Large-scale magnetic fields can explain the baryon asymmetry of the Universe

Helical hypermagnetic fields in the primordial Universe can produce the observed amount of baryon asymmetry through the chiral anomaly without any ingredients beyond the standard model of particle physics. While they generate no $B-L$ asymmetry, the generated baryon asymmetry survives the spharelon washout effect, because the generating process remains active until the electroweak phase transition. Solving the Boltzmann equation numerically and finding an attractor solution, we show that the baryon asymmetry of our Universe can be explained, if the present large-scale magnetic fields indicated by the blazar observations have a negative helicity and existed in the early Universe before the electroweak phase transition. We also derive the upper bound on the strength of the helical magnetic field, which is tighter than the cosmic microwave background constraint, to avoid the overproduction of baryon asymmetry.

Figure 1

The parameter region where the magnetic fields would dominate the energy density of the Universe ${\mathrm{\Omega }}_B>1$ in the solely inverse cascade case (i) is shown as the shaded region. The dashed line represents the electroweak scale. In the case of $B_0>10^{-10}\mathrm{G}$ or ${\lambda }_0>10^{-2}\mathrm{Mpc}$, the magnetic fields must be generated or make the transition from the adiabatic evolution to the inverse cascade regime at the lower temperatures than the electroweak phase transition.

Figure 2

The evolution of baryons asymmetry for $B_0=10^{-13}\mathrm{G}$ in the solely inverse cascade case ($T_{\text{ini}}<T_{\mathrm{TS}}$) is shown. The horizontal axis denotes $x\equiv \sqrt{90/{\pi }^2{g}_{*}}{M}_{\mathrm{pl}}/T$. The initial temperature is taken as $T_{\text{ini}}={10}^7\mathrm{GeV},{10}^6\mathrm{GeV},{10}^5\mathrm{GeV},{10}^4\mathrm{GeV}$, and ${10}^3\mathrm{GeV}$ from left to right. The dotted line shows that the asymptotic behavior at $T<{10}^5\mathrm{GeV}$ is well fitted by $0.3{\gamma }_{\mathrm{y}}/{\gamma }_{e^{11}}$.

Figure 3

The final baryon asymmetry with respect to the present magnetic field is shown in the solely inverse cascade case (i) and the transition case with $T_{\mathrm{TS}}<T_f$. The blue shaded region represents the theoretical uncertainty, which is parametrized by the parameter $\mathcal{C}$. The line shows the observed baryon asymmetry. The gray shaded region is disfavored from the condition ${\mathrm{\Omega }}_B>1$ at $T>T_{\mathrm{EW}}$ (see Fig. 1).

Figure 4

The evolutions of baryon asymmetry for $B_0=10^{-9}\mathrm{G}$ with various $T_{\mathrm{TS}}(<T_{\mathrm{EW}})$ are shown. The transition temperature is taken as $T_{\mathrm{TS}}=10\mathrm{GeV},1\mathrm{GeV},10^{-1}\mathrm{GeV},10^{-2}\mathrm{GeV}$, and $10^{-3}\mathrm{GeV}$. dashed line shows that the asymptotic behavior at $T<{10}^5\mathrm{GeV}$ for $T_{\mathrm{TS}}=10^{-2}\mathrm{GeV}$ is well fitted by $0.3{\gamma }_{\mathrm{y}}/{\gamma }_{e^{11}}$.

Figure 5

The constraints on the comoving strength and the comoving correlation length of the helical magnetic fields. The blue and orange shaded region show the constraints from the CMB and blazar observations, respectively. In the gray shaded region, although the magnetic fields are allowed to exist, the sufficient baryon asymmetry cannot be generated via the chiral anomaly. In the green region, the helical magnetic field causes the overproduction of baryon asymmetry unless the transition from the adiabatic evolution into the inverse cascade is sufficiently late ($T_{\mathrm{TS}}\le T_{\mathrm{TS}}^{\mathrm{b}}$). The white region shows the window in which the observed baryon asymmetry can be successfully generated. The arrow represents the evolution path of the helical magnetic field undergoing the inverse cascade process and eventually reaches the blue thick line [Eq. (10)].