Class of viable modified $f(R)$ gravities describing inflation and the onset of accelerated expansion

0.26 MB·English

by  G. Cognola

Class of viable modified f(R) gravities describing inflation and the onset of accelerated
expansion
G. Cognola1, E. Elizalde2, S. Nojiri3, S.D. Odintsov4, L. Sebastiani1, and S. Zerbini1
1Dipartimento di Fisica, Universit‘a di Trento and Istituto Nazionale di Fisica Nucleare Gruppo Collegato di Trento, Italia
2Consejo Superior de Investigaciones Cient´ıficas ICE/CSIC-IEEC, Campus UAB,
Facultat de Ci`encies, Torre C5-Parell-2a pl, E-08193 Bellaterra (Barcelona) Spain
3Department of Physics, Nagoya University, Nagoya 464-8602. Japan and
4Instituci`o Catalana de Recerca i Estudis Avanc¸ats (ICREA) and Institut de Ciencies de l’Espai (IEEC-CSIC),
Campus UAB, Facultat de Ci`encies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona) Spain
(Dated: February 2, 2008)
A general approach to viable modified f(R) gravity is developed in both the Jordan and the
Einstein frames. A class of exponential, realistic modified gravities is introduced and investigated
8 with care. Special focus is made on step-class models, most promising from the phenomenological
0 viewpoint and which provide a natural way to classify all viable modified gravities. One- and
0 two-steps models are explicitly considered, but the analysis is extensible to N-step models. Both
2 inflationintheearlyuniverseandtheonsetofrecentaccelerated expansionariseinthesemodelsin

n anatural,unifiedway. Moreover,itisdemonstratedthatmodelsinthiscategoryeasilypassalllocal
a tests, including stability of spherical body solution, non-violation of Newton’s law, and generation
J of a very heavy positive mass for theadditional scalar degree of freedom.

0
3 PACSnumbers: 11.25.-w,95.36.+x,98.80.-k

]
h I. INTRODUCTION
t
-
p
Modified gravity models constitute an interesting dynamical alternative to the ΛCDM cosmology in that they are
e
h able to describe with success the current acceleration in the expansion of our Universe, the so called dark energy
[ epoch. Moreover, modified F(R) gravity (for a review, see e.g. [1]) has undergone many studies which conclude

that this gravitational alternative to dark energy [2, 3] is able to pass the solar system tests. The investigation
2
of cosmic acceleration as well as the study of the cosmological properties of F(R) models has been carried out in
v
7 Refs. [1, 2, 3, 4, 5, 6, 7].
1 Recently the importanceofthose modelswasreassessed,namelywith the appearanceofthe so-called‘viable’F(R)
0 models [9, 10, 11, 12]. Those are models which do satisfy the cosmological as well as the local gravity constraints,
4 whichhadcausedanumberofproblemstosomeofthefirst-generationtheoriesofthatkind. Thefinalaimofallthese
.
2 phenomenologicalmodels is to describe a segment as large as possible of the whole history of our universe, as well as
1 to recover all local predictions of Einstein’s gravity, which have been verified experimentally to very good accuracy,
7 at the solar system scale.
0
Letus recallthat, ingeneral(see e.g. [1], fora review),the totalactionforthe modified gravitationalmodels reads
:
v
1
i S = d4x√ g[R+f(R)]+S . (1)
X κ2 − (m)
Z
r
a Here f(R) is a suitable function, which defines the modified gravitationalpart of the model. The generalequation of
motion in F(R) R+f(R) gravity with matter is given by
≡
1 κ2
g F(R) R F′(R) g (cid:3)F′(R)+ F′(R)= T , (2)
2 µν − µν − µν ∇µ∇ν − 2 (m)µν
where T is the matter energy-momentum tensor.
(m)µν
In this paper we investigate two classes of ‘viable’ modified gravitational models what means, roughly speaking,
they have to incorporate the vanishing (or fast decrease) of the cosmological constant in the flat (R 0) limit, and
→
must exhibit a suitable constant asymptotic behavior for large values of R. A huge family of these models, which
we will term first class —and to whom almost all of the models proposed in the literature belong— can be viewed
as containing all possible smooth versions of the following sharp step-function model. To discuss this toy model, at
the distribution level, will prove to be very useful in order to graspthe essentialfeatures that all models in this large
family are bound to satisfy. In other words, to extract the general properties of the whole family in a rather simple
fashion (which will involve, of course, precise distribution calculus).
This simple model reads
f(R)= 2Λ θ(R R ), (3)
eff 0
− −
2
−f(R)
6
2Λeff
-
R
R0
FIG. 1: Typical behavior of f(R) in the one-step model).
where θ(R R ) is Heaviside’s step distribution. Models in this class are characterized by the existence of one or
0
−
more transition scalar curvatures, an example being R in the above toy model (but there can be more, as we will
0
later see).
The other class of modified gravitational models that has been considered contains a sort of ‘switching on’ of the
cosmologicalconstant as a function of the scalar curvature R. A simplest version of this kind reads
f(R)=2Λ (e−bR 1). (4)
eff
−
Here the transition is smooth. The two above models may be combined in a natural way, if one is also interested
in the phenomenological description of the inflationary epoch. For example, a two-steps model may be the smooth
version of
f(R)= 2Λ θ(R R ) 2Λ θ(R R ), (5)
0 0 I I
− − − −
with R <<R , the latter being the inflation scale curvature.
0 I
The typical, smooth behavior of f(R) associated with the one- and two-step models is given, in the smooth case,
in Figs. 1 and 2, respectively. The main problem associated with these sharp models is the appearance of possible
antigravity regime in a region around the transition point and antigravity in a past epoch, what is not phenomeno-
logically acceptable. On the other hand, an analytical study of these models can be easily carried out, as discussed
in the Appendix.
The existenceofviable(or“chameleon”)f(R)theorieswitha phaseofearly-timeinflationisnotknownto us from
the literature. The fact that we are able to provideseveralclassesof models of this kind thatare consistentalso with
the late-time accelerated expansion is thus a novelty, worth to be remarked.
The paper is organized as follows. Next section reviews f(R) gravity in the physical, Jordan frame. Equations of
motion are presented and the solar system tests (absence of a tachyon, stability of the spherical body solution, and
non-violationofNewtonlaw)arediscussed. InSect.III we givea numberofviablemodifiedgravitieswhichmaylead
totheunificationofearly-timeinflationwithlate-timeaccelerationandsatisfythesolarsystemtests. Someproperties
of such viable modified gravity are discussed in detail. Sect. IV is devoted to the presentation of viable models via
conformaltransformation,asa kindof scalar-tensortheorywhichis mathematically equivalentto the originaltheory.
In some examples, the explicit form of the scalar potential is derived. Corrections to Newton’s law are also obtained
for some of the realistic theories here considered. It turns out that these models do pass the stringent Newton law
bounds, since the correctionsto Newton’s law in these cases turn out to be negligible. Some summary and outlook is
givenin the Discussions section. Finally, in an Appendix we show how to evaluate the positions of the corresponding
de Sitter critical points, by considering the sharp version of the one- and two-step models, expressed in terms of the
Heaviside and Dirac distributions.
II. MODIFIED GRAVITY IN THE JORDAN FRAME
Regarding the precise determination of the modifying term, f(R), we here revisit this issue in the Jordan frame
(instead of the Einstein one). Let us recall the two sufficient conditions which often lead to realistic models (see, for
3
−f(R)
6
2ΛI
2Λ0
-
R
R0 RI
FIG. 2: Typical behavior of f(R) in the two-step model.
example [9])
f(0)=0, lim f(R)= α, (6)
R→R1 −
where α is a suitable curvature scale which represents an effective cosmological constant, being R >> R , with
1 0
R > 0, the transition point. The condition f(0) = 0 ensures the disappearance of the cosmological constant in the
0
limit of flat space-time.
By using these conditions, some models in this class are seen to be able to pass the local tests (with some extra
bounds on the theory parameters) and are also capable to explain the observed recent acceleration of the universe
expansion, provided that α=Λ =2H2, H being the Hubble constant at the epoch of reference. However, they do
0 0 0
not incorporate early-time inflation, which comes into play at higher value of R. Thus, one might also reasonably
require that [11]
f(0)=0, lim f(R)= (α+α ), (7)
I
R→R2 −
where α >> α is associated with the inflation cosmological constant, Λ , and where R >> R >> R , R being
I I 2 I 0 I
the corresponding transition large scalar curvature.
Further restrictions, like small corrections to Newton’s law and the stability of planet-like gravitational solutions
needtobefulfilledtoo[3]. AllthosecanbealsoformulatedinthemathematicallyequivalentEinsteinframe. Herewe
presentashortreviewofthemintheJordanframewhichweconsiderasthephysicalone(see,Ref.[6]foradiscussion
of the physical (non-)equivalence of the Einstein and Jordan frames).
The starting point is the trace of the equations ofmotion, whichis trivial in the Einstein theory but givesprecious
dynamical information in the modified gravitationalmodels. It reads
3 2f′(R)=R+2f(R) Rf′(R) κ2T . (8)
∇ − −
The above trace equation can be interpreted as an equation of motion for the non trivial ‘scalaron’ f′(R) (since it is
indeedassociatedwiththecorrespondingscalarfieldintheotherframe). Forsolutionswithconstantscalarcurvature
R∗, the scalaronfield is constant and one obtains the following vacuum solution:
′
R∗+2f(R∗) R∗f (R∗)=0. (9)
−
Furthermore, according to [11], we can describe the degree of freedom associated with the scalaron by means of a
scalar field χ, defined by F′(R) = 1+f′(R) = e−χ. If we consider a perturbation around the vacuum solution of
4
constant curvature R∗, given by R=R∗+δR, where
1+f′(R∗)
δR= δχ, (10)
− f′′(R∗)
then the equation of motion for the scalaron field is
(cid:3)δχ− 31 1+f′′f(R′(R∗)∗) −R∗ δχ=−6(1+κf2′(R∗)T . (11)
(cid:18) (cid:19)
As aresult,inconnectionwiththe localandwiththe planetarytests,the followingeffective massplaysa verycrucial
role:
M2 ≡ 31 1+f′′f(R′(R∗)∗) −R∗ . (12)
(cid:18) (cid:19)
If M2 < 0, a tachyon appears and this leads to an instability. Even if M2 > 0, when M2 is small, it is δR = 0 at
6
long ranges,which generates a large correctionto Newton’s law. As a result, M2 has to be positive and very large in
order to pass both the local and the astronomical tests. This stability condition can be also derived within QFT in
de Sitter space-time (see, for instance, [14]).
Concerning the matter instability [3, 15, 16], this might occur when the curvature is rather large, as on a planet,
as compared with the average curvature of the universe R 10−33eV 2. In order to arrive to a stability condition,
∼
we can start by noting that the scalaronequation can be rewritten in the form [3, 15]
(cid:0) (cid:1)
f′′′(R) (1+f′(R)R 2(R+f(R)) κ2
(cid:3)R+ R ρR+ = T . (13)
f′′(R)∇ρ ∇ 3f′′(R) − 3f′′(R) 6f′′(R)
If we now consider a perturbation, δR, of the Einstein gravity solution R=R = k2T >0, we obtain
e − 2
0 ( ∂2+U(R ))δR+C, (14)
≃ − t e
with the effective potential
F′′′′(R ) F′′′(R )2 R
U(R ) e e R ρR + e
e ≡ F′′(R ) − F′′(R )2 ∇ρ e∇ e 3
(cid:18) e e (cid:19)
F′(R )F′′′(R )R F′(R ) 2F(R )F′′′(R ) F′′′(R )R
e e e e e e e e
+ . (15)
− 3F′′(R )2 − 3F′′(R ) 3F′′(R )2 − 3F′′(R )2
e e e e
If U(R ) is positive, then the perturbation δR becomes exponentially large and the whole system becomes unstable.
e
Thus, the matter stability condition is, in this case,
U(R )<0. (16)
e
Coming back to the vacuum condition (9), we recall that, within the cosmologicalframework, it may be rederived
making use of the dynamical system approach. This consists in rewriting the generalized Friedmann equations of
the modified gravitationalmodel in terms of a first-order differential system and looking for its critical points. For a
modified gravity model, the associated dynamical system can be written as [7, 8] (a(t) being the expansion factor in
a FRW flat space-time):
d
Ω = 2Ω (2 Ω )Ω ) β(1 Ω Ω ),
R R R R F ρ
dlna − − − −
d
Ω = 2Ω (2 Ω )+(Ω Ω )(1 Ω Ω ),
F F R F R F ρ
dlna − − − −
d
Ω = [2(2 Ω ) 3(w+1)+1 Ω Ω ]Ω , (17)
ρ R F ρ ρ
dlna − − − −
where w = p is the usual barotropic constant, being
ρ
R f(R) Rf′(R) χρ
Ω = , Ω = − , Ω = , (18)
R 6H2 F −6H2(1+f′(R)) ρ 3H2(1+f′(R))
5
with
1+f′(R)
β = . (19)
Rf′′(R)
There exists yet another quantity
f˙′(R)
Ω = , (20)
F˙ −H(1+f′(R))
which satisfies the constraint
Ω +Ω +Ω =1. (21)
F˙ F ρ
The critical points are solutions of the (algebraic) system
0 = 2Ω (2 Ω )Ω ) β(1 Ω Ω ),
R R R F ρ
− − − −
0 = 2Ω (2 Ω )+(Ω Ω )(1 Ω Ω ),
F R F R F ρ
− − − −
0 = [2(2 Ω ) 3(w+1)+1 Ω Ω ]Ω . (22)
R F ρ ρ
− − − −
As an example of those, the de Sitter critical points are the ones in the invariant vacuum submanifold Ω = 0.
ρ
These solutions read
ΩR =2, R∗ =12H∗2, (23)
and
′
ΩF =1, R∗ =R∗f (R∗) 2f(R∗). (24)
−
The last equation coincides with Eq. (9). It is a transcendental equation, for all our models, and can be solved only
by iteration (see Appendix) or either numerically by other methods.
The stability condition associated with the de Sitter critical point in this dynamical system framework can be
investigated too, and reads [8]
1+f′(R∗)
1<β(R∗)= R∗f′′(R∗) . (25)
It coincides with the requirement that the effective mass (12) be positive. In the matter-radiation sector, where Ω
ρ
is non-vanishing, other critical points may also exist. In the next section, we give explicit examples of exponential,
viable modified gravity.
III. EXAMPLES OF REALISTIC EXPONENTIAL MODIFIED GRAVITY
After thegeneraldiscussionabove,wewillherepresentsomenewviablef(R)models. Westartwithamostsimple
one
f(R)=α(e−bR 1). (26)
−
Since f(0)=0 and f(R) α for large R, conditions (6) are satisfied. Moreover,
→−
f′(R)= bαe−bR, f′′(R)=b2αe−bR. (27)
−
Wehaveseenthatinthediscussionoftheviabilityofmodifiedgravitationalmodels,theexistenceofvacuumconstant
curvature solutions plays a very crucial role, namely the existence of solutions of Eq. (9). With regard to the trivial
fixed point R∗ =0, this model has the properties
1+f′(0)=1 αb, f′′(0)=αb2. (28)
−
Thus, the effective mass for R∗ =0 is
1 αb
M2(0)= − , (29)
3αb2
6
and then Minkowski space time is stable as soon as αb<1. Such condition is equivalent to 1+f′(0)>0.
In order to investigate the existence of other fixed points, we first have to find the existence conditions and then
make use of Newton’s method or some of its variants (see the Appendix). It is easy to see that for the model (26),
one has the critical point only if αb > 1, namely K′(0) > 1, where the function K(R) defined in Appendix. Then,
one can construct an approximation procedure to solve Eq. (9), in terms of an iteration process, namely
R R f′(R )+2f(R )
n n n n
R =R − . (30)
n+1 n− 1+f′(R ) R f′′(R )
n n n
−
For a starting point, R=R , large enough, f(R) is approximately constant and a few iterations give
1
R∗,1 2α. (31)
≃
This critical point is stable and corresponds to the current acceleration in the universe expansion. This follows from
that fact that the effective mass is
1
M2 e2bα, (32)
≃ 3αb2
namely it is positive and large. However, since αb > 1, for very small R, one has antigravity effects, namely
1+f′(R)<0, as we shall see in Section IV.
Matter instability can also be investigated. Eq. (15) gives
1
U(R ) e2bRe, (33)
e ≃−3αb2
which is negative, thus the matter stability condition is fulfilled.
Themodelwehavediscussedsofardoesnotexhibitasharptransitioncurvature. Buttherearemanymodelswhere
one or more transitions of this kind appear. The Hu-Sawicki (HS) model [9] belongs to this one-step class family of
models. A simple choice for a one-step model is in our case
1+e−bR0 ebR 1
f(R)=α 1 = α − . (34)
1+eb(R−R0) − − ebR+ebR0
(cid:20) (cid:21)
For R very small, we have
αb
f(R) R+O(R2), (35)
≃−1+ebR0
while for suitable values of b, one has the same behavior as in the HS model, where the continuous parameterb plays
the role of the integer n in the above mentioned model. Higher values of b give rise to a sharper transition, occurring
at R from very small values of f(R) towards a constant value α. This model has an effective mass, evaluated at
0
−
R∗ =0,whichturns outto be negative,thus Minkowskispace-timeis unstable in this case,anditmighthappen that
1+f′(R)<0 around the transition.
A simple modification of the above model which incorporates the inflationary era, namely the requirement (7), is
a combination of the two models discussed so far, that is
ebR 1
f(R)=α(e−bR 1) α − , (36)
− − IebR+ebRI
or, as a two-step model,
ebR 1 ebR 1
f(R)= α − α − . (37)
− ebR+ebR0 − IebR+ebRI
Again, f(0) = 0 and, at the value R = R , there is a transition to a higher constant value (α+α ) which can be
I I
−
related to inflation.
We should note that f(R) in (37) is a monotonically decreasing function. Then, when R is very large, f(R) tends
to a constant value, which could correspond to the effective cosmological constant generating inflation. In order to
describe the recent accelerating expansion of the universe, f(R) should remain almost constant, that is, f′(R) = 0,
for sufficiently small values of R corresponding to the curvature of the present universe.
7
The main problem with this models is the appearance of antigravity in connection with the transition point when
the function is sharp(see the discussioninthe Appendix). The appearanceof antigravityin the past, namely around
R , is not acceptable, as we have already commented.
I
A possible modification of the previous model is the following:
ebR 1
f(R)= α(e−bR 1)+cRN − , (38)
− − ebR+ebRI
with N >2 and c>0. In this variant, similarly to the theory [12], during the inflationary era at R>R , f(R), the
I
model acquires also a power dependence on the scalar curvature, which may help to exit from the inflationary stage.
AsdiscussedindetailintheAppendix,forthesharp,thetamodels,besidestheproblemofantigravity,forR <<α
0
and RI <<αI, they posses, generically, two De Sitter critical points, one around the transition point R∗ ≃ 5R40 and
the other being
R∗,2 2α. (39)
≃
We can also investigate the matter instability. For the two-step model (37), we now assume
R R R R . (40)
0 e I
≪ ∼ ≪
Then f(R) in (37) can be approximated as
α bR
f(R) α 1+ 1+e−bR0 e−b(R−R0) I . (41)
∼− − − 1+ebRI
n (cid:0) (cid:1) o
We may assume
α b
I
1 , (42)
1+ebRI ≪
since bR could be very large (see the argument around (78) about antigravity). Then we find
I
eb(Re−R0)
U(R ) , (43)
e ≃−3αb2(1+e−bR0)
which is negative and there is no instability.
We conclude this Section with a variant of the above model which facilitates the analytic computation and the
discussion concerning antigravity. In fact, as a smoothed one-step function, we may consider
b(R R ) bR eb(R−R0) 1 ebR0 1
0 0
f(R)= α tanh − +tanh = α − + − (44)
− 2 2 − eb(R−R0)+1 ebR0 +1
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
When R 0, we find that
→
αbR
f(R) . (45)
→−2cosh2 bR0
2
and thus f(0)=0, as required. On the other hand, when R +(cid:0) ,(cid:1)
→ ∞
bR
0
f(R) 2Λ α 1+tanh . (46)
eff
→− ≡− 2
(cid:18) (cid:18) (cid:19)(cid:19)
If R R in the present universe, Λ plays the role of the effective cosmologicalconstant. We also obtain
0 eff
≫
αb
′
f (R)= , (47)
−2cosh2 b(R−R0)
2
(cid:16) (cid:17)
which has a minimum when R=R :
0
αb
′
f (R )= . (48)
0
− 2
8
Then in order to avoid antigravity, we find
αb
′
0<1+f (R )<1 . (49)
0
− 2
The model given by Eq. (44) is able to describe late acceleration. In order to show that the de Sitter critical points
exist, we cancompute the function K(R)=Rf′(R) 2f(R)of the Appendix and we haveK′(R)=Rf′′(R) f′(R),
K′′(R)=Rf′′′(R), 1 K′(0)=1+f′(0), where − −
−
b(R R )
′ 0 ′
K (R)= bRtanh − +1 f (R), (50)
− 2
(cid:20) (cid:18) (cid:19) (cid:21)
Thus, enforcing the absence of antigravity, one has
′
1+f (0)>0. (51)
namely, K′(0)<1. In this case however K′(R)>0 for R>R , and it is negative for R<R . Therefore, we cannot
0 0
use the argument of the Appendix. For the one step-model, however, one can actually live with antigravity in the
future, thus in the sharp version, the analysis in the Appendix leads again in fact to the existence of two dS critical
points.
As a model which is able to describe both the inflation and the late acceleration epochs, we can consider the
following two-step model:
b (R R ) b R b (R R ) b R
0 0 0 0 I I I I
f(R)= α tanh − +tanh α tanh − +tanh . (52)
0 I
− 2 2 − 2 2
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
We now assume
R R , α α , b b , (53)
I 0 I 0 I 0
≫ ≫ ≪
and
b R 1 . (54)
I I
≫
When R 0 or R R , R , f(R) behaves as
0 I
→ ≪
α b α b
0 0 I I
f(R) + R . (55)
→− 2cosh2 b0R0 2cosh2 bIRI !
2 2
(cid:0) (cid:1) (cid:0) (cid:1)
and find f(0)=0 again. When R R , we find
I
≫
b R b R b R
0 0 I I I I
f(R) 2Λ α 1+tanh α 1+tanh α 1+tanh . (56)
I 0 I I
→− ≡− 2 − 2 ∼− 2
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
On the other hand, when R R R , we find
0 I
≪ ≪
b R α b R b R
0 0 I I 0 0
f(R) α 1+tanh 2Λ α 1+tanh . (57)
→− 0 2 − 2cosh2 bIRI ∼− 0 ≡− 0 2
(cid:20) (cid:18) (cid:19)(cid:21) 2 (cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)
Here we have assumed (54). We also find
α b α b
′ 0 0 I I
f (R)= , (58)
−2cosh2 b0(R−R0) − 2cosh2 bI(R−RI)
2 2
(cid:16) (cid:17) (cid:16) (cid:17)
which has two valleys when R R or R R . When R=R , we obtain
0 I 0
∼ ∼
α b
′ I I
f (R )= α b > α b α b . (59)
0 − 0 0− 2cosh2 bI(R0−RI) − I I − 0 0
2
(cid:16) (cid:17)
9
On the other hand, when R=R , we get
I
α b
′ 0 0
f (R )= α b > α b α b . (60)
I − I I − 2cosh2 b0(R0−RI) − I I − 0 0
2
(cid:16) (cid:17)
Then, in order to avoid the antigravity period, we find
α b +α b <2 . (61)
I I 0 0
The existence of the de Sitter critical points in this two-stepmodel is much more difficult to investigate. However,in
order to get the acceleration of the Universe expansion it is sufficient that ω < 1.
eff −3
WenowinvestigatethecorrectiontotheNewton’slawandthematterinstabilityissue. Inthesolarsystemdomain,
on or inside the earth, where R R , f(R) in (44) can be approximated by
0
≫
f(R) 2Λ +2αe−b(R−R0). (62)
eff
∼−
On the other hand, since R R R , by assuming Eq. (54), f(R) in (52) could be also approximated by
0 I
≪ ≪
f(R) 2Λ +2αe−b0(R−R0), (63)
0
∼−
which has the same expression, after having identified Λ = Λ and b = b. Then, we may check the case of (62)
0 eff 0
only.
We find that the effective mass has the following form
eb(R−R0)
M2 , (64)
∼ 4αb2
which could be very large again, as in the last section, and the correction to Newton’s law can be made negligible.
We also find that U(R ) in (15) has the form
b
1 1
U(R )= 2Λ+ e−b(Re−R0) , (65)
e
−2αb b
(cid:18) (cid:19)
which could be negative, what would suppress any instability.
Thus, we have here presentedseveralrealistic exponentialmodels which naturally unify the inflation with the dark
energy epochs (with a radiation/matter dominance phase between, as in Refs. [11, 12]). In addition, the Newton law
is respected and all spherical body solutions (Earth, Sun, etc) are stable.
IV. VIABLE MODELS IN THE EINSTEIN FRAME
As is well known from previous studies, it is often quite convenient to go from the Jordan (physical) frame to
the mathematically-equivalent Einstein frame description, where f(R) models become scalar-tensor theories with a
suitable potential. In particular, corrections to Newton’s law and the matter instability can be also investigated in
the Einstein frame directly, where the relevant degrees of freedom are a new tensor metric and a scalar field. More
specifically, concerning the inflation issue, the Einstein frame can indeed be very useful. Following e.g. reference [3],
we can introduce the auxiliary field A and rewrite the action (1) as
1
S = d4x√ g (1+f′(A))(R A)+A+f(A) . (66)
κ2 − { − }
Z
From the equation of motion with respect to A, if f′′(A) = 0, it follows that A = R. By using the conformal
transformation g eσg , with σ = ln(1+f′(A)), we ob6 tain the Einstein frame action [3]:
µν µν
→ −
1 3 F′′(A) 2 A F(A)
S = d4x√ g R gρσ∂ A∂ A +
E κ2 Z − ( − 2(cid:18)F′(A)(cid:19) ρ σ − F′(A) F′(A)2)
1 3
= d4x√ g R gρσ∂ σ∂ σ V(σ) , (67)
κ2 − − 2 ρ σ −
Z (cid:18) (cid:19)
A F(A)
V(σ) = eσg e−σ e2σF g e−σ = . (68)
− F′(A) − F′(A)2
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
10
Here g(e−σ) is given by solving σ = ln(1+f′(A)) = lnF′(A), as A = g(e−σ). After the scale transformation
g eσg is done, there appears a −coupling of the scalar field σ with matter. For example, if matter is a scalar
µν µν
→
field Φ, with mass M, whose action is given by
1
S = d4x√ g gµν∂ Φ∂ Φ M2Φ2 , (69)
φ µ ν
2 − − −
Z
(cid:0) (cid:1)
then there appears a coupling with σ (in this Einstein frame):
1
S = d4x√ g eσgµν∂ Φ∂ Φ M2e2σΦ2 . (70)
φE µ ν
2 − − −
Z
(cid:0) (cid:1)
Thestrengthofthecouplingisofthesameorderasthatofthegravitationalcoupling,κ. Unlessthemasscorresponding
to σ, which is defined by
1d2V(σ) 1 A 4F(A) 1
m2 = + , (71)
σ ≡ 2 dσ2 2(F′(A) − (F′(A))2 F′′(A))
isbig,therewillappearalargecorrectiontotheNewtonlaw. Newton’slawhasbeeninvestigatedinthesolarsystem,
as well as on earth, where the curvature is much larger than R . For the model (26), we find
0
ebR
m2 , (72)
σ ∼ 2αb2
which is positive. As we will discuss soon ((74)-(78)), we find 1/b R R, and therefore bR 1, which tell us
0
that m2 could be very large and the correctionto the Newton law w≪ould b≪e very small. For the m≫odel (37), we find
σ
eb(R−R0)
m2 , (73)
σ ∼ 2αb2(1+e−bR0)
which could be very large again, and the correction to the Newton law correspondingly very small.
Eq. (66) also tells that if
′
1+f (A)<0, (74)
then antigravity could appear, since the effective gravitational constant is given by
κ2
κ2 . (75)
eff ≡ 1+f′(A)
In order to avoidit, the condition (75) must be satisfied, at least until present, all the way since the beginning of the
universe. Some remark is in order. In the antigravity region, there is no evolution of the universe with a flat spatial
part. In the usual Einstein gravity, we have the FRW equation, (3/κ2)H2 = ρ, but in the antigravity region, since
the sign of κ2 changes, as κ2 κ2, we get (3/κ2)H2 = ρ. Since the lhs of this equation is always negative and
→ − −
the rhs is alwayspositive,there is no solution,whatshowsthatthere is no time-evolutionofthe universein this case.
We shouldalso note that, evenif 1+f′(A) is negative,the conformaltransformationitself canstill be welldefined, if
we use the absolute value of 1+f′(A), that is, 1+f′(A) g g . In that case, however,in the obtained Einstein
µν µν
| | →
frame action, the sign of the scalar curvature R becomes negative, that is, antigravity again appears. Moreover, the
initial value problem which is formulated in f(R) gravity via conformal transformation [13] is not well defined. This
is the reason why we avoid the consideration of antigravity regimes.
For the simple model (4), the condition (74) reads
1 αbe−bR >0 . (76)
−
Since the scalar curvature R in the past universe could be larger than the curvature R in the present universe, we
0
find 1 αbe−bR >1 αbe−bR0. Therefore, if
− −
1 αbe−bR0 >0 (77)
−
is satisfied, the condition (74) can be satisfied too. Eq. (77) tells us that
1
αb<1 or R . (78)
0
b ≪