k-inflation: Non-separable case meets ACT measurements

Abstract

We investigate a non-separable subset of $k$-essence in which the kinetic and potential sectors interact through an $X^{\rho}V(\phi)$ coupling, implemented via a potential-dependent prefactor $f(\phi)=1+2\mathcal{K}V$. In slow roll, this structure preserves a constant sound speed $c_s^2=1/(2\rho-1)$ while modifying the Hubble flow in a controlled way, thereby shifting the inflationary observables relative to the separable template. For monomial potentials $V=A\phi^n$ (with $n=2$ and $n=2/3$ as representative cases) we derive closed analytic expressions for $n_s(N_\ast)$ and $r(N_\ast)$ to $\mathcal{O}(\epsilon_{\rm mix}^2)$, where $\epsilon_{\rm mix}\propto\mathcal{K}$ encodes the non-separable $X^\rho V$ mixing, and we validate them against exact background integrations. The analytic and numerical predictions agree at the sub-per-mille level for $n_s$ and at the percent level for $r$, confirming the accuracy of the small-mixing expansion. For $\mathcal{K}<0$ the mixing systematically lowers both $n_s$ and $r$ at fixed $N_\ast$, allowing otherwise marginal monomials to fall within the region favored by recent ACT+{\it Planck}+BAO constraints (P--ACT--LB). All solutions shown satisfy the health conditions $f(\phi)>0$, $\rho>\tfrac12$, and the positivity bound $V<1/(2|\mathcal K|)$ (from $f>0$). We also discuss parameter dependence and the expected equilateral-type non-Gaussianity, which remains comfortably within current bounds for the benchmarks considered.