Topological dark energy from black-hole formations and mergers through the gravity-thermodynamics approach

Emmanuel N. Saridakis; Nicholas Petropoulos; Stylianos A. Tsilioukas

REVIEW 4 major objections 6 minor 4 cited by

Topological dark energy from black-hole formations and mergers through the gravity-thermodynamics approach

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

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Pith's one-line read This paper argues that black-hole formation and merger events, by changing the topology of the cosmic apparent horizon, generate an effective dark energy whose equation of state departs from -1 at intermediate redshifts.

desk verdict New twist on topology-change dark energy, but the printed Friedmann equation has factor-of-two coefficient errors and rests on an unvalidated topological conservation law. read the letter →

arxiv 2412.21146 v2 pith:XBYCKU3E submitted 2024-12-30 gr-qc astro-ph.COhep-th

Stylianos A. Tsilioukas , Nicholas Petropoulos , Emmanuel N. Saridakis This is my paper · ORCID

classification gr-qcastro-ph.COhep-th

keywords topologicaldarkenergygravity-thermodynamicsWald-Gauss-BonnetentropyblackholemergersEulercharacteristicstarformationratemodifiedFriedmannequationsphantomandquintessence

topics 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 argues that the formation and merger of black holes can themselves generate dark energy. Using the gravity-thermodynamics conjecture with the Wald-Gauss-Bonnet entropy, whose topological part is the Euler characteristic of a horizon, the authors treat each black-hole birth or merger as a topology change of the cosmic apparent horizon and derive modified Friedmann equations. The resulting effective dark-energy density is proportional to the accumulated difference between black-hole formation and merger rates. Estimating those rates from the observed star-formation history, they find a dark-energy equation of state that sits at the cosmological-constant value $-1$ at early and late times but departs from it at intermediate redshifts, becoming phantom for one sign of the Gauss-Bonnet coupling and quintessence-like for the other. If correct, the result ties cosmic acceleration to the astrophysical black-hole population rather than only to a bare cosmological constant.

What carries the argument

The load-bearing object is the Wald-Gauss-Bonnet entropy, $S_{\rm WGB} = A/4G + (2\pi\tilde{\alpha}/G)\chi(h)$, where $A$ is the apparent-horizon area and $\chi(h)$ is the Euler characteristic of a two-dimensional horizon section; the Chern-Gauss-Bonnet theorem converts the horizon curvature integral in the Wald formula into $4\pi\chi(h)$, making the correction purely topological. The argument then hinges on a bookkeeping rule: since the total boundary $\partial M = H \cup \bigcup_i h_i$ is assumed to keep constant Euler characteristic, each black-hole formation ($\delta\chi(h)=+2$) forces $\delta\chi(H)=-2$ on the apparent horizon and each merger ($\delta\chi(h)=-2$) forces $\delta\chi(H)=+2$. Substituting $\dot{\chi}(H)$ into the Clausius relation $-dE = T\,dS$ at the apparent horizon generates the integral correction in the Friedmann equation, and the star-formation rate enters through the active black-hole number $N=N_{\rm form}-N_{\rm merg}$.

What would settle it

Run a numerical-relativity simulation of a black-hole merger embedded in an expanding FRW spacetime and track the apparent horizon: if its Euler characteristic does not jump by exactly two (upward for a merger, downward for a formation), the modified Friedmann equation (27) is not the consequence of the gravity-thermodynamics setup the paper assumes. Observationally, if tomographic dark-energy measurements show $w_{\rm DE}=-1$ across $z\approx 0.5$-$3$ while star-formation and merger-rate data imply a significant excess of formations over mergers, the mechanism as parametrized is ruled out.

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Extended reading notes

Core claim

The central claim is that in Einstein-Gauss-Bonnet gravity, the first law of thermodynamics applied at the apparent horizon with the Wald-Gauss-Bonnet entropy produces the modified Friedmann equation $$$H^{2}$ = \frac{8\pi G}{3}\rho_m + \frac{k}{$a^{2}$} + \frac{\Lambda}{3} - 8\tilde{\$\alpha$}\int_0^t \left($H^{2}$ + \frac{k}{$a^{2}$}\right)^2 \frac{dN}{dt}\,dt,$$ where $N = N_{\rm form} - N_{\rm merg}$ is the difference between black-hole formation and merger counts. The effective dark-energy density, $$\rho_{\rm DE} = \frac{3}{8\pi G}\left[\frac{\Lambda}{3} - 8\tilde{\$\alpha$}\int_0^t \left($H^{2}$ + \frac{k}{$a^{2}$}\right)^2 \frac{dN}{dt}\,dt\right],$$ is therefore of topological origin. The paper derives the relation $\dot{\chi}(H) = -2(dN_{\rm form}/dt - dN_{\rm merg}/dt)$ from the postulate that the total Euler characteristic of all causal boundaries is conserved, and then estimates $dN/dz$ from the cosmic star-formation rate. The resulting equation-of-state parameter $w_{\rm DE}(z)$ equals $-1$ at early and late times, deviates near $z\approx 2$, and stays within observational bounds across the allowed parameter ranges.

Load-bearing premise

The whole construction rests on the postulate that the total Euler characteristic of all causal boundaries is conserved, so each black-hole formation or merger changes the apparent horizon topology by exactly two units, in opposite directions; if that bookkeeping rule fails, the topological dark-energy term in the Friedmann equation does not follow.

Editorial extensions

If this is right

If no black holes form or merge, $dN/dt=0$, the correction vanishes, and the scenario reduces to standard $\Lambda$CDM with a bare cosmological constant.
The sign of the Gauss-Bonnet coupling decides the behavior: positive $\tilde{\alpha}$ gives phantom-like $w_{\rm DE}<-1$ at intermediate redshifts, negative $\tilde{\alpha}$ gives quintessence-like $w_{\rm DE}>-1$, and both tend to $-1$ at early and late times.
The model reproduces the standard thermal history, with a deceleration-to-acceleration transition at $z\approx 0.6$ and a de Sitter phase in the asymptotic future.
For the observationally allowed ranges of $f_{\rm BH}$, $f_{\rm merge}$, $f_{\rm bin}$, and $\langle m_{\rm prog}\rangle$, the predicted $w_{\rm DE}$ stays inside current observational bounds, while larger $f_{\rm BH}$ or $|\tilde{\alpha}|$ deepens the deviation from $\Lambda$CDM.
Dark matter behaves exactly as in standard cosmology, since it enters as a separate conserved sector, so perturbation and clustering predictions are unchanged.

Reading between the lines

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

An implication the authors leave implicit: if this mechanism is correct, the onset of accelerated expansion is tied to the astrophysical epoch when star formation and black-hole mergers are most active, giving the cosmic-coincidence problem a possible astronomical clock rather than a tuned constant.
The bookkeeping postulate $\delta\chi(\partial M)=0$ could be tested directly in numerical-relativity simulations that track the apparent horizon during a black-hole merger inside a cosmological spacetime; if the Euler characteristic does not actually jump by $\pm 2$, the modified Friedmann equation would not follow.
A sharp, testable extension would be to fold in merger-rate measurements from gravitational-wave observatories at $z\lesssim 1$ and tomographic dark-energy surveys across $z\approx 0.5$-$3$, where the predicted $w_{\rm DE}$ excursion peaks; the model would be distinguished from a constant-$w$ cosmology by the shape of that excursion.
Because the dark-energy density inherits the cosmic star-formation history, the model's predictions are sensitive to high-redshift star-formation and binary-evolution uncertainties; better constraints on those inputs would translate directly into sharper predictions for $w_{\rm DE}(z)$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline: this paper has a genuinely new idea—applying the Wald-Gauss-Bonnet entropy in the gravity-thermodynamics framework to astrophysical black-hole formation and merger rates—but the central derivation as printed contains coefficient errors that shift the final Friedmann equation by a factor of 2. The qualitative mechanism survives; the numbers don't.

What's new: previous papers by the same group used topology-change dark energy from wormholes and spacetime foam. Here they tie it to the Madau-Dickinson star formation rate, converting BH formation/merger counts into a dynamical dark-energy term. That's a real step toward a testable model. The rate estimates are simple but reasonable for a first pass, and the authors are clear about the free parameters and the need for data fitting later. The paper is well organized and the references are relevant, including the second-law-violation issue in Einstein-Gauss-Bonnet mergers, which motivates their conservation postulate.

Where it gets soft: the derivation of the modified Friedmann equations has concrete coefficient mistakes. Differentiating SWGB = A/(4G) + (2π α̃/G)χ gives -2π r_A^4 H/G (Hdot - k/a^2) for the area term and +2π α̃/G χdot for the topological term, not the factors in Eq. (23). An independent first-law integration leads to -4 α̃ in front of the ∫(H^2+k/a^2)^2 dN/dt term, not the -8 α̃ of Eq. (27). Since everything quantitative—Eqs. (30)-(49), all the figures—uses that equation, the precise predictions aren't supported. This looks like a fixable arithmetic slip, but it has to be fixed.

The bigger assumption is δχ(∂M)=0, the conservation of total Euler characteristic of all causal boundaries. It's introduced to avoid second-law violation, and it's not derived or tested. If it fails, the entire topological dark-energy sector disappears. That's a load-bearing postulate, and the paper would be stronger if it acknowledged how speculative that is.

Also minor: the text in Fig. 6 says wDE stays within bounds for fBH=0.001, which should be 0.05; and the observational check is just 'within Planck bounds,' not a real fit. The model covers both phantom and quintessence via the sign of α, so it's flexible enough to match either—that limits predictiveness.

Net: the core idea is worth taking seriously, but not in its current form. I would send it to peer review, with a referee asked to check the coefficients and to push on the conservation assumption. A corrected version could be a useful contribution.

Referee Report

4 major / 6 minor

Summary. The paper proposes a cosmological model in which the Wald-Gauss-Bonnet entropy, including its topological Euler-characteristic contribution, is used in the gravity-thermodynamics derivation of the Friedmann equations. The authors assume that the total Euler characteristic of all causal boundaries is conserved, so that black-hole formation and merger events change the apparent-horizon Euler characteristic by Δχ = −2(N_form − N_merger). This yields a modified Friedmann equation with an effective dark-energy sector sourced by the active black-hole number. Using the Madau-Dickinson star-formation rate and literature ranges for f_BH, f_bin, f_merge, and ⟨m_prog⟩, the authors compute the dark-energy density, equation of state, and deceleration parameter for positive and negative Gauss-Bonnet coupling, finding w_DE → −1 at early and late times with phantom-like or quintessence-like behavior at intermediate redshifts.

Significance. The proposed mechanism is original and falsifiable: it predicts a specific z-dependence of the dark-energy equation of state tied to the cosmic star-formation history, with the sign of the deviation set by the Gauss-Bonnet coupling. The authors make the astrophysical input concrete and survey the allowed parameter ranges, which is a strength. However, the central derivation contains multiple algebraic inconsistencies that propagate into the quantitative predictions, and the key postulate δχ(∂M)=0 is assumed rather than derived. With corrected coefficients the qualitative scenario may survive, but the present numerical results and parameter constraints are not supported by the printed derivation.

major comments (4)

[II C, Eq. (23)] The derivative of the Wald-Gauss-Bonnet entropy is computed incorrectly. From Eq. (15), S_WGB = A/(4G) + (2πα̃/G)χ(H), and with A = 4πr_A^2 and r_A = (H^2 + k/a^2)^(−1/2), direct differentiation gives dS/dt = −2πr_A^4 H/G (Ḡ − k/a^2) + (2πα̃/G)ṇ. The area coefficient printed in Eq. (23), −2πr_A^4/(4G), is a factor of 4 too small, and the topological coefficient πα̃/G is a factor of 2 too small relative to the second term of Eq. (15). This error propagates into every subsequent equation.
[II C, Eqs. (24)-(25)] Eq. (24) does not follow from Eq. (23), and Eq. (25) does not follow from Eq. (24) by the stated integration. Substituting Eq. (23) into the first law with δQ = 4πr_A^3(ρ_m + p_m)Hdt and T = 1/(2πr_A) yields coefficients 1/4 on (Ḡ − k/a^2) and −α̃/2 on the topological term, not the coefficients printed in Eq. (24). Conversely, integrating Eq. (24) as printed with the continuity equation gives H^2 = 8πG/3 ρ_m + k/a^2 + Λ/3 + 8α̃ ∫ (H^2 + k/a^2)^2 ṇ dt, not the 4α̃ of Eq. (25). An independent integration of the corrected first law gives −4α̃ in front of the ∫(H^2+k/a^2)^2 dN/dt dt term in Eq. (27), not −8α̃. The prefactor in Eq. (27) is therefore not established.
[II C, Eqs. (30)-(32)] The dark-energy sector defined by Eqs. (30)-(32) is internally inconsistent. Differentiating Eq. (30) gives ṍρ_DE = −(3α̃/πG)(H^2+k/a^2)^2 dN/dt, so the conservation equation ṍρ_DE + 3H(ρ_DE+p_DE)=0 requires p_DE = −Λ/(8πG) + (α̃/πGH)(H^2+k/a^2)^2 dN/dt + (3α̃/πG)∫(H^2+k/a^2)^2(dN/dt)dt. Equation (31) instead has −2α̃/(πGH) in the middle term, and substituting the printed Eqs. (30) and (31) into the conservation equation leaves a residual −(9α̃/πG)(H^2+k/a^2)^2 dN/dt rather than zero. Equation (32) is also not the equation of state of the fluid defined by Eqs. (30)-(31): those expressions give w_DE + 1 = −16α̃X/[H(Λ − 24α̃I)] with X=(H^2+k/a^2)^2 dN/dt and I=∫(H^2+k/a^2)^2(dN/dt)dt, whereas Eq. (32) gives w_DE + 1 = 8α̃X/[H(Λ − 24α̃I)]. Since Eqs. (35)-(36), (47)-(49), and all figures inherit these expressions, the quantitative results are not supported.
[II B, Eq. (20)] The central mechanism depends on the postulate δχ(∂M)=0, i.e., conservation of the total Euler characteristic of all causal boundaries. This is introduced to avoid the second-law violation discussed for Einstein-Gauss-Bonnet theory, but no physical derivation or independent support is provided beyond an analogy with the holographic principle. Without this postulate the apparent-horizon Euler characteristic does not change, and the topological dark-energy sector vanishes identically. The authors should either justify this assumption more concretely, present the model explicitly as a phenomenological ansatz, or discuss how it could be tested.

minor comments (6)

[Throughout] There are several typographical errors: 'redhsift' in the caption of Fig. 4, 'form' in the title of Section II C should be 'from', and 'NBHMR' is used in Eq. (43) without prior definition.
[Fig. 6] The caption of Fig. 6 states that the highest estimated value is f_BH = 0.001, while Table I and the text give the range extending up to 0.05; this appears to be a typographical error.
[II C, Eq. (45)] The conversion from volume rates to dN/dz in Eq. (45) assumes a flat universe (r_A = 1/H), but Eq. (47) retains the k/a^2 term. Please state whether the analysis is restricted to k = 0 or use the full B-dependent expression for the apparent-horizon volume in nonzero curvature.
[III A, Eq. (39)] The value of f_BH depends on the assumed IMF and the lower mass cutoff; please state the normalization of ξ(m) used in Eq. (39) so that the quoted range 0.001–0.05 is reproducible.
[III B, Eq. (50)] The hypergeometric expression for the integral of ψ(z)/(1+z) should be accompanied by its domain of validity and the chosen branch, since the high-redshift power-law behavior of the integrand is important for the early-time behavior of w_DE.
[IV] The concluding claim that w_DE 'remains within its observational bounds' is based on visual inspection of Figs. 5–11; a quantitative comparison, for example through χ² or confidence contours, would be considerably stronger.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the w_DE(z) shape is derived from the Wald-Gauss-Bonnet entropy plus a stated topology-conservation postulate, not fitted to dark-energy data.

full rationale

The derivation chain is explicit: the Wald-Gauss-Bonnet entropy (15) is differentiated in (23), the first law gives (24), integration with the matter continuity equation gives (25), and the topology relation (21) converts this into the modified Friedmann equation (27). The dark-energy density and w_DE(z) then follow from this equation and the Madau-Dickinson star-formation rate, with astrophysical fractions taken from the literature. The cosmological constant is fixed by the boundary condition Omega_DE0 = 0.69, but this does not fix the redshift dependence of w_DE(z), which is the paper's main prediction. The assumption in Eq. (20) that the total Euler characteristic of causal boundaries is conserved is a stated physical postulate, not a restatement of the desired conclusion. Self-citations [46,47] appear only as background on topology-change dark energy and are not used to derive Eq. (27) or to exclude alternatives. For these reasons, the derivation is not circular. A separate check of the printed algebra suggests coefficient inconsistencies among Eqs. (23)-(27), but that is a correctness issue rather than circularity and does not change the score.

Assumptions & free parameters 7 free parameters · 7 assumptions · 2 invented entities

The model's central claim rests on several astrophysical calibrations and one unvalidated topological conservation law. The free parameters are numerous, and while many have literature priors, none are measured within this paper; the cosmological constant is fit to ΩDE0.

free parameters (7)

Gauss-Bonnet coupling ᾱ = 10^5 to 2x10^5 in H0 units (positive), -10^5 to -2x10^5 (negative)
Completely free parameter; controls the amplitude of the topological dark energy term; the paper scans values without prior constraint.
fBH = 0.025 fiducial; range 0.001-0.05
Fraction of stellar mass ending in BH progenitors; adopted from literature but uncertain; direct effect on wDE amplitude.
fbin = 0.65 fiducial; range 0.5-0.8
Binary fraction of massive stars; chosen in the middle of the cited range.
fmerge = 0.05 fiducial; range 0.01-0.1
Fraction of BH binaries that merge within a Hubble time.
Average progenitor mass ⟨mprog⟩ = 30 M⊙ fiducial; range 25-40 M⊙
Average mass of stars that evolve to form BHs; affects the BH formation rate.
Cosmological constant Λ = Set by imposing ΩDE0≈0.69
The cosmological constant term is fit to the current dark energy density, absorbing the topological contribution at z=0.
Lower integration limit z_i = Unspecified, effectively large
The lower limit in the redshift integrals is never stated; changes in z_i shift Λ and the amplitude of wDE.

assumptions (7)

standard math Chern-Gauss-Bonnet theorem: ∫_h √σ R = 4πχ(h)
Used in Eq. (14) to convert the Ricci integral in Wald entropy to the Euler characteristic.
domain assumption Gravity-thermodynamics conjecture
The first law of thermodynamics on the apparent horizon yields Friedmann equations (Sec. II A).
domain assumption Wald entropy formula applies to the cosmological apparent horizon
Eqs. (11)-(15) are derived for stationary black hole horizons but are applied to the FRW apparent horizon.
ad hoc to paper Total boundary topology is conserved: δχ(∂M)=0
Eq. (20); postulated to restore the second law during BH mergers. It is the load-bearing link between BH events and horizon topology changes.
domain assumption BH formation rate is proportional to star formation rate with constant fBH
Eqs. (39)-(42); ignores metallicity and delay-time effects.
domain assumption BH merger rate is proportional to BH formation rate with constant fbin and fmerge
Eqs. (43)-(44); the paper cites [78] showing the real merger rate deviates from SFR, then proceeds with the simplified relation.
domain assumption Events counted inside the apparent horizon using instantaneous SFR and H(z)
Eqs. (41)-(42); the horizon volume and time conversion depend on the model's H, though H cancels in the final equations.

invented entities (2)

Puncture holes on the apparent horizon
purpose: Provide a concrete realization of Euler characteristic changes (χ=2-2g-b) so that BH formation opens two punctures and merger closes two.
Interpretive device tied to Eq. (22); no observational handle is given.
Topological dark energy fluid
purpose: Effective energy density and pressure absorbing the topology-change term in the Friedmann equations (Eqs. 28-31).
It is an effective fluid, not a fundamental entity; its properties depend on unfixed parameters.

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Cite this review

Pith. "Pith review of Topological dark energy from black-hole formations and mergers through the gravity-thermodynamics approach." pith.science (2026). https://pith.science/paper/XBYCKU3E

@misc{pith2026241221146,
  author       = {Pith},
  title        = {Pith review of: Topological dark energy from black-hole formations and mergers through the gravity-thermodynamics approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBYCKU3E}},
  note         = {Machine review of arXiv:2412.21146}
}

abstract

We apply the gravity-thermodynamics approach in the case of Einstein-Gauss-Bonnet theory, and its corresponding Wald-Gauss-Bonnet entropy, which due to the Chern-Gauss-Bonnet theorem it is related to the Euler characteristic of the Universe topology. However, we consider the realistic scenario where we have the formation and merger of black holes that lead to topology changes, which induce entropy changes in the Universe horizon. We extract the modified Friedmann equations and we obtain an effective dark energy sector of topological origin. We estimate the black-hole formation and merger rates starting from the observed star formation rate per redshift, which is parametrized very efficiently by the Madau-Dickinson form, and finally we result to a dark-energy energy density that depends on the cosmic star formation rate density, on the fraction $f_{\text{BH}}$ of stars forming black holes, on the fraction of black holes $f_\text{merge}$ that eventually merge, on the fraction $ f_{\text{bin}}$ of massive stars that are in binaries, on the average mass of progenitor stars that will evolve to form black holes $ \langle m_{\text{prog}} \rangle $, as well as on the Gauss-Bonnet coupling constant. We investigate in detail the cosmological evolution, obtaining the usual thermal history. Concerning the dark-energy equation-of-state parameter, we show that at intermediate redshifts it exhibits phantom-like or quintessence-like behavior according to the sign of the Gauss-Bonnet coupling, while at early and late times it tends to the cosmological constant value. Finally, we study the effect of the other model parameters, showing that for the whole allowed observationally estimated ranges, the topological dark-energy equation-of-state parameter remains within its observational bounds.

Figures

Figures reproduced from arXiv: 2412.21146 by the authors.

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

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

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Forward citations

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

The cosmic star formation rate density is presented in Fig

Estimating the black-hole formation rate from the star formation rate Since black holes typically evolve from massive stars, it is commonly assumed that the formation rate of BHs (BHFR) is proportional to the cosmic star formation rate (SFR) [69–71]. The cosmic star formation rate density is presented in Fig. 3, and its best fit form is given by Madau and...

[2] [2]

Estimating the black-hole merger rate from the black-hole formation rate The binary black hole merger rate (BHMR) can also be assumed to be proportional to SFR. Nevertheless, ambi- guities arise in this simplified approach as the formation efficiency of compact object is metallicity-dependent and BH formation and binary black hole merging may oc- cur with...

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We elaborate the 8 cosmological equations numerically, imposing Ω DE (z =

˜α >0 case We start our analysis by examining the case where the GB coupling constant ˜α is positive. We elaborate the 8 cosmological equations numerically, imposing Ω DE (z =

[4] [4]

In the upper graph of Fig

≡ ΩDE0 ≈ 0.69 and Ω m(z = 0) ≡ Ωm0 ≈ 0.31 as required by observations [85]. In the upper graph of Fig. 4 we present the evolution of the dimensionless density parameters ΩDE (z) and Ω m(z). We observe that we can acquire the usual thermal history of the Universe, the standard sequence of matter and dark energy epochs, while in the asymptotic future (z → −...

[5] [5]

Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse)

˜α <0 case We now present the cosmological behavior for nega- tive GB coupling constant ˜α < 0. In the upper graph of Fig. 8 we present the evolution of Ω DE (z) and Ωm(z), where one can see that the anticipated thermal history 9 FIG. 8. Upper graph: The evolution of the dimensionless dark energy parameter ΩDE (blue-solid) and the correspond- ing matter d...

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