Constraints on Scalar-Tensor Theories of Gravity

YITP-97-2, KUNS-142, gr-qc/9708030

Constraints on Scalar-Tensor Theories of Gravity from Density Perturbations in In ationary Cosmology 1 Yukawa

Takeshi Chiba1 , Naoshi Sugiyama2 and Jun'ichi Yokoyama1

Institute for Theoretical Physics, Kyoto University, Kyoto 606-01, Japan of Physics, Kyoto University, Kyoto 606-01, Japan (August 19, 1997)

2 Department

Abstract Spectra of density perturbations produced during chaotic in ation in scalartensor theories of gravity are calculated, taking both adiabatic and isocurvature modes into account. Comparing the predicted spectrum of the cosmic microwave background radiation anisotropies with the one observed by the COBE-DMR we calculate constraints on the parameters of these theories, which turn out to be stronger by an order-of-magnitude than those obtained from post-Newtonian experiments. PACS numbers: 04.50+h,98.80.-k

Typeset using REVTEX  e-mail: [email protected]

1

I. INTRODUCTION In ationary cosmology gives a natural explanation for the horizon, atness, homogeneity, and monopole problems [1]. In its simplest form, in ation with a single scalar eld predicts an approximately scale-invariant spectrum of Gaussian adiabatic density perturbations [2]. If other elds are present, however, primordially isocurvature perturbations may also be generated to leave an imprint on the universe today [3,4]. Scalar-tensor theories of gravitation have recently been received a renewed interest. The main reason is that the low-energy eective action of superstring theories [5] generally involves a dilaton coupled to the Ricci curvature. Moreover, as pointed out by Damour and Esposite-Farese [6] in the study of neutron star models, a wide class of scalar-tensor theories not only pass the present weak- eld gravitational tests but also exhibit nonperturbative strong- eld deviations away from general relativity. Detectability of scalar gravitational waves by laser interferometric gravitational wave observatories(LIGO) has been explored in [7,8]. Cosmological consequences of such theories, however, have not been fully examined. Recently, one of us studied the generality of chaotic in ation in scalar-tensor theories in detail and found that the onset of in ation could be greatly aected by the presence of the dilaton in some cases [9]. According to the attractor mechanism of scalar-tensor theories [10], which works only for those theories in which the derivative of the Brans-Dicke coupling is positive, even those theories that are quite dierent from general relativity in the early stage of the universe approach general relativity during the in ationary stage and the matterdominated stage of the universe. Even if this is the case, the deviations in the earlier stage may be imprinted in the spectrum of the cosmic microwave background radiation (CMB). We may constrain model parameters using the CMB anisotropy as a probe of the early universe. In fact, because of the presence of two scalar elds, i.e., the in aton and the dilaton, density perturbations have both adiabatic and isocurvature modes and hence the spectrum may be changed. In this paper, we study formation and evolution of density perturbations during and after chaotic in ation in scalar-tensor theories paying attention to the role of the isocurvature mode. This paper originates from the analysis by Starobinsky and Yokoyama [11], who calculated the spectrum of density perturbations from in ation in Brans-Dicke theory. The scalar-tensor theories we study include those with negative (the derivative of the BransDicke coupling) which have not been studied in the literature [12] whose numerical analysis is limited to positive and the isocurvature mode has been treated incompletely there. These theories exhibit a signi cant deviation from general relativity in the strong eld region [6]. The plan of this paper is as follows: In x2, basic equations of background elds are given. Linear perturbation equations are given and solutions to these equations are derived in x3. We study the evolution of isocurvature perturbations after in ation in x4. In x5, the spectrum of density perturbations are calculated and compared with the COBE observation to constrain the parameters of the theory. Finally x6 is devoted to summary.

2

II. BACKGROUND FIELD EQUATIONS A. scalar-tensor theories of gravitation We consider the in aton, , coupled to scalar-tensor theories of gravity. The action is

p S = d x ?g 161 Z

"

4

R ? !( ) gab

!

;a ;b

#

? 12 gab;a ;b ? V () ;

(2.1)

where is the massless Brans-Dicke dilaton, R is the scalar curvature, !( ) is a dimensionless coupling parameter and V () is the potential of the in aton. Let us consider a conformal transformation, (2.2) gab = 1G gab  e2a( ) gab; with = G?1 or a( ) = 0 today. Then the action can be written as   Z p 1 1 1 ab 2 a ab 4 a 4 S = d x ?g 22 R ? 2 g ';a';b ? 2 e g ;a ;b ? e V () ; where 2  8G and ' is de ned by Z 1 '  d

We also de ne

da ; (')  1 d'

(2.3)

s

!( ) + 23 :

(2.4)

d ; (')  1 d'

(2.5)

which are related with the PPN parameters ( Edd; Edd) and the current constraints on them are given as [13] 2 j (')j 2 2  ( ' ) 4  ( ' ) ? 3 j Edd ? 1j = 1 + 2(')2 < 2  10 ; j Edd ? 1j = 1 + 2(')2 < 6  10?4; (2.6) 0 0 where the subscript 0 indicates the present value. Here the Brans-Dicke theory corresponds to (') = 0 and the general relativity to (') = 0.

B. background equations We consider the spatially at FRW spacetime as a background metric

ds2 = ?dt2 + R(t)2dx2: The equations of motion derived from the action Eq.(2.3) are 3

(2.7)

R_ 2  H 2 = 2 '_ 2 + e2a _ 2 + 2e4a V () ; R 6 ' + 3H '_ =  e2a _ 2 ? 4e4a V () ;  + 3H _ = ?e2a V 0() ? 2'_ ;_ 2 H_ = ? 2 ('_ 2 + e2a _ 2 ); !

h

i

h

i

where _ and 0 denote time and  derivatives, respectively. < j3H _ j; j'j  < j3H '_ j) are written as, The conditions for slow-roll in ation (j j    < e4a V (); max 1 '_ 2 ; 1 e2a _ 2  2 2 < 62V ()2; e?2a V 0()2  < 32V (); e?2a V 00()  < 3; 8(')2  j16(')2 + 4 (')j < 3:

(2.8) (2.9) (2.10) (2.11)

(2.12) (2.13) (2.14) (2.15) (2.16)

When the above inequalities apply, eld equations are simpli ed to 3H 2 ' 2 e4a V (); 3H '_ ' ?4e4a V (); 3H _ ' ?e2a V 0():

(2.17) (2.18) (2.19)

III. LINEAR PERTURBATIONS A. basic equations Now we turn to linear perturbations with a perturbed metric in the longitudinal gauge

ds2 = ?(1 + 2)dt2 + R2(1 ? 2 )dx2:

(3.1)

From the perturbed Einstein equations, each Fourier mode1 satis es the following equations of motion to rst order [11] [12],  = ;  + 3H ' _ + k 2 ' + 4e4a 2 V ()(42 + )' ? e2a 2_ 2 (22 + )' ' R = 4_ '_ + 2e2a _ _ ? 4e4a (V 0 () + 2V ()); 2

k) = (2)?3=2

1 Our convention of the Fourier transformation is Y (

Y.

4

R

(3.2) (3.3)

d3 xe?ikx Y (x) for each variable

2  + 3H _ + k 2  + e2a V 00 () = ?2e2a V 0()( + ')  R _ ? 2 2'' _ + _ (4_ ? 2' _ ) ? 2'_ ; (3.4) 2 _ + e2a _ _ ? e4a V 0 () + (e2a _ 2 ? 4e4a V ())' ; (3.5)  + 4H _ + (H_ + 3H 2) = 2 '_ ' h

i

together with the Hamiltonian and momentum constraints

2h 2 i _ + e2a _ _ + e4a V 0() + (e2a _ 2 + 4e4aV ())' ; (3.6) 3H _ + (H_ + 3H 2) + Rk 2  = ? 2 '_ ' 2 _ + H  = 2 ('' _ + e2a  _ ): (3.7)

B. long wavelength perturbations Since what we need are the non-decreasing adiabatic and isocurvature modes on large scale k  RH , which turn out to be weakly time-dependent as will be seen in the nal result [4], we may consistently neglect _ and those terms containing two time derivatives. Then eqs. (3.3), (3.4), and (3.7) are simpli ed to 0  ' ?2' ? 21 VV ; _ ' ?4 2e4a V '; 3H ' 0 !0 V 2a 0 3H _ ' ?e2a V V  + 2e V ':

Following [11], we nd the last two equations can be integrated to give ' ' 4 Q1 ; 0  ' 12 VV (e?2a Q1 + Q2);

(3.8) (3.9) (3.10)

(3.11) (3.12)

where use has been made of Eqs.(2.17), (2.18) and (2.19). We then nd 0 2 1 V  ' ?8 Q1 ? 22 V (e?2a Q1 + Q2 ); !

2

(3.13)

where Q1 and Q2 are constants of integration.

C. adiabatic and isocurvature modes In order to clarify the physical meaning of the above solutions, we should divide them into adiabatic and isocurvature modes [11] [4]. The primordially isocurvature mode is characterized by its vanishingly small contribution to the curvature perturbation initially, while the 5

growing adiabatic mode can be described by the following expressions in the long wavelength limit k  RH (see Appendix A for a review of its derivation).   Z _ H 0 0 R (t )dt ' ?C1 H2 ; (3.14) ad = C1 1 ? RZ H 'ad = ad = C1 R(t0)dt0 ' C1 ; (3.15) '_ _ R H as a solution of a second-order dierential equation Eq.(A1) without the source term. Here C1 is a constant and the latter approximate equality in each expression is derived by integration by parts during the in ationary stage. As discussed by Starobinsky and Yokoyama [11] when the conditions 0 2 1 V 8  22 V e?2a ; e?2a ' 1 !

(3.16)

2

(3.17)

hold, one can discriminate between adiabatic and isocurvature modes in the nal results by de ning new constants C1 and C3 as C1  ?Q1 ? Q2 and C3  ?Q2 . Using Eqs.(2.17), (2.18) and (2.19) we nd 2

3

0 2 _ H 1 V 2  ' ?C1 2 + C3 ?8 + 2 (1 ? e?2a ) ; H 2 V ' ' C1 ? C3 ; '_ H H  ' C1 ? C3 (1 ? e2a ): _ H H !

4

5

(3.18) (3.19) (3.20)

In the above expressions, terms in proportion to C1 and C3 represent adiabatic and isocurvature modes, respectively if the above conditions hold. In more general scalar-tensor theories (especially in positive theories) Eq.(3.16) and Eq.(3.17) may not hold and consequently terms in proportion to C3 contain the adiabatic mode partially. Then these terms might be better referred to as the non-adiabatic mode. But this is a matter of terminology that does not aect our nal result and we keep using the same expression in this case, too.

D. quantum uctuations We shall next determine the constants C1 and C3 from amplitudes of quantum uctuations of the scalar elds generated during the in ationary stage. Due to the inequalities (2.12),(2.13) and (2.14), Eqs.(3.3) and (3.4) can be approximated by the equation of motion of a free massless scalar eld during in ation for k  RH . The standard quantization gives the well-known result, that is, the Fourier components of the elds can be represented in the form p(tk3) ' (k); (k) = Hp(tk3) e?a  (k); (3.21) '(k) = H 2k 2k 6

where tk is the time when k-mode leaves the Hubble horizon during in ation. Here '(k) and  (k) are classical random Gaussian variables with the following averages:

h'(k)i = h (k)i = 0; hi(k)j (k0 )i = ij (3) (k ? k0 ); i; j = '; :

(3.22)

We thus nd

Hp2(tk ) e?3a  (k) + 1 ? e?2a  (k) ; C1 = = _ '_ tk '_ ' tk 2k3 _  ' Hp2(tk ) e?3a  (k) ? e?2a  (k) : ? = C3 = e?2a H  _ '_ tk '_ ' tk 2k3 _  e?2a H 

"

+ (1 ? e?2a )H '

"

!#

#

#

"

"

#

(3.23) (3.24)

Note that Garcia-Bellido and Wands [12] have incorrectly set C3 = 0 in their analysis which cannot be justi ed since both  and ' are stochastic quantities.

IV. EVOLUTION OF ISOCURVATURE PERTURBATION AFTER INFLATION In order to relate the spectrum of  at the end of in ation to that at decoupling, we have to study the evolution of the isocurvature mode together with the adiabatic mode after in ation. Unlike the adiabatic modes we do not have a universal formula for the isocurvature modes, so we have to calculate them explicitly.

A. isocurvature perturbation in the reheating stage with Brans-Dicke dilaton First let us consider the isocurvature mode during the reheating stage after in ation. In this stage, 2V  e?2a V 00 (see Eq.(2.14)), so that the in aton oscillates coherently. Since equations of motion of perturbed quantities, (3.3) through (3.5), have oscillating coecients, we must worry about possible parametric resonance eect. This issue has been studied by Kodama and Hamazaki [14] recently in the case only the in aton is present and oscillating. They have shown that resonance eect is unimportant at least in the long wavelength regime which is of our interest. In the present case with two scalar elds, however, their approach is not directly applicable, in particular, for the isocurvature mode [15]. Hence we have numerically solved the evolution equations in the long wavelength limit. Figure 1 illustrates an example of the results for the Brans-Dicke theory with ! = 500. We nd that there is no anomalous growth of perturbation variables and that linear analysis suces even in this regime. We also nd qualitatively the same results for more generic scalar-tensor theories.

B. Isocurvature perturbation in the radiation-dominated stage with Brans-Dicke dilaton Next let us consider the isocurvature mode during the radiation dominated era after the reheating stage. In appendix B, a general expression of the evolution equation for  is given in the dilaton-in aton-radiation system. After the in aton has decayed into radiation, Eq.(B37) becomes 7

2  + (4 + 3c2s )H _ + (2H_ + 3(1 + c2s )H 2) + c2s k 2  R   2 (4.1) =  hr hd
d ?
r 3 h 1 + wd 1 + wr where cs is the sound velocity, 2 (4.2) c2s = p__ = 33'_'_ 2++44r =3 ; r with hj  j + pj , wj  pj =j (j = r; d), and h   + p. It is now apparent that terms in the parenthesis in the right hand side of (4.1) are just the entropy perturbation S = 3
? 1
: (4.3) S 4 r 2 d For the isocurvature perturbation, it is nonvanishing. It is known that if we replace d with the baryon density b, the isocurvature mode grows as R [16]. Because the dilaton density d decays as R?6 instead of R?3 , it is clear that the special solution originating from the source term decays as R?2 . On the other hand, the homogeneous part of the growing mode solution is the same as the adiabatic one. Therefore the isocurvature mode in the radiation dominated stage grows no faster than the adiabatic mode. This behavior can be also seen by solving the Eq.(B27) by using the Green's function method. To conclude, the isocurvature mode does not evolve faster than the adiabatic counterpart and so is unimportant. The spectrum at the end of in ation is sucient to compare the observation.

V. SPECTRA OF PERTURBATIONS AND CONSTRAINTS ON PPN PARAMETERS Now we calculate the density uctuation spectra produced by in ation. In order to do so, we have to specify V () and a('). We consider chaotic in ation induced by a mass term V () = m22 =2, of which we have a sensible particle-physics model in the intermediate scale [17]. Here m is determined from the large-angle microwave anisotropies seen by COBE-DMR [18].

A. general remark on initial conditions Initial conditions for ' and  are given in terms of the number of e-foldings from the end of in ation; Z Z Z e2a V () d; 2 NI = ? Hdt ' 4 d' '  (5.1) te 'e (') e V 0 ( ) where the sux e represents the end epoch of in ation. Our present horizon (H0?1 = 3000h?1Mpc) crossed outside the Hubble scale about 60 e-foldings before the end of in ation. We assume that the slow-roll condition is satis ed at NI e-folds from the end of in ation. 8

Then the terms containing the two time derivative can be neglected in Eqs.(3.3-3.7). Initial conditions for H; ';_ _ are given by Eqs.(2.17-2.19). We need to determine 'e to calculate the spectrum of density perturbations. We normalize ' in terms of its present value so that 2(') = 02 (see (2.6)) at present. Then we are able to specify 'e by going back to the end of in ation. In order to connect with the present length scale, we have to estimate the number of efoldings from the end of in ation to the present [19]. After the end of in ation, the oscillating in aton, which behaves like a dust uid for the massive in aton, dominates the universe, and then radiation dominated stage follows. The eective temperature, Trd, and the scale factor Rrd , at the onset of radiation dominated stage is determined from the energy density  3 2 T 4 ; e RRe = ge 30 (5.2) rd rd where e is the energy density at the end of in ation and ge is the eective number of degrees of freedom at temperature T typically taking the value of O(100) at Trd. Assuming the conservation of the entropy in relativistic particles per comoving volume s = 34 R3 =T , we have Rrd3 ge (Trd)Trd3 = (2T 30 + 21 T 3 )R3 4 0 0 43 T 3 R3; (5.3) = 11

0 0 where T0 is the background neutrino temperature. The number of e-folding from the end of in ation to the present is then 0 Ne = ln R Re ' 72 + 13 ln me4 ? 31 ln mTrd ; (5.4) pl pl where mpl is the Planck mass. We set the present value of the scale factor R0 to unity for convenience. Given initial conditions, we solve Eqs.(2.8-2.11) numerically using a fourth-order RungeKutta method. We choose NI = 65 typically. We express all quantities in units of mpl and use x = ln R for the integration variable instead of the cosmic time [19]. The calculations of the perturbed equations Eqs.(3.3-3.7) are begun when k  RH . The initial conditions for ' and  are given by Eq.(3.21). Since the background elds are in _ ; _ , and _ are given by neglecting their slow-rolling phase, the initial conditions for '; terms with the second order time derivative in Eqs.(3.3), (3.4), (3.6) and (3.7). We have to solve Eq.(A1) with or without the source term for the evolution of  so that we can follow the evolution of adiabatic and isocurvature modes separately. The spectral index n de ned by 3 2 n ? 1 = d lndklnhjkj i (5.5) is calculated for the present horizon scale. 9

B. COBE normalization We have to normalize uctuations to compare the spectra among dierent gravitational theories. The COBE-DMR data give the normalization. Following the standard treatment, the temperature uctuations are expanded in terms of spherical harmonics as
T (; ) = X a Y (; ); (5.6) lm lm T lm and we de ne (5.7) halmal m i  Cl ll mm : On COBE-DMR scales, temperature uctuations are dominated by the Sachs-Wolfe eect which is expressed as
T (x)=T = (x)=3 [20] for scalar perturbations. Here we neglect acoustic oscillation terms which may contribute to higher l's. Therefore Cl is written as Cl  hjalmj2i * + 2 Z  2 2 =  dkk 3 jl2(k0 ); (5.8) where 0 = 2H0?1 is the conformal time at present. If we approximate the power spectrum of  by a power law, !n?1 k 3 2 ; (5.9) k hjj i = A H 0 we can write Eq.(5.8) for scalar perturbations as ?(3 ? n)?(l + n?2 1 ) A Cl = 36 ?2( 4?n )?(l + 5?n ) : (5.10) 2 2 The rms quadrupole Qrms?PS used by the COBE-DMR group is written in terms of C2 as 0

0

0

0

s

Qrms?PS = T0 54C2 ; (5.11) where T0 = 2:726K. According to the 4 yr COBE-DMR data [18] Qrms?PS = 18K; (5.12) for n = 1 Harrison-Zel'dovich spectrum. It takes a slightly dierent value if dierent power low indices are considered, but we normalize the spectra by using the above value because the index is varied around n = 1 and furthermore the constraints on the scalar-tensor theories are imposed primarily from the spectral shape or its power-law index rather than from its amplitude. The corresponding amplitude A is 2 ?2( 4?n )?(l + 5?n ) ?PS : A = 1445  ?(3 ?2 n)?(l + n2?1 ) Qrms (5.13) T0 2 Eventually m takes the value ranging from O(10?6mpl ) to O(10?8mpl ). 

10



C. Brans-Dicke theory The Brans-Dicke theory corresponds to  0 and thus

(') = 0; a(') = 0':

(5.14) (5.15)

' = 40 N + 'e; 'e ' 100; 2 2 = 23 e?2a('e ) ? 21 2 (e?2a(') ? e?2a('e ) ):

(5.16) (5.17) (5.18)

Initial conditions for ' and  are

0

In Figs. 2, the spectra k3hjj2i for each 0 or ! are shown. In the Brans-Dicke theory, the conditions Eqs.(2.12) and (2.13) hold and therefore we are justi ed to say that terms in proportion to C3 represent the isocurvature mode. We see that considering the current limit on ! (! > 500) in Eq.(2.16), the contribution of the isocurvature mode is totally negligible in the Brans-Dicke theory in agreement with Starobinsky and Yokoyama [11]. The spectral indices on the comoving horizon scale today are n = 0:962; 0:966; 0:967 for ! = 500; 5000; 50000, respectively. Note that n = 0:967 in general relativity.

D. a class of scalar-tensor theory We take the following simple functional form for a(') in accord with Damour and Nordtvedt [10] a(') = 2 2('2 ? '20 );  const; (5.19) (') = ': (5.20) We choose the origin of the eld ' so that a(') = 0 at present in order to reproduce the correct value of the gravitational constant.2 We have to determine the value of ' at the end of in ation, i.e., 'e, to calculate the spectrum of density perturbation. Since we normalize ' in terms of its present value, we need to go back to the end of in ation to specify 'e. The evolution of ' in the radiation or matter-dominated universe is analyzed by Damour and Nordtvedt [10]. In the radiation-dominated stage ' is hardly changed. We assume for simplicity that 'e is equal to ' at the beginning of matter dominated stage. For j j  1(see Eq.(2.16)), 'e is determined by 'e '  0 e10 : (5.21) 2 The

authors of [12] take () = a1 + a2  and choose the origin so that  = a1 at the end of in ation. However,  deviates from a1 afterwards and may con ict with the limit on 0 .

11

Then ' is given in terms of N as For  with (see Eq.(2.13))

' ' 'ee4 N '  0 e4 N +10 :

(5.22)

V 0 (e) V (e)

(5.23)

e?2a('e )

!2

= 6;

 is given by 20 = h i e 2 ? 2 a ( ' e) (5.24) + 2 Ei(? (')2 ) ? Ei (? ('e)2) ;   ' 3e R where Ei(x) is the exponential integral function de ned by Ei(?x) = ? x1 e?t dt=t. In Figs. 3, the spectra for several with 0 = 10?3 are shown. In some cases the mode coming from C3 part in Eq.(3.18) is not negligible and the spectrum can be dierent from

at one. In scalar-tensor theories with positive , the conditions Eqs.(3.16) and (3.17) do not hold since initially  can be greatly dierent from zero (see Eq.(5.22)), and terms in proportion to C3 contain the adiabatic mode in addition to the isocurvature mode. For theories with negative , however, terms in proportion to C3 represent the isocurvature mode. In Fig. 4 we show that contour plot of n on 0 ? plane. We see that for > 0 n falls down very steeply with increasing . Thus the observed spectrum by COBE-DMR [18], n = 1:2  0:3, strongly constrains . This possibility was pointed out in [12] for > 0. Interestingly, the contour level can be tted by a linear function in log 0. For n  0:9, ! !   0 0 9:0  10?3 log p ? 1:5  10?2   ?8:9  10?3 log p5  10?2 + 5:3  10?3: 5  10?2 (5.25) For n  0:7, ! !   0 0 ? 3 ? 2 ? 3 5:0  10 log p ? 3:7  10   ?9:2  10 log p + 7:3  10?3: ? 2 ? 2 5  10 5  10 (5.26) Since the tensor mode of perturbations does not contribute to the spectrum signi cantly in the above range of the spectral index, the above inequality gives the constraint on . Note that the constraint on has little dependence on 0 as well as on the details of normalization of uctuations, and that the constraint is stronger by an order-of-magnitude than those obtained by post-Newtonian experiments(see Eq.(2.6)). 2 2

VI. SUMMARY We have studied density perturbations produced during chaotic in ation in scalar-tensor theories of gravity taking both adiabatic and isocurvature modes into account. The spectrum 12

of the density perturbation produced by chaotic in ation in a scalar-tensor theory can be a non- at one because of the variable Brans-Dicke coupling. The spectrum observed by COBE-DMR thus constrains such a coupling. The constraint is found to be stronger by an order-of-magnitude than those obtained by post-Newtonian experiments. It is interesting to note that in the case of the simple coupling function (') = ' employed here, our constraint is much more stringent than that obtained by the binary pulsar experiment, > ?5, in a sense. Indeed 0 must be 0 < 10?1000 (!) so that the constraint Eq.(5.25) or Eq.(5.26) is looser than > ?5. Thus we may conclude that cosmological in ation is suited in the framework of general relativity, or more advocatively, in ation favors Einstein. Although our results have been derived by assuming a simple coupling function, we expect that they hold in general classes of coupling functions. In this case, however, in the constraints Eq.(5.25) and Eq.(5.26) may be referred to that at the end of in ation.

ACKNOWLEDGMENTS T.C. would like to thank T.Nakamura and H.Sato for continuous encouragement. Numerical calculations are supported by Yukawa Institute for Theoretical Physics. This work was partially supported by the Japanese Grant in Aid for Science Research Fund of the Ministry of Education, Science, Sports and Culture Nos. 07304033(JY) and 08740202(JY).

APPENDIX A: FORMULAE FOR ADIABATIC MODE In this appendix we review derivation of the formulae for the adiabatic mode Eqs.(3.14) and (3.15). See [4] for a dierent approach. Using Eqs.(3.5), (3.6) and (3.7), we get 2  + (4 + 3c2s )H _ + (2H_ + 3(1 + c2s )H 2) + c2s k 2  R 2 4a k e = (c2s ? 1) R2  + 2 '_ 2 + e2a _ 2 (V 0'_ ? 4e2a V _ )(' _ ? ' _ );

(A1)

where c2s is de ned by 4a (4V '_ + V 0 _ ) 2 e p _ (A2) cs = _ = 1 + 3H ('_ 2 + e2a _ 2) : The left-hand-side of Eq.(A1) is just the same as that of the adiabatic perturbation for hydrodynamical matter and in case of the adiabatic perturbation right-hand-side term vanishes. For long wavelength perturbations right-hand-side vanishes when ' =  : (A3) '_ _ To derive the solution for the adiabatic mode Eq.(3.14), it is convenient to use the constancy of the Bardeen's  2

13

_ +  = const  C1 ;   ? H_ ( + H ?1) (A4) H _ in the which follows from the fact that the left-hand-side of Eq.(A1) is equal to ?H_ =H ~ long-wavelength limit. Substituting  ? C1 = , the above equation becomes a homogeneous form H_ ~ = ?C H: (A5) ~_ + H ~ ? H 1 Noting the left-hand-side is written as H R ~ : ; (A6) R H the solution is immediately given as 0 )dt0 + C H ; R ( t (A7) ~ = ?C1 H 2 R R where C2 corresponds to the amplitude of the decaying mode which is neglected in Eq.(3.14). To derive Eq.(3.15), we use Eq.(3.7) as _ 0 + C H_ _ + H  = ?C1 H Rdt 2 R R 2 2  _ + e2a  _ ) =  ' ('_ 2 + e2a _ 2): (A8) = ('' 2 2 '_ 2





Z

Z

Employing Eq.(2.11) we nd ' =  = 1 C Z Rdt0 ? C  : 2 '_ _ R 1 Neglecting the decaying mode, we get Eq.(3.15).

(A9)

APPENDIX B: LINEAR PERTURBATIONS FOR DILATON-INFLATON-RADIATION SYSTEM Basic equations in the Einstein frame are the Einstein equation   Gab = 2 Tab + @a '@b' ? 12 gabgcd@c'@d ' + e2a @a @b  ? gab( 21 gcde2a @c@d  ? e4a V ()) ; (B1) the dilaton equation of motion 2' = ?(')(T ? gcde2a @c@d  + 4e4a V ()); (B2) the matter equation of motion (Bianchi identity) rbT ba = (T ? gcde2a @c@d  + 4e4a V ())ra'; (B3) and the in aton equation of motion 2 = e2a V 0() ? 2gab';a;b; (B4) where 2 = 8G. 14

1. background equations The matter energy momentum tensor is given by

Tab = r uaub + pr (gab + ua ub): Assuming the at FRW model, the background equations are  2 1  1 2 2 2 a 2 4 a H = 3 r + 2 '_ + 2 e _ + e V () ; 2  H_ = ? 2 r + pr + '_ 2 + e2a _ 2 ; ' + 3H '_ = (?r + 3pr + e2a _ 2 ? 4e4a V ()); (r R3 )_ + pr (R3)_ = (r ? 3pr )R3 a_ (');  + 3H _ = ?e2a V 0() ? 2'_ ;_

(B5) (B6) (B7) (B8) (B9) (B10)

where pr = r =3.

2. linear perturbations We employ the metric perturbation in the longitudinal gauge as

ds2 = ?(1 + 2)dt2 + R2 (1 ? 2)dx2 : Then the perturbed Einstein tensors are [22] h i Gt t = R22 3R_ 2 + 3RR_ _ + k2  ; h i Gt j = R2k2 R_  + R_ Yj ; h i _ : + RR_ _ ji : Gij = R22 (R_ 2 + 2RR) + RR_ _ + (R2) As for the matter components, hydrodynamical perturbations of radiation are

T tt = ?r ; T tj = (r + pr )vYj ; T ij = pr Lji :

(B11) (B12) (B13) (B14) (B15) (B16) (B17)

The dilaton perturbations are expressed as _ T tt = '_ 2 ? '_ '; T tj = Rk ''Y _ j; T ij = ?'_ 2  + '_ '_ ji : h

i

and the in aton perturbations as 15

(B18) (B19) (B20)

T t t = ?e2a (_ _ ? _ 2) ? e2a _ 2 '; T t j = Rk e2a Y _ j; T ij = e2a (_ _ ? '_ 2 ) + e2a _ 2 ' ? e4a V 0() ? 4e4a V ()' ji :

(B21) (B22)

h

i

(B23)

Thus we have the perturbed Einstein equations 2 h _ + e2a _ _ 2 3H_ _ + 3H 2 + Rk 2  = ?2 ?('_ 2 + e2a _ 2 ) + r  + '_ ' "

#

i

+ e2a _ 2 ' + e4a V 0 () + 4e4a V ()' ;   h i R 2 2 a _ 2  + H  =  (r + pr )v k + '' _ + e  _ ;  h i _ + e2a _ _ 2  + 4H _ + (2H_ + 3H 2) = 2 ?('_ 2 + e2a _ 2 ) + 1 r  + '_ ' 3 i + e2a _ 2 ' ? e4a V 0() ? 4e4a V ()' :

(B24) (B25) (B26)

The perturbed equations are rewritten as 2  + (4 + 3c2s )H _ + (2H_ + 3(1 + c2s )H 2) + c2s k 2  R 2 2 2  k   + 2e4a V 0() + 8e4a V ()' = (c2s ? 1) 2  ? R 2 3 r  1 vR 4a 0 4a 2a + h (2Hhr ? 2 V ()_ ? 8e V ()'_ )(hr k + '' _ + e  _ ) ; (B27)

where c2s is de ned by 2 e2a _ 2 ) + 2e4a V 0 ( )' + 8e4a V ( )' c2s = p__ = 4Hr =3 + 3H ('_ 4+H : + 3H ('_ 2 + e2a _ 2 )

r

(B28)

We have arranged the left-hand-side of Eq.(B27) so that it is just the same as that of the adiabatic perturbation for hydrodynamical matter. Now let us examine the structure of the right-hand-side of Eq.(B27) in detail. Before doing so, we de ne the following useful gauge invariant variables [22]. For a general multicomponent system (a ; pa), a and va are de ned by

T tt = ?a a ; T tj = hava Yj ;

(B29) (B30)

where ha = a + pa . The associated gauge invariant variables
a and Va are de ned by
a = a + 3(1 + wa)(1 ? qa ) RH (B31) k VT ; Va = va ; (B32) 16

where wa = pa =a and VT = ha Va=h with h = ha and qa is the source term for each energy momentum tensor. For the case we are considering, i.e., the radiation-in aton-dilaton system, the respective
a (
r ;
i;
d) are written as
r = 3  + 3 RH 1 + (e2a _ 2 ? 4e4a V ())'_ V ; T 1 + wr 4 k 3Hhr
i =  + _ + ' + 3 RH V ; T 1 + wi _ k " # 2a _ 2 ? 4e4a V ( ))'_ _
d =  + ' ( e RH VT : 1 + wd '_ + 3 k 1 ? 3Hhd "

#

(B33) (B34) (B35)

Then Eq.(B27) can be rewritten as 2  + (4 + 3c2s )H _ + (2H_ + 3(1 + c2s )H 2) + c2s k 2  R  2 2 = (c2s ? 1) k 2  ?  1 hr  + 2hr VT RH R 2 2 k ! ! h h vR vR  ' r 2a 0 r 4 a + 2 h e V ()_ _ ? k + 8 h e V ()'_ '_ ? k !# 4a _ '_ '  2 e (B36) + h (V 0()'_ ? 4e2a V ()_ ) _ ? '_ :

When radiation is absent the expression 2agrees with that of in aton-dilaton system. By expanding out the term (c2s ? 1) Rk 2 , we nally have 2  + (4 + 3c2s )H _ + (2H_ + 3(1 + c2s )H 2) + c2s Rk 2      2 2 



 T 2 i d r r = ? 2 3h hr hi 1 + w ? 1 + w + hr hd 1 + w ? 1 + w ? 2RV 3k (_ ? 4V ())'_ r i r d 1 (2V 0 ()e4a _ + 8e4a V ()'_ )(
+ e2a V 0() + 4e4a V ()') + 3Hh ! ! h h vR vR  ' r 2a 0 r + 2 h e V ()_ 3_ ? k + 8 h e4a V ()'_ 3'_ ? k !) 4a _ '_ '  2 e 0 2 a (B37) + h (V ()'_ ? 4e V ()_ ) _ ? '_ :

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FIGURE CAPTION Fig.1. Behavior of the in aton and curvature perturbations during eld oscillation regime in the Brans-Dicke theory with ! = 500. The abscissa is the e-folding number after the end of in ation. Fig.2. The spectra of perturbations in Brans-Dicke theory with ! = 500; 5000; 50000. The abscissa is the wavenumber in unit of hMpc?1. logarithm is base 10. The isocurvature mode is negligible in this theory. Fig.3. The spectra of perturbations in scalar-tensor theories with 0 = 1:0  10?3 and = 0:02; 0:005; ?0:04. For larger j j, the mode coming from C3 part in Eq.(3.18) is not negligible and the spectrum has slope. The respective spectral index is n = 0:640; 0:967; 0:734. Fig.4. The contour plot of spectral index n for scalar-tensor theories. The contour levels n = 0:9; 0:7 are shown. We nd non-zero theories are strongly constrained. The p shaded reagion is the constraint by the post-Newtonian experiment(0 < 5  10?2).

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