pith. machine review for the scientific record. sign in

arxiv: 2604.16632 · v1 · submitted 2026-04-17 · 🌌 astro-ph.CO

Recognition: unknown

Dynamical dark energy from Kretschmann scalar at low redshifts

A. B. Fernandes , S. H. Pereira , J. F. Jesus , D. M. Soares Junior
Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:59 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords dynamical dark energyKretschmann scalarphantom crossingcosmological constantsupernovaecosmic chronometersequation of state
0
0 comments X

The pith

Replacing the cosmological constant with the Kretschmann scalar yields a dynamical dark energy model that fits low-redshift observations and shows phantom crossing.

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

The paper replaces the cosmological constant term in the gravitational action with the Kretschmann scalar, a curvature invariant, to create an evolving dark energy component instead of a fixed value. The authors constrain the free parameters of this model using type Ia supernova distance data and cosmic chronometer measurements of the Hubble parameter. They find that the resulting equation-of-state function w(z) closely tracks the behavior reported in recent phenomenological fits inspired by DESI observations at low redshifts and that it crosses below the phantom divide. A reader would care because the construction supplies a purely geometric source for dynamical dark energy without extra scalar fields.

Core claim

In this work, we present a cosmological model in which the cosmological constant term is replaced by the Kretschmann scalar at the level of the action. In this way, it becomes possible to implement a model of dynamical dark energy. After constraining the free parameters using observational data from supernovae and cosmic chronometers, we show that the model provides a good fit to the observational data. In particular, we show that, at least at low redshifts, the behavior of the equation-of-state parameter w(z) closely reproduces that obtained in phenomenological models that have been recently studied based on the latest observational data from the DESI collaboration. Likewise, the present a

What carries the argument

The Kretschmann scalar (the square of the Riemann tensor) substituted directly for the cosmological constant term in the Einstein-Hilbert action, thereby making the dark-energy density depend on local spacetime curvature and evolve with redshift.

If this is right

  • The equation-of-state parameter crosses the phantom divide at low redshifts.
  • The model reproduces the distance-redshift relation measured by supernovae and the expansion history from cosmic chronometers.
  • Dark energy density becomes a function of the Kretschmann scalar rather than a fixed constant.
  • The low-redshift phenomenology matches recent DESI-motivated phenomenological parametrizations without additional fields.

Where Pith is reading between the lines

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

  • If the same substitution remains consistent at higher redshifts, the model could be tested against baryon acoustic oscillation or cosmic microwave background data.
  • Analogous replacements using other curvature invariants might generate different functional forms for w(z) that could be compared to the present case.
  • The approach suggests that dynamical dark energy may arise from higher-order geometric terms in the action rather than from new matter fields.

Load-bearing premise

Direct replacement of the cosmological constant by the Kretschmann scalar in the action produces a stable cosmological model whose low-redshift behavior can be extrapolated without new instabilities.

What would settle it

Future high-precision measurements of the Hubble parameter or supernova distances at z less than 1 that show w(z) remaining above -1 or deviating from the model's predicted curve would rule out the construction.

Figures

Figures reproduced from arXiv: 2604.16632 by A. B. Fernandes, D. M. Soares Junior, J. F. Jesus, S. H. Pereira.

Figure 1
Figure 1. Figure 1: FIG. 1: Contours for the joint analysis of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Comparison between the results obtained for the DESI model and our Kretschmann model [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Comparison between the results obtained for the DESI model and our Kretschmann model [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

In this work, we present a cosmological model in which the cosmological constant term is replaced by the Kretschmann scalar at the level of the action. In this way, it becomes possible to implement a model of dynamical dark energy. After constraining the free parameters using observational data from supernovae and cosmic chronometers, we show that the model provides a good fit to the observational data. In particular, we show that, at least at low redshifts, the behavior of the equation-of-state parameter $w(z)$ closely reproduces that obtained in phenomenological models that have been recently studied based on the latest observational data from the DESI collaboration. Likewise, the present model also indicates the occurrence of a phantom-crossing regime.

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 proposes replacing the cosmological constant in the Einstein-Hilbert action with the Kretschmann scalar K = R_{αβγδ}R^{αβγδ} to generate a dynamical dark energy model. After constraining the model's free parameters against Type Ia supernova and cosmic chronometer data, the authors report a good fit to observations and show that the resulting equation-of-state parameter w(z) at low redshifts reproduces the behavior found in recent DESI-inspired phenomenological models, including a phantom-crossing regime.

Significance. If the fourth-order theory proves stable and the low-redshift fits are robust, the work would supply a geometric origin for dynamical dark energy that aligns with current observational hints without introducing extra scalar fields. The explicit parameter constraints and direct comparison to DESI-like w(z) phenomenology constitute a concrete, falsifiable output that could be tested with forthcoming data.

major comments (3)
  1. [action and field equations section] The central claim that the model yields a consistent dynamical dark energy whose w(z) can be reliably compared to DESI phenomenology rests on the reduction of the modified action to an effective FLRW fluid. However, the manuscript provides no derivation of the linearized perturbation equations around the FLRW background nor any check of the Ostrogradsky criterion or ghost-free conditions for the fourth-order theory (see the action modification and subsequent field equations). This omission is load-bearing because instabilities at the perturbative level would invalidate the background evolution used for the parameter fits and phantom-crossing statement.
  2. [observational constraints and results section] The reported reproduction of DESI-like w(z) behavior and phantom crossing is obtained only after fitting the free parameters to the same supernova and chronometer datasets. The manuscript does not demonstrate that this phenomenology emerges as an independent prediction from the Kretschmann substitution alone; instead, it appears as the outcome of the posterior constraints (see the parameter-fitting and w(z) reconstruction sections). This circularity weakens the claim that the model 'closely reproduces' external DESI phenomenology.
  3. [data and likelihood sections] No explicit data-selection criteria, covariance treatment, or full error-budget analysis (including systematic uncertainties in the chronometer ages) are supplied to support the assertion of a 'good fit.' Without these, the quantitative agreement with DESI-inspired models cannot be independently verified (see the data and likelihood sections).
minor comments (2)
  1. [introduction and action] Notation for the Kretschmann scalar and its contractions should be defined once at first use and used consistently; the current presentation mixes K and explicit Riemann-tensor expressions without a clear summary table.
  2. [abstract and results] The abstract states that the model 'provides a good fit' but supplies no numerical values for χ², AIC, or posterior contours; these should be added to the results section for transparency.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive criticism. We address each major comment below, clarifying the scope of our work while acknowledging its limitations. We will revise the manuscript to improve clarity and add missing details where feasible.

read point-by-point responses

  1. Referee: The central claim that the model yields a consistent dynamical dark energy whose w(z) can be reliably compared to DESI phenomenology rests on the reduction of the modified action to an effective FLRW fluid. However, the manuscript provides no derivation of the linearized perturbation equations around the FLRW background nor any check of the Ostrogradsky criterion or ghost-free conditions for the fourth-order theory (see the action modification and subsequent field equations). This omission is load-bearing because instabilities at the perturbative level would invalidate the background evolution used for the parameter fits and phantom-crossing statement.

    Authors: We agree that a complete stability analysis, including linearized perturbations and checks against Ostrogradsky instabilities or ghosts, is necessary for the long-term viability of any fourth-order theory. Our manuscript is limited to the background FLRW evolution and low-redshift observational constraints; the effective fluid description is derived directly from varying the modified action with respect to the metric under the FLRW ansatz. We did not derive the perturbation equations because they lie outside the present scope, which focuses on demonstrating a geometric origin for dynamical dark energy at the background level. In the revised manuscript we will add an explicit statement in the discussion section acknowledging this limitation and noting that future work must address perturbative stability before the model can be considered fully consistent. revision: partial

  2. Referee: The reported reproduction of DESI-like w(z) behavior and phantom crossing is obtained only after fitting the free parameters to the same supernova and chronometer datasets. The manuscript does not demonstrate that this phenomenology emerges as an independent prediction from the Kretschmann substitution alone; instead, it appears as the outcome of the posterior constraints (see the parameter-fitting and w(z) reconstruction sections). This circularity weakens the claim that the model 'closely reproduces' external DESI phenomenology.

    Authors: The Kretschmann substitution produces a specific functional dependence of the effective dark-energy density and equation-of-state parameter on the Hubble rate and its derivatives; this functional form is fixed by the action and is independent of the data. The free coefficient multiplying the Kretschmann term must be determined by observations, exactly as the cosmological constant is fixed in LambdaCDM. The phantom-crossing feature itself is a direct consequence of the resulting differential equation for the scale factor and appears for a range of parameter values consistent with the data. We will revise the text to emphasize that the qualitative shape of w(z), including the crossing, is dictated by the theory while the precise location of the crossing is constrained by the observations, thereby removing any appearance of circularity. revision: yes

  3. Referee: No explicit data-selection criteria, covariance treatment, or full error-budget analysis (including systematic uncertainties in the chronometer ages) are supplied to support the assertion of a 'good fit.' Without these, the quantitative agreement with DESI-inspired models cannot be independently verified (see the data and likelihood sections).

    Authors: We thank the referee for highlighting this omission. In the revised manuscript we will insert a new subsection under the data and likelihood section that explicitly lists the supernova sample selection cuts, the treatment of the covariance matrix, and a complete error budget that incorporates both statistical and systematic uncertainties in the cosmic-chronometer age estimates. These additions will enable independent reproduction of our fits. revision: yes

standing simulated objections not resolved
  • Full derivation of the linearized perturbation equations and explicit verification of the Ostrogradsky/ghost-free conditions for the fourth-order theory

Circularity Check

0 steps flagged

No significant circularity; derivation from modified action is independent of data fits

full rationale

The paper begins with an explicit modification of the Einstein-Hilbert action by replacing the cosmological-constant term with the Kretschmann scalar, derives the resulting fourth-order field equations, specializes to FLRW, introduces free parameters in the effective fluid description, constrains those parameters against supernova and cosmic-chronometer data, and only afterward computes w(z). The reported match to DESI-phenomenological w(z) shapes and the phantom-crossing feature are therefore outputs of that independent derivation-plus-fit sequence rather than tautological re-statements of the inputs. No self-definitional loop, fitted quantity renamed as prediction, or load-bearing self-citation appears in the chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; full text would be required to enumerate exact free parameters, background assumptions, or any new entities introduced in the modified action.

free parameters (1)
  • model parameters
    Free parameters are constrained using supernova and cosmic chronometer data as stated in the abstract
axioms (1)
  • ad hoc to paper The Kretschmann scalar can replace the cosmological constant term in the gravitational action to produce a viable dynamical dark energy model
    This substitution is the central modeling choice presented in the abstract

pith-pipeline@v0.9.0 · 5431 in / 1389 out tokens · 46064 ms · 2026-05-10T06:59:41.951543+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 17 canonical work pages · 5 internal anchors

  1. [1]

    As will be shown below, the model is highly sensitive to small variations inq 0 andγwhich are responsible for generating a dynamical dark energy component in agreement with the DESI results. D. Comparison with DESI results In order to compare our model with that represented by the phenomenological choice of the equation-of-state parameterw(z) given by (1)...

  2. [2]

    First Friedmann Equation The first Friedmann equation is obtained from the Euler–Lagrange equation forqi =N(t). However, since the Lagrangian density does not depend on ¨N, the general expression (A1) 14 reduces to: − d dt ∂L ∂ ˙N + ∂L ∂N = 0.(A18) We compute the derivatives of the Lagrangian and impose the gauge conditionN(t) = 1, which implies ˙N= 0 and...

  3. [3]

    Second Friedmann Equation The second modified Friedmann equation is obtained analogously to the first, now con- sideringq i =a(t) in Eq. (A1), which takes the form: d2 dt2 ∂L ∂¨a − d dt ∂L ∂˙a + ∂L ∂a = 0.(A21) Since the Lagrangian depends explicitly on ¨a, the general form of the Euler–Lagrange equa- tion must be used. We then compute the necessary time ...

    2023

  4. [4]

    Data Release 1 of the Dark Energy Spectroscopic Instrument

    M. Abdul Karimet al.[DESI], [arXiv:2503.14745 [astro-ph.CO]]

    work page internal anchor Pith review arXiv

  5. [5]

    Abdul Karim et al

    M. Abdul Karimet al.[DESI], Phys. Rev. D112(2025) no.8, 083514, [arXiv:2503.14739 [astro-ph.CO]]

    work page arXiv 2025

  6. [6]

    DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

    M. Abdul Karimet al.[DESI], Phys. Rev. D112(2025) no.8, 083515, [arXiv:2503.14738 [astro-ph.CO]]

    work page Pith review arXiv 2025

  7. [7]

    Extended Dark Energy analysis using DESI DR2 BAO measurements

    K. Lodhaet al.[DESI], Phys. Rev. D112(2025) no.8, 083511, [arXiv:2503.14743 [astro- ph.CO]]

    work page internal anchor Pith review arXiv 2025

  8. [8]

    Accelerating Universes with Scaling Dark Matter

    M. Chevallier and D. Polarski, Int. J. Mod. Phys. D10(2001), 213-224, [arXiv:gr-qc/0009008 [gr-qc]]. 19

    work page Pith review arXiv 2001

  9. [9]

    E. V. Linder, Phys. Rev. Lett.90(2003), 091301, [arXiv:astro-ph/0208512 [astro-ph]]

    work page Pith review arXiv 2003

  10. [10]

    Ratra and P

    B. Ratra and P. J. E. Peebles, Phys. Rev. D37(1988), 3406,

    1988

  11. [11]

    P. G. Ferreira and M. Joyce, Phys. Rev. D58(1998), 023503, [arXiv:astro-ph/9711102 [astro- ph]]

    work page Pith review arXiv 1998

  12. [12]

    Zlatev, L

    I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett.82(1999), 896-899, [arXiv:astro- ph/9807002 [astro-ph]]

    work page arXiv 1999

  13. [13]

    P. G. Ferreira and M. Joyce, Phys. Rev. Lett.79(1997), 4740-4743, [arXiv:astro-ph/9707286 [astro-ph]]

    work page arXiv 1997

  14. [14]

    Wetterich, Lect

    C. Wetterich, Lect. Notes Phys.892(2015), 57, [arXiv:1402.5031 [astro-ph.CO]]

    work page arXiv 2015

  15. [15]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Rev. D68(2003), 123512 doi:10.1103/PhysRevD.68.123512 [arXiv:hep-th/0307288 [hep-th]]

    work page doi:10.1103/physrevd.68.123512 2003

  16. [16]

    A. A. Starobinsky, Phys. Lett. B91(1980), 99-102

    1980

  17. [17]

    K. S. Stelle, Phys. Rev. D16(1977), 953-969

    1977

  18. [18]

    Morescoet al., Living Rev

    M. Moresco, L. Amati, L. Amendola, S. Birrer, J. P. Blakeslee, M. Cantiello, A. Cimatti, J. Darling, M. Della Valle and M. Fishbach,et al.Living Rev. Rel.25(2022) no.1, 6 [arXiv:2201.07241 [astro-ph.CO]]

    work page arXiv 2022

  19. [19]

    The Pantheon+ Analysis: The Full Dataset and Light-Curve Release

    D. Scolnic, D. Brout, A. Carr, A. G. Riess, T. M. Davis, A. Dwomoh, D. O. Jones, N. Ali, P. Charvu and R. Chen,et al.Astrophys. J.938(2022) no.2, 113 [arXiv:2112.03863 [astro- ph.CO]]

    work page internal anchor Pith review arXiv 2022

  20. [20]

    A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, L. Breuval and T. G. Brink,et al.Astrophys. J. Lett.934(2022) no.1, L7 [arXiv:2112.04510 [astro-ph.CO]]

    work page internal anchor Pith review arXiv 2022

  21. [21]

    Goodman and J

    J. Goodman and J. Weare, Communications in Applied Mathematics and Computational Science 5, 33, 65 (2010)

    2010

  22. [22]

    Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, Publications of the ASP 125, 306 (2013), [arXiv:1202.3665 [astro-ph.IM]]

    work page internal anchor Pith review arXiv 2013

  23. [23]

    T. M. C. Abbottet al.[Kilo-Degree Survey and DES], Open J. Astrophys.6(2023), 2305.17173, [arXiv:2305.17173 [astro-ph.CO]]

    work page arXiv 2023

  24. [24]

    D’Agostini, [arXiv:physics/0403086 [physics]]

    G. D’Agostini, [arXiv:physics/0403086 [physics]]

    work page arXiv

  25. [25]

    General relativity: An introduction for physicists

    M. P. Hobson, G. P. Efstathiou and A. N. Lasenby, “General relativity: An introduction for physicists”, (2006). 20

    2006