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Cosmology and Nongalactic Astrophysics · arXiv:1909.13489 v1 · submitted 30 Sep 2019 · 89 references

Another look at redshift drift and the backreaction conjecture

S. M. Koksbang

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The paper says

“Earlier studies have conjectured that redshift drift is described by spatially averaged quantities and thus becomes positive if the average expansion of the Universe accelerates.”

more of the abstract

This conclusion is reevaluated here by considering exact light propagation in a simple toy-model with average accelerated expansion. The toy-model and light propagation setup is explicitly designed for concordance between spatial averages and averages along light rays. While it is verified that redshift-distance relations are well described by average quantities in this setup, it is found that the redshift drift is not. Specifically, the redshift drift is negative despite the on-average late-time accelerated expansion of the model. This result implies that measuring redshift drift signals at low redshifts gives the potential for directly falsifying the backreaction conjecture. However, the results are based on a toy-model so it is in principle possible that the result is an artifact and that redshift drift is in reality well described by spatially averaged quantities. The result therefore highlights the importance of developing exact solutions to the Einstein equations which exhibit average accelerated expansion without local expansion so that the relation between spatial averages and observations can be firmly established.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 14 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

It can be wrong. Challenge this review or reply to it.

Holds up, with conditions. A clean numerical counterexample to the drift-follows-averages conjecture, but the missing lu-scan for drift leaves the falsification claim under-supported. This line is wrong

2 major objections · 7 minor

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Not checked: figures, code, what the 89 references say.

The paper says, at §4.3 and §5, Figs. 3–4

The review did not quote this passage.

Pith says: Objection 1 of 2 · major

The paper shows in Fig. 3 that the offset between the exact and covariant redshift-distance relations is sensitive to the homogeneity scale: scaling l_u and l_o by 0.1 'substantially decreases' the offset and scaling by 10 increases it. Fig. 4, however, reports the drift only for the fiducial l_u = 50 Mpc.

more of objection 1

Since §5 itself suggests that the drift mismatch may arise from differentiating a small z − z_C offset, the obvious and feasible test is whether the drift mismatch shares this l_u sensitivity. Without that scan the reader cannot distinguish a genuine backreaction effect from a finite-l_u artifact of the alternation rule l_o = l_u (a_o/a_u)^2. The machinery already exists; please add δz for l_u scaled by 0.1 and by 10 (and, if possible, discuss the l_u → 0 limit) and state explicitly whether the sign disagreement at low z persists, shrinks, or grows. Either outcome would materially clarify the interpretation of the headline result.

The paper says, at §5, Fig. 4 close-up; abstract:s5

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

The falsification claim rests on the sign difference at low redshift, but the close-up in Fig. 4 shows both the exact drift and δz_C at the ~10^-11 level for δt_0 = 30 yr, and δz_C necessarily vanishes at z = 0, so the covariant prediction is only marginally positive at small z.

more of objection 2

The manuscript should quote the numerical values of δz_exact and δz_C at, e.g., z = 0.1, 0.5 and 1, and compare them with the ΛCDM drift over the same interval, so that the reader can judge whether the discriminating signal is anywhere near the reach of the surveys cited in the introduction. As written, the statement that low-redshift measurements give 'the potential for directly falsifying the backreaction conjecture' (abstract:s5) is asserted without any estimate of the size of the effect that would have to be measured.

7 minor objections

The paper says, at Abstract vs §5

Pith says: minor objection 1 of 7

The abstract states that 'the redshift drift is negative' while §5 and the Fig. 4 discussion state that it is 'non-positive'. Please use one term consistently, since the distinction matters at high z where the exact curve approaches zero.

The paper says, at §3.1 and §5

Pith says: minor objection 2 of 7

The text states that the results 'are in agreement with those found when using other particular models, obtained by varying f_o', but no such comparison is displayed. Please show at least one additional f_o (for instance the f_o ≈ 0.18 value mentioned as sufficient for realistic acceleration) in a supplementary panel or appendix, so the claimed robustness is visible rather than asserted.

show 5 more

The paper says, at Fig. 4 caption

Pith says: minor objection 3 of 7

The caption should state δt_0 explicitly and identify the curve labeled z_C as the covariant prediction of Eq. (4.7), so the figure is self-contained. It would also help to state the units of the vertical axis in the close-up panel.

The paper says, at Fig. 1

Pith says: minor objection 4 of 7

The sign convention for Ω_DQ should be stated in the caption or the surrounding text, since the definition (3.6) makes Ω_DQ negative for positive Q while the plotted curves are not obviously signed.

The paper says, at Appendix A

Pith says: minor objection 5 of 7

There is a typographical error in the sentence 'a subscripted 2region will be used her instead of D'; 'her' should read 'here'.

The paper says, at Introduction, p. 1

Pith says: minor objection 6 of 7

The phrase 'the drift of the polarization af inverse-Compton scattered CMB photons' contains a typo; 'af' should read 'of'.

The paper says, at References

Pith says: minor objection 7 of 7

Reference [6] gives the year as '(1096)'; it should be 1962. References [57] and [85] appear to be the same entry, as do [69] and [75]; please merge the duplicates.

What would settle it

Recompute the exact redshift drift for lu scaled by 0.1 and by 10 (the same scalings used for the z-D_A offset in figure 3), keeping all else fixed. Plot the exact drift against δz_C for each lu. If at lu × 0.1 the exact drift becomes positive at low z (matching δz_C's sign), the mismatch is a finite-homogeneity-scale artifact and the falsification claim weakens. If the exact drift remains non-positive at all lu, the claim is robust to this check.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 14 Nov 2026

  1. Reader holds up, with conditions confidence moderate
  2. Skeptic partly agrees 3 min 52 s · 18,439 tokens read, 11,421 written
  3. Editor wrote the verdict above 47 s · 19,355 tokens read, 2,273 written
  4. Referee major revision: 2 major objections, 7 minor 2 min 0 s · 18,692 tokens read, 6,086 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

Koksbang's paper is a clean numerical counterexample to a conjecture she herself helped establish: that redshift drift follows spatially averaged quantities. In a two-region toy model (empty void plus closed dust overdensity) tuned so the average expansion accelerates at late times without local acceleration, exact ray tracing gives non-positive drift at all redshifts, while the covariant averaged formula δz_C = [(1+z)H_D0 − HD(te)]δt0 predicts positive drift at low z. The z–DA comparison in Fig. 3 confirms the model is built for concordance with the covariant scheme, so the drift mismatch is not engineered in. That's the interesting part, and it's done carefully: equations in full, three averaging schemes compared, observer-position robustness checked, and the model varied over fo with consistent results. The citation pattern is fine — Räsänen, Buchert, and the timescape literature are cited where they belong, and the template-scheme comparison is a useful side result even if the toy model can't falsify it outright (the author says so). No code is released, so reproduction means reimplementing the ray tracer from Eq. (4.3), which is feasible but not trivial. The soft spot is the one the stress-test flags, and it's real. Fig. 3 shows the z–DA offset between exact and covariant results shrinks when lu is scaled down by 0.1 and grows when scaled up by 10. The drift is reported only for lu = 50 Mpc. If the drift mismatch shares that lu-sensitivity — and the paper's own remark in §5 that drift is 'the derivative of an integral expression' suggests it might — then the sign flip is a finite-scale effect of the toy construction, not evidence that spatial averages fail for drift. The lu → 0 limit is exactly where the covariant scheme's assumptions are best satisfied, so that's the limit the claim needs. A drift-vs-l_u scan is the missing figure, and a referee should ask for it. That said, the paper is honest about the toy-model caveat (abstract:s6, §5), and the abstract's claim is appropriately hedged ('gives the potential for directly falsifying'). The contribution is a numerical counterexample that motivates exact solutions, not a settled result. Who should read it: anyone working on redshift drift forecasts, backreaction, or the observational case for ELT/SKA drift measurements. It's a useful discussion piece even if the central claim needs the lu check. Recommendation: send it out. The lu gap is a referee point, not a desk reject.

The strongest objection

The central claim (abstract:s5) is that the exact drift's non-positivity, in contrast to δz_C's positivity at low z, shows spatial averaging fails for drift and thus low-z measurements can falsify backreaction. This requires the drift mismatch to be a genuine backreaction effect, not an artifact of the toy construction. The paper itself notes (§5) that the mismatch 'may be related to the fact that the redshift drift is the derivative of an integral expression,' i.e., derivative amplification of a small z - z_C offset. Crucially, the paper shows (figure 3) that the z-D_A offset between exact and covariant results is sensitive to the homogeneity scale lu: scaling lu by 0.1 'substantially decreases' the offset, scaling by 10 increases it. But the paper reports the drift only for lu = 50 Mpc and does not show whether the drift mismatch shares this lu-sensitivity. If the drift mismatch shrinks with lu as the z-D_A offset does, then the mismatch is a finite-scale effect of the toy construction, and the claim that drift is 'not described by spatially averaged quantities' is not established — it would be described in the lu → 0 limit, with the finite-lu deviation being a separate (possibly physical) effect. The paper's own derivative-amplification explanation points in this direction. Without the lu-scan for drift, the falsification claim (abstract:s5) is not supported by the presented evidence.

The paper is a careful, narrowly scoped numerical study. The two-region construction, the lo/lu sampling rule, and the piecewise ray tracing are described in enough detail to reimplement. The z-D_A comparison (figure 3) is a genuine validation of the covariant scheme for distance, and the drift comparison (figure 4) is a genuine challenge for drift. The author acknowledges the toy-model limitation (abstract:s6, §5) and does not hide it. The unresolved issue is narrower: the paper shows the z-D_A offset is lu-sensitive but does not show whether the drift mismatch is. Since the paper's own suggested explanation is derivative amplification of a small z - z_C offset, and since the z - z_C offset is lu-sensitive, the natural test is whether the drift mismatch shares that sensitivity. If it does, the mismatch is a finite-scale effect and the falsification claim is weaker than stated. If it does not, the claim is robust. The reader's weakest_assumption (disjointness) is in the right neighborhood but is acknowledged by the author; the sharper, unacknowledged gap is the missing lu-scan for drift. I therefore agree partially with the reader and would keep the verdict at CONDITIONAL.

The referee's report

The question addressed is a real and well-posed one: whether spatially averaged quantities, including the backreaction terms, can be used to describe light-cone observables, and specifically whether redshift drift inherits the sign of the average acceleration. The paper's strengths are that the toy model is deliberately built to satisfy the hypotheses of the covariant scheme, that three averaging prescriptions are compared against exact ray tracing rather than asserted, and that the result is stated as a falsifiable prediction rather than a consistency argument. The explicit and repeated caveat that the model is not an exact Einstein solution (abstract:s6, §5) is appropriate and honest. If the sign disagreement survives the robustness checks requested below, the result would be a useful, concrete counterexample to the conjecture of Refs. [43,44] and would sharpen the motivation for exact inhomogeneous solutions with average acceleration.

Pith's reading of the paper (a machine's summary, not the authors')

The backreaction conjecture says the apparent late-time acceleration of the universe is an artifact of averaging over inhomogeneities rather than a real local acceleration. A natural hope is that redshift drift — the slow change of a source's redshift over decades of observing time — tracks the spatially averaged expansion rate, so that it would turn positive at low redshift whenever the average expansion accelerates. This paper tests that hope by ray-tracing light through a toy universe built from two kinds of disjoint FLRW regions, an empty void and a closed dust overdensity, arranged so that the average expansion accelerates at late times while nothing accelerates locally. The averaged (covariant) prescription reproduces the exact redshift-distance relation well, but the exact redshift drift comes out non-positive at every redshift, in contrast to the positive drift the averaged formula predicts. The author concludes that redshift drift is not reliably described by spatial averages, and that a low-redshift drift measurement therefore has the potential to falsify the backreaction conjecture directly.

What carries the argument. The argument rests on a two-region toy model: an ensemble of disjoint FLRW regions, one empty and one a closed dust-only model, with volume fractions $f_u$ and $f_o$ and a shared development angle $t = t_0(varphi - sin varphi)/(varphi_0 - sin varphi_0)$. The average scale factor $a_D = [(a_u^3 + a_o^3)/(a_{u,0}^3 + a_{o,0}^3)]^{1/3}$ and average Hubble rate $H_D$ inherit a late-time increase from the contracting overdense regions, giving average acceleration without local acceleration. Light rays are propagated alternately through the two region types with comoving lengths chosen so that $l_o = l_u (a_o/a_u)^2$, matching the proper volume fractions and thus deliberately aligning line-of-sight averages with spatial averages. Redshift drift is then computed exactly from the transport equations along the ray, and compared with the covariant averaged formula built from $a_D$ and $H_D$.

Core claim. The paper finds that in a toy model with on-average late-time accelerated expansion and no local acceleration, the exact redshift drift computed along light rays is non-positive at all redshifts, even though the same model's redshift-distance relation is well described by spatially averaged quantities. The averaged prediction, $1+z_C = 1/a_D$ together with $H_D$ and its time derivative, gives $dz_C = [(1+z)H_{D0} - H_D(t_e)]dt_0$, which is positive at low redshift because $H_D$ increases with time there. The exact drift, obtained by propagating rays alternately through the two region types and solving the drift transport equations, is negative at low redshift for observers placed in the void, in the overdensity, and near a boundary. The author reads this as evidence that redshift drift, being a time derivative of a light-cone integral, need not follow spatial averages even when redshift itself does.

The premise it rests on. The load-bearing premise is that an ensemble of disjoint FLRW regions with sharp boundaries is an adequate stand-in for a statistically homogeneous and isotropic exact solution with significant backreaction, an assumption the author acknowledges may fail because smooth transitions would add shear and alter light propagation.

If the paper is right (Pith's inferences, not the authors')

If the result carries over to exact solutions, a measured positive redshift drift at low redshift would no longer be automatic evidence for dark energy, and a measured non-positive drift would count against backreaction-driven acceleration.

Averaged redshift and averaged distance can be good descriptions of observations while a real-time observable such as redshift drift is not, so the two must be tested separately.

The covariant averaging scheme's success at reproducing redshift-distance relations does not extend to time derivatives of those relations, which weakens arguments that build observational predictions on spatial averages alone.

The result motivates constructing exact solutions to the Einstein equations with average but not local accelerated expansion, since only those can settle whether the toy-model behavior is generic.

The failure pattern suggests a broader rule worth testing: observables that are integrals along the light cone (redshift, distance) may average well, while their time derivatives may not, so other real-time signals such as cosmic parallax could inherit the same discrepancy.

Because the toy model's sharp boundaries suppress shear that would be present in smooth transitions, a natural next check is a swiss-cheese or LTB construction where shear and expansion fluctuations partly cancel along the ray; if drift there still disagrees with the averaged formula, the conclusion strengthens considerably.

If drift really departs from the averaged prediction only at low redshift, the practical discriminant is a drift measurement at $z \lesssim 1$, the range where the sign of the averaged prediction is unambiguous and where proposed SKA-class modifications aim.

A useful cross-check would be to compute the drift for the same two-region model using light-cone averaging rather than spatial averaging; agreement between those two would isolate whether the discrepancy is about the foliation or about taking the time derivative.

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The paper

Abstract

Earlier studies have conjectured that redshift drift is described by spatially averaged quantities and thus becomes positive if the average expansion of the Universe accelerates. This conclusion is reevaluated here by considering exact light propagation in a simple toy-model with average accelerated expansion. The toy-model and light propagation setup is explicitly designed for concordance between spatial averages and averages along light rays. While it is verified that redshift-distance relations are well described by average quantities in this setup, it is found that the redshift drift is not. Specifically, the redshift drift is negative despite the on-average late-time accelerated expansion of the model. This result implies that measuring redshift drift signals at low redshifts gives the potential for directly falsifying the backreaction conjecture. However, the results are based on a toy-model so it is in principle possible that the result is an artifact and that redshift drift is in reality well described by spatially averaged quantities. The result therefore highlights the importance of developing exact solutions to the Einstein equations which exhibit average accelerated expansion without local expansion so that the relation between spatial averages and observations can be firmly established.

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.41

“This result implies that measuring redshift drift signals at low redshifts gives the potential for directly falsifying the backreaction conjecture.”

How the main claim is stated: hedged possibility 0.92

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.98 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.93 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v1). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
Earlier studies have conjectured that redshift drift is described by spatially averaged quantities and thus becomes positive if the average expansion of the Universe accelerates. background 1.00 0.03 0.03 speculative 0.89
This conclusion is reevaluated here by considering exact light propagation in a simple toy-model with average accelerated expansion. contribution 0.61 0.87 0.05 asserted 0.87
The toy-model and light propagation setup is explicitly designed for concordance between spatial averages and averages along light rays. method 0.99 0.85 0.03 asserted 0.99
While it is verified that redshift-distance relations are well described by average quantities in this setup, it is found that the redshift drift is not. result 1.00 0.95 0.44 asserted 1.00
Specifically, the redshift drift is negative despite the on-average late-time accelerated expansion of the model. result 1.00 0.95 0.50 asserted 1.00
This result implies that measuring redshift drift signals at low redshifts gives the potential for directly falsifying the backreaction conjecture. interpretation 1.00 0.71 0.50 hedged possibility 0.92
However, the results are based on a toy-model so it is in principle possible that the result is an artifact and that redshift drift is in reality well described by spatially averaged quantities. limitation 1.00 0.46 0.07 hedged possibility 0.99
The result therefore highlights the importance of developing \emph{exact} solutions to the Einstein equations which exhibit average accelerated expansion without local expansion so that the relation between spatial averages and observations can be firmly established. future work 0.99 0.19 0.24 asserted 0.63

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 0.51

What was examined, and what was not

Sources read

  • arXiv listing abstract v1

Not examined

  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers

Levels run

  • P abstract

Not run

  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

How the labels were made

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,271 input tokens.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

Figures (4)
Figure 1 of the paper
Figure 1. Average density parameters of the studied two-region model. Results from one particular model will be studied, but the results are in agreement with those found when using other particular models, obtained by varying fo. The results are shown for a model defined by having fo = 0.5. This choice is made somewhat arbitrarily but with the following considerations in mind: In order to obtain accelerated expansion of the size similar to what is actually observed, only fo ≈ 0.18 is necessary, but a larger value of fo will increase the late-time average accelerated expansion and will thus…
Figure 2 of the paper
Figure 2. Hubble parameters of the studied two-region model. A close-up of the average Hubble parameter, HD, is included in order to show that it increases at late times. similar size as Ω D m and Ω D R at late times. The kinematical backreaction is indeed so significant that it induces a sign change of the curvature during the model’s evolution (a sign change of the spatial curvature is not possible in a pure FLRW model and is thus necessarily a backre￾action effect). The three Hubble parameters Ho, Hu and HD are shown in figure 2. At early times, when the overdense regions make up the larg
Figure 3 of the paper
Figure 3. Redshift-distance relations in toy-model. Results obtained by propagating light through disjoint regions are labeled according to whether the observer was placed in an empty FLRW region (“u”) or the overdense FLRW model (“o”). In both cases, the light ray was propagated the length lu (lo) before being moved into a new region. is the more appropriate one to use. With z C = 1 aD − 1, the average redshift drift is given by an expression analogous to the FLRW expression, i.e. δzC = [(1 + z)HD0 − HD(te)] δt0. (4.7) In the following subsection, this expression is compared to the exact r
Figure 4 of the paper
Figure 4. Redshift drift in toy-model. Results obtained by propagating light through disjoint regions are labeled according to whether the observer was placed in an empty FLRW region (“u”) or an overdense FLRW model (“o”). In both cases, the light ray was propagated the length lu (lo) before being moved into a new region. The light ray labeled “edge” corresponds to an observer placed approximately 1Mpc from the edge of an overdense region so that the light ray was moved into an underdense region quickly after being initialized (at the point of observation). A close-up of the region z ≈ 0 is…

Figures are the authors', reproduced from the paper.

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Cited by 5 Pith papers

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Topics: Dark Energy

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  • This type of two-region models have earlier been used for studying the relation between backreaction and observations (e.g. [ref] ) and form part of the basis for computations in the timescape scenario (see e.g. [ref] for a comparison of timescape predictions with observations).

    Average observational quantities in the timescape cosmology · Model setup · 0.84

  • This type of two-region models have earlier been used for studying the relation between backreaction and observations (e.g. [ref] ) and form part of the basis for computations in the timescape scenario (see e.g. [ref] for a comparison of timescape predictions with observations).

    Redshift drift in an inhomogeneous universe: averaging and the backreaction conjecture · Model setup · 0.84

  • This type of two-region models have earlier been used for studying the relation between backreaction and observations (e.g. [ref] ) and form part of the basis for computations in the timescape scenario (see e.g. [ref] for a comparison of timescape predictions with observations).

    Local and non-local measures of acceleration in cosmology · Model setup · 0.84

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