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Λ is Coming: Parametrizing Freezing Fields Eric V. Linder1,2 1Berkeley Center for Cosmological Physics & Berkeley Lab, University of California, Berkeley, CA 94720, USA 2Energetic Cosmos Laboratory, Nazarbayev University, Astana, Kazakhstan 010000 (Dated: March 22, 2017) Weexplorefreezingdarkenergy,wheretheevolutionofthefieldapproachesthatofacosmological constantatlatetimes. Weproposetwogeneral,twoparameterformstodescribetheclassoffreezing field models, in analogy to ones for thawing fields, here based on thephysics of the flow parameter 7 ′ or the calibrated w–w phase space. Observables such as distances and Hubble parameters are fit 1 to within 0.1%, and the dark energy equation of state generally to within better than 1%, of the 0 exact numerical solutions. 2 r a I. INTRODUCTION II. METHODOLOGY M 1 Cosmic acceleration of the expansion of the universe Freezingfieldsofintereststartinthematterdominated 2 can be treated as arising from an effective scalar field; epoch (by which is meant either nonrelativistic matter thisholdsevenifthephysicaloriginisactually,say,from or radiation) with the field rolling down a steep poten- ] a higher dimensional braneworld. At the level of the tial. At late times (in the future) the field freezes as the O Friedmann equations, the expansion a(t) can be equiva- field enters the shallow region of the potential, and the C lently describedbyadarkenergyequationofstatew(a), fieldactslikeacosmologicalconstant,asymptoticallyap- h. modulo the matter density. proachinga de Sitter state (thoughthe fieldmotionmay never vanish completely at finite times). Potentials of p Thephysicsofanobservationallyviableuniverse–one - with a long period of matter domination in which den- interestdonothaveanonzerominimum, i.e.anintrinsic o cosmologicalconstant. r sity perturbations can grow and then a recent period of t cosmicacceleration–grouptheacceptablescalarfieldbe- Earlyfreezingmodelsdatebacktotheexponentialand s inverse power law potentials [13–16] and were particu- a haviorsintotwoclasses: thawingfieldsandfreezingfields [ thatlieindistinctregionsofthew w phasespace[1–4]. larly interesting for having an attractor behavior that ′ 2 Here a prime denotes d/dlna. Th−ese classes are related broughtthe fieldfromawide varietyofinitialconditions ontoatrackertrajectoryathighredshiftwheretheequa- v tothecompetitionbetweenthedrivingtermofthesteep- tionofstatebecameconstant. Oneproblemwiththiswas 5 ness of the potential and the friction of the Hubble ex- 4 pansion. The separation between the classes is enforced the difficulty of moving the field sufficiently quickly off 4 the tracker so that it could attain a sufficiently negative by avoidance of fine tuning in the field, that the accel- 1 equation of state w 1 near the present. The poten- 0 eration of the field will generically not be so exquisitely tial needed to be mo≈di−fied to allow this to happen, e.g. balanced that it vanishes. . 1 within the supergravity inspired approachof [17]. Acrosstheseclassesahighlysuccessfulgeneraldescrip- 0 An interesting discovery was that higher dimensional tion is provided in terms of two parameters, w and w , 7 0 a braneworldmodels [18]andmorephenomenologicalgen- 1 correspondingtothepresentdarkenergyofstateparam- eralizations by adding a term Hα to the Friedmann ex- : eter and a measure of its time variation. This was first v pansion equation [19] (with the DGP braneworld corre- derived in terms of exact solutions of the Klein-Gordon i spondingtoα=1)actedinaverysimilarwaytofreezing X equation of motion [5] and later quantified as providing scalarfields,andindeedonecouldwritedownaneffective r 0.1% accuracy on observable quantities [4]. For some a potential[20]. Atearlytimesthesebehavelikeaninverse purposes, though, one might be interested in going be- powerlawpotentialV φ n withindexn=2α/(2 α). yondobservablesandseekinganimproveddescriptionof ∼ − − At late times, such freezers approach w = 1 along the the equation of state w(a) itself. − phasespacetrajectoryw =3w(1+w),the lowerbound- ′ An equation of state description for the class of thaw- aryofthefreezingregion. Thiscorrespondstoapotential ing fields has been treated in depth in the literature [6– 11]. Most recently, a one parameter form simplifying 3 V(φ)=V 1+ (φ φ)2 , (1) the general algebraic thawing expression of [7] has been ∞ (cid:20) 8 ∞− (cid:21) demonstratedto have0.3%accuracyinw(a) [12]. Freez- ing field models however have been more problematic, where the field asymptotically freezes to the value φ due to their greater diversity in initial conditions and (while this nominally has a nonzero minimum, it is on∞ly evolution. Here we address the issue of parametrizing an effective potential and there is no true cosmological the equation of state for the class of freezing models. constant, e.g. in Minkowski space). Section II discusses the methodology, and we present Thus the variety of potentials means that there is the results in Sec. III, before concluding in Sec. IV. no expectation that attempting to find a common 2 parametrizationoffreezingfieldpotentialsshouldbesuc- where b is a constant parametrizing the specific freezing cessful. Indeed, there are clear differences between in- model. In the matter dominated epoch this expression verse power law and supergravity inspired potentials. reduces to F = 1/3 as desired, and F again goes to a Thus,weinsteadfocusoncapturingthekeyphysicscom- constantinthefuture. Sincewedonothaveobservations mon to different classes of potentials. inthefuture, wedonotforcebtoaparticularvalue(e.g. One approach is to follow the physics in terms of the for all braneworld models it would be 4/3), but leave it flow parameter [20] as a fitting parameter to observable data. Note that F =1/3 does not fully characterize the ini- 1+w(a) tial conditions as this simply leads to w = 0 at early F(a) , (2) ′ ≡ Ω (a)(V /V)2 times, without determining the specific value of w in the φ φ high redshift tracking regime. This is then an additional where Ωφ(a) is the fractional dark energy density and parameter w , where in the case of early time inverse Vφ = dV/dφ. The flow parameter was shown to be con- power law po∞tential behavior w = 2/(n+ 2). The servedthroughthematterdominatedera,withthevalue two parameters of this ansatz ar∞e then−b and either w 4/27for the thawing class and 1/3 for the freezing class. or n. We will keep the second parameter as w sinc∞e During the present accelerating epoch F slowly deviates this is somewhat more generic. ∞ from these asymptotic values as the dark energy den- To solve the evolution we have a system of coupled sity increases. The flow parameter is directly related to equations of motion, the dark energy phase space evolution and so given a parametrization for F one can derive w(a), from w′ = 3(1 w2) 1 1 (5) − − (cid:20) − √3F(cid:21) 1 w′ =−3(1−w2)(cid:20)1− √3F(cid:21) . (3) Ω′φ =−3wΩφ(1−Ωφ) . (6) One can write a closed form solution for w(a), given by Another approach to consider is based on the phe- C 1 nomenological calibration of the phase space trajecto- w= − (7) ries exhibited in [4]. This showedgreatsuccess in taking C+1 t“hsteredticvheirnsigtypaorfaemqeutaetriso”n(loefasdtiantgetboephaarvtiiocruslaarnvderdseiofinnsinogf C = 1+wi a −6 e2√3Raaidlna′F(a′)−1/2 . (8) 1 w (cid:18)a (cid:19) i i w and w ) that focused the models into narrow bands − 0 a or families. One might parametrize the freezing models If one adopts a perturbative expansion about the high according to band and location along the band. redshift value of the form F(a) = F0 +F1as, e.g. since We now discuss these two approaches – the flow and Ωφ a−3(1+w∞) at high redshift, one can solve this an- ∼ calibration approaches – in more detail. alytically [21] but since we are interested in times when the dark energy density becomes dominant we will work with the form Eq. (4) and the numerical solution. A. Flow Parametrization The flow parametrizationhas the useful property that itiscloselyconnected(muchmoresothanthepotential) towhatwewant: thedarkenergyequationofstatew(a). We know that the physics of the long, matter dominated epoch forces the field evolutioninto certainpaths. Thiscausesthecombinationofdarkenergydensity,equa- B. Calibration Parametrization tionofstate,andsteepnessofthepotentialtobeinterre- lated in such a way as to keep constant the combination in the flow parameter F. If we focus on tracker freezing An even more direct approach is to parametrize w(a) fields, as the mostattractive due to their insensitivity to explicitly. This is of course exactly what the standard initialconditions,thenthe earlyequationofstateiscon- w0–wa form does, but here we are focusing specifically stant, i.e. the dark energydensity ρ a 3(1+w), and so on the freezing class and seek increased accuracy. One φ − ∼ by Eq. (3) we see that F = 1/3 during matter domina- applicationofw0–wadidthiswithaneyeonthedistance- tion. redshiftrelation,in[4],wherethephasespacetrajectories Asthedarkenergydensitygrows,F beginstoincrease of freezing models were calibrated through a stretching from this constant, but slowly – see Fig. 4 of [20] – with relation the deviation proportional to the fractional dark energy density Ωφ. Since dark energy has only become signif- wacalib ≡−w′(a⋆)/a⋆ . (9) icant within the last e-fold of cosmic expansion, and is Adoptingthevalueofthetemporalstretchingparameter notfullydominanttoday,areasonableansatzfortheflow a =0.85wasfoundtocalibratethefreezingfieldmodels parametrization is ⋆ into tight family bands, and the resulting 1 F(a)= 3[1−Ωφ(a)]+bΩφ(a) , (4) w(a)=w0+wacalib,d(1−a) (10) 3 gave accurate reconstruction of the observable distances and Hubble parameters at the 0.1% level. Here we will parametrize the phase space to fit w(a) itself. We will find in the next section that this works very similarly to the distance calibration, but here w is a defined by choosing the stretching to calibrate the entire evolution w(a). This results in the stretching parameter becoming a = 0.82. Thus this approach is ⋆ also a parametrizationwith two free parameters,w and 0 wcalib,w, where the superscript calib,w indicates w is fit a a to w(a) rather than to distance or to w = 2w(z =1) a ′ − or some similar constraint. III. RESULTS To investigate how well the flow parametrization and calibration parametrization can reconstruct the true equationofstate evolutionw(a) we considerseveraltyp- ical freezing models. The inversepowerlaw (IPL)potential[14, 16] V(φ)= V φ ntracksathighredshiftwithadarkenergyequation ⋆ − of state w = 2/(n+2). However, it does not evolve FIG. 1. Flow parametrization (dotted, red) for w(a) com- rapidlytow∞ardw− 1asthedarkenergydensitygrows, paredtoexactresultsforIPL(solid,black),SUGRA(dashed, and so does not ≈acc−ord well with current observations blue),and Hα (dot-dashed,green) cases. unless n 1. ≪ Toamelioratethis,thesupergravityinspired(SUGRA) potential [17] V(φ) = V⋆φ−neφ2/(2Mp2) enables values We repeat the comparison of the exact numerical results with the approximation given by the calibration much closer to w 1 to be attained by the present, ≈ − parametrizationin Fig. 2. In order to reflect the physics despite acting like IPL athighredshift. For the IPL and of the tracker freezing fields at high redshifts, we fix SUGRA potentials, we solve the Klein-Gordon equation w(a < 0.25) = w(a = 0.25). Again the parametriza- of motion numerically (by a fourth order Runge-Kutta tionishighlysuccessful,accurateatthesubpercentlevel, technique) to obtain w(a). andatthe0.01%–0.1%levelontheobservablesofthedis- A more phenomenological model of interest is when tances and Hubble parameters as a function of redshift. thedarkenergycontributiontotheFriedmannexpansion equationtakestheformofaHα modification[19],where Tables I and II list the maximum deviations in w(a), H is the Hubble parameter. The case where α = 1 cor- and in the observables of distance d(z) and Hubble pa- responds to higher dimensionalbraneworldDGP gravity rameterH(z),overallredshifts,fortheflowparametriza- tion and the calibration parametrization respectively. [18]; this has the present equation of state too far from Deviations in the distance to cosmic microwave back- w = 1tobeviable,butw approaches 1forsmallerα 0 − − groundlastscatteringareinallcasesnearthe10 4 level. (withα=0correspondingtothecosmologicalconstant). − The Hα model evolution can be solved analytically, Note that freezing models consistent with observations wouldgenerallyhavew . 0.9;indeedthemodelswith with w = 0.75havedista∞nces−toCMBlastscatteringmore Ωφ 2/[3(2−α)] 1 Ωφ,0 1/3 th∞an 1%−different froma Λ model with the same present a= − (11) (cid:18)Ω (cid:19) (cid:18) 1 Ω (cid:19) matter density (and one would have to shift the matter φ,0 φ − density by 0.04 to get agreement in most cases). 2 α ∼ w = − , (12) −(cid:18)2 αΩ (cid:19) φ − IV. CONCLUSIONS where a subscript 0 denotes the present. Foreachmodelwewillchoosevaluesofw (orequiva- lentlynorα)andthenfindthevaluesofth∞efreeparam- Dark energy evolution and the ensuing cosmic accel- eterbfromEq.(4)thatbestapproximatetheexactw(a) eration is a competition between the steepness of the functions. Figure1illustratesthatthe flowparametriza- (effective) potential and the Hubble friction of the cos- tion provides excellent approximations, better than 1% mic expansion. This, together with the long evolution in w(a) for the viable models not too far from w 1. through the matter dominated epoch, naturally defines ≈ − Even for w = 0.75 cases, w(a) is reconstructed to two classes of dark energy: thawing models that depart better than∞1% ov−er almost all of cosmic history. froma frozen,cosmologicalconstantlike stateandfreez- 4 Model δd/d δH/H δw/w IPL (w∞ =−0.9) 0.01% 0.02% 0.6% SUGRA(w∞ =−0.9) 0.03% 0.04% 0.4% BW (w∞ =−0.9) 0.03% 0.04% 0.3% IPL (w∞ =−0.75) 0.02% 0.03% 0.9% (0.5%z<2) SUGRA(w∞ =−0.75) 0.07% 0.1% 2.4% (0.8%z<3) BW (w∞ =−0.75) 0.05% 0.07% 1.2% (0.6%z<3) TABLE II. Accuracy of calibration parametrization w0- wcalib,winfittingtheexactdistances,Hubbleparameters,and a dark energy equation of state for various dark energy models. Thesenumbersrepresentthemaximumdeviationoverall redshifts, with in some cases parenthetical quantities repre- senting deviations in all but theearly universe. levels of motivation. The flow parametrization builds on the physics of dark energy evolution during the long epoch of matter domination when the flow parameter is constant, and parametrizes its deviation as dark energy grows in influence. For observationally viable models (not too far from w = 1), it achieves better than − 0.5%accuracyonw(a) overallcosmic history,as well as FIG. 2. Calibration parametrization (dotted, red) for w(a) reconstructingobservable distances and Hubble parame- compared to exact results for IPL (solid, black), SUGRA ters to better than 0.06% accuracy. Even for excursions (dashed, blue),and Hα (dot-dashed,green) cases. outtow = 0.75theaccuracyonw(a) remainsnearthe − 1% level and on the observables to 0.1%. Model δd/d δH/H δw/w The second approach is the calibration parametriza- IPL (w∞ =−0.9) 0.03% 0.06% 0.1% tion, using the knownconcept of calibrationof the w–w′ SUGRA(w∞ =−0.9) 0.04% 0.04% 0.5% phase space by stretching the time variable. This gives BW (w∞ =−0.9) 0.02% 0.04% 0.3% the familiar w –w parametrization but here w is de- IPL (w∞ =−0.75) 0.04% 0.04% 0.4% fined by choos0ingathe stretching such that it caalibrates SUGRA(w∞ =−0.75) 0.1% 0.07% 1.4% the entire evolution w(a). Note that the stretching is BW (w∞ =−0.75) 0.08% 0.07% 1.1% chosen to be model independent, i.e. it is fixed for the entire freezing class. This is successful (with the im- TABLEI.Accuracyofflowparametrization infittingtheex- posed model independent leveling at high redshift) at act distances, Hubble parameters, and dark energy equation of state for various dark energy models. These numbers rep- betterthan0.6%accuracyinw(a)and0.04%accuracyin resent themaximum deviation over all redshifts. theobservablesforobservationallyviablemodels,andre- mainsatbetterthan1%accuracyonw(a)forexcursions out to w = 0.75 for z <3. − Both approaches use simple, two parameter fits just ing models that approach cosmological constant behav- like w –w . (Recall that the thawing class actually at- 0 a ior. This description holds over a wide range of models tainedexcellentaccuracyevenwithaoneparameterfit.) for canonical scalar fields and modified gravity that can For the flow parametrization this is w ,b , which pro- be viewed as an effective dark energy. vides information on the high redsh{ift∞, tr}acking state; A general treatment for both classes is given by the for the calibrationparametrization this is w ,wcalib,w , { 0 a } w –w phase space parametrization,demonstratedto be very similar to the standard, general parametrization. 0 a accurate in the observables to 0.1%. One can also focus One can easily derive w = w + 0.75wcalib,w in this 0 a onthe darkenergyequationofstatefunctionw(a) itself. approachas well. ∞ While the thawing class has been relatively tractable in IfouruniverseissuchthatindeedΛiscoming,eitherof suchtreatment,thefreezingclasshasnot. Hereweinves- these approaches gives a way to characterize accurately, tigated parametrizationof the freezing class at the same in a fairly model independent manner, the dark energy accuracyasachievedonw(a) forthe thawingclass. This equation of state evolution as well as the observables. ismorecomplicatedinthatthedarkenergyatearlytimes does not have w = 1, while at present dark energy is not completely domi−nant. Λ is coming, but it is not yet ACKNOWLEDGMENTS here. We demonstratetwoapproachesto parametrizationof I thank Zack Slepian for useful conversations. 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