Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory

Title The Limits of Quintessence

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Author Caldwell, R.R.

Publication Date 2009-09-16

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The Limits of Quintessence

R. R. Caldwell Department of Physics & Astronomy, Dartmouth College, Hanover, NH 03755

Eric V. Linder Physics Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720

This work was supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

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The average energy density and pressure scalar field models leads to a less definite upper bound are ρ = φ˙2/2 + V, p = φ˙2/2 V so that a field stuck in w′ . 0.2w(1 + w) since a red shift z 1 but evolving a local, non-zero minimum of− the potential has w = 1. beyond w′ . w(1 + w) by the present.∼ Again, w . 0.8 To distinguish from an effective cosmological constant,− is a practical limit of applicability of these bounds. − however, we will only consider cases in which the field is We have analyzed the following tracker potentials for evolving towards a zero minimum. their freezing behavior: V = M 4+nφ−n and V = 4+n −n 2 2 In perhaps the simplest such scenario, the field has M φ exp(αφ /MP ) for n > 0 [17, 18, 19, 20, 21]. been frozen by Hubble damping at a value displaced from The latter has an effective cosmological constant, but its minimum until recently, when it starts to roll down has been widely studied and so we consider it nonethe- to the minimum. We call these “thawing” models. At less, provided the field is closer to the origin than the early times the equation-of-state ratio is w 1, but non-zero minimum. Proposed tracker models such as ′ ≈ − 4 grows less negative with time as w w/H˙ > 0. Since V = M exp(MP /φ) do not have a zero minimum, ≡ 4 the Hubble damping limits the scalar field acceleration, and V = M (exp(MP /φ) 1) does not achieve w . − φ¨ < φ/t˙ (3/2)Hφ˙, then the equation of motion im- 0.8, Ωde . 0.8. Other functions have been proposed ≈ ′ plies such models will lie at w < 3(1 + w) in the phase −as tracking potentials, but lack a firm basis in particle plane. The scalar field dynamics suggest a lower bound, theory. too, due to the fact that dark energy is not entirely dom- These distinct, physically motivated “thawing” and inant today, with a fractional energy density Ωde . 0.8. “freezing” behaviors are illustrated in Figure 1, while A study of several classes of thawing models, such as a several examples of specific models are presented in Fig- pseudo Nambu-Goldstone boson (PNGB) [6] or polyno- ure 2. It would be quite useful to determine if one of these ′ mial potentials, indicates the bound w > (1 + w). These classes of physics phenomena is responsible for the dark simple bounds are valid for (1+w) 1, and so w . 0.8 energy accelerating our universe. We see that to distin- ≪ − is a practical limit of applicability. guish thawing from freezing, a measurement resolution We have analyzed the following potentials for thaw- of order σ(w′) (1 + w) is required. 4−n n ≈ ing behavior: Concave potentials with V = M φ are The question of the absolute level of deviation of w ubiquitous, and we have allowed for continuous values of from 1, i.e. the distinction from Einstein’s cosmological the exponent n > 0. The motivation for cases n < 2 is constant,− is less tractable. Certainly, one can obtain a not as straightforward, although n = 1 has been consid- scalar field solution, however unrealistic, at any given ered [12]. Exponential potentials are typical for moduli point in the w w′ phase space. Even for the thawing 4 − or dilaton fields, e.g. [13], with V = M exp( βφ/MP ) and freezing models, parameters may be finely tuned to 19 − where MP 10 GeV is the Planck energy. To avoid keep 1 + w arbitrarily close to zero within the shaded ≈ scaling, which would not provide for the cosmic accel- regions of Figure 1. eration, we restrict β < √24 π [14]. PNGBs, like a dark If the scalar field is prohibited from attaining values 4 2 energy axion, have V = M cos (φ/2f) where f is a sym- exceeding the Planck scale, lest quantum gravitational metry restoration energy scale. We have not included the effects dominate, then there is a lower bound on 1 + w. case f MP since the field rapidly evolves to w 0 Defining a characteristic scale E V/(dV/dφ) we de- ≪ → ≡ | | unless the initial conditions are finely tuned to keep the mand E < MP . Next, for a field rolling down its poten- field balanced upon the top of the potential maxima and tial, we can express the scalar field equation of motion maintain 1 + w 1. ≪ as A second scenario consists of a field which was already rolling towards its potential minimum, prior to the onset ′ 2 MP 3 w = 3(1 w ) + (1 w) Ωde(1 + w). of acceleration, but which slows down and creeps to a − − − E r8π halt as it comes to dominate the universe. For these ′ “freezing” models, initially w > 1 and w < 0. These Taking Ωde 0.7 then thawing models must satisfy 1 + ≈ are essentially tracking models [15],− but may be described w & 0.004 whereas for freezing models 1 + w & 0.01; more generally as vacuumless fields (in the sense that the this may be the limit of quintessence. These margins minimum is attained as φ ) or runaway potentials correspond to a 0.2% difference from a Λ cosmology ∼ characterized by a potential→ with ∞ curvature that slows the in distance to redshift z = 1. Such an absolute precision field evolution as it rolls down towards the minimum. It goal is clearly extraordinarily challenging. follows [16] that there is some value of the field beyond Note that early universe inflation can similarly ap- which the evolution is critically damped by the cosmic proach pure exponential expansion, with its deviations expansion, whence the field is frozen (but, like a glacier broadly characterized by dynamics into models tilted to [16], continues to move) and w 1, w′ 0. The prefer large-scale or small-scale power, and important im- deceleration of the field is limited→ by − the steepness→ of plications in the distinction [22]. The structure we find the potential, roughly φ¨ > dV/dφ, leading to the lower in the canonical scalar-field phase-plane based on simple bound w′ > 3w(1 + w). Investigation of a variety of physical considerations may prove useful, and we have 3 here presented firm targets for a basic test of dark en- damental physics distinction between the thawing and ergy. The language is different from inflation for two freezing regions is challenging but achievable in the next reasons: the dark energy need not be rolling as slowly as generation of experiments [23, 24, 25, 26] if the dynamics the inflaton, and the dark energy is not totally dominant, is sufficiently apparent. In the case 1 + w & 0.05, dedi- unlike the inflaton. cated dark energy experiments now being designed, such Charting the late-time cosmic evolution – through as the Joint Dark Energy Mission, will probe cosmology Type Ia supernovae distances, weak gravitational lensing with sufficient accuracy to be able to decide the issue. probes of large-scale structure evolution, distance ratios This will probably not be the final word on dark energy from baryon acoustic oscillations in galaxy clustering, etc (cf. [27]), but if the answer is not consistent with a cosmo- – is the subject of intense investigations. As a gauge of logical constant then the rewards are obvious in discov- the requisite resolution, a 1% variation in luminosity dis- ering new physics beyond our current standard models. tance to redshift z = 1 distinguishes between: Λ and It is interesting to note that the fate of the Universe is ′ w = 0.95, w = 0; models which evolve along the top very different for the case of a thawing field, as the accel- − and bottom of the thawing region out to w = 0.8; mod- eration is temporary, as compared to a freezing field, for − els which evolve along the top and bottom of the freezing which the acceleration continues unabated. If the result region in to w = 0.95. The goal of making the fun- lies outside the two phase-space regions categorized here − then we may have to look beyond simple explanations, perhaps to even more exotic physics such as a modifica- tion of Einstein gravity. This work is supported by NSF AST-0349213 at Dart- mouth, and by DOE AC03-76SF00098 at LBL. EL thanks Dartmouth for its hospitality.

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