Inflation model building with an accurate measure of $e$-folding

It has become standard practice to take the logarithmic growth of the scale factor as a measure of the amount of inflation, despite the well-known fact that this is only an approximation for the true amount of inflation required to solve the horizon and flatness problems. The aim of this work is to show how this approximation can be completely avoided using an alternative framework for inflation model building. We show that using the inverse Hubble radius, $\mathcal{H}=aH$, as the key dynamical parameter, the correct number of $e$-folding arises naturally as a measure of inflation. As an application, we present an interesting model in which the entire inflationary dynamics can be solved analytically and exactly, and, in special cases, reduces to the familiar class of power-law models.

Figure 1

The $n_S\text{−}r$ plane where $H(\varphi )$ is a randomly generated polynomial of varying order (4, 7 and 10 from left to right). The dark/purple swathes are generated assuming the approximation $N=60$, whereas the light/orange swathes assume the correct definition of $e$-folding: $\tilde{N}=60$.

Figure 2

The ratio between the $(n_S,r)$ values of the two swathes in each panel in Fig. 1, i.e. $r(\tilde{N}=60)/r(N=60)$ plotted against $n_S(\tilde{N}=60)/n_S(N=60)$.

Figure 3

The dynamical evolution of two inflation models (left and right panels). For each mode, we plot the evolution of $\mathrm{\Delta }\varphi$ (field excursion), the Hubble parameter $H$ and the Hubble “slow-roll” parameter, $\epsilon$, as a function of the total number of $e$-folds measured from horizon exit (60 marks the end of inflation). Solid/blue lines use $\tilde{N}$ as the measure of $e$-folds, while dashed/black lines use the approximation $N$. Although the dynamics of the model on the right is essentially unchanged whether $N$ or $\tilde{N}$ is used, the model on the left is inaccurately measured by $N$. In this particular case, $r(\tilde{N}=60)$ is 12% smaller than $r(N=60)$.

Figure 4

The $n_S\text{−}r$ prediction for the Gaussian model $\mathcal{H}\propto \exp (-{\alpha }^2{\varphi }^2)$ shown as the sharp diagonal, with parameter ${\log }_{10}\alpha \in [-0.4,5]$ (from top to bottom of the diagonal), assuming CMB-scale perturbations were generated at physical $e$-folds ${\tilde{N}}_{*}\in [50,70]$ (from left to right). We also show the $2\sigma$ constraints in this plane from Planck [15], using both the temperature and low-$\ell$ polarization data. In fact, the diagonal coincides with the prediction for power-law potentials $V\sim {\varphi }^n$, as discussed in Sec. 3e.

Figure 5

The discrepancy between the physical and slow-roll $e$-folds for the Gaussian model plotted as a function of the parameter $\alpha$, assuming $\tilde{N}=50–70$.