EXACT SOLUTION IN A SCALE INVARIANT MODEL WITH VANISHING COSMOLOGICAL CONSTANT

CHINESE JOURNAL OF PHYSICS VOL. 29, NO. 1 FEBRUARY 1991

Exact Solution In a Scale Invariant Model With Vanishing Cosmolof$cal Constant

W. F. Kao(& 2%)

Department of Electrophysics, Chiao Tung University National Hsinchu, Taiwan 30050, R. 0. C.

(Received Dec. 10, 1990)

We analyze a special scale invariant effective theory with vanishing cosmological constant. We show that, in this theory, the spatial-independence of the inflaton is implied by the equation of motion incorporated with the Robertson-Walker metric. Furthermore, we find that this theory can be solved exactly for k = 0 Robertson-Walker spaces. We also find almost exact solution in k # 0 spaces.

. Scale i n v a r i a n c e ’ has been implemented in string theory to obtain the low energy effective action for massless string mode. It has also been proposed2

to govern the cos-mological evolution of our universe. Much progress3

has been achieved in the past. It was recently shown4 that certain asymptotic boundary condition (ABC) imposed on the scalar field 4, namely I#J(~ -+ -) = u, can be derived as a slow roller solution to the equation of motion. The prescribed spontaneous symmetry breaking effect can thus be derived as a background solution to the theory.

In the meantime, the scale invariant theory bears another completely different solution in the vanishing cosmological constant limit. This different spectrum can be an inflation solution only in an exceptional limit. We will present the details and discuss its implications. We will also present a sketch5 of the proof that the associated scalar field $ must be a function oft only from the symmetry of the Robertson-Walker (RW) metric.6

The RW metric can be shown to describe all Riemannian four spaces with time-foila-tion of homogeneous and isotropic three spaces. Therefore, the observed isotropic and homogeneous universe in the cosmological scale must be a RW type space time. One notes that the Robertson-Walker metric can be read-off from the following definition of invariant scalar measure:

ds2 c gPVcIx@‘dx” = -dt2 + a2(t) ( dr2

I - kr2 + r2dLl) (1)

Here d!A = de2 + sin’0 dlp2 denotes the solid angle. and k = 0. f I stands for a flat. closed or open universe respectively.

\‘e will stud! the foiIoM.ing sole invariant mode!:’

Here $I is a real scalar field. while h and E arc dimensionless coupling constants. Indeed. this theor), can be sh0w.n to be invariant under the scale transformation g’_FL’ = s2guu and c$’ = 5-I $I. with s denoting a constant scale parameter. Moreover. this theory is also known to pro\,ide a natural explanation for a universe with a dimensionful gravitational “constant” and cosmological “constant”.

The solution to this model is very difficult to solve in fact. It can be solved only under a few seemly reasonable assumptions on the initial conditions on C$ and a(r). Recently it was shown4 that the slow roller $ can be used to imply a prescribed spontaneous symmetry breaking mechanism. This dynamical sl’mmetry breaking picture is equivalent to an imposed ABC’ that has been introduced by hand. It depends. however. on the expanding rate of our uni\,erse to determine the symmetry’ breaking scale. This belongs to the class of solutions with h #O. The solution is. however. \ery different for the case with A = 0. In . fact, we will be able to exactly solve the equations of motion in the case X = 0 when k = 0. namely. on the flat three dimensional space-like slice. In the case X-f0. our solution is on11 an approximated solution that requires h 2 >> I. In fact. the solution we obtained satisfies the above assumption fairly well.

Moreover. the h = 0 case prescribes a universe with vanishing cosmological constant. It is beyond the scope of this paper to address on the problem of cosmological constant. We will hereafter assume that the vanishing of the cosmological constant was settled, maybe due to the quantum cosmology approach, well before our model become active as an effective theory that is scale invariant manifestly. It is also beyond our purpose to discuss the physics beyond the Planck scale.

We can derive the classical equations of motion from the least action principle. The results are

Our spatially isotropic and homogeneous universe implies that @(x> must be a function oft and r only, nemely, Q(r) = @(t, I). Otherwise, its anisotropic contributions will affect the

observational data. In fact, we are going to argue that the r dependence of @ can be shown to be absent. This can be proFred by observing that the Euler-Lagrange equations from varying g,i can be reduced to

after inserting the RU’ metric. Moreover the ,, equation of (4) can be regrouped as

w. I.. KAO 3

wiaili/ + a,qa3 = hi,M(a, $, h,,) (6)

Here we have collected all hji proportional terms on the right hand side (RHS) and put the rests on its left hand side (LHS). Here we have also written gji = a2 hli and $ = $*. Further-more, the subscript 3 means that all covariant derivatives on the LHS are to be evaluated on the three dimensional spatial Riemannian sub-manifold (M3, hii).

If we assume that 4(x) = $(t, r), (5) and (6) can be reduced to

wta,3 -

r( 1 - kr*)

aru + (a,.+)2 = 0

(7)

after some algebra regrouping (6). Hereafter we will write $ 3 e e(r~‘) and a = e”lcr) for con-venience. Consequently, (7) and (8) can be written as a system of first order partial differential equations (PDEs) of art9(r,r). These PDEs can thus be solved by observing that (7) and (8) are in fact two total partial derivatives, namely, a,(e-a(f)‘Ils(i,‘)a~s(~,r)) = 0 and a_I(e-Y(,‘)+@(‘.‘)a e(t,r)) = 0. Here p = 1 +&and a,.g(r) = r(t_ir2). Indeed, these equations_I can be integrated accordingly. The result turns out to be very simple:

aro(r, /.) = f&Y(‘)+9(‘)-@(‘.“) _.

(9) Here f3 is an integration constant. It can actually be absorbed by redefining a(t). Note that (9) can be integrated further. The final result reads:

@(r, r) = ,d0/2~ i 1 +f4 [

(’ -

Ik’)

Ii\

Here fj is the appropriate integration constant defined after a(t) is redefined properly. Also. f” as a sign function defined as the sign of the coefficient of r2 term of 11 - kr* I~‘* after dropping the absolute sign. Note that 41 is divergent as r + 00. Indeed. 4kz0 (r --f m) a Yl’@and $X-tO(r +=) 0: rl,*!-‘. Therefore. we prove that $0) = 4(t).

One can therefore obtain the following equations. after inserting the Robertson-Walker metric into (3) and (4),

I, _Q_ + (a’)2 + k = a’$’ & I-($)’ + GF I a2 - . (11) ’ a (I?-) a ” (a’)’ + k 2--+

+,a’$’ +”

a a2 -7 +;I, = 3&-!-I-i($)*,

Here prime denotes the differentiation with respect to t. Furthermore. ( 12)+(13)-2x (1 1) gives a simplified equation

4 EXACT SOLCTION IN A SC4Lt IN\~AKI\NT MODLL WITH \ 4NISHItiG COS,MOLOGI(-~L ~Oyyl.,4y[

+” + ;aijr = o . (14)

a

This is in fact the conserved current equation. DliJp = 0. Here Jp = Gap@ is the Noether current for the scale symmetry. By introducing the integration factor e3a. (14) can be written as (+‘c~~)‘= 0. Hence Eq. (15) can be integrated to give

qye3a = _{c o n s t a n t}

In terms of $ E eecr) and a = ecrcr). ( 12) and (13) become

(15)

1 2crr’+ 3cu’2 -cu’e’ = _-e8’2

%E Also (16) and (17) implies

(17)

a’l+2a’2 =_-&ef2 . (18)

Note that we are considering a set of 3 differential equations. i.e. equations (15, 16. 17). for 2 unknown variables. Even it is in general unlikely to admit a consistent solution without a magic, we will show that (15) actually is derivable from (16) and (18). Hence it is adequate to analyze the system by solving (16) and (18). Note also that we have assumed that k = 0 (or a” >> 1 for k # 0) in deriving (16) and (18).

In fact, (16) is a simple algebraic equation of 8’ and cy’. It can be solved to give the following simple equation

a!1 = -le’ .

or equivalently cy + 18 = constant. Here 1 = lc = ’ “‘;\y-F Note also that I2 We will also write m = 2 +j-& such that m = 3 -+. To be more specifically, rn? In fact, (18) can be written as

(a~erm )’ =

$

This implies e”‘@ = cyO + cyi t. Therefore. rx = ln(cx, + cyi t)“” . e = ln(cu, + (pi t)-l lrtJ + c o n s t a n t Hence (19) =I+&. =3--+. (20) (21)

(22)

W. I’. KAO

I) = l+!lo( 1 + cl!,

f)p’l

a = a,(i +a,t)“”

Here we have used the parameters lc10 G $(t Eq. (15) can be shown to be redundant since

5 (23) (24) 0), a, G a(t = 0) in (23) and (24). Moreover

(25) with $’ and a given by (23) and (24) respectively. Here +b - +‘(t = 0) = --[%$,. Con-sequently, 01, = -I$./. Therefore (23) and (24) can be written as

$ = rl/,(l - ml’L t)-“lmb )

$0 (26)

b

a = a()(1 - m& f)“‘”

$0 (27)

Note that m % 3 and 1, z *G for e << 1. Therefore, the inflationary growth of a(t) will . require (i) I$; < 0 (ii) m-“lLb ’ >‘> 1. There are two possibilities: (i) +’ < 0 if $b < 0, hence # will be decreasing monor;nically. (ii) $J’ > 0 if $b > 0, hence @ will be increasing mono-tonically. Enough inflation can thus be achieved by tuning E, Go and $b properly. The condition m!!$$ << 1 is, however, quite unusual even $ is in fact almost stationary due to the 2 power. In fact, $ is almost static shortly after the inflation turns on. Since (26) and (27) are exact solutions, they will govern the whole domain wherever the action (2) remains effective. It is only when l$b > 0 (such that cyi < 0), the solution can no longer be valid whent>&.

In summary, we analyze a special scale invariant effective theory with vanishing cosmo-logical constant. We show that, in this theory, the spatial-independence of the inflaton is implied by the equation of motion incorporated with the Robertson-Walker metric. Furthermore, we find that this theory can be solved exactly for k = 0 Robertson-Walker spaces. We also find almost exact solution in k # 0 spaces. One remarks here that we consider the action (2) as an effective theory that remains effective only during some era (or energy scale) in the evolution of our universe. In fact, we haven’t found, and we don’t expect there can exist, an all mighty theory that can describe all physics in one.

ACKNOWLEDGEMENT

6 EXACT SOLl:TIOK IN A SCALk INVARIANTMODELWITH \ANISHING COSMOLO~~ICALCONS~ANT

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