Non-minimally coupled loop quantum inflation with inverse-volume corrections

Abstract

We study slow-roll inflation driven by a scalar field non-minimally coupled to gravity within the effective framework of Loop Quantum Cosmology (LQC), including inverse-volume corrections. We consider two physically motivated classes of potentials, a Higgs-like quartic potential $V\proptoφ^{4}$ and string-inspired fractional monomial potentials $V\proptoφ^{p}$ with $p<1$. Working at first order in the slow-roll expansion, we derive analytic expressions for the inflationary observables, namely the scalar spectral index $n_s$, the tensor-to-scalar ratio $r$, and the running $α_s\equiv dn_s/d\ln k$, and then solve the corrected background dynamics numerically to obtain quantitative predictions. Confronting these results with current observational constraints from Planck 2018 and ACT DR6, we find that the model can lie within the allowed region of the $(n_s,r,α_s)$ parameter space, including a mild preference for slightly larger $n_s$, as suggested by recent ground-based measurements. We also compute the probability of achieving sufficient slow-roll inflation in this setting. Although effective LQC replaces the initial singularity with a nonsingular quantum bounce, the likelihood of a sufficiently long inflationary phase depends on the pre-inflationary dynamics and on the inflaton potential. Using the canonical Liouville measure on the effective phase space, we determine the fraction of post-bounce trajectories that yield sufficient inflation and find that the non-minimal coupling parameter $ξ$ substantially enlarges the phase-space volume of favorable initial conditions relative to the minimally coupled case, exhibiting an attractor-like enhancement that saturates at large $ξ$.

Figures (12)

• Figure 1: Inverse-volume correction $D_l(q)$ as a function of $q$, shown for $l=3/4$.
• Figure 2: Evolution of $\delta_D$ with number of e-fold $N$ for $l=3/4$
• Figure 3: Inflationary predictions for $V(\phi)\propto\phi^{4}$ in LQC with inverse-volume corrections. Left panel shows $r$ versus $n_s$ and right panel shows $\alpha_s$ versus $n_s$ for different non-minimal couplings $\xi$, with markers at $N=50$ and $N=70$. We fix the operator-ordering parameters $m=0$, $n=0$, the inverse-volume eigenvalue parameter $l=3/4$, and the initial value $q_i=100$.
• Figure 4: Allowed region in the $(N,\xi)$ plane for the quartic potential $V(\phi)\propto\phi^{4}$ in LQC with inverse-volume corrections, imposing the theoretical bound $r\gtrsim 5 \times 10^{-4}$Franciolini:2018ebs. The left panel uses P-LB-BK18 constraints and the right panel uses P-ACT-LB-BK18 constraints. We fix the operator-ordering parameters $m=0$ and $n=0$, the inverse-volume eigenvalue parameter $l=3/4$, and the initial value $q_i=100$.
• Figure 5: Effect of the non-minimal coupling parameter $\xi$ on the probability of inflation $\mathcal{P}$ for the quartic potential $V(\phi)=\lambda \phi^{4}$, with $\lambda\simeq 6.14\times10^{-13}$ inferred from the scalar power-spectrum amplitude $\mathcal{P}_{s}=2.141\times 10^{-9}$Planck:2018jri. The left panel shows the surface $\mathcal{P}(N,\xi)$ and the right panel shows the corresponding contour map. We fix the inverse-volume eigenvalue parameter to $l=3/4$ and the initial value to $q_i=100$.