View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server MRI-PHY/96-35 hep-th/9611223 Early Universe Evolution in Graviton-Dilaton Mo dels S. Kalyana Rama Mehta Research Institute, 10 Kasturba Gandhi Marg, Allahabad 211 002, India. email: [email protected] ABSTRACT. We present a class of graviton-dilaton mo dels which leads to a singularity free evolution of the universe. We study the evolution of a homogeneous isotropic universe. We follow an approach which enables us to analyse the evolution and obtain its generic features even in the absence of explicit solutions, which are not p ossible in general. We describ e the generic evolution of the universe and show, in particular, that it is singularity free in the present class of mo dels. Such mo dels may stand on their own as interesting mo dels for singularity free cosmology, and may be studied accordingly. They may also arise from string theory. We discuss critically a few such p ossibilities. 1 1. Intro duction General Relativity (GR) is a b eautiful theory and, more imp ortantly, has b een consistently successful in describing the observed universe. But, GR leads inevitably to singularities. In particular, our universe according to GR starts from a big bang singularity. We do not, however, understand the physics near the singularity nor know its resolution. The inevitability and the lack of understanding of the singularities p oint to a lacuna in our fundamental understanding of gravity itself. By the same token, a successful resolution of the singularities is likely to provide deep insights into gravity. In order to resolve the singularities, it is necessary to go beyond GR, p erhaps to a quantum theory of gravity. A leading candidate for such a theory is string theory. However, despite enormous progress [1], string theory has not yet resolved the singularities [2]. Perhaps, one may have to await further progress in string theory, or a quantum theory of gravity, to b e able to resolve singularities. However, there is still an avenue left op en - resolve the singularities, if p ossible, within the context of a generalised Brans-Dicke mo del, and then inquire whether such a mo del can arise from string theory, or a quantum theory of gravity. Generalised Brans-Dicke mo del is sp ecial for more than one reason. It app ears naturally in sup ergravity, Kaluza-Klein theories, and in all the known e ective string actions. It is p erhaps the most natural extension of GR [3], whichmay explain its ubiquitous app earance in fundamental theories: Dicke had discovered long ago the lack of observational evidence for the principle of strong equivalence in GR. Together with Brans, he then constructed the Brans-Dicke theory [4] which ob eys all the principles of GR except that of strong equivalence. This theory, generalised in [5], has b een applied in many cosmological and astrophysical contexts [6]-[10]. Brans-Dicke theory contains a graviton, a scalar (dilaton) coupled non minimally to the graviton, and a constant parameter ! . GR is obtained when ! = 1. In the generalised Brans-Dicke theory, referred to as graviton- dilaton theory, ! is an arbitrary function of the dilaton [3, 5]. Hence, it includes an in nite number of mo dels, one for every function ! . Thus, very likely, it also includes all e ective actions that may arise from string theory or a quantum theory of gravity. This p ersp ective logically suggests an avenue in resolving the singularities: to nd, if p ossible, the class of graviton-dilaton mo dels in which the evolution 2 of universe is singularity free, and then study whether such mo dels can arise from string theory or a quantum theory of gravity. On the other hand, even if not deriveable from string theory or a quantum theory of gravity, such mo dels may stand on their own as interesting mo dels for singularity free cosmology. In either case, one may study further implications of these mo dels in other cosmological and astrophysical contexts. Such studies are fruitful and are likely to lead to novel phenomena, providing valuable insights. Accordingly, in this pap er, we consider a class of graviton-dilaton mo dels where the function ! satis es certain constraints. These constraints were originally derived in [11] in a di erent context, and some of their generic cosmological and astrophysical consequences were explored in [11, 12]. Here we analyse the evolution of universe in these mo dels which, as we will show, turns out to b e singularity free. The essential p oints of the analysis and the results have b een given in a previous letter [13]. We consider a homogeneous isotropic universe, such as our observed one, containing matter. The relevent equations of motion cannot, in general, be solved explicitly except in sp ecial cases [10, 12]. Hence, a di erent approach is needed for the general case which is valid for any matter and for any arbitrary function ! , and which enables one to analyse the evolution and obtain its generic features even in the absence of explicit solutions. We will present such an approach b elow. We rst present a general anal- ysis of the evolution and then apply it to describ e in detail the evolution of universe in the present mo del. We show, in particular, that the constraints on ! ensure that the evolution is singularity free. The universe evolves with no big bang or any other singularity and the time continues inde nitely into the past and the future. An imp ortant question to ask, from our p ersp ective, is whether a function ! as required in the present mo del, can arise from a fundamental theory. We discuss critically a few such p ossibilities in string theory. On the other hand, the present mo del may stand on its own as an interesting mo del for singularity free cosmology and may b e studied accordingly. We mention some issues for future study. This pap er is organised as follows. In section 2, we present our mo del. In section 3, we write its equations of motion in various forms, so as to facilitate our analysis. In section 4, we present the general analysis of the evolution. In section 5, we illustrate our metho d by applying the results of section 4 to describ e the evolution of toy universes and show that their evolution is 3 singularity free in the present mo del. In section 6, we describ e the evolution of our observed universe and show that its evolution is singularity free in the present mo del. In section 7, we discuss further generalisations of our mo del. In section 8, we give a brief summary, discuss critically a few p ossibilities of our mo del arising from string theory, and mention some issues for future study. In the App endix, we derive the suciency conditions for the absence of singularities which are used in the pap er. 2. Graviton-Dilaton Mo del 1 We consider the following graviton-dilaton action in `Einstein frame': Z p 1 1 4 2 () S = d x g R + (r) + S (M;e g ) ; (1) M 16G 2 N where G is the Newton's constant, is the dilaton and () is an arbitrary N function that charcterises the theory. S is the action for matter elds, M denoted collectively by M. They couple to the metric g minimally and to the dilaton through the function (). This function cannot b e gotten rid of by a rede nition of g except when the matter action S is conformally M () invariant - in that case, by de nition, S (M;e g ) = S (M; g ) - M M which is assumed to b e not the case here. We can de ne a metric g = e g ; (2) and write the action (1) in `Dicke frame': Z p 1 1 2 2 4 S = R + + S (M;g ) ; (3) ge 3 1 (r) d x M 16G 2 N d where . If has a nite upp er b ound < 1, then the max d max factor e can be absorb ed into G and the range of can be set to be N max e 0. Also, we will work in the units where G = . N 8 Both forms of the action in (1) and (3) are, however, equivalent [14]. In `Einstein frame', the matter elds feel, b esides the gravitational force, the dilatonic force also which must be taken into account in obtaining the 2 @ g 1 1 In our notation, the signature of the metric is ( + ++) and R = + . 2 @x @x 4 physical quantities. Whereas, in `Dicke frame', the matter elds feel only the gravitational force and, hence, the physical quantities are directly obtained from the metric g . On the other hand, equations of motion are often easier to solve if g is used. De ning a new dilaton eld = e , the action (3) can be written as ! Z p 1 ! () 4 2 d x S = g R + (r) + S (M; g ) ; (4) M 16G N where ! ()isnow the arbitrary function that characterises the theory. This is the form of the action used more commonly and, hence, we will also use it in this pap er in most of what follows. Note that 0 since = e . Furthermore, if has a nite upp er b ound < 1, then the factor max can b e absorb ed into G and the range of can b e set to b e 0 1. max N max Also, we will work in the units where G = .