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arxiv: 2603.21766 · v1 · submitted 2026-03-23 · 🌀 gr-qc · astro-ph.CO· hep-th

Dark energy and accelerating cosmological evolution in a Universe with a Weylian boundary

Tiberiu Harko , Shahab Shahidi This is my paper

Pith reviewed 2026-05-15 00:55 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th
keywords Weyl geometryboundary termsdark energycosmological evolutionFLRW metricgeneral relativity
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The pith

Adding a non-metric Weylian boundary to the Einstein-Hilbert action produces effective geometric dark energy that drives late-time cosmic acceleration.

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

The paper shows that by describing the boundary of the Einstein-Hilbert action with a non-metric Weyl-type geometry, the gravitational field equations gain new terms from the Weyl vector and its derivatives. These additions create an effective, time-dependent dark energy in cosmological settings. When the dark energy is assigned a Barboza-Alcaniz equation of state, the resulting Friedmann equations can be solved numerically to match observations of the Universe's expansion. This geometric approach to dark energy reproduces the predictions of the standard Lambda-CDM model, making it a viable alternative to conventional general relativity.

Core claim

Within the framework of the Einstein-Hilbert action with a Weylian boundary described by non-metric geometry, the field equations are generalized to incorporate the Weyl vector and its covariant derivatives. In a flat FLRW cosmology, these lead to modified Friedmann equations containing extra terms that function as an effective dark energy. Imposing a Barboza-Alcaniz parametrization for the dark energy equation of state allows numerical solutions that align closely with late-time observational data and the Lambda-CDM paradigm.

What carries the argument

The non-metric Weylian boundary term added to the Einstein-Hilbert action, which introduces contributions from the Weyl vector and its derivatives to produce effective dark energy.

Load-bearing premise

The boundary of the Einstein-Hilbert action admits a non-metric Weyl geometry whose vector and derivatives generate the observed dark energy effects.

What would settle it

High-precision measurements of the cosmic expansion history, such as supernova luminosity distances or baryon acoustic oscillation scales, that deviate significantly from the numerical solutions of the generalized Friedmann equations would disprove the model's viability.

Figures

Figures reproduced from arXiv: 2603.21766 by Shahab Shahidi, Tiberiu Harko.

Figure 1
Figure 1. Figure 1: FIG. 1. The corner plot for the values of the parameters [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The behavior of the rescaled Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The behavior of the jerk parameter [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The behavior of the jerk parameter [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The behavior of the matter density abundance Ω [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The behavior of the effective energy density abundance Ω [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The behavior of the effective pressure [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Different regions described by the Barboza-Alcaniz [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Noticing the negative slope of the Om diagram in the range z ∈ (0, 1.5), one can deduce that the effective fluid in the Weyl Boundary model is quintessence-like. However, as can be seen from the Figure, for larger val￾ues of the redshift z ≳ 1.5 the Om diagram has a positive slope indicating that the dark energy behaves like phan￾tom. 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 a … view at source ↗
read the original abstract

We investigate the influence of boundary terms in gravitational field theories, by considering that in the Einstein-Hilbert action the boundary can be described by a non-metric Weyl-type geometry. The gravitational action and the the field equations, are thus generalized to include new geometrical terms, coming from the non-metric nature of the boundary, and depending on the Weyl vector, and its covariant derivatives. The field equations obtained within this framework generalize the standard Einstein equations by including in their mathematical structure the Weyl vector, and its covariant derivatives. As an applications of the general formalism we investigate the cosmological evolution in a flat FLRW geometry. We obtain the generalized Friedmann equations, which contain extra terms depending on the Weyl vector and its derivatives, arising due to the presence of the Weylian boundary, and which describe an effective, time dependent dark energy. By imposing to the dark energy an equation of state parameter of the Barboza-Alcaniz type, the Friedmann equations can be solved numerically. We compare the predictions of the Weylian boundary gravitational theory with late-time observational data and the predictions of the $\Lambda$CDM paradigm. Our results show that the Weylian boundary cosmological models give a good description of the observational data, and they can reproduce almost exactly the predictions of the $\Lambda$CDM paradigm. Hence, the extension of gravitational theories through the addition of Weylian boundary terms, in which dark energy has a purely geometric origin, emerges as a viable alternative to standard general relativity.

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

1 major / 0 minor

Summary. The manuscript proposes generalizing the Einstein-Hilbert action by modeling its boundary with a non-metric Weyl-type geometry. This introduces additional geometric terms depending on the Weyl vector and its covariant derivatives into the field equations. For a flat FLRW cosmology the resulting generalized Friedmann equations contain extra contributions that are interpreted as an effective, time-dependent dark energy. By imposing a Barboza-Alcaniz parametrization on the dark-energy equation of state, the equations are solved numerically and shown to fit late-time observational data while reproducing the predictions of the LambdaCDM model to high accuracy.

Significance. If the central derivation can be completed without external parametrization, the work would supply a concrete geometric mechanism for late-time acceleration that originates entirely from boundary modifications rather than an added cosmological constant or scalar field. The numerical comparison with data and the close match to LambdaCDM already demonstrate phenomenological viability; explicit credit is due for the reproducible numerical solutions and direct data confrontation.

major comments (1)
  1. [Abstract and cosmological applications] Abstract and the section deriving the cosmological equations: the effective dark energy is introduced by imposing a Barboza-Alcaniz equation-of-state parametrization to permit numerical integration, rather than being derived from the evolution of the Weyl vector and its derivatives that appear in the generalized Friedmann equations. Because this step is not shown to follow from the boundary-modified field equations, the claim that dark energy has a purely geometric origin remains unestablished.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the single major comment below and have made partial revisions to improve clarity on the role of the parametrization while preserving the geometric derivation of the effective dark energy terms.

read point-by-point responses

  1. Referee: [Abstract and cosmological applications] Abstract and the section deriving the cosmological equations: the effective dark energy is introduced by imposing a Barboza-Alcaniz equation-of-state parametrization to permit numerical integration, rather than being derived from the evolution of the Weyl vector and its derivatives that appear in the generalized Friedmann equations. Because this step is not shown to follow from the boundary-modified field equations, the claim that dark energy has a purely geometric origin remains unestablished.

    Authors: We agree that the Barboza-Alcaniz parametrization is imposed to enable numerical integration and data comparison rather than being obtained by solving a closed dynamical system for the Weyl vector. The generalized Friedmann equations themselves are derived directly from the boundary-modified action and contain extra geometric contributions proportional to the Weyl vector and its covariant derivatives; these contributions are interpreted as the effective dark energy. The parametrization is adopted as a standard phenomenological ansatz for the equation-of-state evolution of this effective component, allowing us to confront the model with observations in the same manner used in many other modified-gravity cosmologies. In the revised manuscript we have added an explicit statement in the abstract and in the cosmological section clarifying that the parametrization is an effective description of the geometric terms, and we have included a short discussion of possible future routes to deriving the Weyl-vector dynamics from the field equations without external input. This addresses the concern without changing the central claim that the extra terms originate geometrically from the Weylian boundary. revision: partial

Circularity Check

1 steps flagged

Imposition of Barboza-Alcaniz parametrization on effective DE rather than derivation from Weyl vector

specific steps
  1. fitted input called prediction [Abstract]
    "By imposing to the dark energy an equation of state parameter of the Barboza-Alcaniz type, the Friedmann equations can be solved numerically. We compare the predictions of the Weylian boundary gravitational theory with late-time observational data and the predictions of the ΛCDM paradigm."

    The generalized Friedmann equations are stated to contain extra terms from the Weyl vector that describe an effective dark energy. Imposing the Barboza-Alcaniz parametrization on that effective dark energy supplies the functional form needed for numerical solution and data comparison; the resulting agreement with observations is therefore forced by the externally chosen parametrization rather than derived from the boundary geometry.

full rationale

The paper derives generalized Friedmann equations containing extra Weyl-vector terms that are presented as an effective, time-dependent dark energy of purely geometric origin. However, to obtain numerical solutions and compare with data, the equation-of-state parameter of this effective dark energy is externally imposed to be of Barboza-Alcaniz form. This step is not shown to follow from the dynamics of the Weyl vector or the boundary-modified field equations; the subsequent numerical match to observations and to ΛCDM is therefore a direct consequence of the imposed parametrization rather than an independent prediction of the geometric model. The central claim that the Weylian boundary supplies a viable geometric alternative therefore reduces to a fitted functional form.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on assuming a non-metric Weyl boundary, standard FLRW symmetry, and an externally chosen equation-of-state parametrization whose parameters are adjusted to data.

free parameters (2)
  • Weyl-vector amplitude and derivative scales
    These enter the generalized Friedmann equations and must be chosen or fitted to produce the observed acceleration.
  • Barboza-Alcaniz equation-of-state parameters
    Imposed by hand to close the system and enable numerical comparison with data.
axioms (2)
  • domain assumption The boundary contribution to the Einstein-Hilbert action admits a non-metric Weyl geometry
    Invoked to introduce the Weyl vector and its covariant derivatives into the field equations.
  • standard math Cosmological spacetime is described by flat FLRW geometry
    Used to reduce the generalized field equations to Friedmann form.
invented entities (1)
  • Weyl vector arising from the boundary no independent evidence
    purpose: To generate the extra geometric terms that act as effective dark energy
    Postulated as part of the non-metric boundary; no independent falsifiable signature outside the cosmological fit is provided.

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