Duality and scale invariant magnetic fields from bouncing universes

Recently, we numerically showed that, for a nonminimal coupling that is a simple power of the scale factor, scale invariant magnetic fields arise in a class of bouncing universes. In this work, we analytically evaluate the spectrum of magnetic and electric fields generated in a subclass of such models. We illustrate that, for cosmological scales which have wave numbers much smaller than the wave number associated with the bounce, the shape of the spectrum is preserved across the bounce. Using the analytic solutions obtained, we also illustrate that the problem of backreaction is severe at the bounce. Finally, we show that the power spectrum of the magnetic field remains invariant under a two-parameter family of transformations of the nonminimal coupling function.

Figure 1

The power spectra of the magnetic (in blue) and electric (in red) fields, evaluated before the bounce at $\eta =-\alpha {\eta }_0$ using the analytical expressions (12) and (14), have been plotted as a function of $k/k_0$ for $\gamma =3$, $q=1$, $a_0=8.73\times 10^{10}$, and $\alpha =10^5$. Note that the dimensionless quantities ${\mathcal{P}}_{\mathrm{B}}(k)/k_0^4$ and ${\mathcal{P}}_{\mathrm{E}}(k)/k_0^4$ that we have plotted depend only on the combination $k/k_0$. Also, we should mention that our analytical approximations are valid only for scales such that $k\ll k_0/\alpha$. Over this domain, while the spectrum of the magnetic field is strictly scale invariant, the spectrum of the electric field behaves as $k^{-2}$, as is suggested by the spectra (15) and (17) arrived at from the asymptotic forms of the Bessel functions. Needless to add, the question of interest is whether these power spectra will retain their shape after the bounce.

Figure 2

The behavior of ${\eta }_0^2{J}^{\prime \prime }/J$, which depends only on $\eta /{\eta }_0$, has been plotted for $\gamma =3$ (in blue) and $\gamma =5$ (in red). The figure has been plotted over a very narrow range of $\eta /{\eta }_0$ in order to illustrate the presence of a single maximum for $\gamma =3$ and two maxima and one minimum for $\gamma =5$.

Figure 3

The power spectra of the electric (in red) and magnetic (in blue) fields, evaluated at $\eta =\beta {\eta }_0$, with $\beta =10^2$, have been plotted for the same set of values of the parameters as in Fig. 1. Note that the shape of the spectra generated before the bounce is retained for scales such that $k\ll k_0/\alpha$ even after the bounce. We should mention that the values of the parameters we have worked with lead to magnetic fields of observed strengths today corresponding to a few femtogauss. It should also be added that the electric field dominating the strength of the magnetic field is considered to be undesirable (see, for instance, Ref. [36]). This seems inevitable for positive $n$ that we are considering here, but it can be, for example, circumvented by choosing $n$ to be negative (in this context, see Ref. [72]).

Figure 4

The evolution of the ratio of the background energy density to the sum of the energy densities in the electric and magnetic fields for a given mode ($k={10}^{-20}{k}_0$) has been plotted against $e\text{−}\mathcal{N}$-folds. We should mention that we have assumed the same set of values for the various parameters as in Figs. 1 and 3. Evidently, the ratio has to remain large in order to avoid the backreaction problem. However, we find that the energy in the generated electromagnetic field rises sharply as one approaches the bounce, indicating that the problem of backreaction is the most severe at the bounce.

Figure 5

The coupling function $J$ (in blue) and its dual $\tilde{J}$ (in red) have been plotted as a function of $\eta /{\eta }_0$ for $\gamma =3$ and ${\eta }_{*}\to -\infty$ [cf. Eq. (33)]. Also, we have chosen the constant $C$ to be $C/k_0=5.7\times 10^{32}$ so that the dual function $\tilde{J}$ matches the original coupling function $J$ after the bounce.