On the Dynamics of a Quadratic Scalar Field Potential

Abstract

We review the attractor properties of the simplest chaotic model of inflation, in which a minimally coupled scalar field is endowed with a quadratic scalar potential. The equations of motion in a flat Friedmann-Robertson-Walker universe are written as an autonomous system of equations, and the solutions of physical interest appear as critical points. This new formalism is then applied to the study of inflation dynamics, in which we can go beyond the known slow-roll approximation.

Attractor dynamics of inflationary monomial potentials

“…(8), (9), (10), (11), and the definition (12) with the dynamical system of the quadratic model, V (φ ) = (1/2)m 2 φ 2 , presented in Ref. [9], we explicitly see the formal equivalence of all monomial potentials with even powers.…”

“…[9]. As we shall show, when we write the evolution equations as a dynamical system we can find a formal equivalence of all models and also the trajectories that entail useful physical information.…”

“…If we take x = cos(θ | s n ) and y = in(θ )| 1/ , then the equations of motion (8), (9), and (10), can be written as…”

“…However, sometimes the equations of motion result into a non-autonomous dynamical system, and then the properties of the dynamics are more involved and difficult to study. An example is the dynamical system arising from the inflationary model of a scalar field endowed with a quadratic potential [9]. The dynamics in this model was not only classified in terms of the critical points, but also in terms of trajectories of physical interest.…”

A quantum mechanics glimpse to standard cosmology

“…As one can see, the scalar field potential derived from the Morse potential turns out to be λ φ 4 − 32 2 3 λ φ . Surprisingly, it resemble a typical polinomial scalar field used for inflation; and as we show in table 1 the SHO potential, which is one of the most useful and basic potentials of QM, transforms into one of the most useful potentials in this cosmological model, λ φ 2 (see [3], [16], and references there in.) It is even more surprising that other typical QM potentials resemble typical scalar field potentials in standard cosmology; for example, compare the results in [10] and [17] to the ones obtained in Table 1.…”