Weakly model-independent determination of total expansion during inflation

Abstract

We study systematically the total expansion experienced by a certain perturbation mode during single-field inflation, not resorting to explicit models of inflation or reheating. By assuming that during the reheating stage the equation of state w{rh} can be written as a function of e-folds, the unknown dynamics during reheating parametrized by w{rh} is confined within a time integral so that any dependence on the models of inflation and reheating is isolated from model-independent contributions. Especially, the dependence on the reheating dynamics via w{rh} and the reheating temperature T{rh} is dominating. We give two illustrative examples of w{rh} to discuss its impacts on the total expansion, which can be different as much as 10 even for the same reheating temperature, depending on the shape of w{rh}. We also discuss the profile degeneracy of w{rh}, and argue when the degeneracy is lifted.

Figures (3)

• Figure 1: Plots of $w_\text{rh}$ as a function of $n\equiv N/N_\text{rh}$ given respectively by (left) \ref{['eq:w-rh1']}, (middle) \ref{['eq:w-rh2']} and (right) \ref{['eq:w-rh3']}. Here, (solid) $\alpha=0.1$, (dot-dashed) 0.5, (dense dot) 1, (sparse dot) 2 and (dashed) 5 respectively.
• Figure 2: Plots of $N_k$ for (upper panels) \ref{['eq:w-rh1']} and (lower panels) \ref{['eq:w-rh3']} as a function of (horizontal axis) $T_\text{rh}/m_{\rm Pl}$ and (vertical axis) $\alpha$. In both upper and lower panels, the reference scale is set to be (upper rows) $k=0.05$ Mpc$^{-1}$ and (lower rows) $k=0.002$ Mpc$^{-1}$. The value of $\beta$ is set to be (left panels) $\beta = 1.5$, (middle panels) $\beta=10$ and (right panels) $\beta=100$. $T_\text{rh}/m_{\rm Pl}$ is presented up to the value that gives $N_\text{rh}=0$.
• Figure 3: Posterior distribution of $\alpha$ obtained from the MCMC analysis for \ref{['eq:mcmc-toy']}. We have set $N_k^\text{fid} = 55$, $A = 3.0$ and $\sigma = 0.2$. We have taken 50,000 samples with $-1 < \alpha < 1$.