Limit to General Relativity in f(R) theories of gravity

0.19 MB·English

by  Gonzalo J. Olmo

gr-qc/0612047
Limit to General Relativity in f(R) theories of gravity
Gonzalo J. Olmo∗
Physics Department, University of Wisconsin-Milwaukee,Milwaukee, WI 53201, USA
(Dated: December 7th, 2006)
Wediscusstwoaspectsoff(R)theoriesofgravityinmetricformalism. Wefirststudythereasons
tointroduceascalar-tensorrepresentationforthesetheoriesandthebehaviorofthisrepresentation
in the limit to General Relativity, f(R)→R. We find that the scalar-tensor representation is well
behavedeveninthislimit. Thenweworkouttheexactequationsforsphericallysymmetricsources
usingtheoriginal f(R)representation,solvethelinearized equations,andcompareourresultswith
recent calculations of the literature. We observe that the linearized solutions are strongly affected
bythecosmicevolution,whichmakesveryunlikelythatthecosmicspeedupbeduetof(R)models
with correcting terms relevant at low curvatures.
PACSnumbers: 98.80.Es,04.50.+h,04.25.Nx
7
0
0 I. INTRODUCTION [28] for a discussion of Rn models), and others defend
2 the opposite [29, 30, 31, 32, 33, 34] (see also [35] for a
critical discussion of these two positions). An important
n In the last few years, modified theories of gravity
element in this discussion was the identification of f(R)
a of the f(R) type, where R is the Ricci scalar, have
J theorieswithaclassofscalar-tensortheories[22,23,36],
received much attention. These theories have the
9 whichallowedone to reinterpretthe equationsofmotion
ability to generate late-time cosmic acceleration and
and made more accessible the computation of the
also early-time inflation depending on the details of the
2 Newtonian and post-Newtonian limits. More recently,
function f(R) considered. In this sense, they represent
v however, this scalar-tensor approach has been criticized
7 an interesting alternative to the combination of General
and, apparently, shown to break down for theories close
4 Relativity (GR) plus dark energy models [1, 2], and
to GR [34]. It was then claimed that solar system
0 to other explanations for the cosmic speedup [3, 4]. A
tests do not actually rule out theories of the type
2 given gravity lagrangian f(R) can, in addition, lead to
1 f(R)=R+ǫg(R),withǫasmallparameter,becausethe
two completely different theories depending on the vari-
6 corresponding scalar-tensor theory is not well defined in
ational principle used to derive the equations of motion.
0 the limit ǫ 0. This conclusion, on the other hand,
/ Here we will be dealing with the most common choice, has also bee→n criticized recently in [25, 26, 27], where
c the metric variational formalism. The other choice,
q spherically symmetric weak field configurations have
Palatiniformalism,assumesthattheconnectionisafield
- been studied using the original f(R) representation. In
r independentofthemetricandwillnotbeconsideredhere
g those works, it has been found that the post-Newtonian
(see[5,6,7,8,9,10,11,12,13,14,15,16,17]fordetails).
: parameter γ turns out to be γ =1/2, in agreement with
v
theresultsfoundusingthescalar-tensorapproachin[22].
i
X The dominant underlying philosophy in the con- Solar system experiments again seem to rule out f(R)
r struction of f(R) models has been that non-linear theories. However, there is something unclear in those
a curvature terms that grow at low curvatures could have works that should be reanalyzed before accepting their
a negligible effect in stellar systems or galactic scales, conclusions. In [25, 26, 27] the wrong coordinate system
where the curvature is assumed to be relatively high, was used to carry out the calculations and identify the
but a non-trivial one in the cosmic regime, where the post-Newtonian parameter γ. In fact, for the discussion
curvatures involved are orders of magnitude smaller. of solar system experiments and the identification of
This idea has led to many different proposals, among post-Newtonian parameters, one should use isotropic
which the Carroll et al. model f(R) = R µ4/R [18] coordinates instead of Schwarzschild coordinates (see
−
is perhaps the most known (see also [19] for earlier [37], page 1097 for details). This point and others will
models). A rich debate in various arenas has taken be carefully addressed here.
place with some authors focusing on the cosmological
aspects1, and others on the predictions of these theories
In this work we will discuss in some detail the
at smaller scales (Newtonian and post-Newtonian
scalar-tensor representation of f(R) models of the type
regimes). Among the latter, some claim that models
f(R) = R + ǫg(R) and their limit to GR. Firstly,
of the 1/R type are ruled out by solar system and/or
we will motivate the introduction of the scalar-tensor
laboratory experiments [22, 23, 24, 25, 26, 27] (see also
representation and will then show that it is well defined
even in the limit ǫ 0. Secondly, we will consider the
→
weak field limit for spherically symmetric sources using
∗Electronicaddress: [email protected] the original f(R) representation. We will show that
1 Seeforinstance[12,20,21] the same results found in [24] using the scalar-tensor
2
approach are found here using the f(R) form. For and Γα is the connection, which is defined as
βγ
our calculations we use isotropic coordinates. In the
appendix, we also solve the linearized equations using gαλ
Γα = (∂ g +∂ g ∂ g ) (3)
Schwarzschild coordinates and transform the solution to βγ 2 β λγ γ λβ − λ βγ
isotropic coordinates. This will serve as a consistency
Variation of (1) leads to the following field equations for
checkforourcalculations. Inordertoprovideacomplete
the metric
description of the dynamics of f(R) theories, in our
derivation we will take into account the interaction of 1
f′(R)R f(R)g f′(R)+g (cid:3)f′(R)=κ2T
the local system with the background cosmology. This µν−2 µν−∇µ∇ν µν µν
will allow us to better understand the limit to General (4)
Relativity of f(R) theories. We will see how, driven where f′(R) df/dR. According to (4), we see that,
≡
by the cosmic expansion, a theory characterized by a in general, the metric satisfies a system of fourth-order
low curvature scale R goes from a General Relativistic partial differential equations. The trace of (4) takes the
ǫ
phase, in which the post-Newtonian parameterγ 1, to form
≈
ascalar-tensorphase,inwhichγ 1/2. Thoughwewill
be mainly discussing the weak fi→eld regime, we provide 3(cid:3)f′(R)+Rf′(R) 2f(R)=κ2T (5)
−
here the basic concepts and formulas that (hopefully)
This equation will be useful for the physical interpreta-
will be used in a forthcoming work [38], in which the
tion of the field equations.
strong field regime and the contribution of higher-order
corrections to the linearized theory [39] will be studied.
The paper is organized as follows. We first define the A. Physical interpretation
action and derive the equations of motion for f(R) the-
ories. Those equations are then physically interpreted,
Let us consider a generic f(R) theory not necessarily
whichmotivatestheintroductionofthescalar-tensorrep-
close to GR and rewrite (4) in the form
resentation. Wethendiscusstheconditionsfortheequiv-
alence between f(R) theories and scalar-tensor theories 1 κ2 1
andstudy the limit f(R)→R. The nextstep is to write Rµν − 2gµνR = f′(R)Tµν − 2f′(R)gµν[Rf′(R)−f(R)]+
the equations of motion for spherically symmetric sys-
1
tems and obtain the relevant solutions for weak sources. + [ f′(R) g (cid:3)f′(R)] (6)
We then discuss the results obtained and the interaction f′(R) ∇µ∇ν − µν
of the system with the backgroundcosmology. We finish
The right hand side of this equation can now be seen as
with a brief summary and conclusions. In the appendix
thesourcetermsforthemetric. Thisequation,therefore,
we solve the equations using Schwarzschild coordinates
tells us that the metric is generated by the matter and
as a consistency check for our results.
by terms related to the scalar curvature. If we now
wonder about what generates the scalar curvature, the
answer is in (5). That expression says that the scalar
curvature satisfies a second-order differential equation
II. THE THEORY
with the trace T of the energy-momentum tensor of the
matter and other curvature terms acting as sources. We
The action that defines f(R) gravities has the generic
have thus clarifiedthe role of the higher-orderderivative
form
terms present in (4). The scalar curvature is now a
dynamical entity which helps generate the space-time
1
S = d4x√ gf(R)+S [g ,ψ ] (1) metric and whose dynamics is determined by (5).
2κ2 − m µν m
Z
At this point one should have noted the essential
where Sm[g,ψm] represents the matter action, which de- differencebetweenagenericf(R)theoryandGR.InGR
pends on the metric gµν and the matter fields ψm, and theonlydynamicalfieldisthemetricanditsformisfully
κ2 isaconstantwithsuitableunits2. Fornotationalpur- characterized by the matter distribution through the
poses,we remarkthat the scalarR is defined asthe con- equations G =κ2T . The scalar curvature is also de-
µν µν
traction R=gµνRµν, where Rµν is the Ricci tensor terminedbythelocalmatterdistributionbutthroughan
algebraicequation,namely,R= κ2T. Inthe f(R)case
−
Rµν =−∂µΓλλν +∂λΓλµν +ΓλµνΓρρλ−ΓλνρΓρµλ (2) berontehdgbµyν daniffderRenatiraeldeyqnuaamtioicnasl.fiFeuldrtsh,ei.rem.,otrhe,eythaerescgaolavr-
curvature R, which can obviously be expressed in terms
of the metric and its derivatives using (2), now plays
2 We will see later how κ2 is related to the effective Newton’s anon-trivialroleinthedeterminationofthemetricitself.
constant[seeEq.(33)below].
3
B. Scalar-Tensor representation The inverse problem of finding the f(R) theory corre-
spondingtoagivenscalar-tensortheoryoftheformgiven
in (11) also requires an invertibility condition only. In
The physical interpretation given above puts forward
this case, the equations of motion lead to R=dV/dφ. If
the central and active role played by the scalar curva-
this expression can be inverted to obtain φ(R) then the
tureinthefieldequationsoff(R)theories. However,(5)
suggeststhatthe actualdynamicalentityis f′(R)rather corresponding f(R) lagrangianis given by
than R itself. This is so because, besides the metric,
f(R)=Rφ(R) V(φ(R)) (12)
f′(R) is the only object acted on by differential opera- −
tors in the field equations. Motivated by this, we can
We thus conclude thatthe conditionfor the equationsof
introduce the following alternative notation
motion of an f(R) theory to be equivalent to those of
a ω = 0 Brans-Dicke theory is that the function f′(R)
φ f′(R) (7)
≡ be invertible. Conversely, for a given ω =0 Brans-Dicke
V(φ) R(φ)f′ f(R(φ)) (8) theory the condition for the equivalent f(R) theory to
≡ −
exist is that the function dV(φ)/dφ=R be invertible.
and rewrite (4) as
1 κ2 1 C. Limit f(R)→R
R (g) g R(g) = T g V(φ)+
µν µν µν µν
− 2 φ − 2φ
1 Nowthatthescalar-tensorrepresentationoff(R)the-
+ [ φ g (cid:3)φ] (9)
φ ∇µ∇ν − µν ories has been introduced and the conditions for their
equivalence clarified, we will consider the limit f(R) go-
Using the same notation, (5) turns into ing to R to gain some insight on their dynamical prop-
erties. We will also show that the equivalence is, by no
3(cid:3)φ+2V(φ) φdV =κ2T (10) means, broken in this limit.
− dφ Let us concentrate on models of the form f(R) = R+
ǫg(R) with ǫ an adjustable small parameter. The scalar
This slight change of notation helps us identify (9) and field is therefore identified with φ = 1+ǫg′(R) and be-
(10)withthefieldequationsofaBrans-Dicketheorywith comesaconstant,φ=1,inthelimitǫ 0. Therelevant
parameter ω = 0 and non-trivial potential V(φ), whose part of φ is therefore contained in ϕ=→g′(R). By invert-
action takes the form ingthisrelationwefindR=R(ϕ),whichallowsustoex-
pressthescalarpotentialasV(φ)=ǫ[R(ϕ)ϕ g(R[ϕ])]
1 − ≡
S = d4x√ g[φR V(φ)]+S [g ,ψ ] (11) ǫV˜(ϕ). The action (11) then turns into
2κ2 − − m µν m
Z
1
In terms of this scalar-tensor representation our in- S = 2κ2 d4x√−g (1+ǫϕ)R−ǫV˜(ϕ) +Sm[gµν,ψm]
terpretation of the field equations of f(R) theories is Z h i (13)
obvious, since both the matter and the scalar field help
Asexpected,thelimitǫ 0leadssmoothlytotheaction
generate the metric. The scalar field is a dynamical →
of GR. Let us now look at the equation of motion for φ
object influenced by the matter and by self-interactions
to see what happens in that limit. In terms of ϕ, (10)
according to (10).
can be recast as
We would like to remark that, according to the above ǫ[3(cid:3)ϕ+ϕR(ϕ) 2g(R)]=R(ϕ)+κ2T (14)
derivation, the only requirement needed to express the −
field equationsofan f(R)theoryin the formofa Brans- In the limit ǫ 0 the dynamical part of this equation,
Dicke theory is that f′(R) be invertible3, i.e., that R(f′) that involving→(cid:3)ϕ, vanishes and we recover the familiar
exist. This is necessary for the construction of V(φ). (algebraic) relation R = κ2T. This means that ϕ has
Therefore, the differentiability condition f′′(R)=0 that completelydecoupledfrom−the theory. Asaconsequence
6
one finds following other derivations of the scalar-tensor R becomes independent of ϕ and we can state with
representation [22, 23, 34, 36] is just a superfluous con- confidence that the limit to GR is smooth. This shows
dition generated by the particular method used4. that the equivalence between f(R) gravities and their
related Brans-Dicke theories holds even in the limit of
GR, since no inconsistency is found in this limit.
3 The minimum requirement for a function of one variable to be To conclude this section, we mention that from (14)
invertible in an interval is that the function be continuous and we can extract valuable physical information. The
one-to-oneinthatinterval.
4 Iafmfo′n′(gRd)iffvearneinsthebsraantchsoesmteopeoxitnetndR0th,ewseolmutaioynsneRed(f′t)obcehyooonsde fipxareadm, eitnerfrǫo,ntwhoifchthwee(cid:3)nϕowtearsmsummeakteos baeppsamreanllt atnhde
R0. Inother words,R(f′)maynotbeuniqueinthosecases. existence of two regimes. The first one corresponds to
4
a behavior very close to GR, in which R κ2T is a g =η +h ,with ϕ f′ and h 1satisfying
≈ − µν µν µν | |≪| c| | µν|≪
goodapproximation. The secondregime ariseswhen the ϕ 0 and h 0 in the asymptotic region. Note that
µν
→ →
termsontherighthandsidearesmall. Inthatsituation, should the local system represent a strongly gravitating
the contributionfromthelefthandside cannolongerbe system such as a neutron star or a black hole, the per-
neglected. The behaviour of R is then dominated by ϕ, turbative expansion would not be sufficient everywhere.
i.e.,thefulldynamicsof(14)mustbetakenintoaccount, In such cases, the perturbative approach would only be
which means that R receives contributions from T and valid in the far region. Nonetheless, the decomposition
from its self-interactions (or, equivalently, from ϕ). In f′ = f′ +ϕ(x) is still very useful because the equation
c
this situation, the theory behaves as a scalar-tensor for the local deviation ϕ(x) can be written as
theory,with ϕrepresentingthe scalardegree offreedom.
3(cid:3)ϕ+W(f′+ϕ) W(f′)=κ2T, (16)
We will discuss further this point later on. c − c
whereT representsthetraceofthelocalsources,wehave
defined W(f′) R(f′)f′ 2f(R[f′]), and W(f′) is a
≡ − c
slowly changing constant within the adiabatic approxi-
III. APPLICATIONS
mation. In this case, ϕ needs not be small compared to
f′ everywhere, only in the asymptotic regions.
c
Acomplete descriptionofa physicalsystemmusttake
into account not only the system but also its interac-
tion with the environment. In this sense, any physical A. Spherically symmetric solutions
system is surrounded by the rest of the universe. The
relation of the local system with the rest of the universe
The analysisofsphericallysymmetric solutionscanbe
manifests itself in a set of boundary conditions. In our
carried out in several ways. One of them is to write the
case, according to (5) and (6), the metric and the func-
line element using Schwarzschildcoordinates
tion f′ (or, equivalently, R or φ) are subject to bound-
ary conditions, since they are dynamical fields (they are 1
ds2 = B(r˜)dt2+ dr˜2+r˜2dΩ2 (17)
governedbydifferentialequations). Theboundarycondi- − C(r˜)
tionsforthemetriccanbetrivializedbyasuitablechoice
Another possibility, more suitable for the discussion of
of coordinates. In other words, we can make the metric
observableeffects,istouseisotropiccoordinates[37]. We
Minkowskian in the asymptotic region and fix its first
willfollowherethissecondoptionand,forcompleteness,
derivativestozero(seechapter4of[40]fordetails). The
we will discuss the other in the appendix to check the
function f′(R), on the other hand, should tend to the
cosmic value f′(R ) as we move awayfromthe localsys- consistency of our calculations. We define the line ele-
c
ment
tem. The precise value of f′(R ) is obtained by solving
c
theequationsofmotionforthecorrespondingcosmology. 1
ds2 = A(r)e2ψ(r)dt2+ dr2+r2dΩ2 , (18)
Accordingto this, the localsystemwill interactwith the − A(r)
asymptotic(orbackground)cosmologyviatheboundary (cid:0) (cid:1)
value f′(R ) and its cosmic-time derivative. Since the which, assuming a perfect fluid for the sources, leads to
c
the following field equations (see [41] for a different ap-
cosmic time-scale is much larger than the typical time-
proach)
scale of local systems (billions of years versus years), we
can assume an adiabatic interaction between the local 2 5A κ2ρ Rf′ f(R)
r
sgylescttemteramnds stuhcehbaasckf˙g′r(Roucn),dwchosemreodloogtyd.eWnoetecsandetrhiuvastnivee- Arr+Ar(cid:20)r − 4 A (cid:21) = f′ + 2−f′ +(19)
with respect to the cosmic time. + A f′ +f′ 2 Ar
The problem of finding solutions for the local system, f′ rr r r − 2A
(cid:20) (cid:18) (cid:19)(cid:21)
therefore, reduces to solving (6) expanding about the 2 f′ A A2 κ2P Rf′ f(R)
Minkowski metric in the asymptotic region5, and (5) Aψr r + fr′ − Ar − 4Ar = f′ − 2−f′ −
tending to (cid:20) (cid:21)
f′ 2 A
3(cid:3) f′(R )+R f′(R ) 2f(R )=κ2T (15) A r r (20)
c c c c − c c − f′ r − 2A
(cid:20) (cid:21)
wherethesubscriptcdenotescosmicvalue,farawayfrom
where f′ = f′ +ϕ, and the subscripts in ψ ,f′,f′ ,M
the system. In particular, if we consider a weakly grav- c r r rr r
denote derivation with respect to the radial coordinate.
itating local system, we can take f′ = f′ + ϕ(x) and
c Note also that f′ =ϕ and f′ =ϕ . The equation for
r r rr rr
ϕ is, according to (16) and (18),
2
5 Note that the expansion about the Minkowski metric does not Aϕrr = −A r +ψr ϕr − (21)
implytheexistenceofglobalMinkowskiansolutions. Aswewill (cid:18) (cid:19)
see,thegeneral solutionstoourproblemturnouttobeasymp- W(fc′+ϕ)−W(fc′) + κ2(3P ρ)
toticallydeSitterspacetimes. − 3 3 −
5
Equations (19), (20), and (21) can be used to work out where we have defined the effective Newton’s constant
the metric of any spherically symmetric system regard-
κ2 e−mcr
less of the intensity of the gravitational field. For weak G= 1+ (33)
sources,however,itisconvenienttoexpandthemassum- 8πfc′ (cid:18) 3 (cid:19)
ing ϕ f′ and A = 1 2M(r)/r, with 2M(r)/r 1.
| | ≪ c − ≪ and the effective post-Newtonian parameter
The result is
2 κ2ρ 1 2 3 e−mcr
M (r) = +V + ϕ + ϕ (22) γ = − (34)
− r rr f′ c f′ rr r r 3+e−mcr
c c (cid:20) (cid:21)
2 ϕr κ2 This completes the lowest-order solution in isotropic co-
ψ + = P V (23)
r r f′ f′ − c ordinates. The higher order corrections to this solution
(cid:20) c(cid:21) c
2 κ2 will be studied in detail elsewhere [38] (see also [39] for
ϕrr+ rϕr = 3 (3P −ρ)+m2cϕ (24) a recent discussion of those corrections).
where we have defined Had we used the Schwarzschild line element (17), the
result would be (see the appendix for details)
R f′ f(R )
V c c− c (25)
c ≡ 2fc′ ds2 = 1 C2 1+ e−mcr˜ Vcr˜2 dt2+ (35)
f′(R ) R f′′(R ) − − r˜ 3 − 3
m2 = c − c c (26) (cid:18) (cid:20) (cid:21) (cid:19)
c 3f′′(R ) C 1+m r˜ V
c + 1+ 2 1 c e−mcr˜ + cr˜2 dr˜2+r˜2dΩ2
r˜ − 3 3
This expression for m2 was first found in [24] within the (cid:18) (cid:20) (cid:21) (cid:19)
c
scalar-tensor approach. It was found there that m2 > 0 We see that the Newtonian limit, the function in front
c
is needed to have a well-behaved (non-oscillating) New- of dt2, coincides with the result obtained using isotropic
tonian limit. This expressionand the conclusion m2 >0 coordinates (neglecting higher order corrections due to
c
were also reached in [42] by studying the stability of de the change of coordinates). However, the first post-
Sitter space. The same expression has also been found Newtoniancorrectionisnotcorrectlyidentifiedifwejust
more recently in [27, 39]. takethe functioninfrontofdr˜2. Itisnecessarytotrans-
Outside of the sources, the solutions of (22), (23) and form to isotropic coordinates using
(24) lead to
C e−mcr V
r˜ r+ 2 1 cr3 (36)
ϕ(r) = C1e−mcr (27) ≈ 2 (cid:20) − 3 (cid:21)− 12
r
C C V to find the correct expression. Nonetheless, in the limit
A(r) = 1 2 1 1 e−mcr + cr2 (28) m r 1 considered in [25, 26, 27] and neglecting the
− r (cid:18) − C2fc′ (cid:19) 6 Vcr2 ≪terms, the change of coordinates is trivial (dr˜/dr
c
C C V ≈
A(r)e2ψ = 1 2 1+ 1 e−mcr cr2 (29) 1) and (35) agrees with (32). This is, however, just an
− r (cid:18) C2fc′ (cid:19)− 3 accident that leads to the desired result. In general, one
should use isotropic coordinates for the correct identifi-
where an integration constant ψ has been absorbed in
0 cation of the post-Newtonian parameters and take into
a redefinition of the time coordinate. The above so-
account the interaction with the background cosmology
lutions coincide, as expected, with those found in [24]
for the right interpretation of the limit to General Rela-
for the Newtonian and post-Newtonian limits using the
tivity (see next subsection).
scalar-tensor representation. Comparing our solutions
with those, we identify
B. Discussion of the linearized solutions
κ2
C M (30)
2 ≡ 4πf′ ⊙
c We see from (33) and (34) that the parameters G
C 1
1 and γ that characterize the linearized metric depend
(31)
fc′C2 ≡ 3 on the effective mass mc (or inverse length scale m−c1).
Newton’s constant, in addition, also depends on f′.
c
where M⊙ = d3xρ(x). The line element (18) can be Since the value of the background cosmic curvature Rc
written as changes with the cosmic expansion, it follows that f′
R c
and m will also change. The variation in time of f′
2GM V c c
ds2 = 1 ⊙ cr2 dt2+ (32) will induce a time variation in the effective Newton’s
− − r − 3
(cid:18) (cid:19) constant analogous to the well-known time dependence
+ 1+ 2GγM⊙ Vcr2 (dr2+r2dΩ2) that arises in Brans-Dicke theories. Actually, if we use
r − 6 the scalar-tensor representation, the analogy becomes
(cid:18) (cid:19)
6
an identity. The length scale m−1, characteristic of is reduced at the location of the sources, with respect to
c
f(R) theories, does not appears in Brans-Dicke theories the value κ2ρ in the limit of GR, it becomes non-zero
because in the latter the scalar potential is V(φ) 0, in at farther distances from them. In fact, away from the
≡
contrast with (8). sources, (37) can be approximated by
m2κ2M e−mcr
Let us now review how the cosmic expansionproceeds R=V + c ⊙ (38)
in f(R) models with curvature terms that grow at c 4πfc′ r
low curvatures (see also [18, 24, 43] for more details).
which has a smooth 1/r decay when m r 1. Note
During the matter dominated era, the cosmic expansion c ≪
that in the limit of infinite range,m 0, the curvature
takes place as in GR. This is so because Rc and Tc are R does not depend on the local socu→rces. This shows
well above the characteristic scale R of the non-linear
ǫ explicitly that the phylosophythat motivated the design
terms in the lagrangian. In this situation, we find
of f(R) models (see the introduction) was wrong, since
R κ2T , as discussed in the paragraph following
c ≈ − c R not always behaves as in GR.
equation (14). At later times, the sources become very
diluted and R approaches the scale R in which the
c ǫ In summary, we have seen that any theory character-
left hand side of (14) can no longer be neglected. The
izedbynon-lineartermsthatgrowatlowcurvatureswill
scalar curvature then becomes a fully dynamical object
evolve from GR during the matter dominated era to a
and the theory gets into the scalar-tensor dynamical
scalar-tensor theory at later times. This transformation
regime, which triggers the cosmic speedup. Therefore,
takes place at both large (cosmic) and short (solar
the cosmic expansion drives the theory from a General
sytem)scales. Therefore,ifthe cosmicspeedupweredue
Relativistic (decelerated) phase to a scalar-tensor (ac-
tothe growthofnon-linearcorrectionsinthe lagrangian,
celerated) phase. Let us now consider what happens in
today we should be in the scalar-tensor phase, which is
the solar system. During the General Relativistic phase,
characterized by γ 1/2. This fact is in conflict with
tchhaatramctcer≫ized1,bwyhRicch≫leadRsǫ,tofc′e−→mcr1,→fc′′0→rap0id,lwye(vfienrdy s[4o4la])rasnydstemmakoebssietrvva→etriyonusnl(iγkeexlyp t=ha1t+th(e2.c1o±sm2i.c3s)p·e1e0d−u5p,
short scalar interaction range). During this period of
be due to such corrections in the gravity lagrangian.
time, we have γ 1, which coincides with the General
→
Relativistic result and agrees with current observations.
At later cosmic times, when R R , f′′ > 0, the
c ∼ ǫ c
effective mass m becomes finite, decays with time as
c IV. SUMMARY AND CONCLUSIONS
f′′ grows, and eventually leads to e−mcr 1 over solar
c ≈
system scales (very long scalar interaction range). We
In this work we have studied two aspects of modified
then find γ 1/2, which coincides with the value
→ f(R) theories of gravity, namely, their equivalence with
corresponding to the w = 0 Brans-Dicke theory and is
w = 0 Brans-Dicke theories and the weak field solutions
ruled out by observations.
insystemswithsphericalsymmetryusingthef(R)repre-
sentation. We found that the equivalence between f(R)
The change of dynamical regime in local systems can
theories and ω = 0 Brans-Dicke theories only requires
also be seen by looking at the value of R. To illustrate
invertibility of the functions f′(R) and dV/dφ. This re-
this point, we will use the solutions found in [24], which
sult puts forward that other well-known derivations of
arealsovalidwithinthesources. Tolowestorder,wefind
the scalar-tensorrepresentation[22, 23, 36] introduce an
R=V + m2cκ2 d3x′ρ(t, ~x′)e−mc|~x−~x′| (37) artificial condition, f′′(R) 6= 0, that may lead to wrong
c 4πf′ ~x ~x′ conclusions [34] in the limit ǫ 0 in models of the form
c Z | − | f(R)=R+ǫg(R). Weexplicitl→yshowedthatsuchlimitis
Itisnotdifficulttoshow(consideringasmallvolumecen- smooth andthat one recoversGR from the scalar-tensor
tered about ~x ~x′ 0 and integrating by parts) that theory.
thisexpressio|nb−eco|m→esR=κ2ρ(t,~x)inthelimitofGen- We then worked out the spherically symmetric solu-
eral Relativity (R R , f′ 1, f′′ 0, m2 , tions of the linearized field equations. We carried out
c ≫ ǫ c → c → c → ∞
and V 0). As the cosmic expansion takes place and the analysis idependently using two different sets of co-
c
→
R approaches the scale R of the scalar-tensor dynam- ordinates, namely isotropic and also Schwarzschild co-
c ǫ
ical regime (f′ = 1 , f′′ > 0 and growing, m2 finite ordinates. The results are in perfect agreement when
c 6 c c →
anddecreasing)the dependence ofRonthe localmatter one transforms from one coordinate system to the other,
density is softened by the integrationin (37). At a given which confirms the validity of our results. We have also
point ~x, R is then an average over nearby sources with made emphasis on the adiabatic interaction between lo-
weight e−mc|~x−~x′|/~x ~x′ . This shows that the depen- calsystemsandthebackgroundcosmologyviaboundary
| − |
dence of R on the local matter density is less and less conditions. We haveseenthat the parametersthat char-
important as m2 decreases, or equivalently, as the inter- acterizethelinearizedsolutionsdependontheformofthe
c
actionrangel =m−1 ofthescalarfieldgrowsduetothe lagrangianandonthevalueofthecosmiccurvatureR at
c c c
cosmic expansion. Note that thoughthe magnitude of R agivencosmictime. Itisthenstraightforwardtoseethat
7
theGRsolutionisrecoveredinthelimitR R ,where For weak sources it is convenient to expand (A2), (A3),
c ǫ
R representsthecharacteristiclowcurvatur≫escaleofthe and (A4) assuming ϕ f′, D(r˜)=1 2M(r˜)/r˜, with
ǫ | |≪ c −
lagrangian. When R , driven by the cosmic expansion, 2M(r˜)/r˜ 1,andB(r˜)=1 Φ(r˜),withΦ(r˜) 1. The
c
≪ − ≪
approaches the scale R , there is a change of dynamical result is
ǫ
regime and the theory becomes closer to a scalar-tensor
theorythantoGR.Thischangemakesthepredictionsof
2 κ2 1 2
the theory incompatible with solar system observations, M = ρ+V + ϕ + ϕ (A5)
r˜2 r˜ f′ c f′ r˜r˜ r˜ r˜
since γ goes from γ 1 in the GR phase to γ 1/2 c c (cid:18) (cid:19)
∼ →
in the late stages of the scalar-tensor phase. Therefore, 2M(r˜) 2ϕr˜
Φ = +V r˜+ (A6)
since the same mechanism that predicts cosmic speedup r˜ − r˜2 c f′
c
is also responsible for the running of the parameter γ, it 2 κ2
seems very unlikely that this type of f(R) theories may ϕr˜r˜ = −r˜ϕr˜+m2cϕ− 3f′ρ (A7)
be responsible for the observed cosmic acceleration. c
where V and m2 have the same definitions as in the
c c
Appendix A: SCHWARZSCHILD COORDINATES
isotropiccase. The solutionsto the aboveequations lead
to
Using a line element of the Schwarzschildform
1 C˜ e−mcr˜
ds2 = B(r˜)dt2+ dr˜2+r˜2dΩ2 (A1) ϕ(r˜) = 1 (A8)
− D(r˜) r˜
C˜ V
the equations that follow from (6), assuming a perfect M(r) = C˜2− 2f1′[1+mcr˜]e−mcr˜+ 6cr˜3 (A9)
fluid, are c
2C˜ C˜ V
2 f′ D 1 D κ2 Rf′ f(R) Φ(r˜) = 2 1+ 1 e−mcr˜ + cr˜2 (A10)
r˜+ fr˜′ 2r˜ − −r2 = − f′ρ+ 2−f′ + r˜ " 2C˜2fc′ # 3
(cid:20) (cid:21) (cid:20)
D 2
+ f′ + f′ (A2)
f′ r˜r˜ r r˜ Comparing the Newtonian potential Φ(r˜) with the
(cid:18) (cid:19)(cid:21) isotropic solution, we identify C = 2C˜ , C = C˜ . A
D 2 f′ B 1 D κ2 Rf′ f(R) 2 2 1 1
+ r˜ r˜ − = P − transformation from Schwarzschild to isotropic coordi-
2 r˜ f′ B − r2 f′ − 2f′ −
(cid:20) (cid:21) nates[see(36)]showsthatthisidentificationisconsistent
2Df′ with the first post-Newtonian correction, and confirms
r˜ (A3)
− r˜ f′ the validity of our calculations.
where f′ = f′ +ϕ, and the subscripts in D ,B ,f′,f′
c r˜ r˜ r˜ r˜r˜
denote derivation with respect to the radial coordinate.
Note also that f′ =ϕ and f′ =ϕ . The equation for
r˜ r˜ r˜r˜ r˜r˜
ϕ is, according to (16) and (A1),
ACKNOWLEDGMENTS
2 B D
r˜ r˜
Dϕ = D + ϕ (A4)
r˜r˜ − r˜ 2B − 2D r˜− This work has been supported by NSF grants PHY-
(cid:20) (cid:21)
0071044 and PHY-0503366. The author thanks Koji
W(f′+ϕ) W(f′) κ2
c − c + (3P ρ) Uryu for very useful discussions.
− 3 3 −
[1] T. Padmanabhan, AIP Conf.Proc. 861,179-196,(2006), [6] G. Allemandi, A. Borowiec, and M. Francaviglia, Phys.
astro-ph/0603114; see also references therein. Rev.D70, 103503 (2004)
[2] R.R.Caldwell, R.Dave, P.J.Steinhardt,Phys.Rev.Lett., [7] A. Borowiec, W. Godlowski, M. Szydlowski, Phys.Rev.
80,1582 (1998); C.Armendariz-Picon, V.Mukhanov, D74, 043502 (2006); astro-ph/0607639
P.J.Steinhardt, Phys.Rev.D63, 103510 (2001) [8] S. Capozziello, V. F. Cardone, and A. Troisi, Phys.Rev.
[3] T. Padmanabhan, Phys.Rept.380, 235-320 (2003), D71, 043503 (2005)
hep-th/0212290. See references therein. [9] T. Koivisto, Phys. Rev. D73, 083517 (2006);
[4] L.Parker and A.Raval, Phys. Rev.D60, 063512 (1999); Class.Quant.Grav. 23, 4289 (2006)
Phys.Rev.Lett. 86, 749 (2001); L.Parker, W.Komp, and [10] T.KoivistoandH.Kurki-Suonio,Class.Quant. Grav.23,
D.A.T.Vanzella,Ap. J.588, 663 (2003); L.Parker and 2355 (2006)
D.A.T.Vanzella Phys. Rev.D69, 104009 (2004) [11] Bao Li, M.-C. Chu, Phys.Rev. D74, 104010 (2006)
[5] M.Amarzguioui et al.,Astron.Astrophys.454,707 (2006) [12] X.H. Meng and P. Wang,Class.Quant.Grav.20,
8
4949 (2003); Class.Quant.Grav.21, 2029 (2004); [26] Xing-Hua Jin, Dao-Jun Liu, Xin-Zhou Li,
Class.Quant.Grav.22, 23 (2005); Class.Quant.Grav.21, astro-ph/0610854
951 (2004); Phys.Lett. B584, 1 (2004); [27] T. Chiba, T.L. Smith,A.L. Erickcek, astro-ph/0611867
[13] G.J. Olmo, gr-qc/0612002 [28] A.F. Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso,
[14] G.J. Olmo and W. Komp, gr-qc/0403092 Phys.Rev. D74,107101 (2006)
[15] N.J. Poplawski, Phys.Rev. D74, 084032 (2006); [29] S.Nojiri and S.D.Odintsov,Phys.Rev.D74,086005(2006)
Phys.Lett. B 640, 135 (2006); Class.Quant.Grav. 23, [30] S.Capozziello and A.Troisi,Phys. Rev.D72,044022(2005)
4819 (2006); Class.Quant.Grav. 23, 2011 (2006) [31] G.Allemandi, M.Francaviglia, M.Ruggiero, A.Tartaglia,
[16] T.P. Sotiriou, Class.Quant.Grav. 23, 1253 (2006); Gen. Rel. Grav. 37, 1891 (2005)
Phys.Rev. D73,063515 (2006) [32] M.L. Ruggiero and L.Orio, gr-qc/0607093
[17] K.Atazadeh and H.R. Sepangi, gr-qc/0602028 [33] G.Allemandi and M.L. Ruggiero, astro-ph/0610661
[18] S.M.Carroll,V.Duvvuri,M.Trodden,andM.S.Turner, [34] V.Faraoni, Phys.Rev. D74, 023529 (2006)
Phys. Rev.D70, 043528 (2004) [35] T.P. Sotiriou, Gen.Rel.Grav.38, 1407 (2006)
[19] S. Capozziello, Int. J. Mod. Phys.D11, 483 (2002); [36] P.TeyssandierandP.Tourrenc,J.Math.Phys.24,2793(1983)
S. Capozziello, S. Carloni and A. Troisi, Recent Re- [37] C.W.Misner, S.Thorne, and J.A.Wheeler, Gravitation ,
search Developments in Astronomy & Astrophysics, W.H.Freeman and Co., NY,1973.
RSP/AA/21-2003, astro-ph/ 0303041 [38] Gonzalo J. Olmo and Koji Uryu,work in progress.
[20] S. Nojiri and S.D.Odintsov,Phys.Lett.576B,5(2003); [39] I. Navarro, K. Van Acoleyen,gr-qc/0611127
Phys.Rev.D68,123512(2003);Gen.Rel.Grav.36,1765(2004) [40] C.M.Will, Theory and experiment in gravitational
[21] S.M. Carroll et al.,Phys.Rev. D71, 063513 (2005) physics, Cambridge 1993.
[22] T.Chiba, Phys.Lett.576B,5-11 (2003) [41] T.Multamaki and I.Vilja,Phys.Rev.D74,064022(2006)
[23] E.E.Flanagan, Phys.Rev.Lett. 92, 071101 (2004) [42] V. Faraoni and S. Nadeau, Phys. Rev.D 72, 124005
[24] G.J. Olmo, Phys.Rev.Lett.95, 261102 (2005); Phys.Rev. (2005)
D72, 083505 (2005) [43] R.Dick, Gen.Rel.Grav. 36,217-224 (2004).
[25] A.L. Erickcek, T.L. Smith, M. Kamionkowski, [44] B.Bertotti, L.Iess and P.Tortora,Nature 425,374(2003)
astro-ph/0610483