Inflation and the conformal structure of higher-order gravity theories

Volume 214, number 4

PHYSICS LETTERS B

1 December 1988

INFLATION AND T H E CONFORMAL STRUCTURE OF HIGHER-ORDER GRAVITY THEORIES John D. BARROW and S. COTSAKIS Astronomy Centre, Universityof Sussex, Brighton BN1 9QH, UK Received 23 June 1988

We examine gravity theories derived from a gravitational lagrangian that is an analytic function f ( R ) of the scalar curvature R in a space-time of arbitrary dimension D. We show that they are conformally equivalent to general relativity plus a scalar-field matter source with a particular self-interaction potential. The general form of this potential is calculated to be V ( 0 ) = ½[Rf' (R) -f(R) ] [f' (R) ]D/t2-O),where exp(#) = [ f ' (R) ]2/(D--2). Flat potentials arise as ~--, +oo for polynomial lagrangians of leading order fl whenever D = 2ft. Several explicit examples are given and discussed with reference to inflation. We show how our results can lead to singularity theorems for gravity theories derived from a general lagrangian.

It has been known for some time that it is possible to obtain inflationary cosmological models [ 1 ], in ways other than by the explicit presence of a matter field whose energy-momentum tensor violates the strong energy condition [ 2-5 ]. Gravity theories with a gravitational lagrangian quadratic in the scalar curvature of space-time, R, Lo = R + a R 2, ot constant,

Lg = f ( R ) ,

wherefis an arbitrary analytic function of the scalar curvature. The detailed structure of gravity theories arising from the lagrangian (2) in four-dimensional space-time has been discussed in ref. [ 7 ]. In the absence of matter the field equations obtained by the variation of the action associated with (2) with respect to the D-dimensional space-time metric gab are

( 1)

where studied by Starobinskii [6 ] and Barrow and Ottewill [ 7 ] who displayed particular inflationary solutions in which the Hubble expansion rate of the universe is a linear function of time. Subsequently, various aspects of the detailed structure of inflation in the isotropic cosmological models generated from ( 1 ) have been studied [ 8 ]. The reason why inflation occurs in these models in the absence of explicit matter fields that violate the strong energy condition can be seen most clearly from the demonstration by Whitt [ 9 ] (see also Bicknell [ 10 ] ) that the vacuum gravity theory generated by stationary variation of the action associated with ( 1 ) is conformally equivalent to general relativity plus a scalar field. The self-interaction potential of this scalar field was given explicitly for the case ( 1 ) in ref. [ 11 ]. In this letter we shall present a generalization of Whitt's analysis to the case where the vacuum gravity theory in D-dimensional space-time possesses the general lagrangian

(2)

[71 O=f'Rab--!zfg,,b--V,,Vbf' +gab[~f ' = O ,

(3)

where [] =gabVaV b and Va is the covariant differential operator. It is useful to rewrite these equations in the form f ' (Rab -- ½Rgab) + ½gab(Rf' --f) - V a V b f ' +gab[]f'

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=0.

(4)

If we apply a conformal transformation gab = ff'~2gab,

(5)

where 12 is a smooth strictly positive function chosen so that

122= If' (R) ] 2/(~,-2)

(6)

then Rab and R transform into 515

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PHYSICS LETTERS B

"Rab =Rab -- ( D - 1 ) ( D - 2 ) --

--1 (f,)-2V,,f,

Vbf ,

(f' )-lVaVbf' -- (D-- 2 )-~gab(f' )-'ff]f' ,

(7) /1~. 0 c ' )2/(2--D)

× [R+(D-1)(D-2)-t(f')-2(Vf')

-2(0-

1) ( D - 2 ) - t

2

OC' ) - 1 [ - I f ' ] •

(8)

The field equations, (4), for the f ( R ) theory transform under this conformal transformation to become ~ R- _-_- R a b __ l g a b R R-a b - - iI gab

+ ( D - 1 ) ( D - E ) - 1 (f,)-2Vof,Vb f ,

1

(9)

If we introduce a scalar field 0 defined by 0= (D---~-~) i n [ f , (R) ] '

(10)

and use (4) to replace (Rab-- ½R&b) then the conformally transformed field equations (9) become R a b - - ½gab ~ - "

I (D-

1 ) (D-Z)VaOVbO

- ( D - 1 ) (D-2)gab(V~OVcO)/8

-- ½gab(f' )D/(2--D)(Rf' - f ) .

( 11 )

These are the Einstein equations in D-dimensional space-time for a scalar field source with potential V(0) = ½(f')D/~2-o)(Rf' - f ) .

(12)

Using (10) we see that it is possible to write this formally as V= lexp( - ½DO) ×{exp[½(D-Z)OlV(exp[½(D-2)O] )-f},

(12') where, by (10), F = f ' - ~is the inverse function o f f ' a n d f m a y be expressed in terms of 0 via f = I exp[½(D-2)0] d F .

(10')

In practice, we will be able to express (10') or ( 12' ) as an explicit function of 0 only w h e n f ' - ~ can be inverted analytically. For general polynomial lagrangians thi s requires fro be of degree no greater than 516

five although there need not be a real-valued inverse if the polynomial is of even degree. We note also that the transformation ( 10 ) requires f ' (R) > 0 if it is to be non-singular ~1. If it is also required that V(O) be non-negative then additional constraints are placed upon the allowed form o f f ( R ) . If V=0 then the solution of (12) i s f ( R ) = R - general relativity. If Vis constant then the unique form is f ( R ) = ( R - V ) general relativity plus a cosmological constant (in this case eq. (12) just reduces to the general existence condition for maximally symmetric space-times in f ( R ) theories given in ref. [7] ). For the quadratic lagrangian ( 1 ) the potential (12) is V(0) = ~ {1 - e x p [ - ( D - 2 ) 0 / 2 1 }

-- ½( D - 1 ) ( D - 2 ) -lgabf'-Egm"Vmf'Vnf' - f ' - ~ VaVbf' + f ' - ~gabc]f' •

1 December 1988

2

Xexp[(D-4)O/2].

(13)

which agrees with the results of refs. [ 11,12 ] #2 and ref. [ 9 ] for D = 4 and with the results of Maeda [ 14 ] for general D. Maeda first noted the interesting feature that the potential (13) possesses a flat plateau V-+l/8ot conducive to exponential inflation as O+ + oo only for D = 4 ~3. For D # 4 we get a potential of exponential type which leads to power-law inflation of the sort studied in refs. [ 15,16 ]. This feature appears in more general f ( R ) theories since if we t a k e f ( R ) to be a polynomial of order fl f ( R ) =ao + a , R +...+a#R p ,

(14)

then as R ~ oo the potential asymptotes to VocR (D-2fl)/(D-2) OZe x p [ ( D - - Z f l ) O / 2 ( # - 1 ) 1 ,

(15)

and the inflationary plateau arises as 0-~ + ~ whenever D=2fl. Special inflationary solutions will exist in these cases. For the cubic lagrangian f ( R ) = R +aRZ + bR 3

(16)

#l I f f ' < 0 we r e p l a c e f ' (R) by [ f ' (R) I in (10) and the sign of Vin (12) changes. ~2 Similar results have been obtained by Maeda [ 13 ]. ~3 One can see this dependence on D explicitly in the flat Friedman models of the f ( R ) = A + R + c t R 2 theory where a special exact solution [7], with scale-factor increasing as exp ( A t + t z / 7 2 a ) can be found only when D = 4.

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PHYSICS LETTERS B

1 December 1988

we o b t a i n the potential

vie)

1

V(O) = ~ e x p ( - 2 0 ) [ e x p ( 0 ) b + ~a3 exp(-20)

la}

1] z

[ e x p ( 0 ) - 113

(17)

in the D = 4 case a n d 0 ~ 0. The form o f this potential is d i s p l a y e d in fig. 1. The s y m m e t r i c structure o f the potential contributed by higher-order p o l y n o m i a l terms in the lagrangian is apparent. In the D = 4 case, for a p o l y n o m i a l lagrangian o f the form

V I¢~1

k

~ a,R n, k e Z + ,

f(R)=

(18)

n=l

(b)

we expect an equivalent potential o f the form V(0) = A I e x p ( - 2 0 ) k

+ ~, A n e x p ( - 2 0 ) [ 1 - e x p ( O ) ] "

(19)

n=2

for some set o f rational coefficients An(a,). T h e A , can be constructed explicitly for 2 ~ 0, we o b t a i n V(0) = 2 - 3 exp ( 0 ) ( 0 - 1 n 2 - 1 ) ,

(20)

so V> 0 only when 0 > 1 + In 2 a n d there is a single m i n i m u m at 0 = 1 n 2 . F o r the choice f ( R ) = l n ( l + R ) we obtain Vie)

Fig. 1. The form of the potential (17) generated by the cubic lagrangian theory (16) in the D=4 case. We assume that 0 ~ 0 everywhere. Near ~ = 0 we have V(¢) ~02/4a. Power-law inflation is possible in the O--'+ ~ region since V(¢) ~ be~/4a3there.

Fig. 2. (a) The form of the potential (20) derived from the lagrangian f(R) =2 eR, 2> 0, when D=4. Power-law inflation can occur as 0--,+oo, but not as ~ -oo since V<0 for ~< 1+In2. (b) The potential (21) derived from the lagrangian f ( R ) = In( 1+R) for D=4. We have V~<0everywhere and no inflation is possible because the strong energy condition is always satisfied.

V(0) = ( 0 + 1) e x p ( - 2 0 ) - e x p ( - 0 ) ,

(21)

a n d V~<0 with a m a x i m u m at 0 = 0 a n d a single m i n imum. The potentials ( 2 0 ) a n d ( 2 1 ) are d i s p l a y e d in figs. 2a, 2b. We see from ( 1 9 ) that for k = 2 the potential approaches a constant plateau as 0 ~ o o a n d so i f the 0 fields rolls in from + ~ we can o b t a i n exponential inflation. F o r 0--' - o o the potential is typically o f the f o r m e x p ( - 2 0 ) . Previous studies [ 15,16], o f the evolution o f cosmological models containing scalar fields with exponential potentials o f the form exp ( - 20) show that when D = 4 power-law inflation can be o b t a i n e d only when ~4 0 < 22 < 2. W h e n 2 = 0 there is exponential inflation. Thus the 0 ~ - ~ limit found for the p o l y n o m i a l lagrangians gives a poten#4 In D dimensions the corresponding condition for power-law inflationary solutions is 22< 4 / ( D - 2 ) [ 15 ]. 517

Volume 214, number 4

PHYSICS LETTERS B

tial gradient that would not allow power-law inflation to occur. If the polynomial lagrangian is of at least cubic order ( k > 2 ) then V(O)-.exp[ ( k - 2 ) q ~ ] as 0 - , o o and hence power-law inflation can occur only when k = 3 in the D = 4 case. The transformation o f gravity theories derived from (2) into general relativistic theories possessing scalar fields enables some statements to be made about the occurrence o f singularities in f ( R ) lagrangian theories. The singularity theorems [ 17 ] include as their key ingredient the sufficient condition RabKaKb>O, (K a timelike ), for past geodesic incompleteness of the space-time regardless of the space-time dimensionality. General relativity is unique in that its field equations relate the e n e r g y - m o m e n t u m tensor TabtO Rab in a linear fashion. Hence, by Einstein's equations, the sufficient condition on Rab is equivalent to the generalized strong energy condition in D dimensions:

Tab-- Tgab(D-2 )-1> O.

(22)

This condition is not always satisfied by scalar fields with V(¢) > 0 and so one can draw no conclusion as to the existence or non-existence o f singularities (for example in a D = 4 isotropic Friedman universe the scalar field is equivalent to a fluid with density ½62 + Vand isotropic pressure ½62 - Vand so the lefthand side o f (11 ) and (22) equals 2 ( 6 2 - V) and is of indefinite sign if V is positive. However, when V~<0, the energy-momentum tensor of the scalar field obeys (22) for all D and the singularity theorems apply. Hence we see that for gravitational theories derived from a lagrangian f ( R ) that are conformally equivalent to a scalar field with V(#) ~<0 in (12) we can conclude that singularities are inevitable in the v a c u u m theory generated from the lagrangian f ( R ) and also in the presence o f other explicit matter fields obeying (22) because geodesic incompleteness must occur in both conformally-related metrics (5). As an example, see fig. 2b; the theory with f(R)=In( 1 + R) gives rise to the potential (21) which satisfies

518

1 December 1988

V(¢) ~<0 for all # and so geodesic incompleteness follows from the singularity theorems [ 16 ] so long as the other technical conditions o f the theorems are met. We should also note that when V~<0 the fact that the strong energy condition is always satisfied rules out the possibility o f inflation because the mean expansion scale factor of an expanding universe can never accelerate when (22) holds. These results may offer some route towards obtaining more detailed necessary and sufficient conditions for the occurrence o f singularities in lagrangian theories o f gravity. We would like to thank Adrian Burd, Steven Hawthorne, Mark Madsen, Kei-ichi Maeda and Adrian Ottewill for helpful discussions.

References [ 1] A. Guth, Phys. Rev. D 23 ( 1981 ) 347; A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220; A.D. Linde, Phys. Lett. B 108 (1982) 289. [2] J.D. Barrow, Phys. Lett. B 180 (1986) 335. [3] J.D. Barrow, Quart. J. R. Astron. Soc. 22 (1988), to be published. [4] L.F. Abbott and M,B. Wise, Nucl. Phys. B 244 (1984) 541. [ 5 ] F. Lucehin and S. Mataresse, Phys. Rev. D 32 ( 1985) 1316. [6] A.A. Starobinskii, Sov. Astron. Lett. 9 (1983) 302. [7] J.D. Barrow and A.C. Ottewill, J. Phys. A 16 (1983) 2757. [8 ] M.B. Mijie, M.M. Morris and W.-M. Suen, Phys. Rev. D 34 (1986) 2934; A.A. Starobinskii and H. Schmidt, Class. Quant. Grav. 4 (1987) 695; J.D. Barrow, Phys. Lett. B 145 (1987) 285; M. Mijic and J. Stein-Schabes, FermiLab. preprint (1987). [9] B. Whitt, Phys. Lett. B 145 (1984) 176. [ 10] G.V. Bicknell, J. Phys. A 7 (1974) 1061. [ 11 ] J.D. Barrow, Nucl. Phys. B 296 (1988) 697. [12] A.A. Starobinskii, Talk Third CERN-ESO Symp. (Bologna, May 1988). [ 13] K. Maeda, Tokyo preprint (1988). [ 14] K. Maeda, Phys. Rev. D 37 ( 1988) 858. [15]A. Burd and J.D. Barrow, Nucl. Phys. B (1988), to be published. [ 16 ] J.J. Halliwell, Phys. Lett. B 185 ( 1987) 341. [ 17 ] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., Cambridge, 1973).