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On the vertical axis the (common) logarithm of $k/(a H)$ is illustrated as a function of the (common) logarithm of the scale factor. The inflationary stage is followed by the radiation phase and the three different lines illustrate a selectrion of wavenumbers that are all ${\mathcal O}(\mathrm{Mpc}^{-1})$. These wavenumbers become of the order of the comoving Hubble radius when $k/(a\, H) = {\mathcal O}(1)$ close to the elliptic disk that is present for $a>a_{1}$. This range typically corresponds to a conformal time coordinate ${\mathcal O}(10^{-2})\, \,\tau_{eq}$.

In both plots, with the dashed line, we illustrate the same situation described in Fig. \ref{FIGU1} while the full lines account for the case where the post-inflationary expansion rate contains a single (supplementary) phase expanding either faster (i.e. $\sigma>1$) or slower (i.e. $\sigma < 1$) than radiation. If the expansion rate is faster than radiation $k/(a H)$ is systematically larger than in the radiation-dominated phase. Furthermore since the Universe expands faster the total redshift of the post-inflationary phase is larger but the opposite is true when the expansion rate is slower than radiation; in both cases, however, $\tau_{k} = {\mathcal O}(10^{-2}) \,\, \tau_{eq}$. In these plots we have selected, for illustration, $\xi_{r} = 10^{-20}$.

In both plots, with the dashed line, we illustrate the same situation described in Fig. \ref{FIGU1} while the full lines account for the case where the post-inflationary expansion rate contains a single (supplementary) phase expanding either faster (i.e. $\sigma>1$) or slower (i.e. $\sigma < 1$) than radiation. If the expansion rate is faster than radiation $k/(a H)$ is systematically larger than in the radiation-dominated phase. Furthermore since the Universe expands faster the total redshift of the post-inflationary phase is larger but the opposite is true when the expansion rate is slower than radiation; in both cases, however, $\tau_{k} = {\mathcal O}(10^{-2}) \,\, \tau_{eq}$. In these plots we have selected, for illustration, $\xi_{r} = 10^{-20}$.

As in Fig. \ref{FIGU2} with the dashed lines we illustrate the radiation case while the full lines describe the situation where there are {\em two intermediate phases}. In the first case (i.e. $\sigma_{1} < 1$ and $\sigma_{2}> 1$) the expansion rate is first slower and then faster than radiation. In the second case (i.e. $\sigma_{1} > 1$ and $\sigma_{2}< 1$) the expansion rate is first faster and then slower than radiation.

As in Fig. \ref{FIGU2} with the dashed lines we illustrate the radiation case while the full lines describe the situation where there are {\em two intermediate phases}. In the first case (i.e. $\sigma_{1} < 1$ and $\sigma_{2}> 1$) the expansion rate is first slower and then faster than radiation. In the second case (i.e. $\sigma_{1} > 1$ and $\sigma_{2}< 1$) the expansion rate is first faster and then slower than radiation.

In both plots the physical power spectrum is illustrated as a function of $\tau/\tau_{k}$ and for different values of $\gamma$. The leftmost (thick) line corresponds to $\tau = {\mathcal O}(\tau_{1})$ while the rightmost (thick) line denotes $\tau = {\mathcal O}(\tau_{k})$. The labels appearing on the various contours are the common logarithms of $\sqrt{{\mathcal P}_{B}(k,\tau_{0})}$ in nG units and are the same o each curve. The dashed regions correspond to the largest values of the physical power spectra. In the plot at the left $\delta\to 0$ implying that the gauge coupling freezes right after the end of inflation. In the plot at the right $\gamma$ is instead fixed while $\delta$ is very small but it does not vanish. The two plots are compatible in the case $\gamma\to 2$ and this observation justifies, once more, the limit $\delta \to 0$ that has been always assumed in the analytical estimates (see, in this respect, also appendix \ref{APPA}).

In both plots the physical power spectrum is illustrated as a function of $\tau/\tau_{k}$ and for different values of $\gamma$. The leftmost (thick) line corresponds to $\tau = {\mathcal O}(\tau_{1})$ while the rightmost (thick) line denotes $\tau = {\mathcal O}(\tau_{k})$. The labels appearing on the various contours are the common logarithms of $\sqrt{{\mathcal P}_{B}(k,\tau_{0})}$ in nG units and are the same o each curve. The dashed regions correspond to the largest values of the physical power spectra. In the plot at the left $\delta\to 0$ implying that the gauge coupling freezes right after the end of inflation. In the plot at the right $\gamma$ is instead fixed while $\delta$ is very small but it does not vanish. The two plots are compatible in the case $\gamma\to 2$ and this observation justifies, once more, the limit $\delta \to 0$ that has been always assumed in the analytical estimates (see, in this respect, also appendix \ref{APPA}).

The physical power spectrum of the magnetic field is further illustrated with the same notations already established in Fig. \ref{FIGU4}. In both plots $\delta \to 0$ (i.e. the gauge coupling freezes after inflation) and in both cases we consider a long (i.e. $\xi_{r} = 10^{-30}$) post-inflationary expansion rate that differs from radiation. In the plot at the left $\sigma < 1$ (in particular $\sigma =1/2$) while in the plot at the right $\sigma > 1$ (i.e. $\sigma = 2$).

The physical power spectrum of the magnetic field is further illustrated with the same notations already established in Fig. \ref{FIGU4}. In both plots $\delta \to 0$ (i.e. the gauge coupling freezes after inflation) and in both cases we consider a long (i.e. $\xi_{r} = 10^{-30}$) post-inflationary expansion rate that differs from radiation. In the plot at the left $\sigma < 1$ (in particular $\sigma =1/2$) while in the plot at the right $\sigma > 1$ (i.e. $\sigma = 2$).

As in Fig. \ref{FIGU5} we now discuss the possibility of two separate phases both different from radiation. In both plots the first phase is shorter than the second (i.e. $\xi_{1} = 10^{-10}$ and $\xi_{2} = 10^{-20}$). The global duration of the post-inflationary stage coincides with the one analyzed in Fig. \ref{FIGU5} since $\xi_{1} \xi_{2} = \xi_{r} = 10^{-30}$. In the left plot the relevant region with large values of the magnetic power spectrum is comparatively smaller and this is because the overall duration of the phase expanding slower that radiation is smaller.

As in Fig. \ref{FIGU5} we now discuss the possibility of two separate phases both different from radiation. In both plots the first phase is shorter than the second (i.e. $\xi_{1} = 10^{-10}$ and $\xi_{2} = 10^{-20}$). The global duration of the post-inflationary stage coincides with the one analyzed in Fig. \ref{FIGU5} since $\xi_{1} \xi_{2} = \xi_{r} = 10^{-30}$. In the left plot the relevant region with large values of the magnetic power spectrum is comparatively smaller and this is because the overall duration of the phase expanding slower that radiation is smaller.

If the physical power spectrum before the Hubble crossing is evaluated for $\tau= {\mathcal O}(\tau_{k})$ we obtain the plot at the left. Conversely if the power spectrum is evaluated for $ \tau = {\mathcal O}(\tau_{r}) \ll \tau_{k}$ the result is obviously much smaller and it is illustrated with the dashed contours. In both plots of this figure $\sigma > 1$.

If the physical power spectrum before the Hubble crossing is evaluated for $\tau= {\mathcal O}(\tau_{k})$ we obtain the plot at the left. Conversely if the power spectrum is evaluated for $ \tau = {\mathcal O}(\tau_{r}) \ll \tau_{k}$ the result is obviously much smaller and it is illustrated with the dashed contours. In both plots of this figure $\sigma > 1$.

The same analysis illustrated in Fig. \ref{FIGU7} is now discussed for $\sigma = 2$ and it is representative of all the cases $\sigma >1$. The results with what we regard as the correct normalization are illustrated with the full lines while with the dashed lines we consider the case where the value of the physical power spectrum before Hubble crossing is set for $\tau= \tau_{r} \ll {\mathcal O}(\tau_{r})$.

The same analysis illustrated in Fig. \ref{FIGU7} is now discussed for $\sigma = 2$ and it is representative of all the cases $\sigma >1$. The results with what we regard as the correct normalization are illustrated with the full lines while with the dashed lines we consider the case where the value of the physical power spectrum before Hubble crossing is set for $\tau= \tau_{r} \ll {\mathcal O}(\tau_{r})$.

In both plots the late-time power spectrum is illustrated in the presence of an intermediate post-inflationary phase with $\xi_{r}=H_{r}/H_{1}< 1$; the post-inflationary expansion rate is controlled by $\sigma$. The value of the physical power spectrum is evaluated at $\tau_{k}$ and subsequently scaled by following the dominance of the conductivity. Th shaded area corresponds to the region where where the spectral energy density is smaller than $10^{-6}$ and the magnetogenesis requirements are satisfied in their most demanding form (i.e. $\sqrt{{\mathcal P}_{B}(k,\tau_{0})} > 10^{-22}$). In the plot at the left the value of $\gamma$ is fixed to $3/2$ (implying a rather small power spectrum at the magnetogenesis scale). In the plot at the right $\xi_{r} =10^{-30}$.

In both plots the late-time power spectrum is illustrated in the presence of an intermediate post-inflationary phase with $\xi_{r}=H_{r}/H_{1}< 1$; the post-inflationary expansion rate is controlled by $\sigma$. The value of the physical power spectrum is evaluated at $\tau_{k}$ and subsequently scaled by following the dominance of the conductivity. Th shaded area corresponds to the region where where the spectral energy density is smaller than $10^{-6}$ and the magnetogenesis requirements are satisfied in their most demanding form (i.e. $\sqrt{{\mathcal P}_{B}(k,\tau_{0})} > 10^{-22}$). In the plot at the left the value of $\gamma$ is fixed to $3/2$ (implying a rather small power spectrum at the magnetogenesis scale). In the plot at the right $\xi_{r} =10^{-30}$.

The same situation examined in Fig. \ref{FIGU9} is now considered when the power spectrum is evaluated according to the strategy described in Eqs. (\ref{NWR1})--(\ref{NWR2}) and (\ref{NWR3}). As anticipated before on the basis of general arguments the values of the power spectrum at the magnetogenesis scale are much smaller.

The same situation examined in Fig. \ref{FIGU9} is now considered when the power spectrum is evaluated according to the strategy described in Eqs. (\ref{NWR1})--(\ref{NWR2}) and (\ref{NWR3}). As anticipated before on the basis of general arguments the values of the power spectrum at the magnetogenesis scale are much smaller.

We illustrate here the possibility of two post-inflationary phases parametrized by the expansion rates $\sigma_{1}$ and $\sigma_{2}$. In the plot at the left we assumed $\xi_{1}= 10^{-20}$ and $\xi_{2} = 10^{-10}$. In the plot at the right $\sigma_{1} = 1/2$ and $\sigma_{2} = 2$. As before the shaded area denotes the region where the magnetogenesis requirements are satisfied and the spectral energy density is smaller than $10^{-6}$.

We illustrate here the possibility of two post-inflationary phases parametrized by the expansion rates $\sigma_{1}$ and $\sigma_{2}$. In the plot at the left we assumed $\xi_{1}= 10^{-20}$ and $\xi_{2} = 10^{-10}$. In the plot at the right $\sigma_{1} = 1/2$ and $\sigma_{2} = 2$. As before the shaded area denotes the region where the magnetogenesis requirements are satisfied and the spectral energy density is smaller than $10^{-6}$.