Hamilton-Jacobi analysis of noncanonical inflation in $f(R, T)$ gravity: Constraints from Planck/ACT data, and theoretical bounds

TL;DR

This work develops a Hamilton-Jacobi analysis of noncanonical inflation within $f(R,T)$ gravity and confronts two explicit H(phi) forms, a power-law and an exponential, with Planck and ACT DR6 data. The noncanonical kinetic term, controlled by parameters $\alpha$ and the matter-geometry coupling $\lambda$, yields a tunable sound speed $c_s$ that suppresses the tensor-to-scalar ratio and broadens viable parameter space for the power-law case. Reheating and its imprint on the stochastic gravitational-wave background, via constraints on $\Delta N_{eff}$, impose a lower bound on $T_{re}$ and restrict the total number of e-folds to roughly $N\lesssim 64$--$65$, linking inflationary dynamics to post-inflationary history. While swampland criteria are satisfied in the explored region, the Trans-Planckian Censorship Conjecture remains in tension, underscoring the challenges of embedding these scenarios in a UV-complete theory. The framework predicts an enhanced high-frequency PGW spectrum, offering potential observability with future detectors and providing a concrete, testable bridge between early-universe phenomenology and quantum gravity constraints.

Abstract

The latest CMB data from ACT DR6, combined with Planck, DESI, and BICEP/Keck, indicate a slight upward shift in the scalar spectral index, placing several previously favored inflationary models under tension. We study an inflationary scenario within the framework of $f(R, T)$ gravity, featuring a nonminimal matter-curvature coupling, where the inflaton is a noncanonical scalar field with a generalized kinetic energy. Using the Hamilton-Jacobi formalism, we express the Hubble parameter as a function of the scalar field and consider two forms of $H(φ)$, a power-law and an exponential one, deriving the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$. Comparison with ACT DR6 allows us to explore the parameter space, showing that the power-law case is compatible with the data across a wide range, while the exponential form requires a large number of e-folds. We then study reheating, noting its close link with the inflationary dynamics. By imposing the bound on overproduction of primordial gravitational waves encoded in the constraint on $ΔN_{\text{eff}}$, we obtain a lower limit on the reheating temperature, which becomes particularly restrictive for the stiff reheating equation of state $ω_{\text{re}}$. This bound implies that the total number of e-folds should not exceed $N\lesssim 64(65)$. The predicted gravitational-wave spectrum shows an enhanced high-frequency amplitude, potentially observable by future detectors. We also examine consistency with the Swampland conjectures and the Trans-Planckian Censorship Conjecture, finding that combining $f(R, T)$ gravity with noncanonical dynamics provides a rich and testable framework for the early universe.

Figures (8)

• Figure 1: The sound speed $c_s$ as a function of $\lambda$ for different values of $\alpha$. For $\alpha=1$ (canonical case), one has $c_s^2=1$ independently of $\lambda$, while for larger $\alpha$ the sound speed decreases with increasing $\lambda$.
• Figure 2: Predictions of the power-law case in the $r$--$n_s$ plane: (a) trajectories as a function of $\alpha$ for several $n$, and (b) trajectories as a function of $n$ for several $\alpha$. In both panels we take $N=65$ and $\lambda=0.1$. Shaded regions: orange (Planck 2018, $68\%/95\%$ CL) and blue (ACT DR6, 2025, $68\%/95\%$ CL).
• Figure 3: Allowed region in the $(n,\alpha)$ plane consistent with observations. Orange contours: Planck 2018 ($95\%/68\%$ CL). Blue contours: ACT DR6 (2025) ($95\%/68\%$ CL).
• Figure 4: Reheating temperature $T_{\text{re}}$ versus $\omega_{\text{re}}$ for (a) $(n,\alpha)=(0.5,3.5)$ and (b) $(n,\alpha)=(1.2,6)$. Shaded bands: ACT DR6 (2025) $95\%/68\%$ CL in blue and Planck 2018 $95\%$ CL in orange. The red curve indicates the lower bound $T_{\text{re}}^{\rm GW}$ from the PGW constraint $\Delta N_{\rm eff}\leq0.17$ (effective for $\omega_{\text{re}}\gtrsim 0.59$); the region below the red curve is excluded.
• Figure 5: Present-day PGW spectrum $\Omega_{\rm GW}(f)$ for several $\omega_{\text{re}}$ values and two inflationary durations: (a) $N_k=60$, (b) $N_k=65$.