Complex Limiting Velocity Expressions as Likely Characteristics of Dark Matter Particles

Many astrophysical and cosmological observations suggest that the matter in the universe is mostly of the dark matter type whose behavior goes beyond the Standard Model description. Hence it is justifiable to take a drastically different approach to the dark matter particles which is here done through the bicubic equation of limiting particle velocity formalism. The bicubic equation discriminant $D$ in this undertaking satisfy $D\succeq 0 $ determined by the congruent parameter $z$ satisfying $z^{2}\succeq 1$, where formally $z(m)=3\sqrt{3}mv^{2}/2E$, \ with $m$, $v$, and $E$ being respectively, particle mass, velocity and energy. Also nonlinearly related to the the particle congruent parameter $z$ is the particle congruent angle $% \alpha $ . These two dimensionless\ parameters $z$ \ and $\alpha $ simplify expressions in the bicubic equation limiting particle velocity formalism when evaluating the three particle limiting velocities, $c_{1},$ $c_{2}$\ and $c_{3},$ (primary, obscure and normal) in terms of the ordinary particle velocity, $v$. Corresponding to these limiting velocities \ one then deduces, with equal values, dark matter particle energies $E\left(c_{1}\right) $, $E\left( c_{2}\right) $ and $E\left( c_{3}\right) $. The exemplary values of the congruent parameters are in these regions, $1\preceq z\prec 3\sqrt{3}$ $/2$ and $\pi /2\succeq \alpha \succeq \pi /3$ . Already within these ranges of congruent parameters, the bicubic formalism yields for squares of particle limiting velocities that $c_{1}^{2}$ and $c_{2}^{2}$ are complex conjugate to each other, $c_{1}^{2\ast }=c_{2}^{2}$ ,and that $% c_{3\text{ }}^{2}$is real. The imaginary portions of $c_{1}^{2}$ and $% c_{2}^{2}$ do not change the realities of numerically equal to each other dark matter energies $E\left( c_{i}\right) ,i=1,2,3.$ In fact, real $E\left(c_{1,2}\right) $ energies can be equally evaluated with $c_{1,2}^{2}$ or $% \func{Re}$ $c_{1,2}^{2}$ or even with $\func{Im}c_{1,2}^{2}$ so that in new notation, $E\left( _{1,2}^{2}\right) =E\left( \func{Re}c_{1,2}^{2}\right) =E\left( \func{Im}c_{1,2}^{2}\right) $ $=E\left( c_{3}^{2}\right) $ all with the same real values. However, in these notations, the real particle momenta are $\overrightarrow{p}\left( (\func{Re}c_{1,2}^{2}\right) $ and $\\overrightarrow{p}\left( (c_{3}^{2}\right) $, defined with respective energies and, while in similar forms , numerically are different from each other.

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