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arxiv: 2512.03314 · v2 · submitted 2025-12-02 · 🌌 astro-ph.CO · gr-qc· hep-th

Recognition: 2 theorem links

· Lean Theorem

EFT of Dark Energy with Cosmic Chronometers: Reconstructing Background EFT Functions

Fumiya Okamatsu , Kazufumi Takahashi
Authors on Pith no claims yet

Pith reviewed 2026-05-17 01:43 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th
keywords effective field theorydark energycosmic chronometersHubble parameterbackground reconstructionscalar-tensor theoriesquintessencemodel-independent cosmology
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The pith

Cosmic chronometer measurements of the Hubble parameter can be used to reconstruct the background functions of the effective field theory of dark energy without assuming any specific cosmological model.

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

The paper shows how to take the observed expansion history from cosmic chronometers and turn it into the time-dependent functions that appear in the EFT description of dark energy. This reconstruction works directly from the data and therefore applies equally to Lambda CDM, quintessence, and other scalar-tensor models. A reader should care because it supplies a concrete route to test whether the background evolution of dark energy matches the predictions of a given theory. The same reconstructed functions can then be fed into concrete models to derive observational constraints.

Core claim

The background EFT functions can be obtained directly by solving for the time derivatives and combinations of the Hubble parameter H(z) measured by cosmic chronometers; once obtained, these functions serve as a model-independent bridge that lets any scalar-tensor theory be compared with the observed expansion history, including tests of the Lambda CDM limit and explicit quintessence potentials.

What carries the argument

The reconstruction mapping that expresses the EFT background functions in terms of the measured Hubble parameter and its redshift derivatives, without inserting a parametric form for the dark-energy equation of state.

If this is right

  • Any scalar-tensor theory whose background solution can be written in EFT form can be confronted with chronometer data without first choosing a parametric dark-energy model.
  • The Lambda CDM model becomes a special case that can be tested by checking whether the reconstructed functions remain constant or take their expected values.
  • Quintessence models can be constrained by feeding the reconstructed functions into the Klein-Gordon equation and comparing with the same chronometer points.
  • The method supplies a consistency check: if two different theories predict different EFT functions for the same expansion history, the data can discriminate between them at the background level.

Where Pith is reading between the lines

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

  • Combining the reconstructed functions with perturbation-level EFT constraints from large-scale structure could test whether a single set of functions works at both background and perturbation orders.
  • Future chronometer samples at higher redshift would extend the reconstruction into the matter-dominated era and tighten limits on early dark energy.
  • The same mapping could be applied to simulated data from next-generation surveys to forecast how well the EFT functions will be determined once systematics are reduced.

Load-bearing premise

The Hubble parameter values reported by cosmic chronometers are free of large systematic offsets and faithfully trace the true background expansion history.

What would settle it

A statistically significant mismatch between the expansion history predicted by the reconstructed EFT functions (when inserted into a specific model such as quintessence) and an independent measurement of the same history from baryon acoustic oscillations or type Ia supernovae.

Figures

Figures reproduced from arXiv: 2512.03314 by Fumiya Okamatsu, Kazufumi Takahashi.

Figure 1
Figure 1. Figure 1: Summary of 32 CC data used in Ref. [31], taken from the following references, each indicated by a different label: Jimenez et al. (2003) [33], Simon et al. (2005) [34], Stern et al. (2010) [35], Moresco et al. (2012) [36], Zhang et al. (2014) [37], Moresco (2015) [38], Moresco et al. (2016) [39], Ratsimbazafy et al. (2017) [40], and Borghi et al. (2022) [41]. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of H(z) and dH/dz reconstructed by GP with several kernel functions. Each colored band represents the 1σ uncertainty. We used m(z) = 0 in this GP reconstruction. The black dots and error bars in the left panel represent the CC data shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of H(z) and dH/dz reconstructed by GP with several mean functions. Each colored band represents the 1σ uncertainty. We used the Mat´ern kernel with ν = 3.5 in this GP reconstruction. The black dots and error bars in the left panel represent the CC data shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plots of H(z) and dH/dz reconstructed by GP using the method of Ref. [45]. Each colored band represents the 1σ uncertainty. The black dots and error bars in the left panel represent the CC data shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plots of Λ(z)/(M2 PlH2 0 ) reconstructed from CC data using Eq. (3.9) for Ωm0 ∈ {0.25, 0.30, 0.35}. The black dashed line represents the constant value in the ΛCDM model, 3(1 − Ωm0), evaluated at the central Planck 2018 value Ωm0 = 0.315 (±0.007) [2]. 10 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots of c(z)/(M2 PlH2 0 ) reconstructed from CC data using Eq. (3.10) for Ωm0 ∈ {0.25, 0.30, 0.35}. The black dashed line represents the ΛCDM prediction, i.e., c(z) = 0. 3.3 Application: Reconstruction of the Quintessence Model Let us now apply the EFT functions Λ(z) and c(z) obtained in the previous subsection to reconstruct the quintessence model,#6 whose Lagrangian is given by [64–70] L = − 1 2 g µν∂µϕ… view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the reconstructed ϕ(z) and V (ϕ) for Ωm0 = 0.30. In the reconstruction, the upper boundary of the 1σ confidence interval of c(z) is used as a proxy for the parameter c. The shaded regions denote the 1σ uncertainties. The reconstructed scalar field and potential are shown in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plots of the reconstructed Λ(z) using Eq. (A.1) for Ωm0 = 0.30. Each panel corresponds to a different choice of (αM0, s): from left to right, αM0 = −0.1, −0.05, 0.1, and from top to bottom, s = 0.5, 0.7, 1.0, consistent with the Planck 2018 constraints [2]. 15 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Plots of the reconstructed c(z) using Eq. (A.1) for Ωm0 = 0.30. Each panel corresponds to a different choice of (αM0, s): from left to right, αM0 = −0.1, −0.05, 0.1, and from top to bottom, s = 0.5, 0.7, 1.0, consistent with the Planck 2018 constraints [2]. 16 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

The effective field theory (EFT) of dark energy provides a model-independent framework for studying cosmology within scalar-tensor theories. In this work, we explore how the time evolution of the cosmological background, inferred from cosmic chronometer measurements of the Hubble parameter, can be used to reconstruct the relevant EFT functions. Our approach enables the direct determination of these EFT functions from observational data without assuming any specific cosmological model. This makes it possible to test the background evolution of a wide range of dark energy models, including the $\Lambda$CDM model. We further demonstrate how the reconstructed EFT functions can be applied to constrain concrete theories, such as the quintessence model.

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 manuscript proposes reconstructing the time-dependent background EFT functions in the effective field theory of dark energy directly from cosmic chronometer H(z) measurements. It claims this procedure enables model-independent determination of the EFT functions from observational data, allowing tests of the background evolution for a range of dark energy models including ΛCDM and constraints on specific theories such as quintessence.

Significance. If the reconstruction is robust against data systematics and properly validated, the approach could provide a useful data-driven tool for probing scalar-tensor dark energy models at the background level, complementing perturbation-based EFT analyses. The significance is currently limited by the absence of detailed error propagation and mock validation, which are needed to establish that the method yields reliable constraints rather than being dominated by chronometer uncertainties.

major comments (2)
  1. [Reconstruction procedure] Reconstruction procedure (main text, likely §3): The central claim of 'direct determination of these EFT functions from observational data without assuming any specific cosmological model' requires explicit demonstration that residual systematics in cosmic chronometer H(z) (typically 5-10% from stellar population synthesis and selection effects) do not propagate into biased EFT functions. No quantitative propagation or bias assessment is provided, undermining the model-independence assertion.
  2. [Validation and error analysis] Validation and error analysis (likely §4): The manuscript lacks mock-data validation of the reconstruction pipeline and detailed error propagation from H(z) uncertainties to the EFT functions. Without these, it is unclear whether the reconstructed functions can meaningfully constrain models or distinguish them from systematics.
minor comments (2)
  1. [Abstract] The abstract would benefit from specifying the redshift range and number of chronometer data points used in the reconstruction.
  2. [Introduction] Notation for the EFT functions (e.g., time-dependent coefficients) should be defined more explicitly at first use to improve readability for readers unfamiliar with the EFT framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. The points raised regarding robustness to systematics and the need for validation are well taken, and we will strengthen the paper accordingly. We address each major comment below.

read point-by-point responses

  1. Referee: Reconstruction procedure (main text, likely §3): The central claim of 'direct determination of these EFT functions from observational data without assuming any specific cosmological model' requires explicit demonstration that residual systematics in cosmic chronometer H(z) (typically 5-10% from stellar population synthesis and selection effects) do not propagate into biased EFT functions. No quantitative propagation or bias assessment is provided, undermining the model-independence assertion.

    Authors: We agree that a quantitative assessment of systematics propagation is required to fully substantiate the model-independence of the reconstruction. The method derives the background EFT functions directly from the measured H(z) without presupposing any particular dark energy model or parametrization; the model independence refers to the absence of an assumed functional form for the dark energy sector rather than a claim of immunity to data errors. In the revised manuscript we will add an explicit propagation analysis, including sensitivity tests that inject 5-10% systematic shifts into the H(z) data and quantify the resulting bias and uncertainty in the reconstructed EFT functions. revision: yes

  2. Referee: Validation and error analysis (likely §4): The manuscript lacks mock-data validation of the reconstruction pipeline and detailed error propagation from H(z) uncertainties to the EFT functions. Without these, it is unclear whether the reconstructed functions can meaningfully constrain models or distinguish them from systematics.

    Authors: We acknowledge that the current version does not contain mock-data validation or a full error-propagation study. To address this, the revised manuscript will include a dedicated validation section that generates synthetic H(z) catalogs from fiducial EFT models (including ΛCDM and quintessence), applies the reconstruction pipeline, and demonstrates recovery of the input functions within the expected uncertainties. Detailed error propagation from the chronometer covariance matrix to the EFT functions will be provided, using both analytic differentiation and Monte Carlo sampling to produce error bands on the reconstructed quantities. revision: yes

Circularity Check

0 steps flagged

Reconstruction from external chronometer data is independent of fitted EFT outputs

full rationale

The paper derives relations between the EFT background functions and the Hubble parameter H(z) from the standard EFT action for dark energy, then inverts those relations using measured H(z) values from cosmic chronometers as external input. No step reduces a claimed prediction or first-principles result to a fitted parameter by construction, nor does any load-bearing premise rest on self-citation chains or ansatzes imported from prior author work. The output EFT functions are determined directly by the data via the EFT equations rather than being equivalent to the inputs; the method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the validity of the EFT framework for scalar-tensor theories and the assumption that chronometer data directly constrains background evolution; no explicit free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The EFT of dark energy provides a model-independent description of scalar-tensor theories at the background level.
    Invoked as the foundational framework for reconstruction without specific model assumptions.

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · 44 internal anchors

  1. [1]

    A. G. Riess et al.,A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s −1 Mpc−1 Uncertainty from the Hubble Space Telescope and the SH0ES Team,Astrophys. J. Lett.934(2022) L7, [arXiv:2112.04510]. #7A brief comment on this parametrization is in order. In the EFT of cosmological perturbations de- scribed by the action (3.3), on...

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  2. [2]

    Tensions between the Early and the Late Universe

    L. Verde, T. Treu, and A. G. Riess,Tensions between the Early and the Late Universe, Nature Astron.3(2019) 891, [arXiv:1907.10625]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  3. [3]

    J. Renk, M. Zumalac´ arregui, F. Montanari, and A. Barreira,Galileon gravity in light of ISW, CMB, BAO and H 0 data,JCAP10(2017) 020, [arXiv:1707.02263]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  4. [4]

    C. D. Kreisch, F.-Y. Cyr-Racine, and O. Dor´ e,Neutrino puzzle: Anomalies, interactions, and cosmological tensions,Phys. Rev. D101(2020) 123505, [arXiv:1902.00534]

    work page arXiv 2020

  5. [5]

    Braglia, M

    M. Braglia, M. Ballardini, F. Finelli, and K. Koyama,Early modified gravity in light of theH 0 tension and LSS data,Phys. Rev. D103(2021) 043528, [arXiv:2011.12934]

    work page arXiv 2021

  6. [6]

    Cosmological Tests of Modified Gravity

    K. Koyama,Cosmological Tests of Modified Gravity,Rept. Prog. Phys.79(2016) 046902, [arXiv:1504.04623]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    P. G. Ferreira,Cosmological Tests of Gravity,Ann. Rev. Astron. Astrophys.57(2019) 335–374, [arXiv:1902.10503]

    work page arXiv 2019

  8. [8]

    Arai et al.,Cosmological gravity probes: connecting recent theoretical developments to forthcoming observations,PTEP2023(2023) 072E01, [arXiv:2212.09094]

    S. Arai et al.,Cosmological gravity probes: connecting recent theoretical developments to forthcoming observations,PTEP2023(2023) 072E01, [arXiv:2212.09094]

    work page arXiv 2023

  9. [9]

    G. W. Horndeski,Second-order scalar-tensor field equations in a four-dimensional space,Int. J. Theor. Phys.10(1974) 363–384

    work page 1974

  10. [10]

    From k-essence to generalised Galileons

    C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade,From k-essence to generalised Galileons,Phys. Rev. D84(2011) 064039, [arXiv:1103.3260]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  11. [11]

    Generalized G-inflation: Inflation with the most general second-order field equations

    T. Kobayashi, M. Yamaguchi, and J. Yokoyama,Generalized G-inflation: Inflation with the most general second-order field equations,Prog. Theor. Phys.126(2011) 511–529, [arXiv:1105.5723]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  12. [12]

    Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

    D. Langlois and K. Noui,Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability,JCAP02(2016) 034, [arXiv:1510.06930]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  13. [13]

    Extended Scalar-Tensor Theories of Gravity

    M. Crisostomi, K. Koyama, and G. Tasinato,Extended Scalar-Tensor Theories of Gravity,JCAP04(2016) 044, [arXiv:1602.03119]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

    J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order, JHEP12(2016) 100, [arXiv:1608.08135]. 17

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  15. [15]

    "Shadowy" modes in Higher-Order Scalar-Tensor theories

    A. De Felice, D. Langlois, S. Mukohyama, K. Noui, and A. Wang,Generalized instantaneous modes in higher-order scalar-tensor theories,Phys. Rev. D98(2018) 084024, [arXiv:1803.06241]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  16. [16]

    De Felice, S

    A. De Felice, S. Mukohyama, and K. Takahashi,Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories,JCAP12(2021) 020, [arXiv:2110.03194]

    work page arXiv 2021

  17. [17]

    Takahashi, H

    K. Takahashi, H. Motohashi, and M. Minamitsuji,Invertible disformal transformations with higher derivatives,Phys. Rev. D105(2022) 024015, [arXiv:2111.11634]

    work page arXiv 2022

  18. [18]

    Takahashi, M

    K. Takahashi, M. Minamitsuji, and H. Motohashi,Generalized disformal Horndeski theories: cosmological perturbations and consistent matter coupling,PTEP2023 (2023) 013E01, [arXiv:2209.02176]

    work page arXiv 2023

  19. [19]

    Takahashi, M

    K. Takahashi, M. Minamitsuji, and H. Motohashi,Effective description of generalized disformal theories,JCAP07(2023) 009, [arXiv:2304.08624]

    work page arXiv 2023

  20. [20]

    Takahashi,Invertible disformal transformations with arbitrary higher-order derivatives,Phys

    K. Takahashi,Invertible disformal transformations with arbitrary higher-order derivatives,Phys. Rev. D108(2023) 084031, [arXiv:2307.08814]

    work page arXiv 2023

  21. [21]

    Dark Energy and Modified Gravity in Degenerate Higher-Order Scalar-Tensor (DHOST) theories: a review

    D. Langlois,Dark energy and modified gravity in degenerate higher-order scalar–tensor (DHOST) theories: A review,Int. J. Mod. Phys. D28(2019) 1942006, [arXiv:1811.06271]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Horndeski theory and beyond: a review

    T. Kobayashi,Horndeski theory and beyond: a review,Rept. Prog. Phys.82(2019) 086901, [arXiv:1901.07183]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  23. [23]

    Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies

    P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore,Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies,JHEP12 (2006) 080, [hep-th/0606090]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  24. [24]

    The Effective Field Theory of Inflation

    C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore,The Effective Field Theory of Inflation,JHEP03(2008) 014, [arXiv:0709.0293]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  25. [25]

    The Effective Field Theory of Dark Energy

    G. Gubitosi, F. Piazza, and F. Vernizzi,The Effective Field Theory of Dark Energy, JCAP02(2013) 032, [arXiv:1210.0201]. [27]PlanckCollaboration, P. A. R. Ade et al.,Planck 2015 results. XIV. Dark energy and modified gravity,Astron. Astrophys.594(2016) A14, [arXiv:1502.01590]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  26. [26]

    Durakovic, P

    A. Durakovic, P. Hunt, S. P. Patil, and S. Sarkar,Reconstructing the EFT of Inflation from Cosmological Data,SciPost Phys.7(2019) 049, [arXiv:1904.00991]

    work page arXiv 2019

  27. [27]

    Constraining Cosmological Parameters Based on Relative Galaxy Ages

    R. Jimenez and A. Loeb,Constraining cosmological parameters based on relative galaxy ages,Astrophys. J.573(2002) 37–42, [astro-ph/0106145]. 18

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  28. [28]

    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(2020) 82, [arXiv:2003.07362]

    work page arXiv 2020

  29. [29]

    Jalilvand and A

    F. Jalilvand and A. Mehrabi,Model independent estimation of the cosmography parameters using cosmic chronometers,Eur. Phys. J. Plus137(2022) 1341, [arXiv:2209.05782]

    work page arXiv 2022

  30. [30]

    Model-independent determination on $H_0$ using the latest $H(z)$ data

    D. Wang and X.-H. Meng,Model-independent determination on H 0 using the latest cosmic chronometer data,Sci. China Phys. Mech. Astron.60(2017) 110411, [arXiv:1610.01202]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  31. [31]

    Constraints on the equation of state of dark energy and the Hubble constant from stellar ages and the CMB

    R. Jimenez, L. Verde, T. Treu, and D. Stern,Constraints on the equation of state of dark energy and the Hubble constant from stellar ages and the CMB,Astrophys. J. 593(2003) 622–629, [astro-ph/0302560]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  32. [32]

    Constraints on the redshift dependence of the dark energy potential

    J. Simon, L. Verde, and R. Jimenez,Constraints on the redshift dependence of the dark energy potential,Phys. Rev. D71(2005) 123001, [astro-ph/0412269]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  33. [33]

    Cosmic Chronometers: Constraining the Equation of State of Dark Energy. I: H(z) Measurements

    D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, and S. A. Stanford,Cosmic Chronometers: Constraining the Equation of State of Dark Energy. I: H(z) Measurements,JCAP02(2010) 008, [arXiv:0907.3149]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  34. [34]

    Improved constraints on the expansion rate of the Universe up to z~1.1 from the spectroscopic evolution of cosmic chronometers

    M. Moresco et al.,Improved constraints on the expansion rate of the Universe up to z∼1.1from the spectroscopic evolution of cosmic chronometers,JCAP08(2012) 006, [arXiv:1201.3609]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  35. [35]

    Four New Observational $H(z)$ Data From Luminous Red Galaxies of Sloan Digital Sky Survey Data Release Seven

    C. Zhang, H. Zhang, S. Yuan, S. Liu, T.-J. Zhang, and Y.-C. Sun,Four new observational H(z) data from luminous red galaxies in the Sloan Digital Sky Survey data release seven,Research in Astronomy and Astrophysics14(Oct., 2014) 1221–1233, [arXiv:1207.4541]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  36. [36]

    Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z$\sim$2

    M. Moresco,Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z∼2,Mon. Not. Roy. Astron. Soc.450(2015) L16–L20, [arXiv:1503.01116]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  37. [37]

    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 at z∼0.45: direct evidence of the epoch of cosmic re-acceleration,JCAP05(2016) 014, [arXiv:1601.01701]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  38. [38]

    A. L. Ratsimbazafy, S. I. Loubser, S. M. Crawford, C. M. Cress, B. A. Bassett, R. C. Nichol, and P. V¨ ais¨ anen,Age-dating Luminous Red Galaxies observed with the Southern African Large Telescope,Mon. Not. Roy. Astron. Soc.467(2017) 3239–3254, [arXiv:1702.00418]. 19

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  39. [39]

    Borghi, M

    N. Borghi, M. Moresco, and A. Cimatti,Toward a Better Understanding of Cosmic Chronometers: A New Measurement of H(z) at z∼0.7,Astrophys. J. Lett.928 (2022) L4, [arXiv:2110.04304]

    work page arXiv 2022

  40. [40]

    Beckers,An Introduction to Gaussian Process Models,arXiv:2102.05497

    T. Beckers,An Introduction to Gaussian Process Models,arXiv:2102.05497

    work page arXiv

  41. [41]

    V. C. Busti, C. Clarkson, and M. Seikel,Evidence for a Lower Value forH 0 from Cosmic Chronometers Data?,Mon. Not. Roy. Astron. Soc.441(2014) 11, [arXiv:1402.5429]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  42. [42]

    $H_0$ from cosmic chronometers and Type Ia supernovae, with Gaussian Processes and the novel Weighted Polynomial Regression method

    A. G´ omez-Valent and L. Amendola,H0 from cosmic chronometers and Type Ia supernovae, with Gaussian Processes and the novel Weighted Polynomial Regression method,JCAP04(2018) 051, [arXiv:1802.01505]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  43. [43]

    Gaussian Process Cosmography

    A. Shafieloo, A. G. Kim, and E. V. Linder,Gaussian Process Cosmography,Phys. Rev. D85(2012) 123530, [arXiv:1204.2272]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  44. [44]

    K. Liao, A. Shafieloo, R. E. Keeley, and E. V. Linder,A model-independent determination of the Hubble constant from lensed quasars and supernovae using Gaussian process regression,Astrophys. J. Lett.886(2019) L23, [arXiv:1908.04967]

    work page arXiv 2019

  45. [45]

    A. M. Pinho, S. Casas, and L. Amendola,Model-independent reconstruction of the linear anisotropic stressη,JCAP11(2018) 027, [arXiv:1805.00027]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  46. [46]

    Mehrabi and S

    A. Mehrabi and S. Basilakos,DoesΛCDM really be in tension with the Hubble diagram data?,Eur. Phys. J. C80(2020) 632, [arXiv:2002.12577]

    work page arXiv 2020

  47. [47]

    Smoothing Supernova Data to Reconstruct the Expansion History of the Universe and its Age

    A. Shafieloo, U. Alam, V. Sahni, and A. A. Starobinsky,Smoothing Supernova Data to Reconstruct the Expansion History of the Universe and its Age,Mon. Not. Roy. Astron. Soc.366(2006) 1081–1095, [astro-ph/0505329]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  48. [48]

    Model Independent Reconstruction of the Expansion History of the Universe and the Properties of Dark Energy

    A. Shafieloo,Model Independent Reconstruction of the Expansion History of the Universe and the Properties of Dark Energy,Mon. Not. Roy. Astron. Soc.380(2007) 1573–1580, [astro-ph/0703034]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  49. [49]

    Model independent tests of the standard cosmological model

    A. Shafieloo and C. Clarkson,Model independent tests of the standard cosmological model,Phys. Rev. D81(2010) 083537, [arXiv:0911.4858]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  50. [50]

    Hwang, B

    S.-g. Hwang, B. L’Huillier, R. E. Keeley, M. J. Jee, and A. Shafieloo,How to use GP: effects of the mean function and hyperparameter selection on Gaussian process regression,JCAP02(2023) 014, [arXiv:2206.15081]

    work page arXiv 2023

  51. [51]

    K. Aoki, M. A. Gorji, S. Mukohyama, and K. Takahashi,The effective field theory of vector-tensor theories,JCAP01(2022) 059, [arXiv:2111.08119]. 20

    work page arXiv 2022

  52. [52]

    K. Aoki, M. A. Gorji, T. Hiramatsu, S. Mukohyama, M. C. Pookkillath, and K. Takahashi,CMB spectrum in unified EFT of dark energy: scalar-tensor and vector-tensor theories,JCAP07(2024) 056, [arXiv:2405.04265]

    work page arXiv 2024

  53. [53]

    K. Aoki, M. A. Gorji, S. Mukohyama, and K. Takahashi,Effective field theory of gravitating continuum: solids, fluids, and aether unified,JCAP08(2022) 072, [arXiv:2204.06672]

    work page arXiv 2022

  54. [54]

    J. K. Bloomfield, ´E. ´E. Flanagan, M. Park, and S. Watson,Dark energy or modified gravity? An effective field theory approach,JCAP08(2013) 010, [arXiv:1211.7054]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  55. [55]

    Essential Building Blocks of Dark Energy

    J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi,Essential Building Blocks of Dark Energy,JCAP08(2013) 025, [arXiv:1304.4840]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  56. [56]

    The effective field theory of inflation/dark energy and the Horndeski theory

    S. Tsujikawa,The effective field theory of inflation/dark energy and the Horndeski theory,Lect. Notes Phys.892(2015) 97–136, [arXiv:1404.2684]. [59]DESICollaboration, A. G. Adame et al.,DESI 2024 IV: Baryon Acoustic Oscillations from the Lyman alpha forest,JCAP01(2025) 124, [arXiv:2404.03001]. [60]DESICollaboration, A. G. Adame et al.,DESI 2024 VI: cosmol...

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  57. [57]

    A. A. Starobinsky,How to determine an effective potential for a variable cosmological term,JETP Lett.68(1998) 757–763, [astro-ph/9810431]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  58. [58]

    Determining the Equation of State of the Expanding Universe: Inverse Problem in Cosmology

    T. Nakamura and T. Chiba,Determining the equation of state of the expanding universe: Inverse problem in cosmology,Mon. Not. Roy. Astron. Soc.306(1999) 696–700, [astro-ph/9810447]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  59. [59]

    Feasibility of Reconstructing the Quintessential Potential Using SNIa Data

    T. Chiba and T. Nakamura,Feasibility of reconstructing the quintessential potential using SNIa data,Phys. Rev. D62(2000) 121301, [astro-ph/0008175]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  60. [60]

    Fujii,Origin of the Gravitational Constant and Particle Masses in Scale Invariant Scalar - Tensor Theory,Phys

    Y. Fujii,Origin of the Gravitational Constant and Particle Masses in Scale Invariant Scalar - Tensor Theory,Phys. Rev. D26(1982) 2580

    work page 1982

  61. [61]

    Ratra and P

    B. Ratra and P. J. E. Peebles,Cosmological Consequences of a Rolling Homogeneous Scalar Field,Phys. Rev. D37(1988) 3406

    work page 1988

  62. [62]

    Cosmology with x-matter

    T. Chiba, N. Sugiyama, and T. Nakamura,Cosmology with x matter,Mon. Not. Roy. Astron. Soc.289(1997) L5–L9, [astro-ph/9704199]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  63. [63]

    E. J. Copeland, A. R. Liddle, and D. Wands,Exponential potentials and cosmological scaling solutions,Phys. Rev. D57(1998) 4686–4690, [gr-qc/9711068]. 21

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  64. [64]

    P. G. Ferreira and M. Joyce,Cosmology with a primordial scaling field,Phys. Rev. D 58(1998) 023503, [astro-ph/9711102]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  65. [65]

    R. R. Caldwell, R. Dave, and P. J. Steinhardt,Cosmological imprint of an energy component with general equation of state,Phys. Rev. Lett.80(1998) 1582–1585, [astro-ph/9708069]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  66. [66]

    Quintessence, Cosmic Coincidence, and the Cosmological Constant

    I. Zlatev, L.-M. Wang, and P. J. Steinhardt,Quintessence, cosmic coincidence, and the cosmological constant,Phys. Rev. Lett.82(1999) 896–899, [astro-ph/9807002]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  67. [67]

    Effective Field Theory of Cosmic Acceleration: constraining dark energy with CMB data

    M. Raveri, B. Hu, N. Frusciante, and A. Silvestri,Effective Field Theory of Cosmic Acceleration: constraining dark energy with CMB data,Phys. Rev. D90(2014) 043513, [arXiv:1405.1022]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  68. [68]

    Hiramatsu,CMB constraints on DHOST theories,JCAP10(2022) 035, [arXiv:2205.11559]

    T. Hiramatsu,CMB constraints on DHOST theories,JCAP10(2022) 035, [arXiv:2205.11559]. [73]LIGO Scientific, VirgoCollaboration, B. P. Abbott et al.,GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,Phys. Rev. Lett.119 (2017) 161101, [arXiv:1710.05832]. [74]LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium ...

    work page arXiv 2022