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 created solely by dividing by a vanishing memory amplitude.

Jet opening angle limits the gain even before the central-engine energy-release profile is included. With the less restrictive A ≥ 0.01 threshold, the response remains above one-half of its low-frequency value up to at most 5.12 MHz for θj = 0.03 rad, 0.54 MHz for θj = 0.1 rad, and 0.06 MHz for θj = 0.3 rad. A broad top-hat contains many outflow elements with larger α; their longer arrival times and complex polarization factors erase the point-particle Γ−2 scaling after angular integration. Angular integration therefore strongly limits the high-frequency response of broad jets, whereas increasing Γ extends the response to higher frequencies only for narrow jets.

The viewing-angle scan separates three geometric regimes. Exact alignment maximizes arrival-time compression for individual outflow elements but gives zero net memory for an axisymmetric top-hat because all polarization phases are represented equally. Slightly off-axis configurations retain much of the compression while breaking that cancellation, producing spectra with the greatest high-frequency extent at a measurable fraction of the side-on amplitude. Farther from the jet axis, the permanent-memory amplitude can remain substantial, but the characteristic angle α is too large for strong relativistic time compression. The physically useful region is therefore a near-edge compromise between high-frequency extent and amplitude, rather than the formal on-axis maximum of the point-particle estimate.

The acceleration-only spectrum assumes that the entire effective asymmetric energy is released impulsively. A finite central-engine energy-release profile enters through the independent Fourier-domain factor D(f, τeng) in Eq. (4). To isolate this effect, we use the acceleration-only configuration with the largest f1/2 among cases satisfying A ≥ 0.01 as the reference case. Its parameters are Γ = 50, τacc = 30 µs, θj = 0.03 rad, and θv = 0.01 rad. Figure 3(a) compares three acceleration-only spectra with the normalized response of a single smooth 10 µs step. Panel (b) shows that increasing τeng shifts the spectral decline to lower frequencies. For the reference configuration, the frequency at which RE = 1/2 follows fτeng ≃ 0.15111 across τeng = 0.05–250 µs. The central-engine energy-release profile therefore determines where appreciable spectral suppression begins.

Retaining at least one-half of the low-frequency response at 1 MHz requires τeng ≲ 0.15 µs, for which the corresponding light-travel distance is cτeng ≲ 45 m. No observation or global magnetar-flare simulation currently establishes that most of the asymmetric outflow energy is released on such a short timescale. The spectra calculated with submicrosecond τeng assume that the full asymmetric energy participates in the fast component. If the participating fraction is ηfast, the high-frequency memory amplitude and linear EM-response energy scale as ηfast and ηfast2, respectively. For Ekin = 1046 erg, ϵ = 0.1, and a distance of 10 kpc, the permanent-memory amplitude is 5.3 × 10−28 for the 5.12-MHz extreme case. It is 9.5 × 10−28 for the representative Γ = 10, θj = 0.1, and θv = 0.05 case. These are GW-memory amplitudes, not EM-detector strains or fluxes.

We next quantify how the response energy is distributed in frequency for three representative central-engine energy-release times. Figure 4(a) shows RE(f) for τeng = 10, 1, and 0.15 µs. Each spectrum approaches a constant at low frequencies and declines at a frequency set by τeng. Because RE(f) gives the response energy per unit linear frequency, the low-frequency plateau does not imply a preferred zero-frequency response or divergent low-frequency energy. The corresponding normalized energy per logarithmic interval is d(E/Etot)/d ln f = fRE(f) / ∫0∞ RE(f′) df′. This quantity vanishes as f → 0 and integrates to unity over d ln f. In Fig. 4(b), its broad maxima occur near 13 kHz, 130 kHz, and 0.85 MHz for the three values of τeng, respectively. These maxima show a characteristic spectral scale set by τeng, but they do not represent a resonance of the GW–EM conversion.

To characterize the cumulative spectrum without relying on a single chosen level crossing, we define the 90-percent energy frequency f90 by ∫0f90 RE(f) df = 0.9 ∫0∞ RE(f) df. The analytic limit RE(0) = 1 supplies the zero-frequency value. The upper integral is evaluated to 30 MHz, beyond which the tail is negligible for the three cases. Figure 4(c) shows the cumulative energy fraction. For τeng = 10, 1, and 0.15 µs, we obtain f90 = 23.3 kHz, 233 kHz, and 1.51 MHz, respectively. The 10 kHz–1 MHz frequency band contains 45.8%, 94.0%, and 72.9% of the corresponding integrated energy. For the 0.15 µs case, a further 25.8% lies between 1 and 3 MHz, whereas only 0.32% lies above 3 MHz. Thus 10 kHz–1 MHz is the principal frequency band, but 1–3 MHz is a non-negligible extension for a submicrosecond energy-release timescale.

To connect the GW-memory spectrum calculation with the global propagation calculation, we first consider a single 10 µs memory sampled at 121 logarithmic frequencies from 0.1 to 300 kHz with 1200 logarithmic radial shells. The solution is also evaluated directly at ten selected frequencies: 0.1, 1, 5, 10, 20, 30, 50, 74, 100, and 300 kHz. The normalized drive fh̃ is flat for fτ ≪ 1 and falls exponentially for fτ ≳ 1. For the analytic memory spectrum, RE = 1/2 at fτ = 0.15111, corresponding to 15.11 kHz when τ = 10 µs. The response energy retains 73.0, 31.3, and 9.45 percent of its value at 0.1 kHz at 10, 20, and 30 kHz, respectively. It falls to 5.04 × 10−3 at 50 kHz and 1.04 × 10−6 at 100 kHz. These ratios follow the smooth memory rise and are not plasma resonances.

This subsection tests whether the GW-memory-driven EM response reaches the outer boundary rout as an outward-propagating mode, rather than being appreciably attenuated or reflected within the source region. At 0.1 kHz, the global boundary-value solution gives an outgoing EM spectral energy Eglobal = 1.15124 × 1023 erg Hz−1 sr−1 in the model normalization. Both Eglobal(f) and Ecoh(f) denote dEEM/(df dΩ) evaluated at rout. The former is obtained from the global boundary-value solution, whereas the latter is obtained from the coherent local approximation with ∆k = α = 0. Their ratio measures how source-region propagation changes the outgoing EM spectral energy relative to fully coherent outward accumulation. Across the complete 0.1–300 kHz grid, 0.99999998075 ≤ Eglobal/Ecoh ≤ 0.99999999940. Equation (23) shows that the global outgoing energy differs from the coherent result by at most 1.93 × 10−8. The local approximation including dispersion and damping is similarly close to the coherent result. Thus, within the adopted background, cumulative phase mismatch, damping, and reflection remove no measurable fraction of the outgoing response.

We next tested whether a plasma barrier appears at higher frequencies by extending the transfer calculation from 0.1 kHz to 30 MHz. The calculation used 161 frequencies, with convergence checks at 1, 3, 10, and 30 MHz. A 0.05 µs reference spectrum was used to avoid loss of numerical precision from the rapidly decreasing high-frequency source amplitude. In the linear problem, the global-to-coherent ratio is independent of the overall source amplitude. Its largest formal departure from unity is 1.85 × 10−6, below the maximum 1200-to-2400-shell difference of 2.66 × 10−4. The largest radiation-boundary residual is 4.8 × 10−15. The extended transfer ratio is therefore consistent with unity at the numerical resolution of the calculation. Between 0.1 and 300 kHz, the response energy inherited from the GW-memory spectrum varies by more than 22 orders of magnitude, whereas the source-region transfer ratio changes only in the eighth decimal place. The high-frequency decline is therefore inherited from the GW-memory spectrum rather than produced by plasma propagation. Within the adopted magnetar background, the response at every computed frequency from 0.1 kHz to 30 MHz reaches rout as an outgoing EM mode and propagates beyond the modeled source region, with no appreciable source-region suppression or additional plasma-selected cutoff.

The magnitude of the rescaled global field Ψ grows primarily between the stellar surface and approximately 107 cm and is nearly constant farther out. This behavior is shared across the displayed frequencies after normalization. At 50 and 300 kHz the global and coherent radial amplitudes are indistinguishable at line width. The reason is visible in Fig. 6(c): at the selected frequencies the real refractive index differs from vacuum by at most 6.77 × 10−9. A smooth feature near the magnetosphere–wind transition does not accumulate enough phase to suppress the outgoing wave. The normalized outer-boundary residual is below 1.3 × 10−14 in every 0.1–300-kHz convergence run. Comparing 1200 with 2400 radial shells, the largest selected-frequency energy change is 2.65 × 10−4; for 600 shells it is 1.33 × 10−3. The conclusion of negligible source-region suppression is robust because even the conservative refinement change is orders of magnitude below an order-unity spectral filter.

The mechanism is most clearly understood as a sequence rather than as a conversion between two monochromatic waves. Asymmetric relativistic ejecta change the asymptotic metric and leave a viewing-angle-dependent permanent GW memory. Because this memory develops over a finite time, its Fourier transform contains nonzero-frequency components in addition to the zero-frequency distribution associated with the final displacement. The linear plasma equations respond independently to each of these Fourier components. Thus “the same frequency” means that a component of the GW-memory spectrum at f drives the X-mode/fast-magnetosonic response at that same f; it does not assign a single oscillation frequency to the memory as a whole. This construction is related to, but not identical with, vacuum Gertsenshtein conversion. In vacuum the generated field can be described as an electromagnetic wave accumulating coherently along a prescribed magnetic path. Here that magnetic field is embedded in a nonuniform pair plasma, so the initially driven object is a plasma eigenmode with both field and matter content. The global calculation follows that mode until its outward field component reaches the source boundary. Because the background and coupling are linear and stationary, the calculation preserves Fourier frequency; it contains no nonlinear up-conversion. Any subsequent conversion of the escaping kilohertz–megahertz response to radio frequencies would be a separate physical stage.

The GW-memory spectrum calculation and the global propagation calculation play complementary roles. The GW-memory spectrum determines how strongly each frequency drives the plasma response. The global boundary-value problem determines whether the driven disturbance is a local, partly reflected plasma oscillation or an outgoing mode that reaches the edge of the source region. In the adopted background the latter occurs: the outgoing EM response is established mainly inside 107 cm and is transported to rout with a global-to-coherent energy ratio indistinguishable from unity at the accuracy of the calculation. This transfer cannot restore high-frequency components already suppressed in the GW-memory spectrum, and it does not select a narrower preferred band. The broad maxima in Fig. 4(b) are set by the central-engine energy-release timescale rather than by a resonance of the GW–EM conversion.

The GW-memory spectrum calculation also shows why replacing a nominal flare time by τacc/(2Γ2) is insufficient for a complete outflow. Arrival-time compression applies to the acceleration of a relativistic outflow element close to the line of sight. Angular integration adds outflow elements with longer delays and complex polarization factors, while material launched at different engine times retains the uncompressed timescale τeng. Consequently, 10–30 kHz is natural when the effective asymmetric energy is released over several to tens of microseconds. An appreciable response at hundreds of kilohertz or at megahertz frequencies instead requires a submicrosecond component that carries a non-negligible fraction of the asymmetric energy. The quantile f90 describes this shift without introducing an artificial cutoff at 1 MHz. This high-frequency extension is therefore a physical requirement to be tested, not a generic prediction for magnetar giant flares. A dynamical model that provides τeng, τacc, Γ, θj, and the fast participating fraction ηfast can be inserted directly into Eq. (4). A response extending to 1 MHz requires τeng ≲ 0.15 µs and appreciable asymmetric energy released on this timescale. If τeng is set by the light-crossing time of a single causally connected engine region, the corresponding causal scale would be of order 45 m. If global flare calculations instead give τeng ≳ 10 µs, relativistic acceleration by itself cannot prevent the response from declining above the 15-kHz scale.

The top-hat jet is a controlled representation of the net asymmetric outflow, not a complete magnetar-ejecta solution. It assumes one effective lobe, a constant energy per unit solid angle and terminal Lorentz factor inside the cone, a common acceleration time, and separable smooth temporal profiles for acceleration and central-engine energy release. A counterjet, angular structure, a distributed radial acceleration profile, precession, or time-dependent magnetization can change both H and its spectrum. The asymmetry parameter ϵ is likewise an effective projected quantity, and the model does not determine ηfast from first-principles flare dynamics. The parameter scan therefore maps the kinematic conditions required for a GW-memory spectrum extending to high frequencies; it is not an event-population or rate calculation.

Equation (13) removes the local-WKB assumption from the radial propagation problem because the global equation retains both wave directions, reflection, and exact shell phases. The constitutive closure is nevertheless local and deliberately minimal. The wavenumber is obtained from a cold, symmetric, one-dimensional electron–positron pair plasma with a prescribed magnetic field and density. A hot distribution can modify mode content and damping; charge imbalance, oblique propagation, QED vacuum polarization, nonlinear mode coupling, and a flare-created pair fireball are omitted. The scalar source normalization also does not supply the eigenvector overlap and absolute energy normalization of a self-consistent kinetic calculation. The agreement between Eglobal and Ecoh is therefore a statement about source-region transfer within this closure, not proof of high conversion efficiency in every flare environment. A more complete calculation should obtain the outflow timescales, angular structure, and ηfast from relativistic magnetar-flare dynamics and project the GW driving onto normalized plasma eigenmodes. It should then evolve the thermal pair distribution, field geometry, and flare loading before matching the source-region solution to an asymptotically free EM wave. Systematic variations of B∗, κ, Ṅ±, and Γw would identify the backgrounds in which the near-vacuum transfer found here ceases to hold. Equations (4) and (13), together with the angular and radial convergence tests, provide benchmarks for those extensions.

We have followed the complete source-region chain from asymmetric relativistic ejecta to an outgoing electromagnetic response. A magnetar giant flare first produces a step-like GW memory because the unbound outflow changes the asymptotic gravitational field. Representing the net asymmetric component as a relativistic top-hat jet allows the viewing-angle dependence, polarization cancellation, and arrival-time compression to be integrated explicitly. The memory waveform is shaped by two distinct temporal processes: acceleration of each outflow element and release of asymmetric energy by the central engine. The Fourier-domain factors associated with their temporal profiles determine the nonzero-frequency GW spectrum that drives the plasma. The GW-memory spectrum calculation establishes a hierarchy of spectral scales. Angular integration limits the point-particle Lorentz compression, and the uncompressed central-engine energy-release profile sets the high-frequency extent once it is included. For τeng = 10, 1, and 0.15 µs, the angularly integrated calculation gives f90 = 23.3 kHz, 233 kHz, and 1.51 MHz. As τeng decreases, the response shifts from tens of kilohertz toward the megahertz range; for τeng = 0.15 µs, 25.8% of the integrated response energy lies between 1 and 3 MHz. Reaching this extension requires submicrosecond structure in the asymmetric-energy release, rather than relativistic beaming alone; these timescales correspond to light-travel distances of tens of metres.

We next used the GW-memory spectrum as the distributed driver of a transverse X-mode/fast-magnetosonic response in a prescribed magnetized pair plasma. A global non-WKB wave equation, solved with radiative conditions excluding incoming waves, retains both propagation directions and possible reflection. The solution forms an outward mode, is established mainly in the inner source region, and reaches rout = 1013 cm. Across the broader 0.1 kHz–30 MHz frequency range, the global transfer is consistent with the coherent limit at the numerical resolution of the calculation. The modeled source-region plasma therefore does not impose an additional spectral cutoff; the declining high-frequency response is inherited from the acceleration and central-engine energy-release timescales that shape the GW-memory spectrum. This work shows that a step-like GW memory can drive a frequency-resolved EM response that escapes the modeled source region. The central-engine timescale and asymmetric fast-energy fraction determine whether this response is concentrated at tens of kilohertz or extends to megahertz frequencies. The 10 kHz–1 MHz band overlaps the frequency coverage of space-based plasma-wave instruments, so the escaping response may in principle be observable. Whether it reaches the Solar System with measurable amplitude depends on propagation through the outer magnetar environment and intervening plasma, absolute plasma-mode normalization, instrumental sensitivity, and the ambient plasma background. A quantitative assessment of the received flux, signal-to-noise ratio, event rate, and space-based detectability is left to a follow-up study.

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.