Reconstruction of a scalar-tensor theory of gravity in an accelerating universe

4 Pages·0.12 MB·English

by  B. Boisseau

Reconstruction of a scalar-tensor theory of gravity
in an accelerating universe
B. Boisseaua, G. Esposito-Far`eseb,c, D. Polarskia,c, and A.A. Starobinskyd,e
a Lab. de Math´ematique et Physique Th´eorique, UPRES-A 6083 CNRS,
Universit´e de Tours, Parc de Grandmont, F 37200 Tours, France
b Centre de Physique Th´eorique, CNRS Luminy, Case 907, F 13288 Marseille cedex 9, France
c D´epartement d’Astrophysique Relativiste et de Cosmologie,
Observatoire de Paris-Meudon, F 92195 Meudon cedex, France
d Landau Institute for Theoretical Physics, 117334 Moscow, Russia
e Newton Institute for Mathematical Sciences, University of Cambridge, Cambridge CB3 0EH, U.K.
(January 21, 2000)
The present acceleration of the Universe strongly indicated by recent observational data can be
modeled in the scope of a scalar-tensor theory of gravity. We show that it is possible to determine
0 the structure of this theory (the scalar field potential and the functional form of the scalar-gravity
0
coupling)alongwiththepresentdensityofdustlikematterfromthefollowingtwoobservablecosmo-
0
logical functions: theluminositydistanceandthelineardensityperturbationinthedustlikematter
2
component as functions of redshift. Explicit results are presented in the first order in the small

n inverse Brans-Dickeparameter ω−1.
a
J PACS numbers: 98.80.Cq, 04.50.+h

1
2 RecentobservationaldataontypeIasupernovaeexplo- following inequality should be satisfied [4]
sionsathighredshiftsz ≡ a(t0)−1∼1obtainedindepen-
1 a(t) dH2(z)
dently by two groups [1,2], as well as numerous previous ≥3Ω H2(1+z)2 . (1)
v dz m,0 0
6 arguments(seetherecentreviews[3,4]),stronglysupport
6 the existence of a new kind of matter in the Universe Here,H =H(z =0)istheHubbleconstant,Ω isthe
0 m,0
0 whose energy density not only is positive but also domi- present energy density of the dustlike (CDM+baryons)
1 nates the energy densities of all previously known forms matter component in terms ofthe criticaldensity ε =
0 crit
0 of matter [here a(t) is the scale factor of the Friedmann- 3H02/8πG (c = h¯ = 1, and an index 0 stands for
0 Robertson-Walker (FRW) isotropic cosmological model, the present value of the corresponding quantity). Note
/ and t0 is the present time]. This form of matter has a that the inequality (1) saturates when the Λ-term is ex-
c
q stronglynegativepressureandremainsunclusteredatall actly constant. It is not clear from the existing data
- scaleswheregravitationalclusteringofbaryonsand(non- whether (1) is satisfied at all. Actually the opposite
r
g baryonic) cold dark matter (CDM) is seen. Its gravity holds: An attempt to reconstruct U(Φ) from the super-
: resultsinthepresentaccelerationoftheexpansionofthe novaedata[8]andfittingofexistingdatatoamodelwith
v
i Universe: a¨(t0) > 0. In a first approximation, this kind a linear equationof state for the Λ-termpΛ =wεΛ, with
X of matter may be described by a constant Λ-term in the w <−1 [9], shows that the possibility of violation of in-
r gravity equations as first introduced by Einstein. How- equality(1),thoughstronglyrestricted,isnotcompletely
a
ever,a Λ-termcouldalso be slowlyvarying withtime. If excluded. Hence it is natural and important to consider
so,this willbe soondeterminedfromobservationaldata. avariableΛ-terminamoregeneralclassofscalar-tensor
Inparticular,ifweusethesimplestmodelofavariableΛ- theories of gravity where the requirement (1) does not
term (also called quintessence in [5]) borrowed from the arise. Moreover, this generalization of general relativity
inflationaryscenarioofthe earlyUniverse,namely anef- (GR) is inspired by present more fundamental quantum
fective scalar field Φ with some self-interactionpotential theories,likeM-theory. Inthesetheories,thescalarfield
U(Φ) minimally coupled to gravity, then the functional Φis justthe dilatonfield,hence weshallcallit sobelow.
form of U(Φ) can be determined from observationalcos- Thus, we are interested in a universe where gravity is
mological functions: either from the luminosity distance described by a scalar-tensor theory, and we consider the
DL(z) [6,7], or from the linear density perturbation in Lagrangiandensity in the Jordan frame [10]
the dustlike component of matter in the Universe δ (z)
m
for a fixedcomovingsmoothing radius [6]. However,this L= 1 F(Φ) R−gµν∂ Φ∂ Φ −U(Φ)+L (g ) ,
µ ν m µν
model cannot account for any future observationaldata, 2
in particular, for any functional form of DL(z). This (cid:16) (cid:17) (2)
happens because a variable Λ-term in this model should
satisfytheweak-energyconditionε +p ≥0. Intermsof where L describes dustlike matter and F(Φ) > 0.
Λ Λ m
the observable quantity H(z) ≡ a˙(t)/a(t) describing the This corresponds to the Brans-Dicke parameter ω =
evolutionoftheexpandingUniverseatrecentepochs,the F/(dF/dΦ)2 > 0. One may also introduce a function
1
Z(Φ)infrontofthekineticterm(∂ Φ)2,butitcanbeset The effective value of Newton’s gravitationalconstant
µ
either to 1, or to −1 by a redefinition of the scalar field. G in Eqs. (3–4) is given by the formula G = 1/8πF.
N N
Underthe assumptionofabsenceofghostsinthe theory, We shall use its present value G in the definition of
N,0
thesecondpossibilityrequirestheBrans-Dickeparameter the critical density ε . On the other hand, G is not
crit N,0
to lie in the range −3/2 < ω < 0 (see [11] for more de- thequantitymeasuredinlaboratoryCavendish-typeand
tails). Since this clearly contradicts solarsystem tests of solar-system experiments. For a massless dilaton, the
GR either in the absence of U(Φ), or for U(Φ) satisfying effective gravitational constant between two test masses
the condition (8) below for scales of galaxies and clus- is given by
ters of galaxies, we will not discuss this possibility fur-
1 2F +4(dF/dΦ)2
ther. WedonotintroduceanydirectcouplingbetweenΦ
G = . (7)
andL (thoughthispossibilitycouldbeenvisaged,too). eff 8πF 2F +3(dF/dΦ)2
m (cid:18) (cid:19)
Thisguaranteesthattheweakequivalenceprincipleisex-
In our case, the dilaton is massive, so the expression (7)
actly satisfied (universality of free-fall of laboratory-size
will be valid for physical scales R such that
objects), and also that fundamental constants, like e.g.
the fine-structure constant, do not change with time in
d2U d2F
this theory. This is in very good agreement with labora- R−2 ≫max ,H2,H2 . (8)
dΦ2 dΦ2
tory, geophysical and cosmologicaldata [12,13,14]. (cid:18)(cid:12) (cid:12) (cid:12) (cid:12)(cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
Such a scalar-tensortheory was recently consideredas Previously,theexpress(cid:12)ionG(cid:12) waskn(cid:12)ownf(cid:12)romthepost-
(cid:12) (cid:12)eff (cid:12) (cid:12)
a model for a variable Λ-term for some special choices
Newtonianexpansion;belowwerederiveitusingthecos-
of F(Φ) and U(Φ) (see [15]). Our approach is just the
mological perturbation theory.
opposite: We want to derive these functions from obser-
Letus now list the restrictionsof the theory (2)which
vational data. Since we have to determine two functions
follow from solar-system and cosmological tests. The
F(Φ) and U(Φ), we will need both observational func-
post-Newtonian parameters β and γ for this theory are:
tions D (z) and δ (z), in contrast to GR. Then the re-
L m
construction problem can be uniquely solved as will be (dF/dΦ)2
γ =1− , (9)
shown below. Note that the angular diameter as a func- F +2(dF/dΦ)2
tionofz providesthe sameinformationasD (z)(see [4]
L 1 F (dF/dΦ) dγ
and the second reference in [6]). β =1+ . (10)
4 2F +3(dF/dΦ)2 dΦ
It is most appropriate for us to work in the Jor-
dan frame (JF), in which the various physical quanti- Using the upper bounds on (γ − 1) from solar system
ties are those that are being measured in experiments, measurements [16,17], we get
even though the Einstein frame (EF) often provides a
bettermathematicalinsight. Inaddition,thedilatonap- ω−1 =F−1 (dF/dΦ)2 <4×10−4 . (11)
0 0 0
pearstobedirectly coupledtodustlikematterinthe EF
frame, in contrast to the JF. For a flat FRW universe So, GN,0 and Geff,0 coincide with better than 2×10−4
with ds2 = −dt2 +a2dx2, the background equations in accuracy. Onthe otherhand, the difference betweenGN
the JF are then and Geff may be larger at redshifts z ∼ 1 since neither
theupperlimitonβ,northepresentexperimentalbound
3FH2 =ρ + Φ˙2 +U −3HF˙ , (3) |G˙eff/Geff| < 6 × 10−12 yr−1 [17] significantly restrict
m
2 (d2F/dΦ2) . Note that we cannot use the nucleosynthe-
0
−2FH˙ =ρ +Φ˙2+F¨−HF˙ . (4) sisboundonthechangeofG sincethattimeasthebe-
m eff
haviorofG duringtheintermediateperiodisunknown,
eff
Their consequence is the equation for the dilaton itself: unless we make additional assumptions (see below).
Thetheory(2)describesavariableΛ-termwithdesired
dU dF
Φ¨ +3HΦ˙ + −3(H˙ +2H2) =0 . (5) properties if the following three conditions are satisfied:
dΦ dΦ
1) The Λ-term is dynamically important at present,
CombiningEqs.(3)–(4)andchangingtheargumentfrom namely, ΩΛ,0 ∼0.7∼2Ωm,0, or
time t to redshift z, we obtain the following basic equa-
tion for F(z): Φ˙2
+U −3HF˙ ∼0.7ε ∼2ρ . (12)
crit m,0
2
′ !
′′ ′ 4 ′ 6 2(lnH) 0
F + (lnH) − F + − F
1+z (1+z)2 1+z
(cid:20) (cid:21) (cid:20) (cid:21) 2) The Λ-term has a sufficiently large negative pres-
2U H 2 suretoprovideaccelerationofthepresentUniverse. The
0
= +3 (1+z) F Ω , (6)
(1+z)2H2 H 0 m,0 condition a¨0 >0 reads:
(cid:18) (cid:19)
wheretheprimedenotesthederivativewithrespecttoz. 2U0 >(ρm+2Φ˙2+3F¨+3HF˙)0 . (13)
2
3) The dark matter described by the Λ-term remains LetusnowconsidersufficientlysmallscalesR=2πa/k
unclusteredatscalesuptoR∼10h−1(1+z)−1 Mpc and for which the inequality (8) is well satisfied. For exam-
probably even more (here h = H /100 km s−1 Mpc−1). ple, if δ (z) is determined from the abundance of rich
0 m
To achieve this, it is sufficient to assume that the in- clusters of galaxies, then the relevant comoving scale is
equality (8) is satisfied for all scales in question. R∼8h−1/(1+z) Mpc. If the r.h.s. of Eq. (8) is ∼H2,
Thefirststepofourprogramispurelykinematical: we thenthecorrespondingsmallparameterisR2H2 ∼10−5.
0
determine H(z) from D (z) like in GR, Note that we have another parameter, ω−1, which is
L
′ small at the present time, Eq. (11), but it need not be
1 D (z)
= L . (14) so small in the past. Also, this parameter may be larger
H(z) 1+z than a2H2/k2. For this reason, we will first keep it.
(cid:18) (cid:19)
The functionaldependence ofD (z)onthe cosmological The solution of Eqs. (16–19) in the formal short-
L
parameters, like Ω , is of course model dependent. If wavelength limit k → ∞ can be found following the an-
m,0
Ω is already known from other tests, we can find al- alytical method used in [6] in the GR case, confirmed
m,0
readyatthatstageofthe reconstructionaquantity such numerically in [18]. The idea is that the leading terms
as the present effective equation of state of the dilaton in Eqs. (16–19) are either those containing k2, or those
from the formula (cf. [8]): with δm. Then, using (17) and the l.h.s. of Eq.(16), the
standardformoftheequationfordustlikematterdensity
p (2/3)(dlnH/dz) −1
Λ,0 0 perturbation follows:
w ≡ = . (15)
0
ε 1−Ω
Λ,0 m,0 δ¨ +2Hδ˙ +k2a−2φ≃0 . (20)
m m
ε containstheterm−3H F˙ ,sothatΩ +Ω =1.
Λ,0 0 0 m,0 Λ,0
The dilaton equation of state can be determined for Now we consider the solution of Eq. (19) of interest to
z >0, too; one has only to define what should be called us, for which |δ¨Φ| ≪ k2a−2|δΦ|. It corresponds to the
the pressure and the energy density of the dilaton in growing adiabatic mode. So, keeping terms with k2 in
general. Actually, we will show below that Ω is it- Eq.(19)andthenusingthe r.h.s. ofEq.(16),weobtain:
m,0
self self-consistently determined from our approach, so
dF F dF/dΦ
no additional information is required to find w(z). δΦ≃(φ−2ψ) ≃−φ . (21)
dΦ F +2(dF/dΦ)2
IncontrasttoGR,Eq.(6)isnolongersufficienttode-
termine U(z); one should know F(z), too. For this pur- In the GR case, δΦ∝k−2φ in the limit k →∞, so mat-
posewewilluseδ (z). Weconsiderperturbationsinthe
m ter producing the Λ-termis not gravitationallyclustered
longitudinal gauge ds2 =−(1+2φ)dt2+a2(1−2ψ)dx2.
at small scales (physically, due to free streaming). This
WorkinginFourierspace(aspatialdependenceexp(ik.x)
is not so in scalar-tensor gravity: The dilaton remains
with k ≡|k| is assumed), the followingequations areob-
partly clustered for arbitrarily small scales, this cluster-
tained:
ing being small only because ω is large.
Keeping only terms with k2 or δ in Eq. (18), we get
φ=v˙ =ψ−δF/F , (16) m
theexpressionofφthroughδ andδF. Finally,inserting
k2 d(ψ+Hv) m
δ˙ =− v+3 , (17) itintoEq.(20)andusingEq.(21),wearrivetotheclosed
m a2 dt
form of the equation for δ :
m
where the gauge invariant quantity δ ≡ (δρ )/ρ +
m m m
3Hv, and v is the peculiar velocity potential of dustlike δ¨m+2Hδ˙m−4πGeffρm δm ≃0 , (22)
matter. We also get
with G defined in (7) above. In terms of z, (22) reads:
eff
k2
−3F˙φ˙ − 2 F −Φ˙2+3HF˙ φ= (H2)′ H2
a2 H2 δ′′ + − δ′
(cid:18) (cid:19) m 2 1+z m
F˙ k2 F˙2 (cid:18) (cid:19)
=ρmδm+3F δF˙ + a2 −6H2−3F2!δF ≃ 23(1+z)H02GGeff(z) Ωm,0 δm . (23)
N,0
+Φ˙δ˙Φ+3HΦ˙ δΦ+δU, (18)
Eq. (22) does not contain k2 at all. Thus, its solutions,
and the equation for the dilaton fluctuations δΦ: as well as the corresponding expressions for δΦ, do not
oscillatewiththefrequencyk/afork →∞. Thisjustifies
δ¨Φ+3Hδ˙Φ+ k2 −3(H˙ +2H2)d2F + d2U δΦ= the assumption about δ¨Φ made above.
a2 dΦ2 dΦ2
(cid:20) (cid:21) Extracting H(z) (from DL(z)) and δm(z) from ob-
k2 dF servations with sufficient accuracy, we can reconstruct
= (φ−2ψ)−3(ψ¨+4Hψ˙ +Hφ˙)
a2 dΦ Geff(z)/GN,0 analytically. Since, as follows from
(cid:20) (cid:21)
+(3ψ˙ +φ˙)Φ˙ −2φdU . (19) Eteqr.t(h1a1n),0th.0e2q%uaanctciutireascGy,effE,0q.an(2d3G)Nta,0kecnoinactidze=wi0thgbiveets-
dΦ
3
also the value of Ω with the same accuracy. Thus, ACKNOWLEDGMENTS
m,0
in principle, no independent measurement of Ω is re-
m,0
quired. A.S. was partially supported by the Russian Founda-
The resulting equation Geff(z) = p(z), where p(z) is tion for Basic Research, grant 99-02-16224, and by the
a given function following from observational data, can Russian Research Project “Cosmomicrophysics”. Cen-
be transformedinto a nonlinear second order differential tre de Physique Th´eorique is Unit´e Propre de Recherche
equation for F(z) if we exclude dΦ (which appears in 7061.
dF/dΦ) using the background equation (4), which reads
2
Φ′2 =−F′′− (lnH)′+ F′
1+z
(cid:20) (cid:21)
2(lnH)′ H2
+ F −3(1+z) 0F Ω . (24)
1+z H2 0 m,0
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L
will allow implementation of the reconstructionprogram
and determination of the microscopic Lagrangian.
4