Polarization Formalism for Photon-Gravitational Wave Mixing Around Magnetars

Author(s)

Côté, Jean-Simon, Fortin, Jean-François

Abstract

The Gertsenshtein effect can be used to probe the stochastic gravitational wave background at high frequencies, well above the range of standard cosmological sources. In this paper, we revisit the conversion between electromagnetic and gravitational waves in the magnetosphere of magnetars by solving the evolution equations of the associated Stokes parameters. In the process, we point out that the adiabatic approximation usually taken in the literature is not generally justified in the context of the Gertsenshtein effect. To derive analytical results, we focus our attention on two specific geometries where the adiabatic approximation is valid. From these, we derive a lower bound on the stochastic gravitational wave background from the conversion of magnetar electromagnetic emission into gravitational waves, and an upper bound by requiring that the conversion of background gravitational waves into electromagnetic radiation does not exceed the observed magnetar flux in the X-ray band. Our results demonstrate that gravitational waves generated through the Gertsenshtein conversion of magnetar electromagnetic emission produce a negligible stochastic background, as anticipated.

Figures

Geometry for the magnetosphere of a magnetar and the electromagnetic--gravitational wave propagation. The magnetar is centered at the origin, with the magnetic field assumed to follow a dipolar configuration dictated by the magnetic moment $\mathbf{\hat{m}}$. An electromagnetic--gravitational wave propagates along the $\mathbf{\hat{z}}$ direction with an impact parameter $b$, subject to a magnetic field $\mathbf{B}$ with transverse components $\mathbf{B}_t$.
Caption Geometry for the magnetosphere of a magnetar and the electromagnetic--gravitational wave propagation. The magnetar is centered at the origin, with the magnetic field assumed to follow a dipolar configuration dictated by the magnetic moment $\mathbf{\hat{m}}$. An electromagnetic--gravitational wave propagates along the $\mathbf{\hat{z}}$ direction with an impact parameter $b$, subject to a magnetic field $\mathbf{B}$ with transverse components $\mathbf{B}_t$.
Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Caption Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Caption Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Caption Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Caption Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Caption Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Caption Conversion probability and phase shifts induced by the Gertsenshtein effect as functions of the polar angle relative to the magnetic moment $\alpha$ and angular frequency $\omega$. Green curves are obtained from the numerical integration of \eqref{NumericalProbInt} and \eqref{PhotonDiagPhaseshiftExactRadial}, blue curves from the exact expressions \eqref{MixingProbShiftRadialExact}, and red curves from the analytical approximations \eqref{ApproxProbShiftRadial} and \eqref{RadialApproxPhaseShift}. The three results nearly coincide, leaving only the red curves visible. The off-diagonal phase shifts oscillate rapidly due to the large value of $\lambda$, with those of \eqref{ApproxProbShiftRadial} defined modulo $2\pi$ in the interval $(-\pi,\pi)$. Benchmark parameters correspond to the magnetar 1E~1547.0$-$5408, with $r_0=10$ km and $B_0=3.2\times10^{14},\text{G}$. The left panels use $\omega=10,\text{keV}$, while the right panels use $\alpha=\pi/2$.
Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Caption Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Caption Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Caption Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Caption Conversion probability and diagonal phase shift induced by the Gertsenshtein effect as functions of the dimensionless impact parameter $\bar b = \frac*{b}{r_0}$ and angular frequency $\omega$. The green curves are obtained from numerical integration of \eqref{MixingProbShiftParallelExact} and \eqref{PhaseshiftExactParallel}, the red curves correspond to the analytical approximations \eqref{ParallelApproxbsimr0} and \eqref{deltaphihjParallelApproxbsimr0} in the regime $b\sim r_0$, and the blue curves denote the asymptotic approximations \eqref{LargebParallelApprox} valid for $b\gg r_0$, where the conversion probability rapidly decreases. Benchmark magnetar parameters are $r_0 = 10$ km and $B_0 = 3.2\times 10^{14}\, \text{G}$, corresponding to 1E~1547.0-5408. The left panels use $\omega = 10\, \text{keV}$, while the right panels use $\bar b = 200$.
Characteristic strain measured at Earth from gravitational waves produced through Gertsenshtein conversion of radially propagating X-rays in magnetars' magnetospheres.
Caption Characteristic strain measured at Earth from gravitational waves produced through Gertsenshtein conversion of radially propagating X-rays in magnetars' magnetospheres.
Upper bounds on the characteristic strain of the gravitational-wave background derived from Gertsenshtein conversion in magnetars' magnetospheres.
Caption Upper bounds on the characteristic strain of the gravitational-wave background derived from Gertsenshtein conversion in magnetars' magnetospheres.
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