ limits forevery parameter in the ΛCDM+ r model, except for r ,which is quoted with the 2 σ upper limit. We obtain theconstraints on r and n s : r . < .
069 (2 σ ) n s = 0 . +0 . − . (1 σ ) (cid:41) ΛCDM+ r . The result of n s for the primordial power spectrumof scalar perturbations excludes the Harrison-Zel’dovich(HZ) scale-invariant spectrum with n s = 1 at the 7 . σ level.In addition, the constraint on the Hubble constant is H = 68 . +0 . − . km s − Mpc − , which is 2 . σ less thanthe local determination H = 73 . ± .
75 km s − Mpc − .Namely, the direct measurement of H = 73 . ± . − Mpc − is in tension with the fit result derived bythe Planck+BK+BAO+ H data based on the ΛCDM+ r model. As shown in Fig. 2, the green line denotes the one-dimensional posterior distribution for the parameter H in the ΛCDM+ r model using the Planck+BK+BAO+ H data, and the light red band denotes the new local mea-surement of H . Obviously, there is a strong tensionbetween the two results.Next, we consider the extra relativistic degrees of free-dom (i.e., the additional parameter N eff ) in the cosmo-logical model to relieve the tension between the latestmeasurement of H and other observational data. The total radiation energy density in the universe is given by ρ r = (cid:34) N eff (cid:18) (cid:19) / (cid:35) ρ γ , (13)where ρ γ is the energy density of photons. If there areonly three-species active neutrinos in the universe, wehave the standard value of N eff = 3 . N eff = N eff − . > N eff as a freeparameter, varying within its prior range of [0 , N eff < .
046 are less well motivated, because suchvalues would require that standard neutrinos are incom-pletely thermalized or additional photons are producedafter the neutrino decoupling, but we still include thisrange for completeness.The third column of Table I gives the constraint resultsof the cosmological parameters in the ΛCDM+ r + N eff model using the Planck+BK+BAO+ H data. We ob-tain the constraints on r and n s : r . < .
071 (2 σ ) n s = 0 . +0 . − . (1 σ ) (cid:41) ΛCDM+ r + N eff . The value of n s becomes larger than that without consid-ering N eff . The fit result of N eff = 3 . ± .
16 indicatesthat ∆ N eff > . σ level. Due to a pos-itive correlation between n s and N eff , as shown in Fig. 3,∆ N eff > n s .On the other hand, a larger Hubble constant, H =69 . ± .
99 km s − Mpc − , is obtained when the pa-rameter N eff is considered, which is only 1 . σ less thanthe local determination H = 73 . ± .
75 km s − Mpc − .Namely, the tension between H = 73 . ± .
75 km s − Mpc − and other observational data is greatly allevi-ated by introducing the parameter N eff in the cosmo-logical model. As showed in Fig. 2, the constraint on H derived using the Planck+BK+BAO+ H data in theΛCDM+ r + N eff model is much closer to the local mea-surement of H . In addition, when the free parameter N eff is included in the cosmological model, χ decreasesfrom 13616 .
988 to 13612 . χ difference,∆ χ = − . r + N eff model,compared to the ΛCDM+ r model, is more favored bythe current Planck+BK+BAO+ H data. Here we notethat in this paper we compare models through only a χ comparison, because we constrain these models us-ing the same data combination. In this situation, if oneadditional parameter can lead to χ decreasing by morethan 2, then we say that adding this parameter is rea-sonable statistically. Thus, we do not employ Bayesianinformation criterion or Bayesian evidence in this paper,since a χ comparison is sufficient for our task.Furthermore, we consider the inclusion of the run-ning of the spectral index, dn s /d ln k , in the fitto the Planck+BK+BAO+ H data. Figure 4 givesone-dimensional marginalized distributions and two-dimensional contours (1 σ and 2 σ ) for parameters n s , H n s r . N eff H n s r . Λ CDM+r+ N eff FIG. 3: One-dimensional marginalized distributions and two-dimensional contours (1 σ and 2 σ ) for parameters N eff , n s , r . ,and H in the ΛCDM+ r + N eff model using the Planck+BK+BAO+ H data. dn s /d ln k , r . , and H in the ΛCDM+ r + dn s /d ln k model using the Planck+BK+BAO+ H data. We ob-tain the constraints on r , n s and dn s /d ln k (see also thefourth column in Table I): r . < .
077 (2 σ ) n s = 0 . +0 . − . (1 σ ) dn s /d ln k = − . +0 . − . (1 σ ) ΛCDM+ r + dn s /d ln k . We find that dn s /d ln k = 0 is well consistent with thePlanck+BK+BAO+ H data in this case, and the fit re-sult H = 68 . +0 . − . km s − Mpc − is still in tensionwith the direct H measurement. The comparison withthe ΛCDM+ r model gives ∆ χ = − . dn s /d ln k does not effectively im-prove the fit. The comparison with the ΛCDM+ r + N eff model gives ∆ χ = 3 .
14, explicitly showing that N eff ismuch more worthy to be added than dn s /d ln k in thesense of improving the fit.In Fig. 5, we give one-dimensional marginalized dis-tributions and two-dimensional contours (1 σ and 2 σ )for the parameters N eff , n s , dn s /d ln k , r . , and H in the ΛCDM+ r + dn s /d ln k + N eff model using thePlanck+BK+BAO+ H data. We obtain the constraintson r , n s and dn s /d ln k (see also the last column in Ta- ble I): r . < .
074 (2 σ ) n s = 0 . ± . σ ) dn s /d ln k = 0 . +0 . − . (1 σ ) ΛCDM+ r + dn s /d ln k + N eff . We find that the fitting results are almost unchangedcomparing to the case of the ΛCDM+ r + N eff model (al-though the parameter space is slightly amplified), asshown in the third and fifth columns of Table I. The re-sults explicitly show that dn s /d ln k = 0 is in good agree-ment with the current observations. A χ comparisonshows that, when the additional parameter dn s /d ln k isincluded, the χ value decreases only by 1.062 (i.e.,∆ χ = − . dn s /d ln k is not deserved to be consid-ered in the cosmological model in the sense of statisticalsignificance. IV. INFLATION MODEL SELECTION
In this section, we consider a few simple and repre-sentative inflation models and compare them with theconstraint results given in the former section. See alsoRef. [33] for a preliminary research. In what follows,we give the predictions of these inflation models for r H d n s / d l n k r . n s H dn s /d ln k r . Λ CDM+r+ dn s /d ln k FIG. 4: One-dimensional marginalized distributions and two-dimensional contours (1 σ and 2 σ ) for parameters n s , dn s /d ln k , r . , and H in the ΛCDM+ r + dn s /d ln k model using the Planck+BK+BAO+ H data. and n s . For these inflation models, we uniformly takethe number of e -folds N ∈ [50 , N eff modifies the radiation density andthereby changes the post-inflationary expansion history,so that the e -folding number N becomes dependent onthe value of N eff . However, practically it is hard to link N to the actual observations. Thus, the usual treatmentof considering N ∈ [50 ,
60] is of course applicable for ouranalysis.The simplest class of inflation models has a monomialpotential V ( φ ) ∝ φ n [34], which is the prototype of thechaotic inflation model. They have the predictions: r = 4 nN , (14) n s = 1 − n + 22 N , (15)where n is any positive number. We take n = 2 /
3, 1, and2 as typical examples in this work. See also Refs. [35–38]for relevant studies of this class of models.The natural inflation model has the effective one-dimensional potential V ( φ ) = Λ (1 + cos( φ/f )) [39, 40], with the predictions: r = 8( f /M pl ) θ N − cos θ N , (16) n s = 1 − f /M pl ) θ N − cos θ N , (17)where θ N is given bycos θ N (cid:18) − N f /M pl ) (cid:19) . (18)Note that different values of n s and r result from thedifferent decay constant f when the number of e -folds N is set to be a certain value.The spontaneously broken SUSY (SBS) inflationmodel has the potential V ( φ ) = V (1 + c ln( φ/Q )) (where V is dominant and the parameter c (cid:28)
1) [41–45], withthe predictions: r (cid:39) , (19) n s = 1 − N . (20) H n s d n s / d l n k r . N eff H n s dn s /d ln k r . Λ CDM+r+ dn s /d ln k + N eff FIG. 5: One-dimensional marginalized distributions and two-dimensional contours (1 σ and 2 σ ) for parameters N eff , n s , dn s /d ln k , r . , and H in the ΛCDM+ r + dn s /d ln k + N eff model using the Planck+BK+BAO+ H data. n s r . c o n v e x c o n c a v e Planck+BK+BAO (I)Planck+BK+BAO+ H (I)Planck+BK+BAO (II)Planck+BK+BAO+ H (II)Natural inflation R InflationSBS Inflation V ∝ φ / V ∝ φV ∝ φ N = 50N = 60
FIG. 6: Two-dimensional contours (1 σ and 2 σ )for n s and r . using the Planck+BK+BAO andPlanck+BK+BAO+ H data, compared to the theoret-ical predictions of selected inflation models. (I) and(II) correspond to the constraints on the ΛCDM+ r andΛCDM+ r + N eff models, respectively. The Starobinsky R inflation model is described by the action S = M (cid:82) d x √− g ( R + R / M ) (where M denotes an energy scale) [1], with the predictions: r (cid:39) N , (21) n s = 1 − N . (22)In Fig. 6, we plot two-dimensional contours (1 σ and2 σ ) for n s and r . using the Planck+BK+BAO andPlanck+BK+BAO+ H data, compared to the theoret-ical predictions of selected inflation models. The or-ange contours denote the constraints on the ΛCDM+ r model with the Planck+BK+BAO data, the green con-tours denote the constraints on the ΛCDM+ r model withthe Planck+BK+BAO+ H data, the gray contours de-note the constraints on the ΛCDM+ r + N eff model withthe Planck+BK+BAO data, and the blue contours de-note the constraints on the ΛCDM+ r + N eff with thePlanck+BK+BAO+ H data.Comparing the orange and green contours, we find thatwhen the direct measurement of H is included in thedata combination, the constraint on the ΛCDM+ r modelis only changed a little, i.e., a little right shift of n s isyielded, which does not greatly change the result of in-flation model selection (see also Ref. [11] for the case oforange contours). According to both the cases of orangeand green contours, the inflation model with a convexpotential is not favored; both the inflation model with amonomial potential ( φ and φ / cases) and the naturalinflation model are marginally favored at around the 2 σ level; the SBS inflation model is located at out of the 2 σ region; the Starobinsky R inflation model is the mostfavored model in this case.When the parameter N eff is considered in the analy-sis, and if the H measurement is not used (i.e., usingthe Planck+BK+BAO data), we find that the parame-ter space is greatly amplified (mainly for n s ). Comparingthe orange and gray contours, we find that without usingthe H measurement the addition of N eff can only am-plify the range of n s but cannot lead to an obvious rightshift of n s .When the H measurement is also used, comparing thegray and blue contours, we see that the addition of the H prior in the combination of data sets for constrainingthe ΛCDM+ r + N eff model leads to a considerable rightshift of n s (and also a slight shrink of width for the rangeof n s ). In Fig. 3, we explicitly show that H is positivelycorrelated with N eff and N eff is positively correlated with n s , which well explains why the H prior (with a largervalue of H ) will lead to a larger value of n s in a cosmo-logical model with N eff .Next, we compare the green and blue contours, which isfor the comparison of the ΛCDM+ r and ΛCDM+ r + N eff models with the Planck+BK+BAO+ H data, and we seethat using the same data sets including the H measure-ment, the consideration of N eff yields a tremendous rightshift of n s (see also Ref. [12]), which largely changes theresult of the inflation model selection. As discussed inthe last section, the ΛCDM+ r + N eff model is much bet-ter than the ΛCDM+ r model for the fit to the currentPlanck+BK+BAO+ H data, since the inclusion of N eff makes the tension between H measurement and otherobservations be greatly relieved and also leads to a muchbetter fit (i.e., the χ value is largely reduced).We now compare the predictions of the above typicalinflation models with the fit results of ( n s , r ) correspond-ing to the blue contours. We see that, in this case, nei-ther the concave potential nor the convex potential isexcluded by the current data. But, it seems that, whencomparing the two, the inflation model with the concavepotential is more favored by the data. The natural in-flation model is now excluded by the data at more thanthe 2 σ level. For the inflation models with a monomialpotential, we find that the φ model is entirely excluded,the φ model is only marginally favored (at the edge ofthe 2 σ region), and the φ / model is still well consistentwith the current data (located in the 1 σ region). Now,the Starobinsky R inflation model is not well favored,because it is located at the edge of the 2 σ region andactually the N = 50 point even lies out of the 2 σ region.We find that in this case the most favored model is the SBS inflation model, which locates near the center of thecontours.Actually, the brane inflation model is also well consis-tent with the current data in this case (for previous anal-yses of brane inflation, see, e.g., Refs. [46, 47]). We leavea comprehensive analysis for the brane inflation model toa future work.From the analysis in this paper, we have found that theinclusion of the latest local measurement of the Hubbleconstant can exert significant influence on the model se-lection of inflationary models, but one must be aware ofthat the result is dependent on the assumption of dark ra-diation in the cosmological model. Without the additionof the parameter N eff , the H measurement is in tensionwith the Planck observation, and the H prior actuallydoes not greatly influence the fit result of the primor-dial power spectra (see the comparison of the orange andgreen contours in Fig. 6). The H tension can be largelyrelieved provided that the parameter N eff is consideredin the model (the tension is reduced from 2.6 σ to 1.7 σ ).The inclusion of the H measurement in the combinationof data sets, together with the consideration of N eff in thecosmological model, leads to a tremendous right shift of n s (see the comparison of the green and blue contours inFig. 6), which greatly changes the situation of the infla-tion model selection. Future experiments on accuratelymeasuring the Hubble constant and searching for lightrelics (dark radiation) would further test the robustnessof our result in this paper. V. CONCLUSION
In this paper, we investigate how the constraints onthe inflation models are affected by considering the lat-est local measurement of the Hubble constant in the cos-mological global fit. We constrain the primordial powerspectra of both scalar and tensor perturbations by us-ing the current cosmological observations including thePlanck 2015 CMB full data, the BICEP2 and Keck Ar-ray CMB B-mode data, the BAO data, and the directmeasurement of H . In order to relieve the tension be-tween the local determination of the Hubble constantand the other astrophysical observations, we considerthe additional parameter N eff in the cosmological model.We make comparison for the ΛCDM+ r , ΛCDM+ r + N eff ,ΛCDM+ r + dn s /d ln k , and ΛCDM+ r + dn s /d ln k + N eff models.We find that the inclusion of N eff indeed effectivelyrelieves the tension. Comparing the ΛCDM+ r andΛCDM+ r + N eff models, the tension is reduced from 2.6 σ to 1.7 σ . The comparison also shows that the additionof one parameter, N eff , leads to the decrease of χ by4.804. When the running of the spectral index dn s /d ln k is considered, we find that the fit results are basicallynot changed and dn s /d ln k = 0 is well consistent withthe current data. Therefore, it is meaningful to considerthe ΛCDM+ r + N eff model when the latest measurementof the Hubble constant is included in the analysis.We constrain the ΛCDM+ r + N eff model using the cur-rent Planck+BK+BAO+ H data. We find that, in thiscase, the scale invariance is only excluded at the 3.3 σ leveland ∆ N eff > σ level. We then com-pare the obtained 1 σ and 2 σ contours of ( n s , r ) with thetheoretical predictions of some selected typical inflationmodels. We find that, in this case, both the convex andconcave potentials are favored at the 2 σ level, althoughthe concave potential is more favored. The natural in-flation model is now excluded at more than 2 σ level, theStarobinsky R inflation model becomes only favored at around 2 σ level, and the most favored model becomesthe SBS inflation model. Acknowledgments
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