No compelling cosmological models come out of magnetic universes which are based on nonlinear electrodynamics

Here we investigate the cosmic dynamics of Friedmann-Robertson-Walker universes—flat spatial sections—which are driven by nonlinear electrodynamics (NLED) Lagrangians. We pay special attention to check the sign of the square sound speed since, whenever the latter quantity is negative, the corresponding cosmological model is classically unstable against small perturbations of the background energy density. Besides, based on causality arguments, one has to require that the mentioned small perturbations of the background should propagate at most at the local speed of light. We also look for the occurrence of curvature singularities. Our results indicate that several cosmological models which are based in known NLED Lagrangians, either are plagued by curvature singularities of the sudden and/or big rip type, or are violently unstable against small perturbations of the cosmological background—due to the negative sign of the square sound speed—or both. In addition, causality issues associated with superluminal propagation of the background perturbations may also arise.

Figure 1

In the left-hand panel a plot of $F$—solid curve—and of the square sound speed $c_s^2$—dashed curve—vs the cosmic time $t$ is shown for an arbitrarily chosen $\alpha =0.01$, for the model (5) [$\mathcal{L}=-F/4+\alpha F^2$]. For the same theory a plot of $X=\{{\rho }_{\mathrm{B}}(t)—\text{solid},H(t)—\text{dash},\dot{H}(t)—\text{dash dot},a(t)—\text{dots}\}$ is shown in the right-hand panel. Occurrence of a bounce at $t=0$ is evident from this latter figure.

Figure 2

Plots for the model (6) [$\mathcal{L}=-F/4-\gamma /F$] for an arbitrarily chosen value of the parameter $\gamma =0.1$. In the left-hand panel the plot of the cosmic time $t$ vs the scale factor $a$ is shown, while in the right-hand panel we have plotted $X=\{F(a)—\text{solid},{\rho }_{\mathrm{B}}(a)—\text{dash},\dot{H}(a)—\text{dash dot},c_s^2—\text{dots}\}$.

Figure 3

Plot of the time duration of the cosmic history $\mathrm{\Delta }t$ vs the free parameter $\gamma$, for the model (6) [$\mathcal{L}=-F/4-\gamma /F$]. Only the parameter interval $\gamma \in [0.001,1]$ is shown.

Figure 4

Plot of the energy density of the magnetic field ${\rho }_{\mathrm{B}}$ vs $F$—top panels—and of the square sound speed $c_s^2(F)$—bottom panels—for arbitrarily chosen $(\alpha ,\gamma )$ [(0.01, 0.1)—left-hand panels, (0.1, 0.1)—center panels, and (0.1, 0.01)—right-hand panels] for the different NLED models. The solid curve is for the model given by the Lagrangian density (5), while the dashed curve is for the model (6), and the dash dot is for the hybrid model which is based in the Lagrangian density $\mathcal{L}$ of Eq. (7).

Figure 5

Plots of the parametric pressure of the magnetic field $p_{\mathrm{B}}$ vs $F$, for the models (3)—solid curve, and (4)—dashed curve, for chosen values of the free parameters $({\alpha }^2,{\gamma }^2)=(0.01,1)$. In the right-hand panel only $p_{\mathrm{B}}=p_{\mathrm{B}}(F)$ for the model (3) is plotted. For the model (4), since $0\le F\le 1/2{\alpha }^2{\gamma }^2=50$, at the upper bound of the $F$ invariant ($F=50$), the pressure is a finite (negative) quantity. As shown in the right-hand panel, for the model (3), since $0\le F\le F_{+}=51.9258$, as one approaches to $F_{+}$ the absolute value of the (negative) pressure $|p_{\mathrm{B}}|$ unboundedly grows up.

Figure 6

Plots of the energy density of the magnetic field ${\rho }_{\mathrm{B}}$—left-hand panel—for the models (3)—solid curve, and (4)—dashed curve, and of the square sound speed $c_s^2$ (exactly the same for both models)—right-hand panel—for arbitrarily chosen values of the free parameters $({\alpha }^2,{\gamma }^2)=(0.01,1)$. It is seen that, at $F=F_c$, where the energy density of the magnetic field is a maximum, the square sound speed has a vertical asymptote. To the left of the asymptote there is an interval $\mathrm{\Delta }F$, where $c_s^2$ is a negative quantity. Besides, as one approaches to $F_c$ from the left, $c_s^2\to -\infty$. Between the asymptotes $F_c<F<F_{+}$, the square speed of sound $c_s^2>1$.