Pre-reheating magnetogenesis in the kinetic coupling model

Recent blazar observations provide growing evidence for the presence of magnetic fields in the extragalactic regions. While natural speculation is to associate the production with inflationary physics, it is known that magnetogenesis solely from inflation is quite challenging. We therefore study a model in which a noninflaton field $\chi$ coupled to the electromagnetic field through its kinetic term, $-I^2(\chi )F^2/4$, continues to move after inflation until the completion of reheating. This leads to a postinflationary amplification of the electromagnetic field. We compute all the relevant contributions to the curvature perturbation, including gravitational interactions, and impose the constraints from the CMB scalar fluctuations on the strength of magnetic fields. We, for the first time, explicitly verify both the backreaction and CMB constraints in a simple yet successful magnetogenesis scenario without invoking a dedicated low-scale inflationary model in the weak-coupling regime of the kinetic coupling model.

Figure 1

The behavior of $I(a)$ given in Eq. (2.2).

Figure 2

(Left panel) The electromagnetic spectra during inflation. The horizontal axis denotes $K\equiv |k{\eta }_i|$, and we set $n=3.5$. ${\mathcal{P}}_E$ (red lines) and ${\mathcal{P}}_B$ (blue lines) normalized by $H_{\inf }^4$ are shown for $\ln (a/a_i)=1$, 10, 20 from transparent to opaque. The subhorizon modes are shown as the dotted lines. (Right panel) The electromagnetic spectra during the inflaton oscillating phase. We set $n=3.5$ and $N_i\equiv \ln (a_e/a_i)\approx 22$. ${\mathcal{P}}_E$ (red lines) and ${\mathcal{P}}_B$ (blue lines) are shown for $\ln (a/a_e)=1$, 5, 10 from transparent to opaque. One can see that the subhorizon modes oscillate and damp, while the superhorizon ones continue to grow. The modes which did not exit the horizon during inflation and thus are unphysical are shown as the dotted line.

Figure 3

(Left panel) The time evolution of the electric spectrum ${\mathcal{P}}_E$ (red line) and the magnetic spectrum ${\mathcal{P}}_B$ (blue line). We set $n=3.5$, $N_i\equiv \ln (a_e/a_i)\approx 22$, and $K\equiv |k{\eta }_i|=4$, which corresponds to the peak scale of ${\mathcal{P}}_E$ and ${\mathcal{P}}_B$. The horizontal axis denotes the $e$-folding number $N\equiv \ln (a/a_e)$ and inflation ends at $N=0$. (Right panel) The ratio between the magnetic and electric power spectrum, ${\mathcal{P}}_B/{\mathcal{P}}_E$. The parameters are the same as the left panel. The ratio decreases during inflation, but it increases after inflation.

Figure 4

(Left panel) The integrand of $H_n$ defined in Eq. (2.24). Solid lines are for $k_i^{-1}=L=1\mathrm{Mpc}$ and $n=2.1$, 2.5, 3, and 4 from top to bottom. The dashed and dot-dashed lines are for $k_{i}L=10$, 100, respectively, with $n=3$. One can see that the main contribution to $B_{\mathrm{eff}}^2$ comes from $k\sim k_i$ for $n>2.5$. However, for $n<2.5$, the contribution from smaller scales is dominant and $B_{\mathrm{eff}}$ depends on the cutoff scale, $k_{\mathrm{diff}}$. (Right panel) The numerically evaluated $H_n$ for $n>2.5$. The lines correspond to $k_{i}L=1$, 10, 100, and 1000 from top to bottom. $H_n$ shows the logarithmic divergence for $n=2.5$.

Figure 5

(Left panel) The inflation energy scale ${\rho }_{\inf }^{1/4}$ is shown for ${\mathrm{\Omega }}_{\mathrm{em}}({\eta }_r)=1$ (solid lines) and for ${\mathcal{P}}_{\zeta }^{\mathrm{em}}={\mathcal{P}}_{\zeta }^{\mathrm{obs}}$ (dashed lines). We fix $k_i=1$ (blue), 10 (orange), $10^2$ (green), and $10^3{\mathrm{Mpc}}^{-1}$ (red). These lines are the upper bounds on the inflationary energy scale, but the value of ${\rho }_{\inf }^{1/4}$ depends on the given values of ${\mathrm{\Omega }}_{\mathrm{em}}$ and ${\mathcal{P}}_{\zeta }^{\mathrm{em}}$ only weakly, ${\rho }_{\inf }^{1/4}\propto {\mathrm{\Omega }}_{\mathrm{em}}^{1/4n}\propto ({\mathcal{P}}_{\zeta }^{\mathrm{em}}{)}^{1/8n}$. Since we set $T_r=1\mathrm{GeV}$, ${\rho }_{\inf }^{1/4}$ should be larger than 1 GeV, shown as the black dashed line. (Right panel) The peak scale of the electromagnetic field is pushed up to $k_i=1{\mathrm{kpc}}^{-1}$. Again, the solid lines denote ${\mathrm{\Omega }}_{\mathrm{em}}({\eta }_r)=1$ and the dashed lines denote ${\mathcal{P}}_{\zeta }^{\mathrm{em}}={\mathcal{P}}_{\zeta }^{\mathrm{obs}}$. The colors represents different reheating temperature, $T_r=10^3$ (green), $10^2$ (orange), and 10 GeV (blue). The dotted lines show these $T_r$’s, setting the lower bounds on ${\rho }_{\inf }^{1/4}$.

Figure 6

The effective strength of the magnetic field predicted by our model. In this figure, we fix the parameters as $T_r=1\mathrm{GeV}$, $k_i=1{\mathrm{Mpc}}^{-1}$, and $g_{*}=100$. The solid lines represent the cases with ${\mathrm{\Omega }}_{\mathrm{em}}({\eta }_r)=1,{10}^{-1},{10}^{-2},{10}^{-3}$, and ${10}^{-4}$ from top to bottom. The orange shaded region is excluded by the curvature perturbation problem. On the orange dashed line, the curvature perturbation induced by the electric fields is as large as the observed one, ${\mathcal{P}}_{\zeta }^{\mathrm{em}}=2.2\times {10}^{-9}$. The blue dotted line shows the lower bound inferred by blazar observations, $B_{\mathrm{eff}}\gtrsim {10}^{-15}\mathrm{G}$. The viable region in which sufficient magnetic fields can be generated without the backreaction and the curvature perturbation problem exists for $B_{\mathrm{eff}}\lesssim {10}^{-14}\mathrm{G}$.

Figure 7

(Left panel) The case with $k_i=10{\mathrm{Mpc}}^{-1}$. The other parameters and the plot scheme are the same as in Fig. 6. Since the peak scale of the electromagnetic fields becomes smaller, the hierarchy is more reduced and the constraints are weaker than in Fig. 6. In particular, the constraint from the curvature perturbation becomes irrelevant. (Right panel) The case with $k_i=1{\mathrm{kpc}}^{-1}$. The solid lines show ${\mathrm{\Omega }}_{\mathrm{em}}({\eta }_r)=1$ and the reheating temperature is fixed as $T_r=1,10,10^2$, and $10^3\mathrm{GeV}$ from top to bottom. One can see that the reheating temperature cannot exceed 1 TeV for $k_i\le 1\mathrm{kpc}$.

Figure 8

The reheating temperature as a function of $k_i$. Here we fix $B_{\mathrm{eff}}({\eta }_{\text{now}})={10}^{-15}\mathrm{G}$, choose $n=3$, which gives a scale-invariant magnetic spectrum, and show a few cases for different values of ${\mathrm{\Omega }}_{\mathrm{em}}({\eta }_r)$. As can be seen from (4.4), increasing $k_i$ can achieve a higher $T_r$ for a given $B_{\mathrm{eff}}$ and ${\mathrm{\Omega }}_{\mathrm{em}}$.