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This paper shows that small, deliberately introduced inconsistencies among BAO, CMB, and supernova datasets can shift joint dark-energy posteriors enough that mock ΛCDM data appear to exclude the cosmological constant (w0, wa) = (−1, 0).

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2026-08-03 18:45 UTC pith:QGWTD6WH

load-bearing objection The qualitative warning is plausible, but the flagship SNe zero-point result violates the paper's own M marginalization and the quantitative framework is not yet trustworthy. the 6 major comments →

arxiv 2512.04130 v2 pith:QGWTD6WH submitted 2025-12-03 astro-ph.CO

Controlled Tension Forecasting: Quantifying Cross-Probe Biases in ω₀ω_aCDM

Seokcheon Lee This is my paper
classification astro-ph.CO
keywords dynamical dark energyCPL parametrizationpivot equation of statecross-probe tensionsbaryon acoustic oscillationsCMB distance priorssupernova calibrationmock cosmological surveys
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes that cross-probe inconsistencies need not be large to fake dynamical dark energy. Starting from self-consistent mock data drawn from a single ΛCDM cosmology, the author injects controlled offsets in the matter density, the Hubble constant, the supernova absolute magnitude, or the BAO sound-horizon scale, then refits the joint data with a single MCMC pipeline. A supernova zero-point shift of ΔM = −0.09 produces a pivot equation-of-state value wp = −0.863 ± 0.013, a deviation from −1 exceeding 10σ; a CMB-side H0 offset of −5 km/s/Mpc yields (w0, wa) = (−1.178, 1.082) and wp = −1.265. The point is that apparent dynamical-dark-energy signals in combined datasets can arise from calibration mismatches and degeneracy geometry, not from new physics, so probe calibrations must be verified before interpreting deviations from ΛCDM.

Core claim

The paper's central claim, stated fairly, is that the dynamical-dark-energy-like shifts seen in recent multi-probe analyses can be reproduced in simulations whose input cosmology is exactly ΛCDM, by injecting only small cross-probe inconsistencies. In the tension-free baseline, the joint fit returns (Ωm0, H0, w0, wa) = (0.2964, 69.86, −1.014, 0.144), consistent with the fiducial values (0.30, 70, −1, 0). Shifting only the supernova zero point by ΔM = −0.09 moves the joint posterior to (Ωm0, H0, w0, wa) = (0.328, 66.36, −0.910, 0.151) and drives the pivot equation of state to wp = −0.863 ± 0.013, an apparent >10σ exclusion of −1. Shifting only the CMB-side Hubble constant down by 5 km/s/Mpc m

What carries the argument

The central object is the tension vector Δt = (ΔΩm0, ΔH0, ΔM, Δrd), which injects controlled offsets into otherwise self-consistent mock data from BAO, CMB, and supernova surveys. The dark-energy model under test is the CPL parametrization, w(a) = w0 + wa(1 − a), a linear-in-scale-factor equation of state. The framework's key derived quantity is the pivot equation of state wp, the value of w at the scale factor where the w0–wa covariance is minimized; in these runs, wp moves far from −1 even when the input cosmology is exactly a cosmological constant. The propagation of tensions is tracked by a unified MCMC pipeline and by linear-response bias estimates from the likelihood's curvature.

Load-bearing premise

The quantitative results assume that the compressed CMB distance priors (R, ℓA, ωb) with Planck-2018-style covariance respond to shifts in H0 and the dark-energy parameters in the same way the full CMB likelihood would; if that compression is unfaithful, the large phantom-like shifts seen in the CMB-side tension runs would be artifacts of the compression scheme rather than of the injected tensions.

What would settle it

Repeat the CMB-side tension run (ΔH0 = −5 km/s/Mpc) using a full CMB likelihood instead of the compressed three-number distance prior (R, ℓA, ωb). If the recovered (w0, wa) no longer lands near (−1.18, 1.08) with wp ≈ −1.27, the paper's headline result depends on the distance-prior compression, not on the injected tension.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A >10σ pivot shift from a single supernova calibration offset means that statistical significance alone cannot distinguish genuine dark-energy evolution from cross-probe miscalibration; probe absolute calibrations must be validated jointly.
  • The sign of the spurious signal is a fingerprint of the mismatched probe: supernova-side shifts push wp above −1 (quintessence-like), while BAO- and CMB-side shifts push wp below −1 (phantom-like).
  • Linear-response (likelihood-curvature) bias estimates reproduce the direction of the posterior shifts in all runs, confirming the shifts are driven by the geometry of probe degeneracies rather than by numerical or sampling problems.
  • Strong multi-probe misalignment can rotate the w0–wa degeneracy so strongly that the pivot redshift moves to zp ≈ 1.6 and wp flips to −0.52, so the pivot parameters themselves become misleading under large tensions.
  • The 1D tension–bias transfer functions are reliable only for small tensions; for large or correlated tensions, they underestimate the bias by up to an order of magnitude, so nonlinear or multidimensional forecasting is required.

Where Pith is reading between the lines

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

  • A testable consequence: re-analyzing current BAO+CMB+SNe data with the supernova absolute magnitude and the BAO sound horizon jointly self-calibrated (rather than fixed by external priors) should move (w0, wa) back toward (−1, 0) by roughly the amounts predicted by the transfer functions, if the reported dynamical-dark-energy preference is tension-driven.
  • The paper implies a simple consistency diagnostic for future analyses: report the pivot redshift zp alongside wp. An anomalously large |zp|, outside the redshift range actually probed by the data, would flag degeneracy rotation caused by cross-probe inconsistencies rather than genuine dark-energy dynamics.
  • The transfer-function framework could be inverted into a survey design tool: compute, before a survey is built, which combinations of calibration offsets (e.g., a 0.01 mag supernova zero-point drift and a 1% sound-horizon error) would produce a false 3σ or 5σ dark-energy detection, and use that map to set calibration tolerances.
  • Because the 1D linear relations break down far from the fiducial model, any realistic forecast pipeline for next-generation datasets should replace linear scaling with nonlinear emulators trained on injection grids; the paper's 2D maps show the mixing is already non-negligible at small tensions.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 4 minor

Summary. This paper develops a mock-based forecasting framework in which BAO, CMB-compressed-distance-prior, and SNe datasets are generated from a fiducial ΛCDM cosmology and then analyzed under the CPL parametrization after injecting controlled tensions (ΔΩm0, ΔH0, ΔM, Δrd). Seven MCMC runs are presented: a tension-free baseline and six tension-injection scenarios. The paper reports that moderate injected tensions produce posterior biases in (w0, wa) and wp that mimic dynamical dark energy, sometimes at high significance; it then fits one-dimensional transfer functions and a 4×3 bias matrix A to convert tension vectors into predicted parameter shifts, and illustrates the breakdown of the linear map for realistic DESI+CMB shifts.

Significance. If the corrections outlined below are made, the qualitative message is valuable: it adds a controlled, multi-probe mock demonstration to the existing evidence that degeneracy-geometry mismatches can manufacture DDE-like signals. The direct MCMC approach is a strength because it does not depend on the transfer functions for the qualitative conclusion. The transfer-function part is a useful diagnostic skeleton, but its current quantitative content is undermined by the degenerate SNe runs, missing injection amplitudes, and in-sample calibration. The paper explicitly disclaims realism through the compressed CMB prior, which is appropriate, but then the forecasting claims in Sec. V need to be calibrated accordingly.

major comments (6)
  1. [§IV.B, §IV.F, Appendix A3] The stated likelihood samples θ=(Ωm0,H0,w0,wa,rd,M) with flat priors on M and marginalizes it as a nuisance parameter (Sec. III.C.2, Sec. II.C). A constant shift ΔM in the SNe modulus is then exactly degenerate with M: the residual depends on μ_th(z)+ΔM−M = μ_th(z)−(M−ΔM). Consequently run2 and run6 should yield the same joint cosmological posterior as run1; instead Eqs. (15) and (31) report large shifts, and Eq. (38) encodes a huge ΔM response. Appendix A3 concedes that only the redshift-dependent tilt term ΔM1 z cannot be absorbed by a constant M. This invalidates the SNe column of the bias matrix A (Eq. 40) and the paper's run2 demonstration as implemented. Please redesign the injection (e.g., shift the prior on M, fix M to a biased value, or use a redshift-dependent tilt) or remove these runs.
  2. [§IV.F, §IV.G, Fig. 8] Section IV.F does not give the numerical value of ΔM for run6, and Section IV.G specifies run7 only as 'positive shift ΔΩm0>0' and 'negative shift ΔH0<0' with no amplitudes. Despite this, Figure 8 places run6 and run7 as concrete points in the (ΔΩ_inj, ΔH_inj) plane and Sec. V.B constructs 2D maps from their locations. Without exact injection amplitudes, seeds, and the mock covariances, the maps, the 'nonlinear curvature', and the ranking of runs cannot be checked or reproduced. Add exact parameter values and a reproducible generation protocol.
  3. [§IV.B.2, Eq. (16)] Equation (16) defines μ=m−M. A more negative M increases μ for fixed apparent magnitude m, implying larger luminosity distance and hence a lower inferred H0. The text in Sec. IV.B.2 states the opposite: ΔM=−0.09 'moves the SNe-only posterior toward a significantly higher Hubble constant.' The same sign error appears in Sec. IV.F.2. As a result the interpretation of Eq. (15) (H0=66.36 being a BAO+CMB pull against a high-H0 SNe preference) is not physically coherent because, with M free, there is no high-H0 SNe preference. Correct the sign convention.
  4. [§III.C.3, §II.C] Sec. III.C.3 reports emcee settings but no convergence diagnostics (R-hat/ESS, acceptance rates), no random seeds, no code or data release. The paper's headline claims are shifts in wp quoted to ±0.013 and in (w0,wa) to ±0.02–0.04; demonstrating that runs 2–7 are converged to this precision requires diagnostics for every run. Without these, the quantitative posterior means cannot be verified. Provide diagnostics and a public reproducibility package.
  5. [§II.C, §IV.D] The quantitative amplitudes of the CMB-mediated biases are conditional on the compressed distance-prior representation (R, ℓA, ωb) described in Sec. II.C. Run4's wp=−1.265 and run7's extreme pivot rotation are driven by this channel. Compressed priors are well known to have different degeneracy structure and constraining power than the full Planck likelihood; the paper's abstract itself limits the scope, but the transfer functions in Sec. V are presented as a forecasting tool. Add a validation of the compression for CPL parameters (e.g., one run analyzed with a full Planck-like likelihood) or quantitatively bound the compression-induced error.
  6. [§V.B, Eq. (41)] Sec. V.B constructs the linear map A by fitting the slopes of runs 2–7 and then inverts the same map in Eq. (41) to 'predict' biases. This is a circular calibration, not an out-of-sample prediction; the failure of the 1D map for the real-data vector (Sec. V.C) is therefore overinterpreted as evidence of nonlinearity alone. Use a cross-validation or leave-one-out scheme, or an independent set of injection amplitudes, to assess predictive skill.
minor comments (4)
  1. [Throughout] Typos: 'module' for 'modulus' in Secs. IV.B.2 and IV.F.2; 'conter-intuitive' and 'accerlated' in Sec. IV.G.1; 'se;ectively' in Sec. IV; 'a rudeced Ωm0' in Table II; 'as s standard' in Sec. II.C.
  2. [Fig. 8 caption] Runs 2 and 6 are ΔM injections but the axes are (ΔΩ_inj, ΔH_inj). Explain how a ΔM-only run is projected to a point in this plane; otherwise the scatter positions are ambiguous.
  3. [Sec. V.C] The statement that the 1D relations 'correctly identify the direction of the DESI–CMB discrepancy' is not supported by any fit statistic or uncertainty on the inferred (ΔΩ_inj, ΔH_inj, ΔM) from Eq. (41). Report the covariance of the estimated tension vector.
  4. [Sec. III.A, Sec. IV] The abstract and Sec. III mention redshift-dependent SNe tilt ΔM(z) as one of the controlled channels, but no run in Sec. IV exercises this channel. Either add such a run or explicitly state why it is deferred.

Circularity Check

1 steps flagged

No load-bearing circularity in the central MCMC demonstrations; one flagged step: the empirical Delta-M bias channel is calibrated on run2, whose constant zero-point shift the paper's own likelihood says is absorbable by the free nuisance M.

specific steps
  1. other [Sec. IV.B (Run2, Eq. 15/17), Sec. III.C.2, Sec. V.A.3 (Eq. 38), Sec. V.B.1 (Eq. 40), Appendix A3]
    "The sampled cosmological parameter vector is θ=(Ωm0,H0,ω0,ωa,rd,M), with flat priors... Run2 introduces a controlled tension exclusively in the supernova sector... implemented through a coherent negative shift in the SNe absolute magnitude, ΔM=−0.09... The joint MCMC constraints obtained in run2 are Ωm0=0.3283±0.0036, H0=66.362±0.210, ω0=−0.9096±0.0392, ωa=0.1507±0.1150... Injecting a SNe absolute-magnitude shift ΔM yields ΔΩ(post)m0≃−6.9189×10−1 ΔM... [A3] the term proportional to z cannot be absorbed by a constant nuisance parameter M."

    The paper's own likelihood has M as a free, flat-prior nuisance (Sec. III.C.2), and its Appendix A3 concedes that only the redshift-dependent tilt ΔM1z 'cannot be absorbed' by M. A purely constant ΔM is therefore exactly degenerate with M, so the profile/marginal posterior over (Ωm0,H0,w0,wa) should be invariant under run2's injection. Yet run2 reports large (>10σ) shifts in wp and large shifts in (Ωm0,H0,w0,wa), and these run2 outputs are then used to fit the empirical ΔM→bias column (Eq. 38) that enters the composite bias matrix A (Eq. 40), which is subsequently presented as a predictive tension→bias map (Eqs. 39, 41). The ΔM forecasting channel thus reduces to an artifact of the sampling/analysis rather than a genuine propagation of the injected tension; by the model's own construction

full rationale

The paper's central claim—that controlled cross-probe inconsistencies can mimic DDE in CPL fits—is supported directly by the MCMC runs (run2–run7), which are self-contained mock analyses with the baseline run1 recovering the fiducial ΛCDM parameters in-house. That demonstration does not depend on the author's prior papers: Refs. [38] (Paper1) and [43] (Paper2) are cited as motivation (Secs. II.A–II.B), but run1 independently re-establishes the tension-free baseline and the present injections are explicit in Eqs. (8)–(9) and (19)–(25). No uniqueness theorem or external result by the same authors is invoked to force a choice; the CPL parametrization is a stated, conventional assumption. The empirical transfer functions of Sec. V are honest fits to the run outputs, and the paper explicitly disclaims their out-of-sample validity in Sec. V.C, showing the 1D prediction fails by nearly an order of magnitude for DESI DR2+CMB; this is a labeled limitation rather than a hidden prediction. The one genuine concern is run2 itself: the constant zero-point shift is degenerate with the free nuisance M per the paper's own likelihood construction and Appendix A3, so the large reported run2 shifts and the ΔM column of the empirical bias matrix (Eqs. 38, 40) are internally inconsistent with the stated model. This undermines the ΔM channel (runs 2 and 6) and the associated transfer-function column, but the other channels (BAO Ωm0 distortion, CMB H0 shift, combined runs 3–5, 7) are well-defined data-level injections not degenerate with any nuisance. Overall the core result has independent grounding; the flagged step is a partial, channel-specific reduction by construction, not a global circularity. Score 3.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central claim rests almost entirely on the author's own mock construction; no external data are used. The empirical transfer functions and bias matrix are linear fits to the six MCMC runs (free parameters). The mock likelihoods rely on domain assumptions (Gaussianity, compressed CMB priors, survey-matched covariances). No new physical entities are introduced.

free parameters (6)
  • 1D transfer-function slopes and intercepts (Eqs 36–38) = e.g., Δωa slope 7.1200 per unit ΔΩm0; intercept 0.75432
    Linear coefficients fit to the six MCMC runs (run2–run7); no uncertainties reported; no cross-validation.
  • Injected SNe zero-point shift ΔM (run2) = -0.09 mag
    Chosen by hand to mimic a +3σ SH0ES-like H0 preference; drives the headline pivot deviation.
  • Injected BAO matter-density offset ΔΩm0 (run3) = ≈ -0.02 (to shift BAO-preferred Ωm0 to 0.28)
    Chosen by hand to mimic a 2σ BAO-CMB matter-density inconsistency.
  • Injected CMB H0 offset (run4) = -5 km/s/Mpc
    Chosen by hand to mimic a Planck-like low-H0 preference.
  • Injected BAO and CMB offsets (run5) = ΔΩm0=+0.03, ΔH0=-5 km/s/Mpc
    Chosen by hand to create opposite-directed tensions between BAO and CMB.
  • Injection amplitudes for run6 and run7 = Not specified
    The paper does not state the values of ΔM (run6) or the combined offsets (run7); this prevents replication and weakens the empirical maps that use these runs.
axioms (6)
  • domain assumption BAO, CMB, and SNe likelihoods are Gaussian with fixed, mutually independent covariances (Eq 5/10)
    Underlies the unified MCMC analysis; ignores non-Gaussian tails and cross-probe covariance.
  • domain assumption Compressed CMB distance priors (R, ℓA, ωb) calibrated to Planck-2018 faithfully represent the full CMB likelihood's response to H0 and (w0,wa) shifts
    All run4/5/7 CMB-tension results propagate through this compression; abstract explicitly restricts the setup.
  • domain assumption Single mock realization per run is representative; no average over noise draws needed to define bias
    Posterior shifts are compared to a single Gaussian draw (Eq 3); no bootstrap or repeated mocks.
  • domain assumption MCMC chains (128 walkers, 2000 burn-in, 5000 production) have converged for all runs
    No convergence diagnostics (R-hat, autocorrelation times) are reported (Sec III.C).
  • domain assumption Linear-response Fisher bias formula (Eq 12) is a valid local approximation to the posterior shift
    Used for the Fisher-level bias estimates; the paper itself notes nonlinearity dominates for realistic tensions.
  • standard math CPL parametrization w(a)=w0+wa(1-a)
    Standard two-parameter dark-energy model; the space in which all claims are framed (Sec II).

pith-pipeline@v1.3.0-alltime-deepseek · 27845 in / 16131 out tokens · 143088 ms · 2026-08-03T18:45:20.220714+00:00 · methodology

0 comments
read the original abstract

Recent analyses combining DESI DR2 BAO, Planck CMB, and Pantheon+ SNe have reported mild deviations from the \LambdaCDM model. A central challenge is to determine whether these deviations reflect genuine dark-energy evolution or instead arise from cross-probe inconsistencies, prior choices, or mismatches in likelihood construction. Previous work demonstrated that imposing a biased supernova-motivated prior on \Omega_{m0} can artificially displace the BAO-inferred (w_0,w_a) values from the \LambdaCDM expectation. A complementary pedagogic study further showed that differing degeneracy geometries among BAO, CMB, and SNe can generate apparent dark-energy evolution even when the underlying cosmology is exactly \LambdaCDM. Here we present a controlled tension-injection framework designed as a simplified mock-based diagnostic tool for studying how selected probe-level inconsistencies propagate into inferred dark-energy parameters. Self-consistent BAO, CMB, and SNe mock datasets are augmented with parameterized shifts in (\Omega_{m0}, H_0), supernova absolute calibration, and the BAO sound-horizon scale r_d. The resulting datasets are analyzed through a unified MCMC pipeline, enabling a direct assessment of how these controlled tensions propagate into biases in (w_0,w_a) and the pivot equation-of-state parameter w_p. The results should be interpreted within the restricted setup adopted here: the late-time sector is described in the CPL parametrization and the CMB is represented through compressed distance priors. In this sense, the framework is intended primarily as an illustrative and diagnostic device for identifying probe combinations and degeneracy directions that are more vulnerable to tension-induced dynamical-dark-energy-like shifts, rather than as a parametrization-independent or fully realistic prediction tool.

Figures

Figures reproduced from arXiv: 2512.04130 by Seokcheon Lee.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Two-dimensional tension–bias maps for ∆ [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Normalized bias vectors in the ( [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

120 extracted references · 1 canonical work pages · cited by 3 Pith papers

  1. [1]

    These represent miscalibrated data or probe-internal systematics

    Data-level injections The theoretical predictions entering the mock generation are modified prior to adding survey noise dobs X =d th X (θfid) + ∆dX (∆t) +L X g.(8) Examples include SNe zero-point shifts (M→M+ ∆M), BAO-scale distortions(r d →r d + ∆rd), or CMB distance- prior shifts modifying (R, ℓ A) to emulate anH 0 mismatch. These represent miscalibrat...

  2. [2]

    π(Ωm0) =N Ωfid m0 + ∆Ωm0, σ2 Ωm .(9) This corresponds to the prior-bias mechanism identified in Paper 1

    Prior-level injections The mock data remain tension-free, but the likelihood analysis uses a biased external prior, e.g. π(Ωm0) =N Ωfid m0 + ∆Ωm0, σ2 Ωm .(9) This corresponds to the prior-bias mechanism identified in Paper 1

  3. [3]

    Hybrid injections Some realistic scenarios require combining data-level and prior-level offsets. Typical examples include a higher SNe- inferredH 0 induced by ∆Mtogether with CMB priors preferring a lowerH 0, or BAO distance calibrations shifted by ∆rd while CMB keeps the fiducialr d. Hybrid schemes emulate multi-probe inconsistencies linked to distinct p...

  4. [4]

    Likelihood construction Each probe enters through the Gaussian likelihood −2 lnLX (θ) = dobs X −d th X (θ) T C −1 X dobs X −d th X (θ) .(10)

  5. [5]

    Parameter vector and priors The sampled cosmological parameter vector isθ= (Ω m0, H0, ω0, ωa, rd, M), with flat priors for baseline analyses and appropriately biased priors for prior-level tension injections

  6. [6]

    We useemceewith 128 walkers, a 2000-step burn-in, and a 5000-step production run [66]

    Joint sampling Probe combinations (BAO+CMB, BAO+SNe, BAO+CMB+SNe) are sampled by multiplying the individual like- lihoods. We useemceewith 128 walkers, a 2000-step burn-in, and a 5000-step production run [66]. The MCMC output includes marginalized posteriors, median constraints, pivot parameters (a p, zp, wp), and derived summaries of posterior geometry

    2000

  7. [7]

    Fisher-based bias estimates To complement the MCMC forecasts, we compute the Fisher matrix Fαβ = ∂2χ2 ∂θα∂θβ fid ,(11) and derive linear-response bias estimates ∆θ(Fisher) α = X β (F −1)αβ ∂χ2 ∂θβ ∆t .(12) 8 Comparison with the full MCMC results quantifies the accuracy of the linear-response approximation and provides a functional tension–bias mapping for...

  8. [8]

    At first sight these trends appear counterintuitive, since the SNe zero-point shift ∆M <0 drives the SNe-only like- lihood toward largerH 0 and smaller Ωm0

    Posterior Shifts and Degeneracy Geometry The joint MCMC constraints obtained in run2 are Ωm0 = 0.3283±0.0036, H 0 = 66.362±0.210, ω0 =−0.9096±0.0392, ω a = 0.1507±0.1150.(15) Compared with the fiducial cosmology, the joint posterior exhibits a coherent pattern of parameter shifts: the matter density is displaced to higher values, the inferred Hubble const...

  9. [9]

    Tension-Injection Mechanism The tension is injected by applying a uniform shift to the SNe distance modulus fromµ th(z) toµ th(z) + ∆Mwith ∆M=−0.09. Since the distance module is given by µ=m−M= 5 log 10 DL Mpc + 25,(16) where a negative shift inMcorresponds to a brighter absolute magnitude and thus a smaller inferred luminosity distance for a fixed observ...

  10. [10]

    Pivot Equation-of-State The pivot EoS parameters for run2 are ap = 0.6850, z p = 0.4599, ω p =−0.8626±0.0130.(17) The large deviation ofω p from−1 demonstrates that a single-probe calibration bias—here confined entirely to the SNe distance scale—can induce an apparently significant (>3σ) departure from a cosmological constant. This highlights one of the c...

  11. [11]

    Relative to the tension-free baseline, the contours shift coherently along the combined BAO+CMB degeneracy direction

    Joint Posterior Contours Figure 2 shows the full joint posterior for (Ωm0, H0, ω0, ωa) in run2. Relative to the tension-free baseline, the contours shift coherently along the combined BAO+CMB degeneracy direction. Although this displacement would normally be interpreted as evidence for DDE, in this case it arises solely from the imposed SNe zero-point mis...

  12. [12]

    The preferred matter density moves slightly below the fiducial value, whileH 0 shifts upward, lying between the CMB and SNe preferred directions

    Posterior Shifts and Degeneracy Geometry The median and 1σconstraints from the joint BAO+CMB+SNe MCMC analysis are Ωm0 = 0.2874+0.0035 −0.0034, H 0 = 70.562+0.247 −0.251, ω0 =−1.2012 +0.0452 −0.0442, ω a = 0.7412+0.1239 −0.1336.(18) These shifts reflect the characteristic imprint of a BAO-driven low–Ω m0 tension. The preferred matter density moves slightl...

    2012

  13. [13]

    Tension-Injection Mechanism The BAO tension in run3 is introduced by shifting the BAO-preferred matter density through a coherent distortion of the distance-ratio observables,d 1(z)≡D M (z)/rd andd 2(z)≡D H (z)/rd, along their response to Ω m0. Lowering Ωm0 increases the comoving distanceD M (z) and decreases the expansion rateH(z) at the relevant BAO red...

  14. [14]

    For run3, we obtain ap = 0.71, z p ≃0.41, ω p =−1.045±0.018.(21) The pivot value lies moderately below−1, consistent with the phantom-like shift seen in (ω 0, ωa)

    Pivot Equation-of-State Recasting the CPL parameters in terms of the pivot EoS minimises the covariance between the DE amplitude and its slope. For run3, we obtain ap = 0.71, z p ≃0.41, ω p =−1.045±0.018.(21) The pivot value lies moderately below−1, consistent with the phantom-like shift seen in (ω 0, ωa). For comparison, run2—where the injected tension r...

  15. [15]

    Joint Posterior Contours Figure 3 presents the joint posterior for (Ω m0, H0, ω0, ωa). Relative to the tension-free baseline and the SNe-driven tension in run2, the (ω 0, ωa) contours show a pronounced shift toward (ω 0 <−1, ω a >0), consistent with the pivot trend in Eq. (21). In the (Ω m0, H0) plane, the BAO-induced preference for lower Ω m0 is only par...

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    Posterior Shifts and Degeneracy Geometry The median and 1σconstraints obtained from the joint MCMC analysis are Ωm0 = 0.3384±0.0031, H 0 = 69.875±0.166, ω0 =−1.1781±0.0166, ω a = 1.0822±0.0258.(22) These shifts reflect the characteristic imprint of a CMB-driven low–H 0 tension. The inferred matter density increases well above the fiducial value, the resul...

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    Tension-Injection Mechanism The CMB-side tension in run4 is introduced by modifying the distance-prior vector (R, ℓ A) so that it is internally consistent with a lower CMB-preferred Hubble parameter, H (CMB) 0 =H (fid) 0 + ∆H(CMB) 0 = 70−5≃65 km s −1Mpc−1.(23) For fixed (Ωm0, ω0, ωa, rd), a decrease inH 0 increases the sound-horizon angular scaleθ s ≡r s(...

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    The pivot value lies far below−1, indicating a pronounced phantom-like behaviour driven almost entirely by the acoustic-scale inconsistency in the CMB sector

    Pivot Equation-of-State The pivot parameters extracted from the run4 posterior are ap = 1.0806, z p =−0.0746, ω p =−1.2654±0.0186.(27) The negative value ofz p indicates that the pivot direction has rotated pastz= 0, a clear sign that the CMB-induced tension forces the fit to compensate through rapid late-time evolution of the DE EoS. The pivot value lies...

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    The (ω 0, ωa) ellipses are displaced far into the phantom-like quadrant, consistent with the pivot trend in Eq

    Joint Posterior Contours Figure 4 presents the joint posterior contours for (Ω m0, H0, ω0, ωa) obtained in run4. The (ω 0, ωa) ellipses are displaced far into the phantom-like quadrant, consistent with the pivot trend in Eq. (27), while the (Ωm0, H0) contours reflect the combined BAO+CMB ridge that links lowerH 0 with higher Ω m0. Although the underlying ...

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    Posterior Shifts and Degeneracy Geometry The median and 1σconstraints obtained from the BAO+CMB+SNe MCMC analysis are Ωm0 = 0.3371±0.0031, H 0 = 69.611±0.165, ω0 =−1.1533±0.0157, ω a = 1.0563±0.0228.(28) The combined posterior reflects a coherent compromise between the competing BAO and CMB tension directions. The matter density shifts upward by roughly 1...

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    At fixed (H 0, ω0, ωa, rd), increasing Ω m0 raisesH(z) 16 0.35 0.40 Ωm0 −1 0 1 wa −1.35 −1.30 −1.25 −1.20 −1.15 w0 69.5 70.0 70.5 H0 69.5 70.0 70.5 H0 −1.3 −1.2 w0 −1 0 1 wa FIG

    Tension-Injection Mechanism The BAO tension is implemented through a positive shift ∆Ω (BAO) m0 = +0.03, which modifies bothD M (z)/rd and DH (z)/rd along their intrinsic sensitivity to the matter density. At fixed (H 0, ω0, ωa, rd), increasing Ω m0 raisesH(z) 16 0.35 0.40 Ωm0 −1 0 1 wa −1.35 −1.30 −1.25 −1.20 −1.15 w0 69.5 70.0 70.5 H0 69.5 70.0 70.5 H0 ...

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    The enhancement of Ω m0 and the suppression ofH 0 produce a characteristic diagonal orientation in the (ω 0, ωa) plane, reflecting the interference between BAO and CMB degeneracies

    Joint Posterior Contours Figure 5 displays the joint posterior contours for (Ω m0, H0, ω0, ωa) in run5. The enhancement of Ω m0 and the suppression ofH 0 produce a characteristic diagonal orientation in the (ω 0, ωa) plane, reflecting the interference between BAO and CMB degeneracies. The elongation inω a corresponds to the partial cancellation of the ten...

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    Posterior Shifts and Degeneracy Geometry The median and 1σconstraints from the joint MCMC analysis are Ωm0 = 0.2868±0.0033, H 0 = 71.048±0.237, ω0 =−1.0405±0.0427, ω a = 0.1097±0.1460.(31) These shifts display the characteristic fingerprint of a SNe-only zero-point perturbation. The inferred Hubble con- stant moves slightly above the fiducial value, consi...

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    Tension-Injection Mechanism The SNe miscalibration is implemented by shifting the theoretical distance modulus fromµ th(z) toµ th(z) + ∆M with no changes applied to BAO or CMB. Since the distance module is given by µ=m−M= 5 log 10(DL/Mpc) + 25,(32) 19 a brighter absolute magnitude (∆M <0) corresponds to a smaller inferred distance and hence a larger infer...

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    Pivot Equation-of-State The pivot EoS parameters for run6 are ap = 0.7300, z p = 0.3699, ω p =−1.0115±0.0153.(33) This remains essentially indistinguishable fromω p =−1 within the statistical uncertainty, underscoring that a mild SNe zero-point offset has negligible impact on global DE constraints when BAO and CMB data are included. This sharply contrasts...

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    The contours show only modest displacements relative to the tension-free baseline

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    Posterior Shifts and Degeneracy Geometry The median and 1σconstraints from the joint MCMC analysis are Ωm0 = 0.3246±0.0029, H 0 = 70.890±0.167, ω0 =−1.2054±0.0162, ω a = 1.1102±0.0238.(34) Among all runs, run7 produces the most extreme departure from the fiducial ΛCDM values (0.30,70,−1,0). The inferred matter density is significantly higher than the fidu...

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    Tension-Injection Mechanism The multi-probe tension in run7 is generated by combining two independent injections. First, a positive shift ∆Ωm0 >0 is applied to the BAO model predictions, producing directional distortions in bothD M (z)/rd andD H (z)/rd that move the BAO-only posterior toward larger Ω m0 and mildly lowerH 0. Second, a negative shift ∆H 0 i...

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    Joint Posterior Contours Figure 7 shows the joint posterior for (Ω m0, H0, ω0, ωa). The displacement is the largest among all runs examined in this work. The (Ω m0, H0) contours occupy a region that satisfies neither the BAO- nor the CMB-favoured direction individually, while the (ω 0, ωa) contours extend deeply into the (ω 0 <−1.2, ω a >1.1) quadrant. Th...

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