Constraints on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math>-attractor inflation and reheating

We investigate a constraint on reheating followed by $\ensuremath{\alpha}$-attractor-type inflation (the E-model and T-model) from an observation of the spectral index ${n}_{s}$. When the energy density of the Universe is dominated by an energy component with the cosmic equation-of-state parameter ${w}_{\mathrm{re}}$ during reheating, its $e$-folding number ${N}_{\mathrm{re}}$ and the reheating temperature ${T}_{\mathrm{re}}$ are bounded depending on ${w}_{\mathrm{re}}$. When the reheating epoch consists of two phases---where the energy density of the Universe is dominated by uniform inflaton field oscillations in the first phase and by relativistic nonthermalized particles in the second phase---we find a constraint on the $e$-folding number of the first oscillation phase, ${N}_{\mathrm{sc}}$, depending on the parameters of the inflaton potential. For the simplest perturbative reheating scenario, we find the lower bound for a coupling constant of inflaton decay in the E-model and T-model depending on the model parameters. We also find a constraint on the $\ensuremath{\alpha}$ parameter, $\ensuremath{\alpha}\ensuremath{\gtrsim}0.01$, for the T-model and E-model when we assume a broad resonance reheating scenario.

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