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The origin of galactic and extra-galactic magnetic fields is an unsolved problem in modern cosmology. A possible scenario comes from the idea of these fields emerged from a small field, a seed, which was produced in the early universe (phase transitions, inflation, ...) and it evolves in time. Cosmological perturbation theory offers a natural way to study the evolution of primordial magnetic fields. The dynamics for this field in the cosmological context is described by a cosmic dynamo like equation, through the dynamo term. In this paper we get the perturbed Maxwell's equations and compute the energy momentum tensor to second order in perturbation theory in terms of gauge invariant quantities. Two posible scenarios are discussed, first we consider a FLRW background without magnetic field and we study the perturbation theory introducing the magnetic field as a perturbation. The second scenario, we consider a magnetized FLRW and build up the perturbation theory from this background. We compare the cosmological dynamo like equation in both scenarios. PACS numbers: 98.80.-k, 95.30.Qd.

The origin of large scale magnetic fields is one of the most puzzling topics in cosmology and astrophysics. It is assumed that the observed magnetic fields result from the amplification of an initial field produced in the early universe. In this paper we compute the exact power spectrum of magnetic fields created after inflation best known as post inflationary magnetic fields, using the first order cosmological perturbation theory. Our treatment differs from others works because we include an infrared cutoff which encodes only causal modes in the spectrum. The cross-correlation between magnetic energy density with Lorentz force and the anisotropic part of the electromagnetic field are exactly computed. We compare our results with previous works finding agreement in cases where the ratio between lower and upper cutoff is very small. However, we found that spectrum is strongly affected when this ratio is greather than 0.2. Moreover, the effect of a post inflationary magnetic field with a lower cutoff on the angular power spectrum in the temperature distribution of CMB was also exactly calculated. The main feature is a shift of the spectrum's peak as function of the infrared cutoff, therefore analyzing this effect we could infer the value of this cutoff and thus constraining the primordial magnetic fields generation models. PACS numbers: 98.80. 95.30.Qd. 98.80.Cq, 98.70.Vc,

In this paper, we investigate the effects of helical primordial magnetic fields (PMFs) on the cosmic microwave background (CMB) reduced bispectrum. We derive the full three-point statistics of helical magnetic fields and numerically calculate the even contribution in the collinear configuration. We then numerically compute the CMB reduced bispectrum induced by passive and compensated PMF modes on large angular scales. There is a negative signal on the bispectrum due to the helical terms of the fields and we also observe that the biggest contribution to the bispectrum comes from the non-zero IR cut-off for causal fields, unlike the two-point correlation case. For negative spectral indices, the reduced bispectrum is enhanced by the passive modes. This gives a lower value of the upper limit for the mean amplitude of the magnetic field on a given characteristic scale. However, high values of IR cut-off in the bispectrum, and the helical terms of the magnetic field relaxes this bound. This demonstrates the importance of the IR cut-off and helicity in the study of the nature of PMFs from CMB observations.

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