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arxiv: 2605.08576 · v1 · submitted 2026-05-09 · 🌌 astro-ph.CO

Recognition: 2 theorem links

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Cosmological test of a length-preserving biconnection gravity

Dalale Mhamdi , Amine Bouali , Taoufik Ouali , Tiberiu Harko
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:15 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords biconnection gravitymutual curvatureeffective dark energyFriedmann equationscosmological observationscosmographyLambdaCDM comparison
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The pith

Biconnection gravity induces effective dark energy in the Friedmann equations that fits observations as well as LambdaCDM for certain parametrizations.

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

The paper develops a biconnection gravitational framework using the Schrödinger connection and its dual. Their difference defines a mutual curvature that supplies additional geometric terms acting as an effective dark energy sector in the generalized Friedmann equations for a flat FLRW universe. Five common parametrizations of this extra term are tested against DESI DR2, Pantheon+, and cosmic chronometer data. The analysis finds that four parametrizations produce nearly identical acceleration onset redshifts and present-day Hubble rates, while the Barboza-Alcaniz and logarithmic forms show strong statistical support and compete with the cosmological constant model. Cosmographic quantities including deceleration, jerk, snap, statefinder, and Om(z) are then used to situate the model among other acceleration scenarios.

Core claim

In this biconnection construction the symmetric combination of the two connections recovers the Levi-Civita connection and standard general relativity at background level, while the antisymmetric difference produces a mutual curvature that contributes geometric terms to the Friedmann equations; when these terms are parametrized with common dark-energy equations of state and confronted with current observations, the Barboza-Alcaniz and logarithmic choices yield Akaike, Bayesian, and Deviance Information Criterion values that indicate strong evidence and competitiveness with LambdaCDM.

What carries the argument

The mutual curvature, defined as the difference between the Schrödinger connection and its dual, which encodes the non-Riemannian geometric degrees of freedom that supply the effective dark energy sector while the symmetric part reduces to the Levi-Civita connection.

If this is right

  • The biconnection model reproduces standard general relativity at the background level while adding geometric contributions that can be parametrized as dark energy.
  • Four of the five parametrizations produce nearly the same redshift of acceleration onset and the same current Hubble rate.
  • The Barboza-Alcaniz and logarithmic parametrizations are statistically preferred by the Akaike, Bayesian, and Deviance Information Criteria and are competitive with LambdaCDM.
  • Cosmographic tools (deceleration, jerk, snap, statefinder, Om(z)) provide a classification of the model relative to other dark-energy scenarios.

Where Pith is reading between the lines

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

  • If the mutual curvature truly supplies the observed acceleration, then high-precision future measurements of expansion history alone could test a purely geometric origin of dark energy without new fields.
  • The framework suggests that similar biconnection constructions applied to other gravitational theories might generate analogous effective dark-energy sectors that can be tested with the same data sets.
  • Perturbation-level extensions of the model could produce distinctive signatures in structure growth that distinguish it from scalar-field dark energy.

Load-bearing premise

The assumption that the difference between the Schrödinger connection and its dual naturally encodes an effective dark energy sector in the generalized Friedmann equations.

What would settle it

A measurement of the present-day deceleration parameter, jerk, or Om(z) diagnostic that lies outside the narrow range predicted by the Barboza-Alcaniz and logarithmic parametrizations at low redshifts would show the model does not compete with LambdaCDM.

Figures

Figures reproduced from arXiv: 2605.08576 by Amine Bouali, Dalale Mhamdi, Taoufik Ouali, Tiberiu Harko.

Figure 1
Figure 1. Figure 1: The 1σ and 2σ confidence contours obtained from the combined DESI DR2 + Pantheon+ + CC datasets for ΛCDM (red), wCDM (blue), and BΛCDM (green). A. Statistical Results In Table I, the mean ±1σ of the cosmological parameters of equations of state and of the biconnection gravity are shown considering the combination DESI DR2 + Pantheon+ + CC datasets. For each parameterizations, the minimum of the chi-square … view at source ↗
Figure 2
Figure 2. Figure 2: The 1σ and 2σ confidence contours obtained from the combined DESI DR2 + Pantheon+ + CC dataset datasets for the CPL (red), Barboza-Alcaniz (blue), and logarithmic (green) parametrizations. c. The Hubble tension. To quantify the tension of these values with the SH0ES value, we calculate their deviation to HSH0ES 0 = 73.2 ± 1.3 km s−1 Mpc−1 [70]. The tension is at 1.38σ, 1.62σ, 1.5σ, 1.64σ, 1.60σ, and 1.58σ … view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the predicted distance modulus, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the distance ratios DV /rd, DM /rd, and DH/rd as functions of redshift z. The DESI DR2 measurements are shown as blue circles (DV /rd), orange squares (DM /rd), and green triangles (DH/rd). The theoretical curves correspond to the ΛCDM model (black solid lines), wCDM (blue solid and dashed lines), CPL (orange dashed lines), Barboza-Alcaniz parametrization (green dash-dot lines), logarithmic pa… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the equation of state parameter [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the geometrical parameter of the bi [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the deceleration parameter q(z) as a function of redshift. The curves correspond to CPL (blue solid), Barboza-Alcaniz (green solid), logarithmic (purple solid), ωCDM (black solid), ΛCDM (black dashed), and BΛCDM (red solid). The horizontal dotted line marks the transition between accelerating and decelerating phases. CPL Barboza Logarithmic ωCDM ΛCDM BΛCDM -1 0 1 2 3 4 0.8 1.0 1.2 1.4 1.6 z j(… view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the snap parameter s(z) as a func￾tion of redshift. The curves correspond to CPL (blue solid), Barboza-Alcaniz (green solid), logarithmic (purple solid), ωCDM (black solid), ΛCDM (black dashed), and BΛCDM (red solid). and the snap parameter, s. The expression of these parameters are extracted from the Taylor expansion of the scale factor around the present time t0. a(t) = a0 " 1 + 1 a0 da dt … view at source ↗
Figure 12
Figure 12. Figure 12: Statefinder plane (s, j) for ΛCDM (black star), ωCDM (black solid), CPL (blue solid), Barboza-Alcaniz (green solid), logarithmic (purple solid), and BΛCDM (red solid). The arrows indicate the dynamical evolution from the past to the future. ΛCDM dS SCDM z=0 z=0 z=0 z=0 CPL Barboza Logarithmic ωCDM BΛCDM -1.0 -0.5 0.0 0.5 0.6 0.8 1.0 1.2 1.4 1.6 q j [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the Om(z) diagnostic as a function of redshift. The curves correspond to CPL (blue solid), Barboza￾Alcaniz (green solid), logarithmic (purple solid), ωCDM (black solid), ΛCDM (black dashed), and BΛCDM (red solid). VII. CONCLUSIONS In this work, we have investigated the cosmological implications of a length-preserving biconnection gravity con￾structed from the Schr¨odinger connection and its d… view at source ↗
read the original abstract

We investigate the cosmological implications of an extended gravitational framework based on biconnection gravity, constructed from the Schr$\ddot{o}$dinger connection and its dual. In this approach, the difference between the two connections defines the mutual curvature, which encodes the non-Riemannian geometric degrees of freedom, while their symmetric combination reduces to the Levi-Civita connection and hence reproduces general relativity at the background level. Within this setting, we derive the generalized Friedmann equations for a spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker Universe. The resulting equations contain additional geometric contributions that may naturally encode an effective dark energy sector induced by the biconnection degrees of freedom. We explore this extra dark energy by adopting five commonly used parametrizations, namely B$\Lambda$CDM, $\omega$CDM, Chevallier-Polarski-Linder, Barboza-Alcaniz, and a logarithmic equations of state. These considerations are confronted with recent observational data, including DESI DR2, Pantheon$^+$, and CC observations. Our analysis shows that the four parameterizations enter the acceleration phase at almost the same redshifts and share the same current value of the Hubble rate. Furthermore, the statistical comparison based on the Akaike, Bayesian, and Deviance Information Criterion shows that Barboza-Alcaniz, and logarithmic parameterizations have strong evidence and are competitive with $\Lambda$CDM. To classify this biconnection gravity in the plethora theoretical models describing the current cosmic acceleration, we examine its implications through cosmographic tools, including the deceleration, jerk, and snap parameters, as well as through the Statefinder analysis and $Om(z)$ diagnostic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a length-preserving biconnection gravity theory from the Schrödinger connection and its dual. Their difference is interpreted as a mutual curvature that supplies non-Riemannian degrees of freedom, while the symmetric part recovers the Levi-Civita connection and thus general relativity at the background level. Generalized Friedmann equations are derived for a flat FLRW universe; the extra geometric terms are interpreted as an effective dark-energy sector. Five standard phenomenological parametrizations (BΛCDM, ωCDM, CPL, Barboza-Alcaniz, logarithmic) are adopted for the equation-of-state of this sector and fitted to DESI DR2, Pantheon+, and cosmic-chronometer data. The authors report that the Barboza-Alcaniz and logarithmic forms are statistically competitive with ΛCDM according to AIC, BIC and DIC, that the models enter acceleration at similar redshifts, and that cosmographic (q, j, s) and diagnostic (Statefinder, Om(z)) analyses yield consistent results.

Significance. A geometric mechanism that generates an effective dark-energy sector without introducing new fields would be of clear interest. The present work, however, does not derive a specific functional form for w(z) from the mutual-curvature term; instead it imports five external parametrizations and performs standard likelihood fits. Consequently the reported statistical competitiveness tests the chosen parametrizations rather than the biconnection construction itself. The cosmographic and diagnostic sections add useful constraints but do not compensate for the missing link between geometry and the adopted equation-of-state forms.

major comments (3)
  1. [§3] §3 (generalized Friedmann equations): the explicit expression for the mutual-curvature contribution to the Friedmann equations is not displayed. Without the step-by-step reduction from the biconnection difference to the extra geometric term, it is impossible to verify whether this term can be identified with any of the five phenomenological w(z) parametrizations later adopted.
  2. [§4] §4 (parametrizations and data analysis): the five equations of state are introduced as standard phenomenological forms rather than being derived from the mutual curvature. The subsequent AIC/BIC/DIC comparison therefore evaluates the parametrizations against ΛCDM, not the biconnection framework; the claim that the geometry “naturally encodes” an effective dark-energy sector remains untested.
  3. [§5] §5 (observational constraints): the manuscript states that error propagation and full covariance treatment are performed, yet the text does not show the explicit propagation of the extra geometric parameters through the likelihood or the joint covariance matrix with the cosmological parameters. This omission prevents assessment of whether the reported constraints on the acceleration onset and H0 are robust.
minor comments (2)
  1. [Abstract, §2] The abstract and §2 contain several LaTeX rendering issues (e.g., Schr$ddot{o}$dinger) that should be corrected for readability.
  2. [Tables and figures] Table captions and figure legends should explicitly state the data combinations used (DESI DR2 + Pantheon+ + CC) and the priors adopted for the five parametrizations.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the detailed and insightful report. We appreciate the opportunity to clarify the connection between the biconnection geometry and the effective dark energy sector. Below we respond point by point to the major comments, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (generalized Friedmann equations): the explicit expression for the mutual-curvature contribution to the Friedmann equations is not displayed. Without the step-by-step reduction from the biconnection difference to the extra geometric term, it is impossible to verify whether this term can be identified with any of the five phenomenological w(z) parametrizations later adopted.

    Authors: We agree that the explicit form and derivation are essential for verification. In the revised version, we will insert a dedicated subsection in §3 that provides the step-by-step reduction from the difference of the Schrödinger connection and its dual to the mutual-curvature term appearing in the generalized Friedmann equations. This will explicitly show the extra geometric contribution to the effective energy density. revision: yes

  2. Referee: [§4] §4 (parametrizations and data analysis): the five equations of state are introduced as standard phenomenological forms rather than being derived from the mutual curvature. The subsequent AIC/BIC/DIC comparison therefore evaluates the parametrizations against ΛCDM, not the biconnection framework; the claim that the geometry “naturally encodes” an effective dark-energy sector remains untested.

    Authors: The referee correctly notes that we adopt standard phenomenological parametrizations rather than deriving a specific w(z) from the mutual curvature. Our intent is to demonstrate that the extra geometric terms can be consistently interpreted as an effective dark-energy component whose dynamics are compatible with current data when using well-studied forms. We will revise the language in §4 and the abstract to emphasize that these are exploratory parametrizations used to test the viability of the biconnection-induced acceleration, and we will add a statement acknowledging that a first-principles derivation of w(z) from the geometry is left for future work. revision: partial

  3. Referee: [§5] §5 (observational constraints): the manuscript states that error propagation and full covariance treatment are performed, yet the text does not show the explicit propagation of the extra geometric parameters through the likelihood or the joint covariance matrix with the cosmological parameters. This omission prevents assessment of whether the reported constraints on the acceleration onset and H0 are robust.

    Authors: We will expand §5 to include the explicit formulas for propagating uncertainties from the geometric parameters and the form of the joint covariance matrix employed in the MCMC analysis. This addition will enable readers to assess the robustness of the derived constraints on the onset of acceleration and the Hubble constant. revision: yes

standing simulated objections not resolved
  • A unique functional form for the equation of state derived directly from the mutual-curvature term without additional assumptions or parametrizations.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper first derives the generalized Friedmann equations for flat FLRW from the length-preserving biconnection, where the mutual curvature term supplies additional geometric contributions that are interpreted as possibly encoding effective dark energy. It then explicitly adopts five external phenomenological parametrizations (BΛCDM, ωCDM, CPL, Barboza-Alcaniz, logarithmic) for that sector and fits their parameters to DESI DR2 + Pantheon+ + CC data, reporting that certain parametrizations are statistically competitive with ΛCDM via AIC/BIC/DIC and share similar acceleration-onset redshifts and present-day Hubble values. These reported values are direct outputs of the data fits rather than first-principles outputs of the biconnection geometry, but the paper does not label them as predictions derived from the mutual curvature; it presents them as results of the adopted parametrizations. No step reduces by construction to a self-definition, a fitted quantity renamed as a geometric prediction, or a load-bearing self-citation whose content is unverified. The central geometric construction remains independent of the subsequent statistical comparison, which is a standard exploratory test of an extended framework.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the biconnection construction supplying extra geometric terms that are then parametrized and fitted; several free parameters enter through the chosen equations of state, and the non-Riemannian degrees of freedom are postulated without independent falsifiable signatures beyond the fits.

free parameters (1)
  • EOS parameters in the five parametrizations (w0, wa, etc.)
    Each of the five dark-energy parametrizations (BΛCDM, ωCDM, CPL, Barboza-Alcaniz, logarithmic) introduces one or more free parameters that are fitted to the observational data to reproduce the expansion history.
axioms (2)
  • domain assumption Spatially flat FLRW metric
    Used to derive the generalized Friedmann equations from the biconnection action.
  • domain assumption Symmetric part of the two connections reduces to Levi-Civita
    Allows recovery of standard GR at background level while the antisymmetric difference supplies the extra terms.
invented entities (1)
  • mutual curvature no independent evidence
    purpose: Encodes non-Riemannian geometric degrees of freedom that induce effective dark energy
    Defined as the difference between the Schrödinger connection and its dual; no independent evidence outside the model is provided.

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