Deformed Special Relativity with a minimum speed as explanation of the tiny value of the cosmological constant based on the Boomerang experiment in the $\mathrm{\Lambda }CDM$ scenario

In this paper we will show that a new structure of space-time with a minimum speed reveals a connection with Weyl geometry in the approximation of weak-field Newtonian limit. Symmetrical Special Relativity (SSR) has a minimum speed $V$ that plays the role of a preferred reference frame $S_V$ of vacuum that leads to the cosmological constant $\mathrm{\Lambda }$. In order to realize such a connection between $V$ and $\mathrm{\Lambda }$ within a scenario of space-time metric, we will use a model of spherical universe with Hubble radius $R_H$ filled by a low vacuum energy density ${\rho }_{\mathrm{\Lambda }}$ that governs the accelerated expansion of the universe. In doing this, we will show that SSR-metric plays the role of a de-Sitter (dS)-metric with a positive cosmological constant ($\mathrm{\Lambda }>0$). On the other hand, according to the Boomerang experiment as it is shown that the three-dimensional space of the universe is Euclidean and with a slightly accelerated expansion, SSR leads to a dS-metric with an approximation for $\mathrm{\Lambda }<<1$ close to a flat space-time, which is in the $\mathrm{\Lambda }CDM$ scenario where the space is quasi-flat, so that ${\mathrm{\Omega }}_m+{\mathrm{\Omega }}_{\mathrm{\Lambda }}\approx 1$. We have $\mathrm{\Omega }cdm\approx 23\mathrm{\%}$ by representing dark cold matter, ${\mathrm{\Omega }}_m\approx 27\mathrm{\%}$ for matter and ${\mathrm{\Omega }}_{\mathrm{\Lambda }}\approx 73\mathrm{\%}$ for the vacuum energy. Thus, the theory is adjusted for redshift $z=1$, i.e., the time ${\tau }_0$ at which the universe goes over from a decelerating to an accelerating expansion by obtaining the numerical value ${\mathrm{\Lambda }}_0=1.934\times {10}^{-35}{s}^{-2}$, being in good agreement with measurements.