${\varphi }^2$ inflation at its endpoint

In the simplest inflationary model $V=\frac{1}{2}{m}^2{\varphi }^2$, we provide a prediction accurate up to 1% for the spectral index $n_s$ and the tensor-to-scalar ratio $r$ assuming instantaneous reheating and a standard thermal history: $n_s=0.9668\pm 0.0003$ and $r=0.131\pm 0.001$. This represents the simplest and most informative point in the $(n_s,r)$ plane. The result is independent of the details of reheating (or preheating) provided the conversion to radiation is sufficiently fast. A slower reheating or a modified postinflationary evolution (with an equation of state parameter $w\le 1/3$) pushes towards smaller $n_s$ (and larger $r$), so that our prediction corresponds to the maximum $n_s$ (and minimum $r$) for the quadratic potential. We also derive similar results for a general $V\propto {\varphi }^p$ potential.

Figure 1

Energy density as a function of $\log (a)$. Green dot-dashed line: the standard analytical result. Red solid line: analytical result using Eq. (6). The grey region corresponds to numerical solutions between $\mathrm{\Gamma }=m/10$ and $\mathrm{\Gamma }=m/1000$. The improved analytical formula is within 0.3 $e$-folds from the numerical solutions for this range of $\mathrm{\Gamma }$.

Figure 2

Dependence of the expansion of the Universe $\log (a)$ on $\mathrm{\Gamma }$ for fixed initial conditions and fixed final energy density (in radiation dominance). Different curves are obtained for different choices of time at which the inflaton starts to decay.

Figure 3

Evolution of the equation of state (averaged over a period of oscillation) as a function of $\log (a)$ for different values of $\mathrm{\Gamma }$.