Paths of quintessence

Eric V. Linder

Phys. Rev. D 73, 063010 – Published 22 March 2006

Abstract

The structure of the dark energy equation of state phase plane holds important information on the nature of the physics. We explain the bounds of the freezing and thawing models of scalar field dark energy in terms of the tension between the steepness of the potential vs the Hubble drag. Additionally, we extend the phase plane structure to modified gravity theories, examine trajectories of models with certain properties, and categorize regions in terms of scalar field hierarchical parameters, showing that dark energy is generically not a slow roll phenomenon.

Received 17 January 2006

DOI:https://doi.org/10.1103/PhysRevD.73.063010

Authors & Affiliations

Eric V. Linder

Berkeley Lab, University of California, Berkeley, California 94720, USA

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Issue

Vol. 73, Iss. 6 — 15 March 2006

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Images

Figure 1
Contours of the inverse of the characteristic field scale $Φ = - V / V_{, ϕ}$ are plotted in full and zoomed versions, for present matter density $Ω_{m} = 0.28$ . Note that only a slender crescent of the phase plane obeys the conventional slow roll criterion $M_{P} / Φ ≪ 1$ . The dashed red curve shows the null line $V_{, ϕ} = 0$ . Dotted lines show the $w \approx - 1$ limits of the thawing region (upper pair: $w^{'} = 3 (1 + w)$ , ( $1 + w$ )) and freezing region (lower pair: here $w^{'} = w (1 + w)$ , $3 w (1 + w)$ ; not shown in bottom panel). If physics limits the characteristic field scale to be less than the Planck mass, then a barrier forms (second panel) around the cosmological constant, allowing only models in the shaded part of the thawing region.Reuse & Permissions
Figure 2
The physics behind the dynamics becomes more apparent when looking at the ratios of the terms in the Klein-Gordon equation of motion, e.g. field acceleration vs friction vs potential driving. The key general features of the phase plane are the null line $V_{, ϕ} = 0$ , the coasting line $\ddot{ϕ} = 0$ , and the phantom line $w = - 1$ . Thawing (with $w^{'} > 0$ ) and freezing (with $w^{'} < 0$ ) regions are bounded by green, dot-dash curve pairs and are defined through the physical dynamics.Reuse & Permissions
Figure 3
The braneworld model of modified gravity behaves like a freezing scalar field. The solid black line shows its trajectory: starting at high redshift at $w = - 1 / 2$ , $w^{'} = 0$ , with the red crosses indicating $z = 3$ , 2, 1 before the line ends at the present. The dashed blue line shows its future behavior, approaching a cosmological constant, and lying within the quintessence freezing bounds of $w^{'} = 3 w (1 + w)$ and $w^{'} = w (1 + w)$ given by the dotted green curves. A model with a generalized, extra $H^{α}$ term (for the braneworld, $α = 1$ ) in the Friedmann expansion equation looks similar; the magenta, dot-dash curve is for $α = 0.5$ .Reuse & Permissions
Figure 4
Polytropic models, with $β = 1 / α$ in the first panel and with $β = 1 / 2$ in the second panel. Curves are labeled with $α$ and crosses show the present. Dotted curves show the canonical scalar field freezing region (first panel) or thawing region (second panel). In the first panel, models evolve toward a cosmological constant, within the freezing region. In the second panel, the asymptotic state is not deSitter (except for the $α = 2$ curve, which is the same in both panels) and the dynamics is distinct from a canonical scalar field.Reuse & Permissions
Figure 5
Polytropic models as Fig. 4a, with $β = 1 / α$ , but now viewed in the total equation of state phase plane. Curves are labeled with $α$ and crosses show the present. Dotted curves show the dark energy freezing region bounds (light green) and the null line (dark magenta). Note that those models phantom in the dark energy here appear with $w_{tot} > - 1$ but lie below $w_{tot}^{'} = 3 w_{tot} (1 + w_{tot})$ . One should not apply the dark energy dynamics constraints to the total equation of state plane.Reuse & Permissions

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