k-inflation: Non-separable case meets ACT measurements

TL;DR

This work introduces a non-separable $k$-essence model with $P(X,\phi)=f(\phi)[X^{\rho}-V(\phi)]$ and a potential-dependent prefactor $f(\phi)=1+2\mathcal{K}V(\phi)$, yielding a constant sound speed $c_s^2=1/(2\rho-1)$. Focusing on monomial potentials $V(\phi)=A\phi^{n}$, it derives analytic expressions for the inflationary observables $n_s(N_{\ast})$ and $r(N_{\ast})$ up to second order in the small mixing parameter $\epsilon_{\rm mix}$ (with $\epsilon_{\rm mix}\propto\mathcal{K}$) and validates them against exact background integrations. The results show that $\mathcal{K}<0$ systematically lowers both $n_s$ and $r$ at fixed $N_{\ast}$, enabling otherwise marginal potentials to fall within the P--ACT--LB constraints while staying in the slow-roll regime (with $f(\phi)>0$, $\rho>1/2$, and $V<1/(2|\mathcal{K}|)$). The analysis also indicates modest equilateral-type non-Gaussianity within current bounds, and suggests broader future work including full likelihood analyses with ACT+Planck+BAO data and extensions to other potentials and UV completions.

Abstract

We investigate a non-separable subset of $k$-essence in which the kinetic and potential sectors interact through an $X^ρV(φ)$ coupling, implemented via a potential-dependent prefactor $f(φ)=1+2\mathcal{K}V$. In slow roll, this structure preserves a constant sound speed $c_s^2=1/(2ρ-1)$ while modifying the Hubble flow in a controlled way, thereby shifting the inflationary observables relative to the separable template. For monomial potentials $V=Aφ^n$ (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 $\mathcal{O}(ε_{\rm mix}^2)$, where $ε_{\rm mix}\propto\mathcal{K}$ encodes the non-separable $X^ρV$ 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 $f(φ)>0$, $ρ>\tfrac12$, 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.

Figures (3)

• Figure 1: Predictions in the $(n_s,r)$ plane from Eqs. \ref{['epsilonNew']}, \ref{['rr']} and \ref{['nsanalytic']}. Curves correspond to varying $\epsilon_{\rm mix}$ at fixed $(\rho,n,N_\ast)$; contours combine ACT with Planck 2018, BK18 and BAO (P--ACT--LB).
• Figure 2: Allowed region in $(\epsilon_{\rm mix},\rho,N_\ast)$ for $n=2$ (top) and $n=2/3$ (bottom). Shaded points satisfy the P--ACT--LB constraints on $(n_s,r)$. All shown points also obey $f>0$ and $|\mathcal{K}|V<\tfrac{1}{2}$.
• Figure 3: Slices of Fig. \ref{['fig1']} showing the allowed $(\epsilon_{\rm mix},\rho)$ region for fixed $N_\ast$ ($N_\ast=70$ for $n=2$ and $N_\ast=60$ for $n=2/3$). Shaded areas lie inside the P--ACT--LB contour.