A brief review of a modified relativity that explains cosmological constant

The present review aims to show that a modified space-time with an invariant minimum speed provides a relation with Weyl geometry in the Newtonian approximation of weak-field. The deformed Special Relativity so-called Symmetrical Special Relativity (SSR) has an invariant minimum speed $V$, which is associated with a preferred reference frame $S_V$ for representing the vacuum energy, thus leading to the cosmological constant ($\Lambda$). The equation of state (EOS) of vacuum energy for $\Lambda$, i.e., $\rho_{\Lambda}=\epsilon=-p$ emerges naturally from such space-time, where $p$ is the pressure and $\rho_{\Lambda}=\epsilon$ is the vacuum energy density. With the aim of establishing a relationship between $V$ and $\Lambda$ in the modified metric of the space-time, we should consider a dark spherical universe with Hubble radius $R_H$, having a very low value of $\epsilon$ that governs the accelerated expansion of universe. In doing this, we aim to show that SSR-metric has an equivalence with a de-Sitter (dS)-metric ($\Lambda>0$). On the other hand, according to the Boomerang experiment that reveals a slightly accelerated expansion of the universe, SSR leads to a dS-metric with an approximation for $\Lambda<<1$ close to a flat space-time, which is in the $\Lambda CDM$ scenario where the space is quasi-flat, so that $\Omega_{m}+\Omega_{\Lambda}\approx 1$. We have $\Omega{cdm}\approx 23\%$ by representing dark cold matter, $\Omega_m\approx 27\%$ for matter and $\Omega_{\Lambda}\approx 73\%$ for the vacuum energy. Thus, the theory is adjusted for the redshift $z=1$. This corresponds to the time $\tau_0$ of transition between gravity and anti-gravity, leading to a slight acceleration of expansion related to a tiny value of $\Lambda$, i.e., we find $\Lambda_0=1.934\times 10^{-35}s^{-2}$. This result is in agreement with observations.