Cosmological Scaling Solutions of the Tachyon with Multiple Inverse Square Potentials

We investigate cosmological dynamics of multiple tachyon fields with inverse square potentials. A phase-space analysis of the spatially flat FRW models shows that there exists a unique power-law cosmological scaling solution. We study the stability of the solution and find that the solution is a global attractor and a successful inflationary solution which is driven by multiple tachyon fields at lower-Planck energy densities. We briefly discuss the physical consequences of these results. PACS number(s): 98.80.Cq, 04.50.+h ∗e-mail address: [email protected] Cosmological inflation has become an integral part of the standard model of the universe. Apart from being capable of removing the shortcomings of the standard cosmology, the paradigm seems to have gained a fairly good amount of support from the recent observations on microwave background radiation. On the other hand, there have been difficulties in obtaining accelerated expansion from fundamental theories such as string/M-theory. Recently, Sen [1] has constructed a classical time-dependent solution which describes the decay process of an unstable D-brane in the open string theory. During the decay process the tachyon field on the brane rolls down toward the minimum of the potential. The tachyon field might be responsible for cosmological inflation at the early epochs due to tachyon condensation near the top of the effective potential [2], and could contribute to some new form of cosmological dark matter at late times. Several authors have investigated the process of rolling of the tachyon in the cosmological background [3] and in the braneworld scenario [4]. The stability of tachyonic inflation against changes in initial conditions has been studied for an exponential potential [5] and for the inverse power-law potential [6]. Exact solutions for a purely tachyon field with an inverse square potential are known [7], but no solutions exist for multiple tachyon fields, so a dynamical systems approach may be relevant. Interestingly, the inverse square potential plays the same role for tachyon fields as the exponential potential does for standard scalar fields [8]. The potential allows constructing an autonomous system, which gives power-law solutions. Recently, it was pointed out that the tachyon model of inflation is hard to be consistent with observations [9], unless there is a large number of coincident branes [10]. In this letterr, we will consider cosmological dynamics of multiple tachyon fields with inverse square potentials. We have assumed that there is no direct coupling between the inverse square potentials. The only interaction is gravitational. A phase-space analysis of the spatially flat FRW models shows that there exist a cosmological scaling solution which is the unique late-time attractor, and a successful inflationary solution which is driven by multiple tachyon fields at lower-Planck energy densities. We start with more general model with m tachyon fields φi, in which each has an inverse square potential Vi(φi) = V0iφ −2 i . (1) Note that there is no direct coupling of the tachyon fields, which influence each other only via their effect on the expansion. The evolution equation of each tachyon field for a spatially flat FRW model with Hubble parameter H is φ̈i 1− φ̇i 2 + 3Hφ̇i + 1 Vi(φi) dVi(φi) dφi = 0, (2)