Axion-photon conversion in stochastic magnetic fields

TL;DR

This work develops a statistical framework for axion–photon conversion in stochastic, Gaussian cosmic magnetic fields that may possess helicity. By modeling the magnetic field through symmetric and antisymmetric power spectra $P_B(k)$ and $P_{aB}(k)$ and applying the Born approximation, the authors derive the expectation values and variances of the Stokes parameters for photons after conversion, expressing them via four integrals $(oldalpha,oldbeta,oldgamma,olddelta)$ and their kernels. A key result is that unpolarized photons can acquire nontrivial polarization, including a helicity-driven circular polarization peak, and that robust consistency relations among the statistics hold independently of the spectral shapes. The analysis reveals rich frequency- and scale-dependent behavior controlled by the axion mass $m_a$, coupling $g_{a extgamma extgamma}$, correlation length $ extlambda_*$, and propagation distance $d$, with implications for using cosmological photon observations to constrain axion parameters in realistic magnetic-field environments.

Abstract

We investigate axion-photon conversion in stochastic magnetic fields, focusing on the evolution of the photon intensity and polarizations induced by conversion into axions. Assuming Gaussian magnetic fields characterized by the power spectra of their helical/non-helical components, we express the expectation values and variances of the photon intensity and linear/circular polarizations after conversion in terms of these spectra. We find nontrivial dependencies of these statistical quantities on the characteristic magnetic field correlation length, the propagation distance, and the axion mass. Moreover, we find that nontrivial polarizations emerge even if the photons are initially unpolarized, that the variances of these observables become suppressed in specific frequency regions, and that a peak structure arises in the expectation value of the circular polarization in the presence of statistically helical magnetic fields. We also point out consistency relations among these statistical quantities that hold independently of the specific forms of the magnetic field power spectra.

Figures (8)

• Figure 1: Convolution kernels $\mathcal{C}_\alpha$, $\mathcal{C}_\beta$, and $\mathcal{C}_\gamma$ with various values of $\Pi(\omega) d$ are shown as functions of $kd$. The black, red, green, and blue curves correspond to $\Pi(\omega) d = 0.1$, $1$, $10$, and $100$, respectively. The dashed curves indicate that the kernel takes negative values.
• Figure 2: Integrals $\alpha$ (top), $\beta$ (middle), and $\gamma$ (bottom) with the magnetic field power spectra \ref{['eq:PB']} and \ref{['eq:PaB']} are plotted. The black, red, green, and blue curves correspond to the case of $k_* d = 0.1$, $1$, $10$, and $100$, respectively. In each panel, the upper horizontal axis is the dimensionless variable $\Pi(\omega) d$, and the lower horizontal axis is $\omega$ in units of $\omega_\mathrm{eq}\equiv (|m_a^2 - m_\text{pl}^2| / 2\chi_{\mathrm{CMB}})^\frac{1}{2}$, where $(|m_a^2 - m_\text{pl}^2|)^\frac{1}{2}=10^{-11}\mathrm{eV}$, $2\chi_{\mathrm{CMB}}=10^{-42}$, and $d = 100\, \mathrm{Mpc}$ are used. The orange dashed line represents $\omega = \omega_\mathrm{eq}$. The right vertical axis is the value of $\alpha$, $\beta$, or $\gamma$ normalized by $g_{a\gamma\gamma}^2 P_{B*} k_*^2 d / 16\pi^2$. For $\beta$, we set $P_{aB*} = P_{B*}$ assuming the maximally helical case. The left vertical axis is the value of $\alpha$, $\beta$, or $\gamma$ normalized by $g^2_{a\gamma\gamma,10^{-12}\mathrm{GeV}^{-1}}\equiv(g_{a\gamma\gamma}/10^{-12}\mathrm{GeV}^{-1})^2$, $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}} \equiv d / 100\,\text{Mpc}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$. In each row, the left and right panels show the same quantity: the vertical axis is linear in the left, but logarithmic in the right. In the bottom-right panel, the dashed curves indicate that $\gamma$ takes negative values.
• Figure 3: Conversion probability $P_{\gamma \to a} \equiv 1-I$ with the maximally helical ($P_{aB*} = P_{B*}$) magnetic field power spectra \ref{['eq:PB']} and \ref{['eq:PaB']} is plotted for $k_* d = 0.1$, $1$, $10$, and $100$ from top to bottom. In each panel, the black line represents the expectation value $\text{Exp}[1-I]$, and the shaded bands represent the standard deviations $\sqrt{\text{Var}[1-I]}$, $\sqrt{\text{Var}[1-I]/10}$, and $\sqrt{\text{Var}[1-I]/100}$. The upper horizontal axis is the dimensionless variable $\Pi (\omega) d$, and the lower horizontal axis is $\omega$ in units of $\omega_\mathrm{eq}\equiv (|m_a^2 - m_\text{pl}^2| / 2\chi_{\mathrm{CMB}})^{\frac{1}{2}}$ with $(|m_a^2 - m_\text{pl}^2|)^{\frac{1}{2}}=10^{-11}\mathrm{eV}$, $2\chi_{\mathrm{CMB}}=10^{-42}$, and $d = 100\, \text{Mpc}$. The orange dashed line indicates $\omega = \omega_\mathrm{eq}$. The right vertical axis is $1-I$ normalized by $g_{a\gamma\gamma}^2 P_{B*} k_*^2 d / 16\pi^2$. The left vertical axis is $1-I$ normalized by $g^2_{a\gamma\gamma,10^{-12}\mathrm{GeV}^{-1}}\equiv(g_{a\gamma\gamma}/10^{-12}\mathrm{GeV}^{-1})^2$, $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}} \equiv d / 100\,\text{Mpc}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$. In each row, the left and right panels show the same quantity: the vertical axis is linear in the left, but logarithmic in the right.
• Figure 4: Stokes parameter $Q$ (linear polarization) of photons with the maximally helical ($P_{aB*} = P_{B*}$) magnetic field power spectra \ref{['eq:PB']} and \ref{['eq:PaB']} is plotted for $k_* d = 0.1$, $1$, $10$, and $100$ from top to bottom. In each panel, the black line represents the expectation value $\text{Exp}[Q]=0$, and the shaded bands represent the standard deviations $\sqrt{\text{Var}[Q]}$, $\sqrt{\text{Var}[Q]/10}$, and $\sqrt{\text{Var}[Q]/100}$. The upper horizontal axis is the dimensionless variable $\Pi (\omega) d$, and the lower horizontal axis is $\omega$ in units of $\omega_\mathrm{eq}\equiv (|m_a^2 - m_\text{pl}^2| / 2\chi_{\mathrm{CMB}})^{\frac{1}{2}}$ with $(|m_a^2 - m_\text{pl}^2|)^{\frac{1}{2}}=10^{-11}\mathrm{eV}$, $2\chi_{\mathrm{CMB}}=10^{-42}$, and $d = 100\, \text{Mpc}$. The orange dashed line indicates $\omega = \omega_\mathrm{eq}$. The right vertical axis is $Q$ normalized by $g_{a\gamma\gamma}^2 P_{B*} k_*^2 d / 16\pi^2$. The left vertical axis is $Q$ normalized by $g^2_{a\gamma\gamma,10^{-12}\mathrm{GeV}^{-1}}\equiv(g_{a\gamma\gamma}/10^{-12}\mathrm{GeV}^{-1})^2$, $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}} \equiv d / 100\,\text{Mpc}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$. In each row, the left and right panels show the same quantity: the vertical axis is linear in the left, but logarithmic in the right.
• Figure 5: Stokes parameter $-V$ (circular polarization) of photons with the maximally helical ($P_{aB*} = P_{B*}$) magnetic field power spectra \ref{['eq:PB']} and \ref{['eq:PaB']} is plotted for $k_* d = 0.1$, $1$, $10$, and $100$ from top to bottom. In each panel, the black line represents the expectation value $-\text{Exp}[V]$, and the shaded bands represent the standard deviations $\sqrt{\text{Var}[V]}$, $\sqrt{\text{Var}[V]/10}$, and $\sqrt{\text{Var}[V]/100}$. The upper horizontal axis is the dimensionless variable $\Pi (\omega) d$, and the lower horizontal axis is $\omega$ in units of $\omega_\mathrm{eq}\equiv (|m_a^2 - m_\text{pl}^2| / 2\chi_{\mathrm{CMB}})^{\frac{1}{2}}$ with $(|m_a^2 - m_\text{pl}^2|)^{\frac{1}{2}}=10^{-11}\mathrm{eV}$, $2\chi_{\mathrm{CMB}}=10^{-42}$, and $d = 100\, \text{Mpc}$. The orange dashed line indicates $\omega = \omega_\mathrm{eq}$. The right vertical axis is $-V$ normalized by $g_{a\gamma\gamma}^2 P_{B*} k_*^2 d / 16\pi^2$. The left vertical axis is $-V$ normalized by $g^2_{a\gamma\gamma,10^{-12}\mathrm{GeV}^{-1}}\equiv(g_{a\gamma\gamma}/10^{-12}\mathrm{GeV}^{-1})^2$, $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}} \equiv d / 100\,\text{Mpc}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$. In each row, the left and right panels show the same quantity: the vertical axis is linear in the left, but logarithmic in the right.