Abstract
While during inflation a phase of increasing gauge coupling allows for a scale-invariant hyperelectric spectrum, when the coupling decreases a flat hypermagnetic spectrum can be generated for typical wavelengths larger than the effective horizon. After the gauge coupling flattens out the late-time hypermagnetic power spectra outside the horizon in the radiation epoch are determined by the hyperelectric fields at the end of inflation whereas the opposite is true in the case of decreasing coupling. Instead of imposing an abrupt freeze after inflation, we consider a smooth evolution of the mode functions by positing that the gauge couplings and their conformal time derivatives are always continuous together with the background extrinsic curvature. The amplified gauge power spectra are classified according to their transformation properties under the duality symmetry. After clarifying the role of the comoving and of the physical spectra in the formulation of the relevant magnetogenesis constraints, the parameter space of the scenario is scrutinized. It turns out that a slightly blue hyperelectric spectrum during inflation may lead to a quasi-flat hypermagnetic spectrum prior to matter radiation equality and before the relevant wavelengths reenter the effective horizon. In this framework the gauge coupling is always perturbative but the induced large-scale magnetic fields can be of the order of a few hundredths of a nG and over typical length scales between a fraction of the Mpc and 100 Mpc prior to the gravitational collapse of the protogalaxy.
Highlights
Besides the invariance under local gauge transformations, the Weyl [1] and the duality [2, 3] symmetries are relevant for the dynamics of the gauge fields in general relativity and in scalar-tensor theories of gravity
Duality implies that the hypermagnetic power spectra parametrically amplified from quantum fluctuations during a quasi-de Sitter stage of expansion are scale-invariant if the gauge coupling decreases while an increase of the gauge coupling is only compatible with a flat hyperelectric spectrum for wavelengths larger than the effective horizon at the corresponding epoch
The same duality symmetry demands that the late-time gauge spectra do not always coincide with the results obtained at the end of inflation: the late-time hypermagnetic spectra follow directly from the hypermagnetic mode functions at the end of inflation whenever the gauge coupling decreases
Summary
Besides the invariance under local gauge transformations, the Weyl [1] and the duality [2, 3] symmetries are relevant for the dynamics of the gauge fields in general relativity and in scalar-tensor theories of gravity. From the technical viewpoint these results will arise from the discussion of an appropriate transition matrix whose elements have well defined transformation properties under the duality symmetry and control the form of the late-time spectra Using these results we shall investigate the magnetogenesis requirements as well as all other pertinent constraints; we shall conclude that large-scale magnetic fields can be generated during a quasi-de Sitter stage of expansion while the gauge coupling remains perturbative throughout all the stages of the dynamical evolution. All the mode functions and power spectra obtained by the simultaneous evolution of the geometry and of the gauge coupling must be consistent with equation (2.16) both during the inflationary phase and in the subsequent decelerated stages of expansion before the √given scale reenters the effective horizon. The gauge coupling increases when λ decreases and vice versa
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