Equation of State and Duration to Radiation Domination after Inflation

We calculate the equation of state after inflation and provide an upper bound on the duration before radiation domination by taking the nonlinear dynamics of the fragmented inflaton field into account. A broad class of single-field inflationary models with observationally consistent flattening of the potential at a scale $M$ away from the origin, $V(\varphi )\propto |\varphi {|}^{2n}$ near the origin, and where the couplings to other fields are ignored, is included in our analysis. We find that the equation of state parameter $w\to 0$ for $n=1$ and $w\to 1/3$ (after sufficient time) for $n\gtrsim 1$. We calculate how the number of $e$-folds to radiation domination depends on both $n$ and $M$ when $M\sim m_{\mathrm{Pl}}$, whereas when $M\ll m_{\mathrm{Pl}}$, we find that the duration to radiation domination is negligible. Our results are explained in terms of a linear instability analysis in an expanding universe and scaling arguments, and are supported by $3+1$-dimensional lattice simulations. We show that our upper bound on the postinflationary duration before radiation domination reduces the uncertainty in inflationary observables even when couplings to additional light fields are included (at least under the assumption of perturbative decay).

Figure 1

The shape of the inflaton potential studied in this work.

Figure 2

The instability bands and the magnitude of the Floquet exponent [in units of the field dependent effective mass $m(\overline{\varphi })$] are shown as functions of the oscillating condensate amplitude and the dimensionless physical wave number $\kappa =k/am$. The white lines indicate how a given comoving wave number passes through the instability bands as the Universe expands.

Figure 3

The equation of state parameter obtained from the numerical simulations is shown for different values of $n$ and $M$. The orange curve and green curves correspond to initially efficient ($M\approx 7.75\times {10}^{-3}{m}_{\mathrm{Pl}}$) and inefficient resonance ($M\approx 2.45m_{\mathrm{Pl}}$), with $M\sim 2.5\times {10}^{-2}{m}_{\mathrm{Pl}}$ separating the two regimes. The horizontal axes show the number of $e$-folds after the end of inflation for efficient (orange, bottom axis) and inefficient (green, top axis) resonance. The dashed line is drawn at $w=1/3$ and the dotted line denotes the homogeneous equation of state.

Figure 4

A summary for the asymptotic equation of state without coupling to additional fields. The numerical results from lattice simulations are shown as green circles for $M\approx 2.45m_{\mathrm{Pl}}$, and orange squares for $M\approx 7.75\times {10}^{-3}{m}_{\mathrm{Pl}}$. The dotted blue line is the expectation from a homogeneous, oscillating condensate.

Figure 5

Based on our results, the bounds on $\mathrm{\Delta }{N}_{\mathrm{rad}}$ are translated to predictions for $r$ and $n_s$ (filled in colored bands; the black edge is for the upper bound). The narrow width of the filled-in bands corresponds to a change in $\mathrm{\Delta }{N}_{\mathrm{rad}}$ from coupling to other light fields. For comparison, the range of $N_{\star }=50–60$ commonly used to account for reheating related uncertainties is also shown (thin colored lines, and the “dumbbells”); the reduction in uncertainty due to our results is significant. Note that $M\gtrsim m_{\mathrm{Pl}}$ for most of the above plot. For $M\ll m_{\mathrm{Pl}}$ we have $\mathrm{\Delta }{N}_{\mathrm{rad}}\lesssim 1$, and $r\ll {10}^{-3}$ (hence, these are difficult to see in the above observational constraints [1]). For the above plot we have focused on the $\alpha$-attractor models [17, 18, 19], but it can be easily generalized to other models.