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Using Standard Syste PHYSICAL REVIEW D, VOLUME 60, 103513 Preheating in generalized Einstein theories Anupam Mazumdar and Luı´s E. Mendes Astrophysics Group, Blackett Laboratory, Imperial College London SW7 2BZ, United Kingdom ͑Received 10 February 1999; published 26 October 1999͒ We study the preheating scenario in generalized Einstein theories, considering a class of such theories which are conformally equivalent to those of an extra field with a modified potential in the Einstein frame. Resonant creation of bosons from an oscillating inflaton has been studied before in the context of general relativity taking also into account the effect of metric perturbations in linearized gravity. As a natural generalization we include the dilatonic–Brans-Dicke field without any potential of its own and in particular we study the linear theory of perturbations including the metric perturbations in the longitudinal gauge. We show that there is an amplifi- cation of the perturbations in the dilaton–Brans-Dicke field on super-horizon scales (k!0) due to the fluc- tuations in the metric, thus leading to an oscillating Newton’s constant with very high frequency within the horizon and with growing amplitude outside the horizon. We briefly mention the entropy perturbations gen- erated by such fluctuations and also the possibility to excite the Kaluza-Klein modes in the theories where the dilatonic–Brans-Dicke field is interpreted as a homogeneous field appearing due to the dimensional reduction from higher-dimensional theories. ͓S0556-2821͑99͒01820-2͔ PACS number͑s͒: 98.80.Cq I. INTRODUCTION to variations in the Newtonian gravitation ‘‘constant’’ G, and introduces a new coupling constant ␻, with general relativity One of the most important epochs in the history of the recovered in the limit 1/␻!0. The constraint on ␻ based on inflationary Universe is the transition from almost de Sitter timing experiments using the Viking space probe suggests expansion to the radiation dominated universe. Until a few that it must exceed 500 ͓9͔. The JBD theory also mimics the years ago proper understanding of this phenomenon was not effective Lagrangian derived from low energy scale of the well established. Recent developments in the theory of reso- string theory where the Brans-Dicke ͑BD͒ field is called Di- nant particle creation of bosons ͓1͔ as well as fermions ͓2͔ laton and the coupling constant ␻ takes the negative value of due to coherent oscillations of the Bose-condensate inflatons Ϫ1 ͓10͔. It also represents the (4ϩD)-dimensional Kaluza- may explain satisfactorily the emergence of the radiation era Klein theories with an inflaton field which has mainly two ͓ ͔ from the ultracold inflationary universe 1 . Although such subclasses out of which we shall consider the one where the phenomena are hard to reproduce in a laboratory they are the inflaton is introduced in an effective four-dimensional simplest manifestation of a slowly varying scalar field which theory. In this case the BD-dilaton field plays the role of rolls down the potential and oscillates as a coherent source at homogeneous scalar field in 4 dimensions and is related to each and every space-time region. Apart from reheating the the size of the compactification. Most of these models are universe to the temperature ambient for the production of conformally equivalent and can be recast in the form of Ein- light nuclei this phenomena has multifaceted consequences. stein gravity theory ͓11͔. The only difference is that the scal- It is also a well known mechanism to create a nonequilib- rium environment which can be exploited for the generation ing of the fields and their corresponding couplings will be of net baryon antibaryon asymmetry required for baryogen- different for different interpretations of BD-dilaton field. esis ͓3͔. If there exist general chiral fields, then, in particular, In this paper we shall consider the general relativity limit breaking of parity invariance could also lead to different pro- of the JBD theory as well as other variant theories such as duction of left and right fermions. If the rotational invariance string and Kaluza-Klein and for the sake of consistency we ␥ is broken explicitly by an axial background then there would just use one representative of the coupling constant , which be anisotropic distribution of fermions ͓4͔. The same phe- takes different values accordingly. We must say that there is nomena is also responsible for the generation of primordial an obvious advantage in all these theories that they are well magnetic field ͓5,6͔ in the context of string cosmology, constrained at the present day and the evolution of the BD- where the dynamical dilaton field plays the role of an oscil- dilaton field is roughly constant after a period of 60 e fold- lating background. ings of inflation. Here we shall not consider the potential in The preheating scenario has been studied so far in the the dilatonic-BD sector and we assume that the evolution of context of general theory of relativity. It has been well es- the dilaton-BD field during the oscillatory phase is solely due tablished from the present observations that this theory of to its coupling with the inflaton in the Einstein frame. We ␾ ϭ 1 2␾2 gravitation is the correct description of space-time geometry assume the quadratic potential V( ) 2 m for the infla- at scales larger than 1 cm. There exists a class of deviant ton field. It is to be noticed that during the oscillatory phase theories which are scalar-tensor gravity theories, known as the field ␾ decreases in the same way as the density of non- ␳ ϭ 1 ␾˙ 2ϩ 1 2␾2Ϸ Ϫ3 generalized Einstein theories ͑GET͒ of which the Jordan- relativistic particles of mass m: ␾ 2 2 m a , Brans-Dicke ͑JBD͒ theory ͓7,8͔ is the simplest and best- provided the coupling ␥ is very small as in the case of JBD. studied generalization of general relativity. This theory leads For simplicity we discuss the physics of weak coupling and 0556-2821/99/60͑10͒/103513͑7͒/$15.0060 103513-1 ©1999 The American Physical Society ANUPAM MAZUMDAR AND LUI´SE.MENDES PHYSICAL REVIEW D 60 103513 at the end we comment on the strong coupling limits which Ͼ500 so that ␥Ͻ0.09Ӷ1. In the dimensionally reduced we get numerically. In the weak limit the coherent oscilla- theories the original action is different from Eq. ͑1͒ but it is tions of the homogeneous scalar field correspond to a matter still possible to conformally transform it to Eq. ͑2͓͒11͔.In dominated equation of state with vanishing pressure. This this case ␹ and the coupling ␥ are given by suggests that the BD field should evolve according to a well known solution during the dust era ͓12͔. 2͑Dϩ2͒ 1/2 ␹ϭͫ ͬ ln ⌽ , ͑4͒ Apart from the study of nonperturbative creation of D BD bosons and fermions during preheating, metric perturbations have also been studied extensively in Refs. ͓13͔ and ͓14͔.It 2D 1/2 has been shown that the metric perturbations also grow dur- ␥ϭͫ ͬ , ͑5͒ ing preheating in the multiscalar case due to the enhance- Dϩ2 ment of the entropy perturbations. It is also possible to am- ⌽ ϵ D/2 plify the superhorizon modes causally. Such amplification where BD (b/b0) , b is the radius of compactification, requires the linear theory of gravitational perturbations to be and b0 its present value. Notice that in the Kaluza-Klein case supplanted by nonlinear perturbation theory. In Ref. ͓13͔ the it is possible to make U(␹)ϭ0 only when the extra dimen- authors have discussed the two fields case in particular where sions are compactified on a torus which has zero curvature. the first one (␾) represents the inflaton and the second (␴) Compactifying the extra dimensions on a sphere will give represents the newly created bosons with an interaction with rise to a mass to the dilaton corresponding to the curvature of 1 ␾2␴2 ␴ the inflaton of the form 2 g . Perturbations in get am- the sphere. In Sec. III we shall come back to this point again plified along with the metric perturbations. They have argued and discuss what should we expect if such a term is also that all the fields would be possibly amplified except the included in the Lagrangian. The number of extra dimensions inflaton which is subdued by transferring its energy to the is D and ␥ can at most take a value ͱ2. In the superstring other fields. Following their claim it is also possible to am- case ␥ is exactly ͱ2 ͓15͔. As we have already mentioned plify the other fields such as dilaton-BD field as they are also during radiation and matter domination ␹ has to be roughly coupled to the inflaton and can be cause of some concern. In constant in order to comply with the observational data ͓16͔ the concluding section we devote ourselves to the discussion and to reproduce the correct value of the gravitational con- of these issues. stant today. In most of the models the evolution of the extra dimensions is also treated to be constant during the radiation II. THE EQUATIONS era ͓17͔. In JBD theory the action in the Jordan frame ͓7͔ A. Equations of motion ⌽ ␻2 ,⌽ We shall perform our calculation in the Einstein frame ץ ⌽ ץϭ ͵ ͫ BD ˆ ϩ ˆ ␮␯ S ␲R ␲⌽ g ␮ BD ␯ BD 16 16 BD Eq. ͑2͒, for simplicity. The homogeneous equations of mo- tion for a zero-curvature Friedmann universe are 1 ͒ ͑ ␾Ϫ ͑␾͒ͬͱϪ ˆ 4 ץ␾ ץϩ ˆ ␮␯ g ␮ ␯ V g d x, 1 2 ␥ ␹¨ ϩ3H␹˙ ϩ eϪ␥␹␾˙ 2Ϫ2␥eϪ2␥␹V͑␾͒ϭ0, ͑6͒ 2 is transformed into Einstein frame ␾¨ ϩ3H␾˙ Ϫ␥␹˙ ␾˙ ϩeϪ␥␹VЈ͑␾͒ϭ0, ͑7͒ 1 1 ␮␯ ␯␹ϪU͑␹͒ץ␮␹ץ Sϭ ͵ ͫ Rϩ g 2␬2 2 1 1 1 ͫ ␹˙ 2ϩ eϪ␥␹␾˙ 2ϩeϪ2␥␹V͑␾͒ͬϭH2, 1 3 2 2 Ϫ␥␬␹ ␮␯ Ϫ2␥␬␹ ͱ 4 ͒ ͑ ,␯␾Ϫe V͑␾͒ͬ Ϫgd xץ␮␾ץ ϩ e g 2 8 ͑2͒ with an overdot denoting time derivation and a prime denot- ing partial derivative with respect to ␾.
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