Graviton-photon conversion in stochastic magnetic fields

TL;DR

This work analyzes how gravitational waves propagating through stochastic cosmic magnetic fields experience graviton–photon conversion (Gertsenshtein effect) when the field is Gaussian and may be helical. Using a Born-approximation treatment and a density-matrix formalism, the authors compute the GW intensity $I$ and circular polarization $V$ (with $Q=U=0$ for unpolarized initial GWs) after distance $d$, along with their variances, expressing the results in terms of four convolution integrals $ ext{α,β,γ,δ}$ derived from magnetic-field power spectra $P_B(k)$ and $P_{aB}(k)$. They show that nonzero helicity yields a peak in the circular polarization and derive a model-independent consistency relation $ ext{Var}[1-I]+ ext{Var}[V]= ext{Exp}[1-I]^2+ ext{Exp}[V]^2$, valid under their Gaussian-field and Born-approximation assumptions. The analysis identifies regimes of reduced variance and frequency-dependent features controlled by the magnetic-field correlation length $k_*$ and the plasma mass, offering a potential pathway to probe cosmic magnetic fields and their helicity through GW polarization measurements. These results illuminate how stochastic magnetism in the universe could leave detectable imprints on high-frequency gravitational waves and guide future observational strategies.

Abstract

We study graviton-photon conversion in the presence of stochastic magnetic fields. Assuming Gaussian magnetic fields that may possess nontrivial helicity, and unpolarized gravitational waves (GWs) as the initial state, we obtain expressions for the intensity and linear/circular polarizations of GWs after propagation over a finite distance. We calculate both the expectation values and variances of these observables, and find their nontrivial dependence on the typical correlation length of the magnetic field, the propagation distance, and the photon plasma mass. Our analysis reveals that an observationally favorable frequency range with narrower variance can emerge for the intensity, while a peak structure appears in the expectation value of the circular polarization when the magnetic field has nonzero helicity. We also identify a consistency relation between the GW intensity and circular polarization.

Figures (6)

• 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 shown as functions of GW angular frequency $\omega$. 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 $\omega_{\text{eV}} \equiv \omega / \text{eV}$ normalized by $n_{e0} \equiv n_e / \text{m}^{-3}$ and $d_{100\text{Mpc}} \equiv d / (100\,\text{Mpc})$, where $n_e$ is the plasma density and $d$ is the propagation distance. The right vertical axis is the value of $\alpha$, $\beta$, or $\gamma$ normalized by $P_{B*} k_*^2 d / (8\pi^2 M_{\rm P}^2)$. (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 $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$, where $B_{*}$ is the magnetic field strength at the peak scale $k_*$, and $\lambda_{*} \equiv 2\pi/k_*$ is the characteristic length. 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 $\gamma$ is taking negative values.
• Figure 3: Conversion probability $P_{g\rightarrow\gamma} \equiv 1-I$ with the maximally helical ($P_{aB*} = P_{B*}$) magnetic field power spectra \ref{['eq:PB']} and \ref{['eq:PaB']} is shown as a function of GW angular frequency $\omega$ for various values of $k_* d$. From top to bottom, the cases $k_* d = 0.1$, $1$, $10$, and $100$ are shown. 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 $\omega_{\text{eV}} \equiv \omega / \text{eV}$ normalized by $n_{e0} \equiv n_e / \text{m}^{-3}$ and $d_{100\text{Mpc}} \equiv d / (100\,\text{Mpc})$, where $n_e$ is the plasma density and $d$ is the propagation distance. The right vertical axis is $1-I$ normalized by $P_{B*} k_*^2 d / (8\pi^2 M_{\rm P}^2)$. The left vertical axis is $1-I$ normalized by $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$, where $B_{*}$ is the magnetic field strength at the peak scale $k_*$, and $\lambda_{*} \equiv 2\pi/k_*$ is the characteristic length. 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 $-V$ (circular polarization) of GWs with the maximally helical ($P_{aB*} = P_{B*}$) magnetic field power spectra \ref{['eq:PB']} and \ref{['eq:PaB']} is shown as a function of the angular frequency $\omega$ for various values of $k_* d$. From top to bottom, the cases $k_* d = 0.1$, $1$, $10$, and $100$ are shown. 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 $\omega_{\text{eV}} \equiv \omega / \text{eV}$ normalized by $n_{e0} \equiv n_e / \text{m}^{-3}$ and $d_{100\text{Mpc}} \equiv d / (100\,\text{Mpc})$, where $n_e$ is the plasma density and $d$ is the propagation distance. The right vertical axis is $-V$ normalized by $P_{B*} k_*^2 d / (8\pi^2 M_{\rm P}^2)$. The left vertical axis is $-V$ normalized by $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$, where $B_{*}$ is the magnetic field strength at the peak scale $k_*$, and $\lambda_{*} \equiv 2\pi/k_*$ is the characteristic length. 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: Conversion probability $P_{g\rightarrow\gamma} = 1-I$ of GWs with the non-helical ($P_{aB*} =0$) magnetic field power spectrum \ref{['eq:PB']} is shown as a function of GW angular frequency $\omega$ for various values of $k_* d$. From top to bottom, the cases $k_* d = 0.1$, $1$, $10$, and $100$ are shown. 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 $\omega_{\text{eV}} \equiv \omega / \text{eV}$ normalized by $n_{e0} \equiv n_e / \text{m}^{-3}$ and $d_{100\text{Mpc}} \equiv d / (100\,\text{Mpc})$, where $n_e$ is the plasma density and $d$ is the propagation distance. The right vertical axis is $1-I$ normalized by $P_{B*} k_*^2 d / (8\pi^2 M_{\rm P}^2)$. The left vertical axis is $1-I$ normalized by $B_{*\text{nG}}^2 \equiv (B_* / \text{nG})^2$, $d_{100\text{Mpc}}$, and $\lambda_{*\text{Mpc}} \equiv \lambda_{*} / \text{Mpc}$, where $B_{*}$ is the magnetic field strength at the peak scale $k_*$, and $\lambda_{*} \equiv 2\pi/k_*$ is the characteristic length. 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.