Forecasting Constraints on Entropic Holographic Dark Energy via SKA 21 cm Redshift Drift: A Joint Analysis of Galaxy Emission and Damped Lyman-Alpha Systems

Direct kinematic measurements of cosmic acceleration via the cosmological redshift drift (the Sandage–Loeb effect) provide a powerful, model-independent avenue to map the expansion history of the Universe. In this work, which extends the forecasting framework developed in a recent study, we evaluate the capability of the Square Kilometre Array (SKA) to constrain three physically motivated, quantum-gravity inspired holographic dark energy (HDE) paradigms: Standard HDE (with a future event horizon cutoff), Tsallis HDE (THDE), and Rényi HDE (RHDE). Utilizing simulated neutral hydrogen (HI) 21 cm emission lines from a census of over 109 galaxies up to z=1.0 and 21 cm absorption lines in 1.8×103 damped Lyman-α (DLA) systems, we simulate mock velocity drift observations (v˙) at high spectral resolutions (Δν=0.001 Hz and 0.002 Hz) over a semi-annual (Δt=0.5 year) integration baseline. We contrast these forecasts with current real-world constraints obtained from the joint Markov chain Monte Carlo (MCMC) likelihood analysis of the Pantheon SNe Ia and DESI DR2 BAO compilations. Our results reveal a profound limitation of integrated geometric probes: while current SNe Ia and BAO data suffer from severe parameter degeneracies—leaving the non-additive entropy scaling exponents (δ and α) completely unconstrained as open vertical bands—the simulated SKA redshift drift successfully breaks these degeneracies. From emission-line observations at 0.001 Hz spectral resolution, we obtain marginalized 1σ constraints of σc=±0.021 for the standard holographic parameter, σδ=±0.035 for the Tsallis entropy index, and σα=±0.007 for the Rényi parameter. Crucially, we identify and resolve a critical duplicate-plotting mathematical error present in the existing redshift-drift literature by providing the mathematically correct physical scaling and peak structures for both the dimensionless redshift drift (Sz) and the physical velocity drift (Sv). We conclude that the SKA will serve as a premier instrument for testing the holographic principle and non-extensive thermodynamics at cosmological scales.