Table Of ContentQuintessence’s Last Stand?
Eric V. Linder
Berkeley Center for Cosmological Physics & Berkeley Lab,
University of California, Berkeley, CA 94720, USA
(Dated: January 9, 2015)
Current cosmological data puts increasing pressure on models of dark energy in the freezing
class, e.g. early dark energy or those with equation of state w substantially different from −1. We
investigatetowhatextentdatawilldistinguishthethawingclassofquintessencefromacosmological
constant. Sincethawingdarkenergydeviatesfromw=−1onlyatlatetimes,wefindthatdeviations
1+w .0.1 are difficult to see even with next generation measurements; however, modest redshift
5 driftdatacanimprovethesensitivitybyafactoroftwo. Furthermore,technicalnaturalnessprefers
1 specific thawing models.
0
2
n I. INTRODUCTION uedconsistencyofthe datawitha cosmologicalconstant
a Λwillenableallquintessencemodelstobedisfavored,or
J at least relegatedto a regionfine tuned to be close to Λ.
Cosmic acceleration indicates crucial new physics ex-
7 Beyond quintessence, cosmic accelerationmodels with
ists outside the standard model of particle physics and
deviationsin growthof structure relativeto their expan-
] cosmology. Cosmological data imply the expansion his- sion behavior, such as from modified gravity theories, or
O tory of the universe acts like the matter component is
clusteredorcoupleddarkenergy,canalsobeconstrained
C supplemented with a fluid of strongly negative effective
by growth measurements. However within quintessence
h. pressure: darkenergywithaneffectiveequationofstate, – canonical, minimally coupled dark energy – no new
or pressure to energy density, ratio w ≈ −1. One possi-
p signatures arise within growth. Thus the question of
bility is the cosmological constant, with w = −1 for all
- whether we will be able to distinguish thawing dark en-
o times, or redshifts z, and no spatial perturbations. For
ergy,andhencemostoftheremainingviablephasespace
r data sensitive to only the backgroundcosmic expansion,
t of quintessence, from Λ will focus on the expansion his-
s orHubbleparameterH(z),theeffectsarefullydescribed
a tory.
byw(z),evenifthereisnophysicaldarkenergyfluidbut
[ In Sec. II we discuss methods of treating the thaw-
rather a modification of the form of the action.
ing class of dark energy as a whole, drawing a clear dis-
1
v Cosmic microwave background (CMB) measurements tinction with inappropriate slow roll assumptions. We
4 impose two stringent constraints on the dark energy. examine the leverage of future data in separating thaw-
3 From the distance to last scattering, the energy den- ingquintessencefromacosmologicalconstantinSec.III,
6 sity weighted time average of w(z) must be close to −1 highlighting the role of potential redshift drift measure-
1 [1]. From the structure of the CMB temperature power ments. We explore some particle physics implications of
0
spectrum,thedarkenergydensityaroundrecombination fields nearly indistinguishable from Λ, and the virtues of
.
1 (z ≈ 103) is less than about 1% of the critical energy technical naturalness,in Sec. IV and conclude in Sec. V.
0 density [1–3]. Late time observations of the cosmic ex-
5 pansion, such as supernovae or baryon acoustic oscilla-
1
tion distances,alsoindicate that w ≈−1 atrecenttimes II. THAWING QUINTESSENCE
:
v [4, 5]. These results disfavor the first quintessence mod-
i els,trackermodelswithinsensitivitytoinitialconditions,
X The class of thawing quintessence includes common
but moreover most of the freezing region of dark energy potentials such as monomials V(φ) ∼ φn and pseudo-
r
a dynamics (Ref. [6] showedthat the effective dark energy Nambu Goldstone boson (PNGB, or axion) fields with
field either began dominated by Hubble friction – thaw-
V(φ) ∼ [1+cos(φ/f)], where f is the symmetry break-
ingfields –orbythe steepnessofthe potential–freezing
ing energy scale. Two members of this class are of par-
fields, unless fine tuned).
ticular interest since they have shift symmetry protect-
Theremainingdarkenergybehaviorfavoredbydatais ing against high energy radiative corrections. One is the
the thawing class. While thawing fields do not have the PNGB case, long studied in terms of both natural in-
trackers’insensitivity to initial conditions, some of them flation [7] and dark energy [8]. The other is the linear
do have the attractive quality of technical naturalness – potential [9, 10]. Recently the linear potential has gar-
protection against quantum radiative corrections. Since nered renewed interest in connection with both inflation
thawing models only deviate from w ≈−1 at late times, [11] and modified gravity [12].
they are consistent with the data but are also harder We are interested in constraining the thawing class as
to distinguish from a static cosmological constant, de- a whole, rather than individual models. To do this ef-
spite having very different physics implications. Given ficiently it is useful to parametrize its behavior. The
the current state of data, and the advancing plans for standard w –w form for the dark energy equation of
0 a
next generation experiments, we assess whether contin- state, w(a) = w +w (1−a), where a = 1/(1+z) is
0 a
2
the scalefactor,worksfor abroadvarietyofdarkenergy Ref. [16] works quite well:
models,beyondjust the thawingclass. Indeedit wasde-
rivedfromexactsolutionsofthefielddynamics[13](and 1+b 1−p/3
1+w(a)=(1+w )ap , (2)
is completely unrelated to a linear or Taylor expansion). 0 (cid:18)1+ba−3(cid:19)
The w –w form matches exact solutions of the observ-
0 a
ables – distances and Hubble expansion H – to better where the constant b = 0.5. This is derived from the
than 0.1% for a broadrange of currently allowed models physics of how the field evolves upon leaving the matter
[14]. However, because it involves two parameters, con- dominated era and automatically has the correct high
straints from even next generation data have difficulty redshift behavior.
distinguishing thawing models with w0 . −0.9 from a We can turn it into a one parameter form by fixing
cosmologicalconstantat95%confidencelevel,duetoco- p=1, so the final algebraic form we will use is
variances between parameters.
Since we focus our interest on the thawing class, we 1+w(a)=(1+w )a3 3 2/3 . (3)
couldusemorespecializedparametrizations,suchas[15– 0 (cid:18)1+2a3(cid:19)
20]. We emphasize though that from an observational
perspective the w –w parametrization with its 0.1% Note that of course it can fit a cosmological constant as
0 a
matching is wholly sufficient. To obtain tighter model well,withw0 =−1. Boththe (general)algebraicthawed
constraints, we would need a one parameter form. We and the general w0–wa forms have the advantage that
cannot rely on the usual inflationary slow roll field ap- the Hubble parameter H(z) is analytic.
proximation since even thawing dark energy does not Figure 1 demonstrates the fit of the algebraic form to
slowroll,inthe sensethat allterms in the Klein-Gordon the exact solutions for thawing w(z), with w0 = −0.9.
equationforitsdynamicsarecomparable[21]. Forexam- (Note that thawing models with greater present devia-
ple, during the vast majority of e-folds when the field is tion from −1 are already in some tension with current
frozen(e.g.duringthe matter dominatedera),the terms data.) The agreement is better than 0.1% in w(z) (even
are in the ratio of 1:2:−3. Note also that a small field better for the observables like distances and H) for the
assumption is somewhat uncertain: while many thawing favored models of the linear potential and PNGB, and
fields roll a distance ∆φ≈0.24M to get to w =−0.9 better than0.2%and0.3%forthe quadraticandquartic
Pl 0
today, they start at several M from their minima. potentials.
Pl
We cantaketwoapproachesto adoptinga oneparam-
eter thawing equation of state. Ref. [14] showed that
not only was w –w an excellent approximation to ex-
0 a
act solutions of the field evolution, and to observational
quantities, but it could also be derived as a calibration
relation for classes of dark energy dynamics. By appro-
priate choice of w and w , the spread of evolution in
0 a
′
the w–w plane calibrated into narrow tracks for differ-
ent dark energy physics. In particular, a broad range of
thawing models could all be tightly fit by a constrained
w form:
a
w ≈−1.58(1+w ) . (1)
a 0
This combines all the virtues of the general w –w form
0 a
(e.g. 0.1% distance reconstruction) with the leverage of
a one parameter fit.
While this fits the expansion observables (distances
and Hubble parameter) superbly, an alternate approach
is to follow the thawing physics more closely. For exam-
ple, thawing models have w =−1 at high redshift. This
is not a problem for the form Eq.(1) since thawing dark FIG. 1. The one parameter algebraic thawing form (solid
energy density fades quickly into the past and so using black curve, Eq. 3) accurately fits w(z) at all redshifts, for
w(z ≫ 1)= w +w 6= −1 has negligible effect. (Recall the thawing models of linear, quadratic, quartic, and PNGB
0 a
that even accounting for this the distance to high red- potentials.
shifts is accurately matched to better than 0.1%.) If we
feltmorecomfortableapproximatingw(z),despite itnot While monomial potentials indeed have only one pa-
being an observable, then the algebraic thawing form of rameterto describe the equationof state, PNGB models
3
have two: the steepness of the potential, given by the thisroughlyrepresentsasamplefromgroundbasedimag-
symmetry breakingscalef, andthe initialfield position. ing surveys such as the Dark Energy Survey or Large
Forsteeperpotentials,thealgebraicformfitcandegrade Synoptic Survey Telescope (LSST). We also analyze an
to≈1%inw (alternatelywecouldadjustthe valuesofb extended sample inspired by the DESIRE proposal [22]
orpintheoriginalalgebraicformofEq.2,buttherewill for the Euclid satellite, with 7200 SN between z =0.1–1
stillbearangeofPNGBmodelsthatcannotbecaptured and 1000 at z = 1–1.6, with the same systematics con-
to muchbetter than1%in w(z)by onlyone parameter). trol out to z = 1.6. These are simply rough estimates
Despite the success, it is importantto note we are not of potential data. For the CMB we use Planck quality
trying to fit w(z) per se (as it is not an observable), but distancetolastscatteringmeasurementatthe0.2%level
rather to obtain a realistic enough representation that andaprioronthephysicalmatterdensityΩ h2 of0.9%.
m
ourconstraintsondistinguishingthawingmodelsfromΛ
Theparametersforconstrainingthe expansionhistory
in the next section are accurate. Table I demonstrates
include the matter density Ω , dynamical equation of
m
thesuccessofboththealgebraicthawer(Eq.3)andcon-
state parameter w , supernova absolute magnitude M,
0
strainedw –w form(Eq.1)inmatchingtheobservables.
0 a and Hubble constant h. Spatial flatness is assumed. Us-
(Changing the value of, e.g., the matter density would
ing the future data, the constraint on the equation of
slightlydegradetheaccuracyoffittingw(z),butimprove
state parameter is σ(w ) ≈ 0.06. If the thawing dark
0
theaccuracyoffittingdorH sincethereismorefreedom
energy has reachedw =−0.9 today (and a strongerde-
0
allowedfor the fit.) The accuracyis more than sufficient
viation is already disfavored by present data), then this
for next generation observations seeking to distinguish
yields at most a ≈ 1.7σ distinction from Λ. A stronger
thawing dark energy from a cosmological constant Λ.
discriminant would be useful.
Interestingly, the extended distance sample does not
Model d(z) H(z) dlss
Linear potential 0.06% 0.01% 10−5 improve substantially the constraints here. While the
φ2 0.08% 0.02% 10−5 baseline sample gives σ(w0) = 0.063 and 0.062 for the
φ4 0.16% 0.03% 10−5 algebraic thawer and calibrated wa forms, the extended
PNGB (f/MPl =2) 0.03% 0.01% 10−5 sample gives 0.056 and 0.055. The reason for this in-
Constrained wa 0.09% 0.01% 4×10−5 sensitivity is that thawers tend to deviate appreciably
Λ 3.5% 0.7% 0.5% from Λ only at low redshift; for example thawers as rep-
resented by the algebraic form have w(z = 1) = −0.98
TABLE I. Maximum deviations in the observables – the co- while eventually reaching w(z = 0) = −0.9. While su-
moving distance d(z), Hubble parameter H(z), and distance
pernova distances are indeed sensitive to the equationof
toCMBlastscatteringdlss –overallredshifts,relativetothe statevalueatlowredshift,theyarenotsensitiveenough.
algebraic thawing form, are given for the exact solutions of
various thawing dark energy models with w0 = −0.9. The The question then is what observational probe can
matter density and Hubble parameter are fixed (Ωm = 0.3, more keenly test the low redshift equation of state. Re-
h=0.7). cently,cosmicredshiftdrifthasbeenrecognizedashighly
sensitive at low redshift [23]. This constrains the quan-
tity
III. CONSTRAINING THAWERS
z˙ =(1+z)H −H(z) . (4)
0
As Table I shows, the maximum difference between
thawingdarkenergydeviatingfromthecosmologicalcon- Forthe (general)algebraicthawerthe Hubble parameter
stant at late times to w = −0.9 is 3.5% in distance (at can be written analytically as
0
z = 0.5). However, this was with fixing all other cos-
mdisotliongcitciaolnpmaroarmeecthearsllesnogiinngr.eaLliettyucsovsaereiaifncoeusrmtwakoeotnhee- H(z) 2 =Ωma−3+(1−Ωm)e3(1α+pw0)[1−(αa3+β)p/3]
(cid:18) H (cid:19)
parameter thawing forms give data the leverage to dis- 0
(5)
tinguishsuchdynamicaldarkenergyfromacosmological
where α = 1/(1+b), β = b/(1+b). Since redshift drift
constant.
is wholly unproven, we conjecture a single, modest 5%
To test these models with expansion history measure-
measurement at redshift z = 0.5. This one data point
mentsweconsidersupernovadistancesandtheCMBdis-
improves the distinction between thawers and Λ by a
tance to last scattering. (We could use baryon acoustic
factoroftwo,reducingσ(w )to0.030,marginalizedover
oscillationdistances,butsupernovaearesomewhatmore 0
theothercosmologicalparameters,andhence a3.3σ dis-
sensitivetotheequationofstate,especiallyattheneeded
tinction from a cosmologicalconstant.
low redshift.) For supernovae (SN), we consider a future
sampleof150SNatz <0.1,900betweenz =0.1–1,and Of course as w → −1 the dark energy would be in-
0
42 between z = 1–1.7, with a magnitude systematic of creasinglydifficultto distinguishfromΛ. SectionIVdis-
0.02(1+z). Given the systematic control out to z = 1, cussesparticle physicsimplications of suchnear-Λfields.
4
IV. FINE TUNED FIELDS close to Λ can be obtained without fine tuning α, e.g.
w = −0.99 comes from α = 0.25. Furthermore the lin-
0
Thawingfields havea limit inwhichthey areidentical ear potential is substantially protected against quantum
toacosmologicalconstant,i.e.whenw =−1,andsothe corrections. In the future it leads to a cosmic doomsday
0
fullclasscanneverberuledoutifthedataareconsistent [10], actually essential for the recent solution of the cos-
with Λ. We examine whether fields in such a limit are mologicalconstant problem known as the sequester [12].
particularly fine tuned or disfavored in particle physics Shortofdoomsday,thelinearpotentialcanbeextremely
terms. difficulttodistinguishfromΛ,however,andsoupcoming
First consider quadratic and quartic potentials as observationswillnotbeabletoruleoutfullythethawing
thawers. Like all thawers, the fields do not roll very class.
far during their evolution up to the present, as shown For the PNGB model, a shift symmetry protects the
in Fig. 2. However, to attain even only w = −0.9 to- potential form against quantum corrections. Neverthe-
0
day, the fields must start 2.6M and 5.1M respec- less, one might be concerned about the physics valid-
Pl Pl
tively from the minima. The potentials must be suffi- ity if the symmetry energy scale f > MPl. As f de-
ciently steep locally to be able to reach Ω = 0.7 to- creases, and the potential steepens, it becomes harder
de,0
day once they are released from Hubble friction to roll. to attain Ωde,0 = 0.7. The field must be fine tuned to
Values of w0 closer to −1 exacerbate the situation, with an initial position φi closer and closer to the potential
tshpeecfitievledlsyntoeerdeiancgh,tosayst,awrt0 =at−70.9.9M9Palndanhden1c5e.7bMe ePalsriley- mΩdaex,mimaxum∝,fo2u,twohfialellfothrefirxaendgΩed[0e,,0π,f(]φ.i/Ffo)rmafixxe∼deφ−i/1/ff,
confused with Λ. Especially since such potentials have [14]. Thussmallf isproblematic. Toobtainw0 <−0.99
no protection against radiative corrections, we might be with f = 0.5MPl requires φi/f < 0.18, i.e. only allow-
concernedaboutsuchsuperPlanckianvalues(despitethe ing 6% of the full range, while for f = 0.2MPl a 0.5%
field rolling over less than a Planck mass). If we restrict fine tuning is required. Still, a reasonable region near
the field initial position to lie within one Planck mass of f/MPl ≈ 1 is allowed and can give an expansion history
the minimum, these potentials cannot deliver dark en- effectively indistinguishable from Λ.
ergy density Ω >0.45 and 0.17 respectively. Thus we see that nothing prevents thawing fields, and
de
in particular the two technically natural and hence ro-
bust models of the linear potential and PNGB, from ap-
proachingΛbeyondthe abilityofupcomingexperiments
to distinguish.
V. CONCLUSIONS
Greatstrides havebeenmade in constrainingdarken-
ergy physics with ever improving observations. Freezing
darkenergy,including early darkenergy,is substantially
limited by the data as an explanation for cosmic accel-
eration. Probesof the expansionhistory will continue to
become more incisive, and probes of the growth of large
scale structure will impose tighter constraints on more
elaborate theories involving dark energy clustering, cou-
pling, and modified gravity. However even if the data
continue to be consistent with a cosmological constant,
quintessenceremainsaviableoptioninitsthawingclass.
The standard dark energy equation of state
parametrization w(a) = w + w (1 − a) is an ex-
0 a
cellent global approximation over a wide range of
freezer, thawed, or modified gravity models, to the 0.1%
FIG.2. Thedistancethefieldrollsuptothepresentisplotted level in the observables. A constrained, one parameter
vs the present equation of state for several thawer models. form with w = −1.58(1+ w ) can be used to model
a 0
The field only traverses a short distance for, say, w0 = −0.9 specifically the thawing class. Another option is to use
but must start several MPl from the potential minimum. the algebraic thawer form of Eq. (3) that follows the
thawing physics to match w(z) as well as the observ-
The linear potential, however, is robust. For V(φ) = ables. We use these not as fitting forms per se, but to
V (1+αφ),whiletheslopeofthepotentialαV ≈10−120 representthe thawing class as a whole for comparisonto
0 0
asforanyquintessence,the parameterαitselfis oforder a cosmological constant, to evaluate the discriminatory
unity. For w = −0.9 we have α = 0.72, and behavior leverage of forthcoming data.
0
5
We find that next generation distance data can reveal their shift symmetry. Indeed the linear potential has re-
modest, but consistent distinctions of thawers from Λ if cently been found essential in one method for solving
w & −0.9. The leverage in distinguishing the physics the original cosmological constant problem [12]. If fu-
0
doubles with inclusion of a single, 5% measurement by ture data show some definite signature in expansion or
the prospectiveprobeofcosmicredshiftdrift. This gives growth away from Λ, we will be able to focus our efforts
further motivation for its exploration and development. onspecific physicalproperties. EvenifΛ continuesto be
Inparticular,sincethesignatureofthethawingdeviation agoodfit,wewillbe drawneither toexplainingitsmag-
increases at very low redshift – e.g. the deviation is five nitude or to finding the informative high energy physics
times greater at z =0 than at z =1 – then the recently origins of these robust thawing quintessence potentials.
identified power of redshift drift at low redshift [23] is
especially valuable.
While quintessence in the form of thawing fields can- ACKNOWLEDGMENTS
not be ruled out even if the data become increasingly
consistentwitha cosmologicalconstant,despitethe very This work has been supported by DOE grant DE-SC-
different physics, such behavior does point to preferred 0007867andtheDirector,OfficeofScience,OfficeofHigh
models. Both the linear potential and PNGB are tech- EnergyPhysics,oftheU.S.DepartmentofEnergyunder
nically natural, easing high energy quantum effects by Contract No. DE-AC02-05CH11231.
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