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A theoretical prediction for the dipole in nearby distances using cosmography

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims the dipole in luminosity distances of nearby sources can be predicted from gradients of the volume-smoothed expansion rate and matter density, with no background cosmology assumed, and shows the prediction matches fully…

desk verdict A useful, honestly-scoped validation of a smoothed cosmographic dipole prescription; the central averaging step is assumed and scale-tuned, so the 'prediction' label oversells it slightly. read the letter →

arxiv 2507.01095 v3 pith:XQNGOEER submitted 2025-07-01 astro-ph.CO gr-qc

Hayley J. Macpherson , Asta Heinesen This is my paper · ORCID
classification astro-ph.COgr-qc PACS 98.80.-k98.80.Es
keywords cosmographyluminositydistancedipoleanisotropictensionnumericalrelativitycosmologyraytracingaveragingprobleminhomogeneous
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Cosmography estimates cosmic expansion by expanding distance versus redshift in a Taylor series, usually assuming the background is homogeneous; near us, however, structures make distances depend on direction, and this dipole can bias estimates of cosmological parameters. The paper claims the dipole in the luminosity distance, the distance inferred from apparent brightness, can be predicted from the spatial gradients of the volume-averaged expansion rate and matter density, with no background cosmology assumed. The authors derive a third-order expression for the dipole and test it against exact ray-traced distances in fully relativistic simulations. In the smoother simulation the prediction matches the ray-traced dipole amplitude to within about 10 percent for most observers out to $z\approx0.07$, and in a more nonlinear simulation out to $z\approx0.02$; the direction is typically matched to better than 1 percent. The smoothed prediction improves on the unsmoothed local cosmography by up to an order of magnitude, and at large scales the same framework connects the dipole to the observer's peculiar motion.

What carries the argument

The machinery is the generalized cosmographic expansion of the luminosity distance truncated at third order in redshift, combined with a smoothing step. The expansion's coefficients are generalized Hubble, deceleration, curvature, and jerk parameters defined along each line of sight; under quiet-universe assumptions these reduce, at the dipole level, to coefficients constructed from the spatial gradients of the volume-smoothed expansion rate $\langle\theta\rangle$ and density $\langle\rho\rangle$. Smoothing is performed with a Gaussian kernel whose width is matched to the redshift at which the dipole is evaluated, and the paper's test compares the resulting prediction with exact distances from geodesic ray tracing on the same constant-redshift spheres.

What would settle it

Run the same comparison in a simulation that resolves structure below the current fluid cutoff at roughly $8\,h^{-1}\mathrm{Mpc}$; if the predicted dipole from Eq. (8) does not track the ray-traced dipole to within ten percent at $z\lesssim0.02$, or if the residual does not shrink when the Gaussian width is varied across the tested $2\sigma$ to $4\sigma$ range, the smoothing-mapping assumption is falsified.

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Extended reading notes

Core claim

The central claim is a calculational rule: the dipole of luminosity distance on constant-redshift spheres around a given comoving observer is, to third order in redshift, the direction-weighted sum of two gradient terms, $d_{L,\mathrm{dip}}(\mathbf{e},z) = e^\mu (d^{(2)}_{L,\mu} z^2 + d^{(3)}_{L,\mu} z^3)$, with $d^{(2)}_{L,\mu}$ proportional to $D_\mu\langle\theta\rangle/\langle\theta\rangle^3$ and $d^{(3)}_{L,\mu}$ receiving contributions from $D_\mu\langle\theta\rangle$ and $D_\mu\langle\rho\rangle$. These coefficients follow from the generalized cosmography of the luminosity distance in an arbitrary space-time, specialized to a pressureless dust congruence under the quiet-universe approximation. When tested in numerical relativity simulations with fully nonlinear ray tracing, the predicted dipole amplitude agrees with the ray-traced dipole to within about 10 percent for redshifts below 0.07 in a quasi-linear simulation and below 0.02 in a simulation with larger density contrasts; the predicted direction is typically aligned to better than 1 percent. This constitutes up to an order-of-magnitude improvement over the dipole extracted from the local, unsmoothed third-order cosmography.

Load-bearing premise

The load-bearing premise is that volume-smoothed averages of the expansion rate and density can be inserted into the nonlinear dipole coefficients and still predict the observed dipole, even though averaging and non-linear operations do not commute; the prediction also relies on the Gaussian smoothing scale being hand-matched to each redshift.

Editorial extensions

If this is right

  • A measured low-redshift distance dipole can be interpreted as a signal of gradients in the large-scale expansion rate and density, without committing to a particular background cosmology.
  • Because the third-order term is needed for the amplitude while the second-order term already gives the direction, fitting only the second-order dipole coefficient would yield a correct axis but an incorrect magnitude.
  • On large smoothing scales the structure-induced dipole is suppressed and the ray-traced dipole becomes dominated by the observer's velocity relative to the average frame, so scale separates the two physical origins of the dipole.
  • The framework is expected to extend to universe models containing a cosmological constant, since the derivation only requires the quiet-universe assumptions and the field equations, not the absence of dark energy.
  • Applying this prediction to upcoming large low-redshift supernova samples would turn a nuisance anisotropy into a measurable cosmological signal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The optimal match between Gaussian smoothing scale and redshift is not fixed at 3 sigma, since the appendix shows the best match drifts to roughly 3.25-3.5 sigma at low redshift; a scale-dependent matching scheme is a natural extension that may push the 10 percent range farther.
  • A more covariant volume average of the kind used in backreaction studies might repair the breakdown in the nonlinear simulation, because the paper attributes the failure to the naive averaged congruence not capturing kinematics on the averaging scale.
  • The same averaging logic could be adapted to other dipolar observables, such as galaxy number counts or astrometric proper motions, where dipole excesses over the kinematic expectation have also been reported; the paper only hints at these directions.
  • Combining the cosmographic gradient dipole with the kinematic boost dipole in one model could cover the intermediate redshifts $0.1\lesssim z\lesssim0.25$ where neither term alone fits, though the paper notes such a combination would require perturbation theory.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper proposes a method to predict the dipolar anisotropy in the luminosity distance-redshift relation from smoothed cosmography. Using the third-order generalized cosmography of Heinesen, the authors write the dipole as Eq. (6) with coefficients constructed from gradients of the expansion rate and density. They replace the local fields θ and ρ by Gaussian-smoothed averages in Eq. (8), with the smoothing scale matched to the redshift of observation. The prediction is tested against fully relativistic ray-traced distances in two numerical-relativity simulations, one quasi-linear (L=2.56 h^-1 Gpc, N=256, λ_min=400 h^-1 Mpc) and one with more small-scale structure (L=3.072 h^-1 Gpc, λ_min=120 h^-1 Mpc). The authors report approximately 10% accuracy in dipole amplitude for z≲0.07 in the quasi-linear simulation and z≲0.02 in the more nonlinear simulation, with up to an order-of-magnitude improvement over unsmoothed local cosmography. They explicitly flag the assumed θ→⟨θ⟩ mapping as ambiguous and connect the smoothing question to the cosmological fitting problem.

Significance. The numerical comparison is a strength: the ray tracer solves the full geodesic and geodesic-deviation equations, the Richardson extrapolation in Appendix A indicates that numerical error is not the dominant source of mismatch for most observers, and the code is built on open-source components. If the averaging prescription can be justified or its error bounded, the method would provide a useful model-independent route to interpret low-redshift dipole measurements and connect them to expansion and density gradients. The paper is honest about the main caveat, that Eq. (8) assumes commutation of smoothing with nonlinear operations, but this caveat is exactly the load-bearing point. The current manuscript therefore establishes a promising phenomenological prescription rather than a fully derived theoretical prediction.

major comments (3)
  1. [§2.2, Eq. (8)] The central prediction is not derived: the replacement of θ and ρ by ⟨θ⟩ and ⟨ρ⟩ in the nonlinear ratios of Eq. (8) is stated as an assumption, and the text explicitly acknowledges that ⟨θ²⟩ is not equal to ⟨θ⟩². Because no controlled approximation quantifies the error introduced by this commutation, the agreement with the ray-traced dipole in Fig. 2 validates a chosen averaging prescription rather than a derived prediction. I request either a derivation or an estimate of the leading smoothing corrections, for example terms involving ⟨θ²⟩−⟨θ⟩², or an explicit reframing of Eq. (8) as a phenomenological ansatz whose accuracy is to be tested.
  2. [Appendix B; §3.4] The reported approximately 10% accuracy depends on a smoothing-to-redshift matching rule that is calibrated in the same analysis. Appendix B shows that the optimal matching lies between 3.25σ and 3.5σ at low redshift, and §5.1 states that the 3σ choice produces a slight negative bias. Since the same data are used to select the matching rule and to evaluate the prediction, the headline accuracy is partly post-hoc. Please present the dipole comparison for a fixed, prespecified matching rule across the full redshift range, and quantify how the accuracy changes over the 2.5σ to 4σ range shown in Fig. 10.
  3. [§5.3, Fig. 6] The validation is limited to quasi-linear universes: with the L=3.072 h^-1 Gpc simulation, which has typical density contrast around 0.05, the prediction reaches 10% accuracy only for z≲0.02, and the direction is matched to 1% only at even lower redshifts. The paper acknowledges this limitation and relates it to the fitting problem, but the abstract's claim of an order-of-magnitude improvement is based on the smoother simulation. I recommend making this quasi-linear scope explicit in the abstract and conclusions, and stating clearly that no claim is made for the deeply nonlinear regime.
minor comments (4)
  1. [Fig. 1 caption] The caption gives the simulation size as L=2.56 h^-1 Mpc; from Section 3.3 this should be h^-1 Gpc.
  2. [Fig. 5 caption] Typo: 'comsography' should be 'cosmography'.
  3. [Footnote 9, §4.1] The ray-tracing part of mescaline is not yet public; please include a code and data availability statement with a timeline or repository to support reproducibility.
  4. [§3.3] The choice h=0.45, rather than the Planck value used for the power spectrum, is unusual; a sentence explaining that this is a simulation scale choice and does not affect the model-independent prediction would help.

Circularity Check

1 steps flagged · score 4.0 of 10

The smoothed-cosmography dipole prediction is partially calibrated: the 3-sigma smoothing-scale-to-redshift mapping is chosen in Appendix B by comparison with the ray-traced dipole, so the reported ~10% agreement is in-sample for that parameter.

  1. fitted input called prediction [Section 3.4 (Eq. 12-13) and Appendix B]
    "We smooth these quantities using a Gaussian kernel with a 3–σ width that coincides with the redshift of interest (in Appendix B we show that this is the optimal choice for matching smoothing scale with redshift). ... In the main text, we have chosen to match the 3–σ width of the Gaussian kernel to an approximate redshift by comparing the predicted dipole to the ray-traced dipole."

    The smoothing scale is the method's free parameter, and Appendix B fixes it by comparing the predicted dipole to the ray-traced dipole, which is the same benchmark used to claim ~10% agreement. The reported accuracy is therefore not an out-of-sample test of a parameter-free prediction: the 3-sigma matching is calibrated to minimize the discrepancy on the target data. The dipole amplitude itself is still computed from smoothed theta and rho gradients, so this is a partial, not a complete, reduction of the prediction to its inputs.

full rationale

The core dipole formula (8) is constructed from smoothed expansion-rate and density gradients, and the ray-traced luminosity distances provide an external, independent benchmark; the dipole amplitude is not fitted to that benchmark. The main circular element is the choice of the Gaussian smoothing scale: Appendix B explicitly selects the 3-sigma scale-to-redshift mapping by comparing prediction with the ray-traced dipole, so the headline 'within ~10%' accuracy is partially in-sample. The paper also relies on the authors' earlier derivation of the dipole coefficients (7) and on an explicitly assumed, acknowledged-ambiguous replacement of local fields by smoothed averages, since the text notes that <theta^2> is not equal to <theta>^2; these are correctness and provenance concerns rather than circular reductions, because the earlier derivation is not fitted to the present target and the averaging assumption is stated openly. Considering the external validation and the fact that the dipole signal itself is not fitted, the circularity is mild.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The smoothed average congruence is a mathematical construction used in the method, not a new physical entity with independent observable handles.

free parameters (3)
  • Smoothing-to-redshift matching width (n-sigma of Gaussian) = 3 sigma (chosen as optimal over 2 sigma to 4 sigma)
    Appendix B varies n from 2 to 4 and finds 3 sigma gives the smallest mean mismatch; the headline 10 percent accuracy depends on this calibration.
  • Gaussian kernel FWHM in grid cells = 4, 8, 12, 16 dx for the L=2.56 h^-1 Gpc run; 10 grid cells minimum for the L=3.072 h^-1 Gpc run
    The FWHM sets the physical smoothing scale and is translated to a redshift using an EdS distance law with H0=<theta>/3, so the reported accuracy is tied to the chosen scales.
  • Initial power spectrum cutoff lambda_min = 400 h^-1 Mpc for the smooth run, 120 h^-1 Mpc for the more nonlinear run
    A large-scale cutoff removes small modes, keeping the 'smooth' simulation quasi-linear (typical density contrast about 1 percent); the claimed validity range (z<0.07 vs z<0.02) is set by this choice.
assumptions (5)
  • domain assumption Einstein's general relativity holds, with dust matter described by T_mu nu = rho u_mu u_nu at leading order.
    Section 2.2 explicitly assumes GR to derive the dipole coefficients; this is outside the usual cosmographic agnosticism about field equations.
  • domain assumption Quiet universe approximation: pressureless dust, vorticity-free, small shear and electric Weyl compared to expansion, while gradients of shear and electric Weyl can be comparable to gradients of expansion and density.
    Section 2.2 states these conditions; they are needed to reduce the general cosmography to Eqs. (7).
  • ad hoc to paper The mapping theta to <theta> and rho to <rho> is valid at lowest order for predicting observables.
    Section 2.2 says 'we will assume that the mapping... holds'; the paper notes <theta^2> is not <theta>^2, making this an ambiguity.
  • domain assumption Gaussian smoothing on a spatially flat slice is a valid proxy for the average congruence.
    Section 3.4 justifies this via small metric fluctuations (root-gamma fluctuations of order 1e-5), but more accurate geometric averaging is not used.
  • domain assumption The EdS distance-redshift relation with H0=<theta>/3 translates a smoothing scale to a redshift.
    Section 3.4 and Appendix B: the simulations remain close to EdS, but the relation is an extra input to the prediction.

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Pith. "Pith review of A theoretical prediction for the dipole in nearby distances using cosmography." pith.science (2026). https://pith.science/paper/XQNGOEER

@misc{pith2026250701095,
  author       = {Pith},
  title        = {Pith review of: A theoretical prediction for the dipole in nearby distances using cosmography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQNGOEER}},
  note         = {Machine review of arXiv:2507.01095}
}
abstract

Cosmography is a widely applied method to infer kinematics of the Universe at small cosmological scales while remaining agnostic about the theory of gravity at play. Usually cosmologists invoke the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in cosmographic analyses, however generalised approaches allow for analyses outside of any assumed geometrical model. These methods have great promise to be able to model-independently map the cosmic neighborhood where the Universe has not yet converged to isotropy. In this regime, anisotropies can bias parameter inferences if they are not accounted for, and thus must be included for precision cosmology analyses, even when the principle aim is to infer the background cosmology. In this paper, we develop a method to predict the dipole in luminosity distances that arises due to nearby inhomogeneities. This is the leading-order correction to the standard isotropic distance-redshift law. Within a very broad class of general-relativistic universe models, we provide an interpretation of the dipole in terms of the gradients in expansion rate and density which is free from any underlying background cosmology. We use numerical relativity simulations, with improved initial data methods, alongside fully relativistic ray tracing to test the power of our prediction. We find our prediction accurately captures the dipole signature in our simulations to within ~10% for redshifts $z\lesssim 0.07$ in reasonably smooth simulations. In the presence of more non-linear density fields, we find this reduces to $z\lesssim 0.02$. This represents up to an order of magnitude improvement with respect to what is achieved by naive, local cosmography-based predictions. Our paper thus addresses important issues regarding convergence properties of anisotropic cosmographic series expansions that would otherwise limit their applicability to very narrow redshift ranges.

Figures

Figures reproduced from arXiv: 2507.01095 by the authors.

Figure 1
Figure 1. shows all-sky maps of luminosity distances for one observer in the N = 256, L = 2.56 h −1 Gpc simulation. The top row shows distances for a constant-redshift slice of z ≈ 0.0033 and the bottom row for z ≈ 0.0717. The left column shows the full ray-traced luminosity distance, DL, the middle column shows the dipole extracted from the left column, i.e. DL,dip, and the right column shows the predicted dipole from the sm… view at source ↗
Figure 2
Figure 2. — Relative difference between the predicted dipole dL,dip and the ray traced dipole DL,dip as a function of redshift (smoothing scale) for a set of 20 observers in the N = 256, L = 2.56 h−1 Gpc simulation. The left panel shows the absolute value of the difference in amplitude for individual observers (dashed, coloured curves) and the average over all observers (solid, black curves). The right panel shows the alignme… view at source ↗
Figure 3
Figure 3. — Relative difference between the predicted dipole dL,dip and the ray traced dipole DL,dip as a function of redshift. We show the difference with only the second-order contribution to the dipole prediction (i.e. only (8a); dashed curves) and with both second- and third-order contributions (i.e., both (8a) and (8b); solid curves). Individual observers are shown as grey curves and the mean over all observers in black.… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: — Dipole amplitude (normalised by the ray-traced monopole distance) as a function of redshift (smoothing scale) for the ray-traced dipole (solid curve), the predicted dipole from smoothing (dashed curve), and the dipole extracted from the local third-order cosmography …
Figure 5
Figure 5. Figure 5: — Left: absolute value of the relative difference in the dipole amplitude (with respect to the ray traced distance) as a function of redshift for the smoothed prediction (dashed) and the local third order comsography (dotted). Coloured curves show the difference for in…
Figure 6
Figure 6. Figure 6: shows the relative difference between the predicted dipole amplitude (left panel) and direction (right panel) and the ray traced dipole. In both panels, coloured curves show the difference for individual observers and the black curve shows the average over 20 observers…
Figure 7
Figure 7. Figure 7: — Dipole amplitude (normalised by the ray-traced monopole distance) as a function of redshift (smoothing scale) for the ray-traced dipole (solid curve), the predicted dipole from smoothing (dashed curve with crosses), and the dipole from a single boost of the observer …
Figure 8
Figure 8. Figure 8: — Relative difference in the predicted dipole amplitude with respect to the ray traced dipole for all 20 observers and for three redshifts (panels; as indicated in the legends). We show the difference for the third-order prediction only and the error bar is calculated …
Figure 9
Figure 9. Figure 9: — Difference in the dot product of the unit dipole direction vectors from one for the same 20 observers as in [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: — The impact of changing the n–σ matching with a redshift on the predicted dipole accuracy. Each panel shows a different FWHM smoothing scale of the Gaussian kernel (indicated in the legend) in the N = 256 L = 2.56 h−1 Gpc simulation for 20 observers (coloured curves;…
Figure 11
Figure 11. Figure 11: — Hamiltonian constraint violation for the three simulations with L = 2.56 h−1 Gpc and N = 64 (pink), 128 (blue), and 256 (green) for all cells on each simulation final time slice. Light grey shaded region shows the ±1σ spread of the violation for the N = 256, L = 3.0…

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Reference graph

Works this paper leans on

16 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    2022JHEAp34, 49–211

    Abdalla E et al. 2022JHEAp34, 49–211. Abghari A, Bunn E F, Hergt L T, Li B, Scott D, Sullivan R M and Wei D 2024J. Cosmology Astropart. Phys.2024(11),

    work page 2024

  2. [13]

    Cosmology Astropart

    Maartens R, Santiago J, Clarkson C, Kalbouneh B and Marinoni C 2024J. Cosmology Astropart. Phys.2024(9),

    work page 2024

  3. [25]

    Dhawan S, Borderies A, Macpherson H J and Heinesen A 2023 Mon. Not. Roy. Astron. Soc.519(4), 4841–4855. Dunsby P K S, Bassett B A C C and Ellis G F R 1997Classical and Quantum Gravity14(5), 1215–1222. Ehlers J 1993General Relativity and Gravitation 25(12), 1225–1266. Ellis G F R 2009General Relativity and Gravitation 41(3), 581–660. Ellis G F R and Baldwi...

    work page Pith review arXiv 2023

  4. [34]

    Bonvin C, Durrer R and Kunz M 2006Phys. Rev. Lett. 96, 191302. Brown D, Diener P, Sarbach O, Schnetter E and Tiglio M 2009 Phys. Rev. D79(4), 044023. Bruni M, Matarrese S and Pantano O 1995ApJ445,

    work page 2009

  5. [43]

    LSST Science Collaboration, Abell P A, Allison J, Anderson S F, Andrew J R, Angel J R P, Armus L, Arnett D, Asztalos S J, Axelrod T S, Bailey S et al

    Löffler F, Faber J, Bentivegna E, Bode T, Diener P, Haas R, Hinder I, Mundim B C, Ott C D, Schnetter E, Allen G, Campanelli M and Laguna P 2012Classical and Quantum Gravity29(11), 115001. LSST Science Collaboration, Abell P A, Allison J, Anderson S F, Andrew J R, Angel J R P, Armus L, Arnett D, Asztalos S J, Axelrod T S, Bailey S et al. 2009arXiv e-prints...

    arXiv

  6. [53]

    Anchordoqui L A, Di Valentino E, Pan S and Yang W 2021 JHEAp32, 28–64

    Aluri P k, Cea P, Chingangbam P, Chu M C, Clowes R G, Hutsemékers D, Kochappan J P, Lopez A M, Liu L, Martens N C M, Martins C J A P, Migkas K, Ó Colgáin E, Pranav P, Shamir L, Singal A K, Sheikh-Jabbari M M, Wagner J, Wang S J, Wiltshire D L, Yeung S, Yin L and Zhao W 2023Classical and Quantum Gravity40(9), 094001. Anchordoqui L A, Di Valentino E, Pan S ...

    work page 2021

  7. [56]

    Cattoen C and Visser M 2007Class. Quant. Grav.24, 5985–5998. Clarkson C, Ellis G F R, Faltenbacher A, Maartens R, Umeh O and Uzan J P 2012Mon. Not. Roy. Astron. Soc. 426, 1121–1136. Clarkson C, Ellis G, Larena J and Umeh O 2011Reports on Progress in Physics74(11), 112901. Clarkson C and Maartens R 2010Classical and Quantum Gravity 27(12), 124008. Clarkson...

    work page

  8. [67]

    Adamek J, Clarkson C, Durrer R, Heinesen A, Kunz M and Macpherson H J 2024Open J

    Adamek J, Barrera-Hinojosa C, Bruni M, Li B, Macpherson H J and Mertens J B 2020Classical and Quantum Gravity 37(15), 154001. Adamek J, Clarkson C, Durrer R, Heinesen A, Kunz M and Macpherson H J 2024Open J. Astrophys.7, 001c.118782. Adamek J, Daverio D, Durrer R and Kunz M 2016J. Cosmology Astropart. Phys.2016(7),

    work page 2016

Show all 16 references
  1. [68]

    Sanghai V A A and Clifton T 2015Phys. Rev. D91, 103532. [Erratum: Phys.Rev.D 93, 089903 (2016)]. Sarma M, Marinoni C, Kalbouneh B, Clarkson C and Maartens R

    2016

  2. [69]

    arXiv:2510.02510

    Kalbouneh B, Marinoni C, Maartens R, Bel J, Santiago J, Clarkson C, Sarma M and Virey J 2025 p. arXiv:2510.02510. Kalbouneh B, Santiago J, Marinoni C, Maartens R, Clarkson C and Sarma M 2025JCAP02,

    2025

  3. [70]

    MacCallum M A H and Ellis G F R 1970Comm. Math. Phys. 19, 31–64. Macpherson H J 2023J. Cosmology Astropart. Phys. 2023(3),

    2023

  4. [76]

    Koksbang S M 2025Phys. Rev. D111(12), 123516. Kristian J and Sachs R K 1966Astrophys. J.143, 379–399. Lesgourgues J 2011arXiv e-printsp. arXiv:1104.2932. Lewis A and Bridle S 2002Phys. Rev. D66(10), 103511. Lobo F S N, Mimoso J P and Visser M 2020JCAP04,

    arXiv

  5. [111]

    Macpherson H J and Heinesen A 2021Phys. Rev. D104, 023525. [Erratum: Phys.Rev.D 104, 109901 (2021)]. Macpherson H J, Lasky P D and Price D J 2017Phys. Rev. D 95(6), 064028. Macpherson H J, Lasky P D and Price D J 2018Astrophys. J. 865(1), L4. Macpherson H J, Price D J and Lask...

    2021

  6. [112]

    2020A&A641, A6

    Planck Collaboration, Aghanim N, Akrami Y, Ashdown M, Aumont J, Baccigalupi C, Ballardini M, Banday A J, Barreiro R B, Bartolo N, Basak S et al. 2020A&A641, A6. Planck Collaboration, Aghanim N, Armitage-Caplan C, Arnaud M, Ashdown M, Atrio-Barandela F, Aumont J, Baccigalupi C,...

    2019

  7. [279]

    The influence of structure formation on the evolution of the universe

    Umeh O 2013 "The influence of structure formation on the evolution of the universe." PhD thesis University of Cape Town, Faculty of Science, Department of Mathematics and Applied Mathematics. 21 van Elst H 1996 Extensions and applications of 1+3 decomposition methods in genera...

    2013

  8. [2025]

    Rept.984, 1–55

    Schöneberg N, Franco Abellán G, Pérez Sánchez A, Witte S J, Poulin V and Lesgourgues J 2022Phys. Rept.984, 1–55. Secrest N J, von Hausegger S, Rameez M, Mohayaee R and Sarkar S 2022ApJ937(2), L31. Secrest N J, von Hausegger S, Rameez M, Mohayaee R, Sarkar S and Colin J 2021ApJ...

    2023

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