arXiv:astro-ph/9902380v1 26 Feb 1999 QUANTUM CORRECTIONS TO MAXWELL ELECTRODYNAMICS IN A HOMOGENEOUS AND ISOTROPIC UNIVERSE WITH COSMOLOGICAL CONSTANT M. R. de Garcia Maia * , J. C. Carvalho † , and C. S. da Cˆamara Neto ‡ Departamento de F´ısica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal RN Brazil Some cosmological consequences of first order quantum corrections to Maxwell electrodynamics are investigated in the context of a spatially flat homogeneous and isotropic universe driven by a magnetic field plus a cosmological term Λ. The introduction of these quantum corrections may provide a more realistic model for the universe evolution. For a vanishing Λ, we derive the general solution corresponding to the particular one recently found by Novello et al. [gr-qc/9806076]. We also find a general solution for the case when Λ is a non-vanishing constant. Both solutions describe a non-singular, bouncing universe that begins arbitrarilly large, contracts to a minimum non-zero size amin and expands thereafter. However, we show that the first order correction to the electromagnetic Lagrangean density, in which the analysis is based, fails to describe the dynamics near amin, since, at this point, the magnetic fields grows beyond the maximum strength allowed by the approximation used (B ≪ 8.6 × 10 -7 Tesla = 0.0086 Gauss). The time range where the first order approximation can be used is explicitly evaluated. These problems my be circumvented through the use of higher order terms in the effective Lagrangean, as numerical calculations performed by Novello et al. [gr- qc/9809080], for the vanishing Λ case, have indicated. They could also be evaded in some models based on oscillatory behaviour of the fundamental constants. A third general solution corresponding to a constant magnetic field sustained by a time dependent Λ is derived. The temporal behaviour of Λ is univocally determined. This latter solution is capable of describing the whole cosmic history and describes a universe that, although with vanishing curvature (K = 0), has a scale factor that approaches zero asymptotically in the far past, reaches a maximum and then contracts back to an arbitrarilly small size. The cosmological term decays during the initial expansion phase and increases during the late contraction phase, so as to keep B constant throughout. An important feature of this model is that it presents an inflationary dynamics except in a very short period of time near its point of maximum size. PACS number(s): 04.40.Nr, 12.20.-m, 98.80.Cq, 98.80.-k I. INTRODUCTION In a recent paper, Novello and colaborators [1] have analysed the cosmological consequences of quantum corrections to Maxwell electrodynamics. They have considered the quantum effects leading to the production of electron-positron pairs that were first derived by Heisenberg and Euler [2]. The analysis was a semiclassical one (quantum field in a classical general relativistic geometry), made in first order on the effective Lagrangean density (weak field limit) and applied to a spatially flat Friedmann-Robertson-Walker (FRW) model. The more interesting result found in Ref. [1] was the removal of the primordial singularity due to the appearance of a negative pressure in the early stages of the universe. The analytical solution derived shows that the energy density associated with the electromagnetic field vanishes at the point where the scale factor reaches its minimum. The non-singular behaviour of the model is unaffected by the presence of other types of ultrarelativistic matter obeying the equation of state p (ur) = ρ (ur) /3. In a subsequent paper [3], the analysis was extended beyond the first order approximation and the numerical solutions obtained show that the non-singular behaviour is preserved. The conclusion reached by the authors of references [1,3] was that the cosmological singularity of FRW models is a distinguished feature of classical electrodynamics. This problem is overcome when quantum corrections are taken into account, leading to a more realistic description of the universe. In the present paper we extend the analysis developed in [1]. We begin by showing that the analytical solution obtained in that paper is a particular one and write down the corresponding general solution. We then analyse how * Electronic address: mrgm@dfte.ufrn.br † Electronic address: carvalho@dfte.ufrn.br ‡ Electronic address: calistrato@dfte.ufrn.br 1