Closing in on $\alpha$-attractors

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 $\alpha$-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 $\alpha$-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 $\alpha$ that is stronger than the one from the tensor-to-scalar ratio only. We find that $n_s$ is maximised for $\alpha\sim1$, therefore the seven Poincar\'e 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.