Singular $α$-attractors

TL;DR

The paper extends the α-attractor inflationary framework by introducing S-models, whose potentials are singular at the moduli-space boundary. These singular terms uplift the plateau at large field values, allowing a broader range for the scalar spectral index $n_s$ (up toward $1$) while keeping the tensor-to-scalar ratio $r$ within current bounds, thus better accommodating ACT-SPT-DESI results. It systematically analyzes logarithmic and power-law singularities (and SL(2,ℤ) cases), presents concrete T-, E-, P-, and Fibre-inflation realizations, and discusses how S-models can resolve initial-condition problems for single-field α-attractors. The framework links to deformed α-attractors and streamlined supergravity constructions, offering a versatile, geometry-grounded path to reconcile cosmological data with theoretical models.

Abstract

$α$-attractor models naturally appear in supergravity with hyperbolic geometry. The simplest versions of $α$-attractors, T- and E-models, originate from theories with non-singular potentials. In canonical variables, these potentials have a plateau that is approached exponentially fast at large values of the inflaton field $\varphi$. In a closely related class of polynomial $α$-attractors, or P-models, the potential is not singular, but its derivative is singular at the boundary. The resulting inflaton potential also has a plateau, but it is approached polynomially. In this paper, we will consider a more general class of potentials, which can be singular at the boundary of the moduli space, S-models. These potentials may have a short plateau, after which the potential may grow polynomially or exponentially at large values of the inflaton field. We will show that this class of models may provide a simple solution to the initial conditions problem for $α$-attractors and may account for a very broad range of possible values of $n_{s}$ matching the recent ACT, SPT, and DESI data.

Figures (9)

• Figure 1: This figure from Linde:2018hmx shows the axially symmetric $\alpha$-attractor plateau potential bounded by an exponentially steep wall, which emerges because of the singularity of the potential (\ref{['singL']}) at $|Z| = 1$.
• Figure 2: The blue line shows the potential (\ref{['toymodelpotentialhyper']}) for $\alpha = 1$, $\delta = 10^{{-5}}$Linde:2017pwt. The potential height is shown in units of $V_{0}$. For comparison, the yellow line shows the usual $\alpha$-attractor with $\delta = 0$.
• Figure 3: The potential (\ref{['sing1']}) for $\alpha = 1/3$, $m,n=1$, and $\delta = 3\times 10^{-2}$
• Figure 4: The potential (\ref{['sing2']}) shown in units of $V_{0}$ for $\alpha = 1/3$, $m,n=1$, and $\delta = 3\times 10^{-2}$ (upper curve), $\delta=10^{-2}$, $\delta = 2\times 10^{-2}$, $\delta=5\times 10^{-3}$ and $\delta= 10^{-3}$ (lower curve).
• Figure 5: The potential (\ref{['sing2']}) shown in units of $V_{0}$ for $\alpha = 1/3$, $m=1, n=2$, and $\delta = 10^{-3}$. To make its parabolic shape at large $\varphi$ more visible we show the potential for $\varphi < 20$.