Lower $H_0$ within The Theoretical Insights of Special Cosmological Model Pertains to Derived from The Infinite Future

TL;DR

This study proposes a novel method to lower the Hubble constant estimate within a specific cosmological model by combining theoretical and observational data, resulting in values that alleviate the current tension in measurements.

Contribution

It introduces a new approach that integrates a fixed theoretical $H(z)$ value with observational data using Gaussian Processes, leading to refined cosmological parameter constraints and a lower $H_0$ estimate.

Findings

Derived $H_0$ is significantly lower than Planck and Riess estimates.

Incorporating theoretical $H(z)$) reduces tension in $H_0$ measurements.

Refined cosmological parameters improve model consistency.

Abstract

In the realm of the $\omega$CDM cosmological model with quiescence or quintessence as the dark energy, characterized by $\omega>-1$, there exists a fixed value of $H(z)$ at $z=-1$, devoid of dependency on other cosmological parameters. To constrain the Hubble constant, we amalgamated this theoretical $H(z)$ value with the latest 35 observational $H(z)$ data (OHD) using a Gaussian Process (GP) approach that is unrelated to cosmological models but intertwined with kernel functions. Within such a specialized cosmological paradigm, our scrutiny yields $H_0=64.89\pm4.68\ {\rm km\ s^{-1} Mpc^{-1}}$, markedly inferior to the $H_0$ estimate posited by the Planck Collaboration (2018) (exhibiting a tension of $0.53\sigma$), and substantially less than that of \cite{Riess2016A} (manifesting a tension of $1.67\sigma$). Conversely, when solely utilizing the latest 35 OHD, the inferred $H_0=68.77\pm6.24\ {\rm km\ s^{-1} Mpc^{-1}}$ (with a tension of $0.50\sigma$). Leveraging this derived $H_0$, we subsequently engage in $\chi^2$ statistics via the Markov Chain Monte Carlo (MCMC) technique to constrain cosmological parameters. Within the flat $\omega$CDM model, we deduce $\Omega_M=0.32\pm0.02$ and $\omega=-0.80\pm0.05$, whereas in the non-flat $\omega$CDM model, we ascertain $\Omega_M=0.34\pm0.05$, $\Omega_\Lambda=0.76\pm0.12$, and $\omega=-0.78\pm0.07$, magnitudes surpassing those obtained sans the incorporation of theoretical $H(z)$ values.

Figures

Tables

Table 1. Table 1: Tension of H 0 subscript 𝐻 0 H_{0} between different methods.‘CMB’ refers to Planck Collaboration (2015) and ‘SNIa’ refers to Riess et al. (2016). ‘OHD(43)’ refers to Gaussian Processes based on 43 OHD, ‘OHD(43+1)’ refers to Gaussian Processes based on the dataset including the theoretical H ​ ( z ) 𝐻 𝑧 H(z) value in the infinite future. ‘NE’, ‘AE’, and ‘LE’ refer to no error assumption, average error assumption and linear error assumption we adopt to evaluate the error of the theoretical H ​ ( z ) 𝐻 𝑧 H(z) value in the infinite future.

	CMB	SNIa	OHD(43)
CMB	—	$3.42\sigma$	$1.11\sigma$
SNIa	$3.42\sigma$	—	$0.52\sigma$
OHD(43)	$1.11\sigma$	$0.52\sigma$	—
OHD(43+1_NE)	$0.24\sigma$	$1.60\sigma$	$0.72\sigma$
OHD(43+1_AE)	$0.33\sigma$	$1.46\sigma$	$0.64\sigma$
OHD(43+1_LE)	$0.32\sigma$	$1.47\sigma$	$0.65\sigma$

Table 2. Table 2: Constraining results of Ω M subscript Ω 𝑀 \Omega_{M} and ω 𝜔 \omega in flat ω ​ C ​ D ​ M 𝜔 𝐶 𝐷 𝑀 \omega CDM model (column 2&3) and and Ω M subscript Ω 𝑀 \Omega_{M} , Ω Λ subscript Ω Λ \Omega_{\Lambda} and ω 𝜔 \omega in non-flat ω ​ C ​ D ​ M 𝜔 𝐶 𝐷 𝑀 \omega CDM model (column 4-6) . The meanings of the seventh column are the same as Table 1 . MCMC and χ 2 superscript 𝜒 2 \chi^{2} statistics is meaningless for no error assumption to the theoretical H ​ ( z ) 𝐻 𝑧 H(z) value at z = − 1 𝑧 1 z=-1 .

$H_0$ /km $s^{-1}$ ${\mathrm{Mpc}}^{-1}$	${\Omega }_M$	$\omega$	${\Omega }_M$	${\Omega }_{\Lambda }$	$\omega$	Ref. of $H_0$
66.93$\pm 0.62$	0.27$\pm 0.02$	-0.83$\pm 0.08$	0.26$\pm 0.05$	0.74$\pm 0.15$	-0.82$\pm 0.10$	CMB
73.24$\pm 1.74$	0.22$\pm 0.01$	-0.97$\pm 0.03$	0.28$\pm 0.03$	0.91$\pm 0.06$	-0.94$\pm 0.04$	SNIa
71.09$\pm 3.71$	0.24$\pm 0.01$	-0.95$\pm 0.04$	0.28$\pm 0.03$	0.86$\pm 0.09$	-0.91$\pm 0.06$	OHD(43)
67.67$\pm 3.03$	–	–	–	–	–	OHD(43+1_NE)
67.95$\pm 3.15$	0.26$\pm 0.02$	-0.84$\pm 0.06$	0.27$\pm 0.04$	0.79$\pm 0.12$	-0.82$\pm 0.07$	OHD(43+1_AE)
67.98$\pm 3.16$	0.26$\pm 0.02$	-0.85$\pm 0.06$	0.27$\pm 0.04$	0.80$\pm 0.12$	-0.82$\pm 0.07$	OHD(43+1_LE)

Equations

$$H(z=-1)=\{\begin{array}{ll}\infty & \text{ω<−1 (ωCDM)}\\ H_0\sqrt{{\Omega }_{\Lambda }}& \text{ω=−1 (ΛCDM)}\\ 0& \text{ω>−1 (ωCDM)}\end{array}$$

$$k(z,z^{\ast })={\sigma }_f^{2}exp\{-\frac{(z-z^{\ast }{)}^2}{2l^2}\},$$

Full text

1 Introduction

Hubble constant ($H_{0}$) describes the expansion rate of the universe today and plays a important role in the modern cosmology. In recent years, $H_{0}$ tension problem occurs that it shows a 3.4$\sigma$ tension ($n\sigma$ tension measures the discrepancy of two values of Gaussian distribution, given by $n={|\mu_{1}-\mu_{2}|}/{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}$) between the local measurement $H_{0}$=73.24$\pm 1.74\ {\rm km\ s^{-1}Mpc^{-1}}$ (Riess et al. 2016 Riess2016A ) from Type Ia supernovae (SNIa) and the global measurement $H_{0}$=66.93$\pm 0.62\ {\rm km\ s^{-1}Mpc^{-1}}$ (Planck Collaboration 2015 Aghanim2016Planck ) based on Cosmic Microwave Background (CMB) from Planck satellite. Astrophysicists attempt to explain the discrepancy but its astrophysics mechanism still remains unclear now. Therefore, it needs to derive $H_{0}$ from alternative methods different from that above.

One of the simplest way to obtain the value of $H_{0}$ is to use OHD. Busti et al. (2014) Busti2014Evidence proposed a non-parametric method based on GP method and chose the most proper covariance function to determine the correlated points in the reconstructing processes. They used the 19 $H(z)$ measurements by differential age (DA) method and radial baryon acoustic oscillations (BAO) data to reconstruct the $H(z)$ and extrapolate to redshift zero, obtaining $H_{0}$=64.9$\pm 4.2\ {\rm km\ s^{-1}Mpc^{-1}}$ Busti2015The .

In previous works, the dataset including OHD from DA and BAO methods was used to extrapolate to obtain $H_{0}$. However, the error of $H(z)$ is large and redshift range is limited due to observation methods and technology, therefore, the extrapolating to $z=0$ depending on unilateral data is not so reliable. In this letter, we for the first time consider theoretical $H(z)$ value in the infinite future (at redshift $z=-1$, $H(z)=0$ when equation of state of dark energy $\omega>-1$ ) as one point of OHD to figure out its impact on $H_{0}$ and other cosmological parameters. Due to this theoretical $H(z)$ value without any observational error, the reconstruction result of $H_{0}$ can be more accurate, and it can be constrained from both the positive and negative redshift.

2 Methodology

The Hubble parameter describe the expansion rate of the universe and is defined as ${H(t)\equiv{\dot{a}}/{a}}$. Specifically, the Hubble constant $H_{0}$ describes the present ($t=t_{0}$) or local ($z=z_{0}$) value of Hubble parameter. For the universe of $\omega CDM$ model, the Friedmann equation can be written as $H(z)=H_{0}E(z)$ and $E^{2}(z)=\Omega_{M}(1+z)^{3}+\Omega_{R}(1+z)^{4}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}(1+z)^{3(1+w)}$, where $\Omega_{M}$, $\Omega_{R}$, $\Omega_{k}$and $\Omega_{\Lambda}$ are dimensionless cosmological density parameters of matter, radiation, curvature and dark energy (DE), respectively, at present epoch. In this letter, we regard $\Omega_{R}$ as a negligible, and conceptually, for a flat cosmological model, $\Omega_{k}=0$. The constant $\omega$ is the equation of state of DE. For $\omega=-1$, it reduces to a flat $\Lambda CDM$ model, while for the other cases of $\omega>-1$, we can simply get the $H(z)=0$ when $z=-1$. In cosmology, the positive redshift for observed sources represents the past time of the universe while negative redshift means the sources located in the future universe. Furthermore, according to the equation ${1}/{a(t)}=1+z$, from now to infinite future, negative redshift $z$ is getting smaller and smaller. Meanwhile, the scale factor $a(t)$ is getting larger and larger. Extremely, $z=-1$ means $a(t)\rightarrow\infty$ and a infinite future, so the universe expands to the possible maximum and the temperature $T(z=-1)\rightarrow 0$. Therefore, we get:

[TABLE]

We can see from Fig. 1 that $H(z)$ can be taken the same theoretical value at $z=-1$. In this letter, we do not discuss the case where $\omega\leq-1$ since theoretical $H(z)$ value at this point depends on the value of $H_{0}$ and $\Omega_{\Lambda}$ ($\Lambda CDM$ model) or $H(z)$ is diverge to infinity.

Theoretically, redshift is derived from observational spectrum of astrophysical source, $z=(\lambda-\lambda_{0})/{\lambda_{0}}$, where $\lambda$ and $\lambda_{0}$ represent the observed and emitted wavelengths, respectively. Actually, spectrum with negative redshift is not available at present because everything is receding from observers on the earth on the cosmological scale. That is to say, we can not derive electromagnetic spectrum from future universe at present. However, what we can study is the impact of the theoretical value at future universe 2019PhRvD..99b3503L , thus we can presume that $H(z)$ measurements is available from observation at $z=-1$, then we utilize this OHD $H(z=-1)=0$ to constrain $H_{0}$ by GP method.

One advantage of the theoretical $H(z)$ in the infinite future is its definiteness without any observational error ($\sigma_{\rm{NE}}(z=-1)=0$) which can be also used to perform GP. But if we use it to constrain other cosmological parameters with MCMC method, we need to define a hypothetical observational error of $H(z)$ value at $z=-1$. For this, we assume there is a symmetry of observational error between the past light (positive redshift) and the future light (negative redshift). According to the existing OHD data of DA method, for simplicity, we can assume that there is a simple linear relationship between the measurement error and the absolute value of the redshift, shown as Fig. 2, $\sigma(z)=k|z|+b$, then $\sigma_{\rm{LE}}(z=-1)=\sigma[H(-1)]$ can take 17.6177. Besides, we also take average error of exist $H(z)$ data ($\sigma_{\rm{AE}}(z=-1)=\bar{\sigma}=16.7449$) as comparison.

Latest $H(z)$ data can be derived from both DA method 2014RAA….14.1221Z ; 2003ApJ…593..622J ; 2005PhRvD..71l3001S ; 2012JCAP…07..053M ; 2016JCAP…05..014M ; 2017MNRAS.467.3239R ; 2010JCAP…02..008S ; 2015MNRAS.450L..16M and BAO method 2009MNRAS.399.1663G ; 2012MNRAS.426..226C ; 2013MNRAS.431.2834X ; 2012MNRAS.425..405B ; 2013MNRAS.429.1514S ; 2014MNRAS.439…83A ; 2013AA…552A..96B ; 2015AA…574A..59D ; 2014JCAP…05..027F . We add this theoretical value of $H(z)$ in the infinite future to the OHD dataset.

Gaussian Processes is a value-of-function reconstructing method that the reconstruction function $F^{*}\{f(z_{1}^{*}),\\ f(z_{2}^{*}),\dots,f(z_{N}^{*})\}$ for each reconstructed point $Z\{z_{1}^{*},z_{2}^{*},\\ \dots,z_{N}^{*}\}$ is given by a Gaussian distribution. In the GP method of reconstructing $H(z^{*})$, we know the observed $Y\{H(z_{1}),H(z_{2}),\dots,H(z_{N})\}$ data at certain redshift, and use their errors to calculate the covariance matrix, obtaining the function value corresponding to the reconstructed redshift. In this letter, we choose the square exponential covariance function as the covariance matrix, which is

[TABLE]

where $\sigma_{f}^{2}$ and $l$ represent two hyper-parameters related to changes in the function value, and reshift interval to let function value change significantly.

In this letter, we use the public package GaPP (Gaussian Processes in Python) firstly developed by Seikel et al Seikel2012Reconstruction Seikel2013GaPP to achieve the GP. We determine the maximum likelihood value of two hyper-parameters $\sigma_{f}^{2}$ and $l$ and then obtain the the reconstructed function results.

3 Results

We use all the latest OHD to perform GP, and the reconstruction results are shown in Fig. 3(a). We obtain $H_{0}$=71.09$\pm 3.71\ {\rm km\ s^{-1}Mpc^{-1}}$ when extrapolating to the $H(z=0)$, which is consistent with Type Ia supernovae results (Riess et al 2016) within 1$\sigma$ and has a 1.11$\sigma$ discrepancy with Planck Collaboration (2015) results. Fig. 3(b) show the reconstruction results including the theoretical $H(z)$ value in the infinite future, $H_{0}$=67.67$\pm 3.03\ \\ {\rm km\ s^{-1}Mpc^{-1}}$ that is very closed to results we take average error assumption ($H_{0}$=67.95$\pm 3.15\ {\rm km\ s^{-1}Mpc^{-1}}$) and linear error assumption ($H_{0}$=67.98$\pm 3.18\ {\rm km\ s^{-1}Mpc^{-1}}$). It shows a 0.24$\sigma$ discrepancy with Planck Collaboration (2015) results and a 1.60$\sigma$ discrepancy with results from Riess et al. (2016) (see Table 1).

When adding the theoretical $H(z)$ value in the infinite future to the OHD dataset, the $H_{0}$ value from OHD method becomes smaller, which shows a 0.64$\sim$0.72 $\sigma$ (¡ 1 $\sigma$) tension with that from 43 OHD. According to the GP restriction function adopting average or linear error assumption, the $H(z)$ is probably a positive value close to zero and the $1\sigma$ confidence interval at $z=-1$ is much more larger than that at $z=0$. That is to say, though its tinny impact of the theoretical $H(z)$ in the infinite future, it gives a smaller value of $H_{0}$ remarkably and inconsistent with the measurements from Planck Collaboration (2015) and Riess et al. (2016). The comparison of these results from OHD method and two observation results are shown in Fig. 4.

Next, we use the $H_{0}$ results obtained above to constrain the equation of state of DE in $\omega CDM$ model. We assume a uniform distribution of $\omega$ as prior distribution, then use MCMC sampling to compare the GP restruction results with the standard parametric equation, and obtain the cosmological parameters by $\chi^{2}$ statistics analysis Ma2012POWER . For flat $\omega CDM$ model, the constraining results of $\Omega_{M}$ and $\omega$ are shown in Fig. 5, while for non-flat $\omega CDM$ model, the constraining results of $\Omega_{M}$, $\Omega_{\Lambda}$ and $\omega$ are shown in Fig. 6.

In contrast, we also use observed $H_{0}$ value from Riess et al.(2016) and Planck Collaboration (2015) to repeat these processes. The comparison of constraining results are shown in Table 2 and Fig. 7.

Table 2 shows that the values of $\Omega_{M}$ are less sensitive to the changes of $H_{0}$ than and $\Omega_{\Lambda}$ and $w$. The uncertainty of these parameters is larger in non-flat universe. In the results of flat $\omega CDM$ model, the estimated value of $\Omega_{M}$ is about 0.24. However, the estimated value of $\omega$ changes remarkably if the $H_{0}$ alters. The constraining result of $\omega$ is very close to -1.00 (see Fig. 5(a) and Fig. 6(a)), therefore, the constraining results support $\Lambda CDM$ model. But within the frame of $\omega CDM$ model, if the theoretical value of Hubble parameter in the infinite future ($z=-1$) is considered, the value of which closely approximates the theoretical value [math], and the results of $\omega$ change to 0.82$\sim$0.85 and have a deviation from $\Lambda CDM$ model.

Besides, when we consider the theoretical $H(z)$ value in the infinite future, we find the $H_{0}$ value and other cosmological parameters constraining results from flat universe is more close to the Planck Collaboration (2015) results. There may be some cosmological relationship between the observation result from Planck satellite based on CMB and infinite future because both of them are related to global universe.

Comparing the MCMC constraining results including the theoretical $H(z)$ value in the infinite future with others, we find that the $\Omega_{M}$ is about 0.26 with a certainty about 0.02 in flat $\omega CDM$ model and $\Omega_{M}$ is about 0.27 with a large certainty about 0.04 in non-flat $\omega CDM$ model. In the meantime, a smaller $\Omega_{\Lambda}$ about 0.85 is obtained, which suggests a negative $\Omega_{k}$ and a close universe. And the $\omega$ is much more larger when considering infinite future data than that using the $H_{0}$ from other methods or references.

4 Conclusions

We consider the impact of the theoretical $H(z)$ value in the infinite future, presents a model-independent restruction of $H(z)$ and obtain a smaller value of Hubble constant than that without considering $H(z=-1)$. The $H_{0}$ value is in consistent with Planck Collaboration (2015) result ($0.33\sigma$ tension) and in great agreement with latest Planck Collaboration(2018) result ($H_{0}=67.4\pm 0.5\ {\rm km\ s^{-1}Mpc^{-1}}$, 0.09$\sigma$ 2018arXiv180706209P ), but a larger deviation from Riess et al. (2016) ($1.60\sigma$ tension). It relieve the Hubble tension to some extent, but not solve the the problem physically. We also constrain the other cosmological parameters in both flat and non-flat $\omega CDM$ model, obtaining $\Omega_{M}=0.26\pm 0.02$, $\omega=-0.85\pm 0.06$ in flat $\omega CDM$ model and $\Omega_{M}=0.27\pm 0.04$, $\Omega_{\Lambda}=0.80\pm 0.12$, $\omega=-0.82\pm 0.07$ in non-flat $\omega CDM$ model.

We compare our $H_{0}$ result with previous works. Our $H_{0}$ result is very close to that of Macaulay et al. (2018)2019MNRAS.486.2184M , who used the ‘inverse distance ladder’ method with 207 Type Ia supernovae and obtained $H_{0}=67.77\pm 1.30\ {\rm km\ s^{-1}}$ ${\rm Mpc^{-1}}$. And it is also in agreement with the result of Shanks et al.(2018)2019MNRAS.484L..64S , who used Gaia Cepheid parallaxes and ‘Local Hole’ and obtained $H_{0}=68.9\pm 1.6\ {\rm km\ s^{-1}Mpc^{-1}}$. Both of them got a $H_{0}$ value a bit more than Planck Collaboration(2015). However, Feeney et al. (2017)2018MNRAS.476.3861F developed a Bayesian hierarchical model (BHM) that describes the full distance ladder and $H_{0}=72.72\pm 1.67\ {\rm km\ s^{-1}Mpc^{-1}}$. Birrer et al. (2019)2019MNRAS.484.4726B presented a blind time-delay strong lensing (TDSL) cosmographic analysis of the doubly imaged quasar SDSS 1206+4332 and obtained $H_{0}=72.5^{+2.1}_{-2.3}$ ${\rm km\ s^{-1}Mpc^{-1}}$, which was independent of the distance ladder and other cosmological probes. Both of their results shows a large tension with our result and they are in agreement with Riess et al. (2016). It seems that the tension can be relieved thanks to more improved method but it still shows a division into two opposing extremes.

Admittedly, though our work is using a hypothetical observed quantity based on strict cosmological theory, the $H(z)$ value from negative redshift or future universe is still unavailable in astronomical observation. To make it available and meaningful in observation, probably we can re-understand the time relativity of cosmological redshift, and use new ideas to solve the tension problem.

On the one hand, it is believed that the theory of cosmic expansion is more and more conductive to guiding astronomical observation. On the other hand, more precise observation and improved data processing methods are expected to bring more precise Hubble constant values. If the Hubble constant tension is still unsolved, there are probably some new astrophysical mechanisms for understanding of the theory of cosmic expansion. Additionally, finding out a method to obtain the observational data at the negative redshift in the universe is a prospective challenge, which may lead to a revolutionary change in modern observational cosmology.

Acknowledgements

This work was supported by the National Science Foundation of China (Grants No. 11573006, 11528306), and National Key R&D Program of China (2017YFA0402600).