Quintessence in a quandary: Prior dependence in dark energy models

The archetypal theory of dark energy is quintessence: a minimally coupled scalar field with a canonical kinetic energy and potential. By studying random potentials, we show that quintessence imposes a restricted set of priors on the equation of state of dark energy. Focusing on the commonly used parametrization, $w(a)\approx w_0+w_a(1-a)$, we show that there is a natural scale and direction in the $(w_0,w_a)$ plane that distinguishes quintessence as a general framework. We calculate the expected information gain for a given survey and show that, because of the nontrivial prior information, it is a function of more than just the figure of merit. This allows us to make a quantitative case for novel survey strategies. We show that the scale of the prior sets target observational requirements for gaining significant information. This corresponds to a figure of merit $\mathrm{FOM}\gtrsim 200$, a requirement that future galaxy redshift surveys will meet.

Figure 1

Quintessence priors in the $(w_0,w_a)$ plane (out to 95% C.L.), after loose observational priors have been applied. A remarkably tight structure is observed for all physical models. (Red: EFT; Green: axion; Blue: modulus; Yellow: monomial. All have $\mathrm{\Lambda }=0$.) The grey ellipses are predicted constraints for a Dark Energy Task Force Stage IV galaxy redshift survey with a figure of merit of $\sim 600$. Example fiducial models are shown as crosses.

Figure 2

The (log-scaled) density of prior samples in the $(w_0,w_a)$ plane before (top) and after (middle) the observational cuts, for models with a cosmological constant and ${\dot{\varphi }}_i=0$. Models with $\mathrm{\Lambda }=0$ are shown at the bottom, also after observational cuts. Sharp edges in the cut distributions emerge from the fact that $|w(z)|\le 1$ when ${\mathrm{\Omega }}_{\mathrm{DE}}>0$ drives the current accelerated expansion in quintessence models [29].

Figure 3

Relative entropy as a function of figure of merit for a typical future galaxy survey (solid lines), and for the same but with the error ellipse rotated by 90° (dashed lines). The red and grey lines are for likelihoods fixed at given fiducial values of $(w_0,w_a)$, while the black lines are for $\mathrm{\Delta }S$ marginalized over all fiducial values, $⟨\mathrm{\Delta }S⟩$. The thick blue line shows $\mathrm{\Delta }S$ for a uniform prior over $(w_0,w_a)$-space and does not depend on the fiducial point.