Closing in on $α$-attractors

TL;DR

This work investigates whether α-attractor T-models can accommodate larger $n_s$ values hinted by recent CMB data, by allowing a stiff reheating phase governed by $\bar{w}=\frac{p-2}{p+2}$ for even $p$. The authors compute $n_s$ and $r$ across a grid of $p$ and $\alpha$, incorporating reheating duration $\Delta N_{rh}$ up to model- and GW-consistent maximums, and confront these predictions with the P-ACT-LB-BK18 data. They find $n_s$ is maximized near $\alpha\approx1$ and that the largest attainable $n_s$ in T-models is $n_s\approx0.9682$ in the large-$p$ limit; extended reheating enhances $n_s$ but this effect saturates as $\bar{w}\to1$, enabling a potential future falsification of T-models if $n_s$ is measured above this limit. The study highlights that including reheating and GW bounds yields stronger constraints on $\alpha$ than $r$ alone and positions the seven Poincaré models as key candidates for large-$n_s$ inflation, providing a concrete benchmark for upcoming CMB measurements.

Abstract

Recent observations of cosmic microwave background (CMB) anisotropies combined with large-scale structure may point towards higher values of the scalar spectral index, $n_s$. This puts previously preferred inflationary models, such as $α$-attractors, in tension with the new measurements. Pending a resolution of the tension between BAO parameters as determined by CMB datasets and those determined by DESI, we explore in this work the large-$n_s$ regime of $α$-attractor T-models. We show that some T-models can self-consistently produce an extended reheating stage with a stiff equation of state $(\bar w>1/3)$, which allows values for $n_s$ closer to unity. We employ constraints from P-ACT-LB-BK18 data to illustrate what large-$n_s$ observations might imply for T-models. We show that the $n_s$ measurement yields an upper limit on $α$ that is stronger than the one from the tensor-to-scalar ratio only. We find that $n_s$ is maximised for $α\sim1$, therefore the seven Poincaré models are well placed to deliver large $n_s$. However, the ability of a stiff reheating stage to increase the compatibility of T-models with large-$n_s$ measurements saturates as $\bar{w}\to1$. Thanks to this effect, we establish that the largest $n_s$ that T-models can produce is $n_s=0.9682$. T-models are therefore highly predictive in the large-$n_s$ regime and our result provides, under the assumption of perturbative reheating, a benchmark which could be used in the future to rule out T-models.

Figures (10)

• Figure 1: Left panel: For T-models \ref{['eq: T-model potential']} with even $p\in [2,\,20]$ and $10^{-4}\leq \alpha \leq 20$ we represent $\Delta N_\text{rh, max 1}$, obtained from Eq. \ref{['eq: first bound on duration of reheating']}. Right panel: For $p\geq 10$ we show $\Delta N_\text{rh, max}\equiv \min \left( \Delta N_\text{rh, max 1},\, \Delta N_\text{rh, max 2}\right)$ (dashed, black line), and compare it with $\Delta N_\text{rh, max 1}$ (colored lines).
• Figure 2: Top, left panel: For T-models \ref{['eq: T-model potential']} with even $p\in [2,\,20]$ and $10^{-4}\leq \alpha \leq 20$ we represent $\Delta N_\text{rh, max 1}$, obtained from Eq. \ref{['eq: BBN bound on duration of reheating']}. The color legend is the same as in the bottom panel. Top, right panel: For $p\geq 6$ we represent $\Delta N_\text{rh, max}\equiv \min \left( \Delta N_\text{rh, max 1},\, \Delta N_\text{rh, max 2}\right)$ (dashed, black line), and compare it with $\Delta N_\text{rh, max 1}$ (colored lines). Bottom panel: Minimum reheating temperature for models with $p\geq 6$, obtained by using $\Delta N_\text{rh, max}$. For reference, the temperature at the time of BBN is $T_\text{BBN}=0.1 \, \text{MeV}$.
• Figure 3: Predictions in the $(n_s,\,r)$ plane for T-models with potential \ref{['eq: T-model potential']}. Each panel contains results for a different value of $p$ (see the top-right label). For fixed $p$ we produce predictions for $10^{-4}\leq \alpha\leq 20$ (see the top-right panel for the corresponding color legend). For each $(p,\,\alpha)$ pair, we consider 10 reheating scenarios with $0\leq \Delta N_\text{rh}\leq \min\left(\Delta N_\text{rh, max 1}, \, \Delta N_\text{rh, max 2}\right)$, where we have computed $\Delta N_\text{rh, max 1}$ by requiring that reheating is complete before the universe reaches $1\,\text{TeV}$. We distinguish predictions for different $\Delta N_\text{rh}$ by using a different shading of the $\alpha$-identifying color. Lighter (darker) shading indicates a shorter (longer) reheating stage. The pink contours represent the $68\%\,\text{C.L.}$ and $95\%\,\text{C.L.}$ P-ACT-LB-BK18 posterior in the $(n_s,\,r)$ plane ACT:2025tim.
• Figure 4: Same as in Fig. \ref{['fig:predictions in ns and r plane']}, but here we focus on large $p$ values and allow the temperature at the end of reheating to be as low as $T_\text{BBN}$. The dashed, green line indicates the $(n_s,\,r)$ predictions for $\Delta N_\text{rh}=\Delta N_\text{rh, max 1}$, i.e. computed without taking into account the GW constraint in Eq. \ref{['eq: Omega GW Neff bound']}.
• Figure 5: Values of the scalar spectral tilt, $n_s$, computed using Eq. \ref{['eq: ns analytic']} over the $(\alpha, \,p)$ parameter space. For each model we fix the reheating scenario that maximises $n_s$, see main text for more details. The top (bottom) panel contains results obtained by requiring $T_\text{rh}>1\,\text{TeV}$ ($T_\text{rh}>T_\text{BBN}$). Models which are not compatible with the P-ACT-LB-BK18 (2-dimensional) $95\% \, \text{C.L.}$ constraints on $(n_s,\,r)$ are marked with an empty circle. Models excluded by the $95\% \, \text{C.L.}$ upper limit on $r$ only, see Eq. \ref{['eq:r P-ACT-LB-BK18']}, are colored in cyan. For each $p$ value, a green star indicates the value of $\alpha$ that maximises $n_s$. Note that we choose to represent each model with a point, rather than by using a continuous map, to emphasise that $p$ has a discrete prior.