Cosmological Inflation in $f(Q,\mathcal{L}_{m})$ Gravity

TL;DR

The paper investigates cosmological inflation within f(Q, L_m) gravity, a symmetric teleparallel framework in which the non-metricity scalar Q couples directly to the matter Lagrangian. Using a power-law inflaton potential, it derives slow-roll expressions for the linear model $f(Q,L_m) = -α Q + 2 L_m + β$ and the nonlinear model $f(Q,L_m) = -α Q + (2 L_m)^2 + β$, obtaining $n_s$ and $r$ and comparing them to Planck BK15 BAO data. Results show that the linear case is viable for α>0, β with narrow contours in parameter space, while the nonlinear case is viable for α<0, β with predictions clustering around the 68% joint-data region, indicating distinct inflationary regimes within the same theoretical framework. Overall, f(Q, L_m) gravity provides a flexible and competitive alternative to GR for modeling early-universe inflation and motivates further exploration of its phenomenology and observational signatures.

Abstract

Cosmological inflation remains a key paradigm for explaining the earliest stages of the Universe, yet the theoretical limitations of General Relativity (GR) motivate the development of alternative formulations capable of addressing both early and late cosmic acceleration. In this work, we investigate cosmological inflation within the $f(Q,\mathcal{L}_{m})$ gravity framework based on symmetric teleparallel geometry, where the non-metricity scalar $Q$ couples directly to the matter Lagrangian. We formulate the slow-roll dynamics and derive analytical predictions for the scalar spectral index $n_{s}$ and tensor-to-scalar ratio $r$ in both linear and nonlinear non-minimal coupling models, assuming a power-law inflaton potential. Our findings show that the linear case, $f(Q,\mathcal{L}_{m})=-αQ + 2\mathcal{L}_{m}+β$, becomes compatible with Planck+BK15+BAO constraints for positive $α$ and $β$, producing narrow viable contours in parameter space. In contrast, the nonlinear model, $f(Q,\mathcal{L}_{m})=-αQ+(2\mathcal{L}_{m})^{2}+β$, achieves observational viability only for negative $α$ and $β$, and its predictions predominantly fall inside the $68\%$ confidence region of joint data. These results demonstrate that $f(Q,\mathcal{L}_{m})$ gravity produces distinct inflationary regimes, providing a highly competitive alternative to GR.

Figures (4)

• Figure 1: The scalar spectral index $n_{\rm s}$ (top) and tensor-to-scalar ratio $r$ (bottom) for the linear case, shown as functions of the parameters $\alpha$ and $\beta$ for three representative values of $\nu = 10^{-3}$, $10^{-5}$, and $10^{-7}$ with $N = 50$.
• Figure 2: The $(n_{\rm s}, r)$ plane is shown for the linear case with a power-law potential, considering $\nu = 10^{-3}$, $10^{-5}$, and $10^{-7}$ over the range $N \in [50, 60]$ and for different values of $\alpha$ and $\beta$. The solid circles and stars correspond to the predictions for $N = 50$ and $60$, respectively. The shaded regions depict the Planck 2018 observational constraints from various surveys akrami2020planck.
• Figure 3: The scalar spectral index $n_{\rm s}$ (top) and tensor-to-scalar ratio $r$ (bottom) for the nonlinear case, shown as functions of the parameters $\alpha$ and $\beta$ for three representative values of $\nu = 10^{-3}$, $10^{-5}$, and $10^{-7}$ with $N = 50$.
• Figure 4: The $(n_{\rm s}, r)$ plane is shown for the nonlinear case with a power-law potential, considering $\nu = 10^{-3}$, $10^{-5}$, and $10^{-7}$ over the range $N \in [50, 60]$ and for different values of $\alpha$ and $\beta$. The solid circles and stars correspond to the predictions for $N = 50$ and $60$, respectively. The shaded regions depict the Planck 2018 observational constraints from various surveys akrami2020planck.