Constraints on inflation in the Einstein-Brans-Dicke frame

The successful explanation of cosmological puzzles, such as the horizon, flatness, and monopole problems, is achieved by the inflationary scenarios. The inflationary scenarios also predict the spectrum of the density perturbation which seeds the formation of the large scale structure of the Universe. The basic assumption is that, at the early times, the Universe experienced an accelerated expansion, while the Hubble radius changed very slowly. Consequently the wavelength of a quantum fluctuation soon exceeds the Hubble radius. The amplitude of the fluctuation is frozen after the horizon crossing. After the end of inflation, the Hubble radius increases faster than the scale factor; so the fluctuations eventually reenter the Hubble radius during the radiation-dominated (RD) or matter-dominated (MD) eras. The original model, the so-called “old inflation” model [1], is based on the first-order phase transition. It failed because of the “graceful exit” problem. Soon after the “new” and “chaotic” inflationary models were proposed to solve this problem [2]. These models use a simple scalar field as the matter source. The scalar field (inflaton field) slow-rolls down the potential during the inflationary phase. All the above models are based on Einstein gravity. However, inflation may be driven by non-Einstein gravity. “Extended inflation” employs Jordan-Brans-Dicke (JBD) gravity [3]. The introduction of the Brans-Dicke (BD) field slows down the inflation and solves the graceful exit problem. But it was soon found that there was a “big bubble” problem for the original extended inflation [4]. The interaction between the BD field and the inflaton field can change the spectrum of the density perturbation. The spectra of density perturbations were analyzed in the extended new and chaotic inflationary models by several authors [5]. But the density perturbations given by those papers are not correct [6]. The correct density perturbation in BD inflation was given in [6] . In this paper, I use the slow-roll approximation and work with BD gravity in the Einstein frame (let us call it Einstein-Brans-Dicke gravity). There are strong arguments identifying the Jordan frame 2 as the physical one. The possibility of identifying the Einstein frame as the physical one was first raised by Cho [8]. Cho [8] and Damour and Nordtvedt [9] pointed out that only in the Einstein frame does the Pauli metric represent the massless spin-2 graviton and the scalar field represent the massless spin-0 field. In the Jordan frame the graviton is described by both the Pauli metric tensor and the BD scalar field. Cho also pointed out that in the compactification of Kaluza-Klein theory, the physical metric must be identified as the Pauli metric because of the wrong sign of the kinetic term of the scalar field in the Jordan frame. In string theory, the dilaton field appears naturally. The Einstein frame is greatly favored over the string frame although the string frame is chose for the pre-big-bang cosmology. For further discussions about the two frames, see [10] and references therein. Because the dilaton field evolves very slowly during the RD and MD eras, we then assume that the dilaton field at the end of inflation takes the same value as that at present. By this assumption, we can fix the value of the inflaton field at the beginning of inflation. We find that the results are different from those in [6]. Inflationary models based on general scalar-tensor gravity in the Einstein frame were also discussed in [11] and [12]. The BD Lagrangian in Einstein frame is