Dark energy view of inflation

Traditionally, inflationary models are analyzed in terms of parameters such as the scalar spectral index $n_s$ and the tensor to scalar ratio $r$, while dark energy models are studied in terms of the equation of state parameter $w$. Motivated by the fact that both deal with periods of accelerated expansion, we study the evolution of $w$ during inflation, in order to derive observational constraints on its value during an earlier epoch likely dominated by a dynamic form of dark energy. We find that the cosmic microwave background and large-scale structure data is consistent with $w_{\mathrm{inflation}}=-1$ and provides an upper limit of $1+w\lesssim 0.02$. Nonetheless, an exact de Sitter expansion with a constant $w=-1$ is disfavored since this would result in $n_s=1$.

Figure 1

The evolution of $w(\varphi )$ for a range of cases accepted by the CMB likelihood, around the field value ${\varphi }_{*}$ at the pivot scale $k_{*}=0.01/\mathrm{Mpc}$. The red and yellow curves lie within the 95% and 68% confidence regions for the LV formalism; blue and purple give the same information for the flow-equation formalism. From the outside inward, the colored regions are red, yellow, blue, and purple.

Figure 2

The complete evolution of $w(N)$, from the flow-equation results accepted by the CMB likelihood. Inflation is made to end at $N=0$ where $w(N=0)=-1/3$ corresponding to ${ϵ}_H(N=0)=1$. For our choice of priors on the slow-roll parameters at $N=0$, we find that $w$ decreases rapidly towards $-1$ (see inset) and stays close to it during the period when the observable scales leave the horizon ($N\approx 40–60$).

Figure 3

The evolution of $w$ as a function of the comoving scale $k$, using only the 5-yr WMAP CMB data. Red and yellow are the 95% and 68% confidence regions for the LV formalism. Blue and purple are the same for the flow-equation formalism. From the outside inward, the colored regions are red, yellow, blue, and purple.

Figure 4

The evolution of $w$ as a function of the comoving scale $k$, as in Fig. 3, but using in addition the Sloan Digital Sky Survey Data Release 7 Luminous Red Galaxy data. We only show the LV limits.