Dark Energy Models in the w - w' Plane

0.14 MB·English

by  Robert J. Scherrer

Dark Energy Models in the w −w′ Plane
Robert J. Scherrer
Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235
Weexaminethebehaviorof dark energy models in theplanedefinedbyw (theequation of state
parameter for the dark energy) and w′ (the derivative of w with respect to the logarithm of the
scale factor). For non-phantom barotropic fluids with positive squared sound speed, we find that
w′ <3w(w+1), the opposite of the bound on quintessence models previously derived by Caldwell
and Linder. Thus, these barotropic models and quintessence models for the dark energy occupy
disjoint regions inthew−w′ plane. Wealso derivetwonewboundsfor quintessencemodels in the
w−w′ plane: thefirstis ageneral boundforany scalar fieldwith amonotonicpotential, while the
second improves on the Caldwell-Linder bound for tracker quintessence models. Observationally
distinguishing barotropic models from quintessencemodels requiresσ(w′).1+w.
6
0
I. INTRODUCTION proaches zero can be divided naturally into two cat-
0
2 egories, which they dubbed “freezing” and “thawing”
Observationsindicatethatroughly70%ofthetotalen- models,withquitedifferentbehaviorinthew−w′ plane.
n
a ergydensityintheuniverseisintheformofdarkenergy, Here we extend and generalize these results. In the
J which has negative pressure and drives the late-time ac- next section, we determine the behavior of general non-
1 celerationoftheuniverse(seeRef. [1]forarecentreview, phantom barotropic fluids with positive squared sound
3 and references therein). speed in the w−w′ plane, and we show that these mod-
The simplest model for this dark energy is a cosmo- els occupy a regionof phase space which is disjoint from
2 logical constant, but many other alternatives have been that occupied by quintessence models. In Section 3, we
v put forward. One possibility, dubbed quintessence, is a re-examinethe lower bound onw′ from Ref. [17]; we de-
0
model in whichthe dark energyarises froma scalarfield riveanewboundforamoregeneralclassofquintessence
9
8 [2, 3, 4, 5, 6]. In another class of models, the dark en- models, and sharpen the bound from Ref. [17] for the
9 ergy is simply taken to be barotropic fluid, in which the case of tracker potentials. These results are discussed in
0 pressure p and energy density ρ are related by Section4. Ourresultssupporttheargumentthatthebe-
5 haviorofdarkenergymodelsinthew−w′ planeprovides
0 p=f(ρ). (1) a useful discriminator for such models.
/
h
The prototype for this sort of model is the Chaplygin
p
- gas [7, 8], which can be considered a special case of the II. BAROTROPIC FLUIDS
o
generalized Chaplygin gas [9]. Although originally pro-
r
t posedasunifiedmodelsfordarkmatteranddarkenergy, As noted in the previous section, a barotropic fluid is
s
a the Chaplygin gas and generalized Chaplygin gas mod- one for which the pressure is purely a function of the
: els have also been examined as models for dark energy density, p=f(ρ). For such models, we have
v
alone [10, 11, 12, 13, 14]. Interest in barotropic fluids
i
X as dark energy has not been confined to the generalized dw dρ
′
w = . (4)
r Chaplygin gas. Many other models have been proposed, dρ dln(a)
a including,e.g.,theVanderWaalsmodel[15]andthewet
dark fluid model [16]. The factors in this equation are given by
Obviously, it is important to determine whether these
modelscanbedistinguishedbytheirobservationalconse- dw = 1 dp −w , (5)
quences. Inthis vein,CaldwellandLinder[17]examined dρ ρ dρ
(cid:18) (cid:19)
the behavior of quintessence models in the plane defined
′ and
by the quantities w and w for the dark energy. Here w
is the ratio of pressure to density for the dark energy: dρ
=−3(1+w)ρ. (6)
dln(a)
w =p/ρ, (2)
′
′ Then the expression for w becomes
andw isthederivativeofwwithrespecttothelogarithm
of the scale factor a:
dp
w′ =−3(1+w) −w . (7)
′ dw dρ
w = . (3) (cid:18) (cid:19)
dln(a)
Inordertoderiveusefulconstraintsfromequation(7),we
Caldwell and Linder showed that quintessence models need to assume something about the behavior of dp/dρ
in which the scalar field potential asymptotically ap- andw. We willassumefirstthatwearenotdealingwith
2
phantom models, so that 1+w > 0. Second, we note whereA andα areconstants. (The Chaplygingascorre-
thatdp/dρ=c2, andwe willassume further thatc2 >0, sponds to the special case α=1). Then it is straightfor-
s s
to avoidinstabilities in perturbationgrowth. These con- ward to use equation (7) to derive
straintsaresatisfiedbymostbarotropicfluidsofinterest,
′
althoughitisalsopossibleto constructmodelswhichvi- w =3(1+α)w(1+w). (11)
olate both assumptions (see, e.g., Ref. [14]). With these
This behavior is displayed in Fig. 1 for several values of
assumptions, equation (7) gives the limit
′
α. The α = 0 case saturates the upper bound on w ,
′
w <3w(1+w). (8) as it corresponds to the previously-noted p = constant
model, while the original Chaplygin gas model (α = 1)
This limit is shown in Fig. 1, where have used the same
′
gives w =6w(1+w).
scale as Ref. [17] for easy comparison. Note that this is
Another barotropic model is the wet dark fluid model
of Ref. [16], with equation of state
p=w∗(ρ−ρ∗), (12)
where w∗ and ρ∗ are constants. Again, it is straightfor-
ward to use equation (7) to derive the result
w′ =−3(1+w)(w∗ −w), (13)
which is displayed in Fig. 1 for several values of w∗.
Whilewehaveassumedthatthefluidcharacterizedby
′
w and w acts strictly as dark energy, all of the results
derived here can also be applied to unified dark energy
models,inwhichthebarotropicfluidservesasbothdark
matter and dark energy (indeed, this is one of the main
motivations for the Chaplygin gas model and its vari-
ants). However,inthecaseofunifieddarkenergy,w and
′
w refer to the entire fluid, not just to the dark energy
component.
FIG.1: Solidcurvegivestheupperboundonw′ asafunction III. QUINTESSENCE
ofwfornon-phantombarotropicmodelswithpositivesquared
sound speed. Dotted curves give w′ as a function of w for The equation of motion for the quintessence field φ is
Chaplygin gas dark energy with (top to bottom) α = 0.5,
α = 1, α = 2, while α = 0 lies on the solid curve. Dashed dV
curvesgivew′ asafunctionofwforthewetdarkfluidmodel φ¨+3Hφ˙+ dφ =0, (14)
with (top to bottom) w∗ =0.1, w∗ =0.2, w∗ =0.3.
where the Hubble parameter H is
the exact opposite of the limit derived in Ref. [17] for
a˙ ρ 1/2
“freezing” quintessence models: H = = , (15)
a 3
′
w >3w(1+w). (9) (cid:16) (cid:17)
andρisthetotaldensity. (WeassumeanΩ=1universe
Thus, barotropicfluids andfreezing quintessence models
and take 8πG = 1 throughout). Since we are interested
occupynon-overlappingregionsinthew−w′ plane. The
in the late-time evolution of the dark energy, we assume
boundinequation(8)issaturatedwhenc2 =dp/dρ=0,
s thatρ=ρM+ρφ,whereρM isthedensityofthematter,
whichcorrespondstothecasep=constant. Inthiscase, scaling as ρ ∝ a−3, and ρ is the scalar field energy
′ M φ
we have w =3w(1+w). Models of this kind are partic-
density, given by
ularly interesting, since they correspond to a barotropic
fluid that behaves exactly like a mixture of dark mat- φ˙2
ter (dust) andacosmologicalconstant. Suchmodels can ρφ = +V(φ). (16)
2
be constructed in the context of either the generalized
Chaplygin gas [18] or k-essence [19]. The pressure of the scalar field is
Nowconsiderseveralillustrativespecialcases. Forthe
φ˙2
generalized Chaplygin gas, the equation of state is given p = −V(φ). (17)
φ
by 2
A Caldwell and Linder [17] examined models in which φ
p=− , (10)
ρα evolves toward the state V(φ) = 0. They distinguished
3
two cases: “thawing” models, in which w ≈−1 initially,
butw increasesasφrollsdownthepotential,and“freez-
ing”models,inwhichw >−1initially,butwapproaches
−1 as the field rolls down the potential. The bounds de-
rived in Ref. [17] are based on a combination of plau-
sibility arguments and numerical simulations. For the
“thawing” models, these limits are
′
1+w<w <3(1+w), (18)
while for the “freezing” models,
′
3w(1+w)<w <0.2w(1+w). (19)
Asnotedintheprevioussection,theallowedregionfor
barotropic fluids in the w−w′ plane lies adjacent to the
′
lower bound on w for quintessence models proposed in
Ref. [17]. Therefore, it is the lower bound in equation
(19) which is of greatest interest here.
We first generalize the Caldwell-Linder lower bound
to a wider range of potentials. Consider a scalar field
evolving in an arbitrary monotonic potential V(φ), not FIG.2: Solidcurvegivesthelowerboundonw′ asafunction
′ ofwfor“freezing”quintessencemodelsfromRef. [17]. Dotted
necessarily asymptotically zero. We will take V (φ)<0,
curve gives our lower bound on w′ as a function of w for
but of course the argument is identical in the opposite
quintessence models with an arbitrary monotonic potential
case. Now, following Steinhardt et al. [6], we note that (equation 23). Dashed curve gives our lower bound on w′ as
the equation of motion for φ can be rewritten as
afunctionofwfortrackerquintessencemodels(equation31).
′ ′
V 3(1+w) X
− = 1+ , (20)
V s Ωφ (cid:20) 6 (cid:21) constant when ρφ ≪ ρM, but which evolve toward w =
−1 when ρ ≈ ρ . Again, following Ref. [6], we define
φ M
where X is defined as Γ to be given by
1+w
X =ln , (21) V′′V
1−w Γ≡ . (24)
(cid:18) (cid:19) V′2
′
and X is the derivative of X with respect to ln(a), so
that X′ and w′ are related via If wB is the equation of state parameter for the back-
groundfluid,thentrackingbehaviorwithw <w occurs
B
′
′ 2w when Γ is roughly constant with Γ > 1 [6]. Note that Γ
X = . (22)
(1−w)(1+w) is exactlyconstantfor powerlawandexponentialpoten-
tials. In particular, for the potential V ∝ φ−α, we have
Now we note that for a scalar field rolling downhill in Γ=1+1/α. Takingthe derivativeofequation(20)with
a monotonically decreasing potential, the left-hand side respect to φ gives
ofequation(20)isalwayspositive. This,inturn,implies
that 1+X′/6 > 0, which translates into the following 2X′′ 1−w X′
′ Γ = 1− −
bound on w : (1+w)(6+X′)2 2(1+w)6+X′
w′ >−3(1−w)(1+w). (23) 3(wB −w)1−Ωφ
+ . (25)
1+w 6+X′
While not as tight as the Caldwell-Linder bound, equa-
tion (23) applies to a more generalclass ofmodels; it as- Note that this equation differs from the corresponding
sumes nothing about tracking behavior or “freezing” of equation in Ref. [6] because we do not take the limit
the scalar field, nor is it confined to potentials for which Ωφ → 0; all of the interesting evolution occurs when ρφ
V(φ) → 0 asymptotically. This bound is saturated by becomes significant compared to ρM. It does however,
a scalar field rolling on the constant potential V = V0. agreewiththe expressionpreviouslyderivedinRef. [21],
For this case, w evolves from w ≈ +1 to w ≈ −1, with and in the limit where Ωφ →0, equation (25) reduces to
w′ =−3(1−w)(1+w)[20]. Theboundinequation(23), the corresponding equation in Ref. [6].
′
along with the lower bound on w from Ref. [17], are When the tracker solution is reached, and the back-
shown in Fig. 2. ground fluid dominates, w is a constant. We will be
We can tighten this bound considerably for the case interested in the late-time evolution of the quintessence
of “tracker” models, i.e., models for which w is initially field,soweassumethatthebackgroundfluidiscolddark
4
′
matter,withw =0. Thenforthe trackersolution,tak- on X . We first note that Ω <1; applying this limit
B min φ
′′ ′
ing X = X = 0 and Ω = 0 in equation (25) gives (with w <0) to equation (27) yields
φ
2(Γ−1)(1+w)
w = −2(Γ−1) (26) Xm′ in >−6(1−w)+2(Γ−1)(1+w). (28)
2(Γ−1)+1
The right-hand side of equation (28) increases with de-
In this regime, Ωφ increases with time; when Ωφ is no creasing w, so a lower bound on the right-hand side is
longernegligiblewithrespectto1,thetrackersolutionno achieved by assigning the maximum allowed value to w.
longerapplies. Atthispoint,wdecreases,asymptotically Intrackermodels,wdecreasesfromtheinitialvaluegiven
approaching w = −1. (See Ref. [22] for a discussion of by equation (26), so substituting the value for w given
the evolution in this regime). by equation (26) into equation (28) gives a lower bound
Since w′ ≤0 for these trackersolutions,we have X′ ≤ on X′ :
′ min
0. Initially, X = 0, but as Ω becomes nonnegligible,
φ
X′ decreases. It reaches some minimum value, X′ , 12(Γ−1)
min X′ >− . (29)
but then increases back to a value near zero as Ωφ → min 6(Γ−1)+1
1. This behavior is illustrated in Fig. 3, in which we
pshoowwert-hlaewevpooltuetnitoinalfso.rTXhe′ avsalauefuonfcXti′on oifsΩthφefkoerysteoveoruarl fTraokminXg′Xto′ >w′X, wm′eino,batnaidnutshienglimeqituation (22) to convert
min
6(Γ−1)
w′ >− (1−w)(1+w). (30)
6(Γ−1)+1
This is obviously a model-dependent result, since it de-
pends on the value of Γ for which tracking behavior
is achieved. However, we can use it to derive a more
general limit by noting that, since Γ > 1, we have
6(Γ−1)/[6(Γ−1)+1]<1, giving
w′ >−(1−w)(1+w). (31)
′
Equation(31)givesalowerboundonw fortrackermod-
els. This boundisshowninFig. 2. Thisis clearlyavery
conservative limit; for example, it is obvious from Fig. 3
′
that the value of X is often obtained for a value of
min
Ω largerthanis consistentwith observations. However,
φ
equation(31)doesgiveatighterboundthanthatderived
inRef. [17],furtherseparatingquintessencemodelsfrom
thebarotropicfluidmodelsdiscussedintheprevioussec-
tion.
FIG. 3: The evolution of X′ as a function of Ω for the po-
φ
tential V(φ)∝ φ−α with α= 2 (dashed curve), α=1 (solid
curve) and α=0.2 (dotted curve). IV. DISCUSSION
calculation, since it gives a lower bound on w′ through Ourresultsillustratetheusefulnessofthew−w′ plane
equation (22). as a means for observationally distinguishing dark en-
To find this minimum value for X′, we simply take ergymodels. Non-phantombarotropicfluid models with
X′′ = 0 in equation (25), where we assume also that positive squaredsoundspeedand “tracker”quintessence
w =0 and that Γ is nearly constant. This gives models occupy disjoint regions in this plane. Even our
B
less stringent general limit on quintessence models with
(1−Ω )w+2(Γ−1)(1+w) monotonic potentials is inconsistent with a wide class of
X′ =−6 φ . (27)
min (1−w)+2(Γ−1)(1+w) barotropic fluid models, as can be seen by comparing
Figs. 1 and 2.
On the right-hand side of this equation, w, Ω , and Γ SupernovaIameasurements[23,24]havenarrowedthe
φ
′ ′
are to be evaluated at the point at which X = X . range of possible values for w; in particular, they are
min
In general Ω and w evolve in a complex fashion (see, consistent with w ≈ −1. The value of w′ is less well
φ
e.g., Ref. [22]), so equation (27) does not have a simple constrained(see, e.g., Ref. [24]). The results ofRef. [17]
′ ′
solution. However, since we seek a lower bound on X , indicatethattheresolutiononameasurementofw must
it is sufficient to use equation(27) to find a lower bound be of order 1+w in order to distinguish “freezing” and
5
“thawing” quintessence models. We reach a similar con- where Ω is the fraction of the total energy density
DE
clusionwithregardto barotropicfluids andquintessence contributed by the dark energy. Note, however, that the
′
models: observationallydistinguishingbetweenthesetwo mappingbetweenw, w andr, sis non-trivialbecauseof
classes of models also requires that σ(w′).1+w. the factor of Ω in equation (34). For non-constantw,
DE
Finally, we note that there is a relation between the the evolution of Ω depends in a complicated way on
DE
w−w′ parametrizationofRef. [17]andthe“statefinder” both a and w(a). Hence, the statefinder parameters en-
pairintroducedinRef. [25]. Thestatefindersr andsare code somewhat different information about the behavior
′
defined by of the dark energy than do w and w . The statefinder
... parametersaremoredirectlyrelatedtoobservablequan-
r = a , (32) tities, while the w−w′ parametrization describes more
aH3 directly the physical properties of the dark energy.
r−1
s = , (33)
3(q−1/2)
Acknowledgments
where q = −aa¨/a˙2 is the deceleration parameter. Then
′
r and s are related to w and w via [25]
IthankA.A.Sen,R.Caldwell,andE.Linderforhelp-
9 3 ful discussions. I am grateful to E. Babichev and C.
r = 1+ ΩDEw(1+w)− ΩDEw′, (34) Rubano for helpful comments on the manuscript. R.J.S.
2 2
′ wassupportedinpartbytheDepartmentofEnergy(DE-
w
s = 1+w− , (35) FG05-85ER40226).
3w
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