New Barrow holographic dark energy: cosmological dynamics and cosmic chronometer analysis

We formulate new Barrow holographic dark energy (NBHDE) as a second-order expansion of the Barrow entropy correction near the standard holographic dark energy (HDE) limit, adopting the future event horizon as the infrared cutoff. Writing the Barrow factor in exponential form identifies $(\mathrm{\Delta }\mathrm{/}2)\ln S_{\mathrm{B}\mathrm{H}}$, rather than $\mathrm{\Delta }$ alone, as the expansion parameter. In a flat universe containing pressureless matter and NBHDE without a cosmological constant, we derive the background equations, establish consistency with the integral definition of the future event horizon, and recover HDE exactly at $\mathrm{\Delta }=0$. We assess this approximation against the unexpanded Barrow holographic dark energy (BHDE) density, locally and through independently normalized background evolutions. Across $0.50\le c\le 1.20$ and $0\le \mathrm{\Delta }\le 0.003$, the maximum fractional density difference remains below $1\mathrm{\%}$ over $-0.99\le z\le 5$, while the normalized Hubble rates differ by less than $0.074\mathrm{\%}$ over $0\le z\le 5$. Within this domain, the Barrow correction modestly changes the expansion rate and acceleration transition but more strongly affects the dark energy equation of state and phantom crossing. At $c=0.80$, $w_{d0}$ shifts from $-1.031$ in HDE to $-0.900$ at $\mathrm{\Delta }=0.003$. We perform a conditional comparison with 36 spectroscopic cosmic chronometer (CC) measurements. At fixed $H_0=67.4\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$ and ${\mathrm{\Omega }}_{m0}=0.30$, the minimum lies on the HDE boundary at $\mathrm{\Delta }=0$ and $c=0.8457$. Minimizing over $c$ at $\mathrm{\Delta }=0.003$ gives $c_{\min }=0.6879$, with $\mathrm{\Delta }{\chi }_{\mathrm{C}\mathrm{C}}^2\simeq 1.0\times {10}^{-3}$. NBHDE therefore provides a self-consistent, quantitatively controlled framework near the HDE limit, showing that Barrow corrections can alter dark energy behavior even when their effect on $H(z)$ is largely compensated by changing $c$.