Scalar-tensor $σ$-cosmologies

2 Pages·0.06 MB·

by  Spiros Cotsakis

SCALAR-TENSOR σ-COSMOLOGIES
SPIROSCOTSAKIS
GEODYSYC, Department of Mathematics, University of the Aegean, 83200, Samos,
Greece
E-mail: [email protected]
1 JOHN MIRITZIS
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Department of Marine Sciences, University of the Aegean, Mitilene 81100, Greece
0
E-mail: [email protected]
2
n We show that the scalar-tensor σ-model action is conformally equivalent to gen-
a eralrelativitywithaminimallycoupledwavemapwithaparticulartargetmetric.
J Inflationonthesourcemanifoldisthenshowntooccurinanaturalwaydueboth
tothearbitrarycurvaturecouplingsandthewavemapself-interactions.
2

2 1 Scalar-tensor σ-models
v
6
Let ( m,g ) be a spacetime (source) manifold, ( n,h ) Riemannian (target)
8 µν ab
0 manifMold and a ∞ map φ : . We may thiNnk of the scalar fields φa,a =
C M → N
1 1,...,n, as coordinates parametrizing the Riemannian target. Our starting point
1 is the general action functional
0
0 S = L dv , dv =√ gdx, (1)
σ g g
c/ ZM −
q where
-
r L =A(φ)R B(φ)Tr (φ∗h)=A(φ)R B(φ)gµνh ∂ φa∂ φb, (2)
g σ − g − ab µ ν
: whereA,B arearbitrary ∞ functionsofφ. We seethatS hasarbitrarycouplings
v C
to the curvature and kinetic terms and we call it the scalar-tensor σ-model or,
i
X
the scalar-tensor wavemap action. Such a theory includes as special cases many
r of the scalar field models considered in the literature (e.g.,1,2,3). Under compact
a
variations of the families g and φ , s R where ψ˙(s) = [∂ψ /∂s] , the
(s) (s) (s) s=0
∈
Action Principle, S˙ =0, leads to the system,
B 1
G = h φa φb g gρσφaφb (3)
µν A ab ,µ ,ν − 2 µν ,ρ ,σ
(cid:18) (cid:19)
1
+ ( A g 2 A)
µ ν µν g
A ∇ ∇ −
1
2 φa + Γ¯a gµν∂ φb∂ φc+ RAa =0, Γ¯ =Γ(h)+C, (4)
g bc µ ν 2
where A = ∂A/∂φa, Ca = (1/2)(δaB +δaB h Ba) and B = ∂lnB/∂φa.
a bc b c c b− bc a
Without loss of generality we may perform a conformal transformation on the
target metric, h˜ = Bh, to find Γ¯ = Γ(h) and so we set from now on B = 1 in
Eq. (2) and drop the tilde on h. Under a conformal transformation of the source
manifold and the target metric redefinition,
3 B
g˜=A(φ)g, π := A A + h =:Q Q + h (5)
ab 2A2 a b A ab a b 6 ab
gianniscorr: submitted to World Scientific on February 7, 2008 1
∼
(and dropping from the beginning the 2 lnΩ term as a total divergence), the
originalscalar-tensorσ-modelaction(1)-(2)becomesthatofawavemapminimally
coupled to the Einstein term
S˜= L˜ dv , L˜ = g˜ R˜ g˜µνπ ∂ φa∂ φb . (6)
σ g˜ σ ab µ ν
M −
Z p (cid:16) (cid:17)
Thisresultshowsthatallcouplingsofthewavemaptothecurvatureareequivalent.
Varying this conformally related action, S˜˙ = 0, we find the Einstein-wavemap
system field equations for the g˜ metric and involving the π metric namely,
ab
1
G˜ = π φa φb g˜ g˜ρσφaφb (7)
µν ab ,µ ,ν − 2 µν ,ρ ,σ
(cid:18) (cid:19)
∼
2 φa + Dag˜µν∂ φb∂ φc =0, D =Γ(h)+T, (8)
g˜ bc µ ν 6
with T =∂ Q +∂ Q ∂ Q and Q =Q Q .
abc c ab b ac a bc ab a b
−
2 σ-Inflation
Let us now assume that the source manifold ( m,g ) is the 4-dimensional flat
µν
M
FRW model in the original scalar-tensor wavemap theory (1)-(2). After the con-
formal transformation (5), the Friedman equation is H2 = 1T00 , where the 00-
3 WM
component of the energy-momentum tensor of the wavemapis given by
1
T00 = π φ˙aφ˙b. (9)
WM 2 ab
We see that the time derivative of this may change sign and therefore we find that
at the critical points of T00 the universe inflates,
WM
1
a=a exp T00 t . (10)
0 3 WM,crit
r !
This is the simplest example of a general procedure, which we call σ-inflation, in
whichinflation is drivenboth by the couplingA(φ) and the self-interacting(target
manifold is curved!) ’scalar fields’ (φa) which however have no potentials. This
mechanism reduces to the so-called hyperextended inflation mechanism1 when the
targetspace is the realline. On the other hand, when the curvature coupling A(φ)
is equal to one, T00 can have no critical points since it is always positive and
WM
so we have no inflationary solutions. In this case we obtain the so-called tensor-
multiscalar models4. Inflationary solutions become possible in this case by adding
’by hand’ extra potential terms and models of this sort abound.
Details and extensions of the present results will be given elsewhere.
References
1. P.J. Steinhardt and F.S. Accetta, Phys.Rev.Lett. 64, 2740 (1990).
2. T. Damour and G. Esposito-Far`ese,Class.Quant.Grav. 9, 2093 (1992).
3. J.D. Barrow,Phys. Rev. D 47, 5329 (1993).
4. T. Damour and K. Nordtvedt, Phys. Rev. D 48, 3436 (1993).
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