Dynamical Redshift Effect in Cosmology

• Dynamical Redshift Effect is a family of phenomena where redshift arises from evolving gravitational potentials, matter flows, and spacetime dynamics rather than static geometry.
• It plays a key role in cosmological observations, such as the late-time ISW effect in the CMB and redshift-space distortions in galaxy surveys, offering insights into dark energy and modified gravity.
• Observational techniques, including cross-correlation with large-scale structure and angular clustering methods, leverage the effect to constrain cosmological models and the growth of structure.

Searching arXiv for recent and directly relevant papers on the topic and related usages of “dynamical redshift effect.” The term “Dynamical Redshift Effect” is used in several distinct research contexts to denote a redshift or blueshift generated by time-dependent dynamics rather than by a purely static gravitational potential or a simple kinematic recession law. In contemporary cosmology, the most standard usage identifies it with the late-time Integrated Sachs–Wolfe (ISW) effect, in which cosmic microwave background (CMB) photons acquire a net energy shift while traversing evolving gravitational potentials (Krolewski et al., 2021). In other literatures, the same phrase has been applied to the parameterization dependence induced by a pivoting redshift in dynamical-dark-energy inference (Yang et al., 2018), to clock desynchronization under rigid acceleration via gravitational redshift in accelerated frames (Şahin, 2022), to redshift-space distortions generated by peculiar velocities in galaxies and clusters [(Kaeonikhom et al., 2018); (Tsujikawa et al., 2012); (Zheng et al., 2016); (Abdullah et al., 10 Dec 2025); (Shao et al., 24 Jul 2025)], to the redshift drift of FLRW cosmology (Lobo et al., 2022, Lobo et al., 2020), and to frequency shifts produced by spacetime dynamics in gravitational collapse (Koga et al., 14 Mar 2025). The expression is therefore best understood not as a single universal mechanism, but as a family of phenomena in which redshift is sourced by evolving geometry, evolving potentials, evolving dynamical states, or redshift-dependent parameter redefinitions.

1. Cosmological meaning: evolving gravitational potentials and the ISW effect

In the cosmological usage developed most explicitly for late-time structure, the Dynamical Redshift Effect is the net energy change of a CMB photon caused by propagation through a time-varying gravitational potential $\Phi$ (Krolewski et al., 2021). CMB photons redshift and blueshift as they move through gravitational potentials, and if the potential is not constant in time the gains and losses no longer cancel exactly. The result is a net redshift or blueshift identified as the Integrated Sachs–Wolfe effect (Krolewski et al., 2021).

This is distinct from the ordinary Sachs–Wolfe effect at last scattering. The primary Sachs–Wolfe contribution arises from static potentials at recombination and gives

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$

By contrast, the late-time ISW contribution depends on the temporal evolution of the potential along the photon trajectory: $$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$ or equivalently, in the form used in the cited work,

$$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$

In general relativity, $\Phi$ is linked to matter perturbations by

$$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$

During matter domination, linear-theory potentials are approximately constant because $\Phi \propto D(a)/a$ is nearly time independent, so $\dot{\Phi}\approx 0$. Once dark energy becomes dynamically important, or in modified-gravity scenarios where the relation between matter and metric potentials changes, $\dot{\Phi}\neq 0$ and the late-time ISW signal is generated (Krolewski et al., 2021).

The physical intuition is straightforward. During dark-energy domination, cosmic acceleration slows the growth of structure and causes large-scale potentials to decay. A photon falling into a potential well gains energy, but if the well is shallower when it climbs out, the photon retains a net blueshift. A decaying void produces the opposite sign. These shifts accumulate along the line of sight and are sourced predominantly at late times, $z\lesssim 2$, on large angular scales (Krolewski et al., 2021).

2. Growth, kernels, and observational detection in large-scale structure

Because the ISW signal is weak in the primary CMB auto-spectrum and is cosmic-variance limited at low multipoles, robust detection relies on cross-correlation with tracers of large-scale structure (Krolewski et al., 2021). In the cited analysis, the relevant cross-power spectrum is

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$0

with

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$1

and

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$2

The structure of $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$3 shows that the signal is controlled by $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$4, so it peaks around $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$5 and vanishes in matter domination (Krolewski et al., 2021). This is why tomography out to $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$6 and concentration on $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$7 are optimal.

The cited measurement uses Planck 2018 temperature maps cross-correlated with three tomographic unWISE galaxy samples spanning $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$8 (Krolewski et al., 2021). The samples are:

Sample	Mean redshift	Effective bias
Blue	$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$9	$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$0
Green	$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$1	$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$2
Red	$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$3	$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$4

The galaxy–CMB cross-correlation was estimated with a pseudo-$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$5 method using MASTER/NaMaster, with covariance from 300 Planck FFP10 end-to-end simulations and a Hartlap correction applied to the inverse covariance (Krolewski et al., 2021). The analysis explicitly included lensing magnification through

$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$6

where

$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$7

In the unWISE green and red bins, magnification contributes at the 15–20% level to the total ISW signal (Krolewski et al., 2021).

The amplitude was parameterized as

$$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$8

with $$\frac{\Delta T}{T}(\hat{\mathbf{n}})_{\rm ISW} = -\,2 \int_{\eta_*}^{\eta_0} d\eta\;\dot{\Phi}\!\left(\eta,\hat{\mathbf{n}}(\eta_0-\eta)\right),$$9 corresponding to the $$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$0CDM prediction. The results were

$$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$1

$$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$2

$$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$3

and a combined detection

$$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$4

at $$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$5, fully consistent with $$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$6CDM, with $$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$7 for 15 degrees of freedom (Krolewski et al., 2021).

3. Relation to dynamical dark energy and modified gravity

Because the ISW kernel is proportional to $$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$8, the late-time Dynamical Redshift Effect probes the time evolution of structure growth and of the metric potentials, not merely background distances (Krolewski et al., 2021). This makes it complementary to supernovae and baryon acoustic oscillations, which primarily constrain the background expansion.

The cited work studied a phenomenological freezing-quintessence model, the Mocker model, defined by

$$\left(\frac{\Delta T}{T}\right)_{\rm ISW} = -\,2 \int d\chi\; \dot{\Phi}(\chi).$$9

with priors $\Phi$0 and $\Phi$1 (Krolewski et al., 2021). Using CAMBSources with DarkEnergyPPF at fixed Planck 2018 background parameters, the study found that ISW alone constrains the evolution parameter $\Phi$2 most directly, while $\Phi$3 is pinned more strongly by distance probes. Combining unWISE–ISW with Pantheon supernovae and BAO yielded marginalized 95% upper limits

$\Phi$4

and constrained the dark-energy density to be within approximately 10% of a cosmological constant at $\Phi$5 for the Mocker family (Krolewski et al., 2021).

The same measurement also has significance for tests of gravity. ISW responds to the evolution of the metric potentials $\Phi$6 and $\Phi$7; in general relativity without anisotropic stress, $\Phi$8, whereas in modified gravity a gravitational slip $\Phi$9 or a modified Poisson factor $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$0 can alter $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$1 independently of background distances (Krolewski et al., 2021). The cited work did not fit explicit $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$2 or $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$3 parameterizations, but emphasized that the consistency of $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$4 disfavors large deviations in potential evolution on linear scales (Krolewski et al., 2021).

This use of “dynamical redshift” should not be confused with the pivoting-redshift effect in dark-energy parameter estimation. In generalized CPL models,

$$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$5

with

$$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$6

In that context, the “dynamical redshift effect” denotes the dependence of parameter covariance and inferred central values on the chosen pivot redshift $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$7; as $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$8 increases, the $$k^2\,\Phi(k,a) = 4\pi G\,a^2\,\rho_m(a)\,\delta_m(k,a).$$9–$\Phi \propto D(a)/a$0 correlation rotates from negative to positive, with $\Phi \propto D(a)/a$1 yielding near-zero correlation (Yang et al., 2018). This is a parameterization effect, not a photon-propagation effect.

4. Redshift-space distortions as a dynamical redshift phenomenon

A second major usage of the term concerns redshift-space distortions (RSD). In this setting, the observed redshift of a galaxy combines cosmological expansion with a Doppler shift from its line-of-sight peculiar velocity. To first order,

$\Phi \propto D(a)/a$2

and the mapping from real space $\Phi \propto D(a)/a$3 to redshift space $\Phi \propto D(a)/a$4 is

$\Phi \propto D(a)/a$5

This induces anisotropic clustering and is therefore a dynamical redshift effect in the sense that the inferred line-of-sight position is altered by the internal and large-scale dynamics of matter [(Abdullah et al., 10 Dec 2025); (Kaeonikhom et al., 2018); (Tsujikawa et al., 2012)].

In linear theory, the Kaiser approximation gives

$\Phi \propto D(a)/a$6

or for biased tracers,

$\Phi \propto D(a)/a$7

where $\Phi \propto D(a)/a$8 is the growth rate (Kaeonikhom et al., 2018). For dynamical dark energy, $\Phi \propto D(a)/a$9, $\dot{\Phi}\approx 0$0, and the higher-order Lagrangian growth functions $\dot{\Phi}\approx 0$1, $\dot{\Phi}\approx 0$2, $\dot{\Phi}\approx 0$3 become time dependent. The cited Lagrangian-perturbation analysis showed that including these time-dependent growth functions can alter the nonlinear power spectrum by up to $\dot{\Phi}\approx 0$4, with typical changes of $\dot{\Phi}\approx 0$5 over $\dot{\Phi}\approx 0$6–$\dot{\Phi}\approx 0$7 at $\dot{\Phi}\approx 0$8 (Kaeonikhom et al., 2018).

A related analytic framework derives

$\dot{\Phi}\approx 0$9

for general-relativistic dynamical-dark-energy models under the assumptions that the dark-energy sound speed is not much smaller than unity and that $\dot{\Phi}\neq 0$0 does not vary significantly (Tsujikawa et al., 2012). There the growth equation

$\dot{\Phi}\neq 0$1

leads to an analytic expression for $\dot{\Phi}\neq 0$2 in terms of $\dot{\Phi}\neq 0$3 and the growth index $\dot{\Phi}\neq 0$4 (Tsujikawa et al., 2012). This identifies RSD as a probe of dark-energy-driven changes in growth, rather than only of geometry.

On nonlinear scales, random motions inside halos generate the Finger-of-God effect, a further dynamical redshift distortion. In the multi-streaming analysis of RSD, the observed damping can be written

$\dot{\Phi}\neq 0$5

where the multi-streaming component was directly measured in simulations and shown to be non-negligible, with $\dot{\Phi}\neq 0$6 already at $\dot{\Phi}\neq 0$7 at $\dot{\Phi}\neq 0$8 and at $\dot{\Phi}\neq 0$9 at $z\lesssim 2$0 (Zheng et al., 2016). This work argued that an improved understanding of the FoG effect helps break the $z\lesssim 2$1–$z\lesssim 2$2 degeneracy in RSD cosmology (Zheng et al., 2016).

5. Cluster dynamics, angular homogeneity, and practical RSD applications

In spectroscopic cluster studies, the dynamical redshift effect appears as the line-of-sight broadening produced by cluster member peculiar velocities. In VIPERS, cluster candidates were detected as overdense regions in redshift space through the Finger-of-God effect, with the mapping

$z\lesssim 2$3

used implicitly to identify redshift-space elongation (Abdullah et al., 10 Dec 2025). Membership was then assigned in projected phase space $z\lesssim 2$4, where

$z\lesssim 2$5

The same redshift-space dynamics were turned into cluster observables such as $z\lesssim 2$6, $z\lesssim 2$7, and $z\lesssim 2$8 through the projected virial estimator

$z\lesssim 2$9

together with a surface-pressure correction (Abdullah et al., 10 Dec 2025). The resulting $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$00–$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$01 relation,

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$02

with intrinsic scatter $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$03, was consistent with the self-similar expectation and with simulation results (Abdullah et al., 10 Dec 2025). Here the dynamical redshift effect is not a nuisance alone; it is the signal from which the mass proxy is extracted.

A different mitigation strategy appears in angular clustering. The angular correlation dimension $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$04 is inherently less sensitive to small-scale FoG distortions than full 3D statistics, because it is cumulative and angularly projected (Shao et al., 24 Jul 2025). The cited analysis showed that restricting to minimum comoving angular scales of approximately $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$05, corresponding to $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$06 in standard $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$07CDM, significantly reduces FoG systematics (Shao et al., 24 Jul 2025). Applying the method to SDSS DR12 and DR16 LRG data yielded

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$08

fully consistent with CMB analyses (Shao et al., 24 Jul 2025). This suggests that some dynamical redshift effects can be suppressed by statistic design rather than by increasingly detailed nonlinear modeling.

6. Other uses: redshift drift, accelerated frames, and dynamical spacetime

Beyond late-time structure and RSD, the phrase also appears in more formal settings involving explicitly time-dependent spacetimes or accelerated observers.

In FLRW cosmology, redshift drift is the slow time evolution of the observed cosmological redshift between comoving source and observer. The exact result is

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$09

with associated spectroscopic velocity drift

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$10

This is an exact FLRW identity and supplies a direct differential probe of the expansion history rather than an integrated-distance probe (Lobo et al., 2022). In purely cosmographic terms, the low-$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$11 behavior is

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$12

so observation of the drift is directly related to a nonzero deceleration parameter (Lobo et al., 2020). The dynamical redshift effect here is literal time evolution of cosmological redshift on observer timescales comparable to the Hubble time (Lobo et al., 2022, Lobo et al., 2020).

In accelerated-frame relativity, a different dynamical redshift effect was proposed in which gravitational redshift in the accelerating frame is necessary to recover special-relativistic clock desynchronization after rigid acceleration (Şahin, 2022). In Rindler–Kottler–Møller coordinates, the induced metric

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$13

implies

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$14

The accumulated gravitational offset during acceleration,

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$15

cancels the discrepancy between a purely kinematic first-pass derivation and the standard special-relativistic simultaneity offset, yielding

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$16

after the boost (Şahin, 2022). This suggests a strong claim advanced in that work: special-relativistic clock desynchronization, if derived dynamically from the acceleration process, requires an equivalence-principle gravitational redshift term (Şahin, 2022).

In gravitational collapse, the phrase describes a path-integrated energy shift caused by spacetime dynamics rather than by static curvature. For spherically symmetric collapse, the cited work defines a Kodama energy

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$17

and derives the covariant rate

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$18

This was interpreted as the strong-field analogue of the ISW/Rees–Sciama mechanism: time-varying gravitational fields during collapse generally produce redshift, although blueshift can occur in some cases (Koga et al., 14 Mar 2025). The same work argued that such dynamical redshift is crucial for shadow formation in the collapse of a transmissive object, because outgoing rays near horizon formation satisfy $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$19 and hence become infinitely redshifted (Koga et al., 14 Mar 2025).

A plausible implication of this broader literature is that “dynamical redshift effect” functions as an umbrella label for integrated frequency shifts generated by temporal evolution—whether that evolution is in the cosmic background, a gravitational potential, a matter velocity field, or a local spacetime geometry.

7. Conceptual scope, distinctions, and misconceptions

Several distinct mechanisms are often conflated.

First, the late-time ISW effect is not the same as the primary Sachs–Wolfe effect. The former requires $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$20 along the path and is sourced at late times, predominantly $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$21; the latter arises from static potentials on the last-scattering surface (Krolewski et al., 2021).

Second, redshift-space distortions are not gravitational redshifts. They are produced by peculiar velocities superposed on Hubble expansion and are encoded through the real-to-redshift-space mapping of galaxy positions (Kaeonikhom et al., 2018, Abdullah et al., 10 Dec 2025, Zheng et al., 2016). In cluster applications, gravitational-redshift contributions are explicitly noted as subdominant to the $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$22–$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$23 peculiar velocities and negligible for the $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$24-based mass inference performed there (Abdullah et al., 10 Dec 2025).

Third, the pivoting-redshift effect in CPL parameterization is not a propagation effect at all. It is a covariance-rotation effect induced by the choice of the anchor redshift $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$25 in the dark-energy equation of state, with $\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$26 acting as a decorrelation pivot for the data combination studied (Yang et al., 2018).

Fourth, the redshift drift is not an RSD or ISW signal. It is the observer-time derivative of the cosmological redshift itself in FLRW spacetime, and its leading behavior is controlled by the deceleration parameter (Lobo et al., 2022, Lobo et al., 2020).

Finally, some works apply the label to speculative or nonstandard mechanisms. One paper attributes cosmological redshift to a QED-motivated cumulative attenuation law

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$27

rather than to Doppler recession or metric expansion (Hebel, 2011). Another proposes a magnetically induced redshift

$\frac{\Delta T}{T} = -\,\frac{1}{3}\,\Phi .$28

via photon energy loss into gravitational waves in a constant magnetic field (Abdelali et al., 2018). These usages are part of the literature record, but they are conceptually separate from the standard ISW, RSD, and FLRW redshift-drift frameworks.

Taken together, the literature indicates that the most established meaning of Dynamical Redshift Effect is the cosmological one associated with evolving metric potentials and the ISW effect (Krolewski et al., 2021). More broadly, the phrase designates redshift phenomena whose origin lies in dynamical evolution rather than in static geometry alone. This suggests a unifying editorial definition: the Dynamical Redshift Effect is any net frequency shift acquired because the relevant spacetime, gravitational potential, matter flow, or parameter anchoring evolves during propagation, inference, or observation.