Electromagnetic responses driven by gravitational-wave memory in magnetar-flare outflows

An asymmetric relativistic outflow from a magnetar giant flare produces a step-like gravitational-wave (GW) memory. In the magnetized pair plasma surrounding the star, such GW memory drives transverse electromagnetic (EM) X-mode/fast-magnetosonic responses at the corresponding frequencies. We use this GW-memory spectrum to drive a global wave equation for a cold electron–positron pair plasma. With radiative boundary conditions excluding incoming waves, the driven response forms an outgoing EM mode that reaches the outer boundary of the source region. For central-engine energy-release timescales of 10, 1, and 0.15 µs, 90% of the integrated response energy lies below 23.3 kHz, 233 kHz, and 1.51 MHz, respectively. The principal response occupies the 10 kHz–1 MHz frequency band; a substantial extension into the 1–3 MHz frequency band also appears when a submicrosecond asymmetric-energy component is present. The electromagnetic response can escape from the source region and potentially propagate in the interstellar medium.

The central question of this work is whether the GW memory generated by an asymmetric magnetar-flare outflow can drive an electromagnetic response that forms an outgoing wave beyond the source region. This question has two parts. The source problem is to determine the finite-frequency GW-memory spectrum when the outflow acceleration, central-engine energy-release profile, angular delays, and polarization cancellation are treated separately. The propagation problem is to determine whether that spectrum excites only a local plasma disturbance or a transverse X-mode/fast-magnetosonic response that continues outward through the magnetosphere and pair outflow.

We address the two parts in sequence. We first integrate the complex GW memory of a relativistic top-hat jet, represent the acceleration of individual outflow elements and central-engine energy release with separate temporal profiles, and scan the Lorentz factor, acceleration time, opening angle, viewing angle, and energy-release time. We then use the resulting GW spectrum as the distributed driver of a global non-WKB wave equation for a cold electron–positron pair plasma, with radiative boundary conditions that exclude incoming waves. The calculation spans 0.1 kHz–30 MHz, while the physical interpretation focuses on 10 kHz–3 MHz. It identifies the central-engine energy-release time as the main control on the high-frequency extent of the response spectrum and shows that the driven response forms an outgoing EM mode at the source-region boundary. The calculation therefore separates suppression already present in the GW-memory spectrum from filtering during plasma propagation and establishes the source conditions required for an escaping 10 kHz–1 MHz response. The 1–3 MHz extension requires a submicrosecond asymmetric-energy component and is not assumed to be universal.

We represent the asymmetric relativistic outflow by one lobe of a top-hat jet with half-opening angle θj. Here “top-hat” has its standard geometric meaning: the energy per unit solid angle and the terminal Lorentz factor are constant inside the cone and vanish outside it. Retaining one lobe isolates the net anisotropic component that produces memory. It is an effective model for the observed outflow asymmetry, not an assertion that every magnetar flare launches an isolated jet without a counterflow. A symmetric counterjet or a structured angular profile can be incorporated by adding the corresponding complex-memory contribution for its orientation and relative energy.

The jet axis is separated from the line of sight by θv. An infinitesimal angular element of the jet is seen at polar angle α and sky azimuth ϕ. Its complex permanent memory, dH = dh+ + i dh×, is dH = (2GϵEkin/c4r) (β2 sin2 α / (1 − β cos α)) e2iϕ domega/omegaj, where β = (1 − Γ−2)1/2. The effective asymmetric energy is ϵEkin and is distributed uniformly in solid angle. The angular factor is the standard transverse-traceless memory of a relativistic particle. The phase e2iϕ makes polarization cancellation explicit: a perfectly axisymmetric jet viewed exactly on axis has zero net complex memory even though its individual outflow elements are nonzero.

Two physical timescales enter the memory waveform. The first, τacc, is the lab-frame time over which an individual outflow element accelerates. Retarded arrival times compress it to τα = τacc(1 − β cos α). For α ≪ 1 and Γ ≫ 1, τα ≃ τacc(Γ−2 + α2)/2. The second is the central-engine energy-release timescale τeng, the duration over which the engine releases the effective asymmetric outflow energy. Material launched at different engine times carries that delay into the observed memory, so τeng does not receive the same Γ−2 factor. This separation implements the result that jet-memory duration is controlled by both acceleration and the central-engine energy-release profile.

For a smooth step, the normalized Fourier transform of its derivative is D(f, τ) = π2 f τ / sinh(π2 f τ), with D(f, 0) = 1. Representing central-engine energy release and the acceleration of individual outflow elements by separate smooth temporal profiles gives, for f > 0, h̃M(f, r) = −(i/2πf) D(f, τeng) ∫ dH D(f, τα). The GW memory is not a monochromatic signal. In the time domain it approaches the permanent offset H = ∫ dH after the outflow has been established. In the frequency domain the ideal step contains a zero-frequency distribution and a 1/f nonzero-frequency component. The two temporal profiles in Eq. (4) multiply that component by the corresponding Fourier-domain factors: they leave 2πif h̃M constant at low frequency and suppress it once a Fourier period becomes shorter than either physical timescale. We omit the zero-frequency distribution when calculating the propagating response; no unique “memory frequency” is assigned.

We define the normalized frequency-resolved response energy by RE(f) = 2πif h̃M/H2. For comparisons across source models, f1/2 denotes the first frequency at which RE = 1/2. It is an operational marker of appreciable spectral suppression, not a physical cutoff. The linear GW–fast-mode source is proportional to fh̃M and therefore has a low-frequency plateau. A single 10 µs smooth step is recovered when that timescale is the dominant timescale, for which RE = 1/2 at fτ = 0.15111; it is no longer assumed to be a universal flare rise time.

The angular quadrature is uniform in jet azimuth and Gauss–Legendre in µj = cos ϑj over cos θj ≤ µj ≤ 1. Each node is rotated by θv into the observer frame to obtain α and ϕ; the quadrature weights are normalized so that Σn wn = 1. The angular calculation must reproduce two physically distinct limits. As θj → 0 it approaches the relativistic point-particle result, whereas an exactly on-axis axisymmetric jet has vanishing net complex memory because the polarization phases cancel. Both limits are recovered numerically. For the extreme Γ = 50, θj = 0.03, θv = 0.01 case, the 24 × 48, 48 × 96, and 72 × 144 quadratures all give A = 0.099350377 and RE(1 MHz) = 0.973216067 at the reported precision. For the representative Γ = 10, θj = 0.1, θv = 0.05 case, the same grids give A = 0.178494459 and RE(1 MHz) = 0.015053543. Angular refinement therefore changes none of the reported source conclusions at the 1-percent level.

We adopt a magnetar model with radius R∗ = 1.2 × 106 cm, surface field B∗ = 3.0 × 1015 G, and spin period P∗ = 1 s. The prescribed field joins B ∝ r−3 inside the light cylinder to B ∝ r−1 in a radial wind. The inner density is a Goldreich–Julian-like pair density with multiplicity κ = 104; the outer wind has total pair rate Ṅ± = 1038 s−1 and bulk Lorentz factor Γw = 10. The source-domain outer boundary is rout = 1013 cm.

Strongly magnetized neutron-star plasmas support X, Alfvén, and mixed longitudinal branches whose dispersion and polarization depend on field, density, temperature, and streaming frame. Magnetar magnetospheres also sustain relativistic pair flows, and flare-launched Alfvén waves can couple to fast modes. As a minimal constitutive closure we use a cold symmetric electron–positron pair plasma. In the local wind frame, ωp2 = 4πn′total e2/me, ωB = eB′/mec, and S(ω′) = 1 − ωp2 / [(ω′ + iν′)2 − ωB2], k′2c2 = ω′2S. In a symmetric pair plasma the electron and positron contributions to the gyrotropic off-diagonal response cancel, whereas their diagonal transverse response adds. The resulting branch is electromagnetic in polarization and continuously approaches the vacuum light wave as S → 1. In the strong-field limit ωB2 ≫ ω′2, Eq. (8) gives S ≃ 1 + ωp2/ωB2; density and magnetization therefore enter through their competition rather than through a simple unmagnetized plasma cutoff. We use the notation X-mode/fast-magnetosonic response to emphasize this continuous transverse, field-carrying branch across the magnetosphere and wind descriptions. It denotes one modeled response, not the sum of two independent signals. The radial Lorentz transform is applied before solving for the outward, attenuating lab-frame root kX(r, f). The baseline collision fraction is zero. This closure isolates whether smooth dispersion and nonuniformity alone destroy coherence; thermal, kinetic, and flare-induced damping are deferred.

For GW propagation perpendicular to the background magnetic field (θB = π/2), the coupling coefficient is CgF(f, θB) = (2πf sin θB)/(2c), θB = π/2. It has the frequency and angular dependence of uniform-MHD fast-mode driving. With the rescaled magnetic Fourier amplitude U = rB̃, we define q(r, f) = rCgF(f)B0(r)h̃M(f, r). This form exposes the origin of the spectral plateau. At frequencies below both source turnovers, the memory spectrum scales as h̃M ∝ f−1, while the linear coupling contributes one power of f. Their product is therefore approximately independent of frequency. The background magnetic field supplies the dimensional conversion scale and makes the inner magnetosphere dominate the GW–EM coupling. The factor of r removes the leading spherical dilution from the propagated magnetic amplitude. The complex phase of q is retained, so contributions generated at different radii can interfere in the global solution rather than being added as positive local energies.

For comparison, the local transport equation is dU/dr = q − (α/2 + i∆k)U, with ∆k = Re kX − 2πf/c, α = 2 Im kX. Setting ∆k = α = 0 gives the coherent local approximation; retaining both gives the local approximation including dispersion and damping. These local approximations provide useful comparisons with the global calculation, but they retain only the outward-propagating component and cannot describe reflection.

The rescaled global field Ψ = rB̃ satisfies d2Ψ/dr2 + kX2(r, f)Ψ = 2ik0 q(r, f)eik0(r−rin), with k0 = 2πf/c. Unlike Eq. (11), Eq. (13) retains both radial wave directions. A varying kX can therefore generate reflection, and large local wavelengths do not invalidate the propagation method. The constitutive relation kX(r, f) remains local; the calculation is spatially global but not a kinetic plasma simulation. No wave is injected into the computational domain. We impose radiative boundary conditions excluding incoming waves. With the spatial convention eikr for an outward wave, these conditions are Ψ′(rin) + ikX(rin)Ψ(rin) = 0, Ψ′(rout) − ikX(rout)Ψ(rout) = 0. The inner condition permits a source-generated inward wave to leave the domain toward the star but forbids an outward wave from entering through rin; the outer condition admits only an outward wave. These two radiation conditions make the outgoing amplitude an output of the distributed source rather than an imposed initial value.

The radial domain is divided into shells on which kX and q are held at their midpoint values. Within shell j, define the augmented state y = (Ψ, Ψ′, ϕ)T, where ϕ = eik0(r−rin). The inhomogeneous second-order equation becomes the homogeneous first-order system dy/dr = Mj y, with Mj = [[0, 1, 0], [−kj2, 0, 2ik0 qj], [0, 0, ik0]]. The exact shell update is yj+1 = exp(Mj ∆rj)yj. This construction propagates the vacuum phase analytically inside each shell; the radial grid resolves the background and source variation rather than sampling every field oscillation. Let yb be the outer state of the particular solution initialized with (0, 0, 1)T, and let ys be the outer state of the independent homogeneous solution initialized with (1, −ikin, 0)T. The inner complex amplitude required by the outer radiation condition is Ain = −(y′b − ikout yb)/(y′s − ikout ys). Propagating (Ain, −ikin Ain, 1)T then satisfies both radiation conditions. The outgoing complex amplitude at the outer boundary is Ψ(rout)e−ik0(rout−rin). In uniform vacuum, kX = k0 and constant q, the numerical solution reproduces Ψ(rout)e−ik0(rout−rin) = q(rout − rin), recovering the coherent first-order result. A zero source returns zero field. These two analytic limits validate the normalization and boundary signs.

The one-sided source-boundary spectral energies used below are dEEM/(df dΩ) = (2rout2 c/4π)B̃(rout, f)2, dEGW/(df dΩ) = (rout2 c3(2πf)2/8πG)h̃(rout, f)2. The EM energy inherits the normalization of Eq. (9); ratios between the global solution and the local approximations are more robust than its absolute value.

The source and propagation calculations are verified independently before they are combined. The jet calculation is tested against the narrow-jet point limit, exact on-axis axisymmetric cancellation, the f → 0 permanent-memory limit, monotonic energy-release suppression, and angular quadrature refinement. The global calculation is tested against the zero-source and uniform-vacuum analytic limits, vanishing transverse coupling, and the two radiation conditions. The radial convergence study uses 300, 600, 1200, and 2400 radial shells at ten representative frequencies spanning the full frequency range. Because every piecewise-constant shell is advanced by its exact matrix exponential, this refinement tests the radial representation of the prescribed magnetosphere, wind transition, and source profile. It does not impose the unnecessary requirement that the shortest electromagnetic wavelength be resolved cycle by cycle. We additionally monitor radiation-boundary residuals at every selected frequency.

Here and below, “acceleration-only” denotes an impulsive central-engine energy release (τeng = 0), leaving τacc as the only finite source timescale. The source-parameter scan covers 0.1 kHz–30 MHz with 250 unique frequencies, Γ = 5, 10, 20, and 50, τacc = 30, 100, and 300 µs, θj = 0.03, 0.1, and 0.3 rad, and 768 acceleration-only configurations. The angular quadrature uses 48 × 96 nodes. For an exactly on-axis axisymmetric jet, the complex polarization contributions cancel in the angular integral and the net permanent memory vanishes. We therefore interpret the spectral extent only for configurations with nonzero net memory after angular integration and report both the spectral shape and the amplitude ratio A = H / [(2GϵEkin/c4r)β2], where the denominator is the point-particle memory evaluated at α = π/2, which we call the side-on point-particle reference. This fixed normalization removes the common energy–distance scale and the leading β2 dependence, so that A isolates the effects of jet angular extent, viewing geometry, and polarization cancellation.

For τacc = 30 µs and θj = 0.1 rad, increasing Γ extends the acceleration-only response to higher frequencies near the jet while the exact on-axis amplitude remains zero. At Γ = 10, θv = 0.05 rad, the angularly integrated top-hat model has A = 0.178, and its response falls to one-half of the low-frequency value at 0.364 MHz. Across the full scan and requiring A ≥ 0.1, this factor-of-two decline occurs no higher than 0.228, 0.825, 2.469, and 4.989 MHz for Γ = 5, 10, 20, and 50, respectively. These comparison points describe a continuous decline rather than a cutoff. The last value has τacc = 30 µs, θj = 0.03 rad, and θv = 0.015 rad, with A = 0.220. Thus the MHz-scale response is not crea…

Improvements to AI Systems:

1. Multi-timescale physics integration: AI systems can now model systems where two distinct physical timescales (acceleration time τacc and central-engine energy-release time τeng) independently shape an observable spectrum, rather than collapsing them into a single effective timescale. This enables more accurate predictions for transient astrophysical events.

2. Frequency-resolved memory spectrum handling: AI can treat gravitational-wave memory as a continuous spectrum with finite-frequency components rather than a monochromatic or zero-frequency signal. This allows prediction of electromagnetic responses across a broad band (10 kHz–3 MHz) with quantitative energy distribution (e.g., f90 = 23.3 kHz, 233 kHz, 1.51 MHz for τeng = 10, 1, 0.15 µs).

3. Global non-WKB wave propagation: AI can solve boundary-value problems retaining both forward and backward wave propagation, including reflection and phase interference, without assuming local WKB validity. This enables accurate prediction of whether a driven disturbance escapes a source region or remains trapped.

4. Polarization-cancellation-aware angular integration: AI can explicitly integrate over jet angular structure with complex polarization phases, correctly predicting zero net memory for axisymmetric on-axis jets while capturing nonzero memory for off-axis or asymmetric configurations. This prevents overestimation of signals from symmetric sources.

5. Separable source and propagation effects: AI can decompose observed spectral suppression into contributions from the source (GW-memory spectrum) versus plasma propagation (dispersion, damping, reflection). This allows identification of which physical process controls high-frequency cutoff—here, the central-engine timescale, not plasma filtering.

6. Quantitative spectral markers: AI can use robust metrics like f90 (90% energy frequency) and RE(f) (normalized frequency-resolved response energy) instead of arbitrary cutoffs, enabling consistent comparison across models and detection-band assessments.

7. Parameter-space scanning with physical constraints: AI can efficiently scan multi-dimensional parameter spaces (Γ, τacc, θj, θv, τeng) while enforcing physical limits (e.g., A ≥ 0.1 for nonzero memory), identifying which configurations produce observable high-frequency extensions versus those that do not.

8. Validation through analytic limits: AI can be trained to verify numerical solutions against known limits (e.g., point-particle, on-axis cancellation, zero-source, uniform-vacuum), ensuring reliability before application to novel regimes.

What the Improved AI System Can Do:

• Predict the frequency-dependent electromagnetic response from magnetar giant flares, including whether kilohertz-to-megahertz signals escape the magnetosphere, for given outflow parameters.

• Distinguish whether observed high-frequency suppression originates from the central engine’s energy-release profile or from plasma propagation effects.

• Estimate the required submicrosecond asymmetric-energy fraction (ηfast) for a detectable 1–3 MHz extension, and assess detectability with space-based plasma-wave instruments.

• Model other astrophysical systems with step-like memory (e.g., core-collapse supernovae, binary mergers) where asymmetric ejecta and finite-timescale energy release produce broadband electromagnetic counterparts.

• Provide benchmarks for future kinetic plasma simulations by specifying where the cold-plasma approximation holds and where it breaks down (e.g., hot distributions, QED effects, pair fireballs).

• Optimize observational strategies by identifying the frequency bands and source conditions most likely to yield measurable signals, reducing wasted telescope time.