ep - t h / 03 04 04 8 v 2 2 2 M ay 2 00 3 Cosmological Scaling Solutions and Multiple Exponential Potentials

We present a phase-space analysis of cosmology containing multiple scalar fields with positive and negative exponential potentials. In addition to the well-known assisted inflationary solutions, there exist power-law multi-kinetic-potential scaling solutions for sufficiently flat positive potentials or steep negative potentials, which are the unique late-time attractor whenever they exist. We briefly discuss the physical consequences of these results. e-mail address: [email protected] e-mail address: [email protected] Scalar field cosmological models are of great importance in modern cosmology. The dark energy is attributed to the dynamics of a scalar field, which convincingly realizes the goal of explaining present-day cosmic acceleration generically using only attractor solutions [1]. A scalar field can drive an accelerated expansion and thus provides possible models for cosmological inflation in the early universe [2]. In particular, there have been a number of studies of spatially homogeneous scalar field cosmological models with an exponential potential. There are already known to have interesting properties; for example, if one has a universe containing a perfect fluid and such a scalar field, then for a wide range of parameters the scalar field mimics the perfect fluid, adopting its equation of state [3]. These scaling solutions are attractors at late times [4]. The inflation [5, 6] and other cosmological effect [7] of multiple scalar fields have also been considered. The scale-invariant form makes the exponential potential particularly simple to study analytically. There are well-known exact solutions corresponding to power-law solutions for the cosmological scale factor a ∝ t in a spatially flat Friedmann-Robertson-Walker (FRW) model [8], but more generally the coupled Einstein-Klein-Gordon equations for a single field can be reduced to a one-dimensional system which makes it particularly suited to a qualitative analysis [9, 10]. Recently, adopting a system of dimensionless dynamical variables [11], the cosmological scaling solutions with positive and negative exponentials has been studied [12]. Usually there are many scalar fields with exponential potentials in supergravity, superstring and the generalized Einstein theories, thus multi potentials may be more important. In this paper, we will consider multiple scalar fields with positive and negative exponential potentials. We have assumed that there is no direct coupling between the exponential potentials. The only interaction is gravitational. A phase-space analysis of the spatially flat FRW models shows that there exist cosmological scaling solutions which are the unique late-time attractors, and successful inflationary solutions which are driven by multiple scalar fields with a wide range of each potential slope parameter λ. We start with more general model with m scalar fields φi, in which each has an identicalslope potential Vi(φi) = V0i exp (−λκφi) (1) where κ ≡ 8πGN is the gravitational coupling and λ is a dimensionless constant characterising the slope of the potential. Note that there is no direct coupling of the fields, which influence each other only via their effect on the expansion. The evolution equation of each scalar field for a spatially flat FRW model with Hubble parameter H is φ̈i + 3Hφ̇i + dVi(φi) dφi = 0 (2) subject to the Friedmann constraint H = κ 3 m