Modular invariant inflation and reheating

TL;DR

The paper presents a minimal framework where modular symmetry, specifically the finite group $\Gamma_3\cong A_4$, organizes both the lepton flavor sector and early-universe cosmology through a common modulus field that acts as the inflaton. Neutrino masses arise via Type-I seesaw with modular form–determined Yukawas, while inflation proceeds along a modular-invariant trajectory stabilized near fixed points, yielding a flat potential and predictions of $r=\mathcal{O}(10^{-7})$ and $\alpha=\mathcal{O}(-10^{-3})$, with $n_s$ in the Planck range. Reheating is generated from the same modular expansions that govern flavor, with inflaton decays dominantly to right-handed neutrinos; the resulting $T_{\text{rh}}$ can satisfy BBN bounds for sufficiently large $m_\phi$, but non-thermal leptogenesis is challenging in this small-field setup. The work demonstrates that modular symmetry can cohesively address flavor, inflation, and reheating, and points to large-field modular inflation as a promising direction to potentially connect these aspects to the observed BAU.

Abstract

We use modular symmetry as an organizing principle that attempts to simultaneously address the lepton flavor puzzle, inflation, and post-inflationary reheating. We demonstrate this approach using the finite modular group $A_4$ in the lepton sector. In our model, neutrino masses are generated via the Type-I see-saw mechanism, with modular symmetry dictating the form of the Yukawa couplings and right-handed neutrino masses. The modular field also drives inflation, providing an excellent fit to recent Cosmic Microwave Background (CMB) observations. The corresponding prediction for the tensor-to-scalar ratio is very small, $r \sim \mathcal{O}(10^{-7})$, while the prediction for the running of the spectral index, $α\sim -\mathcal{O}(10^{-3})$, could be tested in the near future. An appealing feature of the setup is that the inflaton-matter interactions required for reheating naturally arise from the expansion of relevant modular forms. Although the corresponding inflaton decay rates are suppressed by the Planck scale, the reheating temperature can still be high enough to ensure successful Big Bang nucleosynthesis. The same couplings responsible for reheating can also contribute to generating baryon asymmetry of the Universe through non-thermal leptogenesis. However, the contribution is negligibly small in the current inflationary setup.

Figures (7)

• Figure 1: Shape of the potential along the angular and radial directions with $A = 55.2783$, $\beta = 0.6516$ and $\gamma=0$. The top-left panel depicts the inflation potential with $\rho = 1$, where $\theta = \pi/2$ marks the starting point of inflation. The top-middle panel provides a zoomed-in view of the inflation potential around the desired minimum at $\tau = \tau_0$. Note that $\theta = 2\pi/3$ corresponds to a local minimum, whose potential energy does not vanish, while $\theta \approx 0.661\pi$ represents the global minimum. The top-right panel shows the radial potential with a fixed angular coordinate, where the inflationary trajectory remains at the minimum in this direction. Finally, the bottom panel is a contour plot of the inflation potential, with the red arrow indicating the trajectory of inflation.
• Figure 2: The black lines represent the predictions for $(n_s, r)$ with model parameters: $A = 55.2783$, $\beta = 0.6516$, $\gamma = 0$ (solid line) and $A = 80.2435$, $\beta = 0$, $\gamma = -1.2314$ (dotted line). The yellow shaded region corresponds to constraints from the combined results of Planck 2018, BICEP/Keck 2018, and BAO data BICEP:2021xfz. The small and large red dots indicate $N_e = 50$ and $N_e = 60$, respectively.
• Figure 3: The contour plot of the spectral index $n_s$ in the parameter planes $A-\beta$ for $\gamma=0$ (upper panels) and $A-\gamma$ for $\beta=0$ (lower panels). The e-folds are taken to be 50 and 60 in the left panels and right panels respectively. The dark green regions on the left and dark red regions on the right stand for the $68\%$ CL region of $n_s$ for different values of e-folding. The black solid and dashed lines correspond to the central values of $n_s$ from Planck Planck:2018jri and ACT DR$6$ACT:2025fju respectively.
• Figure 4: Reheating temperature as function of the lightest right handed neutrino mass $M_1$ and inflaton mass $m_\phi$ by considering inflaton two and three body decays.
• Figure 5: Feynman diagrams for inflaton two-body decay.