Can higher curvature corrections cure the singularity problem in f(R) gravity?

0.99 MB·English

by  Tsutomu Kobayashi

WU-AP/295/08
Can higher curvature corrections cure the singularity problem in f(R) gravity?
Tsutomu Kobayashi1∗ and Kei-ichi Maeda1,2†
1 Department of Physics, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan
2 Advanced Research Institute for Science and Engineering,
Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan
Although f(R) modified gravity models can be made to satisfy solar system and cosmological
constraints,ithasbeenshownthattheyhavetheseriousdrawbackofthenonexistenceofstarswith
strong gravitational fields. In this paper, we discuss whether or not higher curvature corrections
can remedy the nonexistence consistently. The following problems are shown to arise as the costs
one must pay for the f(R) models that allow for neutrons stars: (i) the leading correction must
be fine-tuned to have the typical energy scale µ . 10−19 GeV, which essentially comes from the
9 free fall time of a relativistic star; (ii) the leading correction must be further fine-tuned so that it
0 is not given by thequadraticcurvatureterm. Thesecond problem is caused because there appears
0 an intermediate curvature scale, and laboratory experiments of gravity will be under the influence
2 of higher curvaturecorrections. Our analysis thus implies that it is a challenge to construct viable
f(R) models without very careful and unnatural fine-tuning.
n
a
J PACSnumbers: 04.50.Kd,04.40.Dg,95.36.+x

5
1
I. INTRODUCTION However, the potentially viable models of [13, 14, 15]

turnouttohaveaseriousdrawbackinthestronggravity
]
h regime. Thatis,adeep(but notdiverging)gravitational
p The origin of the current accelerated expansion of the potential drives the effective scalar degree of freedom to
-
o Universe [1] is one of the biggest mystery in cosmology. a curvature singularity. This problem was first pointed
r The accelerated expansion may be driven by some un- outbyAppleby andBattyeinacosmologicalsetting[24]
t
s known energy-momentum component. A more intrigu- and then discussed by Frolov in a general context [25].
a ing possibility is that the acceleration could be due to In the previous paper [26], we have studied relativistic
[
long distance modification of gravity. A simple class of starsinf(R)gravityandshownexplicitlythatstarswith

2 modified gravity theories can be constructed by general- stronggravitationalfieldsdevelopcurvaturesingularities
v izingtheEinstein-HilbertLagrangiantosomefunctionof andhenceareprohibited. Thecriticalvalueofthepoten-
4 the Ricci scalar, f(R) [2]. Various models of f(R) grav- tial is typically given by Φ 0.1, implying problematic
6 | |∼
ity have been proposed [3, 4], but inappropriate choices nonexistenceofneutronstarsinthemodelsof[13,14,15].
6
of the function readily cause unwanted instability [5]
5
or gross violation of solar system constraints [6, 7, 8].
.
0 In this paper, we continue our program of study-
The troubles arise due to an extra propagating scalar
1 ing strong gravity aspects of f(R) gravity, and discuss
degree of freedom, and hence viable f(R) models must
8 whether or not higher curvature corrections to the orig-
be constructed in such a way that the dynamics of this
0
inal models can resolve the singularity problem. This is
: scalar field is carefully controlled. This is in principle
v done again by constructing relativistic star solutions. A
possible, and indeed f(R) theories can be made to sat-
i higher curvature correction changes the structure of the
X isfy solar system and laboratory tests by invoking the
effectivepotentialforthescalardegreeoffreedomaround
chameleon mechanism [9, 10, 11]. The key ingredient
r
the singularity [13, 27]. We consider a modified version
a of the chameleon mechanism is the density-dependent
ofStarobinsky’sf(R)[13],addingacorrectiontermpro-
massofthescalarfield;itmediatesashort-rangeforcein
portional to Rm (m 2). We also check whether or not
high density environments such as the solar interior and ≥
the chameleon mechanism works to pass local gravita-
vicinity. (The actual mechanism to hide the chameleon
tional tests in this modified f(R) model. Although we
field is slightly more involved [9].) Concrete examples of
focus on the specific model, our result will hold in the
“chameleon f(R)” are found in [12, 13, 14, 15] (see also
other similar models of this class.
Refs.[16,17,18,19,20]). Theyaretheonlyknownexam-
ples of viable f(R) models that exhibit no problems and
no pathologies in the weak gravity regime [21, 22, 23].1 Thispaperisorganizedasfollows: Inthenextsection,
we describe the field equations of f(R) modified gravity
in terms of a scalar-tensor theory. Then, in Sec. III, we
define the specific theory we consider. Our numerical
∗Email: tsutomu”at”gravity.phys.waseda.ac.jp results are presented in Sec. IV. In Sec. V, we discuss
†Email: maeda”at”waseda.jp
1 Themodelof[12],whichbelongstoadifferentclassofthemod- local tests of gravity in the f(R) model and point out
the problem associated with the high energy correction
els[13,14,15],ishardlydistinguishablefromΛCDMcosmology
becauseoftheverystrongexperimentalconstraints [22]. term. We draw our conclusions in Sec. VI.
2
II. f(R) GRAVITY AS A SCALAR-TENSOR
THEORY
A. Field equations
The action we consider has the form of
f(R)
S = d4x√ g + , (1)
m
− 16πG L
Z (cid:20) (cid:21)
wheref(R)isafunctionofthe RicciscalarR,and is
m
L
the Lagrangian of matter fields. Variation with respect
to metric leads to the field equations2
1 FIG. 1: The potential V. The inset shows the structure
f R f + 2f f g =8πGT , (2)
R µν −∇µ∇ν R R− 2 µν µν around the de-Sitterminimum. The potential of the original
(cid:18) (cid:19) model(withoutRm term)isshownbyabluelineforpurpose
where f := df/dR and T := 2δ /δgµν +g . ofcomparison. Parametersaregivenbyλ=2,n=1,m=2,
R µν − Lm µνLm andε=5×10−4. Thepointχ=1correspondstoacurvature
The trace of Eq. (2) reduces to
singularityintheoriginalmodel,buttheRm termpushesthe
curvaturesingularity toward infinity,χ=∞.
8πG 1
2f = T + (2f f R). (3)
R R
3 3 −
radial profile of χ through Eq. (5), it is useful to note
We now introduce an effective scalar degree of free-
that the equation can be written as
dom, whichsometimes is dubbed “scalaron,”bydefining
χ := fR. Inverting this relation, the Ricci scalar can be d2χ 2dχ dU
expressed in terms of χ: R=Q(χ). In this way Eqs. (2) + = + , (7)
dr2 r dr −dχ F
and (3) are equivalently rewritten as [28]
where
χGν = 8πGT ν + ν δν2 χ χ2V(χ)δν,(4)
µ µ ∇µ∇ − µ − µ dU 2χ3dV
2χ = 8πGT + 2(cid:0)χ3dV , (cid:1) (5) dχ =− 3 dχ (8)
3 3 dχ
and = (8πG/3)T. Here we have ignored the effect of
where the potential V is given by F
the metric for simplicity. (Later we will solve the full set
of the field equations numerically.) Now, by identifying
1
V(χ):= [χQ(χ) f(Q(χ))], (6) r as a time coordinate, Eq. (7) can be regarded as the
2χ2 −
equation of motion in classical mechanics. One can un-
derstand the radial profile of χ intuitively as the motion
and dV/dχ=[2f(Q(χ)) χQ(χ)]/(2χ3).
− ofaparticleinthepotentialU underthetime-dependent
Eqs. (4) and (5) are equivalent to the Jordan frame
force (andthefrictionalforcecorrespondingtothesec-
equationsofmotionintheBrans-Dicketheorywithω =0 F
ondterm inthe left hand side). The mechanicalanalogy
plus a potential V(χ). One can move to the Einstein
isparticularlyusefultocomprehendtheessentialpointof
framebyperformingtheconformaltransformationg˜ =
µν the nonexistence statement for relativistic stars in f(R)
χgµν withχ=exp( 16πG/3φ),whereφisthecanonical gravity [26].
scalarfield. ThepotentialforφisthengivenbyV(χ(φ)).
p
However, we do not work in the Einstein frame in the
following discussion. III. ADDING HIGHER CURVATURE
CORRECTIONS TO f(R) GRAVITY
B. Classical mechanical analogy In the previous paper [26] we studied the strong grav-
ity aspect of Starobinsky’s f(R) theory described by
Wearegoingtoinvestigatestatic,sphericallysymmet- f(R)=R+λR0[(1+R2/R02)−n−1][13]. Thereweshowed
ric stellar solutions in the above system. To study the that stars with strong gravitational fields (e.g., neutron
stars) cannot exist in this model. We argued that this
statement applies to the other similar models [14, 15] as
well. This problem arises due to the dynamics of the ef-
2 In this paper, we focus on the metric approach rather than the fective scalardegree offreedom, χ, in the highcurvature
Palatinione. regime. Therefore, the problem may be cured by adding
3
A de Sitter solution, R = R = constant, minimizes
1
the potential V(χ), and hence is found by solving the
algebraic equation
2f(R ) R f =0. (10)
1 − 1 R|R=R1
We may define the effective “cosmological constant” as
Λ :=R /4.
eff 1
The scalar field χ is written in terms of R as
R R2 −n−1 R m−1
χ=1 2nλ 1+ +m . (11)
− R R2 µ2
0 (cid:18) 0(cid:19) (cid:18) (cid:19)
In the original model without the Rm correction, a cur-
vature singularity R = corresponds to a finite χ
FIG.2: TheeffectivepotentialU. Theinsetshowsthestruc- (χ = 1) and this is very∞close to the de Sitter mini-
turearoundthede-Sitterextremum. Theeffectivepotentialof mum, χ = χ(R ). However, as is clear from Eq. (11),
theoriginalmodel(withoutRm term)isshownbyablueline 1 1
the dangerous curvature singularity now corresponds to
for purpose of comparison. Parameters are given by λ = 2,
n = 1, m = 2, and ε = 5×10−4. The dangerous curvature χ = ∞, and hence one may expect that this model is
singularity is pushedtoward χ=∞ by theRm term. safe. A typical form of the potential V(χ) is shown in
Fig. 1. The effective potential U(χ) is also plotted in
Fig. 2. A straightforwardcalculation shows V R−m+2
forR µ2. Therefore,V const. asR f∝orm=2,
highercurvaturecorrectionsthatmodifythestructureof
≫ → →∞
while V 0 in the same limit for m 3. Similarly, we
thepotentialaroundthelargeRregion,asalreadynoted
→ ≥
have
in the original reference [13] and later discussed in [27].
In general, higher curvature corrections may be writ- dU Rm
otefnthaesale2aRd2in+gao3rRd3er+t·e·r·m,awnidllsboethRe2/mµo2s.t3nTathueraRl2chteoricme 3dχ ≈−R+(m−2)µ2(m−1) (R≫R0). (12)
mayberesponsibleforinflationintheearlyUniverseifµ
From this we see that in the m = 2 case the effective
issettobeaninflationaryscale(e.g.,µ 1012GeV)[29],
potential U becomes steeper as the curvature increases,
∼
butinthispaperwedonotrestrictthemassscalesofthe
leadingfinallytodU/dχ asR + . Form 3,
curvaturecorrectionandassumethatsuchparametersin U has a minimum at R→−∞µ2 and→dU/∞dχ + ≥as
the high energy correction terms take rather arbitrary ∼ → ∞
R + .
values. If the coefficient of the R2 term is highly sup- → ∞
pressed for some reason, then the leading correctionwill
be the form of R3/µ4.4 To make the model simple but
IV. RELATIVISTIC STARS IN f(R) GRAVITY
general enough, let us consider a function WITH HIGH ENERGY CORRECTIONS
R2 −n Rm
f(R)=R+λR 1+ 1 + , (9) We now investigate static and spherically symmetric
0"(cid:18) R02(cid:19) − # µ2(m−1) starswith constantdensities (i.e., a generalizationofthe
Schwarzschild interior solution) in the model defined by
where n(> 0),λ(> 0), R0(> 0), and m( 2) are pa- Eq. (9). We shall work along the lines of the previous
≥
rameters. The present Hubble scale is basically given paper [26]. The basic equations are found there and are
by H02 ∼ O(R0). We define a dimensionless param- replicated in Appendix A. Stars in f(R) gravity have
eter ε := R0/µ2 and assume that ε 1 since the been studied also in Ref. [30].
≪
last term in Eq.(9) is the high energy correction. At Given a density ρ and the central values of the pres-
0
sufficiently low energies we have no cosmological con- sure p and the scalar field χ (or, equivalently, the cen-
c c
sfetn(aRenr)tg,ie≃fs,(RRR)−≃λµR2R,0,t+hweλhlRailse02tn+ftoe1rr/mRR2d0nom≪+in·R·a·t.e≪s.Atµ2vewrye hfiingdh otmrfaetrlhicceuarlslvtyaatrfu,rorrem=Rtch)e,,wrdeeegficuanlanerdincbetyengtpre(artre)E==qs00..(tAo(T3t)hh–ee(Abs6uo)rufnnadcue--
≫ R R
arycondition atthe center is also givenin Appendix A.)
Then, imposing the continuity of the metric functions
N(r)andB(r),thescalarfieldχ,anditsderivativedχ/dr
3 Higher order corrections naturally include terms like RµνRµν, atthestellarsurface,weintegratethevacuumfieldequa-
but in this paper we focus on the f(R)-type modified gravity tions (A4)–(A6) to find the exterior geometry. We are
and hence simplyassume that the corrections arealso given by
looking for a solution such that it is asymptotic to de
afunctionoftheRicciscalar.
4 Notethatthehighercurvaturecorrectionisgivenbythequartic Sitter with Λeff = R1/4 (and hence χ χ1). For fixed
→
termsintypeIIsuperstringtheory. ρ0 and pc, we can find the desired solution (if it exists)
4
FIG. 3: Plots of theRicci scalar R(r)for different ε. Param- FIG. 5: Plots of the metric functions for ε=5×10−9. Solid
etersaregivenbyλ=2,n=1,m=2. Theenergydensityis (dashed) lines correspond to the region inside (outside) the
4πGρ0 =106Λeff andthecentralpressureispc =0.3ρ0. Solid star.
(dashed) lines correspond to the region inside (outside) the
star. These examples typically give GˆM/R≃0.25 – 0.26.
FIG. 6: Plots of R(r) for ε=5×10−10. Parameters are the
sameasthoseinFig.3. Theupper(red)lineisaplotforthe
FIG.4: Plotsofχ(r)fordifferentε. Parametersarethesame solutionwiththecentralcurvatureRc =0.1185×8πGρ0. This
solutionexhibitsthebehavioroffallingtowardthesingularity.
as those in Fig. 3. Solid (dashed) lines correspond to the
Thelower(blue)lineindicatesa“overshooting”solutionwith
region inside (outside) the star.
Rc =0.1184×8πGρ0. Theinsetemphasizestheovershooting
behavior.
by carefully tuning the initial value χ = χ . In the
c crit
mechanical analogy, this solution corresponds to the sit- Bearing the above mechanical picture in mind, let
uation where the particle starts at rest and reaches the us move on to the case with the Rm correction. We
topofthe potential(χ=χ1)in the limitofr . The have carried out numerical calculations for various val-
→∞
particleovershootsthetopofthepotentialforχc <χcrit, ues of ε = R0/µ2 with fixing the other model parame-
while it turns around before it reaches the top and falls ters as λ = 2, n = 1, and m = 2. The profiles of χ
into the singularity for χc >χcrit. and R for regular,asymptotically de Sitter solutions are
Inthepreviouspaper[26]weshowedthatχ becomes shown in Figs. 3 and 4, and the metric functions for the
crit
largerasthe gravitationalpotentialofthe starincreases, ε = 5 10−9 case are plotted in Fig. 5. In these plots
×
getting eventually at χ , above which the slope of the the energy density and the central pressure are given re-
s
potential dU/dχ is greater than the force term and so spectively by 4πGρ0 = 106Λeff and pc = 0.3ρ0, leading
the particle cannot climb up the potential. Thus, if the to the gravitational potential as large as GˆM/ 0.25,
R∼
gravitationalpotential is largerthan a certainvalue, the where Gˆ := G/χ and M := 4πρ 3/3. (The gravita-
c 0
R
desired solution described above does not exist, and we tional potential is controlled by the ratio p /ρ .) Stars
c 0
only have two types of singular solutions, i.e., a “falling- with such large potentials are prohibited in the original
down” type and a “overshooting”one. model without the high energy correction.
5
For ε < 5 10−9, however, we find that the regular
×
solutionceasestoexist. Toseethis,weshowthebehavior
of the Ricci scalar for ε = 5 10−10 in Fig. 6. Taking
×
R = 0.1185 8πGρ [= 1.185 8πG(ρ 3p )], χ goes
c 0 0 c
× × −
toward χ = (R = ). In this case, the pressure is an
∞ ∞
increasing function of r away from the center, and there
is a maximal circumferential radius corresponding to an
infinite proper distance from the center, as can be seen
from Fig. 7. It is not an asymptotically flat space, but a
“cylindrical” shape space with a singularity at the end.
Taking a slightly smaller value, R =0.1184 8πGρ , χ
c 0
×
thenovershootsthetopofthepotentialandrollsdownto
theleft. Duringthisrolling-downphasetheKretschmann
scalar, R Rµνρσ, diverges. Only these two cases are
µνρσ
realized, both of which are unphysical. This is the same
situation one encounters in the model without the high
energy correction [26]. In order for the higher curvature
term to come to the rescue, it must be sufficiently large.
Theminimumvalueofε(orthemaximumvalueofµ2)
that allows for relativistic stars depends on the energy
density. To explore the bound on µ2, we have performed FIG. 7: If the force F is too weak to defeat the potential
numerical calculations for different values of ρ0 ranging slope dU/dχ inside a star and consequently χ(r) and R(r)
from ρ = 104Λ /(4πG) to ρ = 109Λ /(4πG).5 As growwithoutturningback,thepressureshowsanunphysical
0 eff 0 eff
an example, the behavior of χ for ρ =109Λ /(4πG) is behavior. The coordinate choice is not good in this case,
0 eff
shown in Fig. 8. From our numerical results it is con- as can be seen most clearly from the behavior of the metric
firmed that the minimum value of ε is inversely propor- component grr. Plots are for λ=2, n=1, 4πGρ0 =106Λeff,
tionaltotheenergydensity(Fig.9),andweroughlyhave pc =0.3ρ0, and Rc =0.12×8πGρ0.
the bound ε & 10−2R /(8πGρ ) in order for stars with
0 0
stronggravitationalfields ( 0.25)to exist.6 This condi-
∼
tion gives
the central region and g at finite r. To allow for
rr
µ2 <α 8πGρ , α (102). (13) relativistic stars with GˆM/→∞ 0.26, it is required that
× 0 ∼O µ2 <α′ 8πGρ where α′ R ∼(10).
0
Taking ρ ρ 1014g/cm3 10−3GeV4 (nuclear × ∼O
0 nucl
density), on∼e arrive∼sat µ.10−19∼GeV. This result itself Before closing this section, let us comment on the be-
is not surprising because 8πGρ is a natural scale as- havior of the metric for asymptotically de Sitter stellar
nucl
sociatedwithneutronstars. However,purelyfromathe- solutions. A numerical fitting leads to the approximate
oreticalpoint ofview, this providesanunnaturally small expression for the metric outside stars:
energyscale. Obviously,the R2/µ2 termwithsuchsmall
µ cannot be relevant to inflation in the early Universe.
We have also done numericalcalculations to construct N N 1 2c R c2Λ r2 , (14)
stellarsolutionswithGˆM/ 0.26inthem=3model, ≃ ∞ − 1 r − 3 eff
R∼ (cid:18) (cid:19)
and obtained essentially the same result: for sufficiently c
B 1 2c R 4Λ r2, (15)
large ε we can find regular, asymptotically de Sitter so- 3 eff
≃ − r − 3
lutions, while for ε smaller than a certain value we only
have two classes of unphysical solutions. Note here that
for m 3 the structure of the (effective) potential near where c c 1.0 irrespective of ε (&5 10−9), while
≥ 2 ≃ 4 ≃ ×
the curvature singularity is quite different from that of c and c are slightly different for different ε. For ε =
1 3
the m=2model. Nevertheless,the solutioncorrespond- 5 10−9 one finds c 0.29 and c 0.24. This gives
1 3
× ≃ ≃
ingtoχ(r) movingtowardrightshowsanunphysicalna- the post-Newtonian parameter γ c /c 0.81. For
3 1
ture: thepressureisanincreasingfunctionofrawayfrom ε=5 10−6 one has c 0.30,c ≃ 0.21, a≃nd γ 0.72.
1 3
× ≃ ≃ ≃
These results imply that the chameleon mechanism does
not work in the above examples. This is because we are
considering a vacuum exterior. Note that taking into
5 Tomimicaneutronstar,thedensitymustbe8πGρ0∼1044Λeff. accountthe effectofsurroundingmediadoesnotremedy
However,itisdifficulttoimplementsuchanextremelyhighden- the nonexistenceofrelativisticstarsforsmallε: to avoid
sitycontrastinournumericalcomputations.
6 Forgivenεthereisamaximalgravitationalpotential. Therefore, falling downtowardlargeR, χ inevitably overshootsthe
topofthepotentialalsointhepresenceofexteriormatter
theminimumvalueofεwillbedifferentdependingonhowlarge
gravitationalpotentials oneneeds. as we have <0 outside the star.
F
6
convenientto define the effective potentialfortheχ field
as
dV 1 8πG
eff
= [2f(Q(χ)) χQ(χ)]+ T, (16)
dχ 3 − 3
where the energy-momentum tensor of matter is in-
cluded. We consider the regime R R µ2. In this
0
≪ ≪
regime Eq. (11) can be written as
2n+1 m−1
R R
0
χ 1 2nλ +m . (17)
≈ − R µ2
(cid:18) (cid:19) (cid:18) (cid:19)
At the minimum of the effective potential V one finds
eff
dV
eff
=0 R 8πGρ, (18)
dχ ⇒ ≈
FIG. 8: Plots of χ(r) for 4πGρ0 = 109Λeff and pc = 0.3ρ0.
Themodelparametersaregivenbyλ=2,n=1,andm=2. where ρ T is the energy density of nonrelativistic
Intheε=5×10−12 case,thedesiredsolutionisobtainedfor matter. T≈he−mass of the excitation of the χ field around
Rc =0.2245×8πGρ0. However, in thecase of ε=5×10−13 the minimum is given by m2 = d2V /dχ2 .
only singular solutions are found. Two examples are shown: χ eff |R≈8πGρ
one is for Rc =0.1185×8πGρ0 (falling rapidly down to R= Thus, the Compton wavelength λχ = m−χ1 can be com-
∞)andtheotherisforRc =0.1184×8πGρ0 (aovershooting puted as
solution).
λ2 3f
χ ≈ RR|R≈8πGρ
2n+1 m−1
k R k R
1 0 2
+ ,(19)
≈ R R R µ2 (cid:12)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:12)R≈8πGρ
(cid:12)
(cid:12)
where k1 :=6n(2n+1)λ and k2 :=3m(m (cid:12) 1).
−
To evaluate the Compton wavelength, it is important
to note that there is a critical curvature scale defined by
1/(2n+m)
R := R2n+1µ2(m−1) , (20)
∗ 0
(cid:16) (cid:17)
and for R R (respectively, R R ) the first (re-
∗ ∗
≪ ≫
spectively, second) term in Eq. (19) is much greater
than the other. Eq. (20) gives an intermediate curva-
ture scale R R µ2, which implies that the high
0 ∗
≪ ≪
energy correction term comes into play in determining
the Compton wavelength of χ at a much lower scale than
expected. In terms of energy densities, the intermediate
scale ρ :=R /(8πG) may be written as
∗ ∗
ρ := ρ2n+1ρm−1 1/(2n+m), (21)
FIG. 9: The minimum value of ε as a function of ρ0. Points ∗ DE UV
indicate numerical results, showing thescaling relation. where ρ 1(cid:0)0−30g/cm3(cid:1)and ρ . ρ
DE UV nucl
1014g/cm3. F∼or instance, putting n = 1 and m =∼2
yields ρ . 10−19g/cm3. Laboratory experiments are
∗
V. APPEARANCE OF AN INTERMEDIATE usually done at densities much higher than this!
SCALE AND ANOTHER FINE-TUNING
Although the original model is made to satisfy so-
lar system and laboratory tests, the intermediate scale
Assuming that the “UV scale” is given by µ2 brought by a high energy correction term can destroy
∼
8πGρnucl, one may naively expect that higher curvature its success. Indeed, in the above example (n = 1 and
correctionshave no impacton localtests ofgravitysince m=2),themassofχisindependentoflocalenergyden-
relevantdensitiesaremuchsmaller. However,asweshow sities for ρ ρ and so the chameleon mechanism does
∗
≫
below, this expectation is not true. The purpose of this not work in laboratories. Since χ has a gravitational-
section is to point out a new problem brought by the strength coupling and the Compton wavelength is eval-
higher curvature correction. uated as λ µ−1 & 105cm, the m = 2 model is ruled
χ
∼
To discuss the behavior of gravityin laboratories,it is out by the fifth force constraint [31, 32].
7
AsseenfromEqs.(19)and(20),theintermediatescale energy correction completely destroys the success of the
and its consequences are sensitive to the explicit form original f(R) model that passes local tests of gravity.
of the higher curvature correction. For example, in the If the quadratic correction is suppressed relative to the
n=1andm=3caseoneobtainsρ .10−12g/cm3,and other higher curvature terms, possibly this is not always
∗
fordensities higherthanthis the Comptonwavelengthis the case. However, it might be thought of as a prob-
found to be λ &0.1mm (ρ/1g cm−3)1/2. This typi- lem that gravity in the intermediate curvature regime
χ
× ·
callygivesthemarginalscaletestedbylaboratoryexper- (R R µ2) is so sensitive to the explicit form of
0
≪ ≪
iments of gravity. Thus, determining whether or not a UV correctionterms. Inthis sense we need anotherfine-
given high energy correction satisfies local tests requires tuning of the high energy correction.
a more careful study, which is beyond the scope of the To conclude, although there is still a very small room
presentpaper. Wejustemphasizeherethat“highenergy for a possible construction of viable f(R) models that
corrections” play a crucial role above the intermediate evade local tests of gravity and allow for stars with
curvature scale, and the R2/µ2 correction, which seems strong gravitational fields, very careful and unnatural
to appear in natural circumstances, is clearly inconsis- fine-tuningisrequiredforthemodelconstruction,leaving
tent with laboratory tests if one chooses the parameter challenges for f(R) modified gravity.
µ2 so that the theory evades the nonexistence statement
of neutron stars.
Finally, let us comment on cosmology with the Rm
term. In the matter-dominated era, we have an esti- Acknowledgments
mate m2/H2 (µ2/H2)m−1 1, where it is assumed
χ ∼ ≫
that the matter energy density is much greater than ρ .
∗
This work was partially supported by the JSPS under
This implies that the excitation of χ is suppressed, ren-
Contact No. 19-4199, by the Grant-in-Aid for Scientific
dering the field safe for cosmology. Before the time of
Research Fund of the JSPS (No. 19540308) and by the
matter-radiation equality the energy density of nonrel-
ativistic matter is given by ρ = rρ , where ρ is the Japan-U.K. Research Cooperative Program.
m r r
energy density of radiation and r := a/a 1. Since
eq
R 8πGρ and H2 8πGρ /3, one en≪ds up with
m r
m2∼/H2 r−m+2(µ2/H2∼)m−1. Even at nucleosynthesis,
χ ∼ APPENDIX A: SPHERICALLY SYMMETRIC
the ratio m2/H2 is enhanced by the factor r−m+2 ex-
χ STARS IN f(R) GRAVITY
cept for m = 2. Thus, we can approximately reproduce
standard cosmology for H2 .µ2 in models with m 3.
≥ In this appendix we summarize the basic equations
for constructing spherically symmetric stellar solutions
VI. SUMMARY AND CONCLUSIONS in f(R) gravity [26].
In this paper, we have tried to resolve the singularity
problemarisinginthe stronggravityregimeofotherwise
viable f(R) theories. Adding a higher curvature correc- 1. Basic equations
tion in the form of Rm/µ2(m−1), we have studied stars
with strong gravitationalfields which were prohibited in We take the ansatz of a spherically symmetric and
the original models. Our numerical results have shown static metric:
thatthe scaleµ2 cannotbe takentobe aslargeasanin-
flationary energyscale nor a natural UV cutoff scale like dr2
(8πG)−1. Rather, µ2 . (8πGρ), where ρ is the stellar ds2 = N(r)dt2+ +r2 dθ2+sin2θdϕ2 . (A1)
O − B(r)
density and hence is taken to be a nuclear density, is re-
(cid:0) (cid:1)
quiredinordertoremedythe nonexistenceofrelativistic
Theenergy-momentumtensorofmatterfieldsisgivenby
stars. This provides a “high” energy scale as small as
µ . 10−19GeV. This is the first fine-tuning required for
the high energy correction. T ν =diag( ρ,p,p,p). (A2)
µ −
In contrast to the naive expectation, the high energy
corrections come into play at an intermediate curvature From the energy-momentum conservation, T ν = 0,
scale in determining the mass of χ’s excitation around we obtain ∇ν µ
the minimum of the effective potential. If the leading
correction is given by the quadratic curvature term, the N′
intermediate scale is R∗ ∼ R02n+1µ2 1/(2n+2), and the p′+ 2N(ρ+p)=0. (A3)
corresponding energy density is ρ 10−19g/cm3 for
(cid:0) ∗ ∼(cid:1)
n = 1. For densities higher than this, the Compton Here and hereafter a prime denotes differentiation with
wavelength of χ is µ−1 105cm. Therefore, the high respect to r. The (tt) and (rr) components of the field
∼ ∼
8
equations (4) yield, respectively, in the power series of r as
χ
( 1+B+rB′)= 8πGρ χ2V N(r)=1+N r2+..., B(r)=1+B r2+...,
r2 − − − 2 2
B χ′′+ 2 + B′ χ′ , (A4) χ(r)=χc 1+ C2r2+... , (A8)
2
− r 2B (cid:18) (cid:19)
(cid:20) (cid:18) (cid:19) (cid:21) ρ p
χ N′ ρ(r)=ρ + 2r2+..., p(r)=p + 2r2+...,
1+B+rB =8πGp χ2V c 2 c 2
r2 − N −
(cid:18) (cid:19)
2 N′ where χ , ρ and p are the central values of the scalar
B + χ′. (A5) c c c
− r 2N field, the energy density and the pressure, respectively.
(cid:18) (cid:19)
Note that using the scaling freedom of the t coordinate,
The equation of motion for χ [Eq. (5)] gives we set N(0)=1. From Eqs. (A4)–(A6), we obtain
2 N′ B′
B χ′′+ + + χ′ 3B2 = 8πGˆρc χcVc 3C2, (A9)
r 2N 2B − − −
(cid:20) (cid:18) (cid:19) (cid:21) B +2N = 8πGˆp χ V 2C , (A10)
8πG 2χ3dV 2 2 c− c c− 2
= ( ρ+3p)+ . (A6) 8πGˆ 2χ2
3 − 3 dχ 3C = ( ρ +3p )+ cV , (A11)
2 3 − c c 3 χc
We do not integrate the angular components of the field
equations. Instead, we use them to check the accuracy where Gˆ := G/χ , V := V(χ ), and V = dV/dχ .
of our numerical results, because those are derived from These three equactioncs are reacrrangedtχocgive |χ=χc
other equations via the Bianchi identity.
Iftheenergydensityisconstantinsidethestar,ρ=ρ ,
0 8πGˆ χ 2χ2
Eq. (A3) immediately gives B = (2ρ +3p ) cV cV ,(A12)
2 − 9 c c − 3 c− 9 χc
8πGˆ χ χ2
N(r)= ρ0+pc 2. (A7) N2 = 9 (2ρc+3pc)− 3cVc− 9cVχc, (A13)
(cid:20)ρ0+p(r)(cid:21) 8πGˆ 2χ2
C = ( ρ +3p )+ cV . (A14)
In the main text we only consider constant density stars 2 9 − c c 9 χc
for simplicity.
Then, p is derived from the conservation equation:
2
2. Boundary conditions p +N (ρ +p )=0. (A15)
2 2 c c
Let us study the boundary conditions at the center of The Ricci scalar is given by R = R + (r2) with R =
c c
O
a star. Assuming the regularity,we expandthe variables 6(B +N ) near r =0.
2 2
−
[1] A. G. Riess et al. [Supernova Search Team Phys. Lett. B 657, 238 (2007) [arXiv:0707.1941
Collaboration], Astron. J. 116, 1009 (1998) [hep-th]]; S. Nojiri and S. D. Odintsov, Phys.
[arXiv:astro-ph/9805201]; S. Perlmutter et al. [Su- Rev. D 77, 026007 (2008) [arXiv:0710.1738 [hep-th]];
pernova Cosmology Project Collaboration], Astrophys. G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov,
J.517,565(1999)[arXiv:astro-ph/9812133];A.G.Riess L. Sebastiani and S. Zerbini, Phys. Rev. D 77,
et al. [Supernova Search Team Collaboration], Astro- 046009 (2008) [arXiv:0712.4017 [hep-th]]; S. Nojiri and
phys.J. 607, 665 (2004) [arXiv:astro-ph/0402512]. S. D. Odintsov, arXiv:0804.3519 [hep-th]; G. Cognola,
[2] For a review, see, T. P. Sotiriou and V. Faraoni, E.Elizalde, S.D.Odintsov,P.TretyakovandS.Zerbini,
arXiv:0805.1726 [gr-qc];V.Faraoni,arXiv:0810.2602 [gr- arXiv:0810.4989 [gr-qc].
qc]; N.Straumann,arXiv:0809.5148 [gr-qc]. [5] A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1
[3] S. M. Carroll, V. Duvvuri, M. Trodden and (2003) [arXiv:astro-ph/0307285].
M. S. Turner, Phys. Rev. D 70, 043528 (2004) [6] T. Chiba, Phys. Lett. B 575, 1 (2003)
[arXiv:astro-ph/0306438]. [arXiv:astro-ph/0307338].
[4] S. Nojiri and S. D. Odintsov, Phys. Rev. D 68, [7] A.L.Erickcek,T.L.SmithandM.Kamionkowski,Phys.
123512 (2003) [arXiv:hep-th/0307288]; M. C. B. Ab- Rev. D 74, 121501 (2006) [arXiv:astro-ph/0610483].
dalla, S. Nojiri and S. D. Odintsov, Class. Quant. [8] T. Chiba, T.L.Smithand A.L. Erickcek,Phys.Rev.D
Grav. 22, L35 (2005) [arXiv:hep-th/0409177]; S. No- 75, 124014 (2007) [arXiv:astro-ph/0611867].
jiri and S. D. Odintsov, Phys. Lett. B 652, 343 (2007) [9] J. Khoury and A. Weltman, Phys. Rev. Lett. 93,
[arXiv:0706.1378 [hep-th]];S. Nojiri and S. D.Odintsov, 171104 (2004) [arXiv:astro-ph/0309300]; J. Khoury
9
and A. Weltman, Phys. Rev. D 69, 044026 (2004) arXiv:0806.3415 [astro-ph].
[arXiv:astro-ph/0309411]. [24] S.A.ApplebyandR.A.Battye,JCAP0805,019(2008)
[10] D. F. Mota and J. D. Barrow, Mon. Not. Roy. As- [arXiv:0803.1081 [astro-ph]].
tron. Soc. 349, 291 (2004) [arXiv:astro-ph/0309273]; [25] A. V. Frolov, Phys. Rev. Lett. 101, 061103 (2008)
T.Clifton,D.F.MotaandJ.D.Barrow,Mon.Not.Roy. [arXiv:0803.2500 [astro-ph]].
Astron.Soc. 358, 601 (2005) [arXiv:gr-qc/0406001]. [26] T.KobayashiandK.i.Maeda,Phys.Rev.D78,064019
[11] D. F. Mota and D. J. Shaw, Phys. Rev. Lett. 97, (2008) [arXiv:0807.2503 [astro-ph]].
151102 (2006) [arXiv:hep-ph/0606204]; D. F. Mota [27] A. Dev, D. Jain, S. Jhingan, S. Nojiri, M. Sami and
and D. J. Shaw, Phys. Rev. D 75, 063501 (2007) I. Thongkool, arXiv:0807.3445 [hep-th].
[arXiv:hep-ph/0608078]. [28] B. Whitt, Phys. Lett. B 145, 176 (1984); K. i. Maeda,
[12] T.Faulkner,M.Tegmark,E.F.BunnandY.Mao,Phys. Phys. Rev.D 39, 3159 (1989).
Rev.D 76, 063505 (2007) [arXiv:astro-ph/0612569]. [29] A. A.Starobinsky,Phys. Lett. B 91, 99 (1980).
[13] A. A. Starobinsky, JETP Lett. 86, 157 (2007) [30] K.Kainulainen,J.Piilonen,V.ReijonenandD.Sunhede,
[arXiv:0706.2041 [astro-ph]]. Phys. Rev. D 76, 024020 (2007) [arXiv:0704.2729 [gr-
[14] W. Hu and I. Sawicki, Phys. Rev. D 76, 064004 (2007) qc]]; K. Kainulainen and D. Sunhede, arXiv:0803.0867
[arXiv:0705.1158 [astro-ph]]. [gr-qc]; T. Multamaki and I. Vilja, Phys. Rev. D 76,
[15] S. A. Appleby and R. A. Battye, Phys. Lett. B 654, 7 064021 (2007) [arXiv:astro-ph/0612775]; K. Henttunen,
(2007) [arXiv:0705.3199 [astro-ph]]. T. Multamaki and I. Vilja, Phys. Rev. D 77, 024040
[16] J. A. R. Cembranos, Phys. Rev. D 73, 064029 (2006) (2008) [arXiv:0705.2683 [astro-ph]]; T. Multamaki and
[arXiv:gr-qc/0507039]. I. Vilja, Phys. Lett. B 659, 843 (2008) [arXiv:0709.3422
[17] A. W. Brookfield, C. van de Bruck and L. M. H. Hall, [astro-ph]]; A. J. Bustelo and D. E. Barraco, Class.
Phys.Rev.D74,064028 (2006) [arXiv:hep-th/0608015]. Quant. Grav. 24, 2333 (2007) [arXiv:gr-qc/0611149];
[18] I.NavarroandK.VanAcoleyen,JCAP0702,022(2007) E. Barausse, T. P. Sotiriou and J. C. Miller, Class.
[arXiv:gr-qc/0611127]. Quant. Grav. 25, 105008 (2008) [arXiv:0712.1141 [gr-
[19] B.LiandJ.D.Barrow, Phys.Rev.D75,084010 (2007) qc]]; E. Barausse, T. P. Sotiriou and J. C. Miller, Class.
[arXiv:gr-qc/0701111]. Quant. Grav. 25, 062001 (2008) [arXiv:gr-qc/0703132];
[20] L. Amendola and S. Tsujikawa, Phys. Lett. B 660, 125 G. J. Olmo, arXiv:0810.3593 [gr-qc].
(2008) [arXiv:0705.0396 [astro-ph]]. [31] C. M. Will, Living Rev. Rel. 9, 3 (2005)
[21] S. Tsujikawa, Phys. Rev. D 77, 023507 (2008) [arXiv:gr-qc/0510072].
[arXiv:0709.1391 [astro-ph]]. [32] D.J.Kapner,T.S.Cook,E.G.Adelberger,J.H.Gund-
[22] S.CapozzielloandS.Tsujikawa,Phys.Rev.D77,107501 lach,B.R.Heckel,C.D.HoyleandH.E.Swanson,Phys.
(2008) [arXiv:0712.2268 [gr-qc]]. Rev. Lett.98, 021101 (2007) [arXiv:hep-ph/0611184].
[23] P. Brax, C. van de Bruck, A. C. Davis and D. J. Shaw,