Pantheon Sample Research Articles

Testing cosmic anisotropy with the Ep−Eiso (‘Amati’) correlation of GRBs

ABSTRACT We test the possible cosmic anisotropy in 118 long GRBs with the Ep−Eiso (‘Amati’) correlation by employing the dipole fitting (DF) and hemisphere comparison (HC) methods. The distribution of the GRB sample is nearly homogeneous in the sky. The dipole anisotropy is weak in the dipole-modulated $\rm \Lambda$CDM model and the Finslerian cosmological model. The dipole directions from the GRB sample are consistent with ones given by the Pantheon SNe-Ia sample, but with smaller uncertainties. We also investigate whether the GRB sample can reduce the anisotropic signal from inhomogeneous samples like the Pantheon one. The GRB sample is then combined with the Pantheon one, thus providing an SN-G sample. In the dipole-modulated $\rm \Lambda$CDM model, the dipole direction in the SN-G sample shows a considerable change from the one in the Pantheon sample. The angle between the two directions is 26${_{.}^{\circ}}$78. For the HC method, the result of maximum anisotropy level from the G-SN sample is 0.257 ± 0.060 at 68 per cent confidence level (CL) and the corresponding preferred direction is $(l,b)=(82{_{.}^{\circ}}97^{+52{_{.}^{\circ}}73}_{~-61{_{.}^{\circ}}88}, -15{_{.}^{\circ}}09^{+60{_{.}^{\circ}}09}_{~-13{_{.}^{\circ}}54})$. The statistical significance of the $\rm AL_{max}$ is 1.4σ. The angle between the preferred direction and the one from the Pantheon sample is 44${_{.}^{\circ}}$40. Although the amount of data in the GRB sample is about a tenth of that in the Pantheon sample, the GRB sample can considerably impact the results from the Pantheon sample. Our results show that GRBs have the potential to search for a convincing cosmic anisotropy.

On the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030

The difference from 4 to 6 σ in the Hubble constant (H0) between the values observed with the local (Cepheids and Supernovae Ia, SNe Ia) and the high-z probes (Cosmic Microwave Background obtained by the Planck data) still challenges the astrophysics and cosmology community. Previous analysis has shown that there is an evolution in the Hubble constant that scales as f(z)=H0/(1+z)η, where H0 is H0(z=0) and η is the evolutionary parameter. Here, we investigate if this evolution still holds by using the SNe Ia gathered in the Pantheon sample and the Baryon Acoustic Oscillations. We assume H0=70kms−1Mpc−1 as the local value and divide the Pantheon into three bins ordered in increasing values of redshift. Similar to our previous analysis but varying two cosmological parameters contemporaneously (H0, Ω0m in the ΛCDM model and H0, wa in the w0waCDM model), for each bin we implement a Markov-Chain Monte Carlo analysis (MCMC) obtaining the value of H0 assuming Gaussian priors to restrict the parameters spaces to values we expect from our prior knowledge of the current cosmological models and to avoid phantom Dark Energy models with w<−1. Subsequently, the values of H0 are fitted with the model f(z). Our results show that a decreasing trend with η∼10−2 is still visible in this sample. The η coefficient reaches zero in 2.0 σ for the ΛCDM model up to 5.8 σ for w0waCDM model. This trend, if not due to statistical fluctuations, could be explained through a hidden astrophysical bias, such as the effect of stretch evolution, or it requires new theoretical models, a possible proposition is the modified gravity theories, f(R). This analysis is meant to further cast light on the evolution of H0 and it does not specifically focus on constraining the other parameters. This work is also a preparatory to understand how the combined probes still show an evolution of the H0 by redshift and what is the current status of simulations on GRB cosmology to obtain the uncertainties on the Ω0m comparable with the ones achieved through SNe Ia.

Modified emergent dark energy and its astronomical constraints

In this paper, we generalize the Phenomenologically Emergent Dark Energy (PEDE) model, famous for solving the long standing Hubble tension, with our Modified Emergent Dark Energy (MEDE) scenario. This new model introduces one additional parameter ‘[Formula: see text]’ that, depending on the value assumed by the analysis, can recover [Formula: see text]CDM (if [Formula: see text]), PEDE (if [Formula: see text]). We perform a robust observational analysis of MEDE using various observational datasets including cosmic microwave background (CMB) radiation, baryon acoustic oscillations (BAO) distance measurements, a local value of the Hubble constant (R19) and the Pantheon sample of Supernovae Type Ia. We find that when [Formula: see text] is free to vary, the [Formula: see text] tension is alleviated but not solved like in PEDE, and the [Formula: see text]CDM is always preferred by the data when R19 is excluded, as confirmed by the bayesian evidence analysis.

Nonparametric modeling of cosmological database on the χ2 distribution

In the $\Lambda$CDM model, cosmological observations from the late and recent universe reveal a puzzling $\sim 4.5\sigma$ tension in the current rate of universe expansion. In addition to the various scenarios suggested to resolve the tension, non-parametric modeling may provide useful insights. In this paper, we look at three well-known non-parametric methods, the smoothing method, the genetic algorithm, and the Gaussian process. Considering these three methods, we employ the recent Hubble parameters data to reconstruct the rate of universe expansion and supernovae Pantheon sample to reconstruct the luminosity distance. In contrast to the similar studies in the literature, the chi-squared distribution has been used to construct a reliable criterion to select a reconstruction. Finally, we compute the current rate of universe expansion ($H_0$) for each method, provide some discussions regarding the performance of each approach, and compare the results.

Cosmographic Parameters in Model-independent Approaches

Abstract The cosmographic approach, a Taylor expansion of the Hubble function, has been used as a model-independent method to investigate the evolution of the universe in the presence of cosmological data. Apart from possible technical problems like the radius of convergence, there is an ongoing debate about the tensions that appear when one investigates some high-redshift cosmological data. In this work, we consider two common data sets, namely, Type Ia supernovae (Pantheon sample) and the Hubble data, to investigate advantages and disadvantages of the cosmographic approach. To do this, we obtain the evolution of cosmographic functions using the cosmographic method, as well as two other well-known model-independent approaches, namely, the Gaussian process and the genetic algorithm. We also assume a ΛCDM model as the concordance model to compare the results of mentioned approaches. Our results indicate that the results of cosmography compared with the other approaches are not exact enough. Considering the Hubble data, which are less certain, the results of q 0 and j 0 obtained in cosmography provide a tension at more than 3σ away from the best result of ΛCDM. Assuming both of the data samples in different approaches, we show that the cosmographic approach, because it provides some biased results, is not the best approach for reconstruction of cosmographic functions, especially at higher redshifts.

Viscous cosmology in the Weyl-type f(Q,\xa0T) gravity

Bulk viscosity is the only viscous influence that can change the background dynamics in a homogeneous and isotropic universe. In the present work, we analyze the bulk viscous cosmological model with the bulk viscosity coefficient of the form zeta =zeta _0+zeta _1H+zeta _2left( frac{dot{H}}{H}+Hright) where, zeta _0, zeta _1 and zeta _2 are bulk viscous parameters, and H is the Hubble parameter. We investigate the impact of the bulk viscous parameter on dynamics of the universe in the recently proposed Weyl-type f(Q, T) gravity, where Q is the non-metricity, and T is the trace of the matter energy–momentum tensor. The exact solutions to the corresponding field equations are obtained with the viscous fluid and the linear model of the form f(Q, T)=alpha Q+frac{beta }{6kappa ^2}T, where alpha and beta are model parameters. Further, we constrain the model parameters using the 57 points Hubble dataset the recently released 1048 points Pantheon sample and the combination Hz + BAO + Pantheon, which shows our model is good congeniality with observations. We study the possible scenarios and the evolution of the universe through the deceleration parameter, the equation of state (EoS) parameter, the statefinder diagnostics, and the Om diagnostics. It is observed that the universe exhibits a transition from a decelerated to an accelerated phase of the universe under certain constraints of model parameters.

Observational constraint on dark energy from quantum uncertainty * *Supported by Xiaofeng Yang’s Xinjiang Tianchi Bairen project and CAS Pioneer Hundred Talents Program. This work was also partly supported by the National Key R&D Program of China (2018YFA0404602)

We explore the theoretical possibility that dark energy density is derived from massless scalar bosons in vacuum and present a physical model for dark energy. By assuming massless scalar bosons fall into the horizon boundary of the cosmos with the expansion of the universe, we can deduce the uncertainty in the relative position of scalar bosons based on the quantum fluctuation of space-time and the assumption that scalar bosons satisfy P-symmetry under the parity transformation , which can be used to estimate scalar bosons and dark energy density. Furthermore, we attempt to explain the origin of negative pressure from the increasing entropy density of the Boltzmann system and derive the equation for the state parameter, which is consistent with the specific equations of state for dark energy. Finally, we employ the SNIa Pantheon sample and Planck 2018 CMB angular power spectra to constrain the models and provide statistical results for the cosmology parameters.

Using our newest VLT-KMOS HII galaxies and other cosmic tracers to test the Lambda cold dark matter tension

ABSTRACT We place novel constraints on the cosmokinetic parameters by using a joint analysis of the newest Very Large Telescope (VLT)-K-band Multi Object Spectrograph (KMOS) HII galaxies (HIIG) with the Type Ia Supernovae (SNIa) Pantheon sample. We combine the latter data sets in order to reconstruct, in a model-independent way, the Hubble diagram to as high redshifts as possible. Using a Gaussian process, we derive the basic cosmokinetic parameters and compare them with those of Lambda cold dark matter (ΛCDM). In the case of SNIa, we find that the extracted values of the cosmokinetic parameters are in agreement with the predictions of ΛCDM model. Combining SNIa with high-redshift tracers of the Hubble relation, namely HIIG data, we obtain consistent results with those based on ΛCDM as far as the present values of the cosmokinetic parameters are concerned, but find significant deviations in the evolution of the cosmokinetic parameters with respect to the expectations of the concordance ΛCDM model.

A tomographic test of cosmic anisotropy with the recently-released quasar sample

We test the cosmic anisotropy in the dipole-modulated Lambda CDM model and Finslerian cosmological model with the recently-released quasar sample. Based on the redshift tomographic method, the quasar sample is divided into two subsets z le z_{cut} and z > z_{cut} by different cutting redshifts. The dipole amplitudes of the two cosmological models from the subsets z le z_{cut} are very weak. We find that quasars at a higher redshift range may provide more detailed information about the dipole amplitude A_D. The dipole directions of each cosmological model from the subsets z le 1.1 and z > 1.1 are deviated by 1sigma level. The Pantheon sample is combined with the two subsets. The dipole amplitude from the two combined datasets is also very weak. In the dipole-modulated Lambda CDM model, the dipole direction from the combined dataset quasar at z le 1.1 and Pantheon sample is far away from the one given by the Pantheon sample. In the Finslerian cosmological model, the dipole directions from the two combined datasets are close to the one in the Pantheon sample.

Constraints on holographic dark energy model with matter creation in Brans–Dicke theory and thermodynamic analysis

The adiabatic matter creation for a spatially flat, homogeneous and isotropic universe is discussed in holographic dark energy (HDE) in the context of Brans–Dicke theory. Since the HDE density is considered as a dynamical cosmological constant, it is natural to study it in a dynamical frame of Brans–Dicke theory. We show that HDE model with matter creation, whose infrared cutoff is set by the Hubble horizon, leads to solution that unify the early and late time acceleration. The latest observational data from supernovae (Pantheon sample), Hubble H ( z ) and latest local H 0 by SH0ES are used for constraining the model parameters. Using the Markov chain Monte Carlo method, we obtain the best fit values of the parameters for different models. It is found that the HDE model without matter creation does not achieve transition phase when the Hubble horizon is taken as an IR cut-off. However, it achieves the phase transition from decelerated phase to accelerated phase with same IR cut-off with the inclusion of gravitationally induced matter creation. The best fit values of HDE model with matter creation are fitted very well in Hubble error data. Using AIC and BIC, and geometrical diagnostic parameters, we compare the models with Λ CDM model and discuss the viability of the model. A detailed thermodynamic analysis is also carried out.

Constraining the anisotropy of the Universe with the X-ray and UV fluxes of quasars

We test the anisotropy in the Finslerian cosmological model with the X-ray and ultraviolet (UV) fluxes of quasars. The 2015 and 2020 compilations of quasars are used in the cosmological constraints. We find that the dipole direction given by the 2015 quasar compilation is not far away from the one provided by the Pantheon sample and the angular differences are around 30^{circ }. The Pantheon sample is combined with quasars as the “standardized candles” to test the cosmic anisotropy. The results from two combined datasets are consistent. They show that the dipole anisotropy is weak in the Finslerian cosmological model. We investigate the Hubble constant H_0 in the Finslerian cosmological model. Though the central value of H_0 from the combination of six gravitationally lensed quasars, Pantheon sample, and 2020 quasar compilation decreases a little bit, it is consistent with the result from six gravitationally lensed quasars within statistical uncertainties.

Measuring cosmological parameters with a luminosity–time correlation of gamma-ray bursts

ABSTRACT Gamma-ray bursts (GRBs), as a possible probe to extend the Hubble diagram to high redshifts, have attracted much attention recently. In this paper, we select two samples of GRBs that have a plateau phase in X-ray afterglow. One is short GRBs (SGRBs) with plateau phases dominated by magnetic dipole (MD) radiations. The other is long GRBs (LGRBs) with gravitational wave (GW) dominated plateau phases. These GRBs can be well standardized using the correlation between the plateau luminosity L0 and the end time of plateau tb. The so-called circularity problem is mitigated by using the observational Hubble parameter data and Gaussian process method. The calibrated L0 – tb correlations are also used to constrain Lambda cold dark matter (ΛCDM) and w(z) = w0 models. Combining the MD–LGRBs sample from Wang et al. (2021) and the MD–SGRBs sample, we find $\Omega _{\mathrm{ m}} = 0.33_{-0.09}^{+0.06}$ and ΩΛ = $1.06_{-0.34}^{+0.15}$ excluding systematic uncertainties in the non-flat ΛCDM model. Adding Type Ia supernovae from Pantheon sample, the best-fitting results are w0 = $-1.11_{-0.15}^{+0.11}$ and Ωm = $0.34_{-0.04}^{+0.05}$ in the w = w0 model. These results are in agreement with the ΛCDM model. Our result supports that selection of GRBs from the same physical mechanism is crucial for cosmological purposes.

A new comparison between holographic dark energy and standard varLambda -cosmology in the context of cosmography method

The cosmography method is a model-independent technique used to reconstruct the Hubble expansion of the Universe at low redshifts. In this method, using the Hubble diagrams from Type Ia Supernovae (SNIa) in Pantheon catalog, quasars and Gamma-Ray Bursts (GRB), we put observational constraints on the cosmographic parameters in holographic dark energy (HDE) and concordance varLambda CDM models by minimizing the error function chi ^2 based on the statistical Markov Chain Monte Carlo (MCMC) algorithm. Then, we compare the results of the models with the results of the model-independent cosmography method. Except for the Pantheon sample, we observe that there is a big tension between standard cosmology and Hubble diagram observations, while the HDE model remains consistent in all cases. Then we use different combinations of Hubble diagram data to reconstruct the Hubble parameter of the model and compare it with the observed Hubble data. We observe that the Hubble parameter reconstructed from the model-independent cosmography method has the smallest deviation from the Hubble data and the varLambda CDM (HDE) model has the largest (middle) deviation, especially when we keep the observational data point 226^{+8.0}_{-0.8} at redshift z=2.36 in the analysis. On the contrary, in the redshift z <1, we see that the compatibility of varLambda CDM cosmology and observation is even better than the model independent cosmography method.

Do the observational data favor a local void?

The increasing tension between the different local direct measurements of the Hubble expansion rate and that inferred from the Cosmic Microwave Background observation by $\Lambda$-Cold-Dark-Matter model could be a smoking gun of new physics, if not caused by either observational systematics or local bias. We generalize previous investigation on the local bias from a local void by globally fitting the Pantheon sample over all parameters in the radial profile function of a local void described by an inhomogeneous but isotropic Lema\^{i}tre-Tolman-Bondi metric with a cosmological constant. Our conclusion strengths the previous studies that the current tension on Hubble constant cannot be saved by a local void alone.

Touch of neutrinos on the vacuum metamorphosis: Is the H0 solution back?

With the entrance of cosmology in its new era of high precision experiments, low- and high-redshift observations set off tensions in the measurements of both the present-day expansion rate (${H}_{0}$) and the clustering of matter (${S}_{8}$). We provide a simultaneous explanation of these tensions using the Parker-Raval vacuum metamorphosis (VM) model with the neutrino sector extended beyond the three massless Standard Model flavors and the curvature of the universe considered as a model parameter. To estimate the effect on cosmological observables we implement various extensions of the VM model in the standard cosmomc pipeline and establish which regions of parameter space are empirically viable to resolve the ${H}_{0}$ and ${S}_{8}$ tensions. We constrain the parameter space employing the following datasets: (i) the cosmic microwave temperature and polarization data from the Planck mission, (ii) baryon acoustic oscillations (BAO) measurements, and (iii) the Pantheon sample of Supernovae type Ia. We find that the likelihood analyses of the physically motivated VM model, which has the same number of free parameters as in the spatially flat $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model, always gives ${H}_{0}$ in agreement with the local measurements (even when BAO or Pantheon data are included) at the price of much larger ${\ensuremath{\chi}}^{2}$ than $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. The inclusion of massive neutrinos and extra relativistic species quantified through two well-known parameters $\ensuremath{\sum}{m}_{\ensuremath{\nu}}$ and ${N}_{\mathrm{eff}}$, does not modify this result, and in some cases improves the goodness of the fit. In particular, for the original $\mathrm{VM}+\ensuremath{\sum}{m}_{\ensuremath{\nu}}+{N}_{\mathrm{eff}}$ and the Planck+BAO+Pantheon dataset combination, we find evidence for $\ensuremath{\sum}{m}_{\ensuremath{\nu}}=0.8{0}_{\ensuremath{-}0.22}^{+0.18}\text{ }\text{ }\mathrm{eV}$ at more than $3\ensuremath{\sigma}$, no indication for extra neutrino species, ${H}_{0}=71.0\ifmmode\pm\else\textpm\fi{}1.2\text{ }\text{ }\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ in agreement with local measurements, and ${S}_{8}=0.755\ifmmode\pm\else\textpm\fi{}0.032$ that solves the tension with the weak lensing measurements.

Constraining effective equation of state in f(Q,\xa0T) gravity

New high-precision observations are now possible to constrain different gravity theories. To examine the accelerated expansion of the Universe, we used the newly proposed f(Q, T) gravity, where Q is the non-metricity, and T is the trace of the energy–momentum tensor. The investigation is carried out using a parameterized effective equation of state with two parameters, m and n. We have also considered the linear form of f(Q,T)= Q+bT, where b is constant. By constraining the model with the recently published 1048 Pantheon sample, we were able to find the best fitting values for the parameters b, m, and n. The model appears to be in good agreement with the observations. Finally, we analyzed the behavior of the deceleration parameter and equation of state parameter. The results support the feasibility of f(Q, T) as a promising theory of gravity, illuminating a new direction towards explaining the Universe dark sector.

Observational constraints on Myrzakulov gravity

We use data from Supernovae (SNIa) Pantheon sample, from Baryonic Acoustic Oscillations (BAO), and from cosmic chronometers measurements of the Hubble parameter (CC), alongside arguments from Big Bang Nucleosynthesis (BBN), in order to extract constraints on Myrzakulov $F(R,T)$ gravity. This is a connection-based theory belonging to the Riemann-Cartan subclass, that uses a specific but non-special connection, which then leads to extra degrees of freedom. Our analysis shows that both considered models lead to $\sim 1 \sigma$ compatibility in all cases. For the involved dimensionless parameter we find that it is constrained to an interval around zero, however the corresponding contours are slightly shifted towards positive values. Furthermore, we use the obtained parameter chains so to reconstruct the corresponding Hubble function, as well as the dark-energy equation-of-state parameter, as a function of redshift. As we show, Model 1 is very close to $\Lambda$CDM scenario, while Model 2 resembles it at low redshifts, however at earlier times deviations are allowed. Finally, applying the AIC, BIC and the combined DIC criteria, we deduce that both models present a very efficient fitting behavior, and are statistically equivalent with $\Lambda$CDM cosmology, despite the fact that Model 2 does not contain the latter as a limit.

On the Hubble Constant Tension in the SNe Ia Pantheon Sample

Abstract The Hubble constant (H 0) tension between Type Ia supernovae (SNe Ia) and Planck measurements ranges from 4 to 6σ. To investigate this tension, we estimate H 0 in the ΛCDM and w 0 w a CDM (cold dark matter) models by dividing the Pantheon sample, the largest compilation of SNe Ia, into 3, 4, 20, and 40 bins. We fit the extracted H 0 values with a function mimicking the redshift evolution: g ( z ) = H 0 ( z ) = H ˜ 0 / ( 1 + z ) α , where α indicates an evolutionary parameter and H ˜ 0 = H 0 at z = 0. We set the absolute magnitude of SNe Ia so that H 0 = 73.5 km s − 1 Mpc − 1 , and we fix fiducial values for Ω 0 m Λ CDM = 0.298 and Ω 0 m w 0 w a CDM = 0.308 . We find that H 0 evolves with redshift, showing a slowly decreasing trend, with α coefficients consistent with zero only from 1.2 to 2.0σ. Although the α coefficients are compatible with zero in 3σ, this however may affect cosmological results. We measure locally a variation of H 0 ( z = 0 ) − H 0 ( z = 1 ) = 0.4 km s − 1 Mpc − 1 in three and four bins. Extrapolating H 0 ( z ) to z = 1100, the redshift of the last scattering surface, we obtain values of H 0 compatible in 1σ with Planck measurements independent of the cosmological models and number of bins we investigated. Thus, we have reduced the H 0 tension in the range from 54% to 72% for both cosmological models. If the decreasing trend of H 0 ( z ) is real, it could be due to astrophysical selection effects or to modified gravity.

A new measurement of the Hubble constant using Type Ia supernovae calibrated with surface brightness fluctuations

We present a new calibration of the peak absolute magnitude of Type Ia supernovae (SNe Ia) based on the surface brightness fluctuations (SBF) method, aimed at measuring the value of the Hubble constant. We build a sample of calibrating anchors consisting of 24 SNe hosted in galaxies that have SBF distance measurements. Applying a hierarchical Bayesian approach, we calibrate the SN Ia peak luminosity and extend the Hubble diagram into the Hubble flow by using a sample of 96 SNe Ia in the redshift range 0.02 &lt; z &lt; 0.075, which was extracted from the Combined Pantheon Sample. We estimate a value of H0 = 70.50 ± 2.37 (stat.) ± 3.38 (sys.) km s−1 Mpc−1 (i.e., 3.4% stat., 4.8% sys.), which is in agreement with the value obtained using the tip of the red giant branch calibration. It is also consistent, within errors, with the value obtained from SNe Ia calibrated with Cepheids or the value inferred from the analysis of the cosmic microwave background. We find that the SNe Ia distance moduli calibrated with SBF are on average larger by 0.07 mag than those calibrated with Cepheids. Our results point to possible differences among SNe in different types of galaxies, which could originate from different local environments and/or progenitor properties of SNe Ia. Sampling different host galaxy types, SBF offers a complementary approach to using Cepheids, which is important in addressing possible systematics. As the SBF method has the ability to reach larger distances than Cepheids, the impending entry of the Vera C. Rubin Observatory and JWST into operation will increase the number of SNe Ia hosted in galaxies where SBF distances can be measured, making SBF measurements attractive for improving the calibration of SNe Ia, as well as in the estimation of H0.

Hubble constant and sound horizon from the late-time Universe

We measure the expansion rate of the recent Universe and the calibration scale of the baryon acoustic oscillation (BAO) from low-redshift data. BAO relies on the calibration scale, i.e., the sound horizon at the end of drag epoch $r_d$, which often imposes a prior of the cosmic microwave background (CMB) measurement from the Planck satellite. In order to make really independent measurements of $H_0$, we leave $r_d$ completely free and use the BAO data sets combined with the 31 observational $H(z)$ data, GW170817 and Pantheon sample of Type Ia supernovae. In $\Lambda$CDM model, we get $H_0=68.63^{+1.75}_{-1.77}$ km s$^{-1}$ Mpc$^{-1}$, $r_d=146.85^{+3.29}_{-3.77}$ Mpc. For the two model-independent reconstructions of $H(z)$, we obtain $H_0=68.02\pm1.82$ km s$^{-1}$ Mpc$^{-1}$, $r_d=148.18^{+3.36}_{-3.78}$ Mpc in the cubic expansion, and $H_0=68.58\pm1.76$ km s$^{-1}$ Mpc$^{-1}$, $r_d=148.02^{+3.63}_{-3.60}$ Mpc in the polynomial expansion. The values of Hubble constant $H_0$ and sound horizon $r_d$ are consistent with the estimate derived from the Planck CMB data assuming a flat $\Lambda$CDM model, but $H_0$ is in $2.4\sim2.6$ $\sigma$ tension with SH0ES 2019, respectively.