A Better Reconciliation of Hubble Tension in the Dark Energy Scalar Field

Le Fu, Li Chen, Maoyou Yang, Junmei Wang, and Ming-Jian Zhang

1 School of Information and Automation Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China

2 International School for Optoelectronic Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China zhangmj@mail.bnu.edu.cn

Abstract Hubble tension between the local measurement and global observation has been a key problem in cosmology.In this paper, we consider the quintessence scalar field, phantom field and quintom field as the dark energy to reconcile this problem.Different from most previous work,we start from the dimensionless equation of state(w)of dark energy,not a parameterization of potential.The combined analysis shows that observational data sets favor Hubble constant, which can reconcile Hubble tension within 1.20σ.We also perform a Bayes factor analysis using the MCEvidence code, and confirm that the phantom scalar field is still the most effective.To investigate the reason of Hubble tension,we analyze the density parameter.The comparison shows that the scalar fields provide a slightly larger Ωbh2 and smaller Ωch2 than the standard ΛCDM model.We finally analyze a possible reason of Hubble tension from the kinematic acceleration a¨.We find an interesting physical phenomenon.The acceleration a¨ in these scalar fields are similar as the ΛCDM model at about redshift z ＞0.5.However,they increase and deviate from each other at low redshift,especially in the near future.Only the a¨ in phantom scalar field will decrease in the future.

Key words: (cosmology:) dark energy – (cosmology:) cosmological parameters – cosmology: theory

1.Introduction

The secular ΛCDM model in explaining the cosmic acceleration, favors well against the large spans of cosmological data.With the improvement of volume and accuracy of observational data,however,discrepancies of some cosmological parameters in this model become increasingly serious.Especially, tension of Hubble constant H0between the global and local measurement presents a statistical significance.In the latest local measurement(Riess et al.2021), SH0ES Team issued H0=73.2±1.3 km s-1Mpc-1at 68% CL with 1.8% uncertainty (hereafter R20) using the Cepheids observations.However, global temperature spectrum of cosmic microwave background (CMB) for Planck2018 (Aghanim et al.2020) present H0=67.27±0.60 km s-1Mpc-1at 68%CL in the flat ΛCDM scenario.Their differences have increased to be a 4.2σ.This problem is commonly called “Hubble tension.”

Hubble constant is important to our cosmological research.It does not only play a vital influence on determination of cosmic age, but dominate the physical process such as comic nucleosynthesis and growth of cosmic structure.We also have affirmed that Hubble constant inevitably affects dark energy reconstruction (Zhang & Li 2018).Even, Freedman (2017)believed that Hubble tension may indicate a new physics.The Hubble constant,therefore,is an important observational target for a long time.

The observation of Hubble constant is technically difficult.Primarily, it was estimated from the Hubble law v=H0d, a linear relationship between recession velocity of galaxies and distance.The first Hubble constant H0is about 500 km s-1Mpc-1(Hubble & Humason 1931).This large value is due to the confusion of two generations of pulsating stars in calculation of distance standards.Sandage demonstrated this mistake and revised H0down to 75 km s-1Mpc-1.Accurate distance measurement has always been a problem in the Hubble constant program.In 1921,Leavitt&Pickering(1912)found that the period of brightness fluctuation of Cepheid variables is highly regular, i.e., period–luminosity relation.Cepheids thereafter is used as the standard candles.Hubble constant from the SH0ES Team is just based on this method.Till now, a number of other tools are available,such as the Tip of Red Giant Branch,Surface Brightness Fluctuation,Maser in galaxy NGC 4258,gravitational lens time delays, and fashionable gravitational wave (GW).However, there are also differences between them.For example,the updated Tip of Red Giant Branch obtains H0=69.8±0.6(stat)±1.6(sys)km s-1Mpc-1(Freedman 2021).Meanwhile,the GW observations support a larger value.

In the present paper, we would like to return the Hubble tension in the scalar field dark energy.We consider the quintessence field, phantom field and quintom field.For the scalar field study, an inevitable problem is the modeling of potential V(φ) over scalar field φ.Generally, potential V was understood via parameterization, such as power-law potential V(φ)∝φp, exponential potential V(φ)∝e-λφ.However, we note that parameter H0is usually hidden in potential V(φ),which greatly increases the difficulty of numerical calculations.In this paper, we rebuild the potential V from the equation of state (EoS w) of late dark energy, a dimensionless parameter,which can break above difficulties.For the quintessence field φ(t) and phantom field σ(t), we construct a simplified version basing on the Cai et al.(2007a).Our constraints show that one model can reconcile the Hubble tension at a better level.

This article is organized as follows: In Section 2, we introduce the scalar field.In Section 3 we describe the relevant data we use.We present the reconstruction result, and explore the reason in Section 4.Finally, in Section 5 conclusion and discussion are drawn.

2.Scalar Field Theory

In coming section, we will introduce the construction of scalar field, namely, quintessence field, phantom field and quintom field.

We take into account a spatially flat Friedmann–Robertson–Walker universe with matter and scalar field.The dynamical Friedmann equation can be expressed as follows

The EoS parameter for quintessence scalar field is

Combining with Equation (2), we can obtain the potential

Finally, we have the Hubble parameter

For the phantom scalar field σ, its energy density and pressure are

Different from the quintessence field, the minus sign—is in derivative2˙σ.Following the above operation, we obtain the Hubble parameter

Similarly to quintessence scalar field, the Hubble parameter can be solved logically, as long as the scale fields σ and w are available.However, we should notice that EoS is w ＜-1 in this scalar field.

For the quintom field, it is a combination of quintessence field and phantom field.With the cosmic evolution, the quintessence field can transfer into phantom field, or the phantom field transfers into quintessence field.A number of theoretical works were investigated(Zhao et al.2005;Cai et al.2007b, 2010).Similarly, the Hubble parameter can be expressed as

With regard to the reconstruction of scalar field φ(t)and σ(t),Cai et al.(2007a)put forward a solution and studied the cosmic duality in quintom universe.According to Cai et al.(2007a),we draw a simplified version for the scalar field.For the quintessence field, it is given by

3.Observational Data

3.1.Type Ia Supernovae

The latest Type Ia supernova data we use are Pantheon sample from Scolnic et al.(2018), which consists 1048 data points.For these samples, their redshifts have a wide span of 0.01 ＜z ＜2.3.For each SN Ia,the observed distance modulus is given by

Here matrix Dstatis the diagonal part of the statistical uncertainty.The Csysis the systematic covariance matrix between peak magnitude.They are available in the catalogs of Pan-STARRS.3https://archive.stsci.edu/prepds/ps1cosmo/index.html

The theoretical distance modulus is usually estimated as

where Δμ=μobs-μth.

3.2.Baryon Acoustic Oscillations

The BAO data we use here are 15 latest transversal BAO measurements (Nunes et al.2020), θBAO(z).They are obtained through the BAO signal position in the 2PACF, a modelindependent approach(Jassal et al.2005;Blake et al.2011).There is not a fiducial cosmological model assumption,comparing with the traditional BAO measurement.Their inclusion can break the degeneracy of dark energy model parameters, and improve the constraints significantly (Hernández-Almada et al.2021; Motta et al.2021).The theoretical angular scale is evaluated by

where σiis the error of traditional BAO data.

Figure 1.Posterior distributions of the cosmological parameters in quintessence field for fractional form w = w0 + w1 (left) and logarithmic form w = w0 + w1 model (right) using all the data sets.

3.3.Cosmic Microwave Background

The CMB has become one of the most powerful ways to study the cosmology and the physics of early universe.According to the Planck 2018 (Aghanim et al.2020), we use the full temperature and polarization angular power-spectrum data from Planck 2018.Specifically, they are respectively Plik likelihood,a combination of Planck TT,TE,EE spectra at? ＞29, temperature-only Commander likelihood named Planck_lowl_TT at low multipole 2 ≤? ≤29, and low multipole 2 ≤? ≤29 EE likelihood named Planck_lowl_EE from SimAll.

3.4.Observational Hubble Parameter

H(z) is a direct measurement of the cosmic expansion rate,which can be obtained via the differential ages of passively evolving galaxies (Simon et al.2005; Jimenez & Loeb 2008;Stern et al.2010)

This method is also called cosmic chronometer.In our recent work (Zhang & Xia 2016), we used 30 cosmic chronometer data and studied the dark energy,finding a powerful constraint.Cosmological parameters can be constrained by the observational Hubble parameter data via

4.Observational Constraints and Analysis

4.1.Constraints from all Samples

We obtain cosmological parameter constraints using the Einstein–Boltzmann code CLASS-PT (Chudaykin et al.2020)interfaced with the Montepython Monte Carlo sampler(Audren et al.2013; Brinckmann & Lesgourgues 2019).We use the Python module basing on the Markov chain Monte Carlo approach, to perform the corresponding χ2statistics.

Table 1 Constraints of Cosmological Parameters at 68% C.L.for Different Models Using all the Observational Data Sets

For the quintomA and quintomB scalar field,we respectively consider transformation between quintessence field and phantom field, as shown in Figures 3 and 4.For the matter density, parameter Ωm0still can be obtained with a high precision.For the equation of state, parameters w0and w1deviate from the cosmological constant significantly, which is different from the results in quintessence field and phantom field.For the Hubble constant, they present a better constraint than the quintessence field.However,the tension still locates at 1.52σ-1.69σ.

In short, we find that Hubble tension in the phantom scalar field is the smallest.Moreover, it is little affected by the dark energy parameterization.

4.2.Bayesian Evidence

In this section, we would seek which model is more effective,compared with the standard ΛCDM cosmology.This statistical comparison can be realized through the Bayesian evidence.Here we use the publicly available code MCEvidence(Heavens et al.2017a,2017b)to compute the evidence of the model.It is very convenient because of its only usage of MCMC chains.

According to the Bayes’theorem, we can describe the probability that cosmological model M is true as:

Figure 2.The same as Figure 1 but for phantom field.

Figure 3.The same as Figure 1 but for quintomA field.

4.3.A Reason for the Hubble Tension

To test the reason for the Hubble tension, we perform a comparison on the probability density of Ωbh2,Ωch2and Ωm0h2for different dark energy models,as shown in Figure 5.We also analyze a possible physical phenomenon of the low tension from the kinematics, which expect to provide a new understanding of the Hubble tension.

Figure 4.The same as Figure 1 but for quintomB field.

Table 2 Bayes Factor in the Revised Jeffreys’Scale

The matter density has an important effect on the CMB spectra.It affects the amount of lensing in the CMB spectra and the amplitude of the CMB-lensing reconstruction spectrum.From the Planck 2018 release, it is obtained Ωm0h2=0.1432±0.0013 (Aghanim et al.2020).As shown in Figure 24 of this reference, the Planck Collaboration investigated the TT power spectrum residuals over the value of Ωm0h2.They found that a less lensing is allowed by a lower Ωm0h2.Hence a larger oscillatory residual can be given.In Figure 5,we compare the parameters Ωbh2,Ωch2and Ωm0h2for different dark energy models.First, we find that density parameter Ωbh2in these scalar fields are larger than the value in standard ΛCDM model.Especially, density Ωbh2in the quintessence scalar field is farthest from the standard ΛCDM model.Moreover, we find that the phantom field closest to the standard ΛCDM model.Second, in the middle panel of Figure 5, we find that quintessence scalar field still deviates from the standard value farthest.For parameter Ωch2, the phantom field slightly deviates from the standard ΛCDM model.Finally, we find that density parameter Ωm0h2in the phantom field is still closest to the standard ΛCDM model.Therefore, we are in a dilemma.That is, the phantom scalar field can better solve the Hubble tension,but the corresponding density parameters are closest to the standard ΛCDM model.This similarity makes it difficult to distinguish them.

As pointed out in previous work (Linares Cede?o et al.2021), the Hubble tension can be reconciled is because a dark energy model with phantom-like EoS can generate extra acceleration of the universe, when compared with the fiducial ΛCDM model.Their result is consistent with our work.In order to further reveal the reason why Hubble tension can be reconciled in this scenario, we investigate the kinematica¨ in Figure 6.First,we find that accelerationa¨ in these scalar fields is similar at about redshift z ＞0.5.Moreover, they are much similar as the ΛCDM model.However, for low redshift, we should notice that thea¨ deviates from each other,especially in the near future.More importantly, we note that thea¨ for the phantom scalar field decreases in the future,while for the other field,thea¨ increases in the future.In our knowledge,this is the first time discovering this interesting physical phenomenon.

5.Conclusion and Discussion

Hubble tension has become a key problem in cosmology.It even implies a possibility of new physics or the failure of immortal ΛCDM model.In this paper, we consider three scalar fields as the dark energy to reconcile the Hubble tension.The scalar fields we consider are quintessence field,phantom field and quintom field.The observational data sets,SN Ia from Pantheon samples, transversal BAO measurement, CMB power spectra and H(z) data.The constraints indicate that phantom field can reconcile the Hubble tension to 1.20σ.We also perform a model comparison using the Bayes factor from the public code MCEvidence.The comparison shows that phantom scalar field is still the most effective in these models.

Figure 5.Probability density of Ωbh2, Ωch2 and Ωm0h2 for different models.

Figure 6.Comparison of the kinematics a¨ for different models.

To investigate the reason of the Hubble tension,we perform a series of analysis.From the probability density in Figure 5,we find that the scalar fields provide a bigger Ωbh2and a lower Ωch2, when compared with the standard ΛCDM model.Moreover,the phantom field has a Ωbh2closest to the standard ΛCDM model.It can affect the CMB-lensing spectrum(Aghanim et al.2020) and provide an energy transformation between dark matter and dark energy(Di Valentino et al.2020;Yang et al.2020).

A numerous of previous works (Di Valentino et al.2016, 2021b; Yang et al.2019; Alestas et al.2020;Vagnozzi 2020) find that a phantom-like dark energy can reconcile the Hubble tension.To further reveal the reason, we investigate the kinematica¨ in Figure 6.We find that accelerationa¨ in these scalar fields is similar to the standard ΛCDM model at redshift z ＞0.5.However, for low redshift,we should notice that thea¨ deviates from each other,especially in the near future.For the phantom scalar field,we note that thea¨decreases in the future.While for the other field, thea¨increases in the future.This interesting physical phenomenon was discovered for the first time.

Acknowledgments

We thank the anonymous referee whose suggestions greatly helped us improve this paper.M.-J.Zhang thanks Jing-Zhao Qi for the valuable discussion.Ming-Jian Zhang is supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA075).Li Chen is supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2019MA033).