Tsallis holographic inflation in $f(R,T)$ gravity: CMB constraints, reheating, and swampland implications

TL;DR

This paper develops a self-consistent inflationary scenario in which Tsallis holographic dark energy (THDE) density, evaluated at a Granda–Oliveros IR cutoff, acts as the inflaton potential within linear $f(R,T)$ gravity. Slow-roll dynamics, reconstructed $V(φ)$, and observable predictions $n_s$ and $r$ are derived in terms of $(α,β,δ,λ)$ and shown to be compatible with ACT DR6 across broad parameter ranges, with the matter–geometry coupling $λ$ playing a central role in both dynamics and swampland compatibility. Reheating and primordial gravitational waves impose additional constraints, including a BBN bound on $T_{re}$ and a GW-induced bound $ΔN_{eff}$, which together with the stiff-reheating PGW spectrum sharpen the viable region and predict detectable signals for upcoming detectors. The swampland analysis indicates that sub-Planckian excursions and sufficient potential steepness are achieved only for sufficiently large $λ$ (roughly $λ\gtrsim O(10^{2})$), enabling simultaneous satisfaction of distance and gradient criteria. Overall, the study provides a holographic, thermodynamically motivated inflationary framework in $f(R,T)$ gravity that remains consistent with CMB data, swampland bounds, reheating physics, and future gravitational-wave observations.

Abstract

Understanding how early-universe inflation may emerge from generalized holographic energy densities within modified gravity motivates the present analysis. We develop a self-consistent inflationary scenario in which the Tsallis holographic dark energy (THDE) density effectively acts as the inflaton potential in $f(R,T)$ gravity. Using the Granda-Oliveros infrared cutoff, we derive the corresponding slow-roll relations and identify a broad region of the parameter space $(α,β,δ,λ)$ that remains consistent with ACT DR6 (P-ACT-LB) constraints. By exploiting the dependence of the THDE density on the Hubble rate, we reconstruct the inflaton potential $V(φ)$ and show that both the field excursion $Δφ$ and the normalized potential gradient $|V'|/(V M_{p})$ are predominantly controlled by the matter-geometry coupling $λ$. We demonstrate that $λ\gtrsim \mathcal{O}(10^{2})$ suppresses the field excursion below the Planck scale and ensures $|V'|/(V M_{p}) \ge 1$, thereby satisfying both the distance conjecture and the refined de Sitter swampland bound. We also analyze the reheating stage. In addition to the primordial nucleosynthesis requirement $T_{\rm BBN} \approx 4~\mathrm{MeV}$, which sets a lower limit on the reheating temperature, the observational bound $ΔN_{\rm eff} \le 0.17$ imposes an additional constraint from primordial gravitational waves (PGWs). During stiff reheating phases with $ω_{\rm re} > 1/3$, the high-frequency PGW spectrum is significantly enhanced, producing a distinct signature that may fall within the sensitivity of upcoming detectors. Overall, this work provides an observationally consistent realization of holographic inflation in $f(R,T)$ gravity, jointly constrained by CMB data, swampland criteria, reheating physics, and PGW limits.

Figures (8)

• Figure 1: Trajectories in the $(n_s,r)$ plane obtained by varying: (a) $\beta$ at fixed $(\delta,\lambda)=(2.1,3.5)$, (b) $\delta$ at fixed $(\beta,\lambda)=(-10,3.5)$, and (c) $\lambda$ at fixed $(\beta,\delta)=(-10,2.1)$. Along each curve $\alpha$ is scanned over $-0.2\le\alpha\le 0.2$ with $N=65$. The orange band shows the Planck 2018 constraints, and the blue regions correspond to the $68\%$ and $95\%$ confidence intervals of the combined P-ACT-LB data set.
• Figure 2: Parameter-space regions consistent with the combined P-ACT-LB constraints. (a) $(\alpha,\beta)$ plane for $\delta=2.5$; (b) $(\alpha,\delta)$ plane for $\beta=-10$; (c) $(\beta,\delta)$ plane for $\alpha=0.02$. In all cases $\lambda=5$ and $N=65$ are fixed. These slices highlight the combined roles of the GO parameters and the Tsallis index in reproducing the observed $(n_s,r)$ values.
• Figure 3: Running of the scalar spectral index $\alpha_s$ versus $n_s$ for four representative parameter sets $(\alpha,\beta,\delta)$: set1 = $(0.01,-11,2.5)$, set2 = $(0.03,-10,5)$, set3 = $(0.02,-8,4.0)$, and set4 = $(0.005,-9,2.5)$, with $\lambda=5$. Along each trajectory, $N$ varies, and the marked points correspond to $N=65$. In the observationally relevant region, The resulting running is small and negative of the order of $\mathcal{O}(10^{-3})$, consistent with the combined P-ACT-LB bounds.
• Figure 4: Reheating temperature $T_{\mathrm{re}}$ as a function of the effective equation-of-state parameter $\omega_{\mathrm{re}}$ for two representative Tsallis holographic parameter sets, with $(\alpha,\beta,\delta)=(0.01,-11,2.5)$ for (a) and $(0.03,-10,5)$ for (b), and fixed $\lambda=5.0$. The lower horizontal line denotes the BBN limit ($T_{\mathrm{re}} = 4\,\mathrm{MeV}$), while the upper line shows the PGW-induced lower bound $T_{\mathrm{re}}^{\mathrm{GW}}$ derived in Eq. \ref{['eq:TreGW_standard']}, relevant when $\omega_{\mathrm{re}}>1/3$. Allowed reheating histories correspond to temperatures that lie above both lines and are consistent with Eq. \ref{['eq:Tre_closed']}.
• Figure 5: Present-day primordial GW spectrum $\Omega_{\mathrm{GW}}^{(0)}(k)$ for the same parameter sets as in Fig. \ref{['Tre_plots']}. For $\omega_{\mathrm{re}}>1/3$, the high-frequency slope steepens according to the power-law index $(6\omega_{\mathrm{re}} - 2)/(1 + 3\omega_{\mathrm{re}})$ in Eq. \ref{['eq:PGW_spectrum']}, and the turnover scale is set by $k_{\mathrm{re}}$ through the reheating relation in Eq. \ref{['eq:Tre_closed']}. Sensitivity curves of future detectors are included for comparison.