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Home > Constraints on attractor models of inflation and reheating from <math display="inline"><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>k</mi></math>, BICEP/Keck, ACT DR6, and SPT-3G data > Plots
 
 
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 $\varphi/M_P \gtrsim 6$.
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 {\it 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}).
Constraints on $\alpha$-attractor models showing the 68\% and 95\% CL contours from {\it Planck}/BICEP/Keck \cite{BICEP2021} (blue shadings), the P-ACT-LB combination \cite{ACT:2025tim} (purple contours) and the CMB-SPA dataset \cite{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 {\em 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.
Constraints on $\alpha$-attractor models showing the 68\% and 95\% CL contours from {\it Planck}/BICEP/Keck \cite{BICEP2021} (blue shadings), the P-ACT-LB combination \cite{ACT:2025tim} (purple contours) and the CMB-SPA dataset \cite{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 {\em 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.
Plot of $N_*$ as a function of $\trh$ for different values of $k$ in the E- and T-models with $\alpha = 1$.
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 $\alpha \lesssim 14-17$.
As in Fig.~\ref{fig:fullstarok}, showing the constraints on generalized T-model attractors with $V \propto \varphi^k$ minima for $k = 4, 6, 8,$ and $10$. While T-models predict systematically lower $n_s$ than E-models, increasing $k$ substantially improves compatibility with observations. For $k = 10$, even the canonical model with $\alpha = 1$ approaches the observational confidence regions.
As in Fig.~\ref{fig:alpha1}, showing the constraints on deformed E-model attractors (\ref{eq:modstaro}) (left panel) and T-model attractors (\ref{eq:modtmodel}) (right panel) with the deformation parameter $\kappa = 0.9999$. Observational contours and theoretical trajectories for various $\alpha$ values demonstrate how minimal modifications ($|1-\kappa| \sim 10^{-4}$) shift predictions through the centers of observational confidence regions. The canonical model ($\alpha = 1$) with $\kappa = 0.9999$ achieves remarkable agreement with the P-ACT-LB dataset while maintaining consistency with the $r$ constraint. Red-shaded band (E-model) and orange-shaded band (T-model) indicate reheating uncertainty from $T_{\rm EW}$ to $10^{10}$~GeV.
As in Fig.~\ref{fig:alpha1}, showing the constraints on deformed E-model attractors (\ref{eq:modstaro}) (left panel) and T-model attractors (\ref{eq:modtmodel}) (right panel) with the deformation parameter $\kappa = 0.9999$. Observational contours and theoretical trajectories for various $\alpha$ values demonstrate how minimal modifications ($|1-\kappa| \sim 10^{-4}$) shift predictions through the centers of observational confidence regions. The canonical model ($\alpha = 1$) with $\kappa = 0.9999$ achieves remarkable agreement with the P-ACT-LB dataset while maintaining consistency with the $r$ constraint. Red-shaded band (E-model) and orange-shaded band (T-model) indicate reheating uncertainty from $T_{\rm EW}$ to $10^{10}$~GeV.
As in Fig.~\ref{fig:alpha1}, showing the constraints on deformed attractor models with $\kappa = 0.9999$ for generalized potentials with $k = 4$ (left panels) and $k = 6$ (right panels). Both E-models (upper panels) and T-models (lower panels) show significant shift in the predictions for $n_s$ for $\alpha \sim 1$. These results demonstrate that deformations of order $\kappa \sim 10^{-4}$ can fully reconcile attractor models with current observations.
The spectral tilt, $n_s$, as a function of the number of $e$-folds $N_*$ for the $\alpha$-Starobinsky model with $\alpha = 1$. The solid blue line shows the exact numerical solution of the Mukhanov-Sasaki equation~(\ref{eq:MSeq}). The dotted gray line represents the slow-roll approximation using Hubble flow parameters~(\ref{eq:epsetaH}), while the dashed black line uses potential slow-roll parameters~(\ref{eq:epseta}). The shaded region indicates the 68\% CL constraint from \textit{Planck} 2018 data.