$Sp(4,\mathbb{Z})$ modular inflation

TL;DR

This paper develops Siegel modular inflation by extending modular invariance from $SL(2,Z)$ to the genus-2 group $Sp(4,Z)$, introducing three moduli and preserving a hyperbolic Kahler geometry to realize multi-field $\alpha$-attractor dynamics. By employing genus-2 absolute invariants $y_1,y_2,y_3$, the authors construct modular-invariant potentials in targeted two-field and single-field subspaces, achieving E-model and T-model–like plateau inflation and a modified polynomial $\alpha$-attractor to accommodate higher $n_s$ values. In the two-field cases, the potentials yield plateau-like behavior with small tensor-to-scalar ratio $r$ of order $10^{-3}$ and $n_s$ in the Planck/ACT/SPT favored ranges, while the single-field construction reproduces a polynomial-$\alpha$-attractor consistent with ACT/SPT at $N=60$. The results demonstrate a coherent framework in which modular symmetry extends inflationary model-building to multiple inflatons, with Minkowski vacua at fixed points and clear avenues for embedding into string-inspired cosmology, though three-moduli dynamics and isocurvature effects remain to be explored.

Abstract

We investigate inflation models governed by the Siegel modular group $Sp(4,\mathbb{Z})$. The $Sp(4,\mathbb{Z})$ group extends the $SL(2,\mathbb{Z})$ framework from one modulus to three moduli while preserving the hyperbolic geometry of the Kähler potential, allowing for the construction of cosmological $α$-attractor models. In this context, we use genus $g=2$ absolute invariants to construct inflationary potentials within specific subspaces of the Siegel moduli space. These models are driven by the imaginary components of the moduli $τ$ and naturally yield plateau-like potentials consistent with Planck 2018 observations in large field limit. We employ two-dimensional complex subspaces to realize E-model and T-model like two-field inflation scenarios. We explore the subspace of complex dimension one to construct a modified polynomial $α$-attractor model, which can accommodate the larger spectral index $n_s$ favored by recent ACT and SPT data, particularly in the larger $N$ regime.

Figures (5)

• Figure 1: The left panel illustrates the variation of the scalar potential $V$ in Eq. \ref{['Eq:potentialV1']}, as a function of $\varphi_1$ and $\varphi_2$. The right panel displays the corresponding gradient vector field. Here the parameters of potential are set to $\alpha=1/3$ and $\beta=e^{4\pi}$.
• Figure 2: Vector plots of the potential gradients for the potential in Eq. \ref{['Eq:potentialV1']}. The left and right panels correspond to $\alpha=1/3$ and $\alpha=1$, respectively, both with $\beta=e^{4\pi}$. The red region denotes the spectral index range $0.9602 < n_s < 0.9693$Planck:2018vyg. The orange region shows where the tensor-to-scalar ratio satisfies $r \leq 0.036$BICEP:2021xfz, with its boundary indicated by a thin pink line. The green line represents the condition $|\eta_H| = 1$, marking the end of inflation.
• Figure 3: The solid, dashed, and dotted lines represent the theoretical $(n_s, r)$ predictions from the models in Eqs. \ref{['Eq:potentialV1']}, \ref{['eq:tau13potential']}, and \ref{['eq:vact']}, with $\alpha = 1/3$ and distinct initial conditions , respectively. The blue contours denote the $68\%$ and $95\%$ confidence regions obtained from the combined Planck 2018, BICEP/Keck 2018, and BAO data BICEP:2021xfz, whereas the purple contours represent the results from the combined Planck 2018, ACT, BAO, BICEP/Keck 2018 ACT:2025tim. The small orange and large red dots indicate the predictions for $N = 50$ and $N = 60$, respectively.
• Figure 4: The left panel displays the variation of the scalar potential $V$ in Eq. \ref{['eq:tau13potential']} with respect to $\varphi_1$ and $\varphi_2$. The right panel shows the gradient vector field of this potential , where the dashed lines indicate the global minima and the dots denote the local minima. The parameters are set to $\alpha=1/3$ and $\beta=e^{4\pi}$.
• Figure 5: The left panel is the variation of the scalar potential $V$ in Eq. \ref{['eq:vact']} with respect to $\Re(\tau_1)$ and $\Im(\tau_1)$, where $\tau_1$ is the complex modulus. The right panel shows this potential as a function of $\varphi$ for $\theta=0$. Here the potential parameters are taken to be $\alpha=1/3$, $\beta=2$ and $c=0.3$.