Entropy bounds and holographic dark energy: Conflicts and consensus

Cohen, Kaplan, and Nelson’s influential paper established that the UV–IR cut-offs cannot be arbitrarily chosen but are constrained by the relation ${\Lambda }^{2}L\lesssim M_p$. Here, we revisit the formulation of the CKN entropy bound and compare it with other bounds. The specific characteristics of each bound are shown to depend on the underlying scaling of entropy. Notably, employing a non-extensive scaling with the von Neumann entropy definition yields a more stringent constraint, $S_{\text{max}}\approx \sqrt{S_{\text{BH}}}$, where $S_{\text{BH}}$ is the Bekenstein–Hawking entropy. We also clarify distinctions between the IR cut-offs used in these frameworks. Moving to the causal entropy bound, we demonstrate that it categorises the CKN bound as matter-like, the von Neumann bound as radiation-like, and the Bekenstein bound as black hole-like systems when saturated. Emphasising cosmological implications, we confirm the consistency between the bounds and the first laws of horizon thermodynamics. We then analyse the shortcomings in standard Holographic Dark Energy (HDE) models, highlighting the challenges in constructing HDE using ${\Lambda }^{2}L\lesssim M_p$. Specifically, using the Hubble function in HDE definitions introduces circular logic, causing dark energy to mimic the second dominant component rather than behaving as matter. We further illustrate that the potential for other IR cut-offs, like the future event horizon in an FLRW background or those involving derivatives of the Hubble function, to explain late-time acceleration stems from an integration constant that cannot be trivially set to zero. In brief, the CKN relation does not assign an arbitrary cosmological constant; it explains why its value is small.