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Topological dark energy from spacetime foam: A challenge for $\Lambda$CDM

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Pith's one-line read The paper claims that dark energy is a topological effect: gravitational instantons nucleating in spacetime foam generate an effective dynamical cosmological constant, and the non-flat version fits combined cosmic data slightly better…

desk verdict A new ODE for topologically sourced dark energy and its first data constraints, but the derivation is under-specified and the claimed preference over ΛCDM is marginal. read the letter →

arxiv 2507.18389 v1 pith:NKUYDAR6 submitted 2025-07-24 gr-qc astro-ph.CO

Fotios K. Anagnostopoulos , Stylianos A. Tsilioukas , Emmanuel N. Saridakis This is my paper · ORCID
classification gr-qcastro-ph.CO PACS 98.80.-k04.60.-m
keywords topologicaldarkenergyspacetimefoamgravitationalinstantonsEinstein-Gauss-Bonnetgravitydynamicalenergy-darkmatterinteractioncosmologicalmodelselectionPantheon+BAOCCdata
topics Dark EnergyDark Matter
open problems Dark Energy
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 claims that dark energy does not have to be a cosmological constant: it can be a topological consequence of gravitational instantons popping in and out of the spacetime foam. Treating these instantons with Euclidean quantum gravity in Einstein-Gauss-Bonnet gravity gives an effective dark-energy density proportional to the instanton density, and standard vacuum-decay theory turns that into a closed differential equation for the dark-energy evolution. The authors fit the resulting flat and non-flat cosmological models to the Pantheon+ supernova sample, BAO data, and cosmic chronometers, and find that the non-flat model is moderately preferred over flat ΛCDM by the AIC and DevIC criteria, while the flat model is statistically tied with ΛCDM. The same mechanism lets dark energy change sign and interact with dark matter, and it passes Big Bang nucleosynthesis constraints. A general reader should care because this is a parameter-free origin for dark energy that could speak to the cosmological-constant problem and to the H0 and σ8 tensions.

What carries the argument

The load-bearing object is $\Lambda_{\rm eff} = -16\pi^2\alpha \sum_i \delta\chi_i n_i$, the effective topological cosmological constant built from the weighted density of gravitational instantons. It is fed by the standard vacuum-decay rate $n_i = \Gamma_i = A_i e^{-\Delta I_i}$, where $\Delta I_i$ is the Euclidean action difference from Eq. (8), and the closed first-order ODE (Eqs. 9 and 11) obtained by differentiating Eq. (4) then drives the entire Friedmann evolution. This single differential equation is what turns spacetime foam into a concrete, testable dark-energy model.

What would settle it

Measure the dark-energy equation of state $w_{\rm DE}(z)$ at $z\approx 1$--$2$ with a redshift-drift or BAO experiment: TDE predicts a detectable rise from today's value toward matter-like behavior, whereas ΛCDM keeps $w=-1$ at every redshift. A high-significance measurement of $w=-1$ beyond $z=1$ would rule out the TDE evolution equation, since Eq. (14) couples $w_{\rm DE}$ directly to the ODE's solution.

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Extended reading notes

Core claim

In the authors' formulation, adding the Gauss-Bonnet term to the Euclidean action makes the Euler-characteristic change of spacetime a source for the background metric. The result is an effective cosmological constant $\Lambda_{\rm eff}=-16\pi^2\alpha\sum_i\delta\chi_i n_i$ whose sign depends on which instanton species dominate. Combining the semiclassical nucleation rate $n_i=A_i e^{-\Delta I_i}$ with the action difference of Eq. (8) produces a first-order differential equation for $\Lambda_{\rm eff}(z)$; integrating it over cosmic history and imposing the usual normalization at $z=0$ fully determines the dark-energy density with no extra equation-of-state parameter. Fitting this model to Pantheon+/SH0ES, BAO, and cosmic-chronometer data gives the central result: non-flat TDE achieves the lowest AIC and DevIC values among the models compared, while flat TDE sits within $\Delta$IC $<2$ of ΛCDM. The authors conclude that the TDE scenario, especially the non-flat case, is statistically preferred over the concordance model while remaining consistent with BBN.

Load-bearing premise

The model assumes that the rate at which gravitational instantons nucleate, per unit spacetime volume, is the same thing as the instanton density that drives dark energy, and that this rate's prefactor does not change with cosmic time; if that identification is wrong, the central evolution equation has no basis.

Editorial extensions

If this is right

  • Dark energy becomes a deterministic consequence of spacetime topology rather than a tuned constant, so no separate cosmological-constant parameter is needed in the fit.
  • The model predicts a time-varying dark-energy equation of state that moves from matter-like values at early times to $w_{\rm DE}\approx -0.89$ today, so dark energy and dark matter are effectively one interacting sector.
  • The best-fit matter density is lower and $H_0$ higher than in ΛCDM, which is the direction known to relieve the $H_0$ and $\sigma_8$ tensions.
  • The statistical preference is not uniform: flat TDE is statistically indistinguishable from ΛCDM, while the non-flat version is preferred only in AIC/DevIC and not in BIC, so the comparison is data- and criterion-dependent.
  • The BBN consistency check shows that the topological term does not ruin the standard thermal history at $z\sim 10^9$.
  • The model predicts a time-varying dark-energy equation of state that moves from matter-like values at early times to $w_{\rm DE}\approx -0.89$ today, so dark energy and dark matter are effectively one interacting sector.

Reading between the lines

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

  • If the mechanism is right, the same instanton-density reasoning should produce analogous dynamical couplings in gravitational theories with other topological invariants, for instance Pontryagin or Chern-Simons terms; the authors do not explore this.
  • The preference may be driven by the model's lower best-fit $\Omega_{m0}$; a direct geometric measurement of the matter density from lensing or cluster counts would show whether the shift is physical or a parameter artifact.
  • The paper stops at the background level; computing linear perturbations under the same ODE would directly test the claim that the model relieves the $\sigma_8$ tension.
  • The predicted high-redshift approach of $w_{\rm DE}$ to zero is observable in principle: redshift-drift or BAO growth data at $z\gtrsim 1$ should see a departure from $w=-1$ if TDE is real.
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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 presents a model of 'topological dark energy' in which an effective cosmological constant is generated by gravitational instantons in Einstein-Gauss-Bonnet gravity, with the dark-energy density proportional to the instanton density. After writing a differential equation for the evolution of the effective Λ, the authors fit flat and non-flat versions of the model to Pantheon+/SH0ES supernova, BAO, and cosmic-chronometer data, compare with flat ΛCDM using AIC, BIC, and DevIC, and conclude that the non-flat TDE model is moderately preferred. They also perform a BBN consistency check. The central scientific claim is that this preference, if real, would constitute a first-principles challenge to ΛCDM.

Significance. Should the derivation and the statistical claim be correct, the model would be of considerable interest: it would tie dark energy to a quantum-gravitational topological mechanism, predict a time-dependent equation of state with possible dark-energy/dark-matter interaction, and pass basic cosmological data. The observational analysis is competently executed and the paper is transparent about its datasets and information criteria. However, as argued below, the evolution equation is not derived and the reported preference over ΛCDM is not robust across the adopted criteria, so the significance of the result is currently not established.

major comments (4)
  1. [Topological Dark Energy, Eqs. (8)-(9)] The derivation of Eq. (9) is not given. The statement between Eqs. (8) and (9) that differentiating after substituting n_i into Eq. (4) yields Eq. (9) is not correct as it stands: differentiating n_i = A_i e^{-ΔI_i} gives dn_i/dt = -n_i dΔI_i/dt, and dΔI_i/dt contains the time derivative of the integral in Eq. (8), which involves dΛeff/dt and time derivatives of the volume factor. Eq. (9) is an explicit ODE for Λeff, so its right-hand side has eliminated those terms; this requires an additional assumption about how the Euclidean action difference for a nucleated instanton depends on cosmic time, e.g. taking dΔI_i/dt to be the boundary integrand (16πG)^{-1} V_3 [4Λeff - 6(2H^2+Ḣ+k/a^2)]. The paper does not state or justify such an assumption, and the 'Hubble sphere of radius r=1/H' is ambiguous: a physical Hubble radius gives V_3 ∝ H^{-3}, whereas Eq. (9) contains a^3/H^3, corresponding to r=1/H as a comoving coordinate. Since Eq. (11) and all MCMC results are obtained by solving Eq. (9), the central dynamical content of the model is not derived.
  2. [Eqs. (4) and (7)] The identification of the nucleation rate per four-volume, n_i ≡ Γ_i, and the time-independence of the prefactor A_i are unproven modeling choices. In Euclidean quantum gravity, Γ = A e^{-ΔI} is a transition rate for a single tunneling event, while the instanton densities n_i in Eq. (4) are equilibrium densities of topological fluctuations, and their relation to Γ is not automatic. If A_i depends on H or on time, dn_i/dt acquires an extra term and Eq. (9) is modified. The paper's claim that the model follows from 'first principles' therefore requires a derivation of these identifications; Eq. (4) itself is imported from the authors' earlier work [12], so the present paper is not self-contained on this point.
  3. [Table II; Conclusions] The claimed statistical preference is not supported by the quoted numbers. In Table I, flat TDE has exactly the same χ²_min as flat ΛCDM for the two combined datasets that include BAO (1470.78 and 1463.40). Table II shows that the non-flat TDE model is preferred only by AIC and DevIC, with differences of 1.8-2.4 relative to flat ΛCDM, while BIC is worse by 3.0-3.6. On the Jeffreys scale these are weak-to-moderate at best and inconsistent across criteria. Moreover, the comparison is against flat ΛCDM only; the non-flat TDE's improvement comes with an extra parameter Ωk0, and a non-flat ΛCDM model is not considered. The phrase in the Conclusions describing a 'moderate but statistically significant preference' over ΛCDM is therefore an overstatement.
  4. [Eq. (10) and initial condition] The dark-energy density is not predicted from the instanton physics. The normalization E(0)=1 fixes ΩΛeff(0)=1-Ωm0-Ωr0-Ωk0, and the instanton amplitudes and couplings in Eq. (4) are absorbed into this initial condition. Thus the model has the same number of effective parameters as ΛCDM and cannot claim to predict the present dark-energy abundance; the 'parameter-free' characterization in the abstract and Conclusions is misleading.
minor comments (6)
  1. [Throughout] Typos and wording: 'instatons' and 'succesful' in the Introduction, 'Accoustic' before BAO in the Observational Confrontation section, and the caption of Fig. 1 lists the third dataset as 'CC/Pantheon+/SH0ES/BAOs' twice instead of giving the Pantheon+/SH0ES/BAOs combination.
  2. [Table II] The table header uses 'DevIC' and 'ΔDIC' inconsistently for the same criterion; choose one abbreviation.
  3. [Eq. (11)] Equation (11) is presented without an algebraic derivation from Eq. (9); even if the algebra is correct, a brief outline of the intermediate steps would improve reproducibility.
  4. [BBN section] The BBN consistency check is a single ratio at z_BBN ~ 10^9 using best-fit values; the paper should specify the adopted primordial abundances and the BBN code or reference used to assess the thermal-history constraint.
  5. [Observational Confrontation] The MCMC convergence discussion mentions the Gelman-Rubin criterion and autocorrelation-time analysis but does not report the numerical values; they should be given in the text or in an appendix.
  6. [Abstract and Conclusions] The abstract mentions only AIC and DevIC while the text says three criteria are used; BIC should be mentioned consistently wherever the model-selection results are summarized.

Circularity Check

1 steps flagged · score 3.0 of 10

Partial circularity: the central relation Λeff ∝ instanton density is imported from the authors' own prior work, while the subsequent data comparison is external and not circular.

  1. self citation load bearing [Introduction, paragraph 2 (justifying Eqs. (3) and (4))]
    "In a recent work we showed that the consideration of a Gauss-Bonnet term yields an effective cosmological constant of topological origin which is proportional to the density of gravitational instantons [12]."

    Equation (4), Λeff = -16π²α Σ δχᵢ nᵢ, is the foundation of the TDE construction: it feeds into the action difference (8), the differential equation (9), and the claimed 'first principles' origin of the model. This relation is not derived in the present paper; it is imported from the authors' own prior work [12]. No independent, machine-checked, or externally falsifiable derivation is supplied here, so the central premise of the derivation reduces to a load-bearing self-citation. The later MCMC comparison with Pantheon+/BAO/CC data is external and therefore not circular, which limits the overall circularity score.

full rationale

The paper's statistical comparison is a genuine test: the model is integrated using Eq. (9), and the free parameters (Ωm0, Ωk0, h, plus nuisance parameters) are fitted to external datasets, with model selection via AIC, BIC, and DevIC. That part is not circular — the claimed preference over ΛCDM is an empirical fit comparison, not a quantity forced by the model's definitions. However, the physical starting point is not self-contained: the key proportionality Λeff ∝ instanton density is taken from the same authors' earlier paper [12], and the present text explicitly relies on that citation rather than re-deriving the result. A further weakness, noted in the manuscript itself, is that the initial condition ΩΛeff(z=0)=1-Ωm0-Ωr0-Ωk0 fixes the instanton species mix rather than predicting it, so the present DE abundance is imposed by the Friedmann constraint. These issues make the 'from first principles' claim partly dependent on self-citation and on an imposed initial value, but because the evolution equation still has nontrivial dynamical content and is tested against external data, the circularity is only partial. Score 3.

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

The model's predictive equation is the ODE for ΩΛeff, which is independent of the unspecified constants α, A_i, and instanton mix, but those constants and the identification of n_i with Γ_i are assumed. The amplitude of dark energy today is fitted through Ωm0 and Ωk0, not predicted from microphysics.

free parameters (5)
  • Ωm0 = 0.234 ± 0.010 (flat TDE, CC/Pantheon+/SH0ES/BAOs); 0.196 ± 0.022 (non-flat TDE)
    Matter density parameter fitted to the combined cosmological datasets; the TDE model has no independent DE parameter, so Ωm0 partly sets ΩΛeff at z=0.
  • Ωk0 = 0.157 ± 0.082 (non-flat TDE, combined); 0 (flat TDE)
    Curvature density parameter; this extra parameter is what drives the marginal AIC/DevIC preference of non-flat TDE over ΛCDM.
  • h = ≈0.7355 ± 0.010 (combined, flat TDE)
    Dimensionless Hubble constant fitted to data; enters distance moduli and the tiny G H0² terms in Eq. (11).
  • M = -19.25 ± 0.03
    Absolute magnitude nuisance parameter for SNIa.
  • r_d = ≈135.9 ± 2.3 Mpc (combined)
    Sound horizon at baryon drag, fitted in the BAO likelihood.
assumptions (5)
  • domain assumption Euclidean Quantum Gravity spacetime foam generates topologically non-trivial instantons, and Λeff = -16π² α Σ δχ_i n_i.
    Core relation taken from the authors' prior work [12]; no independent derivation here.
  • domain assumption The instanton density per four-volume equals the nucleation rate Γ_i = A_i exp(-ΔI_i) from bubble nucleation theory.
    Eq. (7); this is an analogy from vacuum decay, not a proven statement for gravitational instantons.
  • ad hoc to paper The Euclidean action difference for each instanton is ΔI_i = 2πα χ_i/G + (16πG)^{-1} ∫√g(4Λeff - 6(2H² + dH/dt + k/a²)), with R_inst=4Λeff and the same Hubble-sphere four-volume for background and instanton.
    Eq. (8); the Hubble-sphere cutoff and the use of Λeff in the instanton Ricci scalar are ad hoc.
  • ad hoc to paper The prefactor A_i is independent of cosmic time.
    Assumed after Eq. (9) to obtain the differential equation; not justified.
  • standard math Chern-Gauss-Bonnet theorem applies to compute the topological action ΔI_GB = 2παχ_i/G.
    Standard mathematical theorem, used in Eq. (8).

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Cite this review

Pith. "Pith review of Topological dark energy from spacetime foam: A challenge for $\Lambda$CDM." pith.science (2026). https://pith.science/paper/NKUYDAR6

@misc{pith2026250718389,
  author       = {Pith},
  title        = {Pith review of: Topological dark energy from spacetime foam: A challenge for $\Lambda$CDM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKUYDAR6}},
  note         = {Machine review of arXiv:2507.18389}
}
abstract

Using only the standard considerations of spacetime foam and the Euclidean Quantum Gravity techniques known long ago, we result to a model of Topological Dark Energy (TDE) that outperforms the standard $\Lambda$CDM paradigm with regard to data fitting efficiency. Specifically, it is known that at the foam level, topologically non-trivial solutions such as instantons appear. In the particular case of Einstein-Gauss-Bonnet gravity, we obtain an effective dynamical dark energy term proportional to the instanton density, and the latter can be easily calculated through standard techniques. Hence, we can immediately extract the differential equation that determines the evolution of the topologically induced effective dark energy density. Significantly, this TDE scenario allows for changing sign of dark energy during the cosmic evolution and also exhibits Dark Energy interaction with Dark Matter. We confront the TDE scenario, in both flat and non-flat cases, with Pantheon+/SH0ES Supernovae Type Ia (SNIa), Baryonic Accoustic Oscillations (BAO), and Cosmic Chronometers (CC) datasets. By applying standard model selection methods (AIC and DevIC information criteria), we find a moderate but statistically significant preference over $\Lambda$CDM scenario. Finally, we show that the TDE scenario passes constraints from Big Bang Nucleosynthesis (BBN) and thus does not spoil the thermal history of the Universe.

Figures

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Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
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Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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  2. Observational implications of Wald-Gauss-Bonnet topological dark energy

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    Wald-Gauss-Bonnet topological dark energy is viable against late-universe data but statistically loses to ΛCDM, and its perturbations are nearly indistinguishable from ΛCDM.

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