S. Taghavi, Tayeb Golanbari, Kh. Saaidi
Abstract In this work, we develop an inflationary scenario in which Tsallis holographic dark energy (THDE) provides an effective inflaton potential driving early-universe inflation within $f(R,T)$ gravity. Adopting the Granda–Oliveros infrared cutoff, we derive the slow-roll relations and delineate a viable region of the parameter space $(\alpha ,\beta ,\delta ,\lambda )$ consistent with the Planck/ACT low-band combined likelihood (P-ACT-LB) from ACT DR6. Exploiting the parametric dependence of the THDE density on the Hubble expansion rate, we reconstruct the effective inflaton potential $V(\phi )$ across the observable window. We show that both the inflaton field excursion and the potential gradient $|V^{\prime }|/(V M_p)$ are predominantly controlled by the matter–geometry coupling $\lambda$. In particular, for $\lambda \gtrsim \mathcal {O}(10^{2})$, the field excursion is suppressed below the Planck scale while ensuring $|V^{\prime }|/(V M_p) \gtrsim \mathcal {O}(1)$, consistent with the distance conjecture and the refined de Sitter swampland bound. We further investigate the reheating phase following inflation. In addition to the primordial nucleosynthesis requirement $T_{\rm BBN} \simeq 4\, \mathrm{MeV}$, the observational bound $\Delta N_\text{eff} \le 0.17$ imposes an additional lower limit on the reheating temperature through the integrated contribution of primordial gravitational waves (PGWs). During stiff-reheating stages with $\omega _{\rm re} > 1/3$, the high-frequency PGW spectrum is significantly enhanced, potentially leading to observable signatures in the frequency bands of forthcoming gravitational-wave detectors. For suitable parameter values, the resulting PGW signal may enter the projected sensitivity range of upcoming experiments. Overall, this work presents a unified and observationally consistent realization of holographic inflation in $f(R,T)$ gravity, jointly constrained by cosmic microwave background observations, reheating dynamics, primordial gravitational waves, and swampland bounds.