Dispersion Measure Gradients

• Dispersion Measure Gradients are defined as variations in the line-of-sight free-electron column density, observable as temporal changes or transverse spatial gradients.
• They are measured via chromatic timing offsets from multi-frequency observations, revealing underlying plasma structures in pulsar, FRB, and cosmological contexts.
• Advanced techniques such as scintillometry and joint DM plus achromatic fitting correct timing errors and probe both local and large-scale ionized environments.

Dispersion measure gradients are variations in the line-of-sight free-electron column density that manifest either as temporal changes, usually written $d\mathrm{DM}/dt$, or as transverse spatial gradients, $\nabla_{\perp}\mathrm{DM}$, across neighboring sightlines. Since the cold-plasma group delay obeys $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$, with $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$, DM gradients generate chromatic timing perturbations, refractive astrometric shifts, and, in sufficiently rich samples, anisotropies in DM-based cosmological observables. In pulsar timing they are operationally important because they couple AU-scale plasma structure to microsecond-to-nanosecond timing errors; in fast radio burst and cosmological applications they encode fluctuations in the ionized baryon distribution, host environments, and source-proximate plasma (Keith et al., 2012, Jones et al., 2016, Alonso, 2021).

1. Fundamental definition and kinematic formulation

The dispersion measure is the electron-column integral along the propagation path,

$\mathrm{DM} \equiv \int n_e(l)\,dl,$

with units $\mathrm{pc\ cm^{-3}}$. A transverse DM gradient is the spatial derivative across the plane of the sky,

$$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$

and is often quoted either per AU or per milliarcsecond, depending on the measurement geometry (Baker et al., 21 Aug 2025). The corresponding dispersive delay is

$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$

so a differential gradient sampled across two nearby sightlines separated by $\Delta \boldsymbol{\theta}$ produces

$$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$

In timing data, the operational observable is frequently the temporal derivative $\nabla_{\perp}\mathrm{DM}$0, inferred from epoch-by-epoch DM estimates (Keith et al., 2012, Jones et al., 2016, Baker et al., 21 Aug 2025).

A standard kinematic interpretation connects temporal and spatial behavior. Under a frozen-flow screen approximation, or more generally when transverse motion of the line of sight dominates, temporal drift and spatial structure are related by

$\nabla_{\perp}\mathrm{DM}$1

where $\nabla_{\perp}\mathrm{DM}$2 is the effective transverse velocity formed from the Earth, the pulsar, and any screen motion projected into the screen plane (Jones et al., 2016). For a persistent linear gradient $\nabla_{\perp}\mathrm{DM}$3, the DM time series takes the form

$\nabla_{\perp}\mathrm{DM}$4

and the chromatic timing perturbation is

$\nabla_{\perp}\mathrm{DM}$5

Because $\nabla_{\perp}\mathrm{DM}$6 contains both proper-motion drift and Earth’s annual orbit, a persistent gradient generically produces a linear trend plus an annual modulation (Keith et al., 2012).

This distinction between $\nabla_{\perp}\mathrm{DM}$7 and $\nabla_{\perp}\mathrm{DM}$8 is not merely terminological. The former is cadence- and estimator-dependent, whereas the latter is a property of the plasma screen or integrated medium sampled by the observation. In practice, most classical pulsar-timing analyses measure $\nabla_{\perp}\mathrm{DM}$9 and infer transverse structure indirectly, while scintillometric techniques can access $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$0 directly from a single epoch (Baker et al., 21 Aug 2025).

2. Physical origins of DM gradients

The dominant astrophysical origin of DM gradients in pulsar timing is spatial structure in ionized plasma on AU scales. As the line of sight moves relative to that structure, the sampled electron column changes. The motions explicitly identified as relevant are Earth’s annual orbit, pulsar proper motion, and, in principle, bulk motion of the plasma, although the latter is often neglected in first-pass analyses (Keith et al., 2012). In this picture, annual periodicity is the cleanest signature of a persistent transverse gradient, while monotonic drifts indicate either large-scale gradients sampled by proper motion or longer-term stochastic structure.

Turbulence provides the baseline statistical framework. For Kolmogorov turbulence, the DM structure function obeys $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$1, and equivalent formulations are given for the dispersive-delay structure function $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$2 and its power spectrum $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$3 (Keith et al., 2012, Petroff et al., 2013). However, several data sets show that observed DM variability is often steeper than a Kolmogorov extrapolation from diffractive scintillation scales. In the Parkes Pulsar Timing Array analysis, five pulsars were identified as inconsistent with Kolmogorov extrapolation from diffractive scales, and the behavior was linked to strong linear trends in $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$4 and nearly quadratic structure functions, consistent with dominant large-scale gradients rather than purely stochastic turbulence (Keith et al., 2012). In the six-year Parkes sample of 168 mostly young pulsars, three of the four strongest detections exceeded Kolmogorov expectations and were attributed to local environments rather than the ambient ISM (Petroff et al., 2013).

Localized plasma structures produce sharper signatures. A discrete change in the DM of PSR J1603−7202 of $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$5 lasting about 250 days was interpreted as consistent with an AU-scale high-density plasma structure analogous to an extreme scattering event. Under the assumption of a screen halfway to the pulsar, the line of sight traverses about $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$6 AU during the event, implying a mean electron density of about $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$7 and a transverse gradient magnitude

$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$8

with timing amplitudes of about $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2}$9 at $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$0 MHz and $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$1 at $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$2 MHz (Keith et al., 2012). Such events are qualitatively distinct from smooth annual sinusoids, but they are part of the same underlying category: transverse gradients in the free-electron column.

Solar-wind geometry is an additional, sometimes dominant, source of annual DM structure. In the NANOGrav nine-year data set, several MSPs near the ecliptic showed strong solar-angle correlations, and the adopted spherical $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$3 wind model implies $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$4, where $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$5 is the line-of-sight impact parameter relative to the Sun (Jones et al., 2016). This annual component is operationally separable from interstellar gradients but can mimic them if the solar term is not modeled explicitly.

A persistent misconception is that all DM trends directly trace homogeneous Kolmogorov turbulence. The available data do not support that simplification. Local SNR and PWN environments, solar-wind crossings, discrete ESE-like structures, and gradient-dominated screens all contribute in specific sources, often with amplitudes well above canonical Kolmogorov expectations (Petroff et al., 2013, Jones et al., 2016).

3. Estimation, correction, and inference methodology

High-precision work measures DM gradients through multi-frequency timing residuals, but the estimator itself is nontrivial. In the Parkes PTA analysis, residuals at each epoch and wavelength were modeled as

$$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$6

with chromatic and achromatic terms represented as piecewise-linear time series and solved jointly by generalized least squares. Linear constraints were imposed so that the DM basis was orthogonal to the mean DM and the achromatic common mode was orthogonal to the quadratic timing model and astrometric terms. The piecewise-linear sampling interval $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$7 acts as a low-pass filter with transfer function

$$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$8

and was chosen by comparing the red DM-induced delay spectrum to the white-noise spectrum of the DM estimator (Keith et al., 2012).

The motivation for the joint fit is that a “simple” DM-only correction is biased in the presence of achromatic red processes. When only a $$K \approx 4.148808 \times 10^{3}\ \mathrm{s}\ \mathrm{MHz}^2\ \mathrm{cm}^3\ \mathrm{pc}^{-1}$$9 term is fitted, common-mode signals such as a gravitational-wave background, clock errors, ephemeris errors, or intrinsic pulsar red noise leak into the DM estimate. The Parkes PTA study gave the expectation value of the biased estimator as

$\mathrm{DM} \equiv \int n_e(l)\,dl,$0

showing explicit contamination of the chromatic channel by an achromatic process. Simulations demonstrated that subtracting this biased DM estimate attenuates low-frequency gravitational-wave power, whereas the joint CM+DM fit preserves the achromatic signal and instead pays the unavoidable price of added band-limited white noise below the chosen corner frequency (Keith et al., 2012).

Practical implementations differ by data set. In the NANOGrav nine-year release, epoch-wise DM estimation used the TEMPO DMX parameter with 14-day windows; trend models were then selected among linear, annual, or combined forms by minimizing post-fit reduced $\mathrm{DM} \equiv \int n_e(l)\,dl,$1, with piecewise fits used when trend changes occurred (Jones et al., 2016). In the Parkes sample of 168 pulsars, weighted least-squares linear fits to DM versus time were performed separately for single-band $\mathrm{DM} \equiv \int n_e(l)\,dl,$2-cm and simultaneous multi-frequency series, with detections defined by $\mathrm{DM} \equiv \int n_e(l)\,dl,$3 slopes and “highly significant” detections by $\mathrm{DM} \equiv \int n_e(l)\,dl,$4 plus consistency requirements between the two estimators (Petroff et al., 2013).

Wide-band simultaneity adds another layer of caution. Truly simultaneous LOFAR, Lovell, and Effelsberg observations across $\mathrm{DM} \equiv \int n_e(l)\,dl,$5–$\mathrm{DM} \equiv \int n_e(l)\,dl,$6 MHz showed that the cold-plasma dispersion law is accurate to better than 1 part in 100000 across the band, and that apparent band-dependent DMs can arise from pulse profile evolution rather than any physical chromatic DM gradient. After wide-band, frequency-dependent template modeling, the DMs became consistent across bands for all four pulsars analyzed, whereas static templates had produced spurious band-dependent trends (Hassall et al., 2012). This directly constrains a second misconception: not every frequency-dependent timing residual is evidence for a DM gradient.

The methodological consequence is that DM-gradient inference is inseparable from chromatic calibration. Reliable estimation requires close simultaneity across bands, explicit achromatic common-mode modeling when PTA signals are relevant, and profile-evolution treatment sufficiently accurate that intrinsic pulse-shape changes are not projected into the DM estimator (Keith et al., 2012, Hassall et al., 2012).

4. Observational phenomenology in pulsar samples

Population studies show that DM gradients are common but heterogeneous. In the six-year Parkes study of 168 mostly young pulsars, DM variations were detected at the $\mathrm{DM} \equiv \int n_e(l)\,dl,$7 level in 11 pulsars, with four above $\mathrm{DM} \equiv \int n_e(l)\,dl,$8: J0835−4510, J0908−4913, J1824−1945, and J1833−0827. In the NANOGrav nine-year data set of 37 MSPs, intrinsic variability was found in nearly all pulsars; five showed annual periodicities only, 14 showed monotonic trends only, 13 showed both linear and annual components, and five required piecewise fits because of sign changes or onset/offset behavior (Petroff et al., 2013, Jones et al., 2016).

Several well-characterized sources illustrate the range of behaviors:

Pulsar	$\mathrm{DM} \equiv \int n_e(l)\,dl,$9 $\mathrm{pc\ cm^{-3}}$0 in $\mathrm{pc\ cm^{-3}}$1	Context
J0835−4510	$\mathrm{pc\ cm^{-3}}$2	DM increasing after a 28-year decline
J0908−4913	$\mathrm{pc\ cm^{-3}}$3	Above Kolmogorov expectation; local PWN favored
J1824−1945	$\mathrm{pc\ cm^{-3}}$4	Consistent with Kolmogorov turbulence
J1833−0827	$\mathrm{pc\ cm^{-3}}$5	Largest gradient in sample; local PWN/SNR favored

For J0835−4510, the six-year trend is an increase in DM after decades of gradual decline. The earlier long decline was associated with a filament in the Vela environment, and the later modest increase was argued to be compatible with ambient ISM turbulence after that filament moved out of the line of sight. For J0908−4913 and J1833−0827, the measured gradients exceeded turbulence-based expectations by more than two orders of magnitude and by about an order of magnitude, respectively, which motivated interpretations involving the local bow-shock PWN or SNR environment. J1824−1945, by contrast, remained consistent with a Kolmogorov interpretation (Petroff et al., 2013).

The NANOGrav sample emphasizes timescale diversity. Using a characteristic variation timescale defined from the DM rate or annual amplitude, 18 MSPs had $\mathrm{pc\ cm^{-3}}$6 between $\mathrm{pc\ cm^{-3}}$7 and $\mathrm{pc\ cm^{-3}}$8 days, and 10 more had $\mathrm{pc\ cm^{-3}}$9 shorter than the monthly cadence. Specific examples include J1909−3744 with $$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$0 d, J2317+1439 with $$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$1 d, and B1855+09 with $$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$2 d (Jones et al., 2016). Such values imply that cadence, not just total baseline, can limit DM-gradient recovery.

Structure functions complicate interpretation. In the NANOGrav analysis, five pulsars had very nearly quadratic structure functions, four were roughly Kolmogorov, four were less steep than Kolmogorov, and one had uncertainty too large for classification. The analysis argued that linear trends, discrete structures, finite baseline, and white-noise domination at short lags can steepen or distort inferred SF slopes, so near-quadratic behavior does not by itself imply a non-Kolmogorov medium (Jones et al., 2016). This is an important interpretive caution because gradient-dominated time series can masquerade as unusual turbulence if deterministic components are not separated first.

5. Direct small-scale measurements from scintillometry

Scintillometry offers a distinct route to DM gradients by using multipath propagation itself as the measurement apparatus. For PSR B0834+06, a thin, highly anisotropic scattering screen creates multiple closely spaced effective sightlines, allowing simultaneous differential DM estimation from a single observation. The recovered image-by-image delays satisfy

$$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$3

and Doppler shifts along the main arc map to angular offsets through

$$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$4

with $$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$5 and screen orientation $$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$6 for the B0834+06 screen (Baker et al., 21 Aug 2025).

Two complementary estimators were applied. Image tracking yielded

$$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$7

while arc correlation gave

$$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$8

Conversion using the screen geometry produced an angular gradient

$$\nabla_{\perp}\mathrm{DM} \equiv \frac{\partial \mathrm{DM}}{\partial \boldsymbol{\theta}},$$9

aligned with the screen orientation. Across several weeks, the gradient showed a linear trend

$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$0

and, using the measured arc curvature, a DM time derivative along the scattering direction was inferred: $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$1 These quantities are single-epoch or few-epoch measurements of genuinely transverse DM structure, not just temporal slopes inferred from cadence-limited monitoring (Baker et al., 21 Aug 2025).

The same gradient acts as a plasma prism and therefore has astrometric as well as timing consequences. The measured B0834+06 gradient implies a refractive position shift of

$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$2

and, if the observed change persists, a proper-motion bias of

$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$3

along the screen axis. Timing offsets associated with the chromatic position shift are estimated as $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$4–$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$5 ns across the $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$6–$t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$7 MHz band and up to about $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$8 ns between $t_{\mathrm{DM}}(\nu)=K\,\mathrm{DM}\,\nu^{-2},$9 and $\Delta \boldsymbol{\theta}$0 MHz (Baker et al., 21 Aug 2025). This establishes that DM gradients need not be treated purely as nuisance parameters in TOA fitting; they can also be direct probes of sub-AU plasma structure and of chromatic astrometric bias.

The broader implication is that conventional DM time series and scintillometric gradients are complementary rather than redundant. The former sample motion through plasma along the proper-motion/orbital trajectory; the latter resolve the local transverse derivative along the scattering axis. Combining them plausibly enables a two-dimensional gradient reconstruction on the sky, a possibility explicitly noted for B0834+06 (Baker et al., 21 Aug 2025).

6. FRBs, host environments, and cosmological DM-gradient observables

In FRB and cosmological settings, DM gradients generalize from AU-scale screens to line-of-sight integrals over structured electron fields. In linear cosmological perturbation theory, the mean DM to redshift $\Delta \boldsymbol{\theta}$1 is

$\Delta \boldsymbol{\theta}$2

and the dominant anisotropic fluctuation is the projected free-electron overdensity,

$\Delta \boldsymbol{\theta}$3

with relativistic potential and ISW-like terms present but small. The angular DM-gradient variance is

$\Delta \boldsymbol{\theta}$4

so gradients weight intermediate and high multipoles rather than the very largest angular scales (Alonso, 2021). A central result is that practical FRB DM-gradient cosmology is dominated by electron-density fluctuations, while relativistic effects are negligible except on very low multipoles (Alonso, 2021).

This viewpoint extends to three-dimensional clustering in “DM space.” If the radial coordinate of an FRB is inferred from DM, the mapping is perturbed by integrated electron fluctuations and relativistic effects, producing DM-space distortions analogous to redshift-space distortions. The formalism of DM-space clustering shows that even multipoles are dominated by non-local integrated electron-density terms, while the dipole receives a significant Doppler contribution. For SKA-like samples, even multipoles were forecast at $\Delta \boldsymbol{\theta}$5, with a possible first dipole detection at $\Delta \boldsymbol{\theta}$6 in FRB–galaxy cross-correlations for $\Delta \boldsymbol{\theta}$7 FRBs (Saga et al., 2024). In this sense, cosmological DM gradients are not merely nuisances to the FRB Hubble diagram; they are observables of the large-scale electron distribution.

Host galaxies add a separate, redshift- and impact-parameter-dependent gradient hierarchy. Using IllustrisTNG, the host contribution to observed DM, already divided by $\Delta \boldsymbol{\theta}$8, increases from $\Delta \boldsymbol{\theta}$9 at $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$0 to $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$1 at $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$2, with a fit

$$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$3

The host term also falls with projected distance from the host center: at $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$4, the ratios $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$5 are about $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$6 at $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$7 kpc, $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$8 at $$\Delta t(\nu)=K\,(\nabla_{\perp}\mathrm{DM}\cdot \Delta \boldsymbol{\theta})\,\nu^{-2}.$$9 kpc, $\nabla_{\perp}\mathrm{DM}$00 at $\nabla_{\perp}\mathrm{DM}$01 kpc, and $\nabla_{\perp}\mathrm{DM}$02 at $\nabla_{\perp}\mathrm{DM}$03 kpc (Jaroszynski, 2020). Thus, FRB host gradients are concrete rather than schematic: they introduce a redshift-dependent mean offset, redshift-dependent variance, and strong radial gradients within the host halo/ISM/CGM complex.

A different source-local class of DM gradients appears in magnetar-wind models of repeating FRBs. In the filamentation scenario, the FRB propagates through an ultrarelativistic pair wind and reorganizes the plasma into periodically spaced thin sheets separated by near-vacuum gaps. The resulting dispersion relation is waveguide-like, with effective plasma frequency

$\nabla_{\perp}\mathrm{DM}$04

set by sheet spacing rather than mean density. The effective wind DM can then exceed homogeneous-wind estimates by orders of magnitude and becomes burst-dependent through the radio luminosity, observing frequency, Lorentz factor, and X-ray heating parameters. The paper gives bounds

$\nabla_{\perp}\mathrm{DM}$05

and

$\nabla_{\perp}\mathrm{DM}$06

with the upper bound reaching $\nabla_{\perp}\mathrm{DM}$07 in the parameter regime discussed for FRB 20190520B (Sobacchi et al., 2022). These are radial and burst-to-burst DM gradients in a source-proximate plasma rather than in the ISM or IGM.

A reasonable synthesis is that “DM gradient” denotes a family of observables spanning at least three regimes: AU-scale pulsar screens, galactic/halo structure around FRBs, and large-scale electron-density anisotropies in cosmology. The mathematics changes with scale, but the common object is the spatial derivative of an electron-column integral and its chromatic imprint on pulse propagation (Alonso, 2021, Saga et al., 2024, Jaroszynski, 2020, Sobacchi et al., 2022).