arxiv: 2604.26690 · v1 · submitted 2026-04-29 · 🌌 astro-ph.CO · gr-qc· hep-th

Recognition: unknown

Nonlinear Relativistic Effects on Cosmological Redshift Drift

Pierre B\'echaz , Giuseppe Fanizza , Giovanni Marozzi , Matheus R. Medeiros Silva

Authors on Pith no claims yet

Pith reviewed 2026-05-07 10:58 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th

keywords cosmological redshift driftrelativistic perturbationsgauge-invariant methodssecond-order effectsbispectrumlight-cone coordinatesredshift-space distortions

0 comments

The pith

A gauge-invariant perturbative calculation shows relativistic effects on cosmological redshift drift only at second order, with enhanced nonlinear bispectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the first computation of relativistic effects on the cosmological redshift drift to second order in perturbation theory. The calculation relies on a gauge-invariant approach with specially chosen light-cone coordinates to handle light propagation through an inhomogeneous universe. Redshift-space distortions are found to be a purely second-order phenomenon. The authors derive the leading bispectrum terms on sub-Hubble scales and report that nonlinear contributions are amplified more than the squared linear power spectrum at low redshift for large momenta.

Core claim

Employing light-cone coordinates within a fully gauge-invariant framework, relativistic corrections to the redshift drift are calculated up to second order in cosmological perturbation theory. This reveals that redshift-space distortions occur solely as a second-order effect. Analytical expressions are obtained for the bispectrum of the leading perturbative contributions on sub-Hubble scales, and numerical results show that at low redshift and large momenta the non-linearities in the bispectrum exceed the squared power spectrum.

What carries the argument

Light-cone coordinates that simplify light propagation in an inhomogeneous and anisotropic universe, facilitating a gauge-invariant second-order calculation of redshift drift effects.

If this is right

Redshift-space distortion is absent at linear order but appears at second order.
Analytical bispectrum expressions are derived for leading-order terms on sub-Hubble scales.
Non-linearities in the bispectrum are enhanced relative to the squared power spectrum at low redshift and large momenta.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The coordinate choice may simplify similar second-order calculations for other cosmological observables involving light propagation.
These findings suggest that second-order terms should be accounted for in interpreting future high-precision redshift drift observations to extract accurate cosmological information.
Testing the bispectrum predictions against full numerical ray-tracing in perturbed metrics would validate the analytical approach.

Load-bearing premise

The selected light-cone coordinates allow an accurate description of light propagation without uncontrolled approximations at second order in an inhomogeneous and anisotropic universe.

What would settle it

A direct numerical computation of photon geodesics in a second-order perturbed spacetime yielding a redshift drift bispectrum without the reported nonlinear enhancement would disprove the analytical result.

read the original abstract

Using a fully gauge-invariant approach, we compute for the first time in the literature relativistic effects on the redshift drift up to second order in cosmological perturbation theory. This is achieved by employing a set of light-cone coordinates that simplify the description of light propagation in an inhomogeneous and anisotropic universe. We show that redshift-space distortion occurs only as a second-order effect whereas, as known, it is not present among the linear perturbations. We then derive analytical expressions of the bispectrum for the leading-order perturbative contributions on sub-Hubble scales, providing some numerical evaluations. Our finding is that, at low redshift and for large momenta, the non-linearities in the bispectrum are enhanced more than the squared power spectrum.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

First calculation of second-order relativistic corrections to redshift drift via light-cone coordinates, with analytical bispectrum expressions, but the coordinate consistency at O(2) needs explicit verification.

read the letter

This paper computes second-order relativistic effects on cosmological redshift drift for the first time. The authors use a gauge-invariant perturbation setup with light-cone coordinates to handle light propagation in an inhomogeneous universe, recover that redshift-space distortions only appear at second order, and derive analytical bispectrum expressions for the leading perturbative terms on sub-Hubble scales, plus some numerical evaluations. At low redshift and large momenta the bispectrum nonlinearities come out stronger than the squared power spectrum. The analytical expressions and the clean separation of orders are the parts that stand out as useful. The work follows standard perturbation theory without obvious circularity or invented parameters. The potential weak point is the light-cone coordinate choice itself. The transformation is second-order, so the metric components, null geodesic equation, and redshift definition all need consistent expansion; any missed cross terms between first-order perturbations and coordinate shifts could leak into the bispectrum. The abstract claims simplification but does not show the explicit cancellation checks, so that part of the derivation requires close reading. This is for people working on precision cosmology observables and higher-order relativistic corrections for future surveys. A reader already doing bispectrum or drift calculations will get concrete expressions to build on. It deserves peer review because it supplies new analytical results that fill a documented gap, even if the coordinate details may need tightening.

Referee Report

1 major / 1 minor

Summary. The paper claims to compute relativistic effects on cosmological redshift drift to second order in perturbation theory for the first time, using a fully gauge-invariant approach based on light-cone coordinates that simplify light propagation in an inhomogeneous universe. It shows that redshift-space distortions appear only at second order, derives analytical bispectrum expressions for leading contributions on sub-Hubble scales, and reports numerical evaluations indicating that non-linearities in the bispectrum are enhanced relative to the squared power spectrum at low redshift and large momenta.

Significance. If the central derivation holds, the work supplies the first analytical second-order treatment of relativistic corrections to redshift drift together with explicit bispectrum formulae. These results would be directly relevant to precision measurements of redshift drift with future surveys and would extend the gauge-invariant perturbation toolkit to an observable that is linear at first order but receives distinctive second-order contributions.

major comments (1)

[light-cone coordinate construction and null-geodesic integration] The weakest link is the assertion that the chosen light-cone coordinates render null-geodesic integration and the redshift definition free of uncontrolled O(2) errors while preserving gauge invariance. The coordinate transformation itself is second-order; all cross terms between first-order metric perturbations and first-order coordinate shifts must be shown to cancel or to be absorbed into gauge-invariant combinations in the geodesic equation, the redshift formula, and the observer four-velocity. Without an explicit verification (e.g., by displaying the full O(2) expansion of the connection coefficients or by demonstrating cancellation in the final bispectrum), the derived expressions may contain spurious contributions that undermine the claim of a clean second-order result.

minor comments (1)

[numerical results] The numerical evaluations of the bispectrum should specify the exact range of redshifts, wavenumbers, and cosmological parameters used, together with the integration method and convergence checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address the major comment regarding the light-cone coordinate construction below, and we will revise the manuscript to provide the requested explicit verification.

read point-by-point responses

Referee: [light-cone coordinate construction and null-geodesic integration] The weakest link is the assertion that the chosen light-cone coordinates render null-geodesic integration and the redshift definition free of uncontrolled O(2) errors while preserving gauge invariance. The coordinate transformation itself is second-order; all cross terms between first-order metric perturbations and first-order coordinate shifts must be shown to cancel or to be absorbed into gauge-invariant combinations in the geodesic equation, the redshift formula, and the observer four-velocity. Without an explicit verification (e.g., by displaying the full O(2) expansion of the connection coefficients or by demonstrating cancellation in the final bispectrum), the derived expressions may contain spurious contributions that undermine the claim of a clean second-order result.

Authors: We appreciate the referee's identification of this potential issue. Our light-cone coordinate system is designed from the outset to be aligned with the null geodesics in the perturbed spacetime, allowing us to express all quantities in terms of gauge-invariant perturbations. The first-order coordinate shifts are chosen such that they cancel the gauge-dependent parts at linear order, and at second order, the cross terms are systematically absorbed into the invariant combinations that enter the redshift drift and the bispectrum. Nevertheless, to make this explicit and remove any doubt, we will add a new appendix to the revised manuscript that presents the full second-order expansion of the relevant connection coefficients and demonstrates the cancellation of non-invariant terms in the geodesic equation and redshift formula. This will also include a check that the final bispectrum expressions remain free of spurious contributions. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation proceeds from standard perturbation theory.

full rationale

The paper computes second-order relativistic corrections to redshift drift via a gauge-invariant perturbative expansion on light-cone coordinates. No load-bearing step reduces by construction to its own inputs: the metric perturbations, null geodesic integration, and bispectrum expressions are derived from the Einstein equations and coordinate transformations without self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that substitute for independent verification. The light-cone coordinate choice is an explicit simplifying ansatz whose consistency is asserted within the perturbative framework rather than smuggled or tautologically assumed. This matches the default expectation for a first-principles calculation that remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard assumptions of cosmological perturbation theory without introducing new free parameters or postulated entities.

axioms (2)

domain assumption Validity of cosmological perturbation theory expanded to second order around an FLRW background
Invoked throughout for the gauge-invariant calculation of light propagation effects.
domain assumption Light-cone coordinates adequately describe photon geodesics in inhomogeneous anisotropic spacetimes at the required order
Employed to simplify the description as stated in the abstract.

pith-pipeline@v0.9.0 · 5424 in / 1224 out tokens · 62264 ms · 2026-05-07T10:58:22.398499+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

84 extracted references · 80 canonical work pages · 4 internal anchors

[1]

P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio,The Square Kilometre Array, 8, 2009. – 44 – [4]Euclid Theory Working GroupCollaboration, L. Amendola et al.,Cosmology and fundamental physics with the Euclid satellite, Living Rev. Rel.16(2013) 6, [arXiv:1206.1225]

work page arXiv 2009
[2]

Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report

D. Spergel et al.,Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report,arXiv:1503.03757. [6]PlanckCollaboration, P. A. R. Ade et al.,Planck 2013 results. XVI. Cosmological parameters, Astron. Astrophys.571(2014) A16, [arXiv:1303.5076]. [7]PlanckCollaboration, N. Aghanim et al.,Planck 2018 results. VI. Cosmolo...

work page internal anchor Pith review arXiv 2015
[3]

Lewis and A

A. Lewis and A. Challinor,Weak gravitational lensing of the cmb, Phys. Rept.429(2006) 1–65, [astro-ph/0601594]

work page arXiv 2006
[4]

Durrer, The Cosmic Microwave Background

R. Durrer, The Cosmic Microwave Background. Cambridge University Press, 12, 2020

2020
[5]

V. Bohm, M. Schmittfull, and B. D. Sherwin,Bias to CMB lensing measurements from the bispectrum of large-scale structure, Phys. Rev.D94(2016), no. 4 043519, [arXiv:1605.01392]

work page arXiv 2016
[6]

Marozzi, G

G. Marozzi, G. Fanizza, E. Di Dio, and R. Durrer,CMB-lensing beyond the Born approximation, JCAP1609(2016), no. 09 028, [arXiv:1605.08761]

work page arXiv 2016
[7]

Pratten and A

G. Pratten and A. Lewis,Impact of post-Born lensing on the CMB, JCAP1608(2016), no. 08 047, [arXiv:1605.05662]

work page arXiv 2016
[8]

Lewis and G

A. Lewis and G. Pratten,Effect of lensing non-Gaussianity on the CMB power spectra, JCAP 1612(2016), no. 12 003, [arXiv:1608.01263]

work page arXiv 2016
[9]

Marozzi, G

G. Marozzi, G. Fanizza, E. Di Dio, and R. Durrer,CMB-lensing beyond the leading order: temperature and polarization anisotropies, Phys. Rev.D98(2018), no. 2 023535, [arXiv:1612.07263]

work page arXiv 2018
[10]

Marozzi, G

G. Marozzi, G. Fanizza, E. Di Dio, and R. Durrer,Impact of Next-to-Leading Order Contributions to Cosmic Microwave Background Lensing, Phys. Rev. Lett.118(2017), no. 21 211301, [arXiv:1612.07650]

work page arXiv 2017
[11]

B¨ ohm, B

V. B¨ ohm, B. D. Sherwin, J. Liu, J. C. Hill, M. Schmittfull, and T. Namikawa,Effect of non-Gaussian lensing deflections on CMB lensing measurements, Phys. Rev. D98(2018), no. 12 123510, [arXiv:1806.01157]

work page arXiv 2018
[12]

Bartolo, E

N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto,Non-Gaussianity from inflation: Theory and observations, Phys. Rept.402(2004) 103–266, [astro-ph/0406398]

work page arXiv 2004
[13]

Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond

M. Bruni, S. Matarrese, S. Mollerach, and S. Sonego,Perturbations of space-time: Gauge transformations and gauge invariance at second order and beyond, Class. Quant. Grav.14 (1997) 2585–2606, [gr-qc/9609040]

work page Pith review arXiv 1997
[14]

Matarrese, S

S. Matarrese, S. Mollerach, and M. Bruni,Second order perturbations of the Einstein-de Sitter universe, Phys. Rev. D58(1998) 043504, [astro-ph/9707278]

work page arXiv 1998
[15]

O. Umeh, C. Clarkson, and R. Maartens,Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification: I. Key results, Class. Quant. Grav. 31(2014) 202001, [arXiv:1207.2109]

work page arXiv 2014
[16]

Ben-Dayan, G

I. Ben-Dayan, G. Marozzi, F. Nugier, and G. Veneziano,The second-order luminosity-redshift relation in a generic inhomogeneous cosmology, JCAP1211(2012) 045, [arXiv:1209.4326]

work page arXiv 2012
[17]

O. Umeh, C. Clarkson, and R. Maartens,Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification. II - Derivation, Class. Quant. Grav. 31(2014) 205001, [arXiv:1402.1933]. – 45 –

work page arXiv 2014
[18]

Marozzi,The luminosity distance-redshift relation up to second order in the Poisson gauge with anisotropic stress, Class

G. Marozzi,The luminosity distance-redshift relation up to second order in the Poisson gauge with anisotropic stress, Class. Quant. Grav.32(2015), no. 4 045004, [arXiv:1406.1135]. [erratum: Class. Quant. Grav.32,179501(2015)]

work page arXiv 2015
[19]

Scaccabarozzi and J

F. Scaccabarozzi and J. Yoo,Light-Cone Observables and Gauge-Invariance in the Geodesic Light-Cone Formalism, JCAP1706(2017), no. 06 007, [arXiv:1703.08552]

work page arXiv 2017
[20]

Fanizza, M

G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano,Generalized covariant prescriptions for averaging cosmological observables, JCAP02(2020) 017, [arXiv:1911.09469]

work page arXiv 2020
[21]

Fanizza, B

G. Fanizza, B. Fiorini, and G. Marozzi,Cosmic variance of H0 in light of forthcoming high-redshift surveys, Phys. Rev. D104(2021), no. 8 083506, [arXiv:2102.12419]

work page arXiv 2021
[22]

Magi and J

M. Magi and J. Yoo,Second-order gauge-invariant formalism for the cosmological observables: complete verification of their gauge-invariance, JCAP09(2022) 071, [arXiv:2204.01751]

work page arXiv 2022
[23]

Schiavone, E

T. Schiavone, E. Di Dio, and G. Fanizza,The skewness of the distance-redshift relation in ΛCDM, JCAP02(2024) 050, [arXiv:2307.13455]

work page arXiv 2024
[24]

B´ echaz, G

P. B´ echaz, G. Fanizza, G. Marozzi, and M. R. M. Silva,A Light-Cone Approach to Higher-Order Cosmological Observables, JCAP03(2026) 075, [arXiv:2510.25690]

work page arXiv 2026
[25]

Bertacca, R

D. Bertacca, R. Maartens, and C. Clarkson,Observed galaxy number counts on the lightcone up to second order: I. Main result, JCAP1409(2014), no. 09 037, [arXiv:1405.4403]

work page arXiv 2014
[26]

Bertacca, R

D. Bertacca, R. Maartens, and C. Clarkson,Observed galaxy number counts on the lightcone up to second order: II. Derivation, JCAP1411(2014), no. 11 013, [arXiv:1406.0319]

work page arXiv 2014
[27]

Yoo and M

J. Yoo and M. Zaldarriaga,Beyond the Linear-Order Relativistic Effect in Galaxy Clustering: Second-Order Gauge-Invariant Formalism, Phys. Rev.D90(2014), no. 2 023513, [arXiv:1406.4140]

work page arXiv 2014
[28]

Di Dio, R

E. Di Dio, R. Durrer, G. Marozzi, and F. Montanari,Galaxy number counts to second order and their bispectrum, JCAP1412(2014) 017, [arXiv:1407.0376]. [Erratum: JCAP1506,no.06,E01(2015)]

work page arXiv 2014
[29]

Di Dio, R

E. Di Dio, R. Durrer, G. Marozzi, and F. Montanari,The bispectrum of relativistic galaxy number counts, JCAP1601(2016) 016, [arXiv:1510.04202]

work page arXiv 2016
[30]

gevolution: a cosmological N-body code based on General Relativity

J. Adamek, D. Daverio, R. Durrer, and M. Kunz,gevolution: a cosmological N-body code based on General Relativity, JCAP07(2016) 053, [arXiv:1604.06065]

work page Pith review arXiv 2016
[31]

Fabbian, M

G. Fabbian, M. Calabrese, and C. Carbone,CMB weak-lensing beyond the Born approximation: a numerical approach, JCAP1802(2018), no. 02 050, [arXiv:1702.03317]

work page arXiv 2018
[32]

D. Beck, G. Fabbian, and J. Errard,Lensing Reconstruction in Post-Born Cosmic Microwave Background Weak Lensing,arXiv:1806.01216

work page arXiv
[33]

Sandage,The change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes, Astrophys

A. Sandage,The change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes, Astrophys. J.136(1962)

1962
[34]

McVittie,Appendix to the change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes, Astrophys

G. McVittie,Appendix to the change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes, Astrophys. J.136(1962)

1962
[35]

Trost et al.,The ESPRESSO Redshift Drift Experiment - I

A. Trost et al.,The ESPRESSO Redshift Drift Experiment - I. High-resolution spectra of the Lyman-αforest of QSO J052915.80-435152.0, Astron. Astrophys.699(2025) A159, [arXiv:2505.21615]

work page arXiv 2025
[36]

Trost et al.,The ESPRESSO Redshift Drift Experiment III – The Third Epoch of QSO J052915.80-435152.0,arXiv:2603.02318

A. Trost et al.,The ESPRESSO Redshift Drift Experiment III – The Third Epoch of QSO J052915.80-435152.0,arXiv:2603.02318

work page arXiv
[37]

Jain and S

D. Jain and S. Jhingan,Constraints on Dark Energy and Modified Gravity models by the Cosmological Redshift Drift test, Phys. Lett. B692(2010) 219–225, [arXiv:0910.4825]. – 46 –

work page arXiv 2010
[38]

Heinesen, Phys

A. Heinesen,Redshift drift as a model independent probe of dark energy, Phys. Rev. D103 (2021), no. 8 L081302, [arXiv:2102.03774]

work page arXiv 2021
[39]

Melia,A comparative test of theΛCDM and R h = ct cosmologies based on upcoming redshift drift measurements, EPL148(2024), no

F. Melia,A comparative test of theΛCDM and R h = ct cosmologies based on upcoming redshift drift measurements, EPL148(2024), no. 4 49003, [arXiv:2412.09489]

work page arXiv 2024
[40]

A. C. Alfano and O. Luongo,Mapping the redshift drift at various redshifts through cosmography,arXiv:2604.03167

work page internal anchor Pith review Pith/arXiv arXiv
[41]

Quercellini, L

C. Quercellini, L. Amendola, A. Balbi, P. Cabella, and M. Quartin,Real-time Cosmology, Phys. Rept.521(2012) 95–134, [arXiv:1011.2646]

work page arXiv 2012
[42]

A. G. Kim, E. V. Linder, J. Edelstein, and D. Erskine,Giving Cosmic Redshift Drift a Whirl, Astropart. Phys.62(2015) 195–205, [arXiv:1402.6614]

work page arXiv 2015
[43]

O. F. Piattella and L. Giani,Redshift drift of gravitational lensing, Phys. Rev. D95(2017), no. 10 101301, [arXiv:1703.05142]

work page arXiv 2017
[44]

O. H. Marcori, C. Pitrou, J.-P. Uzan, and T. S. Pereira,Direction and redshift drifts for general observers and their applications in cosmology, Phys. Rev. D98(2018), no. 2 023517, [arXiv:1805.12121]

work page arXiv 2018
[45]

Multipole decomposition of redshift drift – model independent mapping of the expansion history of the Universe,

A. Heinesen,Multipole decomposition of redshift drift – model independent mapping of the expansion history of the Universe, Phys. Rev. D103(2021), no. 2 023537, [arXiv:2011.10048]

work page arXiv 2021
[46]

Bessa, R

P. Bessa, R. Durrer, and D. Stock,Perturbations of cosmological redshift drift, JCAP11(2023) 093, [arXiv:2306.13911]

work page arXiv 2023
[47]

Jeong, F

D. Jeong, F. Schmidt, and C. M. Hirata,Large-scale clustering of galaxies in general relativity, Phys. Rev. D85(2012) 023504, [arXiv:1107.5427]

work page arXiv 2012
[48]

Schmidt and D

F. Schmidt and D. Jeong,Cosmic Rulers, Phys. Rev. D86(2012) 083527, [arXiv:1204.3625]

work page arXiv 2012
[49]

Jeong and F

D. Jeong and F. Schmidt,Large-Scale Structure Observables in General Relativity, Class. Quant. Grav.32(2015), no. 4 044001, [arXiv:1407.7979]

work page arXiv 2015
[50]

Villey, Y

A. Villey, Y. B. Ginat, V. Desjacques, D. Jeong, and F. Schmidt,Large-scale-structure observables in general relativity validated at second order, JCAP10(2025) 105, [arXiv:2506.11260]

work page arXiv 2025
[51]

Gasperini, G

M. Gasperini, G. Marozzi, F. Nugier, and G. Veneziano,Light-cone averaging in cosmology: Formalism and applications, JCAP1107(2011) 008, [arXiv:1104.1167]

work page arXiv 2011
[52]

Fanizza, M

G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano,An exact Jacobi map in the geodesic light-cone gauge, JCAP1311(2013) 019, [arXiv:1308.4935]

work page arXiv 2013
[53]

Fanizza, M

G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano,Observation angles, Fermi coordinates, and the Geodesic-Light-Cone gauge, JCAP01(2019) 004, [arXiv:1812.03671]

work page arXiv 2019
[54]

Fanizza, G

G. Fanizza, G. Marozzi, M. Medeiros, and G. Schiaffino,The Cosmological Perturbation Theory on the Geodesic Light-Cone background, JCAP02(2021) 014, [arXiv:2009.14134]

work page arXiv 2021
[55]

Fanizza, G

G. Fanizza, G. Marozzi, and M. Medeiros,Gauge invariance on the light-cone: curvature perturbations and radiative degrees of freedom, JCAP06(2023) 015, [arXiv:2303.11743]

work page arXiv 2023
[56]

Fanizza, M

G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano,Time of flight of ultra-relativistic particles in a realistic Universe: a viable tool for fundamental physics?, Phys. Lett. B757 (2016) 505–509, [arXiv:1512.08489]

work page arXiv 2016
[57]

Mitsou, G

E. Mitsou, G. Fanizza, N. Grimm, and J. Yoo,Cutting out the cosmological middle man: General Relativity in the light-cone coordinates, Class. Quant. Grav.38(2021), no. 5 055011, [arXiv:2009.14687]

work page arXiv 2021
[58]

M. B. Fr¨ ob and W. C. C. Lima,Cosmological perturbations and invariant observables in geodesic lightcone coordinates, JCAP01(2022), no. 01 034, [arXiv:2108.11960]. – 47 –

work page arXiv 2022
[59]

Fanizza, G

G. Fanizza, G. Marozzi, and M. Medeiros,δNformalism on the past light-cone, Gen. Rel. Grav.56(2024), no. 5 53, [arXiv:2311.08073]

work page arXiv 2024
[60]

Fanizza, M

G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano,A new approach to the propagation of light-like signals in perturbed cosmological backgrounds, JCAP08(2015) 020, [arXiv:1506.02003]

work page arXiv 2015
[61]

Fleury, F

P. Fleury, F. Nugier, and G. Fanizza,Geodesic-light-cone coordinates and the Bianchi I spacetime, JCAP1606(2016), no. 06 008, [arXiv:1602.04461]

work page arXiv 2016
[62]

Sawicki, I

I. Sawicki, I. D. Saltas, L. Amendola, and M. Kunz,Consistent perturbations in an imperfect fluid, JCAP01(2013) 004, [arXiv:1208.4855]

work page arXiv 2013
[63]

Amendola, M

L. Amendola, M. Kunz, M. Motta, I. D. Saltas, and I. Sawicki,Observables and unobservables in dark energy cosmologies, Phys. Rev. D87(2013), no. 2 023501, [arXiv:1210.0439]

work page arXiv 2013
[64]

Amendola, S

L. Amendola, S. Fogli, A. Guarnizo, M. Kunz, and A. Vollmer,Model-independent constraints on the cosmological anisotropic stress, Phys. Rev. D89(2014), no. 6 063538, [arXiv:1311.4765]

work page arXiv 2014
[65]

Sobral-Blanco and C

D. Sobral-Blanco and C. Bonvin,Measuring anisotropic stress with relativistic effects, Phys. Rev. D104(2021), no. 6 063516, [arXiv:2102.05086]

work page arXiv 2021
[66]

Castello, N

S. Castello, N. Grimm, and C. Bonvin,Rescuing constraints on modified gravity using gravitational redshift in large-scale structure, Phys. Rev. D106(2022), no. 8 083511, [arXiv:2204.11507]

work page arXiv 2022
[67]

Fanizza, J

G. Fanizza, J. Yoo, and S. G. Biern,Non-linear general relativistic effects in the observed redshift, JCAP09(2018) 037, [arXiv:1805.05959]

work page arXiv 2018
[68]

S. M. Koksbang,Redshift drift in a universe with structure. II. Light rays propagated through a Newtonian N-body simulation, Phys. Rev. D107(2023), no. 6 063544, [arXiv:2303.06900]

work page arXiv 2023
[69]

Bessa, V

P. Bessa, V. Marra, and T. Castro,Redshift drift fluctuations from N-body simulations, JCAP 12(2025) 046, [arXiv:2409.09977]

work page arXiv 2025
[70]

Oestreicher, C

A. Oestreicher, C. Clarkson, J. Adamek, and S. M. Koksbang,Redshift Drift in Relativistic N-Body Simulations, Open J. Astrophys.8(5, 2025) 2025, [arXiv:2505.04731]

work page arXiv 2025
[71]

Di Dio, R

E. Di Dio, R. Durrer, R. Maartens, F. Montanari, and O. Umeh,The Full-Sky Angular Bispectrum in Redshift Space, JCAP04(2019) 053, [arXiv:1812.09297]

work page arXiv 2019
[72]

Abramowitz, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,

M. Abramowitz, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,. Dover Publications, Inc., USA, 1974

1974
[73]

The Pursuit of Non-Gaussian Fluctuations in the Cosmic Microwave Back- ground

E. Komatsu, The pursuit of non-gaussian fluctuations in the cosmic microwave background. PhD thesis, Tohoku U., 2001.astro-ph/0206039

work page internal anchor Pith review arXiv 2001
[74]

Large-Scale Structure of the Universe and Cosmological Perturbation Theory

F. Bernardeau, S. Colombi, E. Gaztanaga, and R. Scoccimarro,Large scale structure of the universe and cosmological perturbation theory, Phys. Rept.367(2002) 1–248, [astro-ph/0112551]

work page Pith review arXiv 2002
[75]

D. Blas, J. Lesgourgues, and T. Tram,The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes, JCAP07(2011) 034, [arXiv:1104.2933]

work page internal anchor Pith review arXiv 2011
[76]

C. S. Alves, A. C. O. Leite, C. J. A. P. Martins, J. G. B. Matos, and T. A. Silva,Forecasts of redshift drift constraints on cosmological parameters, Mon. Not. Roy. Astron. Soc.488(2019), no. 3 3607–3624, [arXiv:1907.05151]

work page arXiv 2019
[77]

Loeb,Direct Measurement of Cosmological Parameters from the Cosmic Deceleration of Extragalactic Objects, Astrophys

A. Loeb,Direct Measurement of Cosmological Parameters from the Cosmic Deceleration of Extragalactic Objects, Astrophys. J. Lett.499(1998) L111–L114, [astro-ph/9802122]

work page arXiv 1998
[78]

Balbi and C

A. Balbi and C. Quercellini,The time evolution of cosmological redshift as a test of dark energy, Mon. Not. Roy. Astron. Soc.382(2007) 1623–1629, [arXiv:0704.2350]. – 48 –

work page arXiv 2007
[79]

Cosmic dynamics in the era of Extremely Large Telescopes

J. Liske et al.,Cosmic dynamics in the era of Extremely Large Telescopes, Mon. Not. Roy. Astron. Soc.386(2008) 1192–1218, [arXiv:0802.1532]

work page Pith review arXiv 2008
[80]

Martinelli, S

M. Martinelli, S. Pandolfi, C. J. A. P. Martins, and P. E. Vielzeuf,Probing dark energy with the Sandage-Loeb test, Phys. Rev. D86(2012) 123001, [arXiv:1210.7166]

work page arXiv 2012

Showing first 80 references.