I. INTRODUCTION In this work we analize the slow roll approximation in cosmological moels of nonminimally couple theories of gravity (NMC) starting fr

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1 Slow Roll Ination in Non-Minimally Couple Theories: Hyperextene Gravity Approach Diego F. Torres Departamento e Fsica, Universia Nacional e La Plata C.C. 67, 900, La Plata, Buenos Aires, Argentina Abstract The slow roll approximation is stuie for cosmological moels in Hyperextene Scalar-Tensor Theories of Gravity. A proceure to obtain slow roll solutions in non-minimally couple gravity is outline an some examples are provie. An integral conition over the functional form of the non-minimal coupling is impose in orer to obtain intermeiate inationary behavior. torres@venus.sica.unlp.eu.ar

2 I. INTRODUCTION In this work we analize the slow roll approximation in cosmological moels of nonminimally couple theories of gravity (NMC) starting from the complete two free functions lagrangian ensity of Hyperextene Scalar-Tensor Theories (HSTG) []. These HSTG are ierent from both, generalize Brans-Dicke (BD) cases an an NMC cases, when there are non-invertible functions of the el involve in the lagrangian. This give rise to the representation of a singularity in the el transformation which allows the passage from a NMC lagrangian to a BD one []. HSTG may reuce itself either to BD or to NMC whenever convenient. In past work, exact cosmological solutions were obtaine for these moels in the case of vacuum, raiation an sti matter universes []. Concerning ination, exact an approximate solutions were foun for particular scalar-tensor theories []. The slow roll approximation [4] were analize by Garcia Bellio, Line an Line in the case of BD gravity [] an later by Barrow for generalize BD gravity with power law couplings [6]. Here, we are going to present the slow roll approximation in the context of HSTG. We work out, following the metho use by Barrow, examples for non-minimally couple gravity. We impose an integral vinculum over the form of the non-minimal coupling in orer to have intermeiate ination behavior. II. HYPEREXTENDED SCALAR-TENSOR THEORIES AND THE SLOW ROLL APPROXIMATION The hyperextene scalar-tensor el equations are erive from the action: S = p g "6L M + K() ; ; + G() R () From now on, we shall call K() =!() to facilitate the comparison with the BD cases. This must be unerstoo only as a change in the names of the functions. Taking variational erivatives from () with respect to the ynamical variables g an we get the el equations. We are intereste in an isotropical an homogeneous universe. We shall have

3 a metric ene by a Friemann-Robertson-walker one with zero curvature an a stressenergy tensor for matter given by the euce from a lagrangian ensity of another scalar el with self interaction potential V (). Finally, these equations become: H 4 _ +! H G _ G! 6 _ G! + G = G = 8 G _ + V () G G () 8 h 4V () _ i () +H_+V 0 ()=0 (4) where we have ene = 4!! G G! 6 G 4 G! + G G G () an H as usual. These equations reuce to those of GR when equals to a constant an to those of BD when G equals =. In orer to examine the forms of ination that can arise in these kin of moels, we are going to assume the slow roll approximation given by: H _ (6) _ V () (7) H _ H (8) The last conition requires a scalar el which evolves slowly with respect to the expansion of the universe an was stuie an applie in the context of both, BD gravity [] an generalize BD gravity [6]. Conition (8) may be use to x a similar one, suitable to be applie in the more general case. It is possible to see that, for any function f positive ene an with a convergent Taylor serie, conition (8) may be transforme to f() H _ f() H f() (9)

4 in any case in which 9n=n N ^ n _ ' H. In particular, the choice f = G makes possible to x useful simplications on the el equations (-4). Finally, the slow roll el equations become: H _ ' V 0 () (0) _ 4 +H _! H '! _ G + 8 G! + G ' GV () () G V () () G The leaing orer terms referre in the slow roll equations also reuce to the BD ones when G == as expecte (see reference [6] for comparison). From here, an in orer to procee further, it is necessary to ene both,!() an G() together with the form of the potential V () for the inaton. In [6], a complete xing of the problem was obtaine by ening the functional form of the coupling, in that case a power law. We propose now, using the freeom given by having two generic functions instea of one, to sketch how general analytical solutions for the slow roll equations for non-minimally couple theories may be obtaine. From now on, we are going to use equalities even when we only have approximate expressions. III. SLOW ROLL SOLUTIONS OF NON-MINIMALLY COUPLED GRAVITY The NMC theories of gravitywere obtaine in the previous formalism ening! = =. We shall impose that, in orer to recover the Einstein's regime at large times an to be consistent with solar system tests of gravity, (G _ ) 4! 0 when t![7]. This conition is (G ) satise when G tens to a constant without asymptotic variations in the rst erivative. This matching approximation carries the factor in the previous equations to a, given by: "!! G! a = () G 4

5 Finally, the NMC slow roll equations may be written as H _ = V 0 () (4) H = 4 _ G +8GV () () _ G G +H_ 4 G! + G G = V () (6) G A further application of the slow roll conition on (6) allows to write it as H 4 _ G! + G G = V () (7) G Using () in (7) an suppossing that G oes not iverge we obtain an aitional prouct of the slow roll equations: H _ 4 + G G! = Note that the secon term on the left behaves as oes as G G H G (8) 9 G G H (G _ ) while that on the right G G H G an so we are allowe to have another simplication H _ = G G H (9) This last equation may be reaily integrate as G =4ln a a 0 (0) with a 0 a constant of integration; thus giving = (a) for every selection which makes invertible the result of the integral an a = a() always. To n = (t) it is necessary to specify the form of the potential. Let us consier some common cases.

6 A. V () =V 0 =constant The case of constant potential allows complete integration. From the previous paragraph we have H G =8V 0 () Replacing the value of H in (9) we obtain G = G =4 8V0 (t t0 ) () with t 0 a constant ofintegration. Dening the gravitational theory by giving the form of G, () gives t = t() an, in the cases in which inversion is possible, = (t). In such cases, it will be also possible then, to replace in (0) to have a = a(t). We see that the proceure followe here continues the line of that presente by Barrow [6] in the case of given coupling of a BD theory; an as that, this enable to see the form of ination (if any) in a particular gravity theory by integrating an inverting two ierential equations. B. V () =V 0 exp ( ); V 0 ; = constants With this exponential form for the inaton potential the slow roll equations become H _ = V 0 exp () () H G =8V 0 exp ( ) (4) H 4 _ G! + G G = G V 0 exp () () Dening as in [6] a new time coorinate as t = H (6) 6

7 the equations are integrable again: () = lnh i V 0 ( + 0 ) (7) + G G G G = ln [ + 0 ] (8) Deng G an inverting (8) to get as a function of one can use the slow roll equation (4) to nally obtain the behavior of H. C. V () =V 0 r ;V 0 ;r = constant an r 6= With the same time variable, we have now: 0 = rv 0 r (9) an the solutions are: G H G =8V 0 r (0) G! G = G V 0 r () () =[4r( r)v 0 ( 0 )] + G G G G =V 0 [4r( ( r) () r)v 0 ] r r (0 ) ( r) () with 0 a constant of integration. IV. SLOW ROLL SOLUTIONS: EXAMPLES Let us rst show how the formalism works for integrable an invertible examples which o not require numerical recipes an amit comparison with previous works. 7

8 A. G = n, n 6= ;4an V = V 0 = constant We are ealing with the lagrangian ensity given by L = n R + ; ; +6L m (4) Equation (0) may be integrate to give (a) n+ = 4n( n) ln(a=a 0 ) () Equation () may also be integrate: an so (t) n=+ = 4n n + 8V0 t (6) (t) / t n+4 (7) This last relationship, use in () gives a(t)! n+ / exp ( t n+4) (8) a 0 a(t) When n grows, the behavior of the scale factor tens to a 0 / exp [t ]. We have to consier now if these solutions are consistent with the slow roll approximations. We see that _ H for all n greater that 4 an lower than an that the approximation fails in the interval (; 4). The slow roll solutions foun have the form of those of intermeiate ination, in which the scale factor expans more slowly than for De Sitter case but faster than for power law ination []. The consistency of the to a passage may be also teste, we n that G / t n 4 n an so, it is a ecreasing quantity with t. B. G =, V = V 0 = constant The lagrangian ensity given by L = R + ; ; +6L m (9) 8

9 was mainly stuie by the Naples group in relation with scalar potentials an ination [8]. Using equation (0) we get (a) / a 8 a 0 (40) an with equation () (t) =ct (4) with c a constant. It may be verie with these solutions that both, the universe is not in expansion an the slow roll conition is not satise; an so, they must be iscare. C. G = 4, V = V 0 = constant Equation (0) gives in this case a / exp[ ] (4) a 0 while (), / exp[ t] (4) Thus, the behavior of the scale factor with time is given by a / exp [exp[t]] (44) Here, the scale factor appears to exhibit an extreme form of ination. It may beprove that j _ jan G H! 0 when t is large enough. The evolution of () can be foun from equation (4) in each of the cases previously analize since _ / a. D. Spaning of Intermeiate Ination Solutions We are willing now to stuy in a completely general form which kin of couplings may allow intermeiate ination behavior. For simplicity we shall use the case of constant potential. 9

10 In our stuy, we shall have two important equations: (0), which is potential inepenent, an (). From the integration of (0), it may inmeiately be seen that, in orer to have a scale factor evolving as 4 R it is necessary to have t m =, that is: G Deriving with respect to we get: m a(t) / exp [t m ] (4) G = G G G = G m = G G G G which is an integral vinculum over the functional form of G(). Although highly non-linear, equation (47) is very suggestive. It may beprove that G() = with m = (+) 4+ (46) (47) is one of its solutions, as expecte, since we n intermeiate ination behavior for it in the previous section. This case was alreay stuie concerning the fulllment of the slow roll conitions. Also the choice G() =exp [], with a constant, is a solution of (47). For such choice, using the previously erive metho we obtain: an a / exp[t ] (48) / ln [t] (49) It is woth recalling that the behavior of the sclae factor in (48) coincies with that of G = if is large enough. The exponential form of G is such that the slow roll an the! a conitions are fullle. V. DISCUSSION Starting from the hyperextene approach of scalar-tensor theories, we have formulate the slow roll approximation in non-minimally couple gravity. The solutions were given in 0

11 the form of two integrals that may be solve analytically in some cases an numerically in all cases. These integrals give the behavior of as function of a an t. Properly inversion (again analytically or numerically) gives so, a an as functions of time. We stuie constant, power law an exponential forms for the inaton potential. Some examples that show intermeiate ination were presente for power law non-minimally couplings in agreement with the results of power-law couplings in BD theories. An integral vinculum for the functional form of G was stablishe in orer to evelop intermeiate ination behavior an some examples were provie. VI. ACKNOWLEDGMENTS This work was partially supporte by CONICET. Valuable conversations with H. Vucetich are acknolege.

12 REFERENCES [] D.F. Torres an H. Vucetich. At press in Phys. Rev. D. See also N. Sakai an K. Maea, Prog. of Theor. Phys. 90, 00, 99 [] A.R. Lile an D. Wans, Phys. Rev. D 4, 66 (99) [] D. La an P.J. Steinhart, Phys. Rev. Lett. 6, 76 (989); P.J. Steinhart an F.S. Ascetta, Phys. Rev. Lett. 64, 470 (990) J.D. Barrow an J.P. Mimoso, Phys. Rev. D 0, 746 (994); J.D. Barrow, Phys. Lett. B, 40 (990); J.D. Barrow an P. Saich, ibi. 49, 406 (990) J. Garcia Bellio an M. Quiros, Phys. Lett. B 4, 4 (990) [4] See for example: P. Coles an F. Lucchin, Cosmology, John Wiley 99. For a more rigurous approach: A. Lile, P. Parsons an J.D. Barrow, Phys. Rev. D. 0, 7 (994) [] J. Garcia-Bellio, A. Line an D. Line, Phys. Rev. D 0, 70 (994) [6] J.D. Barrow, Phys. Rev. D, 79 (994) [7] S. Capozziello, R. e Ritis an P. Scuellaro, Phys. Lett. A 88, 0 (994). [8] S. Capozziello an R. e Ritis, Phys. Lett. A 9, 48 (994). See also: ibi. 77, (99) an references therein.