A brief review of a modified relativity that explains cosmological constant

Abstract

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 ($Λ$). The equation of state (EOS) of vacuum energy for $Λ$, i.e., $ρ_Λ=ε=-p$ emerges naturally from such space-time, where $p$ is the pressure and $ρ_Λ=ε$ is the vacuum energy density. With the aim of establishing a relationship between $V$ and $Λ$ 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 $ε$ 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 ($Λ>0$). 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 $Λ<<1$ close to a flat space-time, which is in the $ΛCDM$ scenario where the space is quasi-flat, so that $Ω_{m}+Ω_Λ\approx 1$. We have $Ω{cdm}\approx 23\%$ by representing dark cold matter, $Ω_m\approx 27\%$ for matter and $Ω_Λ\approx 73\%$ for the vacuum energy. Thus, the theory is adjusted for the redshift $z=1$. This corresponds to the time $τ_0$ of transition between gravity and anti-gravity, leading to a slight acceleration of expansion related to a tiny value of $Λ$, i.e., we find $Λ_0=1.934\times 10^{-35}s^{-2}$. This result is in agreement with observations.