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REVIEW 4 major objections 6 minor 42 references

Inhomogeneous model with a space dependent Cosmological Constant

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single inhomogeneous dark fluid can reconcile the two measured Hubble constants.

desk verdict A useful Lemaitre drift derivation paired with an H0-bubble claim that currently doesn't satisfy its own perturbativity conditions. read the letter →

arxiv 2501.07968 v1 pith:4QGMZUI6 submitted 2025-01-14 gr-qc

D. Comelli This is my paper · ORCID
classification gr-qc MSC 83F0583C1583C55 PACS 98.80.-k95.36.+x
keywords HubbletensionLemaitremetricspace-dependentcosmologicalconstantunifiedperfectfluidinhomogeneouscosmologyredshiftdriftcosmographicparameterslocalvoid
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

The paper argues that the well-known mismatch between the early-universe value $H_0 = 67.4 \pm 0.5$ km/s/Mpc inferred from the cosmic microwave background and the local value $H_0 = 73.52 \pm 1.62$ km/s/Mpc inferred from distance-ladder measurements can be explained by a local, spherically symmetric bubble embedded in an otherwise homogeneous universe. The bubble is produced by a single perfect fluid whose energy density is a space-dependent cosmological constant $\Lambda(r)$ plus a conserved matter density $n(t,r)$, with pressure $p = -\Lambda(r)$, so dark energy and dark matter are one coupled fluid rather than two separately conserved components. The paper solves the Einstein and null-geodesic equations in two regimes, near the bubble center and perturbatively around a background FRW solution, and shows that with the simplest exponential profiles for the inhomogeneities only two free parameters remain: the dark-energy versus dark-matter fraction $x$ and the bubble size $\Re$. Imposing the early-universe constraints on $\Omega_m$ and $\Omega_\kappa$ leaves a non-empty allowed region of this parameter space, and inside that region the model makes definite predictions for the deceleration parameter, the jerk, effective equations of state, and redshift drift. A sympathetic reader would take the paper's point to be that this kind of local inhomogeneity is a working, testable resolution of the Hubble tension, not merely a consistency puzzle.

What carries the argument

The object that carries the argument is the Lemaitre metric for a spherically symmetric, comoving spacetime, together with the unified perfect-fluid stress-energy tensor with density $\rho = \Lambda(r) + n(t,r)$ and pressure $p = -\Lambda(r)$. The key reduction is that the Einstein equations close on the area radius $R(t,r)$ once the Misner-Sharp mass is integrated to $M = (\Lambda R^3 + m(r))/6$; the transverse expansion then obeys $H_\perp^2 = (\Lambda + m/R^3 + 3E/R^2)/3$, while the longitudinal rate $H_\parallel$ differs from $H_\perp$ through the radial gradient of $\Lambda$ and the curvature function $E$. The paper's working ansatz is the pair of exponential profiles (6.23) for the dark-energy and dark-matter perturbations, which are deliberately minimal: their amplitude is fixed by the observed $\Delta H_0$, and their common width $\Delta$ (equivalently $\Re$) plus the fraction $x$ are the only surviving degrees of freedom. This machinery turns the Hubble tension into a boundary-value problem for two free numbers, with every prediction downstream of those numbers.

What would settle it

Measure the linear-order redshift drift coefficient $\dot z_0/H_0$ together with the linear-order deviations $\Delta H_\perp(z)$ and $\Delta H_\theta(z)$; the model predicts that the combination in eq. (5.30) is zero at order $z$, and a violation of that relation, or a measured $\dot z_0/H_0$ below about $0.55$ while the local $H_0$ offset remains near $6$ km/s/Mpc, would settle against the model.

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Extended reading notes

Core claim

On its own terms, the paper establishes that a single perfect fluid with $\rho(t,r) = \Lambda(r) + n(t,r)$ and $p(r) = -\Lambda(r)$ (eq. 2.6), together with the exponential profiles (6.23), generates a Lemaitre bubble whose central Hubble rate can be set to the local value $73.52 \pm 1.62$ km/s/Mpc while the spacetime asymptotically approaches an FRW universe with $H_0 = 67.4 \pm 0.5$ km/s/Mpc and with effective densities that pass the early-time $\Omega_m$ and $\Omega_\kappa$ bounds. The amplitude of the perturbation is fixed by the observed offset $\Delta H_0 = 6.12 \pm 1.7$ km/s/Mpc, leaving the dark-energy versus dark-matter fraction $x$ and the dimensionless bubble size $\Re$ as the only free parameters. The paper computes the light-cone observables to first order in the two perturbative schemes and identifies an allowed $x$--$\Re$ region; in that region $w_\parallel(0)$ can reach $-1.07$, $w_\perp(0)$ can reach $-0.69$, the deceleration parameter $Q_0$ can be as low as $-1.24$, the jerk $J_0$ as high as $3.75$, and the redshift drift is steeper near $z=0$ and crosses zero earlier than in the baseline dark-energy model.

Load-bearing premise

The load-bearing premise is that a physically realizable single fluid with energy density $\Lambda(r)+n$ and pressure $-\Lambda(r)$ exists; the paper assumes this from an earlier construction and does not rederive it, so if that construction fails the bubble has no known microscopic origin.

Editorial extensions

If this is right

  • If the bubble picture is correct, the early-universe and local Hubble measurements are measuring different expansion rates: the local rate includes the bubble's $\Delta H_0\approx 6$ km/s/Mpc boost, while the asymptotic rate is the CMB value.
  • The model predicts present-day cosmographic parameters that deviate substantially from the flat $\Lambda$CDM values: $Q_0$ can be as negative as $-1.24$ and $J_0$ as positive as $3.75$; low-redshift distance and drift data can test this directly.
  • The redshift drift in the allowed parameter space starts with a steeper slope than in $\Lambda$CDM ($\dot z_0/H_0$ from $0.55$ up to about $0.83$), peaks at a lower redshift, and vanishes at $z_0<2.09$; these features separate the model from homogeneous alternatives.
  • The early-time consistency conditions exclude a purely dark-matter bubble at $1\sigma$ and constrain a dark-energy-dominated bubble to $\Re \lesssim 0.78$ at $2\sigma$, giving a sharp target for microphysical realizations of the fluid.
  • The linear-order consistency relation (5.30), which ties $\Delta H_\theta$, $\Delta H_\perp$, and $\Delta\dot z$ to zero, is a model-internal check that future surveys can apply without assuming the full bubble profile.

Reading between the lines

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

  • A natural extension the author does not carry out is to treat the exponential profiles as one member of a family of radial profiles; the same effective-density matching and consistency relations provide a template for testing arbitrarily shaped bubbles against combined late-time and early-time data.
  • If the unified-fluid origin is taken seriously, the parameter-region bounds could be read as constraints on the microphysical Lagrangian from which the fluid was derived, turning cosmological observations into particle-physics input.
  • The fact that a pure dark-matter bubble is disfavored while a dark-energy-dominated bubble fits suggests that any successful completion must make the dark-energy component, not the matter component, the driver of the local Hubble boost.
  • A future measurement of $\dot z_0/H_0$ that stays at the $\Lambda$CDM value while local $H_0$ remains high would disfavor the bubble class as a whole, not just this parameter choice, because the drift slope and the local Hubble offset are linked in the same geometry.
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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

4 major / 6 minor

Summary. The paper analyzes a spherically symmetric inhomogeneous cosmological model described by a Lemaitre metric with a single perfect fluid whose energy density is ρ(t,r)=Λ(r)+n(t,r) and pressure p(r)=−Λ(r). The author derives perturbative solutions in two regimes: a Taylor expansion around the center (Section 5) and an expansion around a homogeneous FRW background (Section 6). Using the FRW expansion, the model fixes the amplitude of the inhomogeneities by the observed local Hubble constant and then imposes Planck constraints on the effective matter and curvature densities to restrict the two-parameter space (x, ℜ). The paper computes cosmological observables including the longitudinal, transverse, and volume Hubble rates, effective equations of state, deceleration, jerk, and redshift drift. The central assertion is that the bubble model can simultaneously match the Planck early-time Hubble constant and the SH0ES local value while remaining consistent with Planck estimates of Ω_m and Ω_κ, and that it yields specific, testable predictions.

Significance. If the perturbative treatment is valid, the model would be an explicit inhomogeneous alternative to the standard ΛCDM picture in which a radially dependent cosmological constant and a coupled dark-matter component produce a local Hubble excess without altering the early-time cosmological parameters. The paper has several strengths: the Einstein equations are solved analytically for a non-trivial ansatz; the redshift drift is consistently extended to Lemaitre spacetimes (Section 4); a consistency relation between leading-order corrections of Hubble rates and redshift drift, Eq. (5.30), is derived; and the analysis generates falsifiable predictions for the deceleration parameter, jerk, effective equations of state, and redshift-drift shape (Figures 3-9). At the same time, the model is explicitly constructed so that the H0 tension is an input rather than an output, and the advertised parameter-space constraints from Planck require the perturbative expansion to be valid, a condition that is not verified.

major comments (4)
  1. [§6, Eq. (6.10)] The perturbativity conditions in Eq. (6.10) are stated but never checked against the allowed parameter region of Fig. 2. For a representative point in the 1σ region, x=0.5, ℜ=0.5, Eq. (6.22) gives Δλ≈0.23, and the Gaussian profile (6.23) yields aλ1'(r)/(H0^2 r) evaluated at r≈Δ approximately 0.7, while for ℜ=0.4 the same quantity exceeds unity, violating condition (6.10). Since all results in Section 6, including the constraints (6.26)-(6.28) and the predictions plotted in Figures 1-9, rely on the first-order expansion around FRW, a significant part of the claimed parameter space lies outside the regime of validity. The paper should either restrict the analysis to parameter points that satisfy (6.10), or provide a non-perturbative justification for the continued use of the first-order results.
  2. [§6.2, Eqs. (6.21)-(6.24)] The amplitude of the inhomogeneities is fixed by the observed Hubble tension rather than predicted: Eq. (6.22) sets Δλ in terms of ΔH0, and the profiles (6.23) are chosen so that ΔH⊥(1,r)=ΔH0 e^{-r^2/Δ^2}, Eq. (6.24). The abstract itself states that the model 'imposes' an Hubble profile matching both the Planck and SH0ES values. Therefore the central claim that the bubble resolves the H0 tension is, as it stands, a consistency fit rather than an explanation; the model should be presented more carefully, and the paper should clarify which observables are actually predicted rather than fitted. A genuinely predictive test would require the amplitude to be tied to independent parameters or to emerge from the underlying Lagrangian construction.
  3. [§6.2, Eqs. (6.25)-(6.28)] The constraints on the parameter space are obtained by identifying the asymptotic z≫1 expansions of the Hubble functions, Eqs. (6.25)-(6.27), with the Planck-constrained values of Ω_m and Ω_κ in Eq. (6.28). This identification amounts to an assumption that the effective densities measured along the light cone inside the bubble coincide with the background densities of the external FRW region; in an inhomogeneous universe the relationship between local effective parameters and CMB-derived parameters is nontrivial and should be justified or at least explicitly discussed. Without that discussion, the derived bounds on x and ℜ are less robust than stated.
  4. [§1 and §2, Eqs. (2.6)-(2.7)] The physical viability of the unified perfect fluid with ρ=Λ(r)+n and p=−Λ(r) is assumed from the earlier Lagrangian construction in ref. [6], which is not reproduced in this paper. The author explicitly states that the detailed theoretical structure is omitted. This is not by itself an error, but it means that the model's microphysical origin is an imported assumption; if the construction of ref. [6] cannot generate precisely this stress-energy tensor, the bubble solution has no known realization. The main text should identify this as a limitation of the present work rather than presenting the fluid as established.
minor comments (6)
  1. [Abstract and §1] The notation is inconsistent: the same symbol H0 is used for both the Planck value and the local (SH0ES) value, with a note in footnote 1 but not maintained throughout the text. Please use distinct symbols (e.g., H_0^CMB and H_0^loc) in all equations.
  2. [§6.2, Eqs. (6.19)-(6.20)] The units in Eq. (6.20) are written as '#'; this should be written out as 'km/s/Mpc' or a standard unit symbol.
  3. [§6.2, Eq. (6.22)] Equation (6.22) has a missing closing parenthesis in the expression for Δλ; the formula as printed is incomplete.
  4. [Figures 2-9] The figures are only captioned in the text; the actual plots are missing from the manuscript body. The reader cannot verify the claimed contours or the stated parameter-space bounds without seeing the figures.
  5. [§7, Conclusions] The conclusion states a minimum of Q0^min = −1.24, but the contour plot in Figure 5 appears to show values only down to about −1.10. Please check the numerical consistency between the analytic expressions and the plots.
  6. [Appendix B, Eq. (B.3)] The expression for the O(z^2) drift correction is split across a line break with no continuation sign, making the formula ambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the H0 match is an imposed boundary condition, and the derived cosmographic/drift predictions depend on unfitted combinations of the remaining parameters.

full rationale

The paper is transparent that the Hubble-profile matching is an input, not a derived prediction. The abstract states "We impose an Hubble profile that matches the Planck value at early times (H0=67.4±0.5 Km/s Mpc) and the local value (H0=73.52±1.62 Km/s Mpc)." In Section 6.2, eqs. (6.19)-(6.24) fix the perturbation amplitude Delta_lambda from the observed Delta_H0 and then restate that choice as Delta_H_perp(1,r) = Delta_H0 e^{-r^2/Delta^2}. The conclusion explicitly says "the full amplitude of the effect (DE+DM) is fixed by the observed Delta_H0 (see eq. (6.22))", so the H0 'agreement' is a boundary condition, not a concealed fit. The paper's actual predictions - Q0, J0, effective equations of state, redshift drift, and high-z effective densities - are functions of the two remaining free parameters x and R (eqs. 6.23, 6.35, 6.36, and figures), which are not fixed by Delta_H0 alone. The allowed (x,R) region is subsequently constrained by independent Planck limits on Omega_m and Omega_kappa (eq. 6.28 and Fig. 2), so those outputs are not statistically forced by the H0 input. The only self-citations ([6],[7]) motivate the assumed perfect-fluid EMT rho=Lambda(r)+n, p=-Lambda(r), but the paper expressly treats this as a postulate rather than deriving it here ("we omit the detailed theoretical structure of such a models and focus on a single Perfect Fluid (PF) system"). No uniqueness theorem from the authors is invoked, and no ansatz is smuggled in via citation. The perturbativity concern over condition (6.10) is a correctness/validity issue, not circularity. Overall, the derivation chain is self-contained given the stated assumptions, and the claimed H0 matching is correctly labeled as imposed rather than predicted.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on one fitted amplitude (Delta_lambda set by Delta_H0), two free shape parameters (x and R) constrained by early-time data, and an ad hoc Gaussian profile. The physical reality of the fluid is assumed from a self-cited prior paper. No independent code, data, or machine-checked derivation is provided.

free parameters (4)
  • Amplitude Delta_lambda = fixed by Delta_H0 via eq (6.22)
    The size of the DE/DM perturbations is set by the observed 6.12 km/s/Mpc difference between local and Planck H0, making the H0 tension an input.
  • x (DE fraction of perturbation) = constrained to ~0.07-1 at 1 sigma
    Fraction of the perturbation carried by DE vs DM; constrained by matching effective Omega_m and Omega_kappa to Planck early-time data.
  • R (dimensionless bubble size) = 0.4-0.6 for x>=0.3 at 1 sigma
    Dimensionless size of the Gaussian bubble; constrained by early-time Planck data.
  • Gaussian profile shape = e^{-r^2/Delta^2}
    Chosen ad hoc as 'simplest and most economical'; not derived from the dynamics.
assumptions (5)
  • domain assumption The single-fluid EMT (2.6) with rho=Lambda(r)+n and p=-Lambda(r) is realisable as a dark sector
    Based on author's prior work [6], not derived or independently verified in this paper.
  • domain assumption Spacetime is LRS with a center and the observer is at or within a small distance d ~ 0.48 Delta of the center
    Section 3 sets observer at r=0; Section 1 estimates allowed displacement from the CMB dipole.
  • domain assumption Perturbativity conditions (6.10) hold in the allowed parameter region
    Not checked explicitly across the x-R plane; some figures show extreme values that may violate the conditions.
  • domain assumption Early-time FRW matching with Planck Omega_m and Omega_kappa is the correct way to constrain the model
    The model is matched to FRW at a->0 and uses Planck 2018 constraints to set bounds on x and R.
  • standard math General relativity and the Lemaitre metric (2.1)
    Standard background theory.
invented entities (1)
  • Unified dark fluid with rho=Lambda(r)+n and p=-Lambda(r)
    purpose: To source the inhomogeneous bubble geometry and generate the space-dependent acceleration leading to the H0 tension.
    No microphysical derivation or independent observational handle is provided in this paper; the fluid is postulated and grounded only in the author's prior work [6].

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Pith. "Pith review of Inhomogeneous model with a space dependent Cosmological Constant." pith.science (2026). https://pith.science/paper/4QGMZUI6

@misc{pith2026250107968,
  author       = {Pith},
  title        = {Pith review of: Inhomogeneous model with a space dependent Cosmological Constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QGMZUI6}},
  note         = {Machine review of arXiv:2501.07968}
}
abstract

We analyse an inhomogeneous cosmological model featuring a spherically symmetric bubble solution induced by a unified single perfect fluid, comprising spatially dependent Dark Energy (with $w=-1$) and Dark Matter (with $w=0$) components. We impose an Hubble profile that matches the Planck value at early times ($H_0=67.4\pm0.5$ Km/s\,Mpc) and the local value (${\cal H}_0=73.52\pm 1.62$ Km/s\,Mpc). We explicitly derive perturbative solutions in two distinct regimes: one expanded around the center (for small $r\,{\cal H}_0\ll1$) and the other expanded around a homogeneous FRW universe. In both cases, we compute the cosmographic parameters, redshift profiles for the Hubbles expansion rates, and effective equations of state. Furthermore, we investigate the redshift drift behaviour extended to a Lema$\hat{\rm i}$tre metric.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

42 extracted references · 17 canonical work pages

  1. [6]

    A space dependent Cosmological Constant

    D. Comelli, A space dependent Cosmological Constant, JCAP 04 (2024), 080, doi:10.1088/1475-7516/2024/04/080, [arXiv:2311.15866 [gr-qc]]

    work page Pith review arXiv 2024

  2. [1]

    Perlmutter et al

    S. Perlmutter et al. [Supernova Cosmology Project], Measurements of Ω and Λ from 42 High Redshift Supernovae, Astrophys. J. 517 (1999), 565-586, doi:10.1086/307221, [arXiv:astro-ph/9812133 [astro-ph]]

    arXiv 1999

  3. [2]

    Aghanim et al

    N. Aghanim et al. [Planck], Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020), A6, [erratum: Astron. Astrophys. 652 (2021), C4], doi:10.1051/0004-6361/201833910, [arXiv:1807.06209 [astro-ph.CO]]

    arXiv 2020

  4. [3]

    A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, L. Breuval and T. G. Brink, et al. A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team, Astrophys. J. Lett. 934 (2022) no.1, L7, doi:10.3847/2041-8213/ac5c5b, [arX...

    arXiv 2022

  5. [4]

    Di Valentino, O

    E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess and J. Silk, In the realm of the Hubble tension—a review of solutions,” Class. Quant. Grav. 38 (2021) no.15, 153001, doi:10.1088/1361-6382/ac086d, [arXiv:2103.01183 [astro-ph.CO]]

    arXiv 2021

  6. [5]

    J. P. Hu and F. Y. Wang, Hubble Tension: The Evidence of New Physics, Universe 9 (2023) no.2, 94, doi:10.3390/universe9020094, [arXiv:2302.05709 [astro-ph.CO]]

    arXiv 2023

  7. [7]

    Ballesteros, D

    G. Ballesteros, D. Comelli and L. Pilo, Thermodynamics of perfect fluids from scalar field theory, Phys. Rev. D 94 (2016) no.2, 025034, doi:10.1103/PhysRevD.94.025034, [arXiv:1605.05304 [hep-th]]. M. Celoria, D. Comelli and L. Pilo, Intrinsic Entropy Perturbations from the Dark Sector, JCAP 1803 (2018) no.03, 027, doi:10.1088/1475-7516/2018/03/027, [arXiv...

    arXiv 2016

  8. [8]

    P. K. Aluri, P. Cea, P. Chingangbam, M. C. Chu, R. G. Clowes, D. Hutsem´ ekers, J. P. Kochappan, A. M. Lopez, L. Liu and N. C. M. Martens, et al. Is the observable Universe consistent with the cosmological principle?, Class. Quant. Grav. 40 (2023) no.9, 094001, doi:10.1088/1361-6382/acbefc, [arXiv:2207.05765 [astro-ph.CO]]. 16The order O(z2) corrections a...

    arXiv 2023

Show all 42 references
  1. [9]

    G. F. R. Ellis, Dynamics of pressure free matter in general relativity, J. Math. Phys. 8 (1967) 1171, doi:10.1063/1.1705331. J. M. Stewart and G. F. R. Ellis, Solutions of Einstein ’s equations for a fluid which exhibit local rotational symmetry, J. Math. Phys. 9 (1968) 1072, ...

    1967 arXiv

  2. [10]

    Enqvist, Lemaitre-Tolman-Bondi model and accelerating expansion, Gen

    K. Enqvist, Lemaitre-Tolman-Bondi model and accelerating expansion, Gen. Rel. Grav. 40 (2008), 451-466, doi:10.1007/s10714-007-0553-9, [arXiv:0709.2044 [astro-ph]]. K. Enqvist and T. Mattsson, The effect of inhomogeneous expansion on the supernova observations, JCAP 02 (2007),...

    2008 arXiv

  3. [11]

    Garcia-Bellido and T

    J. Garcia-Bellido and T. Haugboelle, Confronting Lemaitre-Tolman-Bondi models with Observational Cosmology, JCAP 04 (2008), 003, doi:10.1088/1475-7516/2008/04/003, [arXiv:0802.1523 [astro-ph]]

    2008 arXiv

  4. [12]

    Bolejko and P

    K. Bolejko and P. Lasky, Pressure gradients, shell crossing singularities and acoustic oscillations - application to inhomogeneous cosmological models, Mon. Not. Roy. Astron. Soc. 391 (2008), 59, doi:10.1111/j.1745-3933.2008.00555.x, [arXiv:0809.0334 [astro-ph]]

    2008

  5. [13]

    Moradi, J

    R. Moradi, J. T. Firouzjaee and R. Mansouri, The spherical perfect fluid collapse with pressure in the cosmological background, doi:10.1142/9789813226609 0593, [arXiv:1301.1480 [gr-qc]]

    arXiv

  6. [14]

    A. A. H. Alfedeel and C. Hellaby, The Lemaitre Model and the Generalisation of the Cosmic Mass, Gen. Rel. Grav. 42 (2010), 1935-1952, doi:10.1007/s10714-010-0971-y, [arXiv:0906.2343 [gr-qc]]

    2010 arXiv

  7. [15]

    Alnes and M

    H. Alnes and M. Amarzguioui, CMB anisotropies seen by an off-center observer in a spherically symmetric inhomogeneous Universe, Phys. Rev. D 74 (2006), 103520, doi:10.1103/PhysRevD.74.103520, [arXiv:astro-ph/0607334 [astro-ph]]

    2006 arXiv

  8. [16]

    Cusin, C

    G. Cusin, C. Pitrou and J. P. Uzan, Are we living near the center of a local void?, JCAP 03 (2017), 038, doi:10.1088/1475-7516/2017/03/038, [arXiv:1609.02061 [astro-ph.CO]]

    2017 arXiv

  9. [17]

    Biswas, A

    T. Biswas, A. Notari and W. Valkenburg, Testing the Void against Cosmological data: fitting CMB, BAO, SN and H0, JCAP 11 (2010), 030, doi:10.1088/1475-7516/2010/11/030, [arXiv:1007.3065 [astro-ph.CO]]

    2010 arXiv

  10. [18]

    Lema ˆitre, The expanding universe, Gen

    G. Lema ˆitre, The expanding universe, Gen. Rel. Grav. 29 (1997) 641, [Annales Soc. Sci. Bruxelles A 53 (1933) 51], doi:10.1023/A:1018855621348

    1997 doi

  11. [19]

    C.Hellaby, Some Properties of Singularities in the Tolman Model , Ph.D.thesis, Queen’s University at Kinston, Ontario, 1985

    1985

  12. [20]

    P. D. Lasky and K. Bolejko, The effect of pressure gradients on luminosity distance - redshift relations, Class. Quant. Grav. 27 (2010) 035011, doi:10.1088/0264-9381/27/3/035011, [arXiv:1001.1159 [astro-ph.CO]]

    2010 arXiv

  13. [21]

    Jimenez, R

    R. Jimenez, R. Maartens, A. R. Khalifeh, R. R. Caldwell, A. F. Heavens and L. Verde, Measuring the Homogeneity of the Universe Using Polarization Drift, JCAP 05 (2019), 048, doi:10.1088/1475-7516/2019/05/048, [arXiv:1902.11298 [astro-ph.CO]]

    2019 arXiv

  14. [22]

    University of Toronto, Toronto(1952)

    Synge,J.L.,Schild,A.:Tensor Calculus. University of Toronto, Toronto(1952). Synge, J.L.: Relativity: The General Theory. Series in Physics. North-Holland Pub. Co., Amsterdam (1960)

    1952

  15. [23]

    de Swardt, P

    B. de Swardt, P. K. S. Dunsby and C. Clarkson, Gravitational Lensing in Spherically Symmetric Spacetimes, [arXiv:1002.2041 [gr-qc]]

    arXiv

  16. [24]

    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, doi:10.1088/0264-9381/31/20/205001, [arXiv:1402.1933 [astro-ph.CO]]. – 27 –

    2014 arXiv

  17. [25]

    S. M. Koksbang and A. Heinesen, Redshift drift in a universe with structure: Lema ˆ ıtre-Tolman-Bondi structures with arbitrary angle of entry of light,Phys. Rev. D 106 (2022) no.4, 043501, doi:10.1103/PhysRevD.106.043501, [arXiv:2205.11907 [astro-ph.CO]]

    2022 arXiv

  18. [26]

    Sandage, The Change of Redshift and Apparent Luminosity of Galaxies due to the Deceleration of Selected Expanding Universes

    A. Sandage, The Change of Redshift and Apparent Luminosity of Galaxies due to the Deceleration of Selected Expanding Universes. Astrophys.J. 136 (1962) 319, doi = 10.1086/147385. G.C. McVittie, Appendix to The Change of Redshift and Apparent Luminosity of Galaxies due to the D...

    1962 doi

  19. [27]

    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, doi:10.1086/311375, [arXiv:astro-ph/9802122 [astro-ph]]

    1998 arXiv

  20. [28]

    Heinesen, Redshift drift cosmography for model-independent cosmological inference, Phys

    A. Heinesen, Redshift drift cosmography for model-independent cosmological inference, Phys. Rev. D 104 (2021) no.12, 123527, doi:10.1103/PhysRevD.104.123527, [arXiv:2107.08674 [astro-ph.CO]]. A. Heinesen, Multipole decomposition of redshift drift – model independent mapping of...

    2021 arXiv

  21. [29]

    F. S. N. Lobo, J. P. Mimoso, J. Santiago and M. Visser, Dynamical Analysis of the Redshift Drift in FLR W Universes, Universe 10 (2024) no.4, 162, doi:10.3390/universe10040162, [arXiv:2210.13946 [gr-qc]]

    2024 arXiv

  22. [30]

    C. M. Yoo, T. Kai and K. I. Nakao, Redshift drift in LTB universes, Phys.Rev.D83:043527,2011, doi:10.1103/PhysRevD.83.043527, [arXiv:1010.0091 [astro-ph.CO]]

    2011 arXiv

  23. [31]

    Mishra and M

    P. Mishra and M. N. C´ el´ erier,Redshift and redshift drift in Λ = 0 quasispherical Szekeres cosmological models and the effect of averaging, Phys. Rev. D 105 (2022) no.6, 063520, doi:10.1103/PhysRevD.105.063520, [arXiv:1403.5229 [astro-ph.CO]]

    2022 arXiv

  24. [32]

    E. G. Chirinos Isidro, C. Zu˜ niga Vargas and W. Zimdahl, Simple inhomogeneous cosmological (toy) models, JCAP 05 (2016), 003, doi:10.1088/1475-7516/2016/05/003, [arXiv:1602.08583 [gr-qc]]

    2016 arXiv

  25. [33]

    Codur and C

    R. Codur and C. Marinoni, Redshift drift in radially inhomogeneous Lema ˆ ıtre-Tolman-Bondi spacetimes, Phys. Rev. D 104 (2021) no.12, 123531, doi:10.1103/PhysRevD.104.123531, [arXiv:2107.04868 [gr-qc]]

    2021 arXiv

  26. [34]

    M. H. Partovi and B. Mashhoon, Toward Verification of Large Scale Homogeneity in Cosmology, Astrophys. J. 276 (1984) 4, doi:10.1086/161588

    1984 doi

  27. [35]

    Adamek, C

    J. Adamek, C. Clarkson, R. Durrer, A. Heinesen, M. Kunz and H. J. Macpherson, Towards Cosmography of the Local Universe, Open J. Astrophys. 7 (2024), 001c.118782, doi:10.33232/001c.118782, [arXiv:2402.12165 [astro-ph.CO]]

    2024 arXiv

  28. [36]

    M. N. Celerier, Do we really see a cosmological constant in the supernovae data?, Astron. Astrophys. 353 (2000), 63-71, [arXiv:astro-ph/9907206 [astro-ph]]

    2000 arXiv

  29. [37]

    Marra and M

    V. Marra and M. Paakkonen, Exact spherically-symmetric inhomogeneous model with n perfect fluids, JCAP 01 (2012), 025, doi:10.1088/1475-7516/2012/01/025, [arXiv:1105.6099 [gr-qc]]

    2012 arXiv

  30. [38]

    S. M. Carroll, M. Hoffman and M. Trodden, Can the dark energy equation-of-state parameter w be less than -1 ?, Phys. Rev. D 68 (2003), 023509, doi:10.1103/PhysRevD.68.023509, [arXiv:astro-ph/0301273 [astro-ph]]

    2003 arXiv

  31. [39]

    Al Mamon and K

    A. Al Mamon and K. Bamba, Eur. Phys. J. C 78 (2018) no.10, 862, doi:10.1140/epjc/s10052-018-6355-2, [arXiv:1805.02854 [gr-qc]]. – 28 –

    2018 arXiv

  32. [40]

    Camarena and V

    D. Camarena and V. Marra, Local determination of the Hubble constant and the deceleration parameter, Phys. Rev. Res. 2 (2020) no.1, 013028, doi:10.1103/PhysRevResearch.2.013028, [arXiv:1906.11814 [astro-ph.CO]]

    2020 arXiv

  33. [41]

    G. F. R. Ellis and H. van Elst, Cosmological models: Cargese lectures 1998, NATO Sci. Ser. C 541 (1999) 1, doi:10.1007/978-94-011-4455-1 1, [gr-qc/9812046]

    1999 arXiv

  34. [42]

    C. A. Clarkson and R. K. Barrett, Covariant perturbations of Schwarzschild black holes, Class. Quant. Grav. 20 (2003) 3855, doi:10.1088/0264-9381/20/18/301, [gr-qc/0209051]. G. Betschart and C. A. Clarkson, Scalar and electromagnetic perturbations on LRS class II space-times, ...

    2003 arXiv

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