Analysis of the ${H}_{0}$ tension problem in the Universe with viscous dark fluid

Analysis of the $H_{0}$ tension problem in the Universe with viscous dark fluid

Emilio Elizalde, Martiros Khurshudyan, Sergei D. Odintsov, and Ratbay Myrzakulov

Phys. Rev. D 102, 123501 – Published 1 December 2020

Abstract

Two inhomogeneous single-fluid models for the Universe, which are able to naturally solve the $H_{0}$ tension problem, are discussed. The analysis is based on a Bayesian Machine Learning approach that uses a generative process. The method here adopted allows for constraint of the free parameters of each model by making use of the model itself only. The observable is taken to be the Hubble parameter, obtained from the generative process. Using the full advantages of our method, the models are constrained for two redshift ranges. Namely, first this is done with mock $H (z)$ data over $z \in [0, 2.5]$ , thus covering known $H (z)$ observational data, which are most helpful to validate the fit results. Then, aiming to extend to redshift ranges to be covered by the most recent ongoing and future planned missions, the models are constrained for the range $z \in [0, 5]$ , too. Full validation of the results for this extended redshift range will have to wait for some years, for when higher-redshift $H (z)$ data become available. This makes our models fully falsifiable. In addition, our second model is able to explain the BOSS reported value for $H (z)$ at $z = 2.34$ .

Received 2 June 2020
Accepted 14 October 2020

DOI:https://doi.org/10.1103/PhysRevD.102.123501

Physics Subject Headings (PhySH)

Cosmological parameters Evolution of the Universe

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Emilio Elizalde^1,2,*, Martiros Khurshudyan ^1,2,†, Sergei D. Odintsov^1,3,4,‡, and Ratbay Myrzakulov^4,§

¹Consejo Superior de Investigaciones Científicas, ICE/CSIC-IEEC, Campus UAB, Carrer de Can Magrans s/n, 08193 Bellaterra (Barcelona) Spain
²International Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia
³ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain
⁴Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Nur-Sultan, 010008, Kazakhstan

^*elizalde@ieec.uab.es
^†khurshudyan@ice.csic.es
^‡odintsov@ieec.uab.es
^§Corresponding author. rmyrzakulov@gmail.com

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Vol. 102, Iss. 12 — 15 December 2020

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Images

Figure 1
Contour map of the model given by Eq. (5) for $z \in [0, 2.5]$ . The best fit values of the model parameters have been found to be $H_{0} = 73.4 \pm 0.1 km / s / Mpc$ , $ω_{0} = 0.608 \pm 0.007$ , $ω_{1} = - 1.64 \pm 0.01$ , $A = - 1.01 \pm 0.01$ , and $n = 0.55 \pm 0.01$ , when a Bayesian Learning approach based on the generative process has been applied.
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Figure 2
Graphical behavior of the Hubble parameter in comparison with known $H (z)$ data. The purple curve is a plot of the Hubble parameter for the best fit values of the model parameters, when $z \in [0, 2.5]$ , while the dashed red curve has been chosen for the case when $z \in [0, 5]$ . The red dots represent the known observational $H (z)$ data, and it is the same as in Ref. [42]. The left-hand side represents the comparison for the model given by Eq. (5), while the right-hand side plot represents the comparison with the model given by Eq. (7). In both cases, only the best fit values for the model free parameters obtained by the Bayesian Learning approach have been used.
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Figure 3
Graphical behavior of the deceleration parameter $q (z)$ and $ω (z) = P / ρ$ for the model given by Eq. (5) is given by the top panel, while the bottom panel represents the graphical behavior of the same parameters when the model is given by Eq. (5). In both cases, only the best fit values of the model parameters have been taken into account. Namely, the purple curve represents the graphical behavior of the deceleration and $ω (z)$ parameters for the best fit values of the model parameters, when $z \in [0, 2.5]$ . On the other hand, the dashed red curve represents the graphical behavior of the same parameters for the best fit values of the model parameters, when $z \in [0, 5]$ has been considered. The best fit values used in these plots, when the model is given by Eq. (5), are: $H_{0} = 73.4 km / s / Mpc$ , $ω_{0} = 0.608$ , $ω_{1} = - 1.64$ , $A = - 1.01$ , and $n = 0.55$ , for $z \in [0, 2.5]$ , while $H_{0} = 73.4 km / s / Mpc$ , $ω_{0} = 0.614$ , $ω_{1} = - 1.55$ , $A = - 0.95$ , and $n = 0.49$ , for $z \in [0, 5]$ . The best fit values used in the plots, when the model is given by Eq. (7), are: $H_{0} = 73.52 km / s / Mpc$ , $ω_{0} = 0.5502$ , $ω_{1} = - 1.605$ , $A = - 0.98$ , and $n = 0.75$ , for $z \in [0, 2.5]$ , while $H_{0} = 73.61 km / s / Mpc$ , $ω_{0} = 0.5105$ , $ω_{1} = - 1.502$ , $A = - 0.98$ , and $n = 0.75$ , for $z \in [0, 5]$ .
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Figure 4
Contour map of the model given by Eq. (7), for $z \in [0, 2.5]$ . The best fit values of the model parameters are $H_{0} = 73.52 \pm 0.15 km / s / Mpc$ , $ω_{0} = 0.5502 \pm 0.0055$ , $ω_{1} = - 1.605 \pm 0.009$ , $A = - 0.98 \pm 0.01$ , and $n = 0.75 \pm 0.01$ , when a Bayesian Learning approach based on the generative process has been employed.
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Figure 5
Contour map of the model given by Eq. (5) for $z \in [0, 5]$ . The best fit values of the model parameters are $H_{0} = 73.4 \pm 0.1 km / s / Mpc$ , $ω_{0} = 0.614 \pm 0.005$ , $ω_{1} = - 1.55 \pm 0.01$ , $A = - 0.95 \pm 0.01$ , and $n = 0.49 \pm 0.01$ , for $z \in [0, 5]$ , where a Bayesian Learning approach based on the generative process has been used.
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Figure 6
Contour map of the model given by Eq. (7) for $z \in [0, 5]$ . The best fit values of the model parameters are $H_{0} = 73.61 \pm 0.15 km / s / Mpc$ , $ω_{0} = 0.5105 \pm 0.0044$ , $ω_{1} = - 1.502 \pm 0.009$ , $A = - 0.98 \pm 0.01$ , and $n = 0.75 \pm 0.01$ , for $z \in [0, 5]$ , where a Bayesian Learning approach based on the generative process has been used.
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