CC BY
41
valuable correspondence. This work was supported by DGICYT under Research Project BMF2002-03758.
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Wang P.,1 Meng X.-H.2 PALATINI FORMULATION OF L(R) GRAVITY
Department of Physics, Nankai University, Tianjin 300071, P.R.China
I. Introduction
It now seems well-established that the expansion of our universe is currently in an accelerating phase. The most direct, evidence for this is from the measurements of type la supernova [1]. Other indirect evidences such as the observations of CMB by the WMAP satellite [2],
large-scale galaxy surveys by 2dF and SDSS [31 also seem supporting this.
But now the mechanisms responsible for this acceleration are not very clear. Many authors introduce a mysterious cosmic fluid called dark energy to explain, this (see Ref. [4] for a review). On the other hand, some authors suggest that maybe there does not exist
1E-mail: pewang@eyou.com
2E-mail:xhmeng@phys,nankai.edu.cn
such mysterious dark energy, but the observed cosmic acceleration is a signal of our first real lack of understanding of gravitational physics [5]. An example is the braneworld theory of Dvali et al. [7]. Recently, there are active discussions in this direction by modifying the action for gravity [7-17]. Specifically, a 1/1? term is suggested to be added to the action [7]: the so called 1/i? gravity. If is interesting that such term may be predicted by string/M-theory [8]. In Ref.
[12], Vollick used Palatini variational principle to derive the field equations for 1/i? gravity. In Ref. [11], Dolgov et al. argued that the fourth order field equations following from the metric variation suffer serious instability problem. If this is indeed the case, the Palatini formulation appears quite appealing, because the second order field equations following from Palatini variation are free of this sort of instability
[13]. However, it is also interesting to note that quantum effects may resolve the instabilities, see Ref.
[14]. In this paper, we will review deriving the full Modified Friedmann equation for Palatini formulation of the L(R') gravity. Then we will discuss the
application of this derivation to 1/i?, R1, l/R+R2 and In R gravity. Recently, Palatini formulation of L(R) gravity was also considered in Ref. [15].
I! The Modified Friedmann Equation
In general, when handled in Palatini formulation, one considers the action to be a functional of the metric
g and a connection which is another
independent variable besides the metric. The resulting modified gravity action can be written as ” 1 2k2
where we use the metric signature {-,+,+,+}, k2 = EkG , Ruv is the Ricci tensor of the connection
, R = Fv4iv > ai,d Sm is the matter action.
Varying the action (1) with respect to g gives
L\m,v-jL(R)gm=K% (2)
where a prime denotes differentiation with respect to R and T is the energy-rnomentum tensor given by
Sig,vSll}=ld4x4^-L-LiR) + Sm
(2)
r - 2 5Sm
(3)
equations. Then the energy conservation equation p + 3H(p + p) = 0 is unchanged.
In the Palatini formulation, the connection is associated with |(1V =L'(k)gpy, which is known from varying the action with respect to f^v. Thus the Christoffel symbol with respect to gpv is given in terms of the Christoffel symbol with respect to g by
=fiv + ^№lW~gllvg'*daL'](4)
Thus the Ricci curvature tensor is given by
+|ar2VML'VvL'-(Lr1^VvL'
2
and
(Lrxxw
R = R- 3(LT WL' + - (L'y2 V.LV» V
(5)
(6)
where R is the Ricci tensor with respect to glK and
R = gilvA5(lv. Note by contracting Eq.(2), we get: L'{R)R — 2L(R) = K2r
(7;
Assume we can solve R as a function of T from Eq. (7). Thus Eqs. (6), do define the Ricci tensor wish respect to /?jiV.
We will consider the general Robertson-Walker metric (note that this is an ansatz for g ami is the
result of the assumed honr and isotropy oi
the universe, thus its form ’ - 1 ni of the gravity
theory):
dsl =-dt2 + a(t)2
dr
■hr
— + r (d&2 + sinz Qd§2 )
(8)
where k is the spatial, curvature and ¿ = -1,0,1 correspond to open, flat and closed universe respectively. The a(t) is called the scale factor of the universe.
From equations (8), and (6), we can get the nonvanishing components of the Ricci tensor:
4 =_3" +!<Z/r20oZ/)2 -|(Lr‘V0V0L'
a 2 2
L = [aä + 2à2 + 2k + (L'ylf(ld0L'
+Z-(L’rlV0V0L%
(9)
(JO)
where SM is the matter action. For a universe filled
with perfect fluid, = [-p, p, p, p]. Note that the
local conservation of energy momentum V 7’|1V — 0 is
a result of the covariance of the action (1) and Noether theorem, thus it is independent of the gravitational field
where a dot denotes differentiation with respect to t.
Substituting equations (9) and (10) into the field equations (2), we can get
6 H2 + 3H(L'T' 30L'+- {L’Y2 (d0L')2 + 6-4
2 c* . ,,,,
tc(p + 3p) + L
¡7
where H =ci! a is the Hubble parameter, p and p are
the total energy density and total pressure respectively.
Using the energy conservation equation p + 3/f (p + p) = 0, we have
'Jk2
a0L'=^-(i-3ci2xp+^)
(12)
where c) = dpi dp is the sound velocity.
Substituting Eq. (12) into Eq. (13) we can get the Modified Friedmann (MF) equation of L(R) gravity in Palatini formulation..
ill. Applications
Iii this section, we review the application of the general Modified Friedmann equation to cosmology.
. :' R gravity
Hie Lagrangian is given by L(R) - R -p.4 IR [7]. It is interesting to note that, in Ref. {8], Nojiri and Odintsov have shown that this action can be derived from string/M theory.
The MF equation follows from eq. (13) [13]
Kp - a
;if!
v a ;
H =
K.
(X
3 a
6 + 3 F
sp
jsp
a
1 + -F\
where the two functions G and F are defined as C(x) = -
F(x} =
G(x)2 + ~
, 1 2
J H— X
4
(13)
(14)
(15)
2k2 (p m +pr) +
H2=-
3p
1 +
2k2P,„ 3p ,
6+3Fn
(
p
K2P„
\\
where the function F0 is given by 2x
F0(x) = ~
1-A
3
(16)
(17)
It is interesting to see from Eq. (16) that all the effects of the R2 ■ term are determined by pm. If pm = 0, Eq. (16) simply reduces to the standard Friedmann equation.
At late cosmological times when K2pm /|3 «1, F ~ 0, the MF equation (16) reduces to the standard Friedmann equation:
ff2=y(P*+Pr)
(18)
In Ref. [13], we have shown that the above MF equation can fit the current SN la data at an acceptable level. However, the effective equation of state it gives shows some pathological behaviors.
B, R2 gravity
It is a well-known result that a R2 term in the Lagrangian can drive an early universe inflation without infiaton [24] (See also Ref. [25] for a comprehensive discussion of R2 gravity).
In the Palatini formulation, the MF equation is [19]
Thus from the BBN constraints on the Friedmann equation [26], p should be sufficiently large so that the
condition K2pm/p«Kl is satisfied in the era of BBN. In typical model of R2 inflation, (5 is often taken to be the order of the Plank scale [24].
Then we will see that whatever we assume on the value of pro during inflation, a R1 driven inflation can nor happen.
Firstly, since inflation, happens in very early universe, where (he temperature is typical of the 10’5 Gev order, if we assume that almost all the matter in the universe at that time is relativistic so that K2p,„/p«:l, thee as we have shown above, the MF equation reduces to the standard Friedmann equation and thus no inflation happens. Note that at the inflation energy scale, all the standard model particles will be relativistic.
Secondly, if there are enough exotic objects other than the standard model particles that, will be non-relativistic during the inflation era so that K2p,„/p 3> 1. Those objects may be primordial black holes, various topological defects which we will not specify here. In this case, from Eq. (17), the MF equation (16) will reduce to
H*- k2p->- ,.. 2Pp-- , 2P
P 7pm 7
(19)
Then we can see that if the [3 term dominates over the other two terms, it will drive an inflation. But this equation is derived under the assumption that (3 « K2pm. Thus no inflation will appear, too. *
Thus, the difference between Palatini and metric formulation of the same higher derivative gravity theory is quite obvious here.
C. 1IR + R2 gravity
In Ref. [17], Nojiri and Odinstov showed that a combination of the 11R and R2 terms can drive both the current acceleration and inflation. The Palatini form of this theory is studied in Ref. [19].
The MF equation reads,
H2 =
K2pm + 2K2p(, + a G(x) 1 + %C(xŸ 3 G(x) 3ß J
, 1 2a ^ N 1 + t -1- G(x) 3 G(x)2 3ß 6+3F(x)j^l+^F(x)j
(20)
, tcp,.
where —
and the two functions G and F are
given by
G(x) = i
F(x) = -7
a
x + :
1 + —JT
4
G{xf + —G(.v)3 + -- UI+-X-
3ß ' 3 AI 4
(21)
(22)
In order to be consistent with observations, we
should have tx«cp. We can see this in two different ways.
Firstly, when K2pm»a, from Eq. (21), G ~ K2pm / a. From the BBN constraints, we know the MF equation should reduce to the standard one in the BBN era. [26]. This can be achieved only when F ~ 0 and from Eq. (22), this can be achieved only when a<s (3 and 1« K2p,n/a <r (¡3/a)1/3.
Secondly, when K2pm «a, we can expand the RHS of Eq. (20) to first order in K2pm / or.
11-q/ß H 2 = 8-4cc/|3
K2p,„ +
3 2 1
-------— k p +-<x
2-oc/ß Hr 2
6 +
4-2a/ß
(1 + a/ß)
K2p„
a
(23)
D. ini? gravity
In Ref. [18] Nojiri and Odintsov proposed using a single In/? term to explain both the current acceleration and inflation. The Palatini formulation of In R gravity is studied in Ref. [27].
The Modified Friedmann equation reads,
_ 0( R , R
KpM + 2icpr -p —-in —
H1*- ■ -
R
1 'A 6 + 3F(x)| l + ^F(x) ' 1
(24)
where x&
F{x) ■
and F is defined as
(Ä(x)/ß) ~2i?(x)/ß
(25)
It can be seen, from equations (24) and (25) that when p -h> 0, the MF equation will reduce continuously to the standard Friedmann equation. Thus, the InR modification is a smooth and continuous modification.
Let us first discuss the cosmological evolution without matter and radiation. Define the parameter n as R0 = ~ae~". Substitute this to the vacuum field equation L(R) = 0, we can get a = e:‘(2n l)p and R0 = ~{2n + I)p. Substitute those to the vacuum MF equadon and sel t = 0, we have p(«. + l)
( i
6 ! 1 + -.1
v In-
j
When a « j} , this will reduces exactly to the first order MF equation in the 1/R theory [13]. Since we have shown there that the MF equation in l/R theory can fit the SN la data at an acceptable level, the above MF equation can not deviate from it too large, thus the condition a<K p should be satisfied.
Thus when a ~ ~ (\0~i}eV)2 and « > -1/2, the
In/? modified gravity can indeed drive a current exponential acceleration compatible with the observation. The role of the parameter p is similar to a cosmological constant or the coefficient of the l/R term in the 1/1? gravity [13].
When the energy density of dust can not be neglected, i.e. xpm /p » 1, F(x)~ 0 and if a satisfies ! ln(Kp,„/a) J« Kpm/p, i.e.
exp(~Kp„, / P) «: a / p « exp(icpm / p), then R - -Kp,„. Then the MF equation (24) reduces to the standard Friedmann equation
H2^(pm+pr) ■ (27)
Thus if exp(-KpK,flW /p) « a/p <r exp('Kp„lJJav /p) where pm BBN is the energy density of dust in the epoch
of BBN, the In R gravity can be consistent with, the BBN constraints on the form of Friedmann equation [26]. One possible choice is a = p, for which the vacuum solution can be solved exactly RQ=-a. Since
P ~ Hi, the condition Kpm /£J »1 breaks down only in recent cosmological time. Thus the universe evolves in the standard way until recently, when In R term begins to dominate and drives the observed cosmic acceleration,
IV. Conclusions and Discussions
In this paper we reviewed the Palatini formulation of L(R) gravity and its applications to cosmology. Hie nature of dark energy is so mysterious that it is promising to seek further the idea that there does not exist such mysterious dark energy, but it is General Relati vity that fails at large scale.
The current “standard theory” of gravitation, Einstein's General Relativity (GR) has passed many tests within Solar system. To reconcile the successful
GR predictions within the solar system, the extended gravity theories may be required to be scale sensitive. It could be challenging and profound to locate the additional curvature terms in our above discussions what form of scale dependence.
Acknowledgements
We would like to express our deep gratitude to Professor Sergei D. Odintsov for numerous invaluable suggestions and comments when the work we described in this paper is performed. We would also like to thank Eanna E. Flanagan, Shin'ichi Nojiri for helpful discussion on those topic.
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