Self-resonance after inflation: oscillons, transients and radiation domination

Homogeneous oscillations of the inflaton after inflation can be unstable to small spatial perturbations even without coupling to other fields. We show that for inflaton potentials $\propto |\phi|^{2n}$ near $|\phi|=0$ and flatter beyond some $|\phi|=M$, the inflaton condensate oscillations can lead to self-resonance, followed by its complete fragmentation. We find that for non-quadratic minima ($n>1$), shortly after backreaction, the equation of state parameter, $w\rightarrow1/3$. If $M\ll m_{pl}$, radiation domination is established within less than an e-fold of expansion after the end of inflation. In this case self-resonance is efficient and the condensate fragments into transient, localised spherical objects which are unstable and decay, leaving behind them a virialized field with mean kinetic and gradient energies much greater than the potential energy. This end-state yields $w=1/3$. When $M\ll m_{pl}$ we observe slow and steady, self-resonace that can last many {\it e}-folds before backreaction eventually shuts it off, followed by fragmentation and $w\rightarrow 1/3$. We provide analytical estimates for the duration to $w\rightarrow 1/3$ after inflation, which can be used as an upper bound (under certain assumptions) on the duration of the transition between the inflationary and the radiation dominated states of expansion. This upper bound can reduce uncertainties in CMB observables such as the spectral tilt $n_{\rm{s}}$, and the tensor-to-scalar ratio $r$. For quadratic minima ($n=1$), $w\rightarrow0$ regardless of the value of $M$.