Effects of R3 and R□R terms on R2 inflation

Volume 245, number 3, 4

Effects of R 3 and

PHYSICS LETTERS B

16 August 1990

RE]R t e r m s o n R 2 i n f l a t i o n

A n d r e w L. B e r k i n 1 a n d Kei-ichi M a e d a 2 Department of Physics, Waseda University, Shinjuka-ku, Tokyo 169, Japan Received 2 March 1990

To study the generality of inflation from lagrangians containing higher orders of curvature, we consider the effects of R 3 and R[]R terms on the R 2 inflationary scenario. Both direct analysis and conformal techniques are used. We find that both terms restrict the initial conditions which will lead to a satisfactory inflationary cosmology.

The inflationary universe scenario has been one of the major cosmological developments of the past decade [1]. Several mechanisms have been used to generate the rapid expansion needed to solve some long standing cosmological problems. One particular mechanism, d u b b e d " R : inflation", adds terms quadratic in the curvature tensors to the standard Einstein-Hilbert action [2,3]. For a certain range of initial parameters, these quadratic terms will generate sufficient expansion without the need for the special matter terms used in most inflationary mechanisms. The justification for addition of quadratic terms is usually twofold. First, renormalization of q u a n t u m field theories in curved spacetime usually mandates such terms [4], although with coefficients much smaller than required for a fully satisfactory inflationary picture. Second, as there is no de facto reason why the action should be only linear, perhaps nature really does contain these terms, which would be observationally undetectable at our present epoch [3]. With both reasons, the question of the influence of even higher order terms in the action arises. The renormalization of higher loop processes introduces higher order corrections to the action, while if one believes that nature is not merely linear in the gravitational action then certainly terms of all orders are possible. Because each higher order is suppressed by the inverse Planck mass squared, their effects will only be significant near the Planck era, in which case not just quadratic but all terms will contribute. In this paper we consider the effects of third order terms on the early universe, where R D R is regarded as third order since two derivatives are dimensionally equivalent to one curvature tensor. If these cubic curvature contributions further increase the a m o u n t of expansion, then the feasibility of inflation from higher order terms seems strengthened. Indeed, these terms have been claimed to enhance the viability of such models by producing effects such as double inflation [5]. However, we will show that these cubic terms restrict the range of parameters which lead to a successful inflationary scenario. Because such terms arise naturally in any theory which admits quadratic contributions, the viability of higher order inflation is weakened. We use a R o b e r t s o n - W a l k e r universe, so to quadratic order only the R 2 term contributes. For simplicity and to facilitate comparison with previous work we consider third order terms which contain only the Ricci scalar and its derivatives ~1. First the effects of an R 3 term and an RI3R term are analyzed separately, then both effects are taken into account. A term of the form V,RV~'R also exists, but u p o n integration by parts it may be 1 E-mail address: [email protected] 2 E-mail address: [email protected] ~1 Second order terms are discussed in ref [4]. All possible third order terms, with coefficients due to quantum corrections, are found in ref. [6]. For terms through fourth order, see ref. [7].

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0370-2693/90/$ 03.50 O 1990- Elsevier Science Publishers B.V. (North-Holland)

Volume 245, number 3, 4

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expressed as a boundary term plus the RI~R term, and hence contains no new information. Our action is then

S=

I d4x ~-~C-~[f(R)+yRDR],

(1)

where f ( R ) = R + a R E + f l R 3 in this case, KE/87r = GN is the gravitational constant, a, fl and y are arbitrary constants, R is the Ricci scalar, and g is the determinant of the metric tensor. Our conventions will be the same as those of Misner, Thorne and Wheeler [8]. Variation of (1) yields the effective field equation

R ~ , ~ - ½ R g ~ = - 2 a ( R R ~ , ~ + D R g ~ - R ; ~ , ~ - ~ R 1 2g~,~) + fl(½R3g~,,, + 6RR;,,~ + 6R;,~R;~ - 6R [] Rg,~ - 6R;,,R;Pg,~ - 3 R E R ~ ) - 2 y [ - ( [ ] R ) ., ~ +

[]

2Rg~,~+R~,~[]R-~R;~,R;~+zR;pR t ~ ;pg~,~].

(2)

We assume the flat Robertson-Walker metric given by ds 2 = - d t 2 + a2(t)(dx 2 + dy 2+ dz2).

(3)

The (00) field equations then give H E= - 2a (6/4H - 3//2 + 1 8 / / H 2) + 12/3 ( - 18/~r/~/H - 36/4H 3 + 6 / / 3 - 45/:/2 H 2 - 108/:/H4 + 12H 6) - 2 y ( - 18jr~H + 1 8 / / ' / / - 1 0 8 ~ H 2 -$/4 2 - 153/~/:/H - 9 0 H H 3 + 72//3 - 153//2H 2 + 216//H4),

(4)

where H =-a/a is the Hubble parameter and an overdot indicates a derivative with respect to time. (I) R 3 contribution (y =0). For y = 0 , we write the (00) equations in the form /:/=~g -2H 2

(5)

/~ = - 3 H 2 ( 1 + 2 a R + 3fiR 2) +½aR E+ fiR 3

(6)

18flH(R + a / 3 f l ) A phase space analysis of (5) and (6) provides much insight (fig. 1). While micro-physical considerations constrain a to be positive [9] (see also below), there are no such restrictions on/3. The behavior of the universe depends strongly on the values of a and ft. Three characteristic cases exist: fl > ] a 2, 0 < / 3 ~ a 2 (fig. la), a universe which enters the classical era with a value near the fixed point D+--- ( H , , RCr) will initially expand like a de Sitter universe. The universe just below H ~ and R~ will slowly evolve towards the origin of phase space, at which point oscillations which couple to matter should reheat the universe, just as in the R E theory [3]. Hence, successful inflationary scenarios do exist. Furthermore, universes which start in the upper right quadrant w i t h / : / > 0, R ~< R~r will eventually evolve towards t h e / : / = 0 curve, at which time they will behave in the manner just described. This mechanism is almost identical with that of R 2 inflation, although we may need an elaborate fine-tuning of initial conditions for successful inflation (see below). A dramatic difference occurs for universes which start with / : / > 0 and R ~> R~. While in the R 2 case these universes eventually evolve to the inflationary regime, here they approach R, H ~ o0. The universe will not attain the current small R value, and there is no successful transition to the standard Hot Big Bang scenario. Hence, this segment of phase space, which inflates when only the curvature squared term is added, will no longer provide a satisfactory cosmology when the cubic term is also considered. For completeness, we have included the diagram for all of phase space. Much as in the R 2 scenario, universes which start with R, H > 0,/~/< 0 will evolve too quickly towards the origin and not undergo sufficient expansion. 349

Volume 245, n u m b e r 3, 4

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Volume 245, number 3, 4

PHYSICS LETTERS B

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H > 0, R < 0 initial conditions either never evolve to R = 0, evolve too quickly to the origin for sufficient expansion to occur, or else reach the H < 0 region where recollapse will occur• Similarly, universes which have H < 0 a n d / - : / < 0 initially will suffer recollapse and are not interesting. Finally, H < 0, /:/> 0 initial conditions evolve too quickly into the origin. For the case 0
(7)

whose potential is given by

laR2+2/3R3 °t3 [ 2~2 ( -~2 \3/2"1 U(~b)=2K 2 (l+2otR_l. 3/3R2)2-27t<2/32 e-224~73,'e - 1 + (1-e4273"*)+ 1 (1-e'/~"*)) J.

(8)

We have excluded the region f'(R)< O, as it cannot evolve into Minkowski space [10]. 351

V o l u m e 245, n u m b e r 3, 4

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16 A u g u s t 1990

The shape of this potential is plotted in fig. 2 for various values of/3. The qualitative difference of the added R 3 term is made obvious by the shape of the potential in the conformal picture. If only an R 2 term is added, the potential consists of a plateau which is virtually flat for large values of the field. This plateau acts as an effective cosmological constant, so that inflation will occur in the conformal frame, which then guarantees inflation in the real world [10]. Eventually, the field rolls to the origin, where it oscillates and reheats the universe, successfully terminating the inflationary era. However, with the added R 3 term, there is now a maximum in the potential at ~Ocr------~/'(Rcr)for /3 > 0 . Any universe which enters the classical regime with a value of R greater than this will no longer slowly roll towards the origin, but rather away towards infinity, and there can be no successful termination of inflation. For negative/3, all expanding universes will evolve into Minkowski space, but inflation will only occur if 1/31 is much less than a2. We easily find from (8) and fig. 2 that the condition Jill < a2 is necessary to find the long plateau which guarantees natural initial conditions for successful inflation. (II) R [] R contribution (13 = 0). We next consider the effects of a term of the form TR [] R. As seen from (2), the contribution is exactly opposite in sign to that found in ref. [5], and will be responsible for the differing behavior that we find. To search for inflationary solutions, we a s s u m e / 2 / < H 2 and ignore all higher derivatives, giving //=

1 36~ + 1443'H 2"

(9)

We will later show that y ~<0 for Minkowski space to be stable. If c~ ~, lylH 2, where a subscript 0 indicates a value at the onset of the classical era, t h e n / : / < 0, H will decrease and the 3' term will never contribute. The usual R 2 inflation occurs, and we find a = ao exp H o t -

.

(10)

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Fig. 2. The p o t e n t i a l for the y = 0 case in the c o n f o r m a l frame, w i t h a = 1 in P l a n c k units a n d v a r i o u s v a l u e s of/3. The R 2 m o d e l is g i v e n b y / 3 = 0.

352

Volume 245, number 3, 4 If a <

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16 August 1990

[3'[Hg, then

a = ao exP(8(_33'),/3 ( t - t o ) 4 / a ) ,

(11)

which certainly is inflationary. However, in this a p p r o x i m a t i o n , / 1 / ~ - 1 / 1 4 4 3 ' H 2 > O. H will grow without end as t 1/3, and the 3' term will continue to dominate. The inflationary era will never terminate, resulting in an unsatisfactory cosmology. Thus, because of the difference in sign of % the double inflation scenario of ref. [5] is ruled out. The analysis in the conformal picture provides further insight. We define the conformal transformation g~,~-~ ~,~ = eZ°'g~,~, where to = ½ln[lf'(R) + 23' [] R[]. We now must introduce two scalar fields [5]: Kqt = x/~ ln[l/' (R) + 23, [] RI],

(12)

K~b = [~y[R.

(13)

This transformation gives an action of S = f d4x~-~(2-- ~ / ~ - ½ ( V O ) 2 - ½ ( s i g n ) e-'/~'*(V~b) 2 - U(~O, ~b)),

(14)

where (sign) is the sign of 3' and the potential is given by 1 ( K~b _- 2,/~/3~,_r{ K~b'~ e_23,/~K,'~ U(q,, 4~) = 2K2\ [x/~y[~ J~ [x/~y[] ,]"

(15)

TO investigate the stability of Minkowski space, which corresponds to 4~ = ~0 = 0, we linearize the above action as in ref. [5], finding the eigenvalues of the mass matrix of 4~ and q, by a rotation. Note that hyperbolic trigonometric functions must be used for y < 0 to preserve the sign of the kinetic part. For 3' > 0 this yields m 2 = (a/23"), ( - 1 +x/1 + 3"/3a2), where m 2 and m 2_ correspond to the rotated ~ and ~ respectively. Since m z_ is negative or complex, Minkowski space is unstable. When 3' < 0 , we find m 2 = (a/23"), (+1 + x / l + 3'/3a2). Although m 2 is again negative (or complex), we must be careful of the signature of the kinetic term for this case. In order for Minkowski space to be stable, the squares of the masses must be opposite in sign to the kinetic terms in the lagrangian. The present condition is then met if a > 0 and - 3 a 2 < 3' < 0, which also guarantees the squares of the masses to be real. The potential is plotted in fig. 3, and shows the hindering effects of the R [] R term on inflation. The 3' = 0 curve, which corresponds to the potential in the conformal frame for the R 2-model, lies along the crest of the

Fig. 3. The potential for the/3 = 0 case in the conformal frame, with a = 1, y = - 1 in Planck units. Beyond the critical point S-=(~s, ~0s), the crest is unstable. Below S the crest is an attractor, with the universe then evolving along the crest as indicated by the arrows. 353

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potential. Although the crest appears unstable, the signature of the kinetic term must be considered. When the kinetic term of the rotated scalar field perpendicular to the crest has an opposite sign, the convex shape of the potential rather guarantees its local stability. There is a critical point (~bs, ffs)~ K-'(3a/[x~y[, ln(v/-60t/1~[)), beyond which the scalar field perpendicular to the crest line has a proper kinetic term, hence the crest points beyond that are unstable. For these values, the universe never experiences inflation unless exceptionally fine-tuned initial conditions are chosen. Below the critical point, we expect the universe to evolve towards the crest. Since the plateau of the potential must be sufficiently long for successful inflation, the critical point cannot be too near the origin. Thus, we find [71 ~ a 2 for successful inflation, with 17[ ,~ a 2 leading to natural initial conditions. Any scheme which considers the effects of curvature squared terms must also logically consider even higher order effects. While an additional R 2 term in the gravitational action does lead to a satisfactory cosmology, albeit with an unnaturally high value of the coupling constant, the presence of third order terms imposes further restrictions. With an added fiR 3 term, we find that [fll ,~ a 2 is necessary to provide a sufficiently long inflationary era, while Ro
References [1] A.H. Guth, Phys. Rev. D 23 (1981) 347; K. Sato, Mon. Not. R. Astron. Soc. 195 (1981) 467; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220; A.D. Linde, Phys. Lett. B 108 (1982) 389. [2] A.A. Starobinsky, Sov. Astron. Lett. 9 (1983) 302. [3] M.B. Mijir, M.S. Morris and W.-M. Suen, Phys. Rev. D 34 (1986) 2934. [4] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space (Cambridge U.P., Cambridge, 1982). [5] S. Gottlrber, H.-J. Schmidt and A. A. Starobinsky, Class. Quant. Grav. 7 (1990) 893. [6] T. Sakai, Trhoku Math. J. 23 (1971) 589; P. B. Gilkey, J. Diff. Geom. 10 (1975) 601; Compositio Math. 38 (1979) 201. [7] P. Amsterdamski, A. L. Berkin and D. J. O'Connor, Class. Quant. Grav. 6 (1989) 1981. [8] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, CA, 1973). [9] K. S. Stelle, Gen. Rel. Grav. 9 (1978) 353; P. Teyssandier and Ph. Tourrenc, J. Math. Phys. 24 (1983) 2793. [10] K. Maeda, Phys. Rev. D 39 (1989) 3159; P. W. Higgs, Nuovo Cimento 11 (1959) 816; B. Whin, Phys. Lett. B 145 (1984) 176; G. Magnano, M. Ferraris and M. Francaviglia, Gen. Rel. Grav. 19 (1987) 465; A. Jakubiec and J. Kijowski, Phys. Rev. D 37 (1988) 1406. [11] K. Maeda, Phys. Rev. D 37 (1988) 858; M. B. Miji6 and J. A. Stein-Schabes, Phys. Lett. B 203 (1988) 353; A. L. Berkin, Waseda University preprint WU-AP/03/89 (1989), Phys. Rev. D (1990), to appear.

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