BICEP2 Research Articles

BICEP/Keck. XVII. Line-of-sight Distortion Analysis: Estimates of Gravitational Lensing, Anisotropic Cosmic Birefringence, Patchy Reionization, and Systematic Errors

We present estimates of line-of-sight distortion fields derived from the 95 and 150 GHz data taken by BICEP2, BICEP3, and the Keck Array up to the 2018 observing season, leading to cosmological constraints and a study of instrumental and astrophysical systematics. Cosmological constraints are derived from three of the distortion fields concerning gravitational lensing from large-scale structure, polarization rotation from magnetic fields or an axion-like field, and the screening effect of patchy reionization. We measure an amplitude of the lensing power spectrum . We constrain polarization rotation, expressed as the coupling constant of a Chern–Simons electromagnetic term g a γ ≤ 2.6 × 10−2/H I , where H I is the inflationary Hubble parameter, and an amplitude of primordial magnetic fields smoothed over 1 Mpc B 1Mpc ≤ 6.6 nG at 95 GHz. We constrain the rms of optical depth fluctuations in a simple “crinkly surface” model of patchy reionization, finding A τ < 0.19 (2σ) for the coherence scale of L c = 100. We show that all of the distortion fields of the 95 and 150 GHz polarization maps are consistent with simulations including lensed ΛCDM, dust, and noise, with no evidence for instrumental systematics. In some cases, the EB and TB quadratic estimators presented here are more sensitive than our previous map-based null tests at identifying and rejecting spurious B-modes that might arise from instrumental effects. Finally, we verify that the standard deprojection filtering in the BICEP/Keck data processing is effective at removing temperature to polarization leakage.

Finite inflation in curved space

We investigate the effects of non-zero spatial curvature on cosmic inflation in the light of cosmic microwave background (CMB) anisotropy measurements from the Planck 2018 legacy release and from the 2015 observing season of BICEP2 and the Keck Array. Even a small percentage of non-zero curvature today would significantly limit the total number of e-folds of the scale factor during inflation, rendering just-enough inflation scenarios with a kinetically dominated or fast-roll stage prior to slow-roll inflation more likely. Finite inflation leads to oscillations and a cutoff towards large scales in the primordial power spectrum and curvature pushes them into the CMB observable window. Using nested sampling, we carry out Bayesian parameter estimations and model comparisons taking into account constraints from reheating and horizon considerations. We confirm the preference of CMB data for closed universes with Bayesian odds of over $100:1$ and with a posterior on the curvature density parameter of $\Omega_{K,0}=-0.051\pm0.017$ for a curvature extension of LCDM and $\Omega_{K,0}=-0.031\pm0.014$ for Starobinsky inflation. Model comparisons of various inflation models give similar results as for flat universes with the Starobinsky model outperforming most other models.

BICEP/ Keck XIV: Improved constraints on axionlike polarization oscillations in the cosmic microwave background

We present an improved search for axion-like polarization oscillations in the cosmic microwave background (CMB) with observations from the Keck Array. An all-sky, temporally sinusoidal rotation of CMB polarization, equivalent to a time-variable cosmic birefringence, is an observable manifestation of a local axion field and potentially allows a CMB polarimeter to detect axion-like dark matter directly. We describe improvements to the method presented in previous work, and we demonstrate the updated method with an expanded dataset consisting of the 2012-2015 observing seasons. We set limits on the axion-photon coupling constant for mass $m$ in the range $10^{-23}$-$10^{-18}~\mathrm{eV}$, which corresponds to oscillation periods on the order of hours to years. Our results are consistent with the background model. For periods between $1$ and $30~\mathrm{d}$ ($1.6 \times 10^{-21} \leq m \leq 4.8 \times 10^{-20}~\mathrm{eV}$), the $95\%$-confidence upper limits on rotation amplitude are approximately constant with a median of $0.27^\circ$, which constrains the axion-photon coupling constant to $g_{\phi\gamma} < (4.5 \times 10^{-12}~\mathrm{GeV}^{-1}) m/(10^{-21}~\mathrm{eV}$), if axion-like particles constitute all of the dark matter. More than half of the collected BICEP dataset has yet to be analyzed, and several current and future CMB polarimetry experiments can apply the methods presented here to achieve comparable or superior constraints. In the coming years, oscillation measurements can achieve the sensitivity to rule out unexplored regions of the axion parameter space.

Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season.

We present results from an analysis of all data taken by the BICEP2, Keck Array, and BICEP3 CMB polarization experiments up to and including the 2018 observing season. We add additional Keck Array observations at 220GHz and BICEP3 observations at 95GHz to the previous 95/150/220 GHz dataset. The Q/U maps now reach depths of 2.8, 2.8, and 8.8 μK_{CMB} arcmin at 95, 150, and 220GHz, respectively, over an effective area of ≈600 square degrees at 95GHz and ≈400 square degrees at 150 and 220GHz. The 220GHz maps now achieve a signal-to-noise ratio on polarized dust emission exceeding that of Planck at 353GHz. We take auto- and cross-spectra between these maps and publicly available WMAP and Planck maps at frequencies from 23 to 353GHz and evaluate the joint likelihood of the spectra versus a multicomponent model of lensed ΛCDM+r+dust+synchrotron+noise. The foreground model has seven parameters, and no longer requires a prior on the frequency spectral index of the dust emission taken from measurements on other regions of the sky. This model is an adequate description of the data at the current noise levels. The likelihood analysis yields the constraint r_{0.05}<0.036 at 95%confidence. Running maximum likelihood search on simulations we obtain unbiased results and find that σ(r)=0.009. These are the strongest constraints to date on primordial gravitational waves.

CMB imprints of non-canonical anisotropic inflation

Effect of non-canonical scalar fields on the CMB imprints of the anisotropic inflation will be discussed in details in this paper. In particular, we are able to obtain the general formalism of the angular power spectra in the scalar perturbations, tensor perturbations, cross-correlations, and linear polarization in the context of the anisotropic inflation model involving non-canonical scalar fields. Furthermore, some significant numerical spectra will be plotted using the most recent data of Planck as well as the BICEP2 and Keck array. As a result, we find a very interesting point that the TT spectra induced by the tensor perturbations as well as by the linear polarization will increase when the speed of sound decreases.

BICEP/Keck XII: Constraints on axionlike polarization oscillations in the cosmic microwave background

We present a search for axionlike polarization oscillations in the cosmic microwave background (CMB) with observations from the Keck Array. A local axion field induces an all-sky, temporally sinusoidal rotation of CMB polarization. A CMB polarimeter can thus function as a direct-detection experiment for axionlike dark matter. We develop techniques to extract an oscillation signal. Many elements of the method are generic to CMB polarimetry experiments and can be adapted for other datasets. As a first demonstration, we process data from the 2012 observing season to set upper limits on the axion-photon coupling constant in the mass range 10−21–10−18 eV, which corresponds to oscillation periods on the order of hours to months. We find no statistically significant deviations from the background model. For periods larger than 24 hr (mass m<4.8×10−20 eV), the median 95% confidence upper limit is equivalent to a rotation amplitude of 0.68°, which constrains the axion-photon coupling constant to gϕγ<(1.1×10−11 GeV−1)m/(10−21 eV), if axionlike particles constitute all of the dark matter. The constraints can be improved substantially with data already collected by the BICEP series of experiments. Current and future CMB polarimetry experiments are expected to achieve sufficient sensitivity to rule out unexplored regions of the axion parameter space.Received 17 November 2020Accepted 7 January 2021DOI:https://doi.org/10.1103/PhysRevD.103.042002© 2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAxionsCosmic microwave backgroundDark matterGravitation, Cosmology & AstrophysicsParticles & Fields

Planck2018 results

We present cosmological parameter results from the final full-missionPlanckmeasurements of the cosmic microwave background (CMB) anisotropies, combining information from the temperature and polarization maps and the lensing reconstruction. Compared to the 2015 results, improved measurements of large-scale polarization allow the reionization optical depth to be measured with higher precision, leading to significant gains in the precision of other correlated parameters. Improved modelling of the small-scale polarization leads to more robust constraints on many parameters, with residual modelling uncertainties estimated to affect them only at the 0.5σlevel. We find good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper), from polarization, temperature, and lensing, separately and in combination. A combined analysis gives dark matter density Ωch2 = 0.120 ± 0.001, baryon density Ωbh2 = 0.0224 ± 0.0001, scalar spectral indexns = 0.965 ± 0.004, and optical depthτ = 0.054 ± 0.007 (in this abstract we quote 68% confidence regions on measured parameters and 95% on upper limits). The angular acoustic scale is measured to 0.03% precision, with 100θ* = 1.0411 ± 0.0003. These results are only weakly dependent on the cosmological model and remain stable, with somewhat increased errors, in many commonly considered extensions. Assuming the base-ΛCDM cosmology, the inferred (model-dependent) late-Universe parameters are: Hubble constantH0 = (67.4 ± 0.5) km s−1 Mpc−1; matter density parameter Ωm = 0.315 ± 0.007; and matter fluctuation amplitudeσ8 = 0.811 ± 0.006. We find no compelling evidence for extensions to the base-ΛCDM model. Combining with baryon acoustic oscillation (BAO) measurements (and considering single-parameter extensions) we constrain the effective extra relativistic degrees of freedom to beNeff = 2.99 ± 0.17, in agreement with the Standard Model predictionNeff = 3.046, and find that the neutrino mass is tightly constrained to ∑mν &lt; 0.12 eV. The CMB spectra continue to prefer higher lensing amplitudes than predicted in base ΛCDM at over 2σ, which pulls some parameters that affect the lensing amplitude away from the ΛCDM model; however, this is not supported by the lensing reconstruction or (in models that also change the background geometry) BAO data. The joint constraint with BAO measurements on spatial curvature is consistent with a flat universe, ΩK = 0.001 ± 0.002. Also combining with Type Ia supernovae (SNe), the dark-energy equation of state parameter is measured to bew0 = −1.03 ± 0.03, consistent with a cosmological constant. We find no evidence for deviations from a purely power-law primordial spectrum, and combining with data from BAO, BICEP2, and Keck Array data, we place a limit on the tensor-to-scalar ratior0.002 &lt; 0.06. Standard big-bang nucleosynthesis predictions for the helium and deuterium abundances for the base-ΛCDM cosmology are in excellent agreement with observations. ThePlanckbase-ΛCDM results are in good agreement with BAO, SNe, and some galaxy lensing observations, but in slight tension with the Dark Energy Survey’s combined-probe results including galaxy clustering (which prefers lower fluctuation amplitudes or matter density parameters), and in significant, 3.6σ, tension with local measurements of the Hubble constant (which prefer a higher value). Simple model extensions that can partially resolve these tensions are not favoured by thePlanckdata.

Optical Characterization of the Keck Array and BICEP3 CMB Polarimeters from 2016 to 2019

The BICEP/Keck experiment (BK) is a series of small-aperture refracting telescopes observing degree-scale Cosmic Microwave Background (CMB) polarization from the South Pole in search of a primordial $B$-mode signature. This $B$-mode signal arises from primordial gravitational waves interacting with the CMB, and has amplitude parametrized by the tensor-to-scalar ratio $r$. Since 2016, BICEP3 and the Keck Array have been observing with 4800 total antenna-coupled transition-edge sensor detectors, with frequency bands spanning 95, 150, 220, and 270 GHz. Here we present the optical performance of these receivers from 2016 to 2019, including far-field beams measured in situ with an improved chopped thermal source and instrument spectral response measured with a field-deployable Fourier Transform Spectrometer. As a pair differencing experiment, an important systematic that must be controlled is the differential beam response between the co-located, orthogonally polarized detectors. We generate per-detector far-field beam maps and the corresponding differential beam mismatch that is used to estimate the temperature-to-polarization leakage in our CMB maps and to give feedback on detector and optics fabrication. The differential beam parameters presented here were estimated using improved low-level beam map analysis techniques, including efficient removal of non-Gaussian noise as well as improved spatial masking. These techniques help minimize systematic uncertainty in the beam analysis, with the goal of constraining the bias on $r$ induced by temperature-to-polarization leakage to be subdominant to the statistical uncertainty. This is essential as we progress to higher detector counts in the next generation of CMB experiments.

Detecting Primordial Gravitational Waves: a forecast study on optimizing frequency distribution of next generation ground-based CMB telescope

Probing primordial gravitational waves is one of the core scientific objectives of the next generation CMB polarization experiment. Integrating more detector modules on the focal plane and performing high accurate observations are the main directions of the next generation CMB polarization telescope, like CMB S4. Also, multi-band observation is required by foreground analysis and reduction, as it is understood that foregrounds have become the main obstacles of CMB polarization measurements. However, ground observation is limited by the atmospheric window and can be usually carried out in one or two bands, like what BICEP or Keck array have done in the south pole. In this paper, we forecast the sensitivity of tensor-to-scalar ratio r that may be achieved by a multi-frequency CMB polarization experiment, basing on which to provide guidance for further expanding frequency bands and optimize the focal plane of a telescope. At the same time, the realization of having two frequency bands in one atmospheric window is discussed. With fixed number of detectors, the simulation results show that, in order to get a good limit, more frequency bands are needed. Better constraints can be obtained when it includes at least three bands, i.e., one CMB channel (95 GHz) + one dust channel (high frequency) and one synchrotron channel (low frequency). For example, 41 + 95 + 220 GHz, which is better than only focusing around the CMB band, like 85 + 105 + 150 GHz, and 95 + 135 + 155 GHz, and this frequency combination is even better than the combination of 41 + 95 + 150 + 220 GHz. As CMB S4 plans to consider two frequency bands in each atmospheric window, and along this way, we find that one CMB band and more bands in synchrotron and dust channels are helpful, for example, 2 bands in lower frequency, 30 + 41 GHz, 2 bands in higher frequency, 220 + 270 GHz, i.e. 30 + 41 + 95 + 220 + 270 GHz, can get better constraints, and in this case, more detectors are asked to be assigned in the CMB channel.

Microwave Multiplexing on the Keck Array

We describe an on-sky demonstration of a microwave-multiplexing readout system in one of the receivers of the Keck Array, a polarimetry experiment observing the cosmic microwave background at the South Pole. During the austral summer of 2018-2019, we replaced the time-division multiplexing readout system with microwave-multiplexing components including superconducting microwave resonators coupled to radio-frequency superconducting quantum interference devices at the sub-Kelvin focal plane, coaxial-cable plumbing and amplification between room temperature and the cold stages, and a SLAC Microresonator Radio Frequency system for the warm electronics. In the range 5-6 GHz, a single coaxial cable reads out 528 channels. The readout system is coupled to transition-edge sensors, which are in turn coupled to 150-GHz slot-dipole phased-array antennas. Observations began in April 2019, and we report here on an initial characterization of the system performance.

BICEP2/Keck Array XI: Beam Characterization and Temperature-to-Polarization Leakage in the BK15 Data Set

Abstract Precision measurements of cosmic microwave background (CMB) polarization require extreme control of instrumental systematics. In a companion paper we have presented cosmological constraints from observations with the BICEP2 and Keck Array experiments up to and including the 2015 observing season (BK15), resulting in the deepest CMB polarization maps to date and a statistical sensitivity to the tensor-to-scalar ratio of σ(r) = 0.020. In this work we characterize the beams and constrain potential systematic contamination from main beam shape mismatch at the three BK15 frequencies (95, 150, and 220 GHz). Far-field maps of 7360 distinct beam patterns taken from 2010–2015 are used to measure differential beam parameters and predict the contribution of temperature-to-polarization leakage to the BK15 B-mode maps. In the multifrequency, multicomponent likelihood analysis that uses BK15, Planck, and Wilkinson Microwave Anisotropy Probe maps to separate sky components, we find that adding this predicted leakage to simulations induces a bias of Δr = 0.0027 ± 0.0019. Future results using higher-quality beam maps and improved techniques to detect such leakage in CMB data will substantially reduce this uncertainty, enabling the levels of systematics control needed for BICEP Array and other experiments that plan to definitively probe large-field inflation.

Search intensifies for primordial gravitational waves

In March 2014, researchers who had been working at the South Pole made a stunning announcement: With a telescope called BICEP2 (Background Imaging of Cosmic Extragalactic Polarization), they had found evidence of primordial gravitational waves, or ripples in the fabric of spacetime produced in the first fractions of a second after the Big Bang. This discovery seemed to be our first glimpse of a critical moment in the birth of the universe (1). Following on the BICEP2 telescope’s failure to find a signature of gravitational waves in the cosmic microwave background, the South Pole telescope (left) and the Keck Array (right) and have joined BICEP2 (middle, small upturned dome) at the South Pole to investigate further. Image credit: Anil Ananthaswamy. The publicity that followed the announcement surprised the team. “All hell broke loose,” recalls observational cosmologist John Kovac, a coleader of the BICEP2 collaboration at Harvard University in Cambridge, MA. Unfortunately, it turned out to be a case of mistaken identity. “The signal was consistent with being just from galactic dust,” says Kovac. But the search has not ended by any means. Since then, the BICEP team has redoubled its efforts, making bigger, better, and faster instruments to search for primordial gravitational waves. Figuring out whether these waves exist is crucial for our understanding of how the universe came to be. “This isn’t just me and a handful of other theorists sitting around in our offices saying this will be a very nice thing to measure,” says Marc Kamionkowski, a theoretical physicist at Johns Hopkins University in Baltimore, MD. If cosmologists are correct, primordial gravitational waves would have been produced during a key epoch in the evolution of the universe called inflation. During inflation, spacetime is thought to have expanded rapidly for a fraction of a second. This theory …

Tachyon inflation in teleparallel gravity

We present a tachyonic field inflationary model in a teleparallel framework. We show that tachyonic coupled with the f(T) gravity model can describe the inflation era in which f(T) is an arbitrary function of torsion scalar T. For this purpose, dynamical behavior of the tachyonic field in different potentials is studied, it is shown that the tachyonic field with these potentials can be an effective candidate for inflation. Then, we discuss slow-roll conditions and show that by the appropriate choice of the parameters, the inflation era can be explained via this model. Finally, we argue that our model not only satisfies the result of BICEP2, Keck Array and Plank for the upper limit of r < .012 but also, the obtained value for spectral index n_{s} is compatible with the results of Plank and also Plank + WMAP + HighL + BAO at the 68% confidence level.

Five years after BICEP2

Five years after BICEP2

Constraints on Primordial Gravitational Waves Using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season.

We present results from an analysis of all data taken by the bicep2/Keck CMB polarization experiments up to and including the 2015 observing season. This includes the first Keck Array observations at 220GHz and additional observations at 95 and 150GHz. The Q and U maps reach depths of 5.2, 2.9, and 26 μK_{CMB} arcmin at 95, 150, and 220GHz, respectively, over an effective area of ≈400 square degrees. The 220GHz maps achieve a signal to noise on polarized dust emission approximately equal to that of Planck at 353GHz. We take auto and cross spectra between these maps and publicly available WMAP and Planck maps at frequencies from 23 to 353GHz. We evaluate the joint likelihood of the spectra versus a multicomponent model of lensed-ΛCDM+r+dust+synchrotron+noise. The foreground model has seven parameters, and we impose priors on some of these using external information from Planck and WMAP derived from larger regions of sky. The model is shown to be an adequate description of the data at the current noise levels. The likelihood analysis yields the constraint r_{0.05}<0.07 at 95%confidence, which tightens to r_{0.05}<0.06 in conjunction with Planck temperature measurements and other data. The lensing signal is detected at 8.8σ significance. Running a maximum likelihood search on simulations we obtain unbiased results and find that σ(r)=0.020. These are the strongest constraints to date on primordial gravitational waves.

Heeding the lessons of BICEP2

Heeding the lessons of BICEP2

Signatures of Higgs dilaton and critical Higgs inflation.

We test the Higgs dilaton inflation model (HDM) using the latest cosmological datasets, including the cosmic microwave background temperature, polarization and lensing data from the Planck satellite (2015), the BICEP and Keck Array experiments, the type Ia supernovae from the JLA catalogue, the baryon acoustic oscillations from CMASS, LOWZ and 6dF, the weak lensing data from the CFHTLenS survey and the matter power spectrum measurements from the latest SDSS data release. We find that the values of all cosmological parameters allowed by the HDM are well within the Planck satellite (2015) constraints. In particular, we determine [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] (at 95.5% c.l.). We also place new stringent constraints on the couplings of the HDM, ξχ <0.00328 and [Formula: see text] (at 95.5% c.l.). We find that the HDM is only slightly better than the w0wa CDM model, with [Formula: see text] Given that the HDM has two fewer parameters, we find Bayesian evidence favouring the HDM over the w0wa CDM model. We also study the critical Higgs inflation model, taking into account the running of both the self-coupling λ(μ) and the non-minimal coupling to gravity ξ(μ). We find peaks in the curvature power spectrum at scales corresponding to the critical value μ that re-enter during the radiation era and collapse to form a broad distribution of clustered primordial black holes, which could constitute today the main component of dark matter.This article is part of the Theo Murphy meeting issue 'Higgs cosmology'.

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

We revisit the constraints on inflation models by using the current cosmological observations involving the latest local measurement of the Hubble constant (H_{0} = 73.00pm 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 3.3sigma level, and Delta N_mathrm{eff}>0 is favored at the 1.6sigma level. Comparing the obtained 1sigma and 2sigma 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 2sigma level, the natural inflation model is excluded at more than 2sigma level, the Starobinsky R^2 inflation model is only favored at around 2sigma level, and the spontaneously broken SUSY inflation model is now the most favored model.

BICEP2 / Keck Array IX: New bounds on anisotropies of CMB polarization rotation and implications for axionlike particles and primordial magnetic fields

We present the strongest constraints to date on anisotropies of cosmic microwave background (CMB) polarization rotation derived from 150 GHz data taken by the BICEP2 & Keck Array CMB experiments up to and including the 2014 observing season (BK14). The definition of the polarization angle in BK14 maps has gone through self-calibration in which the overall angle is adjusted to minimize the observed TB and EB power spectra. After this procedure, the QU maps lose sensitivity to a uniform polarization rotation but are still sensitive to anisotropies of polarization rotation. This analysis places constraints on the anisotropies of polarization rotation, which could be generated by CMB photons interacting with axionlike pseudoscalar fields or Faraday rotation induced by primordial magnetic fields. The sensitivity of BK14 maps (∼3 μK−arc min) makes it possible to reconstruct anisotropies of the polarization rotation angle and measure their angular power spectrum much more precisely than previous attempts. Our data are found to be consistent with no polarization rotation anisotropies, improving the upper bound on the amplitude of the rotation angle spectrum by roughly an order of magnitude compared to the previous best constraints. Our results lead to an order of magnitude better constraint on the coupling constant of the Chern-Simons electromagnetic term gaγ≤7.2×10−2/HI (95% confidence) than the constraint derived from the B-mode spectrum, where HI is the inflationary Hubble scale. This constraint leads to a limit on the decay constant of 10−6≲fa/Mpl at mass range of 10−33≤ma≤10−28 eV for r=0.01, assuming gaγ∼α/(2πfa) with α denoting the fine structure constant. The upper bound on the amplitude of the primordial magnetic fields is 30 nG (95% confidence) from the polarization rotation anisotropies.Received 13 May 2017DOI:https://doi.org/10.1103/PhysRevD.96.102003© 2017 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCosmic microwave backgroundPrimordial magnetic fieldsPhysical SystemsAxionsGravitation, Cosmology & AstrophysicsGeneral Physics

Impact of the latest measurement of Hubble constant on constraining inflation models

We investigate how the constraint results of inflation models are affected by considering the latest local measurement of $H_0$ in the global fit. We use the observational data, including the Planck CMB full data, the BICEP2 and Keck Array CMB B-mode data, the BAO data, and the latest measurement of Hubble constant, to constrain the $\Lambda$CDM+$r$+$N_{\rm eff}$ model, and the obtained 1$\sigma$ and 2$\sigma$ contours of $(n_s, r)$ are compared to the theoretical predictions of selected inflationary models. We find that, in this fit, the scale invariance is only excluded at the 3.3$\sigma$ level, and $\Delta N_{\rm eff}>0$ is favored at the 1.6$\sigma$ level. The natural inflation model is now excluded at more than 2$\sigma$ level; the Starobinsky $R^2$ model becomes only favored at around 2$\sigma$ level; the most favored model becomes the spontaneously broken SUSY inflation model; and, the brane inflation model is also well consistent with the current data, in this case.