Scalar-Tensor Dark Energy Models

0.09 MB·English

by  R. Gannouji

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SCALAR-TENSOR DARK ENERGY MODELS
R.GANNOUJI,D.POLARSKI,A.RANQUET
Lab. de Physique Th´eorique et Astroparticules, CNRS
Universit´e Montpellier II, 34095 MontpellierCedex 05, France
7
A.A.STAROBINSKY
0
0 Landau Institute for Theoretical Physics, Moscow, 119334, Russia
2

n Wepresentheresomerecentresultsconcerningscalar-tensorDarkEnergymodels.These
a modelsareveryinterestinginmanyrespects:theyallowforaconsistentphantomphase,
J the growth of matter perturbations is modified. Using a systematic expansion of the

3 theoryatlowredshifts,werelatethepossibilitytohavephantomlikeDEtosolarsystem
2 constraints.

1 1. Introduction
v
0 The late-time accelerated expansion of the universe is a major challenge for cos-
5 mology. The component producing this acceleration accounts for about two thirds
6
of the total energy density. While this has gradually become a building block of
1
0 our present understanding, the nature of Dark Energy (DE) still remains myste-
7 rious.1–3 The simplest solution is a cosmological constant Λ. A major contender
0
is Quintessence, a minimally coupled scalar field (with canonical kinetic term). We
/
h willconsiderscalar-tensor(ST)DEmodels,amoreelaboratealternativeinvolvinga
p
new physicaldegree offreedom, the scalarpartnerφ of the gravitonresponsiblefor
-
o amodificationofgravity.4–6 Itisnotclearyetwhethersomemodificationofgravity
r
t is required or even preferred in order to explain the bulk of data. The increasing
s
a accuracy of the data, should allow to severely constrain the various viable models.
: ST DE models allow for phantom DE, w < −1, moreover the equation for the
v DE
i growthofmatterperturbationsismodified.4 Wewillreviewhereresultsconcerning
X
their lowz behaviour,inparticularhow the DE equationofstate isrelatedtosolar
ar system constraints.9
2. Scalar-tensor DE models
We consider the microscopic Lagrangiandensity in the Jordan frame
1
L= F(Φ) R−Z(Φ) gµν∂ Φ∂ Φ −U(Φ)+L (g ) . (1)
µ ν m µν
2
(cid:16) (cid:17)
Wedefinewhatwemeanbytheenergydensityρ andthepressurep bywriting
DE DE
the gravitationalequations in the following Einsteinian form :
3F H2 = ρ +ρ (2)
0 m DE
−2F H˙ = ρ +ρ +p . (3)
0 m DE DE
ThiscanbeseenastheEinsteinianform,withconstantG =G (t )=F−1,ofthe
0 N 0 0
gravitationalequationsofSTgravity.Withthesedefinitions,theusualconservation
1
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equation applies, and the equation of state parameter w ≡ pDE plays its usual
DE ρDE
role. Using (2,3), One gets w (z) from the observations through
DE
1+zdh2 −h2+ 1Ω (1+z)2
w (z)= 3 dz 3 k,0 , (4)
DE h2−Ω (1+z)3−Ω (1+z)2
m,0 k,0
if we allow for a nonzero spatial curvature and Ω ≡ ρm .
m 3H2F0
Lookingattheequationsabove,everythinglooksthesameasinGR,STgravity
is hidden inthe definitions ofρ ,p ,andthe variousΩ’s.The conditionforDE
DE DE
to be of the phantom type, w <−1, reads
DE
dh2
<3 Ω (1+z)2+2 Ω (1+z) . (5)
dz m,0 k,0
in the presence of spatial curvature.1,4,7 As first emphasized,4 the weak energy
condition for DE can be violated in scalar-tensor gravity (see also8).
3. General low z expansion of the theory
We investigate now the low z behaviour of the model and the possibility to have
phantom boundary crossing in a recent epoch. For each solution H(z), Φ(z), the
basic microscopic functions F(Φ) and U(Φ) can be expressed as functions of z and
expanded into Taylor series in z:
F(z)
=1+F z+F z2+...>0 , (6)
1 2
F
0
U(z)
≡Ω u=Ω +u z+u z2+... . (7)
3F H2 U,0 U,0 1 2
0 0
From (6,7), all other expansions can be derived, in particular:
w (z)= w +w z+w z2+... , (8)
DE 0 1 2
G˙
H−1 eff = g +g z+g z2+.... . (9)
0 G 0 1 2
eff
A viable ST gravity model must be very close to General Relativity, viz.
6(Ω −Ω −F ) ∆2
ω = DE,0 U,0 1 = >4×104 , (10)
BD,0 F2 F2
1 1
with ∆2 ≡ 6 (Ω −Ω −F ). Therefore, we must have |F | ≪ 1 and ∆2 ≈
DE,0 U,0 1 1
6(Ω −Ω )>0. Moreover,for positive U, ∆2 <6Ω <5
DE,0 U,0 DE,0
1/2
5
|F |< .10−2 . (11)
1
(cid:18)ω (cid:19)
BD,0
It can be shown that the condition |F | ≪ 1 is sufficient to ensure here that solar
1
system constraints are satified.9
We now specialize to the case |F | ≪ 1 yet assuming that other F are not as
1 i
small. Then all expansions simplify considerably and we have in particular,
2F +6(Ω −Ω )
1+w ≃ 2 DE,0 U,0 . (12)
0
3Ω
DE,0
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From (12), the necessary condition to have phantom DE today reads
d2F F
2
= <−1 . (13)
(cid:18)dΦ2(cid:19) 3 (Ω −Ω )
0 DE,0 U,0
HenceF <0isnecessaryforphantomDE,becauseΩ −Ω >0from∆2 >0.
2 DE,0 U,0
InadditionsignificantphantomDErequires|F |∼1.If|F |∼|F |≪1,thepresent
2 1 2
phantomness is very small.
It is actually possible to invert all expansions and to obtain all coefficients in
function of the post-Newtonian parameters γ, β and g . The following results are
0
finally obtained
γ−1
F =g (14)
1 0
γ−1−4(β−1)
β−1
F =−2 g2 (15)
2 0 [γ−1−4(β−1)]2
1 γ−1
Ω −Ω =− g2 (16)
DE,0 U,0 6 0 [γ−1−4(β−1)]2
1 4(β−1)+γ−1
1+w =− g2 (17)
DE,0 3 0 Ω [γ−1−4(β−1)]2
DE,0
The best present bounds are γ −1 = (2.1±2.3)·10−5, β −1 = (0±1)·
PN PN
10−4, G˙eff,0 =(−0.2±0.5)·10−13y−1.Thoughpossibleinprinciple,10theinteresting
Geff,0
possibility to test phantomness in the solar system is very hard while its amount
depends critically on the small quantity g2. In this respect cosmological data are
0
certainly better suited, a conclusion reminiscent of that reachedin6 concerning the
viability of ST DE models with vanishing potential.
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