Dark energy and modified gravities Текст научной статьи по специальности «Математика»

V. 148, Suppt. P. 181, [hep-th/0209261]. Quantum aspects are discussed in Nojiri S., Odintsov S.D., Osetrin K.E. Phys. Rev. 2001. V. 063. P.

084016; Nojiri S., Odintsov S.D., Zerbini S. II Phys. Rev. 2000. V. D82. P. 064008.

3. Brevik i„ Berkje K., Morten J.P. II Gen. Rel. Grav. in press, gr-qc/0310103,

4. Brevik l„ Ghoroku K„ Odintsov S.D., Yahiro M. II Phys. Rev. 2002. V. D66. P. 064016.

5. Brevik I., Milton K.A., Nojiri S., Odintsov S.D. II Nuci. Phys. 2001. V. B599. P. 305.

6. Parker LII Phys. Rev. 1969. V.183. P. 1057.

7. Gherghetta T„ Pomarol A. II Nuci. Phys. 2000. V. B586. P. 141.

8. Nojiri S., Odintsov S.D., Zerbini S. II Class. Quant. Grav. 2000. V.17. P. 4855; Garriga 1, Pujolas O., Tanaka T. II Nuci, Phys., 2001. V, B605.

P. 192; Naylor W., Sasaki M. // Phys. Lett. 2002. V. B542. P. 289; Elizaide E., Nojiri S„ Odintsov S.D., Ogushi S. II Phys. Rev. 2003. V. D 67.

P. 063515.

9. Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products. Academic Press, Inc., New York, 1980, formula 8,778.

10. Parker L. II Phys. Rev. Lett. 1987. V. 59. P, 1369.

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12. Birrell N.D., Davies P.C.W. Quantum Fields in Curved Space. Cambridge University Press, England, 1982,

Nojiri S.1

DARK ENERGY AND MODIFIED GRAVITIES

Department of Applied Physics, National Defence Academy, Hashirimizu Yokosuka 239-8686, JAPAN

PACS numbers; 98.80.-k,04.50.+h,11.1 G.Kk,11,10.Wx

I Introduction

By the recent observations of the universe, we have found that the universe is undergoing a phase of accelerated expansion at the present epoch from about 5x I0iu years ago and the universe is flat.

rr"....^derated expansion of the universe has been

by the observation of the type la ; (SNIa) [11. This, type of supernovae has an aoosiption spectrum of Silicon, which helps us to distinguish the supernovae from other types. An important thing is that the intrinsic luminosities of SNIa are almost uniform and their absolute maginitude is almost -20, from which we can find the distance between the earth and the supernovae. From the distance, we can find how old the SNe are. We can also the speed of the SNe from the redshift. Combining the distance and the redshift, the accelerated expansion of the universe has been established.

The flatness of the universe can be found from the observation of the anisotropies of Cosmic Microwave Background (CMB) observed by the baloons, the BOOMERANG project [2] and the MAXIMA-1 project [3]. The spectrum of the CMB has a characteristic structure and the CMB was produced in the early universe, that is, when the universe becomes transparent or the mean free path of the light becomes infinite. Then the CMB has propagated very long distance, about 10 billion light years and if the uni vers is curved, the structure of the spectrum should be

deformed. The deformation has, however, not observed, which tells that the universe should be flat.

If the universe (the spacial part) is flat, we may assume the metric of tlte flat PRW universe in the following form:

ds1 =-dt2 + u(t)2 ^ (dx"f. (I)

Here a is called a s« universe is expanding ai-.- , • '• the accelerated expansion, Y¥C UilCii

w, which is called as “equation of The present data tell

W = L— i, (2)

P

Here p is the pressure and p is the energy density of the universe. We may explain the relation between w and the accelerated expansion later but if w<-\/3, the universe is accelerating, which Eq. (2) tells. As w is negative, if the energy density p is positive as usual, the pressure p should be negative.

In case of the radiation, we have w~— since the

3

trace of energy-momentum tensor vanishes: I' = -p + 3j? = 0. Usual matters, like baryon, can be regarded as dust with >v = () (p--()). The cosmological term can gives w = ~-l. It is, however unnatural if the cosmological term, which may be suggested from (2), is a part of the fundamental

1 E-mail; snojiri@yukawa.kyoto-u.ac.jp, nojiri@nda.ac.jp

gravity. The cosmological constant should be given by

? a

A - H , where H is the Hubble parameter, H = —.

a

Since the Hubble parameter is given by H -70 kms4 Mpc'1 ~ 10~33 eV, the scale of the cosmological constant is unrealistieally very small compared with the fundamental scale of the gravity, the Planck scale 1019 GeV =10“ eV.

On the other hand, since the universe is flat, the energy density of the universe should be almost critical density : l(T2i> g/cm3. The density of the non-relativistic

matter (baryons, dark matter) is, however, almost ~ of

the critical density. This tells that 70% of the energy density of the universe should come from the matter (?) with negative w . We call the matter (?) as dark energy. Conversely, the density of the dark energy is almost same order with the density of the usual matter. Then maybe it would be more natural to consider that the dark energy would be a kind of matter, rather than that it would come from the fundamental cosmological term in the gravity.

Now we explain the relation between the accelerated expansion and tv. 'The energy conservation

law 0 — V1*

has the following; form in the FRW

iiietrie (1);

0 = p + 3H (fj + p) =: p e

in case lhaf :e (3) i

+ w)H p.

s a constant, since H

', we ca

C*'*-". (4)

The (1st) FRW equation, which is the (l.t) component of the Einstein equation, is given by

6

=P-

(5)

Here k2 =16itG and G is the gravitational constant. Then if w&-l, the scale factor a can be found by integrating (5) by using (4):

(6)

The case w = -1 corresponds to deSitter universe:

a = oae

\e ^ (7)

1 _ 2

Then if w < —, that is, —------------ > 1, the universe

3 3(w + l)

is accelerating, which can be found more directly, by combining 2nd FRW equation,

- = -~-(p + 3p) = ~-p-(l + 3w0p, (8)

Cl 1A 12,

which tells a >0 if w<--. If -1<W<_I ihe

3 ' 3

universe is expanding and accelerating and the

corresponding matter is called as “quintessence” [4]. If w < -1, the universe is accelerating but shrinking in (6) but if we change the direction of time as t ts -1, the universe is expanding (and accelerating). The corresponding matter is called as “phantom” [5].

As clear from (6) after replacing t , when

w < -1, there is a singularity at t = ts, which is called the Big Rip singularity. We will mention more about the singularity later.The present data of the universe seems to tell that w might be less than-I. In the viewpoint of the quantum field theory, however, this is rather strange. As an example, we consider the scalar field f with the potential V . If we assume the FRW metric and (j> only depends on the time coordinate, the enrgy density p and the pressure p are given by

/>=|«i>2-vm (9)

i..

Then if V(<f>) > —cf) , we have negative w but the

value of w should be greater than -1. In order that w < 1, we should change the sign of the kinetic energy:

p = --L^- +V(<j)), p = -~§2 -V((j)), (10)

which tells that the phantom might be ’

negative norm, which should be a serious the quantum field theory1. Conversely w < - f, there should be a new physics.

II Scalar with Exponential Potential

In this section, as a naost simple model, we consider the scalar field theory with the exponential potential This model has been investigated from a very long time ago, say in [7]. The action of the model is given by

S = jd'xyfePjR-ld.m-V®)

(11)

v (<|») = V0e

Usually y>0, but for phantom, we may have y<0. We assume (flat) FRW universe (1) and <{? only depend on time coordinate t. Then the 0 -equation of motion and the FRW equation has the following form:

0 = -t(~| + 3//^]-V'(4»),

dr dt J

6 H'.p, =!(*>

2 {dtj

k2

+v(<S>).

(12)

1 The instability of the vacuum due to the quantum effect in the theories with negative kinetic energy has been investigated in [6].

If we assume eInt, hoc t 1, we can solve the equations in (12) very easily :

<j> = <j>0 In

// =

YK2<t>,

At

1-

3yK2<t>;

(13)

2Vn

Then the scale factor a is given by

r 4

a = a0

Since a = a0t

w = -l +

- „ for general w, we find 8

3yK20o

(14)

(15)

Then if y < G, surely phantom appears,

A very interesting point is that the general solutions of the scalar model with exponential potential can be found. In case y > 0 case [8], if we assume

2 (v-«)

V3Y

2f v-u)

(16)

a =■ e 3 , <|> : j—

tlx = dtj^e ,WJT,

V 8

the Hamiltonian constrain!, and other equations have the following form: dv du dx dx and

d U .. Y r d v . —?.

___ r=(i - )U, j- = (1 - or )v.

dx'

dx2

(17)

(18)

Here

vs«w, U^eil+m'\ a.=

(19)

a~e

4> = -

dx = ±dt,

2i(z-z’)

■spty

2i(z-z*)

a;

X$0yp3y'

Zsse1

(21)

Then the Hamiltonian constraint and other equations have the following form:

—— = 1, -v = (l + a2)Z.

dx dx dx~

(22)

The equations in (17) can be solved very easily and when or < 1, the behavior at /—»+«> coincides with that of the previous solution (13), (14) and w has the following form:

w = -l + or. (20)

2

Then if or < —, the universe is expanding and

accelerating (quintessence).

The case y<0 has been also exactly solved in [9], instead of (16), by using a complex variable z, we assume

'The above equations can be also solved easily. When t 0 (if we replace t->ts-t, t -» ts), the

behaviour corresponds to the one in the previous solution ad we find w is given by m; =—1 — ot2 < — 1, (23)

which surely corresponds to the phantom.

Ill A Modification of Exponential Potential Model

Since the models with negative norm would not be consistent with quantum field theory as the theory may be non-unitary. We consider a modification of the model based on [9].

We may start with Jordan (string) frame action, which is the Brans-Dicke type [10],

S = Id‘‘x4^ \ R - l/«D) j

k _y ' 7 (24'

+ \d,‘xj-g | -~dttx^X~U(x) i

v i-

Here a and y are constant parameters and y can be negative. The scalar field x represents Since x does not interact directly wi equivalence principle would not be violalt

the effective gravitational constant toe 2 depends on f.

2 a

By using the scale transformation: gia = ed~2 gEilv,

the Iordan frame action (24) is transformed in the Einstein frame one:

S=±ld“xfgB(R

\ 2a

d- 2

+ lddxfgE

(25)

da

U(%)

Even if y < 0, when

1+I>0

d -2 2

(26.)

the kinetic energy becomes positive. Especially when

d =4 and y = —1, we have a2 >^. Then the sign of

the kinetic energy depends on the frame.

When d = 4, % = 0 and V(<j>) is exponential type, we can solve the system in a way similar to the last

section. If we define,

f^oc2+j, V(<p)se-“ty(<|>) = V0e *

and assume the FRW metric in Einstein frame : ds\ = —dt\ + aE (tE )2 Y, (dx'f,

/=1,2,3

we obtain the following solution

(27)

(28)

3 ,

\7*5

\ h:o }

, q> = <pQln

tr

(29)

tE0 _ (p0

Vn

27

For the FRW metric in the original Jordan frame in (1), we have

dt - e r dtr, --

«

Pfe

(30)

Then we obtain

: = e

! f

u

0 J

a 1

Pfi

(31)

Then the condition for the acceleration is given by

2J“' + 3

1--

%«

>1,

(32)

2, /or +-

and w is given by

f

foa

W = -l + -

3J

l<p*___________?««...

2, or + -

(33)

fo(2p-9f0) + 8

3(2(5 — 3(p0)(p0

Since the denominator in the above expression can

vanish, by properly choosing the parameters a, %,

(y), we can put w in an arbitrary value. Especially if

cp.,a , 4

======= = 2, we have w = ~l and when % >— we

a>+] 3

find

. ,e 3 i w< 1 if —% >

9a“

a2 + -

r > 2

<Po«

(34)

a* +-

w>-l if

4

Oil the other hand, when q>2 < —,

w < — 1 if 2 >-tJML=>^-(p2 a‘*\ 2 '

r < 2.

if

cp„a 3 , ofl« -~4====r <—())- or —=i==

f. V 2 ! . •>'

!«i+“ " Jor + 1

i 3 V 3

(35)

Even if y > 0, we can have w < - i. For example, if

a2 " \/2 j ’

we have w, which is less than -1: 1 W2 + 203

213

(37)

IV 1/ R -model and generalization

In [11], a modification of Einstein gravity has been proposed. In this section, we consider the model and its generalization (see also [12,13]).

The action of the model in [11] include the inverse power of the curvature:

.it

R

(38)

When curvature is small | R |<s \x2, we have a « t2 2

and w = —-. This model may be called as “c-essence”

(“c” expresses the curvature) [14].

We may consider the generalization of the model as

follows,

S=~j\dAxf^f{K). (39)

Here /(i?) can be an arbitrary function. By introducing the auxilliary fields A,B , we may rewrite the action (39) in the following form:

5 = ^jd*x^{B(R - A) + f(A)}. (40)

By the variation of B, we obtain A = R. By substituting this equation into the above action (40), we obtain the original action. If we consider the variation of A first, we obtain B = f'(A), which can be solved

as A = g(B). Then we obtain

5 = jd*x^[B(R - g(B)) + f(g{B))}.

Instead of A , we may delete B as foil S = -L ^dAXyf-g {f'(A)(R -A) + /(A)},

(41)

Instead of A , we may delete B as follows,

_L

tc2

which can be regarded as the Jordan frame action. By

(42)

using the scale transformation g^ -4 e g(lv with <y = -In f\A), we obtain the Einstein frame action :

f(A)

fw

g^Ad'A-

f(A)_

f'lA) ' f\Af ■

- +

= -V ld4X^g } R -;-gPf3po3 O-l7(0) |,

'tC * I 2

V'(CT) = e°if(e-<,)-eJO/(jf(e'0))

A f(A)

~ f\A) fiAf

Especially, in case of [11), when A(=- R) is small, V(a) behaves as a exponential function:

(43)

(44)

V(a) ■

-e

(45)

Then we can solve the equations as in the previous section. Even for more general case [ 15] as

f(A) - A + yA~n In

(46)

when curvature is small and 0, the potential behaves as an exponential function / -j A _L 1±1 • 1 + — \(-yn)K+l eH+’°

V n)

n +1

(4?)

and we can solve again the equations: MiMsii) 6/i2 +7n-l

a ~ t

w = —

3(n + t)(2n + 2) Then we find that when

n<-

-7 -&sl 2

12

-7 + 6^2

12

w < 0 and when

1

-1 < n < —

2

(48)

(49)

or n >-

i+S -1+V3

n < —■—— or n > -

(50)

2 2 the universe accelerates.

In case that there is no matter and the Ricci tensor R^ is covariantly constant (Vp/^v = 0, which means

R^ = const xg ), the equation of motion is reduced to 0 = 2 f(R)-RfXR), (51)

which is an algebraic equation with respect to R. For

example, when fl

f (R) = R - + bRm, (52)

we have

0 = -r+ + (m _ 2 )bRm. (53)

Especially, in case of [1 !](« = !, a = (J.4, b ~ 0 ), we find

R - ±\/3fi2. (54)

The + sign corresponds to deSitter space, which is an exponentially expanding and accelerating universe. We may regard this solution would correspond to the present universe. We should not that even if b # 0 but m = 2, we obtain the same solution as (54).

Just after the MR -model was proposed, there were several criticism [14,16] and an improve merit was proposed in [17], where the action is modified as

f(R) = R-~ + bR2. ('55;

R

Simultaneously with the acceleration of the present

universe, the model may explain the inflation at the early stage. The models of the inflation using R" -theory has already been investigated in [18,19]. We should note that when R is large, where n goes to 2, from (47), we find w -> -1, which corresponds to the deSitter space.

First, we consider the instability pointed out in 116]. The general equation of motion with matter in f(R) -gravity is given by

^gaJ(R)-R,J'(R)-gilvnf(R) +

(56)

+V(IVv/'(K) = —w

Here 7"(m)IIV is the matter energy-momentum tensor. By multipling the above equation with gllv, we obtain

DStmVlM+ia.

fM(R) p 3fa)(R)

2f(R) K2 3 fm(R) 6 f2}(R)

(57)

-T.

Here T = T(fB)p We consider a perturbation from the solution of the Einstein gravity:

k2

R = R =-------------------T > 0.

0 2

(58)

We should note that T is negative since | |<sc p in the earth and T = -p + 3p —p. Then we assume R = R(}+Rl, (| Rt |«( R01). (59)

Then we find

0=Qi?fl +4r^Vi?0V^ +USlB>-f (R0) 9 3/<2)(i?0)

i?0

2/(i?0)

3/(2i(i?0) 3fa)(Rl})

+aRi + U(Rg}Rt,

J U'o)

U(Rt)) =

fw(R0) fw(R0)

fm(R0) fm(R0f.

/w(W(?)(/W /(1)(i?o)

3/,2)(J?(>)2 " 3/'2)(l?0)

2f(R0)fm(R0)_____

3/a?CI?0) 3/<2>(^)2‘

(60)

system becomes unstable. We i •sumc the matter is a! mo;

we

U(R„

(6)}

In case of the mode! ir? ;

.* ,...

‘ “ 6jr'

If ii corresponds to the acceleration of the present universe as in (54), we obtain

•l(f sec ~ (1 (TB eVf

(62)

This corresponds to the deSitter space which expands exponentlaly. If the present universe expands with power law as in (47), the value of a can be much

larger than the above value.

The value of R0 has been evaluated in [16]:

>3 f _ V

■ (1 O'26 sec) 2

gem

R0 ~ (103 sec)

(63)

gem

Then the system is unstable and would decay in

10'

sec.

In the model (55) [17] with a = \i*, if a

we find

U(R0)-~

R0

•0.

(64)

The system is unstable again but the decay time becomes about 1000 second and macroscopic, that is, it

is improved by 1029. But the assumption 6»—^- is

1*0 I

not so realistic since this means r'«( 10neV)2=(102GeV)2. In [20], it has been

given the constraint for inflation in R2 -theory from COBE-DMR data of CMBR (cosmic microwave background radiation) as

M

m„

—-2.6x10^,

If U(Rt)) is positive, since oR;—c)2!?,, the perturbation I?, becomes exponentially large and the

nRl} ~ 0 if inside the

(65)

V "V

that is b~l ~ (102 GeV)2, which contradicts with the assumption. Anyway instability depends on the details of f (R). Furthermore, if the present universe expands

by the power law, we need not to assume

i

Ii'"2 s a1 -108 sec - (10"33 eV)-1. (66)

In [14], it has been pointed out the problem in the equivalence principle. By the scale transformation, the matter action is transformed as 5(gpv>¥) ^S(eag,v,xif). (67)

Then the matter couples with o, whic e

the equivalence principle dye to the fora y

0, In case of the model in [11], if we assume f66; again., the mass of a becomes Hr” eV --10 GeV ans light, which will violate the equivalence

pri

hi me model [17], in the neighbourhood of R = A - %/3a =■ V-tyr, we obtain

d2V(a)

da2

A-sjla

jf,5toT3 d2V(A)

it dA)

dA

>/3af- + 2W3a

(68)

3

Then if

1

(69)

which means f'(A) ~ 0, the mass of o becomes very large and would not conflict with the equivalence principle. In general, since V(o) = e0g(e^)-~e2c,/(f(e^))

A f(A) (70)

f'(A) f'(A)2 the mass of 0 would become large when f\A) ~ 0. Then if the present acceleration of universe corresponds to deSitter solution R = A^ > 0, if we

consider a model where f\A) ~ 0, the mass of a can be large.

V Big Rip Singularity

In case of phantom (w < -1), the scale factor of the

__2_

universe behaves as a = a0(ts-t)iiyl+l‘. Then a diverges at t —> ts. Then anything like atom or hadron, which has internal structure, should be torn. Then this singularity is called the Big Rip singularity (“rip” meas a long tear or cut). This is the third possibility of the fate in the universe, following the Big Crunch and the eternal exapnsion.

We now consider the origin of the Big Rip. First we note that the energy density p behaves as

p = p0<r3a™>. (71)

Then in case of phantom (w<-l), when a increases, p also increases.

In order to consider any concrete model, we consider the scalar field theory coupled with gravity, whose action is given by

5 =4r [d*xJ^g((72)

KT J ■ V 2 " ' j

If y<0, we have a phantom. By combining the equation given by the variation of f:

I dr dt J

■ y'(<j>). (73)

and the (1st) FRW equation :

\2

+ ¥(#), (74)

6 ff2-n _Yf^ , we find

(75)

which is positive if y < 0, H > 0, and <j> £ 0. Then the energy density of phantom (y< 0) increases with time.

We now consider if the Big Rip singularity really exist or not. First we should note that since y<0, we have < V(§). Then if there is an upper bound in V((j)), there is also an upper bound in . The potential V(cj>) might behave exponentially in the present epoch of the universe, but if there is an upper bound in V (<j>), there does not occur the Big Rip singularity and the space becomes deSitter space asymptotically.

Another possibility to avoid the Big Rip singularity is to include the quantum correction [21] \ Near the Big Rip singularity, since a blows up, cuvatures become large as /? «| f |-2. Since the quantum correction contains power of the curvatures in general, the quantum correction becomes very important. In order to consider such a quantum correction, we include the contribution from conformal anomaly :

T = h\ F + -aR | + b'G + b'nR,

(76)

F

Here F is the square of 4d Weyl tensor and G is the Gauss-Bonnet invariant:

~R2 ~~2RtiRiJ +RutlRIjkl ,

3 J ’ (77)

G= R2 -4RVR‘J +RijklRiJkl,

In case that there are N scalar, Ni;i Dirac spinor, Nt vector fields, N2 (=0 or 1) gravitons, Nm higher derivative conformal scalars, the coefficients b, b', and b' are given by

N + 6Nin +12iV, +61 IN, ~MHD

120(4it)

.,¥ + 1 li¥,,, +62/V, +141 bV2

28A’„

(78;

360(471)' i' = 0.

We should note b > 0 and b~....................

matter except the higher derivative We should note that if can be s renormalization of the local countc . i .

can be arbitrary. By using the co- ■- _• . .

density p„ and pressure pA, TA is given by TA ~-pA +3pA. Then by using the energy conservation in the FRW universe

0 =

■ 4k

dt

+ 3 H (p+ pA),

we may delete pA as

7’ =:-4p

“ * H dt

which gives the following expression of pA :

(79)

(80)

-126

dH

+24 b'\-

dH_

dt

+ H

dt

2dfl

dt

+ Hi

1 Phantom theories with quantum corrections have been discussed in

[22]. The possibility to avoid the Big Rip singularity by the quantum effects has been also considered in [23].

,/2 ..'i I d3H d2H

-6|-b+b

dr

dtl

Then we also have an expression for pA 1 dpA

Pa =-Pa~7-------~

(81)

3 H dt

(dta4HT. ri n< j A

~\2b

dt

df

dr

+4(—)2 +12H1—)

dt dt

1

+2.4b

dm H

Alb

( dH_ dt

(

dH

i v dt j

-6i -b + brH

■h>*L + h*

di

,3

(dH

df

+ 7 H

dfH

dH i

(82)

__ . +J2«-^r til ) dt !

By including the quantum correction in (81), we may write the corrected FEW equation in the following form:

, /

+V№]*9'--

We now assume H = l\.. + 8/i, <[) = (j)0 In

r. -f

+ &j>.

(83)

(84)

and when t-*ts, Sh, &}> are much smaller than the

first terms but

dSh

dt

might be singular. Then (js-

equation of motion reduces as fro

. -t

f

-Y _

V

2Vt2 ^' ()‘l

(ts-t)

f

% j

(85)

- O'

which gives

{/ f2 _ #0 ox 3

Vl =—r-. of = --(ts-t.).

(86)

If we assume—b + b" ^0, we find 3

(2 ir]d28h

_t+4

(87)

Then by substituting (86) and (87) into the corrected FRW equation (83), we find

ts-t \3 ) dt2

+o((ts -?)"'). and

m,

S h = -

2\ |b + b"

(ti ~f)ln

if-t

(88)

(89)

Here t2 is a constant of the integration. Then the scale factor a behaves as

a = a0

...3k.

(ts-rr

b+b* \

xe

There appear logarithmic singularities in

(90)

dH

dt

but the singularity becomes rather mild. Then the universe might develope beyond f: = ;t.

VI Summary and Discussion,

In summary, there are several models to explain the accelerated expansion of the present universe. For the Big Rip singularity, the quantum correction would be very important.

There are many other models, as tachyon (see [24] for an example), the Born-Infeld theory (see [25] for example), Chameleon model [26] (see also [27]), and R + LmilttaRn type model in [28]. There are applications of the modified gravities to the cosmology (for example, [29]). These theories have structures as in the models obtained by the dimensional reduction from the model in the higher dimension. In fact, several models are proposed from the viewpoint of the brane world (for example, see [30]). Nearw = -T, we feel that theories in the extra or higher dimensions (brane world, Kaluza-Klein model) would appear.

Acknowledgments

The author is indebted with S.D. Odintsov for the collaborations. This investigation has been supported in part by the Ministry of Education, Science, Sports and Culture of Japan under grant n. 13135208.

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Timoshkin A. V.

THE ASY!V‘ rOHC CONFORIVIAL INVARIANCE IN CHERN-SIMONS THEORY WITH I./-.; *IN

CURVED SPACE-TIME

Tomsk Pedagogical University, 834041 Tomsk,Russia

1. The study of renormalizable field theories in curved space [1,2,3,4] proved the existance of the phenomenon of asymptotic conformal invariance. In this short review we discuss the asymptotic behaviour of 3d CS theories in curved space. Let us consider renormalizable abelian CS theory with scalar and spinor in three dimensions [5]. The Lagrangian looks like:

L = —epv"AA 9 Ax+1 D O f +

2 (1)

+№DW + af ¥#*# - A(®M>)3.

Here Dil=dtl-ieAu, <&,¥ -complex scalar and dirac spinor consequently, coupling constants e,h, a are dimensiionless. The theory with Lagrangian (I) is multiplicatively renormalizable.

The two-loop RG equation for coupling constants has the form [6,7]:

dt

- 0

(8?t)2 34a(r)e4 (t) - 24/ (t) (2)

dt 3

(8tc)2 = 168ft2 (0 - mh(t)cr(t) + 36e\t) +

dt

+8a (t)e6 (t) + 4a2 (t)e6 (t) - 4a4 (r).

It has been shown in paper [6] that for the theory with Lagrangian (1) exists finite four cases in which the theory is finite at two - loop level: