Abstract
The cosmological model consisting of a nonlinear magnetic field obeying the Lagrangian \(\mathcal {L}= \gamma F^{\alpha },\, F\) being the electromagnetic invariant, coupled to a Robertson-Walker geometry is tested with observational data of Type Ia Supernovae, Long Gamma-Ray Bursts and Hubble parameter measurements. The statistical analysis show that the inclusion of nonlinear electromagnetic matter is enough to produce the observed accelerated expansion, with not need of including a dark energy component. The electromagnetic matter with abundance \(\varOmega _B\), gives as best fit from the combination of all observational data sets \(\varOmega _B=0.562^{+0.037}_{-0.038}\) for the scenario in which \(\alpha =-1, \varOmega _B=0.654^{+0.040}_{-0.040}\) for the scenario with \(\alpha =-1/4\) and \(\varOmega _B=0.683^{+0.039}_{-0.043}\) for the one with \(\alpha =-1/8\). These results indicate that nonlinear electromagnetic matter could play the role of dark energy, with the theoretical advantage of being a mensurable field.
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Acknowledgments
A. M. acknowledges financial support from CONACyT (Mexico) through a Ph.D. Grant. N. B. acknowledges partial support by Conacyt, Project 166581. We also acknowledge to the anonymous referee whose suggestions lead to improve our work.
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Appendices
Appendix 1: Scaling between the scale factor \(a\) and the electromagnetic invariant \(F\)
The energy conservation \(T_{;\mu }^{\mu \nu } = 0\), leads to the equation
which also can be derived from Eq. (6). So, using the expressions of \(\rho \) and \(p\), Eq. (2), in terms of the electromagnetic Lagrangian, the scaling between the scale factor \(a\) and the electromagnetic invariant \(F\) can be determined for a Lagrangian with arbitrary dependence on the two electromagnetic invariants \(\mathcal {L}(F,G)\) as
Now, if one restricts to the case \(G=0\) (i.e. no electric field \(E=0\)), then \(F=2B^2\) and
whose solution, given by \(Fa^4=\mathrm{const}\), is independent of the particular form of \(\mathcal {L}(F)\).
Appendix 2: The scale factor as a function of time
The expressions of Friedmann equations for the nonlinear magnetic terms are
Knowing that \(\mathcal {L}a^{4 \alpha }=\mathrm{const}\), and using the Friedmann equations, the expression for \(a(t)\) can be determined. Let us consider the following derivative,
and substituting Friedmann’s equation, Eq. (25), we realize that the right hand term is constant,
Finally, integrating for \(a(t)\), it is obtained that \(a(t)= \mathrm{const}(t-t_0)^{1/2 \alpha }\).
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Montiel, A., Bretón, N. & Salzano, V. Parameter estimation of a nonlinear magnetic universe from observations. Gen Relativ Gravit 46, 1758 (2014). https://doi.org/10.1007/s10714-014-1758-3
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DOI: https://doi.org/10.1007/s10714-014-1758-3