Dark Energy as Double N-Flation--Observational Predictions

Abstract

We propose a simple model for dark energy useful for comparison with observations. It is based on the idea that dark energy and inflation should be caused by the same physical process. As motivation, we note that Linde's simple chaotic inflation $V=(1/2)m^{2}\phi^{2}$ produces values of $n_{s}=0.967$ and $r=0.13$, which are consistent with the WMAP 1-sigma error bars. We therefore propose $V=(1/2)m_{2}^{2}\phi_{2}^{2}+(1/2)m_{1}^{2}\phi_{1}^{2}$ with $m_{1}\sim10^{-5}$ and $m_{2}\leq10^{-60}$, where $c=1=\hbar$ and the reduced Planck mass is set to unity. The field $\phi_{1}$ drives inflation and has damped by now ($\phi_{1,0}=0$), while $\phi_{2}$ is currently rolling down its potential to produce dark energy. Using this model, we derive the formula $\delta w(z)\equiv w(z)+1=\delta w_{0}(H_0/H(z))^2$ via the slow-roll approximation. Our numerical results from exact and self-consistent solution of the equations of motion for $\phi_2$ and the Friedmann equations support this formula, and it should hold for any slow-roll dark energy. Our potential can be easily realized in N-flation models with many fields, and is easily falsifiable by upcoming experiments -- for example, if Linde's chaotic inflation is ruled out. But if $r$ values consistent with Linde's chaotic inflation are detected then one should take this model seriously indeed.