Exact solution of the transcendental Friedmann equation

Solve the transcendental Friedmann equation generated by the full entropic cosmological model containing the minimal-area quantum terms, the Bekenstein–Hawking term, and the volumetric dark-energy correction, thereby obtaining the Hubble parameter beyond the limiting regimes.

Background

For the full entropy–area relation, the thermodynamic derivation produces a transcendental equation relating the matter density to the Hubble parameter and spatial curvature. Unlike the simpler quantum-bounce case, the equation cannot be algebraically inverted in the paper.

The authors therefore analyze only the two asymptotic regimes corresponding to areas near the minimal area and areas much larger than it. An exact or general solution would enable investigation of the intermediate cosmological regime, including its phenomenology.

References

Unlike the previous cases where one can solve for $\left( H2+\kappa/a2\right)$, we get a transcendental equation, but even if we can not solve for the Hubble parameter we can explore different limits to obtain the corresponding Friedmann equation.

— On entropic cosmology, late time acceleration and the quantum bounce  (2608.19146 - Diaz-Barron et al., 19 Aug 2026) in Section 2, Entropic Origin for Dark Energy