We present evidence that the simplest particle-physics scalar-field models of
dynamical dark energy can be separated into distinct behaviors based on the
acceleration or deceleration of the field as it evolves down its potential
towards a zero minimum. We show that these models occupy narrow regions in the
phase-plane of w and w', the dark energy equation-of-state and its
time-derivative in units of the Hubble time. Restricting an energy scale of the
dark energy microphysics limits how closely a scalar field can resemble a
cosmological constant. These results, indicating a desired measurement
resolution of order \sigma(w')\approx (1+w), define firm targets for
observational tests of the physics of dark energy.Comment: 4 pages, 2 figure

A. D. Linde

Eric V. Linder

P. J. Steinhardt

R. R. Caldwell

R. R. Caldwell

English

arXiv

We present evidence that the simplest particle-physics scalar-field models of dynamical dark energy can be separated into distinct behaviors based on the acceleration or deceleration of the field as it evolves down its potential towards a zero minimum. We show that these models occupy narrow regions in the phase-plane of w and w', the dark energy equation-of-state and its time-derivative in units of the Hubble time. Restricting an energy scale of the dark energy microphysics limits how closely a scalar field can resemble a cosmological constant. These results, indicating a desired measurement resolution of order \sigma(w')\approx (1+w), define firm targets for observational tests of the physics of dark energy

Caldwell, R.R.

Linder, Eric V.

UNT (University of North Texas) Digital Library

    LBNL- 58259    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.  DISCLAIMER  This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California. arXiv:astro-ph/0505494v1  24 May 2005The Limits of QuintessenceR. R. CaldwellDepartment of Physics & Astronomy, Dartmouth College, Hanover, NH 03755Eric V. LinderPhysics Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720(Dated: February 2, 2008)We present evidence that the simplest particle-physics scalar-field models of dynamical dark en-ergy can be separated into distinct behaviors based on the acceleration or deceleration of the field asit evolves down its potential towards a zero minimum. We show that these models occupy narrowregions in the phase-plane of w and w′, the dark energy equation-of-state and its time-derivativein units of the Hubble time. Restricting an energy scale of the dark energy microphysics limitshow closely a scalar field can resemble a cosmological constant. These results, indicating a desiredmeasurement resolution of order σ(w′) ≈ (1 + w), define firm targets for observational tests of thephysics of dark energy.Observations and experiments at the close of the20th century have transformed our understanding of thephysics of the Universe. A consistent picture has emergedindicating that nearly three-quarters of the cosmos ismade of “dark energy” — some sort of gravitationallyrepulsive material responsible for the accelerated expan-sion of the Universe (for reviews see [1, 2, 3]). Proposalsfor the dark energy include Einstein’s cosmological con-stant (Λ), or a dynamical field such as quintessence. Herewe show how scalar field dynamics separates into distinctbehaviors which, through future cosmological measure-ments, can reveal the nature of the new physics acceler-ating our universe.Einstein’s cosmological constant (Λ) is attributed tothe quantum zero-point energy of the particle physicsvacuum, with a constant energy density ρ, pressure pand an equation-of-state w ≡ p/ρ = −1. In contrast,quintessence is a proposed time-varying, inhomogeneousfield with a spatially-averaged equation-of-state w > −1[4, 5, 6, 7, 8]. The simplest physical model consists ofa scalar field, slowly rolling in a potential characterizedby an extremely low mass. (This is similar to inflation,the period of accelerated expansion in the early universe,but at an energy scale many orders of magnitude lower.)Since a scalar field evolving in a very shallow potentialmay be indistinguishable from a Λ, the task of elucidat-ing the physics of dark energy becomes difficult if ob-servations continue to find that w is close to −1, e.g.[9, 10, 11]. In this letter, we examine the likely behav-ior of scalar fields and characterize them into two distinctclasses, based on their evolution in the w−w′ phase space.These results should help define targets for observationaland experimental tests of the physics of dark energy.Our approach is a new take on a familiar system, thescalar field. By emphasizing the dynamics, we discoverrestricted regions of the trajectories of canonical scalarfield models in “position” and “velocity” — the valueof the equation-of-state ratio w and its time variationw′. While there is a myriad of scalar field models mo-tivated by particle physics beyond the standard model,this treatment allows a broad, model-independent assess-ment of a quintessence scalar field slowly relaxing in apotential.The physics is straightforward: the field φ will seek toroll towards the minimum of its potential V , accordingto the Klein-Gordon equation φ̈ + 3Hφ̇ = −dV/dφ. Therate of evolution is driven by the slope of the potentialand damped by the cosmic expansion through the HubbleFIG. 1: The w − w′ phase space occupied by thawing andfreezing fields is indicated by the shaded regions. No strongconstraints on this range of dark energy properties exist atpresent. The fading at the top of the freezing region indicatesthe approximate nature of this boundary. Freezing modelsstart above this line, but pass below it by a red shift z ∼ 1.The short-dashed line shows the boundary between field evo-lution accelerating and decelerating down the potential. Fu-ture cosmological observations will aim to discriminate be-tween these two fundamental scenarios.2parameter H . The average energy density and pressureare ρ = φ̇2/2 + V, p = φ̇2/2 − V so that a field stuck ina local, non-zero minimum of the potential has w = −1.To distinguish from an effective cosmological constant,however, we will only consider cases in which the field isevolving towards a zero minimum.In perhaps the simplest such scenario, the field hasbeen frozen by Hubble damping at a value displaced fromits minimum until recently, when it starts to roll downto the minimum. We call these “thawing” models. Atearly times the equation-of-state ratio is w ≈ −1, butgrows less negative with time as w′ ≡ ẇ/H > 0. Sincethe Hubble damping limits the scalar field acceleration,φ̈ < φ̇/t ≈ (3/2)Hφ̇, then the equation of motion im-plies such models will lie at w′ < 3(1 + w) in the phaseplane. The scalar field dynamics suggest a lower bound,too, due to the fact that dark energy is not entirely dom-inant today, with a fractional energy density Ωde . 0.8.A study of several classes of thawing models, such as apseudo Nambu-Goldstone boson (PNGB) [6] or polyno-mial potentials, indicates the bound w′ > (1+w). Thesesimple bounds are valid for (1+w) ≪ 1, and so w . −0.8is a practical limit of applicability.We have analyzed the following potentials for thaw-ing behavior: Concave potentials with V = M4−nφn areubiquitous, and we have allowed for continuous values ofthe exponent n > 0. The motivation for cases n < 2 isnot as straightforward, although n = 1 has been consid-ered [12]. Exponential potentials are typical for modulior dilaton fields, e.g. [13], with V = M4 exp(−βφ/MP )where MP ≈ 1019 GeV is the Planck energy. To avoidscaling, which would not provide for the cosmic accel-eration, we restrict β <√24π [14]. PNGBs, like a darkenergy axion, have V = M4 cos2(φ/2f) where f is a sym-metry restoration energy scale. We have not included thecase f ≪ MP since the field rapidly evolves to w → 0unless the initial conditions are finely tuned to keep thefield balanced upon the top of the potential maxima andmaintain 1 + w ≪ 1.A second scenario consists of a field which was alreadyrolling towards its potential minimum, prior to the onsetof acceleration, but which slows down and creeps to ahalt as it comes to dominate the universe. For these“freezing” models, initially w > −1 and w′ < 0. Theseare essentially tracking models [15], but may be describedmore generally as vacuumless fields (in the sense that theminimum is attained as φ → ∞) or runaway potentialscharacterized by a potential with curvature that slows thefield evolution as it rolls down towards the minimum. Itfollows [16] that there is some value of the field beyondwhich the evolution is critically damped by the cosmicexpansion, whence the field is frozen (but, like a glacier[16], continues to move) and w → −1, w′ → 0. Thedeceleration of the field is limited by the steepness ofthe potential, roughly φ̈ > dV/dφ, leading to the lowerbound w′ > 3w(1 + w). Investigation of a variety ofscalar field models leads to a less definite upper boundw′ . 0.2w(1 + w) since a red shift z ∼ 1 but evolvingbeyond w′ . w(1 + w) by the present. Again, w . −0.8is a practical limit of applicability of these bounds.We have analyzed the following tracker potentials fortheir freezing behavior: V = M4+nφ−n and V =M4+nφ−n exp(αφ2/M2P) for n > 0 [17, 18, 19, 20, 21].The latter has an effective cosmological constant, buthas been widely studied and so we consider it nonethe-less, provided the field is closer to the origin than thenon-zero minimum. Proposed tracker models such asV = M4 exp(MP /φ) do not have a zero minimum,and V = M4(exp(MP /φ) − 1) does not achieve w .−0.8, Ωde . 0.8. Other functions have been proposedas tracking potentials, but lack a firm basis in particletheory.These distinct, physically motivated “thawing” and“freezing” behaviors are illustrated in Figure 1, whileseveral examples of specific models are presented in Fig-ure 2. It would be quite useful to determine if one of theseclasses of physics phenomena is responsible for the darkenergy accelerating our universe. We see that to distin-guish thawing from freezing, a measurement resolutionof order σ(w′) ≈ (1 + w) is required.The question of the absolute level of deviation of wfrom −1, i.e. the distinction from Einstein’s cosmologicalconstant, is less tractable. Certainly, one can obtain ascalar field solution, however unrealistic, at any givenpoint in the w − w′ phase space. Even for the thawingand freezing models, parameters may be finely tuned tokeep 1 + w arbitrarily close to zero within the shadedregions of Figure 1.If the scalar field is prohibited from attaining valuesexceeding the Planck scale, lest quantum gravitationaleffects dominate, then there is a lower bound on 1 + w.Defining a characteristic scale E ≡ |V/(dV/dφ)| we de-mand E < MP . Next, for a field rolling down its poten-tial, we can express the scalar field equation of motionasw′ = −3(1 − w2) + (1 − w)MPE√38πΩde(1 + w).Taking Ωde ≈ 0.7 then thawing models must satisfy 1 +w & 0.004 whereas for freezing models 1 + w & 0.01;this may be the limit of quintessence. These marginscorrespond to a ∼ 0.2% difference from a Λ cosmologyin distance to redshift z = 1. Such an absolute precisiongoal is clearly extraordinarily challenging.Note that early universe inflation can similarly ap-proach pure exponential expansion, with its deviationsbroadly characterized by dynamics into models tilted toprefer large-scale or small-scale power, and important im-plications in the distinction [22]. The structure we findin the canonical scalar-field phase-plane based on simplephysical considerations may prove useful, and we have3here presented firm targets for a basic test of dark en-ergy. The language is different from inflation for tworeasons: the dark energy need not be rolling as slowly asthe inflaton, and the dark energy is not totally dominant,unlike the inflaton.Charting the late-time cosmic evolution – throughType Ia supernovae distances, weak gravitational lensingprobes of large-scale structure evolution, distance ratiosfrom baryon acoustic oscillations in galaxy clustering, etc– is the subject of intense investigations. As a gauge ofthe requisite resolution, a 1% variation in luminosity dis-tance to redshift z = 1 distinguishes between: Λ andw = −0.95, w′ = 0; models which evolve along the topand bottom of the thawing region out to w = −0.8; mod-els which evolve along the top and bottom of the freezingregion in to w = −0.95. The goal of making the fun-FIG. 2: The evolutionary tracks in w − w′ phase space areshown for a variety of particle physics models of scalar fields.The two broad classes are clear: those that initially are frozenand look like a cosmological constant, starting at w = −1,w′ = 0, and then thaw and roll to w′ > 0, and those that ini-tially roll and then slow to a creep as they come to dominatethe Universe. The sample of thawing models shown have po-tentials V ∝ φn for n = 1, 2, 4 (short-, dot-, and long-dashedcurves) and a PNGB with V ∝ cos2(φ/2f) (solid curves). Theright-most point of the tracks corresponds to the present. Forvariety, the n = 4 model has Ωde = 0.6, and the n = 1model ending at w = −0.8 has Ωde = 0.65. All other modelsend with a fractional energy density Ωde = 0.7. The sampleof freezing models shown have potentials V ∝ φ−n, φ−neαφ2(solid and dashed curves). The line w′ = 1.5w(1 + w) indi-cated by the light, dotted line is a possible lower bound on thefreezing models. The left-most point of the tracks correspondsto the present; the right-most point is at z = 1. For variety,upper and lower close pairs of curves have Ωde = 0.7, 0.8respectively. All other models end with a fractional energydensity Ωde = 0.7.damental physics distinction between the thawing andfreezing regions is challenging but achievable in the nextgeneration of experiments [23, 24, 25, 26] if the dynamicsis sufficiently apparent. In the case 1 + w & 0.05, dedi-cated dark energy experiments now being designed, suchas the Joint Dark Energy Mission, will probe cosmologywith sufficient accuracy to be able to decide the issue.This will probably not be the final word on dark energy(cf. [27]), but if the answer is not consistent with a cosmo-logical constant then the rewards are obvious in discov-ering new physics beyond our current standard models.It is interesting to note that the fate of the Universe isvery different for the case of a thawing field, as the accel-eration is temporary, as compared to a freezing field, forwhich the acceleration continues unabated. If the resultlies outside the two phase-space regions categorized herethen 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. ELthanks Dartmouth for its hospitality.[1] R. R. Caldwell, Phys. World 17, 37 (2004).[2] T. Padmanabhan, Phys. Rept. 380, 235 (2003).[3] P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559(2003).[4] B. Ratra and P. J. Peebles, Phys. Rev. D 37, 3406 (1988).[5] C. Wetterich, Nucl. Phys. B 302, 668 (1988).[6] J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys.Rev. Lett. 75, 2077 (1995).[7] K. Coble, S. Dodelson and J. A. Frieman, Phys. Rev. D55, 1851 (1997).[8] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev.Lett. 80, 1582 (1998).[9] R. A. Knop et al. [The Supernova Cosmology ProjectCollaboration], Astrophys. J. 598, 102 (2003).[10] A. G. Riess et al. [Supernova Search Team Collabora-tion], Astrophys. J. 607, 665 (2004).[11] U. Seljak et al., arXiv:astro-ph/0407372.[12] R. Kallosh, J. Kratochvil, A. Linde, E.V. Linder and M.Shmakova, JCAP 0310, 015 (2003).[13] T. Barreiro, E. J. Copeland and N. J. Nunes, Phys. Rev.D 61, 127301 (2000).[14] P. G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503(1998).[15] I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev.Lett. 82, 896 (1999).[16] P.J. Steinhardt, “Proceedings of the the Pritzker Sym-posium on Status of Inflationary Cosmology,” ed. by M.Turner, (U. Chicago Press, Chicago, 2000).[17] P. Binetruy, Phys. Rev. D 60, 063502 (1999).[18] P. Brax and J. Martin, Phys. Lett. B 468, 40 (1999).[19] A. Masiero, M. Pietroni and F. Rosati, Phys. Rev. D 61,023504 (2000).[20] E. J. Copeland, N. J. Nunes and F. Rosati, Phys. Rev.D 62, 123503 (2000).4[21] A. de la Macorra and C. Stephan-Otto, Phys. Rev. Lett.87, 271301 (2001).[22] W.H. Kinney, E.W. Kolb, A. Melchiorri, and A. Riotto,Phys. Rev. D 69, 103516 (2004).[23] G. Aldering et al., PASP submitted,arXiv:astro-ph/0405232.[24] M. Jarvis, B. Jain, G. Bernstein and D. Dolney,arXiv:astro-ph/0502243.[25] E. V. Linder and R. Miquel, Phys. Rev. D 70, 123516(2004).[26] A. Upadhye, M. Ishak and P. J. Steinhardt,arXiv:astro-ph/0411803.[27] I. Maor and R. Brustein, Phys. Rev. D 67, 103508 (2003).

The Limits of Quintessence

Crossref

Limits of Quintessence

We present evidence that the simplest particle-physics scalar-field models of dynamical dark energy can be separated into distinct behaviors based on the acceleration or deceleration of the field as it evolves down its potential towards a zero minimum. We show that these models occupy narrow regions in the phase plane of w and w′, the dark energy equation of state and its time derivative in units of the Hubble time. Restricting an energy scale of the dark energy microphysics limits how closely a scalar field can resemble a cosmological constant. These results, indicating a desired measurement resolution of order σ(w′)≈(1+w), define firm targets for observational tests of the physics of dark energy

Caldwell, R. R.

Linder, Eric V

Dartmouth Digital Commons (Dartmouth College)

Dartmouth College Dartmouth Digital Commons Open Dartmouth: Published works by Dartmouth faculty Faculty Work 9-28-2005 Limits of Quintessence R. R. Caldwell Dartmouth College Eric V. Linder Lawrence Berkeley National Laboratory Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa  Part of the Cosmology, Relativity, and Gravity Commons Dartmouth Digital Commons Citation Caldwell, R. R. and Linder, Eric V., "Limits of Quintessence" (2005). Open Dartmouth: Published works by Dartmouth faculty. 2014. https://digitalcommons.dartmouth.edu/facoa/2014 This Article is brought to you for free and open access by the Faculty Work at Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: Published works by Dartmouth faculty by an authorized administrator of Dartmouth Digital Commons. For more information, please contact dartmouthdigitalcommons@groups.dartmouth.edu. Limits of QuintessenceR. R. Caldwell1 and Eric V. Linder21Department of Physics & Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA2Physics Division, Lawrence Berkeley Laboratory, Berkeley, California 94720, USA(Received 1 June 2005; published 28 September 2005)We present evidence that the simplest particle-physics scalar-field models of dynamical dark energy canbe separated into distinct behaviors based on the acceleration or deceleration of the field as it evolvesdown its potential towards a zero minimum. We show that these models occupy narrow regions in thephase plane of w and w0, the dark energy equation of state and its time derivative in units of the Hubbletime. Restricting an energy scale of the dark energy microphysics limits how closely a scalar field canresemble a cosmological constant. These results, indicating a desired measurement resolution of orderw0  1 w, define firm targets for observational tests of the physics of dark energy.DOI: 10.1103/PhysRevLett.95.141301 PACS numbers: 98.80.Cq, 98.80.EsObservations and experiments at the close of the 20thcentury have transformed our understanding of the physicsof the Universe. A consistent picture has emerged indicat-ing that nearly three quarters of the cosmos is made of‘‘dark energy’’—some sort of gravitationally repulsivematerial responsible for the accelerated expansion of theUniverse [for reviews, see [1–3] ]. Proposals for the darkenergy include Einstein’s cosmological constant (), or adynamical field such as quintessence. Here we show howscalar-field dynamics separates into distinct behaviorswhich, through future cosmological measurements, canreveal the nature of the new physics accelerating ouruniverse.Einstein’s cosmological constant () is attributed to thequantum zero-point energy of the particle-physics vacuum,with a constant energy density , pressure p, and anequation of state w  p=  1. In contrast, quintes-sence is a proposed time-varying, inhomogeneous fieldwith a spatially averaged equation of state w>1 [4–8]. The simplest physical model consists of a scalar field,slowly rolling in a potential characterized by an extremelylow mass. (This is similar to inflation, the period of accel-erated expansion in the early Universe, but at an energyscale many orders of magnitude lower.) Since a scalar fieldevolving in a very shallow potential may be indistinguish-able from a , the task of elucidating the physics of darkenergy becomes difficult if observations continue to findthat w is close to 1, e.g., [9–11]. In this Letter, weexamine the likely behavior of scalar fields and character-ize them into two distinct classes, based on their evolutionin the w-w0 phase space. These results should help definetargets for observational and experimental tests of thephysics of dark energy.Our approach is a new take on a familiar system, thescalar field. By emphasizing the dynamics, we discoverrestricted regions of the trajectories of canonical scalar-field models in ‘‘position’’ and ‘‘velocity’’—the value ofthe equation-of-state ratio w and its time variation w0.While there is a myriad of scalar-field models motivatedby particle physics beyond the standard model, this treat-ment allows a broad, model-independent assessment of aquintessence scalar field slowly relaxing in a potential.The physics is straightforward: the field  will seek toroll towards the minimum of its potential V, according tothe Klein-Gordon equation  3H _  dV=d. Therate of evolution is driven by the slope of the potentialand damped by the cosmic expansion through the Hubbleparameter H. The average energy density and pressure are  _2=2 V, p  _2=2 V so that a field stuck in alocal, nonzero minimum of the potential has w  1. Todistinguish from an effective cosmological constant, how-ever, we will only consider cases in which the field isevolving towards a zero minimum.In perhaps the simplest such scenario, the field has beenfrozen by Hubble damping at a value displaced from itsminimum until recently, when it starts to roll down to theminimum. We call these ‘‘thawing’’ models. At early timesthe equation-of-state ratio is w  1, but grows less nega-tive with time as w0  _w=H > 0. Since the Hubbledamping limits the scalar-field acceleration, < _=t 3=2H _, then the equation of motion implies such modelswill lie at w0 < 31 w in the phase plane. The scalar-field dynamics suggest a lower bound, too, due to the factthat dark energy is not entirely dominant today, with afractional energy density de & 0:8. Our study of severalclasses of thawing models, such as a pseudo Nambu-Goldstone boson (PNGB) [6] or polynomial potentials,indicates the bound w0 > 1 w. These simple boundsare valid for 1 w  1, and so w & 0:8 is a practicallimit of applicability.We have analyzed the following potentials for thawingbehavior: concave potentials with V  M4nn are ubiq-uitous, and we have allowed for continuous values of theexponent n > 0. The motivation for cases n < 2 is not asstraightforward, although n  1 has been consideredPRL 95, 141301 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending30 SEPTEMBER 20050031-9007=05=95(14)=141301(4)$23.00 141301-1 © 2005 The American Physical Society[12,13]. We restrict attention to power-law potentials withindex n < 6; for higher powers the field rolls too quicklydown the potential, with w>0:8 by today. Also, suchpotentials can lead to tracking behavior [14] whereby thescalar field evolves with a positive equation of state andthere is no cosmic acceleration. Exponential potentials aretypical for moduli or dilaton fields, e.g., [15], with V M4 exp=MP where MP  1019 GeV is the Planckenergy. To avoid scaling, which would not provide for thecosmic acceleration, we restrict <24p[16]. PNGBs,like a dark energy axion [e.g., [17,18] ], have V M4cos2=2f where f is a symmetry restoration energyscale. We have not included the case f MP since thefield rapidly evolves to w! 0 unless the initial conditionsare finely tuned to keep the field balanced upon the top ofthe potential maxima and maintain 1 w 1.A second scenario consists of a field which was alreadyrolling towards its potential minimum, prior to the onset ofacceleration, but which slows down and creeps to a halt asit comes to dominate the Universe. For these ‘‘freezing’’models, initially w>1 and w0 < 0. These are essentiallytracking models [19], but may be described more generallyas vacuumless fields (in the sense that the minimum isattained as! 1) or runaway potentials characterized bya potential with curvature that slows the field evolution as itrolls down towards the minimum. It follows [20] that thereis some value of the field beyond which the evolution iscritically damped by the cosmic expansion, whence thefield is frozen [but, like a glacier [20], continues to move]and w! 1, w0 ! 0. The deceleration of the field islimited by the steepness of the potential, roughly >dV=d, leading to the lower bound w0 > 3w1 w.Our investigation of a variety of scalar-field models leadsto a less definite upper bound w0 & 0:2w1 w since aredshift z	 1 but evolving beyond w0 & w1 w by thepresent. Again, w & 0:8 is a practical limit of applica-bility of these bounds.We have analyzed the following tracker potentials fortheir freezing behavior: V  M4nn and V M4nn exp2=M2P for n > 0 [21–25]. The latterhas an effective cosmological constant, but has beenwidely studied and so we consider it nonetheless, providedthe field is closer to the origin than the nonzero minimum.Proposed tracker models such as V  M4 expMP= donot have a zero minimum, and V  M4expMP=  1does not achieve w & 0:8, de & 0:8. Other functionshave been proposed as tracking potentials, but lack a firmbasis in particle theory.These distinct, physically motivated thawing and freez-ing behaviors are illustrated in Fig. 1, while several ex-amples of specific models are presented in Fig. 2. It wouldbe quite useful to determine if one of these classes ofphysics phenomena is responsible for the dark energyaccelerating our universe. We see that to distinguish thaw-ing from freezing, a measurement resolution of orderw0  1 w is required.Complicated potentials that have a built-in cosmologicalconstant, for which V 0 vanishes, can violate the inequalitieswe have established in the w-w0 phase plane. If we add aconstant potential energy to a quintessence field, the equa-tion of state w is shifted closer to 1. Although the Klein-Gordon equation is insensitive to this addition until thedark energy drives the evolution of H, the magnitude of w0changes. We find that our inequalities continue to hold forfreezing models. For thawing fields, w0 gets smaller, andthe phase-plane trajectory can just cross the line w0  1w by the present.The results of this study strictly apply only to canonicalscalar-field theories, in particular, with w  1.Preliminary investigation indicates that at least some otherdark energy models also follow the dynamics discussedhere (work in progress). Also note that thawing modelswould appear to have an averaged, or enforced constant,value of w very close to 1, indicating the importance oflooking for dynamics in the form of w0.The question of the absolute level of deviation ofw from1, i.e., the distinction from Einstein’s cosmological con-stant, is less tractable. Certainly, one can obtain a scalar-field solution, however unrealistic, at any given point in thew-w0 phase space. Even for the thawing and freezingFIG. 1 (color online). The w-w0 phase space occupied bythawing and freezing fields is indicated by the shaded regions.No strong constraints on this range of dark energy propertiesexist at present. The fading at the top of the freezing regionindicates the approximate nature of this boundary. Freezingmodels start above this line, but pass below it by a redshift z	1. The short-dashed line shows the boundary between fieldevolution accelerating and decelerating down the potential.Future cosmological observations will aim to discriminate be-tween these two fundamental scenarios.PRL 95, 141301 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending30 SEPTEMBER 2005141301-2models, parameters may be finely tuned to keep 1 warbitrarily close to zero within the shaded regions of Fig. 1.If the scalar field is prohibited from attaining valuesexceeding the Planck scale, lest quantum gravitationaleffects dominate, then there is a lower bound on 1 w.Defining a characteristic scale E  jV=dV=dj we de-mand E<MP. Next, for a field rolling down its potential,we can express the scalar-field equation of motion asw0  31 w2  1 wMPE38de1 ws:Taking de  0:7 then thawing models must satisfy 1w * 0:004 whereas for freezing models 1 w * 0:01;this may be the limit of quintessence. These marginscorrespond to a 	0:2% difference from a  cosmologyin distance to redshift z  1. Such an absolute precisiongoal is clearly extraordinarily challenging.Note that early universe inflation can similarly approachpure exponential expansion, with its deviations broadlycharacterized by dynamics into models tilted to preferlarge-scale or small-scale power, and important implica-tions in the distinction [26]. The structure we find in thecanonical scalar-field phase plane based on simple physicalconsiderations may prove useful, and we have here pre-sented firm targets for a basic test of dark energy. Thelanguage is different from inflation for two reasons: thedark energy need not be rolling as slowly as the inflaton,and the dark energy is not totally dominant, unlike theinflaton.Charting the late-time cosmic evolution—through TypeIa supernovae distances, weak gravitational lensing probesof large-scale structure evolution, distance ratios frombaryon acoustic oscillations in galaxy clustering, etc.—isthe subject of intense investigations. As a gauge of therequisite resolution, a 1% variation in luminosity distanceto redshift z  1 distinguishes between:  and w 0:95, w0  0; models which evolve along the top andbottom of the thawing region out to w  0:8; modelswhich evolve along the top and bottom of the freezingregion in to w  0:95. The goal of making the funda-mental physics distinction between the thawing and freez-ing regions is challenging but achievable in the nextgeneration of experiments [27–30] if the dynamics issufficiently apparent. In the case 1 w * 0:05, dedicateddark energy experiments now being designed, such as thejoint dark energy mission, will probe cosmology withsufficient accuracy to be able to decide the issue.This will probably not be the final word on dark energy[cf. [31] ], but if the answer is not consistent with a cos-mological constant then the rewards are obvious in discov-ering new physics beyond our current standard models. It isinteresting to note that the fate of the Universe is verydifferent for the case of a thawing field, as the accelerationis temporary, as compared to a freezing field, for which theacceleration continues unabated. If the result lies outsidethe two phase-space regions categorized here then we mayhave to look beyond simple explanations, perhaps to evenmore exotic physics such as a modification of Einsteingravity.This work is supported by NSF AST-0349213 atDartmouth, and by DOE AC03-76SF00098 at LBL. E. L.thanks Dartmouth for its hospitality.[1] R. R. Caldwell, Phys. 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