Constraints on Attractor Models of Inflation and Reheating from Planck, BICEP/Keck, ACT DR6, and SPT-3G Data

TL;DR

This work confronts α-attractor inflation—specifically E-model and T-model potentials derived from no-scale supergravity—with the latest CMB constraints from Planck 2018, BK18, ACT DR6, and SPT-3G. By incorporating reheating physics through the reheating temperature $T_{RH}$ and an effective equation of state, the authors translate observables into constraints on the number of $e$-folds $N_*$ and the model parameter $\alpha$, extending the analysis to generalized minima with $V(\varphi)\propto\varphi^k$ and to deformed attractors with a deformation parameter $\kappa$. They find that canonical $\alpha=1$ Starobinsky-like models are under tension with ACT DR6, while higher $\alpha$ or higher $k$ (e.g., $k\ge6$) and small deformations ($\kappa\approx0.9999$) can bring predictions into agreement with multiple data sets. The results highlight a flexible landscape where attractor models remain compatible with current observations, and point to future tests by upcoming missions (e.g., LiteBIRD) to decisively probe $r$ and discriminate among attractor realizations. Overall, the paper systematically maps how reheating, generalized minima, and deformations affect $(n_s,r)$ predictions and their compatibility with Planck, BK18, ACT DR6, and SPT-3G data, advancing our understanding of the inflationary paradigm within no-scale supergravity.

Abstract

We analyze the latest cosmic microwave background (CMB) constraints on the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$ from Planck 2018, BICEP/Keck 2018, the Atacama Cosmology Telescope Data Release 6 (ACT DR6), and the South Pole Telescope (SPT-3G) data, focusing on their implications for attractor models of inflation. We compare systematically observational bounds with theoretical predictions for both E-model ($α$-Starobinsky) and T-model potentials. The observational constraints accommodate E-models with $α\lesssim 25$, with the canonical Starobinsky model ($α= 1$) predicting $n_s = 0.958-0.963$ for reheating temperatures between $100 - 10^{10}$ GeV, in good agreement with Planck 2018 data and within the 95% CL region determined by the Planck-ACT-SPT combination, but below the 95% confidence region of the Planck-ACT-DESI combination. Higher reheating temperatures from near-instantaneous reheating improve the compatibility. T-models predict slightly lower $n_s$ values (0.956-0.961), in some tension with Planck 2018 data, and we find an upper limit of $α\lesssim 11$ in these models. We extend our analysis to generalized $α$-attractors with monomial potentials $V(\varphi) \propto \varphi^k$ near the minimum, demonstrating that models with $k \geq 6$ naturally predict $n_s \simeq 0.965 - 0.968$ for typical number of $e$-folds, in better agreement with the ACT DR6 data. We also consider deformed E- and T-models, which allow significantly higher values of $n_s$ for low values of $α\simeq 1$.

Figures (9)

• Figure 1: Deformed attractor potentials with $\alpha = 1$ and $k = 2$. Left panel: Deformed Starobinsky (E-model) potential for $\kappa = 1$, $0.9999$, $0.9998$, and $0.9997$. Right panel: Deformed T-model potential for the same $\kappa$ values. The deviations from $\kappa = 1$ introduce observable modifications to the inflationary plateau, and become substantial for $\varphi/M_P \gtrsim 6$.
• Figure 2: The relation between $N_*$ (defined for a pivot scale of $0.05$ Mpc$^{-1}$) and the reheating temperature in the E-model (upper panels) and the T-model (lower panels), for $\alpha = 1$ (left panels) and $\alpha = 10$ (right panels). The horizontal shadings corresponds to the 95% CL bounds from Planck 2018, CMB-SPA, and P-ACT-LB. The vertical shadings show the constraints of $T_{\rm RH}$ from Big Bang Nucleosynthesis (BBN) and gravitino production in supersymmetric models. Note that the limits on $n_s$ (right vertical axis) depend on $\alpha$, as the calculated value of $r$ depends on $\alpha$ as seen in Eq (\ref{['eq:gentmodr']}).
• Figure 3: Constraints on $\alpha$-attractor models showing the 68% and 95% CL contours from Planck/BICEP/Keck BICEP2021 (blue shadings), the P-ACT-LB combination ACT:2025tim (purple contours) and the CMB-SPA dataset SPT-3G:2025bzu (brown rectangles). Note that the latter provides only an $r$-independent limit on $n_s$. All pivot scales are taken as $k_*=0.05$ Mpc$^{-1}$ except for the Planck pivot scale for $r$, conventionally chosen to be $k_*=0.002$ Mpc$^{-1}$. Left panel: $\alpha$-Starobinsky (E-model) predictions in the $(n_s, r)$ plane. Solid lines indicate reheating temperatures from $T_{\rm BBN}$ (4 MeV), $T_{\rm EW}$ (100 GeV), $10^{10}$ GeV (the gravitino bound), and $2 \times 10^{15}$ GeV (instantaneous reheating with $\Gamma_\varphi = H$). The dashed line shows $N_* = 50$ for reference. Right panel: As in the left panel for the T-model predictions.
• Figure 4: Plot of $N_*$ as a function of $T_{\rm RH}$ for different values of $k$ in the E- and T-models with $\alpha = 1$.
• Figure 5: As in Fig. \ref{['fig:alpha1']}, showing the constraints on generalized $\alpha$-Starobinsky (E-model) attractor models with $V \propto \varphi^k$ minima for $k = 4, 6, 8,$ and $10$. Shaded bands indicate the range of $N_*$ allowed by reheating temperatures from $T_{\rm EW}$ to $10^{10}$ GeV. The curve for $T_{\rm BBN}$ is degenerate with that shown for $T_{\rm EW}$ when fragmentation effects are included (see Fig. \ref{['fig:NvsTvsk']}). For $k>4$, the curve for instantaneous reheating is to the left of the shaded band. Higher $k$ values systematically shift predictions toward larger $n_s$, improving consistency with the ACT DR6 data. The constraint on $r$ limits all models to $\alpha \lesssim 14-17$.