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This paper argues that Wald-Gauss-Bonnet topological dark energy, tied to black-hole formation and mergers, gains ~3σ support from combined early and late cosmological data, boosting H0 to 69.8 km/s/Mpc and reducing the Hubble tension by ~0

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2026-08-01 02:40 UTC pith:AFNFA2Y4

load-bearing objection Solid first CMB constraints on WGB dark energy, but the ~3σ preference is SH0ES-dependent and not shown to favor the black-hole mechanism over a generic phantom fluid. the 4 major comments →

arxiv 2607.26101 v1 pith:AFNFA2Y4 submitted 2026-07-28 gr-qc astro-ph.COhep-th

Early- and late-time constraints on Wald-Gauss-Bonnet topological dark energy and implications for the H₀ and S₈ tensions

Stylianos A. Tsilioukas , Fotios K. Anagnostopoulos , Spyros Basilakos , Andronikos Paliathanasis , Emmanuel N. Saridakis This is my paper
classification gr-qc astro-ph.COhep-th MSC 83F0583D05 PACS 95.36.+x98.80.-k
keywords Wald-Gauss-Bonnet dark energytopological dark energyhorizon thermodynamicsblack-hole mergersHubble tensionS8 tensionphantom dark energycosmological constraints
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.

The paper claims that a 'topological dark energy' generated by the horizon thermodynamics of black-hole formation and mergers can act as a viable late-time dark energy. Using a single dimensionless parameter, the model is implemented as a dynamical dark-energy fluid and constrained with cosmic microwave background, baryon acoustic oscillation, and supernova datasets. The full combination modestly favors the new contribution at about 3σ, with a phantom-like equation of state. This raises the inferred Hubble constant from 68.5 to 69.8 km/s/Mpc, reducing the H0 tension by ~0.9σ while slightly increasing the S8 clustering amplitude. Late-time data alone are consistent with ΛCDM, so the preference emerges only when early and late probes are joined.

Core claim

The central discovery is that a cosmological model in which the Gauss-Bonnet correction to the Wald entropy of the apparent horizon is sourced by the astrophysical history of black-hole formation and mergers fits the combined early- and late-Universe data better than ΛCDM, with the coupling parameter Cn = 0.435^{+0.150}_{-0.132}, a ~3σ phantom-side deviation from zero. The mechanism preserves the standard acoustic scale, leaving the primary CMB nearly unchanged, while the late-time expansion is enhanced enough to shift H0 upward by about 1.3 km/s/Mpc. The cost is a modest rise in S8, and independent growth probes (lensing and redshift-space distortions) are slightly worse-fit than in ΛCDM. T

What carries the argument

The load-bearing object is the Wald-Gauss-Bonnet entropy S_WGB = A/4G + 2πα̃/G χ(h), whose Euler-characteristic correction χ(h) becomes dynamical by relating changes in the apparent horizon to the net number of black-hole formations minus mergers, δχ = -2(δN_form - δN_merg). This tie, approximated by the cosmic star-formation history, enters the modified Friedmann equation as a single dimensionless parameter Cn = α̃C/H0², which controls the effective dark-energy equation of state, with positive Cn giving phantom behavior. The implementation treats the sector as a smooth, non-clustering dark-energy fluid with no new propagating degrees of freedom, so all growth and lensing effects stem from t

Load-bearing premise

The model's distinctive redshift dependence rests on the assumption that changes in the Euler characteristic of the cosmic horizon are proportional to the net number of black-hole formations minus mergers as traced by the cosmic star-formation rate; if that mapping is wrong, the fitted Cn is just a generic phantom dark-energy parameter and the black-hole mechanism is not established.

What would settle it

A measurement of the black-hole formation/merger rate at 0 < z < 2 that disagrees with the adopted star-formation history would invalidate the mapping between horizon topology change and black-hole population history, since Cn would then absorb a different functional shape. Alternatively, a future BAO+lensing analysis that constrains the H(z) crossover near z ≈ 0.55 and finds no suppression of the expansion rate at intermediate redshifts at >3σ confidence would rule out the mechanism while leaving a generic phantom fluid viable.

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

If this is right

  • If the model is correct, the Hubble tension is partially geometric: a phantom late-time epoch that leaves the sound horizon unchanged can raise H0 by ~1.3 km/s/Mpc without altering early-Universe physics.
  • The predicted enhancement of CMB lensing and small-scale clustering (2-4% and up to ~4.5%, respectively) is a testable signature; the accompanying mild rise in S8 is a generic consequence of the phantom branch.
  • The preference is dataset-dependent: only the full CMB+BAO+SN combination with the local distance ladder yields moderate Bayesian evidence (lnB ≈ 2.4-2.8); late-time-only combinations remain consistent with ΛCDM.
  • The crossover in H(z) at z ≈ 0.55, where the WGB expansion rate becomes lower than ΛCDM at intermediate redshifts, provides a distinctive background signature that BAO and geometric probes can check.
  • Improved constraints on black-hole binary fractions and merger rates would sharpen the prior on Cn and either consolidate or weaken the physical interpretation.

Where Pith is reading between the lines

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

  • The mechanism's redshift dependence is fixed by the star-formation history; if that proxy fails, the fitted Cn would collapse into a generic one-parameter phantom fluid, and the astrophysical interpretation would lose its status as an explanation rather than a parametrization.
  • The near-perfect Cn-H0 degeneracy in CMB-only data means that future Stage-IV lensing and BAO measurements, which break the degeneracy, will be decisive: they can either confirm the WGB crossover or rule it out while a generic phantom fluid survives.
  • The S8-enhancing side of the model hints at a deeper lesson for late-time DE: attempts to raise H0 by phantom dynamics generically cost growth consistency, so a combined solution to both tensions may require either an early-time modification or a different physical channel.
  • If the preference strengthens with future data, it would suggest that cosmic black hole populations—through horizon thermodynamics—leave an imprint on the expansion history that is measurable at the percent level.

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

4 major / 4 minor

Summary. The paper presents the first early- and late-Universe constraints on Wald-Gauss-Bonnet (WGB) topological dark energy, implemented as an effective fluid in a modified CLASS solver. It analyzes five combinations of Planck/ACT/SPT-3G CMB data (via the candl likelihood), DESI DR2 BAO, and Pantheon+ supernovae with and without the SH0ES calibration. The headline result is a ~3σ preference for a positive WGB amplitude C_n = 0.435^{+0.150}_{-0.132} in the full early+late combination including SH0ES (Run 5), with an improved fit of Δχ² ≈ -7.2 and moderate support from AIC/DevIC and an independent k-NN Bayes factor. The preferred solution raises H0 from 68.5 to 69.8 km/s/Mpc, reducing the Hubble tension by ~0.9σ, at the cost of a mild S8 increase. The paper reports convergence checks, public code, information criteria, and an out-of-sample RSD comparison.

Significance. If the WGB mechanism is correct, this is a novel and physically motivated late-time phantom dark-energy scenario arising from horizon thermodynamics and black-hole formation/merger history, with a distinctive redshift dependence set by the star-formation rate. The paper provides the first CMB-era constraints, a public Boltzmann implementation, and a transparent set of model-comparison diagnostics. The caution is that the empirical evidence for the WGB-specific ψ(z) shape is not isolated from a generic one-parameter phantom fluid, and the headline preference appears only when the SH0ES calibration is included in the likelihood. The paper's strengths are its reproducibility, the multiple consistency checks, and the honest discussion of prior-limited and degeneracy-driven runs.

major comments (4)
  1. [§IV.A, §IV.B, Tables IV and VI] The headline ~3σ preference for C_n>0 and the moderate Bayes factor (lnB10=2.80) occur only for Run 5, which includes the SH0ES local-H0 calibration in the PPS likelihood. In the otherwise identical uncalibrated Run 3 (CMB-SPA+PP+DESI), C_n=0.209±0.121/0.144 (~1.5σ), ΔAIC>0, and lnB10=-1.91 favoring ΛCDM. Since C_n is a free parameter fitted to the same SH0ES-inclusive data, the statement that WGB 'raises H0' is partly a consequence of the fit rather than an independent prediction. The paper should present the uncalibrated combination as the primary evidence for the model, or at least prominently frame the SH0ES dependence as a prior-driven effect rather than a data-driven discovery.
  2. [§II.B, Eqs. (10)–(13), §IV.B] No comparison is made against a generic one-parameter phantom dark-energy model (e.g., constant w0 or CPL) using the same five dataset combinations. The perturbation sector is standard GR with a smooth dark-energy fluid, so all observational differences between WGB and ΛCDM enter through the background expansion history encoded in w_DE(a) with the fixed Madau-Dickinson ψ(z) shape. A constant-w phantom with one free parameter could plausibly reproduce the same fit, in which case the data would not discriminate the WGB black-hole mechanism from any smooth phantom fluid. Adding such a baseline and reporting ΔAIC/ΔBIC and Bayes factors for w0CDM would directly test whether the ψ(z) shape, rather than merely the presence of a late-time phantom component, is favored.
  3. [§II.A, Eqs. (4)–(8), §IV.B] The physical interpretation 'topological dark energy from black holes' rests on the uncalibrated assumptions δχ = -2(δN_form - δN_merg) and the Madau-Dickinson proxy ψ(z), with all astrophysical uncertainties absorbed into C_n. No independent astrophysical prior is imposed on C_n; the adopted flat prior C_n∈[-2,5] is broad and the Bayes factor is prior-width dependent. The paper correctly notes that a tighter prior would strengthen the evidence, but it should also test robustness of the headline conclusion to the prior range and, if possible, fold in current estimates of f_BH, f_bin, f_merge, and <m_prog> from Eq. (6). Without this, the statistical preference for C_n>0 cannot be separated from a generic phantom-fluid preference, and the mechanism-specific claim is underdetermined.
  4. [§IV.C, Savage-Dickey estimate] The Savage-Dickey Bayes factor uses a skew-normal fit to the marginal posterior because the KDE tail near C_n=0 is sparsely sampled. This is a delicate boundary-density estimate. The k-NN evidence is an independent estimator, but it is computed from the same MCMC chains, so the two methods are not fully independent. Reporting the k-NN value as the primary evidence, with the Savage-Dickey as a cross-check, would be more appropriate; the current wording slightly overstates the independence.
minor comments (4)
  1. [§II, around Eq. (2)] Typo: 'General Relativivy' should be 'General Relativity'.
  2. [References] References [14] and [15] duplicate [11] and [12]; also the CosmoVerse white paper appears as both [73] and [81]. These should be consolidated.
  3. [Table III] The late-time-only runs have several parameters marked as prior-dominated; it would be clearer to omit these columns or to separate them into a supplemental table, as the current layout may mislead readers into interpreting prior-dominated values as constraints.
  4. [§V.B, Fig. 9] The statement that RSD data 'mildly prefer ΛCDM' is supported by Δχ²=+2.9 for Run 5, but the absolute χ²/N<0.6 for both models might be more visible. Consider reporting the reduced χ² explicitly in the figure caption to avoid over-interpreting the small difference.

Circularity Check

0 steps flagged

No significant circularity: the WGB preference is a data-driven model-comparison result, and the out-of-sample fσ8 check is genuinely external.

full rationale

The paper's central claim—that the full CMB+BAO+SN dataset prefers a non-zero WGB amplitude Cn—is a standard Bayesian model-comparison result obtained by fitting Cn to external data, not a prediction derived from the model's own inputs. The model's redshift shape ψ(z) is fixed by an explicit astrophysical assumption (Eqs. 4-5, citing external Refs. [86-88]), while Cn is openly a fitted parameter that absorbs astrophysical uncertainties (Eqs. 6-8). The paper does not present the resulting H0 shift as an independent prediction; it reports the MAP posterior of a fit ('The preferred solution raises H0 from 68.5 to 69.8 km/s/Mpc'). The one quantity explicitly labeled a genuine prediction, fσ8(z), is out-of-sample: 'These measurements are not included in the likelihood and therefore constitute a genuine prediction rather than a fit' (Sec. V.B, Fig. 9). The reconstructed CMB lensing and matter-power enhancements are posterior evaluations of the same fitted model, not out-of-sample confirmations, and are described as reconstructions rather than independent tests. The self-citations to Refs. [74] and [80] supply the model framework and companion code, but the statistical preference is determined by the external Planck/ACT/SPT/DESI/Pantheon+ likelihoods, so no load-bearing argument reduces to a self-citation. The absence of a generic-phantom comparison is a model-discrimination limitation, not circularity: it concerns whether the WGB-specific ψ(z) shape is empirically favored, not whether the fitted result is equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

No new particle, field, or fundamental entity is introduced. The WGB 'fluid' is an effective description of a horizon-thermodynamics correction; its amplitude is the free parameter C_n listed above, and no independent falsifiable handle (e.g., a predicted mass or fifth force) is provided.

free parameters (1)
  • C_n = 0.435^{+0.150}_{-0.132} (Run 5); 0.209^{+0.144}_{-0.121} (Run 3)
    Amplitude of the WGB term, combining Gauss-Bonnet coupling and the astrophysical factor C (f_BH, f_bin, f_merge, <m_prog>). It is not fixed by independent BH/merger data; it is marginalized and is the only model parameter beyond ΛCDM. The central ~3σ preference is the significance of this fitted parameter.
axioms (5)
  • domain assumption Gravity-thermodynamics conjecture applied to the cosmic apparent horizon yields the modified Friedmann equation (Eq. 7).
    Foundation of the WGB scenario, invoked in Section II A via refs [82-85]. If the thermodynamics/gravity connection for the apparent horizon fails, the modified background has no basis.
  • ad hoc to paper Wald entropy with a Gauss-Bonnet topological correction, S_WGB = A/(4G) + 2π α̃ χ(h)/G (Eq. 3), applies to the cosmological apparent horizon.
    Introduced for this model in refs [74,77-79] by overlapping authors; no direct observational confirmation. Load-bearing because the correction term's size and sign enter Eq. (7).
  • ad hoc to paper Horizon Euler-characteristic changes are sourced only by black-hole formation and mergers, δχ = −2(δN_form − δN_merg), with rates traced by the Madau-Dickinson star-formation rate ψ(z) (Eqs. 4-5).
    This fixes the redshift dependence of the WGB fluid, the only feature distinguishing it from a generic phantom DE. No independent check of this proxy is provided; it is the weakest astrophysical link.
  • domain assumption The 4D Gauss-Bonnet term is topological, so field equations and linear perturbations remain those of GR; WGB affects only the background expansion (G_eff ≃ G, non-clustering PPF fluid).
    Stated in Section II B. If topological corrections sourced perturbations or modified G_eff, the CMB lensing/S8 predictions and constraints would change materially.
  • domain assumption Standard cosmological baseline: spatially flat FRW, one massive neutrino with Σmν = 0.06 eV, HMcode-2020 nonlinear modeling, standard recombination/thermal history.
    Standard analysis assumptions shared with the ΛCDM baseline; not specific to WGB but required for the quoted constraints.

pith-pipeline@v1.3.0-alltime-deepseek · 25640 in / 17911 out tokens · 179777 ms · 2026-08-01T02:40:38.807401+00:00 · methodology

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The persistent $H_0$ and $S_8$ tensions motivate the search for new dark-energy mechanisms capable of modifying the late-time expansion history while preserving the successful early-Universe predictions of $\Lambda$CDM scenario. Wald-Gauss-Bonnet (WGB) topological dark energy provides a physically motivated realization of this possibility, where the effective dark-energy sector emerges from cosmic horizon thermodynamics and the black-hole formation and merger history. We present the first early- and late-Universe analysis of WGB cosmology, implementing the model as an effective fluid in a modified \texttt{CLASS} solver and constraining it against CMB data from \textit{Planck}, ACT~DR6 and SPT-3G, DESI~DR2 BAO, and Pantheon+ supernovae. While late-time data alone are consistent with $\Lambda$CDM, the full dataset prefers a non-zero WGB contribution, $\Cn=0.435^{+0.150}_{-0.132}$, corresponding to a $\sim3\sigma$ phantom-like deviation and an improved fit. The preferred solution raises $H_0$ from $68.5$ to $69.8~\kmsmpc$, reducing the Hubble tension by $\sim0.9\sigma$, at the cost of a mild increase in $S_8$. The reconstructed cosmological observables show that WGB leaves the primary CMB almost unchanged while enhancing lensing and small-scale clustering, revealing a characteristic $H_0$-$S_8$ trade-off. WGB dark energy therefore emerges as a physically motivated and observationally viable late-time mechanism for partially alleviating the Hubble tension without introducing new early-Universe physics.

Figures

Figures reproduced from arXiv: 2607.26101 by Andronikos Paliathanasis, Emmanuel N. Saridakis, Fotios K. Anagnostopoulos, Spyros Basilakos, Stylianos A. Tsilioukas.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The improvement is driven primarily by the Pan￾theon+ & SH0ES supernova likelihood (∆χ 2 = −4.71), together with the SPT-3G MUSE ϕϕ+EE (−2.87), the SPT-3G TnE likelihood (−2.53), and the ACT DR6 tem￾perature and polarization spectra (−2.10). These gains are partly offset by the DESI DR2 BAO data (+2.71), whose residuals reveal the expected modest deterioration associated with the higher-H0 solution, and by… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: , which highlights the trade-off between the increase [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

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