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Scale invariance and inflation
Volume 208, number 2
PHYSICS LETTERS B
14 July 1988
SCALE I N V A R I A N C E A N D I N F L A T I O N B. G R A D W O H L and G. K A L B E R M A N N Racah Institute of Physics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel Received 16 December 1987
A scale invariant model for early universe inflationary cosmology is developed. In order to realize dilatation invariance and spontaneous symmetry breaking we introduce two scalar fields, a dilaton and an inflaton. The scale invariant theory encompasses the Brans-Dicke and induced-gravity models as limiting cases. The model is solved numerically for a wide class of initial conditions. We find that the inflationary epoch is generically characterized by a two phase evolution of the universe: A single or double exponential era and a power-law expansion. Onset of gravity triggers double exponential evolution of the scale factor. We further examine inflation in the Brans-Dicke theory and find that scale invariance is restored in the course of spontaneous symmetry breaking.
Scale invariance (SI) as a fundamental symmetry of nature has been addressed in many contexts. Dicke [ 1 ] introduced the idea of arbitrary scale-transformations (ST) on spacetime metric in connection to a scalar-tensor theory of gravity, namely the BransDicke theory [2]. A large body of literature on this topic - and its extensions to conformal invariance has built up in the intervening years [3]. Recently, SI has been called upon when dealing with early universe cosmology. Lucchin et al. [ 4 ] considered an SI action for the inflationary scenario [ 5 ], but chose to solve the non-SI case o f non-minimally coupled scalar field; although, as these authors point out, SI theories are essential for renormalizability purposes. Their work originates from a paper by Accetta et al. [6] (see also ref. [7] ), in which a scalar field (identifiable with the Brans-Dicke field) was responsible for both, inducing gravity and inflation through a spontaneously broken symmetry (SSB) with a G i n z b u r g - L a n d a u type o f potential. In neither of the above mentioned works SI was kept as an exact symmetry at the classical level. This is due to the fact that one cannot fulfill SI and SSB simultaneously with a single scalar field. It is therefore appealing to consider a model in which two scalar fields take part: ( 1 ) a dilaton, generating global SI, and (2) an inflaton, such that SI is an exact symmetry. Several other options, for which SI is either partially or absolutely 198
conserved, are at hand. If SI is kept for the gravitational fields (scalar-tensor) and omitted for the matter fields we confront a Brans-Dicke type of theory. In this case, the dilaton is connected to the BransDicke scalar qb. The addition of an SSB potential for may allow for the elimination of the inflaton, although breaking scale invariance ("induced-gravity models" [ 8 ] ). An alternative way that conserves SI consists o f introducing a (~4 type of potential [ 4 ], but without SSB. Peccei et al. [ 9 ] suggested an action to implement SI and SSB with the aim o f adjusting dynamically the cosmological constant. The breaking of dilatation invariance ( D I ) at the q u a n t u m level then was sought to produce a finite range interaction identified by them as a possible "fifth force". We here use their action as a prototype model to serve the purpose stated above, namely SI and inflation through the conventional inflaton (Higgs field). We define the SI action to be [ 9 ] d = f d4x x / ~
[(M2~/16n) e x p ( Z S / M ) R
+½0uSO~'Sexp(2S/M)+½0~,¢OuO+-V]
,
(la)
where S is the dilaton field, ¢ the complex inflaton field, R the Ricci scalar [ 10 ] and M is the characteristic scale of the DI, taken here to be of the order of the Planck mass (Mp~).
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 208, number 2
PHYSICS LETTERS B
V = i ~ [ 0 * q - 0 o2 exp(2S/M)12 ,
(2)
is the SI potential with SSB, where 0o is a constant connected to the v a c u u m expectation value (VEV) of the inflaton. (We use units where fi=c= 1.) Eq. (2) shows the dynamical nature of the VEV, originating from the d e m a n d o f scale invariance. Both the dilaton S and the inflaton ~ may acquire a non-vanishing value in the vacuum. Depending on the relative dominance o f the dilaton on the inflaton, or vice versa, V becomes an effective potential for the prevailing field. Therefore, this kind of interaction encompasses both the limiting cases of induced gravity [8] (0 kept at a constant value) and standard inflationary models [ 5 ] (S frozen). We can alternatively write the action ofeq. ( l a ) as
[11
J = f d4x ~
14 July 1988
and dots indicate time derivatives. We transformed the complex Higgs field to radial and angular degrees of freedom, 0 = P eie, and rescaled the fields to dimensionless variables by the substitutions
v=p/Mp|,
u=S/M,
M=aMpl , 0 o = y M N ,
~=elM~, . By varying the action of eq. ( 1a) with respect to the fields and the metric we obtain the classical field equations:
i;+ 3 (a/a) b+ 4ev( v 2 - yee 2u) - v0 2 = 0 ,
( 7a )
~'+ ( 3a/a+ 2b/v)O=O ,
(Vb)
a+u2+3(a/a)i~-4e(y2/o~2) ( v 2 - ~ 2 e 2.) - ( 3 / 4 h a 2 ) ( 1/a 2) (/~a + a 2 + k ) = 0 ,
(7c)
2iUa+ (a/a)2 +k/a2 +4(a/a)i~+4i~2 + 2ii
[ (M~,/1 6n)/~
+ 8 g e - 2 U [ l ~a 2 u. 2 e 2u + ~1.2 v + ~1v 2 0' 2
+ (1 + 6M2/16rim 2 )O,,S&S exp ( - 2 S / M ) + exp ( - 4 S / M ) ( ½0~,0&O + - V) ] ,
__8(/)2
( 1b)
where
(Vd)
(d/a)2+k/a2+2(d/a)it - ~ 8n e --2u [ ~1 a 2"u 2 e2U+/1)2+ -~/)202
Ru, = exp ( 2 S / M ) g~,, ,
( 3)
a n d / ~ is the Ricci scalar obtained from g. As we will show later, we can choose the "physical" metric to be dictated either by g or by R without changing the content of the theory. The action o f eq. ( l a) is invariant under global scale reparametrizations
x*~---,e-~x ~', 0 ~ e ~ 0 ,
S--+S+oeM.
(4)
For the action of eq. ( 1b) we have to add the condition ~uv-+ e 2~ guy. For a Brans-Dicke theory the potential becomes V=~(¢~-0--
)y2 e2U)2] = 0 ,
0o2) 2 .
(5)
The Brans-Dicke scalar is related to the dilaton by q ~ = M exp ( S / M ) . In the following we focus on the action ofeq. ( l a ) . For the purpose of treating inflationary cosmology we use the Robertson-Walker metric. The Ricci scalar becomes R = -- ( 6 / a 2) (da+a2+k) ,
(6)
where a is the expansion factor, k the curvature index
+ e ( v 2 - - y 2 e2") 2 ] = 0 .
(7e)
Eq. (7d) can as well be obtained by varying the action with respect to the expansion factor a. It corresponds also to a straightforward derivation ofeq. (7e) together with eqs. ( 7 a ) - (7c). It is clear from the above set of equations that the original lagrangian was not diagonal in the variables a and u. In order to maintain SI we set k = 0. Let us now consider approximate analytical solutions of the equations of motion for two limiting cases. If the dynamics of the dilaton u can be neglected we may retrieve the "standard" inflationary scenario. Redefining the inflaton to be ~= v e - " and dropping /~ a n d / / t e r m s we can identify the effective density and pressure to be p~--- ½~ 2 + Ceff(~ 2 -- ~'2 )2 ,
(8a)
p = ½~2-e~rf(ff2- y2)2 ,
(8b)
where eerr=e e 2". Under the assumption of slow rollover, ~'/~H<<~/¢H~4e~fr(~2-y2)/3H2<< 1 with H= a/a (the Hubble constant) we obtain the needed condition for (exponential) infation, n a m e l y p = - p . 199
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P H Y S I C S LETTERS B
Note, however, that the effective coupling constant is affected by the value of the dilaton, enhancing or diminishing the inflationary yield (see below). We may obtain a powerful initial inflationary expansion with small e and u > 0. A small bare coupling constant e may be advantageous if the dilaton decreases significantly by the time the fluctuations responsible for galaxy formation appear. A more interesting case is that of a frozen inflaton and a fast moving dilaton, simulating induced gravity. Taking v constant we find a solution to eq. (7c) with u linear in time. In this case we obtain a double exponential evolution of the scale factor a (superinflation ), ln(a)
=C
1
"~C2 exp(c3t) ,
(9)
with cb c2, c3 constants. The latter result is reminiscent of Kaluza-Klein [ 11 ] inflationary cosmologies [ 12 ]. Here and in ref. [ 12 ], c3 < 0, implying an initial value for u <<0. If this is indeed the case, a straightforward inspection of our SI lagrangian, eq. ( 1 ), shows us that gravity is initially turned off. The increase of u with time brings about the "onset of gravity". This shows the striking similarity that exists between our SI model and the Kaluza-Klein theory [ 11 ] as applied to inflation. Omitting all matter fields (p, 0) and defining A ( t ) = e u as the time-dependent scale factor of the extra dimensions enables us to identify all terms in eqs. (7) (up to coefficients) with the corresponding ones in Einstein's equations for Kaluza-Klein cosmological models [ 13 ]. The dilaton therefore plays a similar role to the Hubble constant in the extra dimensions. This characteristic can be attributed to the analogy between scale transformations and global gauge transformations. As stated by Bailin and Love [ 14], KaluzaKlein theories can be cast, upon dimensional reduction, in four-dimensional Brans-Dicke form [ 2 ]. The latter reduces to SI models in the absence of matter fields. We now proceed to show that the above characteristics do not depend on the choice of the "physical" metric tensor. The transformation of eq. (3) can instead be realized by a redefinition of time and the scale factor a, d F = e x p ( S / M ) dt , From eq. (7a) 200
gt=exp(S/M) a .
(10)
v2 - -
y2e2U-~
14 July 1988
--
[3(it/a)b+i)] 4ev - Q'
( 11)
where we have dropped the 0 field for the sake of simplicity. Inserting eq. ( 11 ) in the field equations ( 7 c ) (7e) we obtain ii+ ft2 + 3( d / a ) f t - 4 a ( T Z / o ~ 2 ) Q - (3/4~rc~ 2) ( 1/a 2) (/~a + a 2) = 0 ,
(12a)
2ii/a+ ( a/a)2 + 4 ( d / a ) i t + 4it2 + 2ii + 87~ e - Z u [ 10/2b12 eZu-t- 1/)2-- e ( u 2 - - ~2
e2") Q] - - 0 , (lZb)
(a/a)2+2(a/a)ft 1 "~ ,~(U2 -- ~2 -- 38 ~ ,~-2ur "" t21.ta,24,2 . . ~2u.l . ~/)2
eZ")Q] = 0 . (12c)
The above three independent equations for u, a and v are manifestly invariant under the time transformation ofeq. (10). Furthermore, the transformation of the scale factor a, does not affect the behavior of the analytical solutions discussed above. The frozen dilaton case remains unaltered, the double-exponential solution of eq. (9) transforms to an identical equation for the "tilted" variables and appropriately modified constants. Other cases follow the same pattern and therefore the choice of "physical" metric does not change the qualitative features of the model. We have found numerically that the azimuthal degree of freedom 0 is completely insignificant during inflation. Its relevance may become apparent through processes that break the azimuthal symmetry. In this case 0 becomes a massive pseudo Goldstone boson and may give rise to sizeable isothermal fluctuations [ 15 ], in close analogy to the invisible axion case [ 16 ]. In the following we show the results by ignoring the 0 field. The equations above neglect the decay of the scalar fields, an essential process in the post-inflationary epoch. At the end of inflation the fields oscillate around their VEVs behaving like pressureless fluids, then decaying and thus reheating the universe to the temperature needed for nucleosynthesis and baryosynthesis [17]. Oscillations are indeed observed in our numerical solutions in both the inflaton and dilaton fields. The particular mechanism causing the decay is highly model dependent. Although it is customarily mocked up by means of a phenomeno-
Volume 208, number 2
PHYSICS LETTERS B
14 July 1988
8O /
//
4-,
a
/
4O
i
'
b I
O-
- -~ -f,/. . . . . . . . . . .
0
2
4
6
,/
I
i
/ 0/ 0
8
/
!1
i
2
4
Time
6
8 Time
Fig. 1. (a) Field configurations as a function of time for "new" inflation. Inflaton 0 (full line), dilaton u (dashed line) and VEV evolution (small dash line; VEV=Te"). All initial velocities (#, u) are set to zero; e= I. (b) Power dependence of expansion factor, q = Ht (full line), and e-folds count Z (dashed line ) as a function of time. logical friction term in the equations o f motion, we prefer to o m i t this detail from our treatment. F o r the dilaton field there exists yet a n o t h e r m e c h a n i s m leading to its presently almost vanishing value, namely the standard post-inflationary evolution o f the universe [ 18 ]. We solved the field equations, eqs. ( 7 a ) - ( 7 d ) , numerically and checked the accuracy o b t a i n e d using eq. (7e). Figs. 1 a n d 2 display typical results for new a n d chaotic inflation. We chose for our runs c e = 7 = 1 a n d k = 0 . Variation o f the e p a r a m e t e r a m o u n t s merely to a rescaling o f the time unit, irrelevant for SI theories; a distinctive feature that m a y become i m p o r t a n t when dealing with q u a n t u m fluctuations. Moreover, SI reduces the n u m b e r o f i n d e p e n d e n t
choices o f initial conditions (v, b, u,/~) to three. Fig. l a corresponds to an almost frozen dilaton (the first analytical solution we considered a b o v e ) . We indeed observe an increase in the inflationary gain due to the dilaton. A measure o f the yield is generally taken to be the n u m b e r o f e-foldings Z = f H dt. Note that this quantity is invariant u n d e r the transformations o f eq. (10). Fig. l b s h o w s z and t l = H t as a function o f t i m e for the new inflationary scenario. Clearly, constant H yields exponential growth, whereas constant ~/ reflects power-law inflation with aoct'L In o r d e r to realize enough inflation Z~>60-70 [5]. Power-law inflation needs t/>~2 [ 19 ]. The exponential growth era is followed by a short power-law phase. Fig. 2
8O
5
4 f
/ o?
f
J
z tf /
4O
a
b 2o
C
-5
-10 0
o 5o
25 Time
x lOOO
-4 25
50 Time
x 1000
25
50 Time
Fig. 2. (a) and (b) Same as fig. 1 but for chaotic inflation; e= 5 × 10- ,0. (c) Double exponential evolution of the scale factor: log (H) as a function of time.
201
Volume 208, number 2
PHYSICS LETTERS B
14 July 1988
80 ¸
j J J /"
/
a
--8i~
o
//
40
b
0¸
5
10 Time
15
o
5
10 Time
x I00
x 1 O0
Fig. 3. Same as fig. 1 but for Bran s-Dicke theory [ 2 ] (VEV = Y); e = 10 - 4 For very short times (not appearing in the graph ) the inflation is "double-exponential". shows the results o f t h e c h a o t i c SI inflation. F o r v e r y short t i m e s it c o r r e s p o n d s to s u p e r - i n f l a t i o n ( t h e seco n d analytical s o l u t i o n a b o v e , fig. 2c). A n i n t e r e s t i n g effect e m e r g i n g f r o m fig. 2 is a d y n a m i c a l l y g e n e r a t e d s y m m e t r y b r e a k i n g for the i n f l a t o n field d u e to the d i l a t o n , w h i c h brings the V E V o f v f r o m z e r o to a finite value. O n c e u b e c o m e s a p p r o x i m a t e l y 0 the exp a n s i o n changes s u d d e n l y to a p o w e r - l a w i n f l a t i o n w i t h a p o w e r q ~ 6.5 for t h e case displayed. T h i s twophase i n f l a t i o n a r y e v o l u t i o n is s i m i l a r to d o u b l e - i n flation [ 20 ]. By a careful n u m e r i c a l study o f the i n d u c e d - g r a v ity case [ 6 ], we f o u n d a similar two-stage process e v e n for n e w inflation. T h e d o u b l e - e x p o n e n t i a l era, acc o u n t i n g for m o s t o f the inflation, lasts a c o m p a r a b l y short t i m e a n d p a s s e d u n n o t i c e d to A c c e t t a et al. [ 6 ] in t h e i r a n a l y t i c a l - n u m e r i c a l a p p r o a c h d u e to t h e i r r e s t r i c t i v e a s s u m p t i o n o f slow rollover. T h e p o w e r law d e p e n d e n c e that we f i n d for the s e c o n d stage coincides w i t h t h e i r results. F o r the sake o f c o m p l e t e n e s s we h a v e also i n v e s t i g a t e d the possibility o f an i n f l a t i o n a r y e v o l u t i o n in a B r a n s - D i c k e t h e o r y [ 2 ] . F o r this p u r p o s e we introd u c e d the n o n - S I p o t e n t i a l o f eq. ( 4 ) . Scale i n v a r i ance emerges here as a b y p r o d u c t o f SSB. Fig. 3 shows the results o f the a b o v e r e p l a c e m e n t for n e w inflation. F r o m the t i m e e v o l u t i o n o f the e - f o l d i n g Z, fig. 3b, we n o t i c e a r e s e m b l a n c e to i n d u c e d - g r a v i t y : short b u t strong d o u b l e - e x p o n e n t i a l phase f o l l o w e d by a p r o l o n g e d p o w e r - l a w era. We w o u l d like to t h a n k F. Englert a n d T. P i r a n for fruitful discussions. 202
References [1] R.H. Dicke, Phys. Rev. 125 (1962) 2163. [2] C. Brans and R.H. Dicke, Phys. Rev. 124 ( 1961 ) 925. [3] C.M. Will, Theory and experiment in gravitational physics (Cambridge U.P., Cambridge, 1981 ). [ 4 ] F. Lucchin, S. Matarrese and M.D. Pollock, Phys. Lett. B 167 (1986) 163. [5] A.D. Linde, Rep. Prog. Phys. 47 (1984) 925, and references therein. [6] F.S. Accetta, D.J. Zoller and M.S. Turner, Phys. Rev. D 31 (1985) 3046. [7] B.L. Spokoiny, Phys. Len. B 147 (1984) 39. [8] A. Zee, Phys. Rev. Lett. 42 (1979) 417. [9] R.D. Peccei, J. Sol~i and C. Wenerich, Phys. Lett. B 195 (1987) 183. [ 10 ] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Pergamon, New York, 1983). [ 11 ] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin Math. Phys. K1 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [ 12] J.D. Barrow, A.B. Burd and D. Lancaster, Class. Quantum Grav. 3 (1986) 551. [ 13 ] E. Alvarez and M. Bel6n Gavela, Phys. Rev. Lett. 51 ( 1983 ) 931. [ 14 ] D. Bailin and A. Love, Rep. Prog. Phys. 50 (1987) 1087. [ 15] D. Seckel and M.S. Turner, Phys. Rev. D 32 (1985) 3178. [ 16] J. Kim, Phys. Rev. Lett. 43 (1979) 103; M. Dine, W. Fischer and M. Srednicki, Phys. Lett. B 104 (1981) 199. [ 17] See e.g.P.J. Steinhardt and M.S. Turner, Phys. Rev. D 29 (1984) 2162. [ 18 ] R.T. Singh and S. Deo, Intern. J. Phys. 26 (1987) 901. [ 19 ] F. Lucchin and S. Matarrese, Phys. Rev. D 32 ( 1985 ) 1316. [ 20 ] J. Silk and M.S. Turner, Phys. Rev. D 35 ( 1987 ) 419.
PHYSICS LETTERS B
14 July 1988
SCALE I N V A R I A N C E A N D I N F L A T I O N B. G R A D W O H L and G. K A L B E R M A N N Racah Institute of Physics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel Received 16 December 1987
A scale invariant model for early universe inflationary cosmology is developed. In order to realize dilatation invariance and spontaneous symmetry breaking we introduce two scalar fields, a dilaton and an inflaton. The scale invariant theory encompasses the Brans-Dicke and induced-gravity models as limiting cases. The model is solved numerically for a wide class of initial conditions. We find that the inflationary epoch is generically characterized by a two phase evolution of the universe: A single or double exponential era and a power-law expansion. Onset of gravity triggers double exponential evolution of the scale factor. We further examine inflation in the Brans-Dicke theory and find that scale invariance is restored in the course of spontaneous symmetry breaking.
Scale invariance (SI) as a fundamental symmetry of nature has been addressed in many contexts. Dicke [ 1 ] introduced the idea of arbitrary scale-transformations (ST) on spacetime metric in connection to a scalar-tensor theory of gravity, namely the BransDicke theory [2]. A large body of literature on this topic - and its extensions to conformal invariance has built up in the intervening years [3]. Recently, SI has been called upon when dealing with early universe cosmology. Lucchin et al. [ 4 ] considered an SI action for the inflationary scenario [ 5 ], but chose to solve the non-SI case o f non-minimally coupled scalar field; although, as these authors point out, SI theories are essential for renormalizability purposes. Their work originates from a paper by Accetta et al. [6] (see also ref. [7] ), in which a scalar field (identifiable with the Brans-Dicke field) was responsible for both, inducing gravity and inflation through a spontaneously broken symmetry (SSB) with a G i n z b u r g - L a n d a u type o f potential. In neither of the above mentioned works SI was kept as an exact symmetry at the classical level. This is due to the fact that one cannot fulfill SI and SSB simultaneously with a single scalar field. It is therefore appealing to consider a model in which two scalar fields take part: ( 1 ) a dilaton, generating global SI, and (2) an inflaton, such that SI is an exact symmetry. Several other options, for which SI is either partially or absolutely 198
conserved, are at hand. If SI is kept for the gravitational fields (scalar-tensor) and omitted for the matter fields we confront a Brans-Dicke type of theory. In this case, the dilaton is connected to the BransDicke scalar qb. The addition of an SSB potential for may allow for the elimination of the inflaton, although breaking scale invariance ("induced-gravity models" [ 8 ] ). An alternative way that conserves SI consists o f introducing a (~4 type of potential [ 4 ], but without SSB. Peccei et al. [ 9 ] suggested an action to implement SI and SSB with the aim o f adjusting dynamically the cosmological constant. The breaking of dilatation invariance ( D I ) at the q u a n t u m level then was sought to produce a finite range interaction identified by them as a possible "fifth force". We here use their action as a prototype model to serve the purpose stated above, namely SI and inflation through the conventional inflaton (Higgs field). We define the SI action to be [ 9 ] d = f d4x x / ~
[(M2~/16n) e x p ( Z S / M ) R
+½0uSO~'Sexp(2S/M)+½0~,¢OuO+-V]
,
(la)
where S is the dilaton field, ¢ the complex inflaton field, R the Ricci scalar [ 10 ] and M is the characteristic scale of the DI, taken here to be of the order of the Planck mass (Mp~).
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 208, number 2
PHYSICS LETTERS B
V = i ~ [ 0 * q - 0 o2 exp(2S/M)12 ,
(2)
is the SI potential with SSB, where 0o is a constant connected to the v a c u u m expectation value (VEV) of the inflaton. (We use units where fi=c= 1.) Eq. (2) shows the dynamical nature of the VEV, originating from the d e m a n d o f scale invariance. Both the dilaton S and the inflaton ~ may acquire a non-vanishing value in the vacuum. Depending on the relative dominance o f the dilaton on the inflaton, or vice versa, V becomes an effective potential for the prevailing field. Therefore, this kind of interaction encompasses both the limiting cases of induced gravity [8] (0 kept at a constant value) and standard inflationary models [ 5 ] (S frozen). We can alternatively write the action ofeq. ( l a ) as
[11
J = f d4x ~
14 July 1988
and dots indicate time derivatives. We transformed the complex Higgs field to radial and angular degrees of freedom, 0 = P eie, and rescaled the fields to dimensionless variables by the substitutions
v=p/Mp|,
u=S/M,
M=aMpl , 0 o = y M N ,
~=elM~, . By varying the action of eq. ( 1a) with respect to the fields and the metric we obtain the classical field equations:
i;+ 3 (a/a) b+ 4ev( v 2 - yee 2u) - v0 2 = 0 ,
( 7a )
~'+ ( 3a/a+ 2b/v)O=O ,
(Vb)
a+u2+3(a/a)i~-4e(y2/o~2) ( v 2 - ~ 2 e 2.) - ( 3 / 4 h a 2 ) ( 1/a 2) (/~a + a 2 + k ) = 0 ,
(7c)
2iUa+ (a/a)2 +k/a2 +4(a/a)i~+4i~2 + 2ii
[ (M~,/1 6n)/~
+ 8 g e - 2 U [ l ~a 2 u. 2 e 2u + ~1.2 v + ~1v 2 0' 2
+ (1 + 6M2/16rim 2 )O,,S&S exp ( - 2 S / M ) + exp ( - 4 S / M ) ( ½0~,0&O + - V) ] ,
__8(/)2
( 1b)
where
(Vd)
(d/a)2+k/a2+2(d/a)it - ~ 8n e --2u [ ~1 a 2"u 2 e2U+/1)2+ -~/)202
Ru, = exp ( 2 S / M ) g~,, ,
( 3)
a n d / ~ is the Ricci scalar obtained from g. As we will show later, we can choose the "physical" metric to be dictated either by g or by R without changing the content of the theory. The action o f eq. ( l a) is invariant under global scale reparametrizations
x*~---,e-~x ~', 0 ~ e ~ 0 ,
S--+S+oeM.
(4)
For the action of eq. ( 1b) we have to add the condition ~uv-+ e 2~ guy. For a Brans-Dicke theory the potential becomes V=~(¢~-0--
)y2 e2U)2] = 0 ,
0o2) 2 .
(5)
The Brans-Dicke scalar is related to the dilaton by q ~ = M exp ( S / M ) . In the following we focus on the action ofeq. ( l a ) . For the purpose of treating inflationary cosmology we use the Robertson-Walker metric. The Ricci scalar becomes R = -- ( 6 / a 2) (da+a2+k) ,
(6)
where a is the expansion factor, k the curvature index
+ e ( v 2 - - y 2 e2") 2 ] = 0 .
(7e)
Eq. (7d) can as well be obtained by varying the action with respect to the expansion factor a. It corresponds also to a straightforward derivation ofeq. (7e) together with eqs. ( 7 a ) - (7c). It is clear from the above set of equations that the original lagrangian was not diagonal in the variables a and u. In order to maintain SI we set k = 0. Let us now consider approximate analytical solutions of the equations of motion for two limiting cases. If the dynamics of the dilaton u can be neglected we may retrieve the "standard" inflationary scenario. Redefining the inflaton to be ~= v e - " and dropping /~ a n d / / t e r m s we can identify the effective density and pressure to be p~--- ½~ 2 + Ceff(~ 2 -- ~'2 )2 ,
(8a)
p = ½~2-e~rf(ff2- y2)2 ,
(8b)
where eerr=e e 2". Under the assumption of slow rollover, ~'/~H<<~/¢H~4e~fr(~2-y2)/3H2<< 1 with H= a/a (the Hubble constant) we obtain the needed condition for (exponential) infation, n a m e l y p = - p . 199
Volume 208, n u m b e r 2
P H Y S I C S LETTERS B
Note, however, that the effective coupling constant is affected by the value of the dilaton, enhancing or diminishing the inflationary yield (see below). We may obtain a powerful initial inflationary expansion with small e and u > 0. A small bare coupling constant e may be advantageous if the dilaton decreases significantly by the time the fluctuations responsible for galaxy formation appear. A more interesting case is that of a frozen inflaton and a fast moving dilaton, simulating induced gravity. Taking v constant we find a solution to eq. (7c) with u linear in time. In this case we obtain a double exponential evolution of the scale factor a (superinflation ), ln(a)
=C
1
"~C2 exp(c3t) ,
(9)
with cb c2, c3 constants. The latter result is reminiscent of Kaluza-Klein [ 11 ] inflationary cosmologies [ 12 ]. Here and in ref. [ 12 ], c3 < 0, implying an initial value for u <<0. If this is indeed the case, a straightforward inspection of our SI lagrangian, eq. ( 1 ), shows us that gravity is initially turned off. The increase of u with time brings about the "onset of gravity". This shows the striking similarity that exists between our SI model and the Kaluza-Klein theory [ 11 ] as applied to inflation. Omitting all matter fields (p, 0) and defining A ( t ) = e u as the time-dependent scale factor of the extra dimensions enables us to identify all terms in eqs. (7) (up to coefficients) with the corresponding ones in Einstein's equations for Kaluza-Klein cosmological models [ 13 ]. The dilaton therefore plays a similar role to the Hubble constant in the extra dimensions. This characteristic can be attributed to the analogy between scale transformations and global gauge transformations. As stated by Bailin and Love [ 14], KaluzaKlein theories can be cast, upon dimensional reduction, in four-dimensional Brans-Dicke form [ 2 ]. The latter reduces to SI models in the absence of matter fields. We now proceed to show that the above characteristics do not depend on the choice of the "physical" metric tensor. The transformation of eq. (3) can instead be realized by a redefinition of time and the scale factor a, d F = e x p ( S / M ) dt , From eq. (7a) 200
gt=exp(S/M) a .
(10)
v2 - -
y2e2U-~
14 July 1988
--
[3(it/a)b+i)] 4ev - Q'
( 11)
where we have dropped the 0 field for the sake of simplicity. Inserting eq. ( 11 ) in the field equations ( 7 c ) (7e) we obtain ii+ ft2 + 3( d / a ) f t - 4 a ( T Z / o ~ 2 ) Q - (3/4~rc~ 2) ( 1/a 2) (/~a + a 2) = 0 ,
(12a)
2ii/a+ ( a/a)2 + 4 ( d / a ) i t + 4it2 + 2ii + 87~ e - Z u [ 10/2b12 eZu-t- 1/)2-- e ( u 2 - - ~2
e2") Q] - - 0 , (lZb)
(a/a)2+2(a/a)ft 1 "~ ,~(U2 -- ~2 -- 38 ~ ,~-2ur "" t21.ta,24,2 . . ~2u.l . ~/)2
eZ")Q] = 0 . (12c)
The above three independent equations for u, a and v are manifestly invariant under the time transformation ofeq. (10). Furthermore, the transformation of the scale factor a, does not affect the behavior of the analytical solutions discussed above. The frozen dilaton case remains unaltered, the double-exponential solution of eq. (9) transforms to an identical equation for the "tilted" variables and appropriately modified constants. Other cases follow the same pattern and therefore the choice of "physical" metric does not change the qualitative features of the model. We have found numerically that the azimuthal degree of freedom 0 is completely insignificant during inflation. Its relevance may become apparent through processes that break the azimuthal symmetry. In this case 0 becomes a massive pseudo Goldstone boson and may give rise to sizeable isothermal fluctuations [ 15 ], in close analogy to the invisible axion case [ 16 ]. In the following we show the results by ignoring the 0 field. The equations above neglect the decay of the scalar fields, an essential process in the post-inflationary epoch. At the end of inflation the fields oscillate around their VEVs behaving like pressureless fluids, then decaying and thus reheating the universe to the temperature needed for nucleosynthesis and baryosynthesis [17]. Oscillations are indeed observed in our numerical solutions in both the inflaton and dilaton fields. The particular mechanism causing the decay is highly model dependent. Although it is customarily mocked up by means of a phenomeno-
Volume 208, number 2
PHYSICS LETTERS B
14 July 1988
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Fig. 1. (a) Field configurations as a function of time for "new" inflation. Inflaton 0 (full line), dilaton u (dashed line) and VEV evolution (small dash line; VEV=Te"). All initial velocities (#, u) are set to zero; e= I. (b) Power dependence of expansion factor, q = Ht (full line), and e-folds count Z (dashed line ) as a function of time. logical friction term in the equations o f motion, we prefer to o m i t this detail from our treatment. F o r the dilaton field there exists yet a n o t h e r m e c h a n i s m leading to its presently almost vanishing value, namely the standard post-inflationary evolution o f the universe [ 18 ]. We solved the field equations, eqs. ( 7 a ) - ( 7 d ) , numerically and checked the accuracy o b t a i n e d using eq. (7e). Figs. 1 a n d 2 display typical results for new a n d chaotic inflation. We chose for our runs c e = 7 = 1 a n d k = 0 . Variation o f the e p a r a m e t e r a m o u n t s merely to a rescaling o f the time unit, irrelevant for SI theories; a distinctive feature that m a y become i m p o r t a n t when dealing with q u a n t u m fluctuations. Moreover, SI reduces the n u m b e r o f i n d e p e n d e n t
choices o f initial conditions (v, b, u,/~) to three. Fig. l a corresponds to an almost frozen dilaton (the first analytical solution we considered a b o v e ) . We indeed observe an increase in the inflationary gain due to the dilaton. A measure o f the yield is generally taken to be the n u m b e r o f e-foldings Z = f H dt. Note that this quantity is invariant u n d e r the transformations o f eq. (10). Fig. l b s h o w s z and t l = H t as a function o f t i m e for the new inflationary scenario. Clearly, constant H yields exponential growth, whereas constant ~/ reflects power-law inflation with aoct'L In o r d e r to realize enough inflation Z~>60-70 [5]. Power-law inflation needs t/>~2 [ 19 ]. The exponential growth era is followed by a short power-law phase. Fig. 2
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50 Time
x 1000
25
50 Time
Fig. 2. (a) and (b) Same as fig. 1 but for chaotic inflation; e= 5 × 10- ,0. (c) Double exponential evolution of the scale factor: log (H) as a function of time.
201
Volume 208, number 2
PHYSICS LETTERS B
14 July 1988
80 ¸
j J J /"
/
a
--8i~
o
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40
b
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10 Time
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Fig. 3. Same as fig. 1 but for Bran s-Dicke theory [ 2 ] (VEV = Y); e = 10 - 4 For very short times (not appearing in the graph ) the inflation is "double-exponential". shows the results o f t h e c h a o t i c SI inflation. F o r v e r y short t i m e s it c o r r e s p o n d s to s u p e r - i n f l a t i o n ( t h e seco n d analytical s o l u t i o n a b o v e , fig. 2c). A n i n t e r e s t i n g effect e m e r g i n g f r o m fig. 2 is a d y n a m i c a l l y g e n e r a t e d s y m m e t r y b r e a k i n g for the i n f l a t o n field d u e to the d i l a t o n , w h i c h brings the V E V o f v f r o m z e r o to a finite value. O n c e u b e c o m e s a p p r o x i m a t e l y 0 the exp a n s i o n changes s u d d e n l y to a p o w e r - l a w i n f l a t i o n w i t h a p o w e r q ~ 6.5 for t h e case displayed. T h i s twophase i n f l a t i o n a r y e v o l u t i o n is s i m i l a r to d o u b l e - i n flation [ 20 ]. By a careful n u m e r i c a l study o f the i n d u c e d - g r a v ity case [ 6 ], we f o u n d a similar two-stage process e v e n for n e w inflation. T h e d o u b l e - e x p o n e n t i a l era, acc o u n t i n g for m o s t o f the inflation, lasts a c o m p a r a b l y short t i m e a n d p a s s e d u n n o t i c e d to A c c e t t a et al. [ 6 ] in t h e i r a n a l y t i c a l - n u m e r i c a l a p p r o a c h d u e to t h e i r r e s t r i c t i v e a s s u m p t i o n o f slow rollover. T h e p o w e r law d e p e n d e n c e that we f i n d for the s e c o n d stage coincides w i t h t h e i r results. F o r the sake o f c o m p l e t e n e s s we h a v e also i n v e s t i g a t e d the possibility o f an i n f l a t i o n a r y e v o l u t i o n in a B r a n s - D i c k e t h e o r y [ 2 ] . F o r this p u r p o s e we introd u c e d the n o n - S I p o t e n t i a l o f eq. ( 4 ) . Scale i n v a r i ance emerges here as a b y p r o d u c t o f SSB. Fig. 3 shows the results o f the a b o v e r e p l a c e m e n t for n e w inflation. F r o m the t i m e e v o l u t i o n o f the e - f o l d i n g Z, fig. 3b, we n o t i c e a r e s e m b l a n c e to i n d u c e d - g r a v i t y : short b u t strong d o u b l e - e x p o n e n t i a l phase f o l l o w e d by a p r o l o n g e d p o w e r - l a w era. We w o u l d like to t h a n k F. Englert a n d T. P i r a n for fruitful discussions. 202
References [1] R.H. Dicke, Phys. Rev. 125 (1962) 2163. [2] C. Brans and R.H. Dicke, Phys. Rev. 124 ( 1961 ) 925. [3] C.M. Will, Theory and experiment in gravitational physics (Cambridge U.P., Cambridge, 1981 ). [ 4 ] F. Lucchin, S. Matarrese and M.D. Pollock, Phys. Lett. B 167 (1986) 163. [5] A.D. Linde, Rep. Prog. Phys. 47 (1984) 925, and references therein. [6] F.S. Accetta, D.J. Zoller and M.S. Turner, Phys. Rev. D 31 (1985) 3046. [7] B.L. Spokoiny, Phys. Len. B 147 (1984) 39. [8] A. Zee, Phys. Rev. Lett. 42 (1979) 417. [9] R.D. Peccei, J. Sol~i and C. Wenerich, Phys. Lett. B 195 (1987) 183. [ 10 ] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Pergamon, New York, 1983). [ 11 ] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin Math. Phys. K1 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [ 12] J.D. Barrow, A.B. Burd and D. Lancaster, Class. Quantum Grav. 3 (1986) 551. [ 13 ] E. Alvarez and M. Bel6n Gavela, Phys. Rev. Lett. 51 ( 1983 ) 931. [ 14 ] D. Bailin and A. Love, Rep. Prog. Phys. 50 (1987) 1087. [ 15] D. Seckel and M.S. Turner, Phys. Rev. D 32 (1985) 3178. [ 16] J. Kim, Phys. Rev. Lett. 43 (1979) 103; M. Dine, W. Fischer and M. Srednicki, Phys. Lett. B 104 (1981) 199. [ 17] See e.g.P.J. Steinhardt and M.S. Turner, Phys. Rev. D 29 (1984) 2162. [ 18 ] R.T. Singh and S. Deo, Intern. J. Phys. 26 (1987) 901. [ 19 ] F. Lucchin and S. Matarrese, Phys. Rev. D 32 ( 1985 ) 1316. [ 20 ] J. Silk and M.S. Turner, Phys. Rev. D 35 ( 1987 ) 419.