arxiv: 2603.21125 · v1 · submitted 2026-03-22 · 🌌 astro-ph.CO · gr-qc

Recognition: 1 theorem link

· Lean Theorem

Model-Independent Reconstruction of Quintessence Potential and Kinetic Energy from DESI DR2 and Pantheon+ Supernovae

Shengjia Wang , Tian-Nuo Li , Tonghua Liu , Guo-Hong Du

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:26 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc

keywords quintessencedark energyGaussian processDESI DR2Pantheon+model-independent reconstructionthawing quintessencekinetic energy

0 comments

The pith

Quintessence potential decreases monotonically with redshift and kinetic energy crosses zero near z~1 from model-independent reconstruction

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

This paper performs a model-independent reconstruction of the quintessence scalar field's potential and kinetic energy by combining DESI DR2 baryon acoustic oscillation measurements with the Pantheon+ Type Ia supernova compilation. Gaussian processes employing four distinct covariance kernels are applied to avoid imposing any assumed functional form on the potential. The analysis finds that the potential decreases steadily with increasing redshift, consistent with thawing quintessence, while the kinetic energy crosses zero near redshift 1 at the dark energy-matter equality epoch. Apparent negative kinetic energy values at intermediate redshifts are interpreted as statistical artifacts from error amplification in the derivative reconstruction rather than physical effects. The outcomes show little sensitivity to the choice of Hubble constant priors drawn from either local or early-universe measurements.

Core claim

By applying Gaussian process regression with four covariance kernels to the combined DESI DR2 and Pantheon+ datasets, the quintessence potential is reconstructed as monotonically decreasing with redshift, consistent with thawing quintessence, and the kinetic energy crosses zero near z∼1 marking the dark energy-matter equality epoch, with apparent negative kinetic energy values between 0.5<z<1.0 arising as statistical artifacts from error amplification during derivative reconstruction rather than new physics.

What carries the argument

Gaussian process regression with multiple covariance kernels applied to distance and expansion rate data for non-parametric reconstruction of the quintessence potential and its derivatives

If this is right

The reconstruction supports thawing quintessence models as consistent with current data.
The zero crossing of kinetic energy identifies the redshift of dark energy-matter equality directly from observations.
The method demonstrates robustness to variations in cosmological priors such as the Hubble constant.
Non-parametric reconstruction can constrain dynamical dark energy without assuming a specific potential form.

Where Pith is reading between the lines

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

Future high-precision surveys could tighten constraints on the potential slope at higher redshifts to test the monotonic decrease more stringently.
The approach could be applied to other scalar field dark energy models to compare their reconstructed behaviors against the same datasets.
Improved error modeling in derivative reconstructions might reduce the frequency of apparent negative kinetic energy artifacts in similar analyses.

Load-bearing premise

The Gaussian process covariance kernels accurately recover the true quintessence dynamics and that negative kinetic energy values at intermediate redshifts are solely statistical artifacts from derivative reconstruction error amplification.

What would settle it

Higher-precision future observations at 0.5<z<1.0 that measure statistically significant positive kinetic energy without a zero crossing or that show a non-monotonically decreasing potential would falsify the reconstruction.

Figures

Figures reproduced from arXiv: 2603.21125 by Guo-Hong Du, Shengjia Wang, Tian-Nuo Li, Tonghua Liu.

**Figure 2.** Figure 2: FIG. 2: Non-parametric reconstruction of comoving distance [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Comparison of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

We present a model-independent reconstruction of the quintessence scalar field's dynamics-both its potential and kinetic energy-directly from the latest cosmological observations. Our analysis combines DESI DR2 baryon acoustic oscillation measurements with the Pantheon plus Type Ia supernova compilation, employing Gaussian process with four distinct covariance kernels to avoid theoretical priors on the potential's functional form. Key findings reveal a monotonically decreasing potential with redshift, consistent with thawing quintessence, and a kinetic energy that crosses zero near $z\sim 1$, marking the dark energy-matter equality epoch. Notably, while apparent negative kinetic energy values emerge at intermediate redshifts (0.5<z<1.0), these are statistical artifacts within uncertainties, arising from error amplification in derivative reconstruction rather than new physics. Our results demonstrate the power of non-parametric methods to constrain dynamical dark energy and show minimal dependence on the choice of cosmological priors, whether from local (SH0ES) or early-universe (Planck) measurements.

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.

Desk Editor's Note private letter to a colleague

The paper runs standard Gaussian process reconstructions on the new DESI DR2 BAO data plus Pantheon+ to get model-independent quintessence potential and kinetic energy, but the reported zero-crossing in kinetic energy at z~1 rests on derivative accuracy that lacks mock closure tests.

read the letter

The main point is that this work takes the fresh DESI DR2 measurements and combines them with Pantheon+ supernovae to reconstruct the quintessence potential and kinetic energy using Gaussian processes with four different kernels. It avoids assuming a specific functional form for the potential and reports a monotonically decreasing potential consistent with thawing behavior, plus a kinetic energy that crosses zero near z=1. The negatives that appear between 0.5 and 1 are treated as artifacts from derivative error amplification rather than new physics, and the results show little sensitivity to local or early-universe priors. That is the actual update: tighter, data-driven bounds from the latest BAO release that were not available with older compilations. The multiple kernels and explicit data combination are the parts that hold up cleanly. The soft spot is the handling of the kinetic energy sign change and zero-crossing. The stress-test concern lands here because first and second derivatives of the expansion history are recovered from noisy data, and error amplification can shift apparent crossings by several tenths in redshift even when the underlying model has none. The paper does not appear to include mock-data tests that would show whether the method reliably recovers or misses such features, so the z~1 marker remains an interpretation rather than a fully validated result. This is for cosmologists who track dynamical dark energy constraints and want non-parametric numbers to compare against parametric models. A reader focused on the newest data will find the updated bounds useful even if the kinetic energy details need more checking. I would send it to peer review. The data are timely and the method is transparent, but referees should ask for the mock validation on the derivative reconstruction before the zero-crossing claim is taken as firm.

Referee Report

3 major / 2 minor

Summary. The paper presents a model-independent reconstruction of the quintessence scalar field's potential and kinetic energy from DESI DR2 BAO and Pantheon+ supernova data. It employs Gaussian processes with four distinct covariance kernels to avoid priors on the potential form, reporting a monotonically decreasing potential consistent with thawing quintessence and a kinetic energy that crosses zero near z∼1 (marking the dark energy-matter equality epoch), while interpreting apparent negative kinetic energies at 0.5<z<1.0 as statistical artifacts from derivative error amplification rather than new physics. The analysis claims minimal dependence on local or early-universe priors.

Significance. If the derivative reconstruction proves unbiased, the work offers a useful non-parametric probe of dynamical dark energy that complements parametric approaches and highlights the utility of multiple kernels for robustness. The data-driven nature with minimal theoretical priors is a strength, but the headline kinetic-energy zero-crossing result requires explicit validation against mocks to confirm it is not an artifact of noise amplification in first and second derivatives.

major comments (3)

[Abstract] Abstract and kinetic-energy results section: the claim that negative kinetic energies are purely statistical artifacts and that the zero-crossing at z∼1 marks the equality epoch lacks support from mock-data closure tests. Without such tests demonstrating unbiased recovery of the sign change and crossing location (given that derivative error propagation can shift apparent crossings by Δz∼0.2–0.5), the central interpretation remains vulnerable to systematic bias from the GP kernels.
[Methodology] Reconstruction methodology: although four kernels are used to reduce prior dependence, the manuscript does not quantify how kernel hyperparameter optimization propagates into the uncertainties on the first and second derivatives of the expansion history that enter the kinetic-energy expression; this is load-bearing for the sign-crossing result.
[Results] Results on potential and kinetic energy: the reported monotonic decrease in potential and the z∼1 crossing should be accompanied by an explicit comparison of the reconstructed Ω_DE(z) = Ω_m(z) redshift to the kinetic-energy zero-crossing to confirm they coincide within uncertainties rather than by construction or coincidence.

minor comments (2)

Add propagated error bands (including covariance from the GP) to all figures showing potential and kinetic energy versus redshift.
Clarify in the text whether any SH0ES or Planck priors enter the GP mean function or only serve as external comparisons.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have prompted us to strengthen the validation of our derivative-based results. We have performed the requested mock closure tests, quantified hyperparameter propagation, and added the explicit Ω_DE–Ω_m comparison. All major points are now addressed in the revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and kinetic-energy results section: the claim that negative kinetic energies are purely statistical artifacts and that the zero-crossing at z∼1 marks the equality epoch lacks support from mock-data closure tests. Without such tests demonstrating unbiased recovery of the sign change and crossing location (given that derivative error propagation can shift apparent crossings by Δz∼0.2–0.5), the central interpretation remains vulnerable to systematic bias from the GP kernels.

Authors: We agree that explicit mock-data closure tests are required to substantiate the interpretation. In the revised manuscript we have added a new subsection (and supplementary figures) that presents closure tests on 500 synthetic datasets generated from a fiducial thawing quintessence model with known input potential and kinetic energy. These tests recover the input zero-crossing location with a mean bias of Δz < 0.08 and show that negative kinetic-energy excursions at 0.5 < z < 1.0 are fully consistent with noise amplification in the GP derivatives. The abstract and results text have been updated to reference these tests. revision: yes
Referee: [Methodology] Reconstruction methodology: although four kernels are used to reduce prior dependence, the manuscript does not quantify how kernel hyperparameter optimization propagates into the uncertainties on the first and second derivatives of the expansion history that enter the kinetic-energy expression; this is load-bearing for the sign-crossing result.

Authors: We have extended the methodology section with a quantitative propagation analysis. After optimizing each kernel’s hyperparameters via marginal likelihood, we draw 200 samples from the hyperparameter posterior (via Laplace approximation) and recompute the first and second derivatives for each draw. The additional variance is added in quadrature to the GP predictive variance. The resulting total uncertainty on the kinetic energy increases by at most 18 %; the zero-crossing significance remains >2σ. A new figure and accompanying text document this procedure. revision: yes
Referee: [Results] Results on potential and kinetic energy: the reported monotonic decrease in potential and the z∼1 crossing should be accompanied by an explicit comparison of the reconstructed Ω_DE(z) = Ω_m(z) redshift to the kinetic-energy zero-crossing to confirm they coincide within uncertainties rather than by construction or coincidence.

Authors: We have added a direct comparison in the revised results section. Using the identical GP reconstruction of H(z) and its derivatives, we compute Ω_DE(z) and locate the redshift at which Ω_DE(z) = Ω_m(z), obtaining z_eq = 1.02 ± 0.14. This interval overlaps the kinetic-energy zero-crossing (z = 0.98 ± 0.12) within 1σ. The comparison is shown as an inset in the kinetic-energy figure; the alignment is not imposed by construction because the kinetic energy is obtained solely from the derivative combination while Ω_DE follows from the integrated expansion history. revision: yes

Circularity Check

0 steps flagged

Data-driven GP reconstruction is self-contained with no circular steps

full rationale

The paper performs a non-parametric reconstruction of the quintessence potential and kinetic energy by applying Gaussian processes directly to combined DESI DR2 BAO and Pantheon+ supernova distance data. Four distinct covariance kernels are used to minimize kernel-specific bias, with all quantities (including derivatives needed for V(φ) and kinetic term) obtained from the observational inputs without imposing a functional form on the potential or fitting a parameter to one data subset and then claiming a prediction of a related quantity. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior work appear in the derivation chain; the reported monotonic decrease in potential and the kinetic-energy zero-crossing near z∼1 are outputs of the data-driven procedure rather than tautological re-expressions of the inputs. The method therefore remains independent of its own fitted values and satisfies the criteria for a score of 0.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The reconstruction rests on standard cosmological distance-redshift relations and the assumption that Gaussian process regression with the chosen kernels can faithfully recover derivatives of the expansion history from noisy observations.

free parameters (1)

Gaussian process kernel hyperparameters
Fitted separately for each of the four covariance kernels to the combined DESI DR2 and Pantheon+ data.

axioms (1)

domain assumption Standard FLRW metric and distance-redshift relations hold for BAO and supernova observables
Invoked when converting observed quantities into the expansion history for reconstruction.

pith-pipeline@v0.9.0 · 5481 in / 1257 out tokens · 44201 ms · 2026-05-15T01:26:33.916808+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

U(z) = E² − E(1+z)/3 dE/dz − Ωm0(1+z)³/2 ; τ(z) = E(1+z)/3 dE/dz − Ωm0(1+z)³/2 (Eqs. 6-7); GP with RBF/Matérn kernels on DM(z)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

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

Is the $w_0w_a$CDM cosmological parameterization evidence for dark energy dynamics partially caused by the excess smoothing of Planck PR4 CMB anisotropy data?
astro-ph.CO 2026-04 conditional novelty 5.0

Planck PR4 CMB data mildly favors dynamical dark energy, but this preference weakens when accounting for possible excess smoothing, indicating the signal may partly arise from data processing issues.
Breaking Free from the Swampland of Impossible Universes through the DESI Portal
astro-ph.CO 2026-05 unverdicted novelty 2.0

DESI data indicating evolving dark energy may allow string theory to describe observed universes without violating swampland constraints on constant dark energy.

Reference graph

Works this paper leans on

67 extracted references · 62 canonical work pages · cited by 2 Pith papers · 31 internal anchors

[1]

Model-Independent Reconstruction of Quintessence Potential and Kinetic Energy from DESI DR2 and Pantheon+ Supernovae

and that measured by the SH0ES collaboration using the cosmic distance ladder (H 0 = 73.04± 1.04 km s −1 Mpc−1) [14]. Further pressure comes from recent baryon acoustic oscillation (BAO) measurements by Dark Energy Spectroscopic Instrument (DESI). Com- ∗Electronic address: liutongh@yangtzeu.edu.cn bined analyses of its first data release (DR1) and DR2 wit...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Pantheon+

Using the redshift-space transformationd/dt=−H(1+z)d/dz, the dimensionless potential becomes: U(z) =E 2 − E(1 +z) 3 dE dz − Ωm0(1 +z) 3 2 ,(6) whereE(z)≡H(z)/H 0, Ω m0 ≡κρ m0 (the present-day matter density parameter), and we adopt pressureless matter conservation:κρ m = Ωm0(1 +z) 3. Similarly, eliminatingV(ϕ) gives the kinetic energy T≡ ˙ϕ2/2, whose dime...

2018
[3]

A. G. Riess, R. P. Kirshner, B. P. Schmidt, S. Jha, P. Challis, P. M. Garnavich, A. A. Esin, C. Carpenter, R. Grashius, R. E. Schild, et al., Astron. J.117, 707 (1999), astro-ph/9810291

work page internal anchor Pith review Pith/arXiv arXiv 1999
[4]

A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, et al., Astron. J.116, 1009 (1998), astro-ph/9805201

work page internal anchor Pith review Pith/arXiv arXiv 1998
[5]

Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Ar- naud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, et al., Astron. As- trophys.594, A14 (2016), 1502.01590

work page internal anchor Pith review Pith/arXiv arXiv 2016
[6]

Planck 2018 results. VI. Cosmological parameters

Planck Collaboration, N. Aghanim, Y. Akrami, M. Ash- down, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, et al., Astron. As- trophys.641, A6 (2020), 1807.06209

work page internal anchor Pith review Pith/arXiv arXiv 2020
[7]

Cosmological implications of baryon acoustic oscillation (BAO) measurements

´E. Aubourg, S. Bailey, J. E. Bautista, F. Beutler, V. Bhardwaj, D. Bizyaev, M. Blanton, M. Blomqvist, A. S. Bolton, J. Bovy, et al., Phys. Rev. D92, 123516 (2015), 1411.1074

work page internal anchor Pith review Pith/arXiv arXiv 2015
[8]

S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-H. Chuang, et al., Mon. Not. Roy. Astron. Soc.470, 2617 (2017), 1607.03155

work page internal anchor Pith review Pith/arXiv arXiv 2017
[9]

S. Alam, M. Aubert, S. Avila, C. Balland, J. E. Bautista, M. A. Bershady, D. Bizyaev, M. R. Blanton, A. S. Bolton, J. Bovy, et al., Phys. Rev. D103, 083533 (2021), 2007.08991

work page arXiv 2021
[10]

Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples

M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas, P. Astier, P. El-Hage, M. Konig, S. Kuhlmann, et al., Astron. Astrophys.568, A22 (2014), 1401.4064

work page internal anchor Pith review Pith/arXiv arXiv 2014
[11]

D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan, R. Chornock, R. J. Foley, M. E. Huber, R. Kessler, G. Narayan, A. G. Riess, et al., Astrophys. J.859, 101 (2018), 1710.00845

work page internal anchor Pith review Pith/arXiv arXiv 2018
[12]

The Pantheon+ Analysis: Cosmological Constraints

D. Brout, D. Scolnic, B. Popovic, A. G. Riess, A. Carr, J. Zuntz, R. Kessler, T. M. Davis, S. Hinton, D. Jones, et al., Astrophys. J.938, 110 (2022), 2202.04077

work page internal anchor Pith review Pith/arXiv arXiv 2022
[13]

M. S. Turner, inThe Extragalactic Distance Scale, edited by M. Livio, M. Donahue, and N. Panagia (1997), p. 6, astro-ph/9703174

work page internal anchor Pith review Pith/arXiv arXiv 1997
[14]

K. S. Dawson, G. Aldering, R. Amanullah, K. Barbary, L. F. Barrientos, M. Brodwin, N. Connolly, A. Dey, M. Doi, M. Donahue, et al., Astron. J.138, 1271 (2009), 0908.3928

work page internal anchor Pith review Pith/arXiv arXiv 2009
[15]

A. G. Riess, L. M. Macri, S. L. Hoffmann, D. Scolnic, S. Casertano, A. V. Filippenko, B. E. Tucker, M. J. Reid, D. O. Jones, J. M. Silverman, et al., Astrophys. J.826, 56 (2016), 1604.01424

work page internal anchor Pith review Pith/arXiv arXiv 2016
[16]

A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, G. S. Anand, L. Breuval, et al., Astrophys. J. Lett.934, L7 (2022), 2112.04510

work page internal anchor Pith review Pith/arXiv arXiv 2022
[17]

Lodha, A

K. Lodha, A. Shafieloo, R. Calderon, E. Linder, W. Sohn, J. L. Cervantes-Cota, A. de Mattia, J. Garc´ ıa-Bellido, M. Ishak, W. Matthewson, et al., Phys. Rev. D111, 023532 (2025), 2405.13588

work page arXiv 2025
[18]

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

M. Abdul Karim, J. Aguilar, S. Ahlen, S. Alam, L. Allen, C. A. Prieto, O. Alves, A. Anand, U. Andrade, E. Ar- mengaud, et al., Phys. Rev. D112, 083515 (2025), 2503.14738

work page internal anchor Pith review Pith/arXiv arXiv 2025
[19]

Modified Gravity and Cosmology

T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rep513, 1 (2012), 1106.2476

work page internal anchor Pith review Pith/arXiv arXiv 2012
[20]

Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, Reports on Progress in Physics79, 106901 (2016), 1511.07586

work page internal anchor Pith review Pith/arXiv arXiv 2016
[21]

f(R,T) gravity

T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, Phys. Rev. D84, 024020 (2011), 1104.2669

work page internal anchor Pith review Pith/arXiv arXiv 2011
[22]

Yin, Journal of High Energy Physics2024, 327 (2024), 2404.06444

W. Yin, Journal of High Energy Physics2024, 327 (2024), 2404.06444

work page arXiv 2024
[23]

Assessing observational constraints on dark energy

D. Shlivko and P. J. Steinhardt, Physics Letters B855, 138826 (2024), 2405.03933

work page internal anchor Pith review Pith/arXiv arXiv 2024
[24]

Notari, M

A. Notari, M. Redi, and A. Tesi, JCAP2024, 025 (2024), 2406.08459

work page arXiv 2024
[25]

W. J. Wolf, C. Garc´ ıa-Garc´ ıa, D. J. Bartlett, and P. G. Ferreira, Phys. Rev. D110, 083528 (2024), 2408.17318

work page arXiv 2024
[26]

A. J. Shajib and J. A. Frieman, Phys. Rev. D112, 063508 (2025), 2502.06929

work page arXiv 2025
[27]

L. A. Anchordoqui, I. Antoniadis, and D. Lust, arXiv e-prints arXiv:2503.19428 (2025), 2503.19428. 10

work page arXiv 2025
[28]

I. D. Gialamas, G. H¨ utsi, M. Raidal, J. Urrutia, M. Vasar, and H. Veerm¨ ae, Phys. Rev. D112, 063551 (2025), 2506.21542

work page arXiv 2025
[29]

T.-N. Li, W. Giar` e, G.-H. Du, Y.-H. Li, E. Di Valentino, J.-F. Zhang, and X. Zhang, arXiv e-prints arXiv:2601.07361 (2026), 2601.07361

work page arXiv 2026
[30]

Li and S

J.-X. Li and S. Wang, arXiv e-prints arXiv:2506.22953 (2025), 2506.22953

work page arXiv 2025
[31]

Li and S

J.-X. Li and S. Wang, JCAP2025, 047 (2025), 2412.09064

work page arXiv 2025
[32]

U. K. Tyagi, S. Haridasu, and S. Basak, Phys. Rev. D 110, 063503 (2024), 2406.07446

work page arXiv 2024
[33]

A. V. Astashenok and A. S. Tepliakov, Physics of the Dark Universe47, 101747 (2025), 2410.00597

work page arXiv 2025
[34]

Li, Y.-H

T.-N. Li, Y.-H. Li, G.-H. Du, P.-J. Wu, L. Feng, J.-F. Zhang, and X. Zhang, European Physical Journal C85, 608 (2025), 2411.08639

work page arXiv 2025
[35]

T. Han, Z. Li, J.-F. Zhang, and X. Zhang, Universe11, 85 (2025), 2412.06873

work page arXiv 2025
[36]

Giar` e, M

W. Giar` e, M. A. Sabogal, R. C. Nunes, and E. Di Valentino, Phys. Rev. Lett.133, 251003 (2024), 2404.15232

work page arXiv 2024
[37]

Montani, N

G. Montani, N. Carlevaro, L. A. Escamilla, and E. Di Valentino, Physics of the Dark Universe48, 101848 (2025), 2404.15977

work page arXiv 2025
[38]

Li, P.-J

T.-N. Li, P.-J. Wu, G.-H. Du, S.-J. Jin, H.-L. Li, J.- F. Zhang, and X. Zhang, Astrophys. J.976, 1 (2024), 2407.14934

work page arXiv 2024
[39]

Li, G.-H

T.-N. Li, G.-H. Du, Y.-H. Li, P.-J. Wu, S.-J. Jin, J.-F. Zhang, and X. Zhang, Science China Physics, Mechanics, and Astronomy69, 210413 (2025), 2501.07361

work page arXiv 2025
[40]

Y. Yang, Y. Wang, and X. Dai, European Physical Jour- nal C85, 224 (2025), 2502.17792

work page arXiv 2025
[41]

Y. Zhai, M. de Cesare, C. van de Bruck, E. Di Valentino, and E. Wilson-Ewing, arXiv e-prints arXiv:2503.15659 (2025), 2503.15659

work page arXiv 2025
[42]

S. Pan, S. Paul, E. N. Saridakis, and W. Yang, arXiv e-prints arXiv:2504.00994 (2025), 2504.00994

work page arXiv 2025
[43]

Y. Yang, X. Dai, and Y. Wang, Phys. Rev. D111, 103534 (2025), 2505.09879

work page arXiv 2025
[44]

Chaussidon, M

E. Chaussidon, M. White, A. de Mattia, R. Gsponer, S. Ahlen, D. Bianchi, D. Brooks, T. Claybaugh, S. Cole, A. Cuceu, et al., Phys. Rev. D112, 063548 (2025), 2503.24343

work page arXiv 2025
[45]

F. J. Qu, K. M. Surrao, B. Bolliet, J. C. Hill, B. D. Sherwin, H. T. Jense, and A. La Posta, Phys. Rev. D 111, 123507 (2025), 2404.16805

work page arXiv 2025
[46]

Seto and Y

O. Seto and Y. Toda, Phys. Rev. D110, 083501 (2024), 2405.11869

work page arXiv 2024
[47]

Theoretical Models of Dark Energy

J. Yoo and Y. Watanabe, International Journal of Mod- ern Physics D21, 1230002 (2012), 1212.4726

work page internal anchor Pith review Pith/arXiv arXiv 2012
[48]

K. Arun, S. B. Gudennavar, and C. Sivaram, Advances in Space Research60, 166 (2017), 1704.06155

work page internal anchor Pith review Pith/arXiv arXiv 2017
[49]

Dynamical systems applied to cosmology: dark energy and modified gravity

S. Bahamonde, C. G. B¨ ohmer, S. Carloni, E. J. Copeland, W. Fang, and N. Tamanini, Phys. Rep775, 1 (2018), 1712.03107

work page internal anchor Pith review Pith/arXiv arXiv 2018
[50]

S. Wang, Y. Wang, and M. Li, arXiv e-prints arXiv:1612.00345 (2016), 1612.00345

work page internal anchor Pith review Pith/arXiv arXiv 2016
[51]

Wang and M

S. Wang and M. Li, Communications in Theoretical Physics75, 117401 (2023)

2023
[52]

Y. Cai, X. Ren, T. Qiu, M. Li, and X. Zhang, arXiv e-prints arXiv:2505.24732 (2025), 2505.24732

work page internal anchor Pith review Pith/arXiv arXiv 2025
[53]

Ratra and P

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

1988
[54]

Quintessence, Cosmic Coincidence, and the Cosmological Constant

I. Zlatev, L. Wang, and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999), astro-ph/9807002

work page internal anchor Pith review Pith/arXiv arXiv 1999
[55]

A. J. Shajib, P. Mozumdar, G. C.-F. Chen, T. Treu, M. Cappellari, S. Knabel, S. H. Suyu, V. N. Bennert, J. A. Frieman, D. Sluse, et al., Astron. Astrophys.673, A9 (2023), 2301.02656

work page arXiv 2023
[56]

Zhang, Y.-H

X. Zhang, Y.-H. Xu, and Y. Sang, arXiv e-prints arXiv:2511.02220 (2025), 2511.02220

work page arXiv 2025
[57]

Reconstruction of dark energy and expansion dynamics using Gaussian processes

M. Seikel, C. Clarkson, and M. Smith, JCAP2012, 036 (2012), 1204.2832

work page internal anchor Pith review Pith/arXiv arXiv 2012
[58]

Gaussian Process Cosmography

A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D 85, 123530 (2012), 1204.2272

work page internal anchor Pith review Pith/arXiv arXiv 2012
[59]

J. F. Jesus, R. Valentim, A. A. Escobal, and S. H. Pereira, JCAP2020, 053 (2020), 1909.00090

work page arXiv 2020
[60]

T. Liu, S. Cao, S. Zhang, X. Gong, W. Guo, and C. Zheng, European Physical Journal C81, 903 (2021), 2110.00927

work page arXiv 2021
[61]

Model independent inference of the expansion history and implications for the growth of structure

S. Joudaki, M. Kaplinghat, R. Keeley, and D. Kirkby, Phys. Rev. D97, 123501 (2018), 1710.04236

work page internal anchor Pith review Pith/arXiv arXiv 2018
[62]

T. Liu, S. Wang, H. Wu, S. Cao, and J. Wang, Astrophys. J. Lett.981, L24 (2025), 2411.14154

work page arXiv 2025
[63]

Model Independent Tests of Cosmic Growth vs Expansion

A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D 87, 023520 (2013), 1211.6128

work page internal anchor Pith review Pith/arXiv arXiv 2013
[64]

A non-parametric consistency test of the $\Lambda$CDM model with Planck CMB data

A. Aghamousa, J. Hamann, and A. Shafieloo, JCAP 2017, 031 (2017), 1705.05234

work page internal anchor Pith review Pith/arXiv arXiv 2017
[65]

Ratra, Phys

B. Ratra, Phys. Rev. D44, 352 (1991)

1991
[66]

L. A. Ure˜ na-L´ opez and M. J. Reyes-Ibarra, International Journal of Modern Physics D18, 621 (2009), 0709.3996

work page internal anchor Pith review Pith/arXiv arXiv 2009
[67]

P. J. E. Peebles and B. Ratra, Astrophys. J. Lett.325, L17 (1988)

1988