Stability issues in the modified starobinsky model

Abstract

We discuss the stability of the anomaly-induced inflation (modified Starobinsky model) with respect to the arbitrary choice of initial data and with respect to the small perturbations of the conformal factor and tensor modes of the metric in the later period of inflation and, partially, in the present Universe.The basic principles of the anomaly-induced inflation has been explained in [ 1] (see also references and notations therein).The advantage of this inflationary model is that it requires smaller amount of the cosmological phenomenology than the usual inflaton models. To some extent, this model is mainly based on the principles of quantum field theory. In particlar, the simplest inflationary solution follows from the anomaly-induced quantum correction to the vacuum actionΓ[g µν ] which is a solution of the equationThe coefficients w, b, c are defined in [ 1], C 2 is a square of the Weyl tensor and E is an integrand of the Gauss-Bonnet topological term. This equation admits an explicit solution [ 2]. It is easy to see that this solution contains an ambiguity because an arbitrary conformal functional S c [g µν ] plays the role of the "integration constant" for the Eq. (1). However, this "integration constant" does not affect the equation for the conformal factor of the metric and in this respect the initial inflationary solution follows from the exact effective action. The tempered form of expansion which is observed at the later inflationary phase is not exact, but it has quite a robust background.The stability of the inflationary solution from the initial stage until the graceful exit and the stability of the classical solution in the theory with loop corrections represent a strong consistency test of the model. Let us start from the initial stage of inflation, when the particle content N 0 , N 1/2 N 1 (number of scalar, fermion and vector fields) of the theory provides the stabilityof the exponential solutionof Starobinsky [ 3]. According to [ 4], the condition (2) is independent on the cosmological constant and on the choice of the metric k = 0 or k = ±1. For the sake of simplicity we consider k = 0 and also assume that the cosmological constant is small during inflation, such that the last is driven by the quantum effects only. The original Starobinsky model deals with the unstable case. The initial data are chosen very close to the exponential solution (3) such that the inflation lasts long enough. Using the 0-0 component of the Einstein equations with quantum correction, Starobinsky constructed the phase diagram of the theory. This phase diagram represent several distinct attractors, FRW behaviour is one of them and others represent physically unacceptable run-away type solutions. In the modified version of the model [ 1], the inflation starts in the stable phase (2). In this case the phase diagram, dual to the one of [ 3], has the form shown at the Figure1. This phase portrait indicates to the unique stable solution (3). Therefore the anomaly-induced inflation does not depend on the choice of initial d...

From stable to unstable anomaly-induced inflation

“…(9) may have different signs, depending on the particle content of the theory. For small perturbations σ (t) → σ (t) + δσ (t) around inflationary exponential solution, it is easy to check that it is stable for c > 0 and unstable for c < 0 [15,43,44]. Assuming (9), the stability condition c > 0 boils down to the relation…”

“…Equation (16) has important particular solutions with constant Hubble parameter [15,16] (for = 0 the solution was obtained in [43,44]),…”

“…A relevant question is why the energy scale of stable inflation is decreasing, such that the gravitational decoupling could take place. The solution to this problem has been suggested in [42][43][44]. The stable anomaly-induced inflation is due to the quantum effects of massless conformal fields, and is strictly exponential, such that the Hubble parameter is constant.…”

Effect of the trace anomaly on the cosmological constant

“…If one includes a cosmological constant, the theory of anomaly induced inflation is plagued by instabilities, which we will also come to address. The Modified Starobinsky Model as advocated by [41,42,43], takes advantage of these instabilities to account for a graceful exit from inflation. It is argued that supersymmetry breaking changes the degrees of freedom such that it destabilises the quantum anomaly driven attractor and simultaneously stabilises the classical de Sitter attractor.…”

“…In the literature (see e.g. [43]), one only has considered an empty Universe with a cosmological term and trace anomaly. In reality, one solved for a radiation dominated Universe, for only radiation (ω = 1/3) does not contribute to the trace of the Einstein field equations.…”