Table Of ContentCrossing the Phantom Divide
Martin Kunz∗ and Domenico Sapone†
D´epartement de Physique Th´eorique, Universit´e de Gen`eve,
24 quai Ernest Ansermet, CH–1211 Gen`eve 4, Switzerland
(Dated: January 27, 2007)
Weconsiderfluidperturbationsclosetothe“phantomdivide”characterisedbyp=−ρanddiscuss
the conditions under which divergencies in the perturbations can be avoided. We find that the
behaviour of the perturbations depends crucially on the prescription for the pressure perturbation
δp. Thepressureperturbationisusuallydefinedusingthedarkenergyrest-frame,butweshowthat
thisframebecomesunphysicalatthedivide. Ifthepressureperturbationiskeptfiniteinanyother
frame, then the phantom divide can be crossed. Our findings are important for generalised fluid
darkenergyusedindataanalysis(sincecurrentcosmologicaldatasetsindicatethatthedarkenergy
is characterised by p ≈−ρ so that p<−ρ cannot be excluded) as well as for any models crossing
7
the phantom divide, like some modified gravity, coupled dark energy and braneworld models. We
0
also illustrate theresults by an explicit calculation for the“Quintom” case with two scalar fields.
0
2
PACSnumbers: 98.80.-k;95.36.+x
n
a
J I. INTRODUCTION limitation that prevents an effective fluid from crossing
7 the phantom divide.
2 The paper is organised as follows: We start with a
The discoveryby the supernovasurveys[1, 2] that the
short recapitulation of the first order perturbations in
2 expansion of the universe is currently accelerating came fluids,whichalsoservestodefineournotation. Insection
v as a greatsurprise to cosmologists. Within the standard
III we study the behaviour of barotropic fluids close to
0 cosmological framework of a nearly isotropic and homo-
p = ρ, concluding that this class of fluids cannot cross
4 geneous universe and an evolution described by General −
0 the phantom divide self-consistently. We allow for non-
Relativity, this behaviour requires a component with a
9 adiabatic perturbations in section IV and show that the
negative pressure p< ρ/3, commonly dubbed dark en-
0 − phantom divide can now be crossed as long as we define
6 ergy. However, it is very difficult to understand why the the pressure perturbationin a frame that stays physical.
0 dark energy should appear at such a low energy scale,
WethenillustratetheresultswiththeQuintommodelof
/ and why it should start to dominate the overall energy
h two scalar fields before presenting our conclusions. The
density just now. Explaining its nature is correspond-
p appendices finally discuss the calculation of the effective
- ingly regardedas one ofthe most importantproblems in Quintom perturbations in more detail.
o observational cosmology.
r
t Current limits on the equation of state parameter
s
a w =p/ρofthedarkenergyseemtoindicatethatp ρ II. FIRST ORDER PERTURBATIONS
: [3,4],sometimeseventhatp< ρ[5],oftencalledp≈ha−n-
v
−
i tom energy[6]. Althoughthereisnoproblemtoconsider In this paper we use overdots to refer to derivatives
X
w < 1 for the background evolution, there are appar- with respect to conformal time η which is related to the
−
r ent divergencies appearing in the perturbations when a physical time t by dt = adη. We will denote the phys-
a
model tries to cross the “phantom divide” w = 1[7]. ical Hubble parameter with H and with the confor-
− H
Eventhough this regionmay be unphysicalat the quan- malHubble parameter. For simplicity, we considera flat
tum level [8, 9], it is still important to be able to probe universe containing only (cold dark) matter and a dark
it, not least to test for alternative theories of gravity or energy fluid, so that the Hubble parameter is given by
higher dimensional models which can give rise to an ef-
fective phantom energy [10, 11, 12, 13, 14]. It would 1da 2
H2 = =H2 Ω a−3+(1 Ω )f(a) (1)
certainly be unwise to build a strong bias like w 1 (cid:18)a dt(cid:19) 0 m − m
≥ − (cid:2) (cid:3)
intoouranalysistoolsaslongasexperimentsdonotrule
it out. In this paper we consider the evolution of the where f(a) = exp 3 a 1+w(u)du , implying that the
perturbations for models where w crosses 1. We find scale factor today his−aR1= 1.u We wiill assume that the
− 0
that inmany realistic casesthe divergenciesareonly ap-
universe is filled with perfect fluids only, so that the en-
parent and can be avoided. At the level of cosmological
ergy momentum tensor of each component is given by
first order perturbation theory, there is no fundamental
Tµν =(ρ+p)uµuν +p gµν. (2)
where ρ and p are the density and the pressure of the
∗Electronicaddress: [email protected] fluid respectively and uµ is the four-velocity. This is po-
†Electronicaddress: [email protected] tentially a strong assumption for the dark energy, but a
2
more general model should lead to more freedom in the Asw 1thetermscontaining1/(1+w)willgenerally
→−
dark energy evolution, not less. diverge. This can be avoided by replacing θ with a new
Wewillconsiderlinearperturbationsaboutaspatially- variable V defined via V = (1+w)θ. This corresponds
flat backgroundmodel, defined by the line of element: to rewriting the 0i component of the energy momentum
tensor as ik Tj = ρ¯V which avoids problems if Tj =
ds2 = a2 (1+2A)dη2+2Bidηdxi+ 0 when p¯ =j 0ρ¯. Replacing the time derivatives b0y6a
− −
+((cid:2)(1+2H )δ +2H )dx dxj (3) derivative with respect to the scale factor a (denoted by
L ij Tij i
a prime), we obtain:
(cid:3)
where A is the scalar potential; B a vector shift; H
i L
is the scalar perturbation to the spatial curvature; Hij V 1 δp
T δ′ = 3(1+w)ψ′ 3 wδ (14)
is the trace-free distortion to the spatial metric, see e.g. − Ha2 − a(cid:18) ρ¯ − (cid:19)
[15] for more details.
V k2 δp k2
Thecomponentsoftheperturbedenergymomentten- V′ = (1 3w) + +(1+w) ψ.(15)
− − a Ha2 ρ¯ Ha2
sor can be written as:
T0 = (ρ¯+δρ) (4) Inthisformeverythinglooksperfectlyfiniteevenatw =
0 − 1. But in order to close the system, we need to give
T0 = (ρ¯+p¯)(v B ) (5) −
j j − j an expression for the pressure perturbations. A priori
Ti = (ρ¯+p¯)vi (6) this is a free choice which will describe some physical
0
Ti = (p¯+δp)δi +p¯Πi. (7) properties of the fluid. In the following we will consider
j j j
several possible choices and discuss how they influence
Here ρ¯and p¯are the energy density and pressure of the the behaviour of the fluid at the phantom divide. We
homogeneousandisotropicbackgrounduniverse,δρisthe will see how the specification of δp determines if a fluid
density perturbation, δp is the pressure perturbation, vi can cross the divide or not.
is the vector velocity and Πi is the anisotropic stress Inadditiontothefluidperturbationequationsweneed
j
perturbation tensor [15]. to add the equation for the gravitational potential ψ. If
We want to investigate only the scalar modes of the thereareseveralfluidspresent,thentheevolutionofeach
perturbationequations. WechoosetheNewtoniangauge of them will be governed by their own set of equations
(also known as the longitudinal gauge) which is very fortheirmattervariables δ ,V ,linkedbyacommonψ
i i
{ }
simple for scalar perturbations because they are char- (whichreceivescontributionsfromallthefluids)[29]. We
acterised by two scalar potentials ψ and φ; the metric use [16],
Eq. (3) becomes:
1 δ ρ¯ k2 ψ
ds2 =a2 (1+2ψ)dη2+(1 2φ)dx dxi (8) ψ′ = i i i +1 . (16)
(cid:2)− − i (cid:3) −2a(cid:18)P iρ¯i (cid:19)−(cid:18)3H2a2 (cid:19) a
where we have set the shift vector B = 0 and Hij = P
i T The evolution of a fluid is therefore governed by
0. The advantage of using the Newtonian gauge is that
Eqs. (14) and (15), supplemented by a prescription for
the metric tensor g is diagonal and this simplifies the
µν the internal physics (given by δp) and the external
calculations [16].
physics (through ψ and ψ′). There are two points worth
Theenergy-momentumtensorcomponentsintheNew-
emphasising. Firstly,thedarkenergyfluidperturbations
tonian gauge become:
cannot be self-consistently set to zero if w = 1. Even
6 −
T0 = (ρ¯+δρ) (9) if δ = V = 0 on some initial hypersurface, it is unavoid-
0 − ablethatperturbationswillbegeneratedbythepresence
ik Ti = ik T0 =(ρ¯+p¯)θ (10)
i 0 − i i of ψ. As this function describes physics external to the
Ti = (p¯+δp)δi (11) fluid, it cannot be controlled directly. Conceivably δp
j j
could be chosen to cancel the external source of either δ
where we have defined the variable θ =ikjvj which rep- or V, but not of both. Even then such a δp would have
resents the divergence of the velocity field and we have to be incredibly fine-tuned as ψ and ψ′ depend on the
alsoassumedthattheanisotropicstressvanishes,Πi =0 evolution of the other fluids as well.
j
(implying φ=ψ). Secondly, we see that as w 1 the external sources
The perturbation equations are [16]: areturnedoff. Afluidcanther→efo−remimicacosmological
constant as then (and only then) δ = V = δp = 0 is
a˙ δp
δ˙ = (1+w) θ 3ψ˙ 3 wδ (12) a solution. This corresponds to an energy momentum
− (cid:16) − (cid:17)− a(cid:18) ρ¯ − (cid:19) tensor (2) with Tµν = p¯gµν, so that Λ ρ¯. However, in
∼
a˙ w˙ general the perturbations do not vanish even if w = 1.
θ˙ = (1 3w)θ θ+ −
−a − − 1+w A perfect fluid is therefore in general not a cosmological
δp/ρ¯ constant even if ρ¯= p¯.
+k2 +k2ψ. (13) −
To calculate perturbations in different gaugeswe need
1+w
3
0.6
to introduce the coordinate transformation:
ρ
η = η˜+T (17) -p
xi = x˜i+Li. (18) 0.5
the gaugetransformationofthe mattervariablesis then:
0.4
a˙
δ˜ρ = δρ+3 (1+w)ρ¯T (19)
a
δ˜p = δp +3a˙ (1+w)c2aT (20) 0.3
p¯ p¯ a w
u˜ = u+L˙ (21)
0.2
wherewehaveintroducedanewquantityc2 =p¯˙/ρ¯˙,called
a
adiabatic sound speed. 0 0.2 0.4 0.6 0.8 1
a
III. BAROTROPIC FLUIDS FIG. 1: Energy density (black solid line) and pressure (red
dashed line) as functions of thescale factor. w crosses −1 at
We define a fluid to be barotropicif the pressurep de- a× =0.5, sothatρisminimalatthispointwhile pdecreases
monotonically there(but has a maximum earlier).
pends strictly only on the energy density ρ: p = p(ρ).
These fluids have only adiabatic perturbations, so that
they are often called adiabatic. We can write their pres- so that dw/dN 3/2C2ρ¯ near the crossing. The full
×
sure as ≃−
solution (with N =N at w= 1) is
×
−
dp
p(ρ)=p(ρ¯+δρ)=p(ρ¯)+ dρ(cid:12)(cid:12)ρ¯δρ+O(cid:0)(δρ)2(cid:1). (22) w(N)= 1 C√ρ¯×tan 3C√ρ¯×(N N×) , (26)
(cid:12) − − (cid:18) 2 − (cid:19)
Here p(ρ¯) = p¯ is the pressure o(cid:12)f the isotropic and ho-
mogeneous part of the fluid. Introducing N lna as a clearly not a realistic solution for our universe as there
≡
newtimevariable,wecanrewritethebackgroundenergy are divergencies in the finite past and future. Of course
conservation equation, ρ¯˙ = 3 (ρ¯+p¯) in terms of w, we could modify the example (24), but we are mostly
− H
interestedatthebehaviournearthecrossing. Inourcase
dw dρ¯ dw dw
= = 3(1+w)ρ¯ . (23) 1+w(a) (a a×), ie. we observea linear behaviour.
dN dN dρ¯ − dρ¯ This is ac∝tu−ally−the limiting case: if we choose(1+w)
∝
(ρ¯ ρ¯ )ν close to the crossing, we find
We see that the rate of change of w slows down as w ×
−
→
1,andw = 1isnotreachedinfinite time [17],except
− − d(1+w)
ifdw/dρ¯diverges(orρ¯,butthatwouldleadtoasingular C(1+w)(2ν−1)/ν (27)
dN ≃−
cosmology). The physicalreasonis that we demand p to
be a unique function of ρ, but at w = 1 we find that
for small 1+ w. Correspondingly we cross with zero
−
ρ¯˙ = 0. If the fluid crosses w = 1 the energy density | |
slope for ν > 1/2 and with infinite slope for ν < 1/2.
−
ρ will first decrease and then increase again, while the
However, if dw/dN = 0 at w = 1 then the system will
pressure p will monotonically decrease (at least near the −
turn into a cosmologicalconstantandstay there forever.
crossing), Fig. 1. It is therefore impossible to maintain
On the other hand, this is unstable against small per-
a one-to-one relationship between p and ρ, see Fig. 2
turbations towards p¯ < ρ¯, so that the crossing might
(noticethatthemaximumofpandtheminimumofρ do −
eventually be completed anyway.
not coincide).
Let us have a closer look at the perturbations. The
Anexamplethatcanpotentiallycrossw = 1isgiven
− second term in the expansion (22) can be re-written as
by
(1+w(ρ))2 =C2(ρ¯ ρ¯×). (24) dp = p¯˙ =w w˙ c2 (28)
− dρ(cid:12) ρ¯˙ − 3 (1+w) ≡ a
Starting with w > 1 and ρ¯ > ρ¯× both will decrease (cid:12)(cid:12)ρ¯ H
− (cid:12)
until w = 1 and ρ¯ = ρ¯×. If it is possible to switch whereweusedtheequationofstateandtheconservation
−
from the branch with w > 1 to the other one at this equation for the dark energy density in the background.
−
point, then ρ¯can start to grow again while w continues We notice that the adiabatic sound speed c2 will neces-
a
to decrease. Using the evolution equation (23) for w, we sarily diverge for any fluid where w crosses 1.
find Foraperfectbarotropicfluidtheadiabatic−soundspeed
dw = 3 C2ρ¯×+(1+w)2 (25) pc2aerttuurrbnastioounts.toItbsehotuhledpthhyerseicfoarlepnroevpeargabteiolnarsgpereetdhaonf
dN −2
(cid:2) (cid:3)
4
-0.2
The pressure perturbation behaves like
αa
δp ρ¯ (a a )α−1+ ((a a )α) . (33)
-0.22 ∝ (cid:16) 3 − × O − × (cid:17)
We need α 1, otherwise the pressure perturbation will
≥
diverge as a power-law at the crossing. Unfortunately
p-0.24
this condition is not sufficient. Looking at the second
perturbation equation (15) we see that although noth-
-0.26 ing diverges, there is no reason for V to go to zero at
the crossing,and in generalit will not, except if we fine-
tune it with infinite precision. As δ vanishes at crossing
-0.28 andcansopotentiallycancelthedivergenceinthesound
0.22 0.24 0.26 0.28 0.3 0.32 speed, this term is no longer necessarilydominant in the
ρ
differential equation for δ, Eq. (31). We also need to
takeintoaccounttheothercontributions. Thetermcon-
FIG. 2: The pressure as function of the energy density. The taining ψ′ is sufficiently suppressed by the 1+w factor
graph of p(ρ) shows that p is not a single valued function of to neglect it. Taking V to be constant at a (to lowest
×
the energy density, and that there is a point with an infinite order) V(a )=V , we find:
× ×
slope(correspondingtotheminimumofρandthedivergence
othfec2am).axTimheumpoionftpwahnedreapv(aρn)ihshaisnagzc2aer.o slope corresponds to δ′ =−C×+ 1+w′wδ (34)
withC =V /(H(a )a2). Thesolutiontothisequation
× × × ×
the speed of light, otherwise our theory becomes acausal is:
[18, 19]. The condition c2 1 for (1+w)>0 (our point
of departure) is equivalenat≤to δ =(cid:26)δ(a×(−aa×a)×(δ)×α−+C(a×loag×|a)C−×a/×(1|) α) αα==11 (35)
− − − 6
dw
3(1+w)(1 w). (29) Wenoticethatforα 1thedensityperturbationsvanish
− dN ≤ − ≥
at crossing. However, for α = 1 they do not vanish fast
No barotropic fluid can therefore pass through w = 1 enough. Instead of the behaviour given in Eq. (33), the
±
from above without violating causality. Even worse, the pressure perturbation exhibits now a logarithmic diver-
pressure perturbation gence since the sound speed cancels the factor (a a ).
×
−
Even though our derivation here is not rigorous, numer-
w˙
δp=c2δρ= w δρ (30) icalcalculations confirm this behaviour,also if the back-
a (cid:18) − 3 (1+w)(cid:19) ground is not completely matter dominated, see Fig. 3.
H
Although a full solution may be possible in some cases,
will necessarily diverge if w crosses 1 and δρ = 0. Us-
− 6 it would turn out to be a special function, obscuring the
ing the gauge transformation Eqs. (19) and (20) we see
structure of the result while still showing essentially the
that the relation between pressure and energy density
same simple behaviour.
perturbations is gauge invariant. If the pressure pertur-
Wefindthereforethreepossiblebehavioursfortheper-
bationdivergesinoneframe,itwilldivergeinallframes.
turbations,depending onhoww crossesthe phantomdi-
The only possible way out would be to force δρ 0 fast
→ vide:
enoughatthe crossing. Let us study the behaviourclose
to w = 1 in some detail now: 1. α < 1: In this case w crosses 1 with an infi-
First−welookonlyatthedominantcontributiontothe nite slope,leadingtoapower-law−divergenceofthe
right hand side of Eq. (14). Near w = 1 this is clearly: pressure perturbation.
−
3 w′ 2. α = 1: w crosses 1 with a finite, but non-zero
δ′ = c2δ =+ δ. (31) −
−a a 1+w slope. δρvanishes,V staysfiniteandgenerallynon-
zero and δp diverges logarithmically.
Assuming that near the crossing the equation of state
behaves like 1+w λ(a a )α with α>0 we find the 3. α>1: w crosses 1withazeroslope. δρ vanishes,
×
≈ − −
solution: V and δp stay finite and generally non-zero.
δ (a a )α (32) The only acceptable case is the last one, but as dis-
×
∝ − cussed at the beginning of this section, we expect the
(independentofλwhichdropsoutoftheequation). This system to get stuck at w = 1 and to never cross.
−
solution goes to zero at the crossing. If the sign had Barotropic perfect fluids therefore fail to cross either at
been different, we would have found instead the solution the backgroundorperturbationlevel. Evenifthe fluctu-
δ (a a )−α which diverges. ations can lead to a crossing with zero slope due to the
×
∝ −
5
the pressure perturbation. However, there are two rel-
evant variables that we could compare δp to, the fluid
velocityV andtheperturbationintheenergydensityδρ.
Clearlyit wouldbe counterproductiveto replacea single
free function by two free functions, and it would lead to
degeneracies between the two. Another problem is that
the perturbation variables depend on the gauge choice.
Butinthiscasethetwoproblemscanceleachother,lead-
ing to a simpler solution: Going to the rest-frame of the
fluid both fixes the gauge and renders the fluid velocity
physically irrelevant, so that we can now write [20]:
δpˆ=cˆ2δρˆ, (36)
s
where a hat denotes quantities in the rest-frame. The
physical interpretation is that cˆ2(k,t) is the speed with
s
which fluctuations in the fluid propagate, ie. the sound
speed. Again, some physical models lead to simple pre-
scriptions for the sound speed. The barotropic models
discussed in the last section have cˆ2(k,t) = c2(t), and
s a
the perturbations in a scalar field correspond (to linear
order) exactly to those of a fluid with cˆ2(k,t)=1.
s
FIG. 3: This figure shows the logarithmic divergence of Therest-frameischosensothattheenergy-momentum
δ/(a−a×) near a× = 0.5. The black curve is the numeri- tensor looks diagonal to an observer in this frame. In
cal solution while the red dashed line shows the approxima-
terms of the gauge transformations this amounts to
tion given by Eq. (35) for α = 1. The approximate formula
choosing B = θ . However, we notice immediately a
describesthestructureofthedivergencequitewell. Thepres- DE
potential flaw in this prescription close to w = 1: The
sureperturbationδpexhibitsaverysimilardivergenceinthis −
off-diagonal entries of the energy-momentum tensor are
case.
actually(ρ¯+p¯)θsothatdemandingθ B =0isastronger
−
conditionthanrequired. Inotherwords,theconditionto
instability of the cosmological constant solution in some beatrestwithrespecttotheflowofρ¯+p¯cannotbemain-
cases, the model is still not realistic. The perturbations tained at ρ¯+p¯= 0. However, as ρ¯> 0, we could define
seem to propagate acausally, and generically after the insteadthesoundspeedinaframewherethereisnoflow
transition c2 < 0 for a period, leading to classical insta- of energy density.
a
bilities with exponential growthof the perturbations. In To see this, let us calculate the pressure perturbation
thenextsectionwewillrelaxthe barotropicassumption, defined by Eq. (36) in the conformal Newtonian frame,
whichallowsfor entropyperturbations that canstabilise following [20]: Breaking the single link between ρ and
the system and keep the sound speed below the speed of p amounts to the introduction of entropy perturbations.
light. A gauge invariant entropy perturbation variable is Γ
δp c2aδρ [20, 21]. By using ≡
p¯ − w ρ¯
IV. NON-ADIABATIC FLUIDS p˙ δp δρ
wΓ= (37)
ρ(cid:18) p˙ − ρ˙ (cid:19)
The discussion of the barotropic fluid shows that we
have to violate the constraint that p be a function of ρ andtheexpressionforthegaugetransformationofδρ[21],
alone. At the level of first order perturbation theory,
V
this amounts to changing the prescription for δp which δρˆ=δρ+3 ρ¯ (38)
now becomes an arbitrary function of k and t. This H k2
problem is conceptually similar to choosing the back-
we find that the pressure perturbation is given by
ground pressure p¯(t), where the conventional solution is
to compare the pressure with the energy density by set- V
ting p¯(t) = w(t)ρ¯(t). In this way we avoid having to δp=cˆ2sδρ+3H cˆ2s−c2a ρ¯k2. (39)
deal with a dimensionfull quantity and can instead set (cid:0) (cid:1)
w, which has no units (up to a factor of the speed of Asc2 atthe crossing,itisimpossiblethatallother
a →∞
light)andsoisgenericallyoforderunity andoften hasa variables stay finite except if V 0 fast enough. Again
→
simpleform,likew=1/3foraradiationfluidorw = 1 wewillshowthatthisisnotingeneralthecase,exceptif
−
for a cosmologicalconstant. cˆ2 0 or w′ 0 at crossing. However, we will then see
s → →
It certainly makes sense to try a similar approach for thatthisisnotrequired,indeedwewillargueinthenext
6
sectionthatthe more genericsolutionis to let cˆ2 diverge for the pressure perturbations has singled out the one
s
at the crossing in order to cancel the divergence of c2! frame which we cannot use for fluids crossing the phan-
a
To this end, we consider again the structure of the tomdivide. The reasonis thatthe gaugetransformation
perturbation equations near w = 1. Inserting the ex- relatingthepressureperturbationsinthedifferentgauges
−
pressionforδpintooursystemofperturbationequations is:
we find
V
δ′ = 3(1+w)ψ′ k2 +3H(cˆ2 c2) V δpˆ=δp+3Hρ¯c2ak2. (44)
−(cid:18)Ha2 s− a (cid:19)k2
If V does not vanish fast enough at the crossing then
3
cˆ2 w δ (40) the pressure perturbation has to diverge in at least one
−a s−
(cid:0) (cid:1) frame. As we have just discussed, the dark-energy rest-
V′ 1+3(c2 w cˆ2) V
= a− − s + framebecomesunphysicalatcrossing. Itisclearlybetter
k2 − a k2 to specify a finite pressure perturbation in a different
1+w cˆ2 frame.
+ ψ+ s δ. (41)
Ha2 Ha2 Oneproblemistofindawayofcharacterisingthepres-
sure perturbations in a physical way – in the Quintom
Thespoilerhereisthecontinuedpresenceofc2 whichwe
a example of the next section, we find that the additional
knowtodivergeatcrossing. Letusstartbyconsideringa
contributions diverge but in such a way that we end up
finite andconstantcˆ2. Forallreasonablechoicesofk the
s with a finite result for δp. As another example, we just
dominanttermintheV equationwillbetheonecontain-
choose δp proportionalto δρ,
ing c2. We find that this time the velocity perturbation
a
is driven to zero at crossing:
δp(k,t)=γδρ(k,t) (45)
w′
V′ = V. (42) in the conformal Newtonian gauge. This will work as
1+w
long as δρ = 0, otherwise it forces δp to vanish in the
6
Proceeding as in the last section, we find that now V same place as δρ, which is not general enough. If we
(a−a×)α,cancellingthedivergenceinc2a forα≥1. No∝w insert this expression into eqs. (14) and (15), we obtain:
δρ will in general not vanish at crossing, and we have to
include that term in the differential equation for V. The V 1
δ′ = 3(1+w)ψ′ 3 (γ w)δ (46)
termwithψ issuppressedby1+wandisofhigherorder. − Ha2 − a −
As in the last section for δ, the solutions for V are now V k2 k2
V′ = (1 3w) + γδ+(1+w) ψ.(47)
to lowest order: − − a Ha2 Ha2
(a a )(V D log a a ) α=1
V = − × ×− × | − ×| As w 1 none of the terms diverge, so that δ and V
(cid:26)V×(a−a×)α+(a−a×)D×/(1−α) α6=1 stay in→ge−neral finite and non-zero at crossing. We show
(43)
in Figs. 4 and 5 a numerical example for two choices of
with D being cˆ2k2δ/(Ha2), evaluated at crossing.
× s γ where it is impossible to see that the phantom divide
Againtherearenodivergencesappearingintheenergy
has been crossed at a =0.5.
×
momentumtensoronlyifα>1,ie. ifwcrosses 1witha It is ofcourse possible to expressγ in terms of cˆ2, and
zeroslope. Apossible wayto getaroundthis fi−ne-tuning cˆ2 in terms of γ. Using the expression for the prsessure
is to demand that cˆ2 = 0 at crossing, as in this case s
s perturbationin the rest-framegivenby Eq.(36) and the
the logarithmically divergent term disappears. We also
gauge transformation given by Eqs. (19) and (20), we
noticethattheusualvelocityperturbationθ V/(1+w)
≡ obtain
doesdivergeinallcases,eitherlogarithmicallyifα=1or
as(a a )1−α ifα>1. Foranobserverintherest-frame V V
− × γδρ+3 c2ρ¯ =cˆ2 δρ+3 ρ¯ . (48)
where B = θ this means that the metric perturbations H a k2 s(cid:18) H k2(cid:19)
become large – at crossing the metric is even singular.
At the very least, perturbation theory is no longer valid In general γ will therefore be scale dependent even if
for such an observer. cˆ2 is not (even though of course cˆ2 will also in general
s s
Another way to see that the rest-frame is ill-defined is dependonk andt),orviceversa. Also,toreproducethe
to look at the energy momentum tensor (2). For p= ρ evolution with finite γ shown in Figs. 4 and 5 we would
the first term disappears, leaving us with Tµν = p g−µν. havetosubstituteadivergentcˆ2 tocancelthedivergence
s
Normallythefour-velocityuµ isthetime-likeeigenvector in c2 (cf. figure 7 which shows how the apparent sound
a
of the energy momentum tensor, but now suddenly all speed diverges in the Quintom example). We also notice
vectors are eigenvectors. The problem of fixing a unique that on very small scales where k 1 we find γ cˆ2 for
≫ ≈ s
rest-frame is therefore no longer well-posed. finite c2, which is the usual result that on small scales
a
However, by construction the pressure perturbation gaugedifferencesbecomeirrelevant(forphysicalgauges).
looks perfectly fine for precisely the observer in the rest- Finally we would like to emphasise again that for a
frame, as δpˆ = cˆ2δρˆ does not diverge. Our prescription general,finite pressureperturbation in any gaugeexcept
s
7
FIG. 4: This figure shows the density contrast δDE (red and FIG.5: Thegravitationalpotentialψ inthesamescenarioas
blue lines, negative for small scale factor a) and the velocity Fig. 4(withγ =0.2for theblack solid lineand γ =1forthe
perturbationVDE (cyanandmagentalines, positiveforsmall red dashed line). Again, the crossing of the phantom divide
a)foradarkenergycomponentwith apressureperturbation at a× =0.5 (dotted green vertical line) is not apparent. The
givenintheconformalNewtoniangaugethroughγ(asdefined gravitational potential is constant during matter domination
in Eq.(45)). The solid lines show the results for γ =0.2 and (small a) and starts to decay as the dark energy begins to
thedashedlinesforγ =1.Thecrossingofthephantomdivide dominate and the expansion rate of the universe accelerates.
at a× = 0.5 (dotted green vertical line) is not apparent in Thecontributionofthedarkenergyperturbationstoψisvery
these figures, everythingis finite. small (∼<0.1%) so that there is no visible difference for the
two choices of γ.
the fluid rest-frame, there is no problem with the per-
two fluid energy momentum tensors. This leads to:
turbation evolution across the phantom divide. Also, in
general the perturbations, including V, will not vanish
ρ¯ = ρ¯ +ρ¯ (49)
eff 1 2
at crossing.
p¯ = p¯ +p¯ (50)
eff 1 2
w ρ¯ +w ρ¯
1 1 2 2
w = (51)
eff
ρ¯ +ρ¯
1 2
ρ¯ δ +ρ¯ δ
1 1 2 2
δ = (52)
V. THE QUINTOM MODEL AS EXPLICIT eff ρ¯ +ρ¯
1 2
EXAMPLE
(1+w )ρ¯ θ +(1+w )ρ¯ θ
1 1 1 2 2 2
θ = . (53)
eff
(1+w )ρ¯ +(1+w )ρ¯
1 1 2 2
To clarify some of the above points we consider an
explicitexampleofamodelcrossingthephantomdivide, Re-expressingtheperturbationequationsinthesevari-
the Quintom model [22, 23], and compute the pressure ableswefindpreciselythenormalperturbationequations
perturbation ab initio at crossing (see also [24]). for a single fluid (14) and (15) with p¯ and δp re-
eff eff
placing p and δp. p¯ is simply given by w ρ¯ , but
The original Quintom model considered two scalar eff eff eff
the density perturbationis moreinvolved. Starting from
fields. For us it is advantageous to use instead two flu-
δp =δp +δp we can write it as
ids with a constant and equal rest-frame sound speed eff 1 2
cˆ2 = 1. At the level of first order perturbation theory
s δp = cˆ2 δρ +δp +δp
the two models are exactly equivalent, see appendix A. eff s,eff eff rel nad
As inthe originalmodel,weuse twofluids withconstant V
+3 cˆ2 c2 ρ¯ eff (54)
equations of state parameters w1 and w2. We will have H s,eff − a eff k2
(cid:0) (cid:1)
tomapthetwofluidsontoasingleeffectivefluid. Tothis
endwedefinetheeffectiveparametersinsuchawaythat where the (effective) adiabatic sound speed is as always
the effective energy momentum tensor is the sum of the c2 = p¯˙ /ρ¯˙ and where cˆ2 = 1 for two scalar fields.
a eff eff s,eff
8
The other terms are the relative pressure perturbation
(w w )ρ¯˙ ρ¯˙
2 1 1 2
δp = − S (55)
rel 3 ρ¯˙ 12
eff
H
given by the relative density perturbation of the two
scalar fields,
δ δ
1 2
S = (56)
12
1+w − 1+w
1 2
(which correspondsto a gauge invariantrelative entropy
perturbation[25]) as well as the non-adiabatic term
ρ¯˙ ρ¯˙ 3
1 2
δp = (w w ) S + H∆θ (57)
nad − 2− 1 3 ρ¯˙ (cid:20) 12 k2 12(cid:21)
eff
H
givenby the relative motion of the two scalar fields with
∆θ =θ θ (see appendix C).
12 1 2
−
We notice that the relative perturbations act as inter-
nal perturbations which couple to the effective variables
purely through the behaviour of the pressure perturba-
tion. Understanding the physics behind the dark energy
FIG.6: Thisfigureshowsthedifferentdivergentcontributions
in such a case will therefore require a precise measure-
tothepressureperturbation,Eq.(54),multipliedby109. The
mentofthepressureperturbationsaswellastheircareful
relativepressureperturbationisshownasreddottedline,the
analysis, to uncover the different internal contributions.
non-adiabaticpressureperturbationasgreendash-dottedline
Even though the effective sound speed cˆ2 = 1 in
s,eff and the contribution from the gauge transformation to the
Eq.(54) is finite inthe rest-frame,the transformationto conformal Newtonian frame as blue dashed line. Each of the
any other frame will lead to a divergence as discussed in contributionsdivergesatthephantomcrossing,buttheirsum
the previous section. This divergence then needs to be (shown as black solid line), and so δp,stays finite.
cancelled by the two additional terms, δp and δp .
rel nad
In our example, both diverge, cancelling together the di-
vergentcontributionfromthesingulargaugetransforma-
tion as well as their own divergencies, see Fig. 6. This
behaviour,whichlooksextremelyfine-tuned,isautomat-
icallyenforcedinthismodel. Suchacancellationmecha-
nism is requiredfor any model in order to cross w = 1.
−
We also see that although the effective sound speed
remains simply cˆ2 = 1, this is only true if we know that
s
thereareinternalrelativeandnon-adiabaticpressureper-
turbations (as well as their form). But in general we
would try to parametrise the pressure perturbation as:
V
δp =c2δρ +3 c2 c2 ρ¯ eff. (58)
eff x eff H x− a eff k2
(cid:0) (cid:1)
where the apparent sound speed c2 is now a mixture of
x
the real effective sound speed together with the relative
and the non-adiabatic pressure perturbations. In this
casewenolongerfindasimpleformforthesoundspeed.
InFig. 7weplotc2 asafunctionoftheequationofstate
x
parameterwforseveralwavevectorsk. Aspredicted,the
apparentrest-framesoundspeeddivergesatthecrossing.
In the Quintom case the effective perturbations do not FIG. 7: We plot the apparent sound speed c2x defined by
vanish at w = 1. Eq. (58) for three different wave vectors, k = 1/H0 (black
We think th−at the lessons learned from the Quin- solid line), k = 10/H0 (red dashed line) and 100/H0 (blue
tom model are applicable also to more general models dotted line). Although the real sound speed is just cˆ2s = 1,
the apparent sound speed diverges at w = −1 and can even
with multiple fields, non-minimally coupled scalarfields,
become negative.
brane-world models and other modified gravity models
9
thatcanberepresentedbyaneffectivedarkenergyfluid. to an apparent sound speed which diverges. It is only
In all these models, as in the Quintom case, w = 1 the sum of all contributions to δp which remains finite.
eff
−
is not a special point in their evolution. There is no rea- Amorespeculativeconclusionisthatitseemsdifficult
sonto expectthat the modelwill adjustits behaviourat for “fundamental” fields to cross w = 1 as their ap-
−
this point, as the crossing of the phantom divide is in- parentrest-framesoundspeed,definedthroughEq.(36),
cidential. In representing these kinds of models with an wouldgenerally be the actual propagationspeed of their
effective fluid we thereforeexpect that the perturbations perturbations,whichmustremainsmallerthanthespeed
willnotvanish,andthattheapparentsoundspeedwould of light. From the discussion in section IV we learn that
have to diverge. fieldswithawelldefinedrest-frameandsoundspeedhave
One importantdifference between the Quintommodel only two routes to phantom crossing:
and more general modified gravity models is that the
latter seem to require generically a non-zero anisotropic dw/da=0atcrossing: Thislooksratherfine-tuned
•
stress as φ = ψ [26]. In this case cˆ2 = 0 does not allow asthefieldneedstobeawareofthepresenceofthe
crossing the6 phantom divide, sincesD in Eq. (43) con- barrier. On the other hand, a normal minimally
×
tains an additional non-zero term due to the anisotropic coupled scalar field reaches w = 1 when φ˙ = 0,
−
stress. This reinforces our view that the Quintom-like and it does so with zero slope. A scenario where
crossing,where all perturbations stay non-zero and δp phantom crossing of this kind is realised could be
eff
remainsfinite inspite ofdivergencesinmostconvention- built by changing the sign in front of the kinetic
ally expected terms, is more generic. This means that term whenever φ˙ = 0. As an example, a cosine
thereisnoobviouswaytopredicttheformofδp ingen- potential then leads to a “phaxion” scenario, see
eff
eral for modified gravity models. On the positive side, if Fig. 8. On average, the phaxion behaves like a
thedarkenergyisnotjustacosmologicalconstant,andif cosmological constant, but oscillates around w =
we are able to measure δp then we may hope that this 1. However,it is unclear how to constructsuch a
eff −
will provide us with clues about the physical mechanism scenario in a covariant way.
thatiscausingthe acceleratedexpansionoftheuniverse.
vanishing sound speed: A field with a vanishing
•
sound speed can also avoid the logarithmic diver-
genceofδp. Thisrequireseitheracouplingbetween
VI. CONCLUSIONS
1+w and cˆ2 or else the soundspeed mightbe zero
s
at all times, see Fig. 9. In the latter case the pres-
As long as the cosmological data indicates the pres- sureofthefieldwouldneedtoremaincloseto ρat
ence of a dark energy with an effective equation of state alltimesinordertopreventthe small-scalepe−rtur-
p ρ it will be necessary to consider models with the bations from growing too quickly. Also, the sound
≈ −
same equation of state. In general, we will have to al- speeds needs to be exactly zero, otherwise the log-
low the equation of state to cross the phantom divide, arithmic divergence reappears. This may require a
w = 1. Even though such a fluid model may not be symmetryenforcingcˆ2 =0asevenaverysmallbut
− s
viable at the quantum level, it is possible that this be- non-zerosoundspeedwouldrenderthisscenarioin-
haviour is only apparent, or due to a modification of viable.
General Relativity or the existence of more spatial di-
mensions. In analysing the data we therefore have to be In view of these difficulties, the Quintom family of mod-
able to use general self-consistent models at the level of els, although more complicated than a single field, may
linear perturbation theory. prove to be the conceptually simplest way to cross w =
Inthispaperwehavestudiedthebehaviouroftheper- 1withamodeldefinedthroughanaction. Theyalsoil-
−
turbations in general perfect fluid models close to w = lustratethattheeffectivepressureperturbationmaypro-
1. We have shown that although models with purely vide a kind of fingerprint of the mechanism behind the
−
adiabatic perturbations cannot cross w = 1 without accelerated expansion of the universe if it can be mea-
−
violating important physical constraints (like causality sured.
or smallness of the perturbations), it is possible to rec- Concerning the use of phantom-crossing fluids for the
tifythesituationbyallowingfornon-adiabaticsourcesof purpose of data analysis, it is straightforward to imple-
pressure perturbations. However,the parametrisationof ment them by just avoiding the usual parametrisation
δp interms ofthe rest-frameperturbationsofthe energy of the pressure perturbations in terms of the rest-frame
densitycannotbe usedasthisframebecomesunphysical sound speed. However, there does not seem to be a
at w = 1. By parameterising δp instead in any other canonicalwaytochoosethepressureperturbationsifone
−
frame the divergencies are avoided. isnotallowedtousethedarkenergyrest-frame. Theaim
WealsocomputedallquantitiesintheQuintommodel for the far future will be to directly measure the pres-
which provides an explicit example of the above mecha- sure perturbations of the dark energy in order to gain
nism. In this model, even though the propagationspeed insight into the physical origin of the phenomenon. For
of sound waves remains finite and constant, the addi- now,wehaveto ensurethatthe definitionofδp doesnot
tional internal and relative pressure perturbations lead lead to unphysical situations, while preserving the usual
10
FIG.8: Thephaxion: Thesigninfrontofthekinetictermofa
normalscalarfieldisflippedeverytimeφ˙ passesthroughzero FIG. 9: The pressure perturbation δp for three fluids with
w(shoicthhaitsφ˙b≥oun0)d.edAsboptohtenfrtoimal waebouvseeaVn(dφ)b=elo1w+. coTsh(φe/fime0ld) ddiaffsehr-ednotttreedstl-ifnrea)m,ecˆ2ss=ou0n.d01sp(meeidd,dlcˆe2scu=rv0e.,1bl(uteopdacsuhrevde,linreed)
thenmovestolargevalues(topleftgraph),whilewoscillates and cˆ2s = 0 (lowest, solid line). Only the last case does not
havealogarithmicdivergenceinthepressureperturbationat
around −1 (top right). The energy density ρ also oscillates
thephantom divide.
so that the field mimics an effective cosmological constant
(bottom left). As the phantom divide is crossed with zero
slope,thereisnodivergenceinthepressureperturbation(bot-
tomright),butthetime-dependenteffectivemasscanleadto spectrum for λ=1/100 and λ=10−4 we find a relative
strong growth of theperturbations in spite of cˆ2s =1. differenceintheCℓ oflessthan1%onlargescales,which
rapidly drops to 10−6 by ℓ=50, well below cosmic vari-
ance. We expect detectable differences only for strongly
parametrisation in terms of the rest-frame sound speed clustering dark energy with a sound speed that is close
as much as possible far away from the phantom divide. to zero or even negative. Although it is reassuring that
Maybethesimplestwayoutistoregularisetheadiabatic directly measureable quantities are not sensitive to the
sound speed c2 which appears in Eq. (39) because of the precise value ofλ in Eq.(59), this also shows that it will
a
gauge transformation into the rest-frame. While any fi- be very difficult to distinguish between different models
nite choice of δp is a physically acceptable choice, it is of dark energy if w 1.
preferable to modify Eq. (39) in a minimal way so that ≈−
the usual interpretation of cˆ2 is preserved for w = 1.
s 6 −
We propose to use Acknowledgments
w˙(1+w)
c˜2 =w (59) M.K. and D.S. are supported by the Swiss NSF. It
a − 3 [(1+w)2+λ]
H is a pleasure to thank Bruce Bassett, Camille Bonvin,
where λ is a tuneable parameter which determines how Takeshi Chiba, Anthony Lewis, Andreas Malaspinas,
close to w = 1 the regularisation kicks in. A value Norbert Straumann, Naoshi Sugiyama, Jochen Weller,
of λ 1/1000−should work reasonably well, as shown PeterWittwerandespeciallyFilippoVernizzi,RuthDur-
in Fig≈. 10. In this case δp is well-defined and there are rer and Luca Amendola for helpful and interesting dis-
no divergencies appearing in the perturbation equations cussions.
(40) and (41). Although the differences in δp for the dif-
ferent choices of λ in Fig. 10 look important, we have to
remember that, firstly, the perturbations in the dark en- APPENDIX A: EQUIVALENCE BETWEEN
ergy arenormally small(especially close to the phantom SCALAR FIELDS AND FLUID MODELS
divide) and that secondly they are only communicated
to the other fluids via the gravitational potential ψ. As The aim of this appendix is to show that at the level
in Fig. 5, we find also here that the dark energy pertur- of first-order perturbation theory a scalar field behaves
bations are subdominant. Computing the CMB power just like a non-adiabatic fluid with cˆ2 = 1. To this end
s
