Robust approach to f(R) gravity

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by  Luisa G. Jaime

Robust approach to f(R) gravity
Luisa G. Jaime1,2,∗ Leonardo Patin˜o2,† and Marcelo Salgado1‡
1Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico, A.P. 70-543, M´exico D.F. 04510, M´exico
2 Facultad de Ciencias, Universidad Nacional Auto´noma de M´exico, A.P. 50-542, M´exico D.F. 04510, M´exico
(Dated: February 2, 2011)
We consider metric f(R) theories of gravity without mapping them to their scalar-tensor coun-
terpart, but using the Ricci scalar itself as an “extra” degree of freedom. This approach avoids
then the introduction of a scalar-field potential that might be ill defined (not single valued). In
ordertoexplicitlyshowtheusefulnessofthismethod,wefocusonstaticandsphericallysymmetric
spacetimesanddealwiththerecentcontroversyabouttheexistenceofextendedrelativisticobjects
1 in certain class of f(R) models.
1
0 PACSnumbers: 04.50.Kd,04.40.Dg,95.36.+x
2

n I. INTRODUCTION gravity (STT) (cf. [7]). Under this mapping, the Ricci
a
J scalar has a behavior of the sort R 1/(χ χ0), where
∼ −
1 Modified f(R) theories of gravity are metric theories χ := ∂Rf is the scalar-field associated with the corre-
3 whichpostulateanaprioriarbitraryfunctionoftheRicci sponding STT and χ0 = const. The key point is to de-
scalar R as the Lagrangian density. These theories have termine if the dynamics of χ leads it or not to the value

] beenproposedtoexplainthelatetimeacceleratedexpan- χ=χ0 withinthespacetimegeneratedbytherelativistic
c
q sion of the Universe as well as a mechanism to produce object. Irrespectiveofthe differentresultsandconfusing
- inflation in the early Universe in terms of geometry in- explanations obtained in [4–6], we will argue that their
gr steadofintroducinganydarkenergyideasorascalarin- conclusions are rather questionable due to the fact that
[ flaton[1]. Thankstotheaboveproperties,thesetheories the mapping to the STT is ill defined. To be more spe-
havebecomeoneofthemostpopularalternativetheories cific,thescalar-fieldpotentialusedtostudythedynamics
3
of gravity. In recent years, several specific f(R) models of χ is not single valued and possesses pathological fea-
v
7 have been analyzed in different settings (see Ref. [2], tures. Since similar kind of singularities were also found
4 for a review). Although the early models were ruled in the cosmological setting [8], it is then worrisome that
7 out for failing several tests (like the Solar System ex- the ill-defined potential play such a crucial role in those
5 periments) new proposals were put forward to overcome analyses (cf. Refs. [9, 10] for further criticisms. See also
6. such drawbacks. Nevertheless, a considerable amount of [11]and referencesthereinfor a more detaileddiscussion
0 analysis and observational confrontation is still required on cosmologicalsingularities in f(R) theories).
0 in order to compare the preliminary successes of such The goal of this communication is threefold: 1) to
1 theories with the great achievements of general relativ- propose a straightforward and robust approach which
v: ity (GR). In particular, the models that were claimed consists in recasting the field equations in a more suit-
i to pass several cosmological and local tests have been able way without mapping the original f(R) theory to
X
scrutinizedinthe stronggravityregimeonly recently. In anyscalar-tensorcounterpart. This methoddispensesus
r fact, the first studies concerning their ability to describe from dealing with ill-defined quantities that might arise
a
relativistic extended objects, like neutron stars, seem to when performing such transformation; 2) to reanalyze
givecontradictoryresults. Usinga f(R)modelproposed carefully the issue about the existence of relativistic ex-
by Starobinsky [3], Kobayashiand Maeda [4] found that tended objects using our approach; 3) to stress that the
suchrelativisticobjectsweredifficulttoconstructsincea analysisoff(R)theoriesbasedontheSTTanaloguewith
curvaturesingularitydevelopedwithintheobject. Later, ill-definedquantitiesisnottrustworthy,andthatincases
this issue was reanalyzed by several authors [5, 6], who wherethe STTapproachiswelldefined(thisdependson
foundthatsingularity-freerelativisticobjectscanindeed the specific f(R)model) it turns out to be ratherconvo-
be constructed, arguing that the conclusion reached in lutedandnotveryinsightful. Sincef(R)theoriesarestill
[4] was due to a bad choice of the matter model (i. e. undercloseexamination,asoundapproachisrequiredto
the equation of state, hereafter EOS) [5], while in [6] it treat them appropriately. This is the first step in that
wasclaimedthata“chameleon”istheresponsibleforthe direction.
existence of such “stars,” regardless of the EOS.
A common feature to all of the aforementioned
works [4–6] is that the analysis was performed by map- II. f(R) THEORIES OF GRAVITY
ping the Starobinsky model to a scalar-tensor theory of
Thegeneralactionforaf(R)theoryofgravityisgiven
by
∗ [email protected]
† [email protected] f(R)
‡ [email protected] S[gab,ψ]= √ gd4x+Smatt[gab,ψ], (1)
2κ −
Z
where R stands for the Ricci scalar, κ := 8πG0, and variables will also be functions of r. The analysis of this
ψ represents collectively the matter fields (we use units kind of spacetimes is interesting in many ways;however,
where c = 1). The field equation arising from varying we will focus here only on the issue about the existence
the action Eq. (1) with respect to the metric is (or absence thereof) of well behaved compact objects.
Equation (4) yields
1
f R fg ( g 2)f =κT , (2) 1
R ab− 2 ab− ∇a∇b− ab R ab R′′ = m(κT +2f Rf ) 3f R′2
R RRR
3f − −
RR
where fR := ∂Rf and 2 = gab∇a∇b. It is straightfor- + mh′ n′ 2 R′ . i (6)
ward to write the above equation in the following way
2m − 2n − r
(cid:18) (cid:19)
fRGab fRR a bR fRRR( aR)( bR) (where ′ := d/dr). From the t t, r r, and θ θ
− ∇ ∇ − ∇ ∇ − − −
1 components of Eq. (5) and using also Eq. (6) we find
+g (Rf f)+f 2R+f ( R)2 =κT ,(3)
ab R RR RRR ab
2 − ∇
(cid:20) (cid:21) m′ = m 2f (1 m) 2mr2κTt
where ( R)2 := gab( aR)( bR). Taking the trace of r(2fR+rR′fRR)( R − − t
∇ ∇ ∇
this equation yields mr2 rR′f mr2
RR
+ (Rf +f +2κT)+ (2Rf f +κT)
R R
2R= 1 κT 3f ( R)2+2f Rf , (4) 3 fR h 3 −
RRR R
3f − ∇ −
where T :=TRaR.hFinally, using Eq. (4) in (3), wie find −κmr2(Ttt+Trr)+2(1−m)fR+2rR′fRR ), (7)
a i
Gab = f1R fRR∇a∇bR+fRRR(∇aR)(∇bR) n′ = r(2fR+nrR′fRR) mr2(f −RfR+2κTrr)
g h h
− 6ab RfR+f +2κT +κTab . (5) +2fR(m−1)−4rR′fRR , (8)
(cid:16) (cid:17) i 2nm 1 i R′
Equations (4) and (5) are the basic equations for f(R) n′′ = κTθ (Rf +f +2κT)+ f
f θ− 6 R rm RR
gravity that we propose to treat in every application, R
instead of transforming them to STT. Now, several im- n h m′ n′ rn′ m′ n′ i
+ 2 + + . (9)
portant remarks are in order regarding this set of equa- 2r m − n n m n
tions. First, aside from the case f(R) = R, where the h (cid:18) (cid:19) (cid:18) (cid:19)i
Note that Eqs. (8) and (9) are not independent. In fact,
field equations reduce to those of GR, one is to consider
onehasthefreedomofusingoneortheother(seeSec.IV
functions f(R) such that f ,f > 0 (i. e. monoton-
R RR below). Now, from the usual expressionof R in terms of
ically growing and convex f(R) functions) in order to
the Christoffel symbols one obtains
avoid potential blowups in the field equations. These
two conditions have been consideredpreviously; the first R= 1 4n2m(m 1)+rnm′(4n+rn′)
oneleadstoaapositivedefinitegravitational“constant” 2r2n2m2 −
Geff =G0/fR when regardingf(R) theories as effective 2rnm(2hn′+rn′′)+r2mn′2 . (10)
theories. The condition f >0 was suggested to avoid −
RR
i
gravitational instabilities [12]. Second, Eq. (5) supplies As one can check by a direct calculation, using
a second orderequationfor the metric providedthat the Eqs.(7) (9)inEq.(10)leadstoanidentityR R. This
− ≡
Ricci scalar is considered as an independent field. This result confirms the consistency of our equations. When
is possiblethanksto Eq.(4)whichprovidesthe informa- defining the first order variables Q =n′ and Q :=R′,
n R
tion needed to solve Eq. (5) for the metric alone (given Eqs. (6) (9) have the form dyi/dr = i(r,yi), where
a matter source). yi = (m,−n,Q ,R,Q ), and therefore caFn be solved nu-
n R
As will become evident below, this approach is rather merically. As far as we are aware, the system (6) (9)
−
clean and free of the pathologies that can arise when has not been considered previously (see Ref. [13] for an
mapping f(R) gravity to STT. alternative approach). These equations can be used to
tackle several aspects of SSS spacetimes in f(R) grav-
ity. Finally, we observe that for f(R) = R the above
III. STATIC AND SPHERICALLY SYMMETRIC
equationsalsoreducetothe wellknownequationsofGR
SPACETIMES
for SSS spacetimes. When dealing with extended ob-
jects,notablythosedescribedbyperfectfluids,theabove
In order to give some insight to the usefulness of equations are supplemented by the matter conservation
our approach, we consider a static and spherically sym- equations. TheBianchiidentitiesimply the conservation
metric (SSS) spacetime with a metric given by ds2 = of the effective energy-momentum tensor [which corre-
n(r)dt2 + m(r)dr2 + r2 dθ2+sin2θdϕ2 , where the sponds to the right-hand-side (rhs) of Eq. (5)] which to-
−
metric coefficients n and m are functions of the coor- gether with Eqs.(4) and (5) lead to the equationof con-
dinate r solely. The Ricci(cid:0)scalar as well a(cid:1)s the matter servation aT =0forthematteralone. Soforaperfect
ab
∇
2
fluidwithT =(ρ+p)u u +g p,theconservationequa- different from the scalar-fieldpotential that arises under
ab a b ab
tionleadstop′ = (ρ+p)n′/2n[wheren′ isgivenexplic- the STT map. Furthermore, V(R) is as well defined as
−
itly by the rhs of Eq. (8)]. This is the modified Tolman- the function f(R) itself.
Oppenheimer-Volkoffequationofhydrostaticequilibrium A technical difficulty that one faces when integrating
to be complemented by an EOS. the equations for neutron star models that asymptoti-
Inordertosolvethedifferentialequationssomebound- cally match realistic values of the cosmological constant
ary conditions must be supplied, which in this case are is that a fortiori two completely different scales are in-
rather regularity and asymptotic conditions. Regular- volved. On one hand ρ(0) ρnuc 1014gcm−3; on the
∼ ∼
ity (smoothness) at r = 0 implies the following ex- other Λ˜ 10−29gcm−3 (Λ˜ = Λ/G0). That is, there are
pansion near r = 0: φ(r) = φ0 + φ1r2/2 + (r4) around 4∼3 orders of magnitude between the typical den-
(where φ stands for m,n, or R). This implies Om′ = sity within a neutron star and the averagedensity of the
0 = n′ = R′ at r = 0. We choose m(0) = 1 (local- Universe. This ratiobetweendensities naturallyappears
flatness condition) and n(0) = 1. The coefficients φ0 in the equations since the parameters which define the
and φ1 associated with m,n,R, and the matter vari- specific f(R) theory are of the order of Λ˜, while the ap-
ables(andwhichcorrespondtothe valuesofthese quan- propriate dimension within neutron stars is ρnuc. So in
tities and its second derivatives evaluated at r = 0, units of ρnuc, the cosmological constant turns out to be
respectively) are related to each other. For instance, ridiculouslysmall,whileinunits ofΛ˜, ρ(r)andp(r)turn
from the above power expansion and from Eqs. (9), (6) out to be ridiculously large within the neutron star. So
and (10), we find n′′(0) = f0−2fR0R0−4kT0+18kTθθ0 and far, the authors that have studied relativistic objects in
9fR0 f(R) gravity [4–6, 9] have faced this kind of technical
R′′(0) = 2f0−f9R0fR00+kT0, where the quantities at the rhs problem, and, in order to avoid it, they have only con-
RR
are evaluated at r =0 1. structedobjects whichare far fromrepresentingneutron
Now,asconcernstheasymptoticconditions,wearein- stars embedded in a realistic de Sitter background. This
terested in finding solutions that are asymptotically de means that either the background is realistic while the
Sitter, since the solutions are supposed to match a cos- objects are ridiculously large (several orders of magni-
mologicalsolutionthatrepresentstheobservedUniverse. tude largerthanarealneutronstar)andnotverydense,
Therefore, we demand that asymptotically R R1, or the objects are realistic but Λ˜eff is far too large. Such
→
whereR1 is acriticalpoint(maximumorminimum) ofa objects are relativistic in the sense that their pressure is
potential whichis definedbelow. The value R1 allows to of the same order of magnitude as their energy-density,
define the effective cosmologicalconstantas Λeff =R1/4 andtheratiobetweenasuitablydefinedmassandradius
(like in GR with Λ)2. We mention that at the end of G0 / is not far from 4/9.
M R
the numericalintegration,werescalen(r) inorderto get
the canonical asymptotic behavior n(r) 1 Λeffr2/3.
This rescalingamounts toredefine the t c∼oord−inate. The IV. NUMERICAL RESULTS
asymptotic behavior R R1 depends on the value of
→
R(0). The details of the correlation between R(0) and In order to test our method, we used first
R1 depend on the matter model, and once this latter is f(R) = R αR ln(1+R/R ) (α,R are positive
∗ ∗ ∗
fixed (for instance, an incompressible fluid with a given constants; R−sets the scale), which was proposed in
∗
centralpressure)thevalueR(0)canbefoundbyashoot- Ref. [9]. In that analysis the authors mapped the theory
ing method [14]. Since we are interested in finding an totheSTTcounterpart. However,unliketheStarobinsky
exterior solution of Eq. (6) with R R1 (and R′ 0) model (see below), in this case the resulting scalar-field
→ →
asymptotically, we observe that such a solution exists potential turns to be single valued and the authors of
if R1 corresponds to a critical point of the “potential” [9] did not find any singularity within the object. Under
R
V(R) = Rf(R)/3+ f(x)dx. That is, R1 is a point our approach, we associate to this f(R) the potential
−
wfRhRer(eR1d)V=(R)0/dinRE=q.(R(26f)]−.fRTRh)is/3povtaennitsihalesis[arsasduimcailnlyg V(R˜)= R62∗ (1+R˜)(R˜+6α−1)−2α(3+2R˜)ln(1+R˜) ,
6 where R˜ =nR/R . This potential has several criticoal
∗
points. Figure 1 (top panel, right inset) depicts the
potentialforα=1.2. Equations(6) (8)wereintegrated
1 NotethatinGRthefieldequationsimplythealgebraicrelation- using a four-order Runge-Kutta alg−orithm by assuming
ship R=−kT, which determine the value of R in terms of the an incompressible fluid with the same density and
matter content solely. In f(R) theories this relationship is dif-
central pressure as in [9]. By implementing a shooting
ferential [cf. Eqs. (4) and (6)], and therefore the conditions for
R(0), R′′(0) andeven forn′′(0) arenotfixed inadvance bythe method [14], we found a solution for R that goes from
mattercontentonly. WecanrecovertheusualexpressionsofGR the globalminimum V(R˜1) of the potential to a positive
2 wOhneenctaankienagsifly(Rs)ho=wRt.hat the system Eqs. (6)−(9) in vacuum value. The minimum at R˜1 corresponds to the de
has the exact de Sitter solution n(r)=m(r)−1 =1−Λeffr2/3, Sitter value and gives rise to the effective cosmological
R = R1 = const. with Λeff = R1/4 and R1 = 2f(R1)/fR(R1), constant Λeff = R1/4. Figure 1 (top panel) depicts
whichcorrespondspreciselywheredV(R)/dR=0,asdefined in the solution for R, the pressure (top panel, left inset)
themaintext. and the metric potentials (bottom panel) plotted up to
3
the value where n(r) reaches the cosmological horizon oscillatory behavior around R0 seems a priori not ade-
given by n(r ) = 0. We checked that asymptotically quate to produce the required de Sitter background; we
h
the metric potentials matched perfectly well the de plan to analyze those solutions in more detail elsewhere.
Sitter solution n(r) = m(r)−1 = 1 Λeffr2/3. We also Here, we are only interested in showing a solution for
−
verified that our solution corresponds to the one found which R goes to R1 (a local minimum). Figure 2 shows
in [9] and also confirmed that no singularity whatsoever the numericalsolutionwhich is asymptotically de Sitter.
was encountered. Furthermore, we checked that the As one can appreciate, no singularity in the Ricci scalar
numericalsolutions foundusing the systemEqs.(6) (8) was encountered in this solution and R never crosses
−
andthesystemEqs.(6),(7)and(9)gavethesameresults the two real-valued zeros of f (which correspond to
RR
withinarelativeerrornolargerthan 10−6andthatfor R = R /√3) nor those of f at which blowups in the
∼ ± ∗ R
both systems the identity Eq. (10) was satisfied within equations can be produced. Even if for the static and
a relative error of 10−10. In this way, we ensured spherically symmetric solutions, one can avoid the zeros
∼
that no mistakes were introduced in the code during of f and f , the Starobinsky model does not satisfy
R RR
the typing of the equations. We emphasize that for this the conditions f ,f > 0 in general, and so, in other
R RR
model the condition fRR >0 is satisfied by construction settings (e. g. cosmology) one has to keep this in mind
but the condition fR > 0 is not in general. However, when solving the equations.
for the solutions we found, this latter is always satisfied,
particularly at R1. Having analyzed the solutions for
this f(R), we turned our attention to the Starobinsky
modelgivenbyf(R)=R λR 1 1+(R/R )2 −β
∗ ∗
− −
with β = 1. This is the mnodel (cid:2)analyzed in (cid:3)[4–6o]
using the transformation to STT. For such a model,
the associated scalar-field potential is multiple-valued
(cf. Fig. 1 in [8] and Fig. 2 in [4]). Despite the
potential being ill-defined, the authors in [4] argue
that in the particular region where the scalar-field is
evaluated, the potential is nevertheless well defined. It
seems suspicious that in [5, 6] the odd features of the
potential are not mentioned at all. Even if one tried
to ignore the region where the potential is pathological
and consider only the single-valued region, there is no
guarantee that the ill-defined region will not play any
role in more general settings than the very specific case
of static and spherically symmetric compact objects.
Modifications of the Starobinsky model by adding a
quadratic term do not cure this pathology, as one
can see in Fig. 5 of [15]. As we stressed before, the
solutions found in [4–6] gave rise to a controversy
FIG. 1. (Top) Ricci scalar and pressure (left inset) as func-
since in [4] the authors found a singularity in the Ricci tions of r taking ρ = ρ0 = const. = 5×107R1/G0, p(0) =
scalar within the extended object, while in [5, 6] the 0.3ρ0, and R1/R∗ ∼ 1.405 as in [9]. The stars depict the
authorsdidnot. Followingourapproach,wegetV(R˜)= location of the object’s surface. Potential V(R˜) (right inset)
R2∗ R˜ R˜ 4λ 2λ 1+R˜2 −1 +3λarctan(R˜) . in units of R∗2, for the model f(R) = R−αR∗ln(1+R/R∗)
3 2 − − withα=1.2. (Bottom)metricpotentialsassociated withthe
Thi(cid:26)s po(cid:20)tential has a(cid:16)rich st(cid:17)ruct(cid:21)ure depending o(cid:27)n the solution for R.
value of λ (such structure arises also for the potential
discussed above for different values of α). Figure 2 (top
panel, left inset) depicts the potential for three values
V. CONCLUSIONS
of λ. There is a value λcrit = 8/√27 below which the
potential has only one critical point (the global mini-
We have devised a straightforward and robust ap-
mum) located at R = 0. In this regime (λ < λcrit), we
proachto treat f(R) metric gravitywithout resorting to
have found only oscillatory solutions around the global
the usual mapping to scalar-tensor theories. With this
minimum. For λ > λcrit, three critical points appear,
method, it is possible to analyze in a rather transparent
with a minimum always located at R = 0 (R0). In view
and well defined way several aspects of these alterna-
of this structure, several kinds of solutions are possible
tive theories. In particular, we focused on the existence
foranygivenvalue ofλ>λcrit. We havefoundsolutions
of relativistic extended objects embedded in a de Sitter
where R asymptotically approachesa minimum of V(R)
background and concluded that for some f(R) models
at R1 =0, the local maximum, and also solutions where
6 such objects can be constructed without ambiguity and
R oscillates asymptotically around R0. An asymptotic
without resorting to any dubious explanations based on
4
the use of ill-defined quantities. It seems that the anal-
ysis of the solutions presented here as well as other (i.
e. solutions that admit different de Sitter backgrounds,
even possibly Minkowski backgrounds) can only be car-
ried out easily following our method since our potential
V(R) depicts many features in a rather clear and clean
fashionthatallowsustoidentifythecriticalpointswhich
R can reach asymptotically.
Building realistic neutron stars with a realistic de Sit-
ter background in f(R) gravity still remains a technical
challenge. In the near future, we plan to study in more
detailthis issue alongwith other aspects off(R)gravity
using our approach.
ACKNOWLEDGMENTS
Special thanks are due to S. Jor´as and D. Sudarsky.
This work was partially supported by CONACyT Grant
FIG. 2. Similar as Fig. 1 for the Starobinsky model, with No. SEP-2004-C01-47209-F and by DGAPA-UNAM
λ=1.56andβ=1takingρ0=106R1/(16πG0),p(0)=0.1ρ0 Grants No. IN119309-3, No. IN115310, and No. IN-
with R1/R∗ ∼1.983. 108309-3.
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