Eur. Phys. J. C (2017) 77:882 https://doi.org/10.1140/epjc/s10052-017-5454-9 Regular Article - Theoretical Physics Constraints on inflation revisited: an analysis including the latest local measurement of the Hubble constant Rui-Yun Guo1, Xin Zhang1,2,a 1 Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China 2 Center for High Energy Physics, Peking University, Beijing 100080, China Received: 21 June 2017 / Accepted: 8 December 2017 / Published online: 18 December 2017 © The Author(s) 2017. This article is an open access publication Abstract We revisit the constraints on inflation models by current cosmological observations can be used to explore using the current cosmological observations involving the the nature of inflation. For example, the measurements of latest local measurement of the Hubble constant (H0 = CMB anisotropies have confirmed that inflation can provide 73.00 ± 1.75 km s −1 Mpc−1). We constrain the primordial a nearly scale-invariant primordial power spectrum [5–8]. power spectra of both scalar and tensor perturbations with the Although inflation took place at energy scale as high as observational data including the Planck 2015 CMB full data, 1016 GeV, where particle physics remains elusive, hundreds the BICEP2 and Keck Array CMB B-mode data, the BAO of different theoretical scenarios have been proposed. Thus data, and the direct measurement of H0. In order to relieve the selecting an actual version of inflation has become a major tension between the local determination of the Hubble con- issue in the current study. As mentioned above, the primor- stant and the other astrophysical observations, we consider dial perturbations can lead to the CMB anisotropies and LSS the additional parameter Neff in the cosmological model. We formation, so comparing the predictions of these inflation find that, for the CDM+r+Neff model, the scale invariance models with cosmological data can provide the possibility to is only excluded at the 3.3σ level, and Neff > 0 is favored identify the suitable inflation models. at the 1.6σ level. Comparing the obtained 1σ and 2σ con- The astronomical observations measuring the CMB tours of (ns, r) with the theoretical predictions of selected anisotropies have provided an excellent opportunity to inflation models, we find that both the convex and the con- explore the physics in the early universe. The Planck col- cave potentials are favored at 2σ level, the natural inflation laboration [9] has measured the primordial power spec- model is excluded at more than 2σ level, the Starobinsky R2 trum of density perturbations with an unprecedented accu- inflation model is only favored at around 2σ level, and the racy. Namely, the spectral index is measured to be ns = spontaneously broken SUSY inflation model is now the most 0.968 ± 0.006 (1σ ), ruling out the scale invariance at more favored model. 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 tem- perature data combined with the Planck lensing likelihood. 1 Introduction The constraint on the tensor-to-scalar ratio is r0.002 < 0.11 at the 2σ level, also derived by using the Planck tempera- Inflation is the leading paradigm to explain the origin of the ture data combined with the Planck lensing likelihood. In primordial density perturbations and the primordial gravi- addition, the Keck Array and BICEP2 collaborations [10] tational waves, which is a period of accelerated expansion released a highly significant detection of B-mode polariza- of the early universe. It can resolve a number of puzzles tion with inclusion of the first Keck Array B-mode polar- of the standard cosmology, such as the horizon, flatness, and ization at 95 GHz. These data were taken by the BICEP2 monopole problems [1–4], and offer the initial conditions for and Keck Array CMB polarization experiments up to and the standard cosmology. During the epoch, inflation can gen- including the 2014 observing season to improve the cur- erate the primordial density perturbations, which seeded the rent constraints on primordial power spectra. The constraint cosmic microwave background (CMB) anisotropies and the on the tensor-to-scalar ratio is r0.05 < 0.09 at the 2σ level large-scale structure (LSS) formation in our universe. Thus, 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 a e-mail: [email protected] 123 882 Page 2 of 11 Eur. Phys. J. C (2017) 77 :882 BICEP2/Keck Array B-mode data are combined with the impacts of the latest measurement of H0 on the selection of Planck CMB data plus other astrophysical observations. the inflation model. A conclusion is given in Sect. 5. The baryon acoustic oscillation (BAO) data can effec- tively break the degeneracies between cosmological param- eters and further improve the constraints on inflation mod- 2 Slow-roll inflationary scenario els (see, e.g., Refs. [11–15]). In this paper, we employ the latest BAO measurements including the Date Release 12 In this paper, we only consider the simplest inflationary sce- of the SDSS-III Baryon Oscillation Spectroscopic Survey nario within the slow-roll paradigm, for which the acceler- (BOSS DR12) [16], the 6dF Galaxy Survey (6dFGS) mea- ated expansion of early universe is driven by a homogeneous, surement [17], and the Main Galaxy Sample of Data Release slowly rolling scalar field φ. According to the energy density ˙ 7 of Sloan Digital Sky Survey (SDSS-MGS) [18]. of the inflaton ρφ = φ2/2 + V (φ), the Friedmann equation Recently, Riess et al. [19] reported their new result of a becomes direct measurement of the Hubble constant, H0 = 73.00 ± −1 −1 1 1 1.75 km s Mpc , which is 3.3σ higher than the fitting H 2 = φ˙2 + V (φ) , (1) −1 −1 2 result, H0 = 66.93 ± 0.62 km s Mpc , derived by the 3Mpl 2 Planck collaboration [20] based on the CDM model assum- =˙/ ing mν = 0.06 eV using the Planck TT, TE, EE+lowP where H a a (with a the scale√ factor of the universe) is = / π data. The strong tension between the new measurement of the Hubble parameter, Mpl 1 8 G is the reduced Planck (φ) H0 and the Planck data may be from some systematic uncer- mass, V is the inflaton potential, and the dot denotes the tainties in the measurements or some new physics effects. derivative with respect to the cosmic time t. In order to reconcile the new measurement of H0 and the The equation of motion for the inflaton satisfies Planck data, one can consider the new physics by adding φ¨ + φ˙ + (φ) = , some extra parameters, such as the parameters describing a 3H V 0 (2) dynamical dark energy [21,22], extra relativistic degrees of freedom [19,23–26] and light sterile neutrinos [23,24,27– where the prime is the derivative with respect to the inflaton φ φ˙2 φ¨ 31]. Due to the slow-roll approximation, 0 and 0, Although there are strong tensions between the new mea- Eqs. (1) and (2) can be reduced to (φ) surement of H0 and other cosmological observations, the 2 V −1 −1 H ≈ , (3) result of H0 = 73.00 ± 1.75 km s Mpc can play an 2 3Mpl important role in current cosmology due to its reduced uncer- φ˙ ≈− (φ). tainty from 3.3 to 2.4%. In this paper, we combine the new 3H V (4) measurement of H0 with the Planck data, the BICEP2/Keck Usually, the inflationary universe can be characterized with Array data and the BAO data to constrain inflation mod- the slow-roll parameters, which can be defined as els. The aim of this work is to investigate whether the local 2 2 = . ± . −1 −1 Mpl V (φ) determination H0 73 00 1 75 km s Mpc will have = , (5) V (φ) a remarkable influence on constraining the primordial power 2 spectra of scalar and tensor perturbations. In order to relieve V (φ) η = M2 , (6) the tension between the local determination of the Hubble pl V (φ) constant and other astrophysical observations, we decide to M4 V (φ)V (φ) 2 pl consider dark radiation, parametrized by Neff (defined by ξ = , (7) V 2(φ) Neff −3.046), in the cosmological model in our analysis. The (φ) constraint results of (ns, r) will be compared with the theo- and so on. The inflaton slowly rolls down its potential V retical predictions of some typical inflation models to make as long as 1 and |η|1. a model selection analysis. The tensor-to-scalar ratio, which is defined to be the ratio The structure of the paper is organized as follows. In of the tensor spectrum Pt(k) to the scalar spectrum Ps(k), Sect. 2, we briefly introduce the single-field slow-roll infla- can be given by the slow-roll approximation as tionary scenario. In Sect. 3, we report the results of the con- P (k) straints on the primordial power spectra with the combina- r = t = . ( ) 16 (8) tion of the Planck data, the BICEP2/Keck Array data, the Ps k BAO data and the latest measurement of H . In Sect. 4,we 0 Similarly, according to the slow-roll approximation, we compare the constraint results of (n , r) with the theoretical s can obtain the spectral index, predictions of some typical inflationary models and show the ns = 1 − 6 + 2η, (9) 123 Eur. Phys. J. C (2017) 77 :882 Page 3 of 11 882 and the running spectral index, 2 2 dns/d ln k = 16 η − 24 − 2ξ .