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Cognola G„ Zerbini S. Minisuperspace approach of generalized gravitational models
Cognola G.,' Zerbini S,2
MINISUPERSPACE APPROACH OF GENERALIZED GRAVITATIONAL MODELS
Dipartimento di Fisica, Université di Trento and istituto Nazionale di Fisica Nucleate Gruppo Coilegato di Trento, Italia
11ntroduction
It is well known that recently there has been found strong evidence for an accelerate expansion of our universe, apparently due to the so called dark energy. With regard to this issue, here we would like to make some considerations involving general relativistic theories of gravitation. In fact, recently alternative and geometric descriptions for the dark energy in modern cosmology have been proposed and discussed in several related issues [1,2]. Such models are higher derivative gravitational theories, thus they may contain instabilities [3] and deviation from Newton gravity [4], However, if one takes quantum effects into account, one can get a viable theory [5]. The Palatini method has also been applied in consistent way [6,7,8] and the evaluation of the black hole entropy within these models has been investigated in [9].
To this aim, we shall consider a general relativistic theories, (see for example [10,11]), namely let us assume that our model is described by the action
(1.1)
f(R) =
R-
f(R) = R + R'
y+pin
R
(1.5)
where y, P and p. are suitable constants.
2 Minisuperspace approach
Our aim in this section will be the issue of a minisuperspace Lagrangian description, in order to investigate classical and quantum aspects, like the stability and canonical quantization. For these reasons, one has to restrict to FRW isotropic and homogeneous metrics with constant spatial section. We choose a spatial flat metric, namely
ds2 = a2 CnX-A'2 (r\)dVi2d2x), (2,1)
where T) is the conformal time, a(f|) the cosmological
factor and N(r\) an arbitrary lapse function, which describes the gauge freedom associated with the
reparametrization invariance of the minisuperspace gravitational model. For the above metric, the scalar curvature reads
' a'N'
R = 6
tï'/Y*'
1 6KG
with f(R) depending only on the scalar curvature. As first example, let us consider the Lagrangian [i]
Ml"
R (1.2)
where (,i is a new cosmological parameter [ 1 ]. As is well known , there exist constant curvature de Sitter and AdS vacuum solutions such that J?02 = 3|x4. (1.3)
Another well known example, is given by the choice
f(R) = R + jR2 -2A, (1.4)
where the the other possible quadratic term giving by the Weyl invariant has been omitted because is vanishing for space-time we are dealing with (see, for example [10]).
As a third example, let us consider an effective Coleman-Weinberg like model
a'N'
in which ' stands for
(2.2)
d _
if one plugs this expression in the Eq. obtains, an higher derivative Lagrangian thee derivative Lagrangian theory may b canonically by means the Ostrogadski method (■: . 1 example [12] and references cited therein).
Here we have found more convenient to follow the method outlined in ref. [13]. To deal with a non standard higher derivatives Lagragian system, we make use of a Lagrangian multiplier Pi and we write
S =■
V,
\6itG
¡dl\Na*
a'N' '
(2.3)
1 1 1 a3N2 a N
Making the variation with respect to R , one gets df(R)
dR
(2.4)
Thus, substituing this value and making a standard integration by part, one arrives at the Lagrangian, which will be our starting point
1 cognolaiscience.unitn.it
2 zerbiniiscience.unitn.it
TSPU Vestnik. 2004, Issue 7(44). Volume: NATURAL ANDEXACT SCIENCES
+Na4\ f(R) - R
N dR
df(R)' dR
N dR1
(2.5)
= 0,
(2.7)
N- dR
N*
dR'
(2.8)
^{fim^Myo,
and we may choose, for example, the gauge N = 1.
(2.9)
¡'R'a d''f(R)
(2.10)
N dR N
4f(R
dR2
(2.11)
Afc4 /<*)-*
dR
R=R0, a0 -—, A Ra = 12. 11
J?o = 3|i4, (2.14)
while for the Lagrangian (1.4), Eq. (2,13) gives R0= 4A, (2.15)
and for the Coleman-Weinberg like model gives
it should be noted that N appears as "eiiibein" .^tingian multiplier, as it should be, reflecting the
\ 11 ue'rization invariance of the action. In fact the x.,«gi4gian is quasi-invariant with respect to the infinitesimal gauge transformation
Sa = e(x)a'(x), 8R = €(x)R'(x), m = —[e(T)M'x)].
d%
(2.6)
As a consequence, we have the (energy) constraint
d_L
dN
namely
tan df(R) t to'uM'd2f(R)
K0=±, R0= 0.
(2.16)
It is easy to check that such kind of solutions are physically ones, because we have
E =
6A
1-
2f(R0)
R«Tr{RO\
(2.17)
namely they satisfy identically the energy constraint E = 0. Thus, the condition (2.13) turns out to be a necessary and sufficient condition in order to have physical constant curvature solutions.
In order to investigate the Hamiltonian formalism, it is convenient to make the following change of variables [13]: N N, a—>q and defined
by
éim=b y
dR
(2.18)
a = qe"9. (2.19)
In the first, B is a suitable constant, fixed by means
of
.20)
and R , as a function of the new variable <>, is defined implicitely R = R($).
For example, for the choice (1.2), one has
' dR N dR5
The conserved quantity is the energy, computed with the standard Legendre transformation
£ = ./aR' d2f(R)
df(R]
dR and
ft
R ■
R =
(B2e-1 )"2 '
For the Lagrangian (1.4), one obtains B2e2^ -1 "
E is vanishing on shell due to the Eq. of motion for the einbein N.
We shall be interested in models which admit solution with constant 4-dimensional curvature of the de Sitter type, namely
/1
2y
(2.21)
(2.22)
(2.23)
In the case of Coleman-Weinberg like model, one only has
B2e2tf =l + 2i?
y + Pin4- l + PR-
F
(2.24)
(2.12)
If we plug this particular solutions in the above Eqs. of motion, we get the condition [10]
2 f(Ro) = R*~^(Ro%
which may be used to find the constant curvature R0, For the model defined by Eq. (1.2), Eq. (2.13) leads again to the condition
Thus, it is not possibile to obtain explicetily R as a
function of the new variable <j>.
The de Sitter like solution corresponds to
(2.13) ^0=°. A Ra =12.
A direct calculation leads to Lagrangian
L = ~{(2q- q-? f2 -qn}~ NV% q), N
(2.25)
(2.26)
Cognola G., Zerbini S. Mmimperspace approach of generalized gravitational models
in which the potential reads
9 „-4,
a
ClK
(2.27)
For example, for the Lagrangian (1.2), we have (see alsof 1,2,3])
4 _
B2
[BV*-lf
while for the Lagrangian (1.4), one obtains (BV^-i)2'
D
2A+-
4y
The conserved energy reads
E = JL[{2q - q2 _ ] + NV($, q),
N
(2.28)
(2.29)
(2.30)
and it is vanishing on shell. The corresponding Hamiltonian is
H = N
1
(2.31)
1 -44
q e *
3 I?2
(Iq-qftf + Kq-qqW-
-2f{R) + R
df(R)
= 0.
(2,32)
(2.33)
6i?2 ' L ' ' dR
If the Lagrangian satisfies the condition (2.13), it is easy to check that the one has again the de Sitter solution (2.25).
We conclude this Section writing down the equations for the small disturbances around the de Sitter solution, namely
q = q0+8q, f]> = 5fj). (2.34)
Taking Eq. (2.13) into account again, the equations for the small disturbance around de Sitter solution turn out to be
d\ W
(2.35)
2---
T| J d~ri r| ^ t| J dr\
A3
ÏÏ
2 f(R0) d2f
d%
(2.36)
00 = 0.
q is universal, namely it does not depend explicitly on function f(R), but we remind that the constant value R() depends on it. Third, the general solution of the first equation is not hard to find and reads
bq = Cjff + e2Tf
(2.37)
2A(2q-q2) * 24"'
The Lagrangian equation of motion, in the gauge N =1,read
However, the perturbed solution q - q0 + 5«j must satisfy the energy constraint E = 0 and this leads to 8E = 0, (2.38)
namely, around the de Sifter solution <$¡$-<¿59 = 0. (2.39)
As a result c, = 0 and we have
T)[ Ail _
Recall that the relation between the conformal time and the cosmological time t may be written and 1.2A = J?2, R0 being the de Sitter curvature and
t = 0 corresponds to %.
!
(2.41)
and 12A = R2;, R0 being the de Sitter curvature and
i ~ 0 corresponds to t)0.
Thus, the solution remains small with respect the de Sitter one for
Tp
^ 12llo%o
1.42)
Some remarks are in order. First the small disturbance equations are decoupled in the conformal time. Second, the equation associated with the variable
R(l being the de Sitter curvature.
Along the same lines, one may investigate the Starobinski model [14] and its generalization including the Brans-Dicke field [15] and its brane-world
generalization [16].
3 Conclusions
In this paper, we have presented a miriisuperspace approach for general relativistic pure gravitational models. The inclusion of the matter can be easily taken into account. A canonical approach has been presented by means of the methods of ref. [13]. These models are, in general, instable, due to the presence, at the beginning, of higher derivative terms [3]. However, the inclusion of quantum effects may resolve the problem [5].
Acknowledgments
We would like to thank E. Elizalde, S. Nojiri and S D. Odintsov for stimulating collaboration. ,,
References
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TSPU Vestnik. 2004. Issue 7 (44). Volume: NATURAL AND EXACT SCIENCES
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Gamttmi R.1
cmmm ENERGY ANP THE- ' OSMOLOGICAL CONSTANT
Université degli Studi di Bergamo, Facoità di Ingegneria, Viaie Marconi 5, 24044 Dalmine (Bergamo) ITALY.
I. Introduction ;.d as the bare cosmological
V Ul U» C". | L.tii). . .. " ■ '
t s'. - , ; ■ 1 ■ .rtG(p). C6)
(I)
where At, is the cosmological constant, G is the gravitational constant and T is the energy-momentum tensor. By redefining A.,
7' lot __ nr
1 M-v ~= *■ f.tv "*' "
BnG
(2)
one can regain the original form of the field equations
(3)
^v --g^R=8itcr;:' = 8KG(I;1v +'0,
at the prize of introducing a vacuum energy density and vacuum stress-energy tensor
A.,
Pa =-
, » hiv "Pa<?u.v
(4)
EnG *
Alternatively, Eq. (1 ) can be cast into the form, 2'
where we have included the contribution of the vacuum energy density in the form Tm = -<p)#(JV. In this case
^RV ^ + ^eff 8,,v = 0'
(5)
-) i> OftV-
(7jiy
2 + m2 =
(7)
• = JO71 GeV4
16k2
This gives a difference of about 118 orders [1]. The approach to quantization of general relativity based on the following set of equations
2k
jfV -&(R-2AC)
2k
(8)
and
-2V(.Jtff»F[ftf] = 0, (9)
where R is the three-scalar curvature, Ac is the bare cosmological constant and k = 8jiG , is known as Wheeler-De Witt equation (WDW) [2]. Eqs. (8) and (9) describe the wave function of the universe. The WDW equation represents «variance under time reparametrization in an operatorial form, while Eq. (9)
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