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arxiv: 2606.12464 · v1 · pith:AO6EGDBVnew · submitted 2026-06-09 · ⚛️ physics.gen-ph

Kinematic Probes of Type-II MMG: Pad\'e Cosmographic Analysis of VCDM

Soumya Kanta Bhoi , Sai Swagat Mishra , P.K. Sahoo This is my paper

Pith reviewed 2026-06-27 10:50 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords VCDM modelcosmographyPadé approximationΛCDMcosmic chronometersBAOType Ia supernovaejerk parameter
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The pith

VCDM reproduces the ΛCDM background expansion under Padé cosmographic reconstruction from current data.

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

The paper examines the late-time expansion of the Universe in the VCDM model, a minimal Hamiltonian modification of gravity that generates a time-dependent vacuum sector without extra degrees of freedom. It applies a Padé P(2,1) approximation to the Hubble parameter and luminosity distance so that cosmographic parameters can be written directly in terms of the VCDM parameters. A Bayesian MCMC analysis with cosmic chronometers, DESI BAO, and Type Ia supernova data (Union3, Pantheon+, DESY5) yields tight parameter constraints, a jerk parameter near its ΛCDM value of 1, and no evidence of a previously reported transition feature. The results indicate that VCDM mimics ΛCDM at the background level and that the transition is sensitive to the reconstruction parametrization rather than required by the data.

Core claim

Within the VCDM framework, a Type-II MMG realization that preserves the successes of General Relativity while extending beyond constant vacuum energy through a minimal Hamiltonian modification without introducing additional degrees of freedom, the Padé P(2,1) cosmographic reconstruction of the expansion history from joint datasets yields parameters consistent with ΛCDM, with j0 ≈ 1 and no observed transition feature, indicating that such a feature is sensitive to parametrization choice rather than a robust data requirement.

What carries the argument

Padé P(2,1) approximation for the Hubble parameter and luminosity distance, which expresses cosmographic parameters directly in terms of VCDM model parameters for data-driven reconstruction of the expansion history.

If this is right

  • The VCDM model parameters are tightly constrained and consistent across different dataset combinations.
  • The jerk parameter remains very close to its ΛCDM value, indicating no significant deviation at higher-order cosmography.
  • The transition feature previously reported in VCDM is absent, showing it is not a robust requirement of current observational data.
  • Model-independent reconstructions are important for assessing whether alternative cosmological scenarios differ from ΛCDM at the background level.

Where Pith is reading between the lines

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

  • Other reconstruction methods or higher-order terms might reveal features that the P(2,1) choice suppresses.
  • The same cosmographic approach could be applied to additional modified-gravity models to test for similar background-level mimicry.
  • Future datasets with higher precision at intermediate redshifts could distinguish VCDM from ΛCDM even under this reconstruction.

Load-bearing premise

The Padé P(2,1) approximation for the Hubble parameter and luminosity distance is sufficient to reconstruct the expansion history without biasing against detection of the transition feature reported in prior VCDM work.

What would settle it

A clear transition feature appearing in the VCDM expansion history when the same datasets are analyzed with a different cosmographic parametrization or higher-order Padé approximant would falsify the claim that the feature is not robust.

Figures

Figures reproduced from arXiv: 2606.12464 by P.K. Sahoo, Sai Swagat Mishra, Soumya Kanta Bhoi.

Figure 1
Figure 1. Figure 1: FIG. 1: Joint posterior distributions of the parameter space [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evolution of the Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Distance modulus [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Constraints in the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

We study the late-time expansion history of the Universe within the VCDM model, a Type-II MMG realization that preserves the successes of General Relativity while extending beyond constant vacuum energy through a minimal Hamiltonian modification, generating a time-dependent vacuum sector without introducing additional degrees of freedom. We investigate this framework within a cosmographic approach by employing a Pad\'e $P_{(2,1)}$ approximation for the Hubble parameter and luminosity distance, allowing the cosmographic parameters to be expressed directly in terms of the underlying VCDM model parameters and enabling a data-driven reconstruction of the expansion history. The model is constrained within a Bayesian framework using the MCMC technique, implemented via the affine-invariant ensemble sampler, with a joint analysis of cosmic chronometers, DESI BAO, and Type Ia supernova datasets (Union3, Pantheon+, and DESY5). We find that the model parameters are tightly constrained and consistent across different dataset combinations, with the jerk parameter remaining very close to its $\Lambda$CDM value, $j_0 \simeq 1$, indicating no significant deviation at the level of higher-order cosmography. Furthermore, the transition feature previously reported in VCDM is not observed within the Pad\'e $P_{(2,1)}$ cosmographic reconstruction, suggesting that it is not a robust requirement of current observational data but is sensitive to the choice of parametrization. Overall, our results indicate that the VCDM model effectively mimics $\Lambda$CDM at the background level when constrained through a cosmographic approach, underscoring the importance of model-independent reconstructions in assessing alternative cosmological scenarios.

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

2 major / 2 minor

Summary. The paper claims that a Padé P(2,1) cosmographic reconstruction of the Hubble parameter and luminosity distance in the VCDM (Type-II MMG) model, when constrained via MCMC on cosmic chronometers, DESI BAO, and Type Ia supernova datasets (Union3, Pantheon+, DESY5), yields tightly constrained parameters with jerk parameter j0 ≃ 1, shows no evidence for the transition feature reported in prior VCDM work, and indicates that VCDM effectively mimics ΛCDM at the background level; this is taken to imply that the transition is parametrization-sensitive rather than data-driven.

Significance. If the central claim holds, the work demonstrates the value of model-independent cosmographic methods for testing minimal modifications to GR that preserve background successes while altering the vacuum sector. Credit is due for the joint analysis across named datasets (cosmic chronometers + DESI BAO + three SN compilations) using an affine-invariant ensemble sampler, which supports the consistency conclusion at the level of current data.

major comments (2)
  1. [§3] §3 (Padé P(2,1) construction for H(z) and d_L(z)): the central claim that the transition feature 'is not a robust requirement of current observational data but is sensitive to the choice of parametrization' is load-bearing on the approximant being flexible enough to recover the feature if present. A (2,1) Padé has only three free coefficients after normalization; the manuscript should demonstrate, e.g., via mock-data injection of the prior-reported VCDM transition, that the form can accommodate it before concluding non-detection implies absence.
  2. [§4] §4 (MCMC implementation and priors): the abstract and results sections report tight constraints and consistency across dataset combinations, but no explicit statement of priors, convergence diagnostics (e.g., Gelman-Rubin statistic), or error propagation from the cosmographic coefficients to the VCDM parameters is provided; this limits verification that the posterior truly rules out the transition rather than being prior- or sampler-dominated.
minor comments (2)
  1. Abstract: the description of the MCMC setup omits priors and convergence checks; moving a brief statement of these to the abstract would improve reproducibility without lengthening the text.
  2. Notation: the subscript notation P_{(2,1)} is used inconsistently with standard P(2,1) in some equations; uniform adoption of one form would aid clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Dear Editor, We thank the referee for the constructive and detailed report. The two major comments identify genuine gaps in the current manuscript that we will address through targeted revisions. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [§3] §3 (Padé P(2,1) construction for H(z) and d_L(z)): the central claim that the transition feature 'is not a robust requirement of current observational data but is sensitive to the choice of parametrization' is load-bearing on the approximant being flexible enough to recover the feature if present. A (2,1) Padé has only three free coefficients after normalization; the manuscript should demonstrate, e.g., via mock-data injection of the prior-reported VCDM transition, that the form can accommodate it before concluding non-detection implies absence.

    Authors: We agree that the flexibility of the P(2,1) approximant must be explicitly verified before interpreting the non-detection as evidence that the transition is parametrization-dependent rather than data-driven. In the revised manuscript we will add a dedicated mock-data section: we will generate synthetic datasets that include the transition feature reported in earlier VCDM analyses, with noise levels matching the real chronometer, DESI BAO and supernova compilations, and then demonstrate that the same P(2,1) reconstruction recovers the injected transition at high significance. This test will be presented alongside the real-data results to substantiate the claim. revision: yes

  2. Referee: [§4] §4 (MCMC implementation and priors): the abstract and results sections report tight constraints and consistency across dataset combinations, but no explicit statement of priors, convergence diagnostics (e.g., Gelman-Rubin statistic), or error propagation from the cosmographic coefficients to the VCDM parameters is provided; this limits verification that the posterior truly rules out the transition rather than being prior- or sampler-dominated.

    Authors: We acknowledge these omissions. The revised manuscript will include: (i) the full set of prior distributions adopted for the cosmographic coefficients and the derived VCDM parameters; (ii) Gelman–Rubin R-hat statistics (and any additional convergence checks) for all chains; and (iii) an explicit description of how uncertainties on the fitted Padé coefficients are propagated to the VCDM parameters. These additions will be placed in a new subsection of §4 and will be referenced in the abstract and results sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constraints are data-driven with external datasets

full rationale

The paper's central results derive from MCMC constraints on VCDM parameters using independent observational datasets (cosmic chronometers, DESI BAO, Union3/Pantheon+/DESY5 SNIa). The Padé P(2,1) form is adopted as an approximation to express cosmographic quantities in terms of model parameters, but the posterior and the non-detection of the transition feature are outputs of the fit rather than reductions by construction. No self-citations are invoked as load-bearing uniqueness theorems, no fitted inputs are relabeled as predictions, and the derivation chain remains self-contained against external benchmarks without definitional loops.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only access prevents enumeration of specific free parameters in the VCDM model (such as any time-dependent vacuum parameters) or background axioms; the model is described as a minimal Hamiltonian modification generating time-dependent vacuum without additional degrees of freedom.

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discussion (0)

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Reference graph

Works this paper leans on

66 extracted references · 55 canonical work pages · 28 internal anchors

  1. [1]

    in whichj 0 is introduced only as a free kinematic quantity. d.Physical Implications:The present results indicate that, at the background level, VCDM remains observa- tionally close toΛCDM, similar to other modified gravity scenarios constructed to mimic the concordance expansion history while modifying the underlying dynamics [60, 61]. This is not a weak...

  2. [2]

    A. G. Riesset al.(Supernova Search Team), Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J.116, 1009 (1998), arXiv:astro-ph/9805201

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  3. [3]

    Cosmology from Type Ia Supernovae

    S. Perlmutteret al.(Supernova Cosmology Project), Cosmology from Type Ia supernovae, Bull. Am. Astron. Soc.29, 1351 (1997), arXiv:astro-ph/9812473

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  4. [4]

    Planck 2018 results. VI. Cosmological parameters

    P . Collaboration, Planck 2018 results. vi. cosmological parameters, A&A (2020), 1807.06209

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    Rosenberg,CMB analysis with ACT and Planck, Ph.D

    E. Rosenberg,CMB analysis with ACT and Planck, Ph.D. thesis, Institute of Astronomy, Cambridge U. (2023)

    2023

  6. [6]

    Beyond $\Lambda$CDM: Problems, solutions, and the road ahead

    P . Bullet al., BeyondΛCDM: Problems, solutions, and the road ahead, Phys. Dark Univ.12, 56 (2016), arXiv:1512.05356 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    Weinberg, The Cosmological Constant Problem, Rev

    S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys.61, 1 (1989)

    1989

  8. [8]

    Zlatev, L

    I. Zlatev, L. Wang, and P . Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant, Physical Review Letters82, 896 (1999)

    1999

  9. [9]

    A. G. e. a. Riess, A comprehensive measurement of the local value of the hubble constant, ApJ Letters (2022), arXiv:2112.04510

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  10. [10]

    Verde, T

    L. Verde, T. Treu, and A. G. Riess, Tensions between the early and late universe, Nature Astronomy 10.1038/s41550-019-0902- 0 (2019)

    work page doi:10.1038/s41550-019-0902- 2019

  11. [11]

    In the Realm of the Hubble tension $-$ a Review of Solutions

    E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, In the realm of the Hubble tension—a review of solutions, Class. Quant. Grav.38, 153001 (2021), arXiv:2103.01183 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  12. [12]

    N. S. Kavya, S. Swagat Mishra, and P . K. Sahoo, f(Q) gravity as a possible resolution of the H0 and S8 tensions with DESI DR2, Sci. Rep.15, 36504 (2025)

    2025

  13. [13]

    Modified Gravity and Cosmology

    T. Clifton, P . G. Ferreira, A. Padilla, and C. Skordis, Modified Gravity and Cosmology, Phys. Rept.513, 1 (2012), arXiv:1106.2476 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  14. [14]

    Beyond the Cosmological Standard Model

    A. Joyce, B. Jain, J. Khoury, and M. Trodden, Beyond the Cosmological Standard Model, Phys. Rept.568, 1 (2015), arXiv:1407.0059 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  15. [15]

    The Geometrical Trinity of Gravity

    J. Beltr ´an Jim ´enez, L. Heisenberg, and T. S. Koivisto, The Geometrical Trinity of Gravity, Universe5, 173 (2019), arXiv:1903.06830 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  16. [16]

    S. S. Mishra, S. Patel, and P . K. Sahoo, BBN to late-time acceleration in f(T,Lm) gravity, Phys. Lett. B872, 140098 (2026), arXiv:2603.11760 [gr-qc]

    work page arXiv 2026

  17. [17]

    A theory of type-II minimally modified gravity,

    A. De Felice, A. Doll, and S. Mukohyama, A theory of type-II minimally modified gravity, JCAP09, 034, arXiv:2004.12549 [gr-qc]

    work page arXiv 2004

  18. [18]

    Akarsu, A

    ¨O. Akarsu, A. De Felice, E. Di Valentino, S. Kumar, R. C. Nunes, E. ¨Oz ¨ulker, J. A. Vazquez, and A. Yadav, Cosmological constraints onΛsCDM scenario in a type II minimally modified gravity, Phys. Rev. D110, 103527 (2024), arXiv:2406.07526 [astro-ph.CO]

    work page arXiv 2024

  19. [19]

    $\Lambda_{\rm s}$CDM cosmology from a type-II minimally modified gravity

    ¨O. Akarsu, A. De Felice, E. Di Valentino, S. Kumar, R. C. Nunes, E.¨Oz ¨ulker, J. A. Vazquez, and A. Yadav,ΛsCDM cosmology from a type-II minimally modified gravity, Mon. Not. Roy. Astron. Soc.546, staf2276 (2026), arXiv:2402.07716 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  20. [20]

    Dynamical dark energy parametrizations in VCDM,

    S. Arora, A. De Felice, and S. Mukohyama, Dynamical dark energy parametrizations in VCDM, Phys. Rev. D112, 123518 (2025), arXiv:2508.03784 [gr-qc]

    work page arXiv 2025

  21. [21]

    Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(John Wiley and Sons, New York, 1972)

    S. Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(John Wiley and Sons, New York, 1972)

    1972

  22. [22]

    Visser, Jerk, snap and the cosmological equation of state, Classical and Quantum Gravity21, 2603 (2004)

    M. Visser, Jerk, snap and the cosmological equation of state, Classical and Quantum Gravity21, 2603 (2004)

    2004

  23. [23]

    Cosmography in f(T)-gravity

    S. Capozziello, V . F. Cardone, H. Farajollahi, and A. Ravanpak, Cosmography in f(T)-gravity, Phys. Rev. D84, 043527 (2011), arXiv:1108.2789 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  24. [24]

    Zimdahl and D

    W. Zimdahl and D. Pavon, Statefinder parameters for interacting dark energy, Gen. Rel. Grav.36, 1483 (2004), arXiv:gr- qc/0311067

    work page arXiv 2004

  25. [25]

    A. C. Alfano and O. Luongo, Cosmic distance duality after DESI 2024 data release and dark energy evolution, Phys. Dark Univ.51, 102205 (2026), arXiv:2501.15233 [astro-ph.CO]

    work page arXiv 2024

  26. [26]

    S. S. Mishra, N. S. Kavya, P . K. Sahoo, and K. Bamba, DESI DR2 meets cosmography: a comparative study of Pad ´e, Cheby- shev, and Taylor expansions, Mon. Not. Roy. Astron. Soc.546, stag197 (2026)

    2026

  27. [27]

    Rodrigues, R

    G. Rodrigues, R. de Souza, and J. Alcaniz, Cosmography with DESI DR2 and SN data, Phys. Rev. D112, 103519 (2025), arXiv:2506.22373 [astro-ph.CO]

    work page arXiv 2025

  28. [28]

    S. S. Mishra, S. K. Bhoi, and P . K. Sahoo, Exploring late-time cosmic acceleration in VCDM cosmology, Eur. Phys. J. C86, 192 (2026), arXiv:2603.27178 [astro-ph.CO]. 12

    work page arXiv 2026

  29. [29]

    The Hubble series: Convergence properties and redshift variables

    C. Cattoen and M. Visser, The Hubble series: Convergence properties and redshift variables, Class. Quant. Grav.24, 5985 (2007), arXiv:0710.1887 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  30. [30]

    High-Redshift Cosmography

    V . Vitagliano, J.-Q. Xia, S. Liberati, and M. Viel, High-Redshift Cosmography, JCAP03, 005, arXiv:0911.1249 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    The Pantheon+ Analysis: Cosmological Constraints

    D. Broutet al., The Pantheon+ Analysis: Cosmological Constraints, Astrophys. J.938, 110 (2022), arXiv:2202.04077 [astro- ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  32. [32]

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

    D. Scolnicet al., The Pantheon+ Analysis: The Full Data Set and Light-curve Release, Astrophys. J.938, 113 (2022), arXiv:2112.03863 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  33. [33]

    Cosmographic analysis of the equation of state of the universe through Pad\'e approximations

    C. Gruber and O. Luongo, Cosmographic analysis of the equation of state of the universe through Pad ´e approximations, Phys. Rev. D89, 103506 (2014), arXiv:1309.3215 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  34. [34]

    Cosmological Constant, Fine Structure Constant and Beyond

    H. Wei, X.-B. Zou, H.-Y. Li, and D.-Z. Xue, Cosmological Constant, Fine Structure Constant and Beyond, Eur. Phys. J. C77, 14 (2017), arXiv:1605.04571 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  35. [35]

    J. P . Hu and F. Y. Wang, High-redshift cosmography: Application and comparison with different methods, Astron. Astrophys. 661, A71 (2022), arXiv:2202.09075 [astro-ph.CO]

    work page arXiv 2022

  36. [36]

    S. S. Mishra, N. S. Kavya, P . K. Sahoo, and T. Harko, Pad´e cosmography and its insights into teleparallel gravity, Mon. Not. Roy. Astron. Soc.543, 2816 (2025)

    2025

  37. [37]

    Rational approximations of $f(R)$ cosmography through Pad\'e polynomials

    S. Capozziello, R. D’Agostino, and O. Luongo, Rational approximations off(R)cosmography through Pad ´e polynomials, JCAP05, 008, arXiv:1709.08407 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    Moresco, R

    M. Moresco, R. Jimenez, L. Verde, A. Cimatti, and L. Pozzetti, Setting the Stage for Cosmic Chronometers. II. Impact of Stellar Population Synthesis Models Systematics and Full Covariance Matrix, Astrophys. J.898, 82 (2020), arXiv:2003.07362 [astro-ph.GA]

    work page arXiv 2020

  39. [39]

    A. G. Adameet al.(DESI), DESI 2024 VI: cosmological constraints from the measurements of baryon acoustic oscillations, JCAP02, 021, arXiv:2404.03002 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  40. [40]

    Shahet al.(DES), Constraints on compact objects from the Dark Energy Survey 5-yr supernova sample, Mon

    P . Shahet al.(DES), Constraints on compact objects from the Dark Energy Survey 5-yr supernova sample, Mon. Not. Roy. Astron. Soc.536, 946 (2024), arXiv:2410.07956 [astro-ph.CO]

    work page arXiv 2024

  41. [41]

    Union Through UNITY: Cosmology with 2,000 SNe Using a Unified Bayesian Framework

    D. Rubinet al., Union Through UNITY: Cosmology with 2,000 SNe Using a Unified Bayesian Framework, Astrophys. J.986, 231 (2025), arXiv:2311.12098 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  42. [42]

    R. L. Arnowitt, S. Deser, and C. W. Misner, The Dynamics of general relativity, Gen. Rel. Grav.40, 1997 (2008), arXiv:gr- qc/0405109

    work page arXiv 1997

  43. [43]

    De Felice and S

    A. De Felice and S. Mukohyama, Weakening gravity for dark matter in a type-II minimally modified gravity, JCAP04, 018, arXiv:2011.04188 [astro-ph.CO]

    work page arXiv 2011

  44. [44]

    Comparison of two theories of Type-IIa minimally modified gravity,

    A. De Felice, K.-i. Maeda, S. Mukohyama, and M. C. Pookkillath, Comparison of two theories of Type-IIa minimally modified gravity, Phys. Rev. D106, 024028 (2022), arXiv:2204.08294 [gr-qc]

    work page arXiv 2022

  45. [45]

    De Felice, S

    A. De Felice, S. Mukohyama, and M. C. Pookkillath, AddressingH 0 tension by means of VCDM, Phys. Lett. B816, 136201 (2021), [Erratum: Phys.Lett.B 818, 136364 (2021)], arXiv:2009.08718 [astro-ph.CO]

    work page arXiv 2021

  46. [46]

    A. Ganz, P . Martens, S. Mukohyama, and R. Namba, Bouncing cosmology in VCDM, JCAP04, 060, arXiv:2212.13561 [gr-qc]

    work page arXiv

  47. [47]

    Cosmography: Cosmology without the Einstein equations

    M. Visser, Cosmography: Cosmology without the Einstein equations, Gen. Rel. Grav.37, 1541 (2005), arXiv:gr-qc/0411131

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  48. [48]

    Capozziello, R

    S. Capozziello, R. D’Agostino, and O. Luongo, High-redshift cosmography: auxiliary variables versus Pad ´e polynomials, Mon. Not. Roy. Astron. Soc.494, 2576 (2020), arXiv:2003.09341 [astro-ph.CO]

    work page arXiv 2020

  49. [49]

    emcee: The MCMC Hammer

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The MCMC Hammer, Publ. Astron. Soc. Pac.125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  50. [50]

    Constraining Cosmological Parameters Based on Relative Galaxy Ages

    R. Jimenez and A. Loeb, Constraining cosmological parameters based on relative galaxy ages, Astrophys. J.573, 37 (2002), arXiv:astro-ph/0106145

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  51. [51]

    A 6% measurement of the Hubble parameter at $z\sim0.45$: direct evidence of the epoch of cosmic re-acceleration

    M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro, and D. Wilkinson, A 6% measurement of the Hubble parameter atz∼0.45: direct evidence of the epoch of cosmic re-acceleration, JCAP05, 014, arXiv:1601.01701 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  52. [52]

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

    M. Abdul Karimet al.(DESI), DESI DR2 results. II. Measurements of baryon acoustic oscillations and cosmological con- straints, Phys. Rev. D112, 083515 (2025), arXiv:2503.14738 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  53. [53]

    UNITY: Confronting Supernova Cosmology's Statistical and Systematic Uncertainties in a Unified Bayesian Framework

    D. Rubinet al.(Supernova Cosmology Project), Unity: Confronting Supernova Cosmology’s Statistical and Systematic Un- certainties in a Unified Bayesian Framework, Astrophys. J.813, 137 (2015), arXiv:1507.01602 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  54. [54]

    Vincenziet al.(DES), Comparing the DES-SN5YR and Pantheon+ SN cosmology analyses: investigation based on ‘evolv- ing dark energy or supernovae systematics’?, Mon

    M. Vincenziet al.(DES), Comparing the DES-SN5YR and Pantheon+ SN cosmology analyses: investigation based on ‘evolv- ing dark energy or supernovae systematics’?, Mon. Not. Roy. Astron. Soc.541, 2585 (2025), arXiv:2501.06664 [astro-ph.CO]

    work page arXiv 2025

  55. [55]

    De Felice, S

    A. De Felice, S. Mukohyama, and M. C. Pookkillath, Static, spherically symmetric objects in type-II minimally modified gravity, Phys. Rev. D105, 104013 (2022), arXiv:2110.14496 [gr-qc]

    work page arXiv 2022

  56. [56]

    De Felice, K.-i

    A. De Felice, K.-i. Maeda, S. Mukohyama, and M. C. Pookkillath, Gravitational collapse and formation of a black hole in a type II minimally modified gravity theory, JCAP03, 030, arXiv:2211.14760 [gr-qc]

    work page arXiv

  57. [57]

    De Felice, A

    A. De Felice, A. Doll, F. Larrouturou, and S. Mukohyama, Black holes in a type-II minimally modified gravity, JCAP03, 004, arXiv:2010.13067 [gr-qc]. 13

    work page arXiv 2010

  58. [58]

    A. F. Jalali, P . Martens, and S. Mukohyama, Spherical scalar collapse in a type-II minimally modified gravity, Phys. Rev. D 109, 044053 (2024), arXiv:2306.10672 [gr-qc]

    work page arXiv 2024

  59. [59]

    A. Ganz, P . Martens, S. Mukohyama, and R. Namba, Bispectrum from inflation/bouncing Universe in VCDM, Int. J. Mod. Phys. A40, 2540007 (2025), arXiv:2407.02882 [gr-qc]

    work page arXiv 2025

  60. [60]

    A kinematical approach to dark energy studies

    D. Rapetti, S. W. Allen, M. A. Amin, and R. D. Blandford, A kinematical approach to dark energy studies, Mon. Not. Roy. Astron. Soc.375, 1510 (2007), arXiv:astro-ph/0605683

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  61. [61]

    Kolhatkar, P

    A. Kolhatkar, P . Dalal, and P . K. Sahoo, Structural constraints on interacting dark sector inΛCDM-mimicking f(Q) gravity, Phys. Lett. B876, 140397 (2026)

    2026

  62. [62]

    S. S. Mishra, A. Kolhatkar, and P . K. Sahoo, Big Bang Nucleosynthesis constraints on f(T,T) gravity, Phys. Lett. B848, 138391 (2024), arXiv:2312.07558 [astro-ph.CO]

    work page arXiv 2024

  63. [63]

    Kolhatkar, S

    A. Kolhatkar, S. S. Mishra, and P . K. Sahoo, Implications of cosmological perturbations of √Qin STEGR, Eur. Phys. J. C85, 656 (2025)

    2025

  64. [64]

    Joint Constraints on Neutrinos and Dynamical Dark Energy in Minimally Modified Gravity

    A. Ladeira, R. C. Nunes, S. Pan, and W. Yang, Joint constraints on neutrinos and dynamical dark energy in minimally modified gravity, Phys. Rev. D113, 083503 (2026), arXiv:2601.02077 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  65. [65]

    Linke, Euclid – The Dark Universe detective, in58th Rencontres de Moriond on Cosmology(2024) arXiv:2405.01037 [astro- ph.CO]

    L. Linke, Euclid – The Dark Universe detective, in58th Rencontres de Moriond on Cosmology(2024) arXiv:2405.01037 [astro- ph.CO]

    work page arXiv 2024

  66. [66]

    N. E. Chisariet al.(LSST Dark Energy Science), Core Cosmology Library: Precision Cosmological Predictions for LSST, Astrophys. J. Suppl.242, 2 (2019), arXiv:1812.05995 [astro-ph.CO]

    work page arXiv 2019