On classical anisotropies in models of open inflation

In the simplest model of open inflation there are two inflaton fields decoupled from each other. One of them, the tunneling field, produces a first stage of inflation which prepares the ground for the nucleation of a highly symmetric bubble. The other, a free field, drives a second period of slow roll inflation inside the bubble. However, the second field also evolves during the first stage of inflation, which to some extent breaks the needed symmetry. We show that this generates large supercurvature anisotropies which, together with the results of Tanaka and Sasaki, rule out this class of simple models (unless, of course, Ω0 is sufficiently close to one.) The problem does not arise in modified models where the second field does not evolve in the first stage of inflation. Open inflation has recently received some attention as a viable model for the early universe [1]–[5]. The first version of the open universe model was based on the theory of one scalar field [1, 2]. However, no realistic models of that type have been proposed so far. Then Linde and Mezhlumian proposed a class of models involving two scalar fields [3]. The simplest model discussed in [3] describes a tunneling field σ, responsible for bubble nucleation, and a free field φ of mass m that undergoes slow rollover. The two fields are decoupled from each other, except of course gravitationally. When the field σ is in its false vacuum, it dominates the energy density and the universe is in a de Sitter phase with constant Hubble rate H1. The true vacuum of the the field σ has vanishing energy density, and when a bubble nucleates, the slowly rolling field φ drives a second period of inflation in its interior, with hubble rate H2 = (m φ). In order for this model to work, the field φ has to evolve very slowly outside the bubble. Otherwise the surfaces of constant φ would not be well synchronized with the hyperboloids of constant σ inside the bubble, and large anisotropies could be expected. The danger of this effect was already realised in [3], therefore Linde and Mezhlumian also suggested the possible modifications of the simplest model where synchronization was exact and no such problems appear. However since the simplest model looks more natural, it would still be interesting to clarify whether it is compatible with observations for some range of the parameters or not. Outside the bubble of the field σ, we have φ(x, t) ≈ φs ≡ Ae −αH1 , (1) where t̂ is the cosmological time in the flat Friedmann Robertson Walker (FRW) chart, and α = m2/3H1 2 << 1 is the slow-rollover parameter. We shall now concentrate on fluctuations which will arise because of the t̂ dependence of φ, so we take A to be constant over the region of interest. The true and false vacua for the field σ are strongly non-degenerate, which means we are in the thick wall regime.. For simplicity, we shall take the solution (1) to be valid everywhere outside the forward light-cone from the nucleation event. This approximation is clearly valid if the bubble size at the moment of nucleation is small compared with H 1 . Inside the light-cone, the spacetime can be covered with the open chart ds = −dt + a(t)[dr + sinh r(dθ + sin θdφ)]. (2) The scale factor a obeys the Friedmann equation