Poincar\'e-De Sitter Flow and Cosmological Meaning

We introduce the Poincar'e-de Sitter flow with real numbers $\{r,s\}$ to parameterize the relativistic quadruple ${\frak Q}{PoR}=[{\cal P}, {\cal P}2, {\cal D}+,{\cal D}-]{M/M\pm/D_\pm}$ for the triple of Poincar'e/\dS/\AdS
group $\mathcal{P}\mathrm{/}{\mathcal{D}}_{+}\mathrm{/}{\mathcal{D}}_{-}$ invariant special relativity. The dual Poincar'e group ${\mathcal{P}}_2$-invariant degenerated Einstein manifold $M_{\pm }$ of ${\mathrm{\Lambda }}_{\pm }=\pm 3l^{-2}$ is for the space/time-like domain ${\dot{R}}_{\pm }$ of the compact lightcone ${\bar{C}}_O$ associated to the common space/time-like region $R_{\pm }$ of the lightcone $C_O$ at common origin on Minkowski/\dS/\AdS
spacetime $M\mathrm{/}{D}_{+}\mathrm{/}{D}_{-}$. Based on the principle of relativity with two universal constants $(c,l)$, there are the law of inertia, coordinate time simultaneity and so on for the flow on a Poincar'e-\dS\ symmetric Einstein manifold of ${\mathrm{\Lambda }}_s=3s{l}^{-2}$. Further, there is Robertson-Walker-like cosmos of the flow for the propertime simultaneity. The \dS\ special relativity with double $[{\mathcal{D}}_{+},{\mathcal{P}}_2{]}_{D_{+}\mathrm{/}{M}_{+}}$ can provide a consistent kinematics for the cosmic scale physics with an upper entropy bound $S_R=k_B\pi g^{-2}, g^2:=(\ell_P/R)^2 \simeq 10^{-122}$,for$R\simeq (3/\Lambda)^{1/2}\sim 13.7 Gly$.