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\begin{document}

\title{Study of the Hubble tension in the framework of a phantom dark energy model}
\author{Safae Dahmani}
\email{dahmani.safae.1026@gmail.com}
\author{Amine Bouali}
\email{a1.bouali@ump.ac.ma}
\author{Imad El Bojaddaini}
\email{i.elbojaddaini@ump.ac.ma}
\author{Ahmed Errahmani}
\email{ahmederrahmani1@yahoo.fr}
\author{Taoufik Ouali}
\email{t.ouali@ump.ac.ma}

%\date{\today }
\affiliation{Laboratory of Physics of Matter and Radiation, Mohammed I University, BP 717, Oujda, Morocco}

\begin{abstract}

The discrepancy between Planck data and direct measurements of the current expansion rate $H_0$ is one of the most intriguing puzzles in cosmology nowadays. In fact, the value predicted at high-redshifts i.e. by cosmic microwave background measurements assuming the $\Lambda$CDM model, is $H_0=67.37\pm 0.54$ km s$^{-1}$ Mpc$^{-1}$ \cite{PL}, while the one determined by the Cepheid calibrated supernovae Ia, is $H_0=73.04\pm 1.04$ km s$^{-1}$ Mpc$^{-1}$ \cite{R21}. This discrepancy between these two values of $H_0$ is at about $\sim 5\sigma$. This tension could be evidence of new physics beyond the standard model of cosmology. In this work \cite{safae}, we study a phantom dynamical dark energy model as an alternative to $\Lambda$CDM. This phantom model is characterised by one extra parameter, $\Omega_{pdde}$, compared to the standard model and was proposed proposed for the first time, as one of the mildest models of the possible apocalypse in the future ~\cite{LSBR1} and is dubbed as the Little Sibling of the Big Rip (LSBR). For $\alpha>0$. The model is considered to be a phantom dark energy. For $\alpha<0$ the Universe rebounds at some time in the future and the model is considered to be a quintessence dark energy. Several works have studied this model in different contexts e.g. the quantum cosmology  and $f(R)$ cosmology \cite{LSBRH}. Moreover, this model has been shown to be supported by observation data for the first time by \cite{LSBR2}. Most importantly, its proximity to the Standard Model, $\Lambda$CDM, makes it one of the models that perfectly describe the current acceleration of the Universe \cite{LSBR2}. In our analysis, we obtain a strong positive correlation between $H_0$ and $\Omega_{\mathrm{pdde}}$, for all data combinations.  Using Planck18 measurements together with BAO and Pantheon, we find that the $H_0$ and  the $S_8$ tensions are $ 3.4\sigma$ and $ 2.75\sigma$, respectively. By introducing a prior on the absolute magnitude, $M_\mathrm{B}$, of the SN Ia, the $H_0$ tension decreases to $2.49\sigma$ with $H_0 = 69.7^{+0.83}_{-0.86} $ km s$^{-1}$ Mpc$^{-1}$ and the $S_8$ tension reaches the value $2.51 \sigma$ with $S_8 = 0.8265^{+0.011}_{0.012}$ \cite{safae}.
\begin{thebibliography}{}

\bibitem{PL}
Planck Collaboration: N. Aghanim et al.,``Planck 2018 results. VI. Cosmological parameters,'' Astron. Astrophys. {\bf 641} (2020) A6, [\href{https://arxiv.org/abs/1807.06209}{	arXiv:1807.06209[astro-ph.CO]}].


\bibitem{R21}
A. G. Riess et al., `` A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team, ''ApJL {\bf 934} (2022) L7, [\href{https://arxiv.org/abs/2112.04510}{	arXiv:2112.04510 [astro-ph.CO]}].


 
 \bibitem{safae}
S. Dahmani, A. Bouali, I. el Bojaddaini, A. Errahmani, T. Ouali,
``Smoothing the  $H_0$ tension with a dynamical dark energy model'',
Phys. Dark. Univ. {\bf 42} (2023) 101266.
  [\href{https://www.sciencedirect.com/science/article/pii/S2212686423001000}{https://doi.org/10.1016/j.dark.2023.101266}].
\bibitem{LSBR1}
M. Bouhmadi-Lopez, A. Errahmani, P. Martin-Moruno, T. Ouali, Y. Tavakoli, ``The little sibling of the big rip singularity, '' Int. J. Mod. Phys. D. {\bf 24} (2015) 1550078, [\href{https://arxiv.org/abs/1407.2446v2}{arXiv:1407.2446 [gr-qc] }].
\bibitem{LSBRH}
T B. Vasilev, M. Bouhmadi Lopez and P. Martin-Moruno, ``The classical and quantum fate of the Little Sibling of the Big Rip in f(R) cosmology
'', Phys. Rev. {\bf D 100} (2019) 084016 [\href{https://arxiv.org/abs/1907.13081}{ 	arXiv:1907.13081 [gr-qc]}].
 
 \bibitem{LSBR2}
A. Bouali, I. Albarran, M. Bouhmadi-López, T. Ouali,
``Cosmological constraints of phantom dark energy models'',
Phys. Dark. Univ. {\bf 26} (2019) 100391.
 [\href{https://arxiv.org/abs/1905.07304}{arXiv:1905.07304 [astro-ph.CO]}].
 

\end{thebibliography}
\end{abstract}



\maketitle


\end{document} 








