Pith. sign in

REVIEW 2 major objections 8 minor 260 references

One master Vainshtein equation lets a single N-body code simulate many dark-energy and modified-gravity models.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-31 05:23 UTC pith:I373WXWP

load-bearing objection Solid infrastructure paper: one master Vainshtein solver plus public code and multi-model P(k) checks; the large-scale linear mismatch is real but not cleanly proven physical, and novelty is moderate given the ECOSMOG lineage and a parallel release. the 2 major comments →

arxiv 2607.24940 v1 pith:I373WXWP submitted 2026-07-27 astro-ph.CO gr-qcphysics.comp-ph

EFT-Ramses: a code to simulate the effective field theory of dark energy

Nathaniel Ota Woodcock , Sownak Bose , Yunhao Gao , Baojiu Li This is my paper
classification astro-ph.CO gr-qcphysics.comp-ph
keywords effective field theory of dark energyN-body simulationsVainshtein screeningmodified gravityHorndeski theorymatter power spectrumcubic GalileonDGP
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Stage-IV surveys will map cosmic structure so precisely that background expansion alone cannot tell dynamical dark energy from modified gravity. The authors therefore built EFT-RAMSES, an extension of an existing adaptive-mesh N-body code that embeds the effective field theory of dark energy. Diverse scalar- and vector-tensor theories are reduced to one master Vainshtein equation controlled by three time-dependent coefficients. High-resolution runs of normal-branch DGP, cubic Galileon, generalised cubic covariant Galileon and generic EFT models recover the expected screening turnover and match legacy and independent codes to roughly one-to-two percent on nonlinear scales. The result is a single, publicly available engine that can specialise to many models without rewriting the solver, ready for precision tests with forthcoming survey data.

Core claim

By consolidating the normal and self-accelerating DGP branches, cubic scalar and vector Galileons, the generalised cubic covariant Galileon and generic EFT parameterisations into a single master Vainshtein equation whose physics is carried by three time-dependent background functions, EFT-RAMSES produces nonlinear matter power spectra that agree with both its parent code and an independent approximate code while correctly recovering Vainshtein screening.

What carries the argument

The master Vainshtein equation: a single nonlinear elliptic equation for the scalar-field perturbation, written in code units, whose model-specific content is entirely encoded in three time-dependent coefficients α, β and R_c².

Load-bearing premise

The quasi-static and weak-field approximations drop all time derivatives of the scalar and metric perturbations, so the equations never see dark-energy sound-speed operators and cannot describe models in which dark energy itself clusters.

What would settle it

Run the same initial conditions with a solver that retains the dropped time derivatives or the M_4^{2} operator; if the nonlinear matter power spectrum then differs from EFT-RAMSES by more than the claimed one-to-two percent on the scales validated here, the master-equation reduction is incomplete.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A single public code base can generate nonlinear predictions for nDGP, cubic Galileons, GCCG and generic α-basis EFT models without model-specific solvers.
  • Background expansion and fifth-force effects can be cleanly separated by comparing full runs against QCDM counterparts that share the same expansion history.
  • The same pipeline can produce halo mass functions and weak-lensing maps deep into the nonlinear regime for Stage-IV survey forecasts.
  • Parameter-space scans over α_M and α_B become feasible because only the three background coefficients need updating.

Where Pith is reading between the lines

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

  • Once the quasi-static restriction is lifted, the same master-equation architecture could be reused for clustering dark-energy models that current Stage-IV analyses may need.
  • The observed large-scale power suppression relative to linear theory, shown to be driven by nonlinear mode coupling, may itself become a diagnostic of Vainshtein-type screening in survey data.
  • Vector-tensor models already covered by the master equation open a route to testing whether longitudinal vector modes leave distinct nonlinear signatures compared with pure scalar Galileons.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 8 minor

Summary. The authors present EFT-RAMSES, an extension of the ECOSMOG/RAMSES N-body code that embeds the EFTofDE framework (in both the original EFT-function and α-basis parameterisations) into a single "master" Vainshtein equation (Eq. 27) with three time-dependent coefficients (α, β, R_c²). Under the quasi-static and weak-field approximations, this one equation covers nDGP/sDGP, cubic scalar and vector Galileons, GCCG, and generic EFT models, so a single multigrid Newton–Gauss–Seidel solver serves all of them. The code is validated with 1024³-particle, 1 Gpc/h simulations of nDGP, csG, GCCG, and two EFT models, compared against legacy ECOSMOG-V, an independent GCCG ECOSMOG variant, and the independent HiCOLA code, finding ~1–2% agreement on nonlinear scales and the expected Vainshtein screening turnover. A persistent ~1% scale-independent suppression relative to linear theory at k≲0.05 h/Mpc is interpreted in Appendix A as physical mode-coupling driven by the nonlinear Vainshtein term, based on diagnostics that rescale R_c² and vary relaxation parameters.

Significance. If the validation holds, this is a useful and timely tool: a publicly released, model-agnostic nonlinear solver for the full class of Vainshtein-screening Horndeski theories, which removes the need to hard-code model-specific scalar-field equations and is directly relevant to Stage-IV survey modelling. Strengths worth naming: the master-equation reduction (Eqs. 27–28) is a genuine reparameterisation validated against independent solvers rather than against quantities derived from itself; the validation is multi-code (legacy ECOSMOG-V, an independent GCCG ECOSMOG implementation, HiCOLA) and multi-redshift; QCDM counterpart runs cleanly separate background-expansion from fifth-force effects; the csG comparison uses paired inverted-phase ICs to control sample variance; and the code is publicly available. The explicit treatment of the cvG→csG limit (§3.2.4) and the EFT-basis mappings (Eqs. 62–64) add to the paper's usefulness as a reference.

major comments (2)
  1. [§4.3.1 and Appendix A (Figs. A1–A2)] The ~1% scale-independent suppression at k≲0.05 h/Mpc relative to linear theory is asserted to be 'inherent mode-coupling' and the text states the authors 'can confidently rule out it as a code error'. The evidence shown does not support that confidence. The Appendix A diagnostic rescales R_c² and shows the mismatch tracks the strength of the nonlinear term; but a discretization error in the operator-split finite-difference representation of (∇²φ)²−(∂i∂jφ)² (the Chan & Scoccimarro splitting inherited from ECOSMOG-V) would also scale with R_c² at leading order, so this test cannot distinguish continuum mode-coupling from a term-proportional mesh error. The codes exhibiting the effect (legacy ECOSMOG-V, EFT-RAMSES, ecosmog-eft, mg-glam) all share the grid-relaxation/operator-splitting algorithm class, while the one fully independent code, HiCOLA, does not show the anomaly — a fact the pape
  2. [§4.3.1, Fig. 1 discussion] The mismatch of the longest-wavelength (lowest-k) mode is attributed to 'some version difference that affects the new and old ΛCDM runs, where EFT-RAMSES shows a better behaviour.' This is too vague for a validation paper: the ΛCDM runs should be the most trivially reproducible part of the comparison, and an unexplained discrepancy in the baseline undermines confidence in the ratios built on it. Please identify the cause (IC generator difference — MPGRAFIC vs 2LPTic is a candidate given Table 1 —, background integration, or binning) and state it explicitly.
minor comments (8)
  1. [§3.2.3] Equations (36)–(38) for the cvG model duplicate the equation numbers already used for the nDGP parameters in §3.2.2 (Eqs. 36–38). This is a numbering error that will confuse readers; please renumber.
  2. [§1, final paragraph] The description of Section 4 is duplicated verbatim ('Section 4 details our computational methodology, outlining the architecture...'). One of the two sentences should be removed or rewritten.
  3. [Fig. 2 caption] Typo: 'the csG (red)z and QCDM (orange) models' — stray 'z'.
  4. [§4.1] Typo: 'the first efficientlympi parallelised N-body simulation code' — 'efficientlympi' should be 'efficiently MPI-parallelised'.
  5. [§1] 'the Large Survey of Space Telescope (Collaboration 2012)' should read 'the Large Synoptic Survey Telescope (LSST)'.
  6. [§2.3] The statement that the validity of the combined QSA+WFA 'has been firmly established by previous counterpart works that successfully benchmarked these approximations against full, un-approximated dynamical simulations' is given without citations at that point; please cite the specific benchmarking studies (e.g., Winther & Ferreira 2015; Barreira et al. 2013b are cited only later, in §5).
  7. [§5 / Data Availability] The conclusions quote a 1–2% bound on nonlinear-scale deviations, but the figures show only qualitative agreement; a short quantitative statement (e.g., maximum fractional residual between EFT-RAMSES and each comparator over a stated k-range) would make the headline number verifiable. Also, given that the code is public, consider depositing the power-spectrum data in a public repository rather than 'upon request'.
  8. [Table 1] All models except csG use a single realisation. Since code-to-code comparisons use identical ICs this is acceptable, but the linear-theory comparisons at low k (the basis for the Appendix A anomaly claim) are subject to sample variance for the single-realisation models; a caveat to this effect would be appropriate.

Circularity Check

0 steps flagged

No significant circularity: master equation is a genuine reparameterisation; validations include independent HiCOLA and linear theory, not quantities forced by construction.

full rationale

EFT-RAMSES is a numerical-methods paper. The load-bearing derivation (Horndeski/EFT under QSA → N-body equation Eqs. 19–21 → master Vainshtein equation Eqs. 27–28 with coefficients α, β, R_c²) is an algebraic reparameterisation of known field equations into a single solver form; the coefficients are fixed by each model’s background functions, not fitted to the comparison spectra. Validation P(k) ratios are compared to legacy ECOSMOG, an independent approximate code (HiCOLA), and linear growth factors—none of which are defined from the same fit as the outputs. Self-citations are to the parent ECOSMOG multigrid/AMR infrastructure being extended, which is normal for a code paper and is cross-checked externally. The Appendix A large-scale mismatch vs linear theory is a correctness/interpretation issue (shared-lineage artifact vs physical mode-coupling), not a circular reduction of a claimed prediction to its inputs. Score 1 only for mild intra-family validation weight; central claim remains independently contentful.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 1 invented entities

The result is a numerical methods paper. Load-bearing content is standard GR+Horndeski/EFT machinery, the quasi-static reduction, GW170817-motivated α_T=0, neglect of M₄², tracker backgrounds for Galileons, and the code-unit master-equation rewrite. No new physical entity is postulated; free parameters are user-chosen model inputs for the validation runs, not fitted to manufacture agreement.

free parameters (5)
  • Ω_rc (nDGP crossover) = 0.25
    Chosen validation value Ω_rc=0.25 in Table 1; sets fifth-force strength and R_c² for nDGP runs.
  • c_B (EFT braiding amplitude) = ±1.0
    User coefficient in α_B=c_B Ω_DE(a); set to ±1 for EFTP/EFTN demonstration runs.
  • w0, wa (CPL dark-energy EoS) = w0=-1.0, wa=-0.2
    Background EFT expansion parameters in Table 1; fixed to (-1.0, -0.2) for the EFT examples.
  • s, q (GCCG parameters) = s=0.649, q=1.061
    Background and braiding parameters for GCCG validation (s=0.649, q=1.061).
  • f_φ, β̃_3 (cvG/csG) = f_φ=1 (csG); β̃_3 tiny for csG limit
    Scalar fraction and cubic vector coupling; csG recovered as β̃_3→0 (e.g. 1e-6) with f_φ=1 in the csG runs.
axioms (7)
  • domain assumption Quasi-static and weak-field approximations: |∂_i δQ| ≫ |δQ̇| ~ |H δQ| and neglect of first-derivative and time-derivative perturbation terms in the scalar EoM.
    Section 2.3; required to close the N-body equation without M₄² and without full time-dependent scalar dynamics.
  • domain assumption α_T = 0 from GW170817, so G_{4X}=G_5=0 and only generalised cubic Galileon operators remain.
    Sections 2.1–2.2; truncates Horndeski to the class the master equation covers.
  • domain assumption Dark-energy clustering operator M₄² is negligible (c_s ~ c); N-body equations fully fixed by α_M, α_B and background.
    Explicitly stated after Eqs. 20–21 and in conclusions; defines the scope of EFT-RAMSES.
  • domain assumption Background functions K, G3, G4 are not strongly nonlinear, so coefficients may be evaluated on the homogeneous background (δφ ≪ φ̄).
    Paragraph after Eq. 24; fails for clustering DE with highly nonlinear G3.
  • domain assumption Galileon models follow tracker solutions that fix c2, c3 (or equivalent) from cosmological parameters.
    Section 3.2.4 (csG) and 3.2.6 (GCCG); removes extra free functions for the validation models.
  • domain assumption Longitudinal vector mode dominates cvG clustering; transverse modes negligible, allowing map onto scalar master equation.
    Section 3.2.3, citing Becker et al. 2020a.
  • standard math Standard FLRW Newtonian-gauge perturbation theory and Horndeski/EFT operator dictionary (Bellini & Sawicki; Gleyzes et al.).
    Sections 2.1–2.2; background literature assumed throughout.
invented entities (1)
  • Master Vainshtein equation (Eq. 27) with coefficients α, β, R_c² independent evidence
    purpose: Single discrete nonlinear PDE that specialises to nDGP, sDGP, csG, cvG, GCCG and generic EFT by changing only background functions.
    Not a new physical field; it is a code-oriented rewrite of the already-derived N-body equation (19–21). Independent evidence is the cross-code power-spectrum agreement.

pith-pipeline@v1.2.0-grok45-kimik3 · 35159 in / 4110 out tokens · 78876 ms · 2026-07-31T05:23:20.216346+00:00 · methodology

0 comments
read the original abstract

While the standard $\Lambda$CDM paradigm is in excellent agreement with most current cosmological observations, theoretical challenges surrounding the cosmological constant ($\Lambda$) have strongly motivated the exploration of dynamical dark energy (DE) and modified gravity (MG) models. Investigating the physical nature of the cosmic acceleration requires N-body simulations to probe the non-linear growth of cosmic structure and prepare for the high-precision data from Stage-IV surveys. In this paper, we present EFT-RAMSES, a comprehensive extension of the ECOSMOG cosmological simulation code designed to explore non-linear structure formation in DE and MG scenarios. We embed the effective field theory of dark energy (EFTofDE) framework into this new numerical pipeline, utilising the $\alpha$-basis parameterisation to provide a versatile, model-agnostic, computational engine. By consolidating diverse scalar and vector-tensor theories---including the normal and self-accelerating Dvali-Gabadadze-Porrati (DGP) models, cubic Galileons (cubic scalar Galileon (csG), cubic vector Galileon (cvG) and generalised cubic covariant Galileon (GCCG)), and generic effective field theory (EFT) parameterisation---into a single "master" Vainshtein equation, this pipeline bypasses the need for model-specific solvers and easily specialises to any particular model. As validations, we perform high-resolution N-body simulations for the normal-branch DGP (nDGP), csG, GCCG, and EFT models, comparing the resulting matter power spectra against dependent and independent codes such as legacy ECOSMOG and HiCOLA, as well as linear theory, and find excellent agreement. EFT-RAMSES provides a robust and versatile computational tool for precision cosmological tests of DE and MG using upcoming cosmological surveys. The code is available for download from the GitHub EFT-RAMSES repository.

Figures

Figures reproduced from arXiv: 2607.24940 by Baojiu Li, Nathaniel Ota Woodcock, Sownak Bose, Yunhao Gao.

Figure 1
Figure 1. Figure 1: presents the non-linear matter power spectrum (top panel) and its enhancement relative to the ΛCDM baseline (bottom panel), for the nDGP model, at three distinct redshifts: 𝑧 = 1.0, 0.5, and 0). We compare eft-ramses (dashed lines) against an older, established version of the ecosmog-V (solid lines) code. The latter differs from [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The fractional enhancement of the non-linear matter power spec￾trum at 𝑧 = 0 for the csG (red)z and QCDM (orange) models. The solid lines denote the eft-ramses simulations, which have been averaged over two inverted-phase realisations to suppress sample variance at low 𝑘. The dashed lines show the predictions from the HiCOLA code for comparison. The dotted horizontal lines indicate the linear-theory expect… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the non-linear matter power spectra at 𝑧 = 0 for the QCDM and GCCG cosmologies. The ecosmog-QCDM and ecosmog￾GCCG data are preliminary results from work in preparation by Atayde et al. (2026). Top panel: The absolute matter power spectra, 𝑃(𝑘). To improve visual clarity and prevent overlapping, the QCDM curves have been artificially shifted downward by a decade (×10−1 ). The outputs from ecos… view at source ↗
Figure 4
Figure 4. Figure 4: The fractional enhancement of the non-linear matter power spectrum for the EFT and QCDM models relative to the ΛCDM counterpart. Results are shown across three redshifts: 𝑧 = 1.0 (blue), 𝑧 = 0.5 (green), and 𝑧 = 0.0 (red). The left panel displays the EFTP model (𝑐B = 1.0), while the right panel displays the EFTN model (𝑐B = −1.0). Solid lines denote the full N-body simulations from eft-ramses, and dashed l… view at source ↗
Figure 5
Figure 5. Figure 5: The fractional enhancement of the non-linear matter power spectrum for the EFT model relative to the QCDM counterpart. Results are shown across three redshifts: 𝑧 = 1.0 (blue), 𝑧 = 0.5 (green), and 𝑧 = 0.0 (red). The left panel displays the EFTP model (𝑐B = 1.0), which suppresses structure formation relative to QCDM. The right panel displays the EFTN model (𝑐B = −1.0), which enhances structure formation. S… view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

260 extracted references · 4 canonical work pages

  1. [1]

    Alestas, George and Antoniou, Savvas and Perivolaropoulos, Leandros , title =. Phys. Rev. D , volume =. 2024 , doi =. 2309.15781 , archivePrefix =

    Pith/arXiv arXiv 2024

  2. [3]

    Mlapm - a c code for cosmological simulations

    Knebe, Alexander and Green, Andrew and Binney, James. Mlapm - a c code for cosmological simulations. Mon. Not. Roy. Astron. Soc. 2001. doi:10.1046/j.1365-8711.2001.04532.x. arXiv:astro-ph/0103503

    arXiv 2001

  3. [4]

    Structure Formation by Fifth Force I: N-Body vs

    Li, Baojiu and Zhao, Hongsheng. Structure Formation by Fifth Force I: N-Body vs. Linear Simulations. Phys. Rev. D. 2009. doi:10.1103/PhysRevD.80.044027. arXiv:0906.3880

    Pith/arXiv arXiv 2009

  4. [5]

    Astrophys

    Structure Formation by Fifth Force: Power Spectrum from N-Body Simulations. Astrophys. J. , keywords =. doi:10.1088/2041-8205/712/2/L179 , archivePrefix =. 0910.3207 , primaryClass =

    Pith/arXiv arXiv 2041

  5. [6]

    , keywords =

    Structure formation by the fifth force: Segregation of baryons and dark matter. , keywords =. doi:10.1103/PhysRevD.81.104047 , archivePrefix =. 1001.3152 , primaryClass =

    Pith/arXiv arXiv

  6. [7]

    N-body Simulations for f(R) Gravity using a Self-adaptive Particle-Mesh Code

    Zhao, Gong-Bo and Li, Baojiu and Koyama, Kazuya. N-body Simulations for f(R) Gravity using a Self-adaptive Particle-Mesh Code. Phys. Rev. D. 2011. doi:10.1103/PhysRevD.83.044007. arXiv:1011.1257

    Pith/arXiv arXiv 2011

  7. [8]

    N-Body Simulations for Coupled Scalar Field Cosmology

    Li, Baojiu and Barrow, John D. N-Body Simulations for Coupled Scalar Field Cosmology. Phys. Rev. D. 2011. doi:10.1103/PhysRevD.83.024007. arXiv:1005.4231

    Pith/arXiv arXiv 2011

  8. [9]

    Astrophys

    N-body Simulations for Extended Quintessence Models. Astrophys. J. , keywords =. doi:10.1088/0004-637X/728/2/109 , archivePrefix =. 1009.1400 , primaryClass =

    Pith/arXiv arXiv

  9. [10]

    Nonlinear Structure Formation with the Environmentally Dependent Dilaton

    Brax, Philippe and van de Bruck, Carsten and Davis, Anne-Christine and Li, Baojiu and Shaw, Douglas J. Nonlinear Structure Formation with the Environmentally Dependent Dilaton. Phys. Rev. D. 2011. doi:10.1103/PhysRevD.83.104026. arXiv:1102.3692

    Pith/arXiv arXiv 2011

  10. [11]

    and Winther, Hans A

    Davis, Anne-Christine and Li, Baojiu and Mota, David F. and Winther, Hans A. Structure Formation in the Symmetron Model. Astrophys. J. 2012. doi:10.1088/0004-637X/748/1/61. arXiv:1108.3081

    Pith/arXiv arXiv 2012

  11. [12]

    and Li, Baojiu

    Becker, Christoph and Eggemeier, Alexander and Davies, Christopher T. and Li, Baojiu. Proca-stinated cosmology. Part II. Matter, halo, and lensing statistics in the vector Galileon. J. Cosmol. Astropart. Phys. 2021. doi:10.1088/1475-7516/2021/06/014. arXiv:2011.01719

    Pith/arXiv arXiv 2021

  12. [13]

    Vainshtein

    A.I. Vainshtein. To the problem of nonvanishing gravitation mass. Physics Letters B. 1972. doi:https://doi.org/10.1016/0370-2693(72)90147-5

    doi:10.1016/0370-2693(72)90147-5 1972

  13. [14]

    A new multiprobe analysis of modified gravity and evolving dark energy , year =

    Lu, Zhiyu and Simon, Th. A new multiprobe analysis of modified gravity and evolving dark energy , year =. 2511.10616 , archivePrefix =

    arXiv

  14. [15]

    and others , collaboration =

    Riess, Adam G. and others , collaboration =. Observational evidence from supernovae for an accelerating universe and a cosmological constant , journal =. 1998 , doi =. astro-ph/9805201 , archivePrefix =

    Pith/arXiv arXiv 1998

  15. [18]

    2025 , month =

    DESI 2024 VI: Cosmological constraints from the measurements of baryon acoustic oscillations , journal =. 2025 , month =. doi:10.1088/1475-7516/2025/02/021 , eprint =

    doi:10.1088/1475-7516/2025/02/021 2024

  16. [19]

    Bertschinger, Edmund and Zukin, Phillip , title =. Phys. Rev. D , volume =. 2008 , doi =. 0801.2431 , archivePrefix =

    Pith/arXiv arXiv 2008

  17. [20]

    Gubitosi, Giulia and Piazza, Federico and Vernizzi, Filippo , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , doi =. 1210.0201 , archivePrefix =

    Pith/arXiv arXiv 2013

  18. [22]

    Pogosian, Levon and others , title =. Phys. Rev. D , volume =. 2010 , doi =. 1002.2382 , archivePrefix =

    Pith/arXiv arXiv 2010

  19. [23]

    Frusciante, Noemi and Perenon, Louis , title =. Phys. Rept. , volume =. 2020 , doi =. 1907.03150 , archivePrefix =

    Pith/arXiv arXiv 2020

  20. [24]

    Horndeski, Gregory Walter , title =. Int. J. Theor. Phys. , volume =. 1974 , doi =

    1974

  21. [25]

    Astrophys

    Ma, Chung-Pei and Bertschinger, Edmund , title =. Astrophys. J. , volume =. 1995 , doi =. astro-ph/9506072 , archivePrefix =

    Pith/arXiv arXiv 1995

  22. [27]

    Andrade, Uendert and Capistrano, Abra. Mon. Not. Roy. Astron. Soc. , year =. doi:10.1093/mnras/stae402 , eprint =

    work page doi:10.1093/mnras/stae402

  23. [28]

    and Langlois, D

    Gleyzes, J. and Langlois, D. and Piazza, F. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =. 2015 , doi =. 1408.1952 , archivePrefix =

    Pith/arXiv arXiv 2015

  24. [29]

    and Papadomanolakis, G

    Frusciante, N. and Papadomanolakis, G. and Silvestri, A. , title =. J. Cosmol. Astropart. Phys. , volume =. 2016 , doi =. 1601.04064 , archivePrefix =

    Pith/arXiv arXiv 2016

  25. [30]

    and Langlois, D

    Gleyzes, J. and Langlois, D. and Vernizzi, F. , title =. Int. J. Mod. Phys. D , volume =. 2015 , doi =. 1411.3712 , archivePrefix =

    Pith/arXiv arXiv 2015

  26. [31]

    and others , title =

    Amendola, L. and others , title =. Living Rev. Rel. , volume =. 2018 , doi =. 1606.00180 , archivePrefix =

    Pith/arXiv arXiv 2018

  27. [33]

    Kodama, Hideo and Sasaki, Misao , title =. Prog. Theor. Phys. Suppl. , volume =. 1984 , doi =

    1984

  28. [34]

    Essential building blocks of dark energy , journal =

    Gleyzes, J. Essential building blocks of dark energy , journal =. 2013 , doi =. 1304.4840 , archivePrefix =

    Pith/arXiv arXiv 2013

  29. [35]

    Lombriser, Lucas and Dalang, Charles and Kennedy, Joseph and Taylor, Andrew , title =. J. Cosmol. Astropart. Phys. , volume =. 2019 , doi =. 1811.08460 , archivePrefix =

    Pith/arXiv arXiv 2019

  30. [36]

    and Mirmelstein, M

    Carron, J. and Mirmelstein, M. M. and Lewis, A. , title =. J. Cosmol. Astropart. Phys. , volume =. 2022 , doi =. 2206.07783 , archivePrefix =

    Pith/arXiv arXiv 2022

  31. [37]

    and others , title =

    Popovic, B. and others , title =. 2025 , eprint =

    2025

  32. [38]

    and Eifler, T

    Krause, E. and Eifler, T. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2017 , doi =. 1601.05779 , archivePrefix =

    Pith/arXiv arXiv 2017

  33. [39]

    and Rogozenski, P

    Miranda, V. and Rogozenski, P. and Krause, E. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2021 , doi =. 2009.14241 , archivePrefix =

    Pith/arXiv arXiv 2021

  34. [40]

    Sharma, R. K. and Lesgourgues, Julien , title =. 2025 , eprint =

    2025

  35. [41]

    and Poulin, V

    Simon, Th\'eo and Zhang, P. and Poulin, V. , title =. Phys. Rev. D , volume =. 2023 , doi =. 2208.05929 , archivePrefix =

    Pith/arXiv arXiv 2023

  36. [42]

    and Poulin, V

    Simon, Th\'eo and Zhang, P. and Poulin, V. , title =. J. Cosmol. Astropart. Phys. , volume =. 2023 , doi =. 2210.14931 , archivePrefix =

    Pith/arXiv arXiv 2023

  37. [43]

    doi:10.1088/1475-7516/2025/10/057 , url =

    Yunhao Gao and Baojiu Li and Jie Wang , title =. doi:10.1088/1475-7516/2025/10/057 , url =

    work page doi:10.1088/1475-7516/2025/10/057 2025

  38. [44]

    and Baugh, Carlton M

    Barreira, Alexandre and Li, Baojiu and Hellwing, Wojciech A. and Baugh, Carlton M. and Pascoli, Silvia , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , doi =. 1306.3219 , archivePrefix =

    Pith/arXiv arXiv 2013

  39. [45]

    hi\_class: Horndeski in the Cosmic Linear Anisotropy Solving System , journal =

    Zumalac. hi\_class: Horndeski in the Cosmic Linear Anisotropy Solving System , journal =. 2017 , doi =. 1605.06102 , archivePrefix =

    Pith/arXiv arXiv 2017

  40. [46]

    and Kopp, Michael and Padilla, Antonio and Saffin, Paul M

    Copeland, Edmund J. and Kopp, Michael and Padilla, Antonio and Saffin, Paul M. and Skordis, Constantinos , title =. Phys. Rev. Lett. , volume =. 2019 , doi =. 1810.08239 , archivePrefix =

    Pith/arXiv arXiv 2019

  41. [47]

    and Jimenez, Raul and Verde, Licia , title =

    Bellini, Emilio and Cuesta, Antonio J. and Jimenez, Raul and Verde, Licia , title =. J. Cosmol. Astropart. Phys. , volume =. 2016 , doi =. 1509.07816 , archivePrefix =

    Pith/arXiv arXiv 2016

  42. [48]

    and Zumalac

    Alonso, David and Bellini, Emilio and Ferreira, Pedro G. and Zumalac. Phys. Rev. D , volume =. 2017 , doi =. 1610.09290 , archivePrefix =

    Pith/arXiv arXiv 2017

  43. [49]

    Gleyzes, J. J. Cosmol. Astropart. Phys. , volume =. 2016 , doi =. 1509.02191 , archivePrefix =

    Pith/arXiv arXiv 2016

  44. [50]

    Chudaykin, Anton and Kunz, Martin , title =. Phys. Rev. D , volume =. 2024 , doi =. 2407.02558 , archivePrefix =

    Pith/arXiv arXiv 2024

  45. [51]

    Abbott, B. P. and others , collaboration =. Astrophys. J. Lett. , volume =. 2017 , doi =. 1710.05834 , archivePrefix =

    Pith/arXiv arXiv 2017

  46. [53]

    and Komatsu, Eiichiro , title =

    Kreisch, Christina D. and Komatsu, Eiichiro , title =. J. Cosmol. Astropart. Phys. , volume =. 2018 , doi =. 1712.02710 , archivePrefix =

    Pith/arXiv arXiv 2018

  47. [55]

    Chevallier, Michel and Polarski, David , title =. Int. J. Mod. Phys. D , volume =. 2001 , doi =. gr-qc/0009008 , archivePrefix =

    Pith/arXiv arXiv 2001

  48. [56]

    , title =

    Linder, Eric V. , title =. Phys. Rev. Lett. , volume =. 2003 , doi =. astro-ph/0208512 , archivePrefix =

    Pith/arXiv arXiv 2003

  49. [58]

    and others , collaboration =

    Aghanim, N. and others , collaboration =. Astron. Astrophys. , volume =. 2020 , doi =. 1907.12875 , archivePrefix =

    Pith/arXiv arXiv 2020

  50. [60]

    and others , collaboration =

    Abdul Karim, M. and others , collaboration =. Phys. Rev. D , volume =. 2025 , doi =. 2503.14739 , archivePrefix =

    Pith/arXiv arXiv 2025

  51. [61]

    and others , title =

    Bennett, Charles L. and others , title =. Astrophys. J. Suppl. , volume =. 2013 , doi =. 1212.5225 , archivePrefix =

    Pith/arXiv arXiv 2013

  52. [62]

    Aiola, Simone and others , collaboration =. J. Cosmol. Astropart. Phys. , volume =. 2020 , doi =. 2007.07288 , archivePrefix =

    Pith/arXiv arXiv 2020

  53. [63]

    Choi, Soonyeong and others , collaboration =. J. Cosmol. Astropart. Phys. , volume =. 2020 , doi =. 2007.07289 , archivePrefix =

    Pith/arXiv arXiv 2020

  54. [64]

    Astrophys

    Qu, Feng and others , collaboration =. Astrophys. J. , volume =. 2023 , doi =. 2301.06219 , archivePrefix =

    Pith/arXiv arXiv 2023

  55. [65]

    and others , collaboration =

    Madhavacheril, Mathew S. and others , collaboration =. Phys. Rev. D , volume =. 2023 , doi =. 2301.06254 , archivePrefix =

    Pith/arXiv arXiv 2023

  56. [66]

    Balkenhol, Leonhard and others , collaboration =. Phys. Rev. D , volume =. 2023 , doi =. 2212.05642 , archivePrefix =

    Pith/arXiv arXiv 2023

  57. [67]

    and others , title =

    Ross, Ashley J. and others , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2015 , doi =. 1409.3242 , archivePrefix =

    Pith/arXiv arXiv 2015

  58. [68]

    Alam, Shadab and others , collaboration =. Phys. Rev. D , volume =. 2021 , doi =. 2007.08991 , archivePrefix =

    Pith/arXiv arXiv 2021

  59. [71]

    Amendola, Luca and Kunz, Martin and Sapone, Domenico , title =. J. Cosmol. Astropart. Phys. , volume =. 2008 , doi =. 0704.2421 , archivePrefix =

    Pith/arXiv arXiv 2008

  60. [72]

    and Linder, Eric V

    Daniel, Samuel F. and Linder, Eric V. and Smith, Tristan L. and Caldwell, Robert R. and Cooray, Asantha and Leauthaud, Alexie , title =. Phys. Rev. D , volume =. 2010 , doi =. 1002.1962 , archivePrefix =

    Pith/arXiv arXiv 2010

  61. [73]

    Scolnic, D. M. and others , title =. Astrophys. J. , volume =. 2018 , doi =. 1710.00845 , archivePrefix =

    Pith/arXiv arXiv 2018

  62. [74]

    and others , collaboration =

    Akrami, Y. and others , collaboration =. Astron. Astrophys. , volume =. 2020 , doi =. 2007.04997 , archivePrefix =

    Pith/arXiv arXiv 2020

  63. [75]

    and Mirmelstein, M

    Carron, J. and Mirmelstein, M. and Lewis, A. , title =. J. Cosmol. Astropart. Phys. , volume =. 2022 , doi =. 2206.07773 , archivePrefix =

    Pith/arXiv arXiv 2022

  64. [76]

    and Lewis, A

    Carron, J. and Lewis, A. and Fabbian, G. , title =. Phys. Rev. D , volume =. 2022 , doi =. 2209.07395 , archivePrefix =

    Pith/arXiv arXiv 2022

  65. [77]

    and others , title =

    Pellejero-Ibanez, M. and others , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2017 , doi =. 1607.03155 , archivePrefix =

    Pith/arXiv arXiv 2017

  66. [78]

    Sabogal, M. A. and Nunes, R. C. , title =. J. Cosmol. Astropart. Phys. , volume =. 2025 , doi =. 2505.24465 , archivePrefix =

    arXiv 2025

  67. [79]

    and others , title =

    Zhou, R. and others , title =. J. Cosmol. Astropart. Phys. , volume =. 2023 , doi =. 2309.06443 , archivePrefix =

    Pith/arXiv arXiv 2023

  68. [80]

    and others , collaboration =

    Zhou, R. and others , collaboration =. Astron. J. , volume =. 2023 , doi =. 2208.08515 , archivePrefix =

    Pith/arXiv arXiv 2023

  69. [81]

    and Lesgourgues, J

    Audren, B. and Lesgourgues, J. and Benabed, K. and Prunet, S. , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , doi =. 1210.7183 , archivePrefix =

    Pith/arXiv arXiv 2013

  70. [82]

    and Lesgourgues, J

    Brinckmann, T. and Lesgourgues, J. , title =. Phys. Dark Univ. , volume =. 2019 , doi =. 1804.07261 , archivePrefix =

    Pith/arXiv arXiv 2019

  71. [83]

    and others , collaboration =

    Abdul-Karim, M. and others , collaboration =. Phys. Rev. D , volume =. 2025 , doi =. 2503.14738 , archivePrefix =

    Pith/arXiv arXiv 2025

  72. [84]

    Astrophys

    Astron. Astrophys. , volume =. 2025 , doi =

    2025

  73. [85]

    2012 , eprint=

    Large Synoptic Survey Telescope: Dark Energy Science Collaboration , author=. 2012 , eprint=

    2012

  74. [86]

    Abbott, B. P. and others , collaboration =. Astrophys. J. Lett. , volume =. 2017 , doi =. 1710.05833 , archivePrefix =

    Pith/arXiv arXiv 2017

  75. [87]

    Abbott, T. M. C. and others , collaboration =. Phys. Rev. D , volume =. 2026 , doi =. 2503.06712 , archivePrefix =

    arXiv 2026

  76. [88]

    and others , collaboration =

    Aghanim, N. and others , collaboration =. Astron. Astrophys. , volume =. 2020 , doi =. 1807.06209 , archivePrefix =

    Pith/arXiv arXiv 2020

  77. [89]

    Alimi, J. -M. and Fuzfa, A. and Rasera, Y. and Courtin, J. and Corasaniti, P. -S. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2010 , doi =. 0903.5490 , archivePrefix =

    Pith/arXiv arXiv 2010

  78. [90]

    and Bellini, E

    Baker, T. and Bellini, E. and Ferreira, P. G. and Lagos, M. and Noller, J. and Sawicki, I. , title =. Phys. Rev. Lett. , volume =. 2017 , doi =. 1710.06394 , archivePrefix =

    Pith/arXiv arXiv 2017

  79. [91]

    and Arnold, C

    Becker, C. and Arnold, C. and Li, B. and Heisenberg, L. , title =. J. Cosmol. Astropart. Phys. , volume =. 2020 , doi =. 2006.02422 , archivePrefix =

    Pith/arXiv arXiv 2020

  80. [92]

    and Jimenez, R

    Bellini, E. and Jimenez, R. and Verde, L. , title =. J. Cosmol. Astropart. Phys. , volume =. 2015 , doi =. 1504.04341 , archivePrefix =

    Pith/arXiv arXiv 2015

Showing first 80 references.