Generation of scale invariant magnetic fields in bouncing universes

Abstract

Abstract. We consider the generation of primordial magnetic fields in a class of bouncing models when the electromagnetic action is coupled non-minimally to a scalar field that, say, drives the background evolution. For scale factors that have the power law form at very early times and non-minimal couplings which are simple powers of the scale factor, one can easily show that scale invariant spectra for the magnetic field can arise before the bounce for certain values of the indices involved. It will be interesting to examine if these power spectra retain their shape after the bounce. However, analytical solutions for the Fourier modes of the electromagnetic vector potential across the bounce are difficult to obtain. In this work, with the help of a new time variable that we introduce, which we refer to as the e-N -fold, we investigate these scenarios numerically. Imposing the initial conditions on the modes in the contracting phase, we numerically evolve the modes across the bounce and evaluate the spectra of the electric and magnetic fields at a suitable time after the bounce. As one could have intuitively expected, though the complete spectra depend on the details of the bounce, we find that, under the original conditions, scale invariant spectra of the magnetic fields do arise for wavenumbers much smaller than the scale associated with the bounce. We also show that magnetic fields which correspond to observed strengths today can be generated for specific values of the parameters. But, we find that, at the bounce, the backreaction due to the electromagnetic modes that have been generated can be significantly large calling into question the viability of the model. We briefly discuss the implications of our results.

Cross-correlations between scalar perturbations and magnetic fields in bouncing universes

“…Therefore, the solution has to be utilized to evaluate the power spectrum of the magnetic field in the domain of its validity. Moreover, the numerical analysis of the modes suggest that the analytical solutions that we have obtained would be invalidated well before the condition k 2 = J ′′ /J is satisfied after the bounce [64]. For these reasons, we shall choose to evaluate the power spectrum after the bounce at η = β η 0 , with β chosen to be about 10 2 .…”

“…Under these conditions, we have J ′′ /J ≃γ (γ − 1)/η 2 , which is of the same form as in the inflationary case that we had discussed in the previous sub-section. Therefore, in the first domain, we can express the electromagnetic modesĀ k in terms of the Hankel function H (1) ν (x) as follows [64,65]:…”

“…In these models, the universe goes through a period of contraction followed by an expanding phase, with a 'bounce' connecting the two epochs. Lately, there have been some efforts towards understanding the generation of magnetic fields in the bouncing scenarios [62][63][64][65][66][67][68].…”

Viable tensor-to-scalar ratio in a symmetric matter bounce

“…In order to describe the completely symmetric bouncing universe of our interest, we shall introduce a new time variable N , in terms of which the scale factor is given by [65,66] …”

“…We shall also work with a new time variable that we have introduced in an earlier work on bouncing scenarios, viz. e-N -folds, which we denote as N [65,66].…”