Underdetermination of dark energy

There is compelling evidence that the Universe is undergoing a late phase of accelerated expansion. One of the simplest explanations for this behavior is the presence of dark energy. A plethora of microphysical models for dark energy have been proposed. The hope is that, with the ever increasing precision of cosmological surveys, it will be possible to precisely pin down the model. We show that this is unlikely and that, at best, we will have a phenomenological description for the microphysics of dark energy. Furthermore, we argue that the current phenomenological prescriptions are ill-equipped for shedding light on the fundamental theory of dark energy.

Figure 1

Current constraints on the $(w_0,w_a)$ parameter space. Broadly speaking, this parameter space can be divided up into a few regions. “Freezing” quintessence corresponds to the region of the parameter space where $w_a>0$ and refers to models where the dark energy equation of state is evolving asymptotically towards $w_{\mathrm{DE}}\simeq -1$, while “thawing” quintessence corresponds to the region where $w_a<0$ and refers to models where the dark energy equation of state is evolving away from $w_{\mathrm{DE}}\simeq -1$ to less negative values. The region given by $w<-1$ is known as the phantom region. It requires more exotic physics to describe and will not be a focus of this study. As we can see, most of the viable parameter space is locating in the thawing region.

Figure 2

$w(a)$ evolution for different choices of model parameters for the slow-roll/positive quadratic model and the hilltop/negative quadratic model. The parameter $m{t}_0$ was varied and $X$ was chosen so that ${\tilde{w}}_0\simeq .90$. For the slow-roll models, all converge on the same evolutionary track and will have the same slope in the $({\tilde{w}}_0,{\tilde{w}}_a)$ plane. All the hilltop models have very different evolutions and the trajectories become steeper as $m{t}_0$ is increased. Consequently, they will be represented differently in the $({\tilde{w}}_0,{\tilde{w}}_a)$ plane.

Figure 3

${\tilde{w}}_0\equiv w(a=1)$ and ${\tilde{w}}_a\equiv -dw/da(a=1)$ assuming matter domination for both the slow-roll/positive quadratic model and the hilltop/negative quadratic model. The slow-roll model was randomly sampled for parameters $m{t}_0\in [.01,1]$ and $X\in [.1,2]$ and the hilltop model was randomly sampled for parameters $m{t}_0\in [.01,6]$ and $\epsilon \in [.0002,2]$. The hilltop model can arbitrarily sweep across huge swaths of the broader thawing (${\tilde{w}}_a<0$) region when compared to more standard slow-roll models that only live along a narrow strip of this region given by ${\tilde{w}}_a\sim -3({\tilde{w}}_0+1)$.

Figure 4

$w(a)$ numerical integration for both the slow-roll/positive quadratic model and the hilltop/negative quadratic model in a mixed dark matter/dark energy universe compared to their respective matter dominated analytic solution for the same choice of parameters. In both cases, we see that the numerical solution does not evolve quite as much. This is because the Hubble friction term is enhanced when dark energy becomes a significant part of the scalar field equation of motion when compared with the matter dominated solution.

Figure 5

This is the result for ${\tilde{w}}_0\equiv w(a=1)$ and ${\tilde{w}}_a\equiv -dw/da(a=1)$ determined after numerically integrating Eq. (30) for a universe with ${\mathrm{\Omega }}_{\mathrm{DE}}\simeq .7$ for both the slow-roll/positive quadratic model and the hilltop/negative quadratic model. Both models are qualitatively similar in their behavior, with the slow-roll model now living on ${\tilde{w}}_a\sim -1.5({\tilde{w}}_0+1)$ while the hilltop model takes values roughly between $-1.5({\tilde{w}}_0+1)\lesssim {\tilde{w}}_a\lesssim -10({\tilde{w}}_0+1)$ and sweeps across the parameter space.

Figure 6

The light blue shaded region depicts the result for $w_0$ and $w_a$ determined by numerically integrating $w(a)$ for the hilltop model in a universe with ${\mathrm{\Omega }}_{\mathrm{DE}}\simeq .7$ and finding the best fit for Eqs. (31) and (32) over $z\in [.15,1.85]$. This is overplotted against the ${\tilde{w}}_0\equiv w(a=1)$, ${\tilde{w}}_a\equiv -dw/da(a=1)$ result (dark gray shaded region) for the same hilltop model and choice of parameters depicted in Fig. 5. This indicates that there is an ambiguity in how these hilltop models are represented in the $(w_0,w_a)$ plane as the exact size of the swept region (for the same choice of parameters) will sensitively depend on the range of redshifts that one fits over due to the highly nonlinear evolution of $w(a)$ in these hilltop models. Fitting over a more restricted range of more recent redshifts would cause the fitted results to more closely resemble the $a=1$ results. By contrast, the slow-roll models (dark blue line) still lie in the same narrow strip as their location is insensitive to the choice of fitting procedure as their evolution is highly linear.

Figure 7

Comparison between the $w_a$ and $w_0$ values depicted in the $(w_0,w_a)$ plane of Fig. 6. This shows how the values determined by fitting $H(z)$ through Eqs. (31) and (32) and those determined by ${\tilde{w}}_0$ and ${\tilde{w}}_a$ map onto each other. In other words, this fitting procedure squeezes the representation of the hilltop model in the $(w_0,w_a)$ plane. The exact degree to which the swept region is squeezed depends on the range of redshifts included in the fit.

Figure 8

Evolution of $w(a)$ compared with two different fits for the $w(a)=w_0+w_a(1-a)$ parametrization. The yellow line was determined by fitting $H(z)$ through Eqs. (31) and (32) for $z\in [.15,1.85]$, while the green line was determined by fitting from $z\in [0.15,0.25]$. We can see that fitting over the more recent range of redshifts better captures the $w(a)$ evolution for the steeper hilltop models considered here. Consequently, we can understand why the swept area of Fig. 6 is squeezed when fit over a large range of redshifts: the fit for $w_0$ and $w_a$ in that case does not fully capture the steeper part of the $w(a)$ trajectory that occurs at the more recent redshifts and consequently maps these model into a less steep part of the $(w_0,w_a)$ plane. The smaller and more recent the range of redshifts included in the fit, the closer that the fitted $(w_0,w_a)$ values will be to the $({\tilde{w}}_0,{\tilde{w}}_a)$ values determined at $a=1$.