A new parametrization of Hubble parameter and Hubble tension

Abstract

We present a new Hubble parameterization method and employ observational data from Hubble, Pantheon, and Baryon Acoustic Oscillations to constrain model parameters. The proposed method is thoroughly validated against these datasets, demonstrating a robust fit to the observational Hubble, Pantheon, and BAO data. The obtained best-fit values are $H_0 = 67.5^{+1.3}_{-1.6}$ $\text{km s}^{-1} \text{Mpc}^{-1}$, $\Omega_{\rm{m0}} = 0.2764\pm{0.0094}$, and $\alpha = 0.33\pm{0.22}$, consistent with the Planck 2018 results, highlighting the existence of Hubble tension.

Full text

arXiv:2405.17212v1 [gr-qc] 27 May 2024
A new parametrization of Hubble parameter and Hubble tension
Tong-Yu He,1Jia-Jun Yin,1Zhen-Yu Wang,2Zhan-Wen Han,1,2 and Rong-Jia Yang a 1, 3, 4, 5, †
1College of Physics Science and Technology, Hebei University, Baoding 071002, China
2Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China
3Hebei Key Lab of Optic-Electronic Information and Materials, Hebei University, Baoding 071002, China
4National-Local Joint Engineering Laboratory of New Energy Photoelectric Devices, Hebei University, Baoding 071002, China
5Key Laboratory of High-pricision Computation and Application of Quantum
Field Theory of Hebei Province, Hebei University, Baoding 071002, China
We present a new Hubble parameterization method and employ observational data from Hubble,
Pantheon, and Baryon Acoustic Oscillations to constrain model parameters. The proposed method is
thoroughly validated against these datasets, demonstrating a robust fit to the observational Hubble,
Pantheon, and BAO data. The obtained best-fit values are H0= 67.5+1.3
−1.6km s−1Mpc−1,Ωm0 =
0.2764±0.0094, and α= 0.33±0.22, consistent with the Planck 2018 results, highlighting the existence
of Hubble tension.
I. INTRODUCTION
Hubble constant (H0) is a fundamental parameter that can quantify the rate of cosmic expansion. It is frequently
employed to elucidate the motion of celestial bodies, such as galaxies, relative to the observer’s position. In recent
decades, measurements of the Hubble constant have garnered attention within the scientific community due to no-
table discrepancies among results obtained from diverse measurement methodologies and data sources [1]. Early
measurements of the Hubble constant often entailed observations of primordial cosmic signals, such as the cosmic
microwave background radiation (CMB). For instance, using the ΛCDM standard cosmological model, a Hubble
constant of H0= (67.4±0.5) km s−1Mpc−1was derived [2]. Integrated Baryon Acoustic Oscillations (BAO) measure-
ments with cosmic microwave background (CMB) data from WMAP resulted in H0= (67.63 ±1.30) km s−1Mpc−1
[3]. Early Hubble constant measurements consistently indicate values lower than 70 km s−1Mpc−1[4, 5]. The age of
an old quasar APM 08279+5255 at z= 3.91 also tends to support a lower Hubble constant [6]
Later Hubble constant measurements primarily rely on astrophysical observations such as supernovae and galax-
ies. A result of H0= (73.04 ±1.04) km s−1Mpc−1was derived based on the Cepheid-SN (Cepheid-supernova)
sample [7]. According to [8], they employed a joint analysis of six strongly lensed gravitational lensing events with
measured time delays, providing a Hubble constant estimate of H0= 73.3+1.7
−1.8km s−1Mpc−1. The detection of gravi-
tational wave events from neutron star mergers by LIGO and Virgo yielded an estimate of H0= 70+12
−8km s−1Mpc−1
in 2017. Using the Hubble Space Telescope, the Hubble constant was directly measured as H0= (74.03 ±1.42)
km s−1Mpc−1through the distance ladder method, providing calibration for the magnitude-redshift relation for 253
Type Ia supernovae [9]. It is evident that numerous late-time measurements favor H0>70 km s−1Mpc−1with a
minority reporting a value of H0≈70 km s−1Mpc−1.
Observations of the early universe, including CMB data, typically yield lower values for the Hubble constant
(H0). Conversely, measurements of celestial bodies at closer distances, such as supernovae and other large-scale
cosmic structures, result in higher values for H0. The inconsistency between the results obtained from these two
methods has captured the attention of researchers. This disparity is referred to as the Hubble tension [4, 10, 11],
and the statistical significance of these differences surpasses the range of measurement errors, prompting significant
discussion and research. The existence of Hubble tension suggests the possibility of unknown physical processes or
issues in observational systems concerning the evolution and nature of the universe [4, 12–15]. Efforts to address
this issue include improvements in data analysis, reduction of systematic errors, the adoption of new observational
methods, and a re-examination of cosmological models.
aCorresponding author
†yangrongjia@tsinghua.org.cn

2
In addressing the Hubble tension, Lin et al. [16] proposed a potential resolution in the Early Dark Sector (EDS),
where dark matter mass depends on the Early Dark Energy (EDE) scalar field. They explored a Plank-suppressed
EDE coupled with dark matter, finding that this Triggered Early Dark Sector (tEDS) model naturally resolves the
coincidence problem of EDE on the background level. Fitting the current cosmological data, including local distance
gradients and low-redshift amplitudes of fluctuations, they obtained a Hubble constant of H0= 71.2km s−1Mpc−1.
In the presence of non-standard cosmology, a reconciliation between CMB and local measurements has yielded
H0= 70−74 km s−1Mpc−1[17]. According to [18], they explored a novel dark fluid model known as the Exponential
Acoustic Dark Energy (eADE) model to alleviate the tension in the Hubble telescope. Comparisons with the standard
model resulted in H0= 70.06+1.13
−1.09 km s−1Mpc−1. Some dynamical dark energy models, see for example [19–34],
could also reduce the Hubble tension.
In this paper, we proposed a novel Hubble parameterization method and constrained the model parameters us-
ing observations from Hubble parameter (Hubble for short later), Pantheon, and BAO data. The results indicate
that the Hubble tension may also exist between the Hubble+Pantheon+BAO data and the measurements from local
Cepheid–type Ia supernova distance ladder.
The script is structured as follows: In Sec. II, we have proposed a new parameterisation method for Hubble
parameters. In Sec. III, utilizing the Markov Chain Monte Carlo (MCMC) method, we constrains the cosmologi-
cal model parameters, namely H0,Ωm0 , and α, using the Hubble dataset, the Hubble+Pantheon dataset, and the
Hubble+Pantheon+BAO dataset. Sec IV presents the result and Sec V is the conclusion of the study.
II. A NEW PARAMETRIZATION OF HUBBLE PARAMETER
According to the Planck 2018 results, the spacetime is spatially flat: ΩK0 = 0.001 ±0.002 [2], so here we consider a
flat Friedmann-Robertson-Walker-Lemaˆıtre (FRWL) spacetime
ds2=−dt2+a2(t)hdr2+r2(dθ2+ sin2θdφ2)i,(1)
where a(t)is the scale factor. We use the unit c= 1 here. The Friedmann equations take the form
H2≡˙a
a2
=8πG
3ρ=H2
0hΩm0(1 + z)3+ Ωx(z)i,(2)
where Ωm0 is the matter density at present time and Ωx0 represents contributions from dark energy and can be
expressed as
Ωx(z) = Ωx0 exp 3Zz
0
1 + wx(z′)
1 + z′dz′,(3)
where wx(z) = px/ρxis the equation of state (EoS) of dark energy. For ΛCDM model, wx(z) = −1and Ωx(z) =
1−Ωm0. For a constant EoS wx(z)6=−1, we have Ωx(z) = (1 −Ωm0 )(1 + z)3(1+wx)and
H2≡˙a
a2
=8πG
3ρ=H2
0hΩm0(1 + z)3+ (1 −Ωm0 )(1 + z)3(1+wx)i.(4)
There are usually two way to parameterize dark energy: one way is to parameterize the EoS (such as the widely
used CPL parameterization model [35, 36]), the other is to directly parameterize the Hubble parameter. Here we
adopt the latter approach. Since the Hubble tension mainly rises from Planck 2018 (based on ΛCDM model) and
Cepheid calibrated supernovae Ia measurements [37], we consider a parameterization slightly different from ΛCDM.
Furthermore, since the degeneracy between parameters could affect the fitting results, we consider the model with
as few parameters as possible. The Hubble parameter we suggest takes the following form
H2=H2
0hΩm0(1 + z)3+ (1 −Ωm0 )(1 + z)αi,(5)
where αis a constant. This parameterization model is simperer than Eq. (4), though they are equivalent via wx=
−1 + α/3.
In the next section of our investigation, we employed multiple observational data, including Hubble parameter
data, Pantheon samples, and BAO data. We utilized the MCMC method to constrain the model parameters H0,Ωm0,
and α.

3
III. OBSERVATIONAL DATA AND METHODOLOGY
In the previous section, we discussed a novel Hubble parameterization method, and now we aim to validate
whether the approximate values of the model parameters can effectively describe the current universe based on
observational data. We primarily employed three datasets, namely the Hubble dataset with 62 data points, the
Pantheon dataset with 1701 data points, and a set of six Baryon BAO datasets. For numerical analysis and parameter
constraints using the mentioned datasets, we utilized the emcee code [38]. Additionally, to understand the outcomes
of the MCMC study, we employed 64 walkers and 2000 steps across all datasets. And the final results will be
discussed in the form of 2D contour plots with 1-σand 2-σerrors.
A. Observational Hubble data
Utilizing Hubble observational data to constrain cosmological models is a significant methodology. This approach
involves measuring the historical expansion of the universe to derive constraints on cosmological parameters. The
Hubble parameter, denoted as H(z), characterizes the rate of cosmic expansion and serves as a fundamental cosmo-
logical quantity. Its dependence on redshift (z) provides essential insights into the cosmic evolution [39, 40].
H(z) = −1
1 + z
dz
dt .(6)
In this part, dz is obtained through spectroscopic surveys, making a measurement of dt a means to derive the model-
independent value of the Hubble parameter. The H(z)data set we use consists of 34 H(z)measurements obtained by
calculating the differential ages of galaxies, which is called cosmic chronometer [41–46], and 28 H(z)measurements
inferred from the BAO peak in the galaxy power spectrum [47–59], as collected in [60]. As the data provided by the
DA method is independent of cosmological models, it can be employed to explore alternative cosmological models.
The range of these data points is 0 < z < 1.965. Furthermore, we have adopted an intermediate value of H0= 70
km/s/Mpc for our analysis [61]. To determine the mean values of the model parameters H0,Ωm0 and α(using
maximum likelihood analysis), we employed the chi-square function as follows [62]:
χ2
H=
62
X
i=1
[Hth
i(H0,Ωm0, α)−Hobs
i(zi)]2
σ2
H(zi).(7)
The theoretical value of the Hubble parameter is denoted as Hth
i, the observed value as Hobs
i, and σ2
Hrepresents the
standard error of the observed H(z)values at redshift zi.
As shown in Figure 1, we obtained the best-fitting values for the model parameters H0,Ωm0, and α, along with 1-σ
and 2-σconfidence level contours. The best-fitting values are H0=65.4+2.0
−2.2km s−1Mpc−1,Ωm0 =0.266 ±0.013 and α
=0.71 ±0.31 at 1-σand confidence level. Additionally, in Figure 2, we present error bar plots for the aforementioned
Hubble data, compared with the ΛCDM model (H0= 67.4km/s/Mpc and Ωm0 = 0.315) [2]. The model proposed
here effectively captures the observed Hubble dataset.
B. Observational Pantheon data
SNe Ia commonly known as standard candles, serve as powerful distance probes for studying the cosmological
dynamics of the universe. Over the past two decades, the sample size of SNe Ia datasets has steadily increased. We
employed the largest supernova Ia sample to date, Pantheon+, which amalgamates data from various surveys such
as the Sloan Digital Sky Survey (SDSS), the SNe Legacy Survey (SNLS), the Hubble Space Telescope (HST) survey,
and others, comprising 1701 confirmed supernovae from 18 different surveys [7, 63]. The Pantheon+ dataset spans
a redshift range of z ∈(0.0012, 2.2614), with a notable increase in the number of SNe at low redshifts. We need to
fit the model parameters by comparing the theoretical distance modulus µth values with the observed µobs values.
Each distance modulus can be computed using the following formula:
µth(z) = 5 log10
dL(z)
Mpc + 25,(8)

4
60 65 70
H
0
−0.5
0.0
0.5
1.0
1.5
α
0.22
0.24
0.26
0.28
0.30
Ω
m
0
H
0
= 65.4
+2.0
−2.2
0.24 0.26 0.28 0.30
Ω
m
0
Ω
m
0
= 0.266± 0.013
−0.5 0.0 0.5 1.0 1.5
α
α
= 0.71± 0.31
Hubble data
FIG. 1. The best-fit values for the model parameters H0,Ωm0 , and αwith 1−σand 2−σconfidence level contours obtained from
Hubble data.
0.5 1.0 1.5 2.0 2.5
z
50
75
100
125
150
175
200
225
250
H(z)
Our model
ΛCDM
Hubble data
FIG. 2. The relationship between the Hubble function H(z)and the redshift zconstrained from Hubble data. The red line
represents the model proposed here and the solid black line corresponds to the ΛCDM model.
The luminosity distance dL(z)is
dL(z) = (1 + z)Zz
0
dz′
H(z′),(9)

5
The chi-square function χ2
SN about the Pantheon data is
χ2
SN =
1701
X
i,j=1
∆µiC−1
SN ij ∆µj.(10)
In this expression, CSN represents the covariance matrix, as introduced by Suzuki [64]. Additionally, ∆µi =
µth(zi, H0,Ωm0 , α)−µobsis defined as the disparity between the observed distance modulus value, derived from
cosmic data, and its theoretical counterpart generated from the model using the parameter space H0,Ωm0 and α.
62 64 66 68 70 72
H
0
0.0
0.5
1.0
α
0.24
0.26
0.28
0.30
Ω
m
0
H
0
= 67.2± 1.7
0.24 0.26 0.28 0.30
Ω
m
0
Ω
m
0
= 0.264± 0.012
0.0 0.5 1.0
α
α
= 0.47 ± 0.26
Pantheon + Hubble data
FIG. 3. The best-fit values for the model parameters H0,Ωm0 , and αwith 1−σand 2−σconfidence level contours obtained from
Pantheon+Hubble data.
Taking χ2
H+χ2
SN as the minimum constraint for the model parameters H0,Ωm0, and α, we obtained the best-fit
values for these parameters using the Hubble parameter and Pantheon data, as shown in Figure 3 with 1-σand
2-σconfidence level contours. The best-fit values are H0=67.2±1.7km s−1Mpc−1,Ωm0 =0.264 ±0.012 and α
=0.47 ±0.26 at 1-σconfidence level. Additionally, in Figure 4, we present the error bar plot for the aforementioned
supernova data, comparing our model with the ΛCDM model (H0= 67.4km/s/Mpc and Ωm0 = 0.315) [2]. The
model proposed here demonstrates a good fit to the Hubble+Pantheon dataset.
C. Observational Baryon Acoustic Oscillations data
BAO arises from acoustic density waves in the early universe’s primordial plasma, causing fluctuations in the
observable baryonic matter density in the cosmos. Detecting BAO involves large-scale surveys and redshift mea-
surements to gather information about the large-scale structure of the universe. BAO detectors offer highly precise
measurements of large-scale structures, largely unaffected by uncertainties in the nonlinear evolution of the matter
density field and other systematic errors. They are considered a standard ruler for measuring the cosmological back-
ground evolution [65]. To improve statistical significance, broaden the redshift range, and gain more comprehensive

6
0.0 0.5 1.0 1.5 2.0 2.5
z
32
34
36
38
40
42
44
46
48
μ
(z)
1.25 1.26 1.27 1.28 1.29
44.6
44.8
45.0
Our model
ΛCDM
Patheon data
FIG. 4. The relationship between the distence modulus µ(z)and the redshift zfor our model and ΛCDM constrained from
Pantheon+Hubble data.
zBAO 0.106 0.2 0.35 0.44 0.6 0.73
dA(z∗)
DV(zBAO)30.95 ±1.46 17.55 ±0.60 10.11 ±0.37 8.44 ±0.67 6.69 ±0.33 5.45 ±0.31
TABLE I. Values of dA(z∗)/DV(zBAO)for distinct values of zBAO.
cosmological insights, we utilize the combined data from six different BAO measurements at various redshifts [66–
68]. The information taken from the BAO peaks in the matter power spectrum can be used to determine the Hubble
parameter H(z)and the angular diameter distance dA(z)which takes the form
dA(z) = Zz
0
dz′
H(z′).(11)
The combination of the angular diameter distance and the Hubble parameter, DV(z), is given by [69]
DV(z) = hdA(z)2z/H (z)i1/3.(12)
The chi-square function χ2
BAO about BAO is given by the following expression
χ2
BAO =XTC−1
BAOX, (13)
where
X=












dA(z⋆)
DV(0.106) −30.95
dA(z⋆)
DV(0.2) −17.55
dA(z⋆)
DV(0.35) −10.11
dA(z⋆)
DV(0.44) −8.44
dA(z⋆)
DV(0.6) −6.69
dA(z⋆)
DV(0.73) −5.45












.
The inverse covariance matrix C−1
BAO is represented in [68]. The six BAO datasets are provided in Table I. At
z∗≃1091, photon decoupling occurred, allowing the CMB to propagate through the universe, eventually becom-
ing the observed cosmic microwave background radiation today. This redshift value is derived through detailed
observations and analysis of the CMB [70].

7
64 66 68 70 72
H
0
−0.5
0.0
0.5
1.0
α
0.25
0.26
0.27
0.28
0.29
0.30
0.31
Ω
m
0
H
0
= 67.5
+1.3
−1.6
0.26 0.28 0.30
Ω
m
0
Ω
m
0
= 0.2764± 0.0094
01
α
α
= 0.33 ± 0.22
Bao + Pantheon + Hubble data
FIG. 5. The best-fit values for the model parameters H0,Ωm0 , and αwith 1−σand 2−σconfidence level contours obtained from
the Hubble+Pantheon+BAO data.
By minimizing χ2
H+χ2
SN +χ2
BAO to constrain the model parameters H0,Ωm0 , and α, we determined the best-fit
values using the Hubble+Pantheon+BAO dataset, as illustrated in Figure 5. The resulting values are H0= 67.5+1.3
−1.6
km s−1Mpc−1,Ωm0 = 0.27684 ±0.0094, and α= 0.33 ±0.22, at 1-σconfidence level. Table II presents the best-
fit values and their corresponding uncertainties obtained from three independent simulations of the model using
different datasets. We observed that with an increasing volume of data, the effectiveness of our model fitting in
constraining parameters improves progressively.
Data H0(km/s/Mpc) Ωm0 α
Prior (64.9,76.8)
Hubble 65.4+2.0
−2.20.266 ±0.013 0.71 ±0.31
Hubble+Pantheon 67.2±1.7 0.264 ±0.012 0.47 ±0.26
Hubble+Pantheon+BAO 67.5+1.3
−1.60.2764 ±0.0094 0.33 ±0.22
TABLE II. A summary of the results derived from the analysis of three datasets.
IV. RESULTS
In this section, We discussed the results in Sec III. We leveraged multiple observational datasets, including Hub-
ble, Hubble+Pantheon samples and Hubble+Pantheon+BAO data [71]. Employing the MCMC method, we finally
constrained the model parameters H0,Ωm0, and α.
Firstly, the Hubble dataset, consisting of 62 data points, is utilized to quantify the historical expansion of the uni-
verse. In Figure 1, the best-fitting values for the model parameters are determined as H0= 65.4+2.0
−2.2km s−1Mpc−1,

8
Ωm0 = 0.266 ±0.013 , and α= 0.71 ±0.31. It can be observed that the fitting parameters H0have relatively large
uncertainties, and a significant contributing factor is the limited amount of data. In Figure 2, we can find that our
model is relatively close to the ΛCDM model, and it fits well with the observational data. Next, in order to reduce
the uncertainties in the parameters, we have incorporated 1701 observational data points from the Pantheon dataset
[61]. Figure 3 presents the fitting results: H0= 67.2±1.7km s−1Mpc−1,Ωm0 =0.264 ±0.012 and α=0.47 ±0.26. It
is observed that the error in H0has decreased. In Figure 4, our model is seen to fit well with the observational data
from Pantheon and is close to the ΛCDM model. Next, we included an additional 6 BAO data points and obtained
the fitting results: H0= 67.5+1.3
−1.6km s−1Mpc−1,Ωm0 = 0.2764 ±0.0094, and α= 0.33 ±0.22. In Table II, we can
find with the increase in the observational data size, the constraints on the parameters H0,Ωm0, and αbecome pro-
gressively more accurate. To validate the model’s efficacy, we compared error bar plots for Hubble and Pantheon
datasets with the ΛCDM model. Consistently, the model fits the observed dataset very well.
V. CONCLUSIONS
In conclusion, we proposed a novel Hubble parameterization method and constrained the model parameters using
observations from Hubble, Pantheon, and BAO. This model is validated with Hubble, Pantheon, and BAO data,
providing best-fit values for H0= 67.5+1.3
−1.6km s−1Mpc−1,Ωm0 = 0.2764 ±0.0094, and α= 0.33 ±0.22, consistent
well with the Planck 2018 data (H0= 67.4±0.5Km/s/Mpc [2]), but deviating from Cepheid-supernova observation
(H0= 73.04 ±1.04 km s−1Mpc−1[7]) by more than 4 σ. The Hubble tension was mainly triggered with the higher
Hubble constant (H0= 74.0±1.04 km s−1Mpc−1[7]) estimated from the local Cepheid–type Ia supernova distance
ladder being at odds with the lower value extrapolated from CMB data, assuming the standard ΛCDM cosmological
model (H0= 67.4±0.5km s−1Mpc−1[2]). However, our analyses indicate that the Hubble tension may also exist
between the Hubble+Pantheon+BAO data and the measurements from local Cepheid–type Ia supernova distance
ladder if taking the parametrization (5). These results may contribute to our understanding of the current universe
and its cosmodynamics.
Future research directions may encompass the expansion of datasets, exploration of additional cosmological mod-
els, and investigation into the impact of observational technologies on parameter constraints. The continuous ad-
vancement in observational methods is anticipated to refine our understanding of the universe’s evolution.
ACKNOWLEDGMENTS
This study is supported in part by National Natural Science Foundation of China (Grant No. 12333008) and Hebei
Provincial Natural Science Foundation of China (Grant No. A2021201034).
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