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 22 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 22Ϸ 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 22 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␥V͑͒ϭ0, ͑6͒ 2 is transformed into Einstein frame
¨ ϩ3H˙ Ϫ␥˙ ˙ ϩeϪ␥VЈ͑͒ϭ0, ͑7͒ 1 1 ϪU͑͒ץץ Sϭ ͵ ͫ Rϩ g 22 2 1 1 1 ͫ ˙ 2ϩ eϪ␥˙ 2ϩeϪ2␥V͑͒ͬϭH2, 1 3 2 2 Ϫ␥ Ϫ2␥ ͱ 4 ͒ ͑ ,Ϫe V͑͒ͬ Ϫgd xץץ ϩ e g 2 8
͑2͒ with an overdot denoting time derivation and a prime denot- ing partial derivative with respect to . We use natural units through the conformal transformation where 2ϭ1. We do not pay much attention to the field equations during inflation, since much has been studied in ϭ⍀2 ˆ g g , this area ͓15,18͔, rather we concentrate on the coherent os- cillations of the inflaton. Before we turn to linear perturba- 2 ͱϩ tion theory we should mention that we have to introduce the ⍀2ϵ ⌽ ϵe( / 3/2), ͑3͒ 8 BD bosons resonantly produced from the nonperturbative decay of inflaton field which we shall denote by . We consider the where 2ϭ8G, ␥ is a constant related to . and are coupling of particles with the inflaton to be g/222, the BD and inflaton fields respectively. For our concern ␥ where g is the coupling strength. Hence, the old potential is ϭ1/ͱϩ3/2 and U()ϭ0 ͑see, however, the discussion be- modified during oscillations due to the presence of massless low and on Sec. III͒. Observational constraints give particles:
103513-2 PREHEATING IN GENERALIZED EINSTEIN THEORIES PHYSICAL REVIEW D 60 103513
1 1 2 2 2 2 2 k V͑,͒ϭ m ϩ g . ͑9͒ ␦¨ ϩ͑3HϪ␥˙ ͒␦˙ ϩͫ ϩg2eϪ␥ͬ␦ 2 2 a2
ϩ Ϫ␥␦Ϫ␥˙ ␦˙ Ϫ␥ 2 Ϫ␥␦ In the Heisenberg representation the equation of motion for 2g e g e the scalar field can be expressed in terms of the temporal ϭ ⌽˙ ˙ Ϫ 2 Ϫ␥⌽ ͑ ͒ part of the mode with comoving momentum k, 4 2g e , 16 where ␦, ␦, and ␦ are gauge invariant perturbations in 2 the respective fields. At this point the evolution equations for k Ϫ␥ ¨ ϩ3H˙ Ϫ␥˙ ˙ ϩͫ ϩg2e ϪRͬϭ0, ͑10͒ the background are not determined by Eqs. ͑6͒–͑8͒; rather 2͑ ͒ a t they are modified by the presence of particles. For the Friedmann equation ͓Eq. ͑8͔͒ we now have where is the nonminimal coupling to the curvature, which 1 1 1 1 we shall not take into account. This is the equation of great ͫ ˙ 2ϩ eϪ␥˙ 2ϩ eϪ␥˙ 2ϩeϪ2␥V͑,͒ͬϭH2. importance for the study of resonant creation of ͑details 3 2 2 2 can be seen in Ref. ͓1͔͒. This particular equation takes the ͑17͒ form of the well known Mathieu equation after redefining the ͑ ͒ field. This equation has very rich properties, and its solutions Similarly, the homogeneous equations for and , Eqs. 6 ͑ ͒ can fall into two categories, either stable or unstable depend- and 7 are to be modified accordingly with V( ) replaced Ј ing on the choice of two parameters ͓A(k),q͔, defined later by V( , ) and V ( ) replaced by V( , ), . It is worth ͓19͔. mentioning that in the absence of field and the metric perturbation it is possible to express the perturbation in the BD field in a simpler form. Let us further note that in an B. Linear perturbations expanding universe without BD field the behavior of the in- Ϸ Linear perturbations can be taken into account in a gauge flaton field is sin(mt)/mt and the scale factor goes as Ϸ 2/3 invariant fashion using the longitudinal gauge. We write the a(t) a0(t/t0) . In the presence of the BD field this is Ϸ (2ϩ3)/(3ϩ4) perturbed metric as modified to a(t) a0(t/t0) , such that in the large limit it reduces to the previous one. We expect simi- 2ϭ ϩ ⌽͒ 2Ϫ 2 ͒ Ϫ ⌿͒␦ i j ͑ ͒ ds ͑1 2 dt a ͑t ͑1 2 ijdx dx , 11 lar modification in the behavior of an inflaton field. For our calculation’s sake we can assume that the modified evolution of the inflaton for Ͼ500 and ␥Ͻ0.09 is where ⌽ and ⌿ are gauge invariant metric potentials. The
Fourier modes satisfy the following equations of motion sin͑mt͒ which are derived perturbing the Einstein’s equations Ϸ͓1ϩO͑␥͔͒ . ͑18͒ mt ͓20,21͔. For our purpose the exact numerical expression for O(␥) ⌽ϭ⌿, ͑12͒ does not matter because we are retaining the lowest order in ␥. Simplifying the perturbed BD field Eq. ͑14͒ with the help of Eq. ͑18͒ and noting that ␦ can be rescaled by introducing 3/2 1 Ϫ␥ Ϫ␥ uϭa (t)␦(t) and zϭmt,weget ⌽˙ ϩH⌽ϭ ͓˙ ␦ϩe ˙ ␦ϩe ˙ ␦͔, ͑13͒ 2 d2u k2 3 ␥2 5 ␥2 ϩͫ ϩ Ϫ cos͑2z͒ͬuϭ0. ͑19͒ dz2 a2m2 4 z2 4 z2 k2 ␥2 ␦¨ ϩ3H␦˙ ϩͫ Ϫ eϪ␥˙ 2ϩ4␥2eϪ2␥ V͑,͒ͬ␦ a2 2 • The above equation resembles the Mathieu equation ͓19͔:
ϩ␥ Ϫ␥˙ ␦˙ Ϫ ␥ Ϫ2␥ ͒ ␦ Љϩ͓ ͑ ͒Ϫ ͔ ϭ ͑ ͒ e 2 e V͑ , , x A k 2q cos 2z x 0 20 Ϫ ␥ Ϫ2␥ ͒ ␦ 2 e V͑ , , with
˙ Ϫ2␥ ϭ4⌽˙ ϩ4e V͑,͒⌽, ͑14͒ k2 3 ␥2 A͑k͒ϭ ϩ , ͑21͒ 2 2 2 a m 4 z 2 ¨ k Ϫ␥ ␦ϩ͑3HϪ␥˙ ͒␦˙ ϩͫ ϩe V͑,͒ ͬ␦ 2 2 , , 5 ␥ a qϭ . ͑22͒ 8 z2 ϩ Ϫ␥ ͒ ␦Ϫ␥˙ ␦˙ Ϫ␥ Ϫ␥ ͒ ␦ e V͑ , ,, e V͑ , , Since qӶ1 we are never in the broad resonance regime ϭ ⌽˙ ˙ Ϫ Ϫ␥ ͒ ⌽ ͑ ͒ 4 2e V͑ , , , 15 which is essential to amplify the modes during the oscilla-
103513-3 ANUPAM MAZUMDAR AND LUI´SE.MENDES PHYSICAL REVIEW D 60 103513
␦ 2 2 ϭ ϫ 3 ϭ ␦ 2 2 ϭ ϫ 3 ϭ FIG. 1. The evolution of ln for g( 0/m) 9 10 k 0, and FIG. 2. The evolution of ln for g( 0/m) 9 10 , k 0 ␥ϭ1.22 corresponds to Dϭ6 extra compactified dimensions. and ␥ϭ1.22 corresponds to Dϭ6 extra compactified dimensions. while the corresponding time derivatives are set to zero.1 The tions of the scalar field in presence of expansion. In this case initial condition for is that of a plane wave solution and the we are always in the narrow resonance regime ͓1͔. BD-dilaton field is assumed to be very small Ӷ1 to match During preheating the resonant modes should grow as the present strength of the gravitational constant ͑see Figs. 1 ␦ ϰ k exp( kt), where k is the Floquet index and its analyti- and 3͒. cal estimation has been done in Ref. ͓1͔ for the bosonic We have plotted the fluctuations in the BD-dilaton field Ϫ 2 Ϸ ϩ (k/ag 0) Ϫ ϭ ͑ fields, k ln(1 2e Q), where Q is contributed and the perturbations in the field for k 0 see Figs. 2 and by the initial and final quantum states of the created bosons, 4͒. In the weak limit for ␥ϭ0.09, the perturbations do not which in the case of preheating are not thermal but have the grow much and they saturate much earlier in comparison characteristics of a squeezed state ͓1͔. As it has been pointed with the strong coupling limit, such as ␥ϭ1.22, which, in the out in Ref. ͓13͔, the resonant production of newly created case of a Kaluza-Klein theory corresponds to six extra di- bosons grows for kϭ0, so the super-Hubble modes are mensions compactified to a six-dimensional torus. In the present even in the absence of metric perturbation. In this strong limit case the nonlinearity is achieved as soon as the case, however, the BD modes are not amplified. We recog- metric perturbation ⌽ϭ1. It is obvious that for Kaluza-Klein nize that in the Einstein equations as we perturb the matter theories and for string motivated theories the linear theory sector we automatically perturb the gravitational sector, and soon becomes invalid unless the backreaction is taken into gravity being a source of negative heat capacity it can only account. It is important to notice that our q parameter is not transfer energy to the metric perturbation which acts back to the same as in general relativity. Instead of defining the q ϭ 2 2 the matter sector enhancing the resulting perturbation in the parameter to be q g( 0/m), where 0 is the initial am- matter fields. Energetically it is easier to transfer energy to plitude of the inflaton field, which is 0.08M Pl in our case, in ⌽ the lowest modes so most of the energy is transfered to kϭ0 presence of the BD field q is modified by an exponential ϭ ϭ 2 2 Ϫ␥ and that is the reason why k 0 is favored. factor q g( 0/m)e . The negative sign in the exponent 2 2 ϭ t drags down the q parameter. For the figures g( 0/m) 9 ϫ103. There is some difference between the perturbations in and as visible from the plots ͑see Figs. 1–4͒.Itisim- C. Numerical result portant to note that has as such no potential with minima Our analytical approximation breaks down as we increase unlike and fields which oscillate around the minima of the numerical value for the coupling constant ␥ and the per- their respective potentials and for the modes outside the ho- turbation equations become intractable as we introduce the rizon the perturbed field almost stops oscillating and starts metric perturbations. Hence we solve the system of 14 first growing exponentially. The increment in the perturbations in order differential equations numerically. Here while solving the strong coupling limit can be understood qualitatively. Let the homogeneous equation for we consider only the zero us first see how the scale factor behaves in these two cases. mode contribution, hence Eq. ͑10͒ can be treated as a homo- As we have already discussed the oscillations in the inflaton geneous background equation for the linear perturbation in field on average correspond to zero pressure. This suggests with kϭ0. By doing so we neglect the zero mode contribu- that the BD-dilaton field on average evolves similar to the tion from the metric potential and including the zero mode matter dominated era, and in presence of the BD-dilaton Ϸ (2ϩ3)/(3ϩ4) ␥ metric contribution can only lead to enhancement in the am- a(t) a0(t/t0) . For large and small , the plification in the matter and the BD-dilaton sector as dis- growth in a(t) is roughly the same as in general relativity, cussed in Ref. ͓13͔. For our numerical calculation we have assumed that the perturbation in and in contains the generic, scale invariant spectrum produced during inflation. 1We have chosen our initial conditions to be scale-invariant, i.e., ⌽ ϭ Ϫ5 We set k(t0) 10 and we also consider the same initial independent of k. There are, however, other subtleties involved in conditions for the fluctuations in the BD-dilaton and the fluc- deciding the initial conditions. For a recent discussion we refer to ␦ ϭ␦ ϭ␦ ϭ Ϫ5 ͓ ͔ tuations in the matter fields kϭ0 kϭ0 kϭ0 10 Ref. 13 .
103513-4 PREHEATING IN GENERALIZED EINSTEIN THEORIES PHYSICAL REVIEW D 60 103513
␦ 2 2 ϭ ϫ 3 ϭ ␦ 2 2 ϭ ϫ 3 ϭ FIG. 3. The evolution of ln for g( 0/m) 9 10 , k 0 and FIG. 4. The evolution of ln for g( 0/m) 9 10 , k 0 ␥ϭ0.09 corresponds to ϭ500 in JBD theory. and ␥ϭ0.09 corresponds to ϭ500 in JBD theory.
but for small and ␥ close to ͱ2, the departure of the Ref. ͓18͔ that at the end of inflation when is very close to behavior of the inflaton oscillations from general relativity is zero the metric perturbation is dominated by the adiabatic ϩ ϩ quite significant. Since ϳ1/tϳ1/a(3 4)/(2 3), for small contribution and at the time of horizon crossing the fractional , the amplitude of the inflaton decreases slowly compared energy density fluctuations in are much smaller than the to the general relativity limit, therefore leading to a remark- adiabatic ones. This suggests that the isocurvature fluctua- able growth in the production of particles. tions in the JBD model are negligible. If we assume that the BD field is a candidate for the cold dark matter then one can D. Perturbing Newton’s constant estimate its energy density from the exact solution of the BD field during the dust era, which is a good approximation In ͓22͔ the authors have considered the oscillations in the during the oscillatory phase when the average pressure be- Newton’s constant by introducing a potential for the BD field comes zero and energy density falls as 1/t. Since explicitly and concluded that G oscillates about its mean value and for reasonable values of the mass of the BD field 1 2 2 (mу1 GeV͒, the oscillations have very high frequency ϭ ˙ Х␥ , ͑23͒ Ϫ 2 1Ӷ1 s, compared to the Hubble expansion. In their case, however, the amplitude of the oscillations is exponentially small, and the oscillation energy is dissipated through where is the post inflationary energy density of the uni- Hubble redshift. Considering our scenario, where we do not verse, it is obvious that the energy density of the BD field is have to invoke the potential for the BD explicitly, the per- negligible when compared to the total. Usually for multiple turbations in the BD field will cause the oscillations in the fields the difference between the relative perturbations are ͓ ͔ Newton’s constant in the Jordan frame even if the modes are defined as entropy perturbation S1,2 24 by well within the horizon. Such oscillations will not only have ␦ ␦ a temporal but also a spatial variation. As we approach to ϭ 1 Ϫ 2 ͑ ͒ ϭ S1,2 ϩ ϩ , 24 k 0 mode, gradually the amplitude of such oscillations also 1 p1 2 p2 increases exponentially in time and causes a large variation in the Newton’s constant. Such oscillations are certainly per- where 1 , 2 , p1 , p2 are respectively the energy densities missible but whether they would withstand all known tests and pressures. For nonadiabatic initial conditions ␦/˙ from cosmology and general relativity is not known at the Þ␦/˙ Þ␦/˙ , S , S , and S would not vanish, moment. It is expected that the classical fluctuations will , , , thus producing entropy perturbation inside as well as outside lead to nonlinearity after some time and the linear theory of the horizon. Thus for superhorizon modes isocurvature fluc- perturbation will break down when ⌽ϭ1. Such provocative tuations would evolve though adiabatic fluctuation would interpretation of the Newton’s constant and the nondecayable have frozen. The growth in the isocurvature perturbation out- property of the BD field could be the source of dark matter in side the horizon gives the tilt in the spectrum, which is usu- our universe. Had we introduced a potential with a minimum ally blueshifted and roughly estimated by the ratio of effec- for the BD-dilaton field, superimposed on the exponential tive mass square of the cold dark matter ͑CDM͒ field and growth, we should also see small oscillations with frequency square of the Hubble expansion. If is treated as a CDM and amplitude depending on the exact form of the potential. field then the effective mass is roughly ␥2V(,), the main contribution coming from the large coupling 0.5g22, but E. Entropy perturbations the coupling is subdued by the presence of ␥2, provided we During inflation the presence of BD field renders the are in a weak limit JBD theory. In string motivated and in three-curvature of comoving hypersurfaces in terms of Kaluza-Klein theories the effective mass increases a lot. Dur- Bardeen’s gauge invariant quantity time varying on super- ing preheating the Hubble expansion is very small compared horizon scales, thus producing not only adiabatic but also to the effective mass and thus results in an extreme blue tilt isocurvature perturbations ͓18,23͔. It has been estimated in specially in string motivated theories.
103513-5 ANUPAM MAZUMDAR AND LUI´SE.MENDES PHYSICAL REVIEW D 60 103513
F. Shaking the Kaluza-Klein modes ondary phase of inflation. Such a phase is also desirable to As we have described, depending on the numerical value smoothen the large superhorizon resonances as mentioned in ͓ ͔ of the coupling constant ␥, the role of also changes, and in Ref. 13 . fact it covers a wide range of theories from effective action ⌽ III. CONCLUSIONS for superstring models to JBD theories. If BD is interpreted as a homogeneous field which appears as an effective mass term for the Kaluza-Klein modes ͓25͔ in the effective four- We have studied the preheating scenario in the context of 2 ⌽ dimensional Kaluza-Klein action defined as M l ( BD) then, scalar-tensor theories. We have discussed the linear pertur- as mentioned before, the BD-dilaton can be related to the bation theory and showed that the perturbations in BD- radius of the compactified D dimensions if the compactifica- dilaton field grow outside the horizon. The interpretation of tion is done on a D dimensional torus or on sphere. In terms such modes is not very clear at the moment but one may of : hope that by including backreaction in a consistent way may solve this problem. Nevertheless, including perturbations in b the BD-dilaton field introduces new physics well within the ϭ ͫ ͬ ͑ ͒ 0 ln , 25 horizon such as temporal and spatial variation in Newton’s b0 constant. Such perturbations could play an important role during structure formation and it is important to see whether D͑Dϩ2͒ ϭͱ ͑ ͒ such variations in G could be detected or not. In the strong 0 . 26 2 coupling regime fluctuations in field grow faster than the fluctuations in the case of weak coupling ͑JBD theory͒ for If the extra dimensions are compactified on a sphere then the the zero mode and can also excite the Kaluza-Klein modes mass M l( ) of the four-dimensional Kaluza-Klein field lm by lowering the mass of l,m . Such excitations can give rise in terms of to highly noninteracting quanta which can be a very good candidate for the present dark matter. ͑ ϩ Ϫ ͒ We have also discussed the entropy perturbations during l l D 1 Ϫ ϩ 2͑͒ϭ (D 2) / 0 ͑ ͒ M l 2 e . 27 reheating which are solely due to the fact that there is more b0 than one scalar field and the initial conditions in the relative density fluctuations in the respective components are nona- In the usual case, is assumed to be very close to zero after diabatic. In general relativity the reason behind such super inflation and its variation is assumed to be negligible in ra- amplification in the fluctuations is due to the enhancement in diation and matter dominated eras. That leads to very high the entropy perturbations outside the horizon. In our case mass of these modes, very close to Planck scale and propor- such amplification is even stronger especially in the strong tional to the inverse square of the radius of the extra dimen- coupling case because we have an extra field which contrib- sions. During reheating the perturbation in grows exponen- utes its fluctuations to the entropy perturbation. Isocurvature tially in time and that causes fluctuations in the mass of the fluctuations will be generated with a large tilt in string mo- Kaluza-Klein modes. The perturbations effectively give rise tivated and Kaluza-Klein theories provided BD-dilaton is to the Casimir force as a fluctuating boundary condition for treated as a CDM field. the D-dimensional sphere. This in fact can stabilize the po- Here we must point out that we have intentionally ne- tential for the homogeneous scalar field, in our notation glected the potential term coming from the dilaton sector. ⌽ BD . Though we have neglected such potentials in our Such a scheme is possible provided the extra dimensions are analysis since we have concentrated on the compactification compactified on a torus and not on a sphere. Compactifying on a torus, in this case the Kaluza Klein mass scale is still on a sphere gives rise to a term proportional to the curvature inversely proportional to the square of the size of the com- of the sphere and such a term acts as a potential for the pactification radius. If the compactification is done on a dilaton. However, as we noted before, the curvature term sphere and if a potential term for the dilaton is taken into lacks the global minima in the direction of the dilaton field account then the effect of resonance will be even more pro- and one has to invoke the first order Casimir corrections at a nounced. However, the potential lacks a local minima and in one-loop level to give rise to a global minimum. Parametric order to stabilize it one needs to invoke the Casimir effects at excitations of the Kaluza-Klein modes in such a potential a one-loop level. The issue raises a few questions such as have already been discussed in Ref. ͓25͔ but the author has particle creation in the external dimensions due to fluctuating not taken the metric perturbations into account. Inclusion of boundary and their stability requires a detailed study. the dilaton potential may even enhance the metric perturba- If the Kaluza-Klein modes are excited their excitations tion as it acts as an extra source term for Eq. ͑15͒. Concrete may also give rise to some undesirable particles such as the predictions requires a detailed study and it is worth investi- pyrgon ͓26͔, whose energy density decreases as aϪ3, giving gating this point as a separate issue. We must mention that ϳ during the radiation era pyrgon / rad a. Their stringent re- recently there has been an intense discussion upon the initial quirement to decay ͓26͔ can enhance their lifetime and they conditions for the perturbations in the matter and in the BD- can be a candidate for dark matter. Overproduction of such dilaton fields. We have chosen our initial conditions to be particles could have overcome the critical density, but their scale invariant. There have been other proposals such as the number density could have been diluted by invoking a sec- commonly used quantum plane wave initial conditions for
103513-6 PREHEATING IN GENERALIZED EINSTEIN THEORIES PHYSICAL REVIEW D 60 103513 the fluctuations in the fields. This, however, favors small k leave them for future investigation. We also point it out that modes more than the large k modes. Another attractive pro- our analysis has ignored the the contribution coming from ⌽ posal can be evolving the BD-dilaton from 60 e foldings kϭ0 to the background evolution equations. Inclusion of before the end of inflation when aHӶk until the horizon such terms will certainly help to amplify the fluctuations crossing and then subsequently evolving the fields during the even faster and we can expect the fluctuations to become oscillatory phase. These are open issues which require fur- nonlinear before. The aspects of nonlinearity in the gravita- ther investigation. Our analysis was mainly restricted to the tional sector is least understood and the further evolution of k→0 mode but we should also expect the same physical the nonlinear modes will become important leaving an im- consequences for other modes excited during preheating al- print on the cosmic microwave background radiation which though with a much smaller magnitude. will be the ultimate verification for the validity of such The issue of backreaction becomes important after a few claims. inflaton oscillations and the subject becomes almost intrac- table as we increase the number of fields gradually. Backre- ͓ ͔ action has been considered before by many authors 1 , but ACKNOWLEDGMENTS in our situation we have to worry about the contribution coming from the BD-dilaton sector also. A self-consistent A.M. was supported by the Inlaks foundation. L.E.M. is study of the inflaton and the particles should take into supported by FCT ͑Portugal͒ under Contract No. PRAXIS account the effective change in the mass of the inflaton but in XXI BPD/14163/97. We are grateful to Andrew Liddle for this case such corrections will be larger due to the contribu- discussions on various aspects of perturbation theory. We tion from the BD-dilaton sector. Similarly, the BD-dilaton also thank Juan Garcı´a-Bellido and Ian Grivell for stimulat- will also acquire an effective mass. To make a complete ing discussions. L.E.M acknowledges discussions with Al- picture one must answer all these relevant issues and we fredo Henriques and Gordon Moorhouse.
͓1͔ L. A. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. ͓12͔ S. Weinberg, General Relativity and Cosmology ͑Wiley, New Lett. 73, 3195 ͑1994͒; Phys. Rev. D 56, 3258 ͑1997͒;Y.Sh- York, 1972͒. tanov, J. Traschen, and R. H. Brandenberger, ibid. 51, 5438 ͓13͔ B. A. Bassett, D. I. Kaiser, and R. Maartens, Phys. Lett. B 455, ͑1995͒; D. Boyanovsky et al., ibid. 51, 4419 ͑1995͒. 84 ͑1999͒; B. A. Bassett, F. Tamburini, D. I. Kaiser, and R. ͓2͔ J. Baacke, K. Heitmann, and C. Patzold, Phys. Rev. D 58, Maartens, hep-ph/9901319. 125013 ͑1998͒; P. B. Greene and L. Kofman, Phys. Lett. B ͓14͔ F. Finelli and R. Brandenberger, Phys. Rev. Lett. 82, 1362 448,6͑1999͒. ͑1999͒; M. Parry and R. Easther, Phys. Rev. D 59, 061301 ͓3͔ E. W. Kolb, A. Linde, and A. Riotto, Phys. Rev. Lett. 77, 4290 ͑1999͒. ͑1996͒; D. S. Salopek, J. R. Bond, and J. M. Bardeen, Phys. ͓15͔ A. L. Berkin and K. Maeda, Phys. Rev. D 44, 1691 ͑1991͒. Rev. D 40, 1753 ͑1989͒; B. A. Bassett and F. Tamburini, Phys. ͓16͔ T. Damour and B. Pichon, Phys. Rev. D 59, 123502 ͑1999͒;A. Rev. Lett. 81, 2630 ͑1998͒. R. Liddle, A. Mazumdar, and J. D. Barrow, ibid. 58, 083508 ͓4͔ A. L. Maroto and A. Mazumdar, Phys. Rev. D 59, 083510 ͑1998͒. ͑1999͒. ͓17͔ M. Gasperini and M. Giovannini, Phys. Rev. D 47, 1519 ͓5͔ M. Gasperini, M. Giovannini, and G. Veneziano, Phys. Rev. ͑1993͒. Lett. 75, 3796 ͑1995͒. ͓18͔ A. A. Starobinsky and J. Yokoyama, gr-qc/9502002. ͓6͔ M. Giovannini, Phys. Rev. D 56, 631 ͑1997͒. ͓19͔ N. W. McLachlan, Theory and Application of Mathieu Func- ͓7͔ C. Brans and R. H. Dicke, Phys. Rev. 124, 925 ͑1961͒;J.D. tions ͑Dover Publications, New York, 1961͒. Barrow, gr-qc/9711084. ͓20͔ J. M. Bardeen, Phys. Rev. D 22, 1882 ͑1980͒. ͓8͔ C. M. Will, Theory and Experiment in Gravitational Physics ͓21͔ V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, ͑Cambridge University Press, Cambridge, 1993͒. Phys. Rep. 215, 203 ͑1992͒. ͓9͔ R. D. Reasenberg et al., Astrophys. J. Lett. Ed. 234, L219 ͓22͔ P. J. Steinhardt and C. M. Will, Phys. Rev. D 52, 628 ͑1995͒. ͑1979͒. ͓23͔ J. Garcı´a-Bellido and D. Wands, Phys. Rev. D 52, 6739 ͓10͔ C. G. Callan, D. Friedan, and M. J. Pollock, Nucl. Phys. B262, ͑1995͒. 593 ͑1985͒. ͓24͔ H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78,1 ͓11͔ G. Magnano, M. Ferraris, and M. Francaviglia, Gen. Relativ. ͑1984͒; Int. J. Mod. Phys. A 1, 265 ͑1986͒. Gravit. 19, 465 ͑1987͒; A. Jakubiec and J. Kijov, Phys. Rev. D ͓25͔ S. Mukohyama, Phys. Rev. D 57, 6191 ͑1998͒. 37, 1406 ͑1988͒. ͓26͔ E. W. Kolb and R. Slansky, Phys. Lett. 135B, 378 ͑1984͒.
103513-7