Pith. sign in

REVIEW 2 major objections 8 minor 1 cited by

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

T0 review · 2 major / 8 minor · reviewed 2026-07-31 · grok-4.5

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

Pith's one-line read One master Vainshtein equation lets a single N-body code simulate many dark-energy and modified-gravity models.

desk verdict 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. read the letter →

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

Nathaniel Ota Woodcock , Sownak Bose , Yunhao Gao , Baojiu Li This is my paper · ORCID
classification astro-ph.COgr-qcphysics.comp-ph
keywords effectivefieldtheoryofdarkenergyN-bodysimulationsVainshteinscreeningmodifiedgravityHorndeskimatterpowerspectrumcubicGalileonDGP
topics Dark Energy
open problems Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

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.

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

Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, 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 · score 1.0 of 10

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.

Assumptions & free parameters 5 free parameters · 7 assumptions · 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.
assumptions (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.

how reviews work

0 comments
Cite this review

Pith. "Pith review of EFT-Ramses: a code to simulate the effective field theory of dark energy." pith.science (2026). https://pith.science/paper/I373WXWP

@misc{pith2026260724940,
  author       = {Pith},
  title        = {Pith review of: EFT-Ramses: a code to simulate the effective field theory of dark energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I373WXWP}},
  note         = {Machine review of arXiv:2607.24940}
}
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 the authors.

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. 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 expectations bas… view at source ↗
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 ecosmog (thin … view at source ↗
Figures from the paper (2 more)
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…
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…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. N-body Simulations of Large-Scale Structure in the Generalized Cubic Covariant Galileon Model

    astro-ph.CO 2026-08 conditional novelty 6.0 of 10

    In the GCCG modified-gravity model, late-time structure formation is enhanced by up to about 7 percent at z=0, and the fast halo-model reaction method matches simulations to percent level on mildly nonlinear scales.

Reference graph

Works this paper leans on

260 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

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

    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

    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 =

    arXiv

  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 =

    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

    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

    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 =

    arXiv

Show all 260 references
  1. [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

    2011 arXiv

  2. [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

    2012 arXiv

  3. [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

    2021 arXiv

  4. [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

    1972 doi

  5. [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 =

  6. [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 =

    1998 arXiv

  7. [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 =

    2024 doi

  8. [19]

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

    2008 arXiv

  9. [20]

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

    2013 arXiv

  10. [22]

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

    2010 arXiv

  11. [23]

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

    2020 arXiv

  12. [24]

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

    1974

  13. [25]

    Astrophys

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

    1995 arXiv

  14. [27]

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

    doi

  15. [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 =

    2015 arXiv

  16. [29]

    and Papadomanolakis, G

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

    2016 arXiv

  17. [30]

    and Langlois, D

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

    2015 arXiv

  18. [31]

    and others , title =

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

    2018 arXiv

  19. [33]

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

    1984

  20. [34]

    Essential building blocks of dark energy , journal =

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

    2013 arXiv

  21. [35]

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

    2019 arXiv

  22. [36]

    and Mirmelstein, M

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

    2022 arXiv

  23. [37]

    and others , title =

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

    2025

  24. [38]

    and Eifler, T

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

    2017 arXiv

  25. [39]

    and Rogozenski, P

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

    2021 arXiv

  26. [40]

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

    2025

  27. [41]

    and Poulin, V

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

    2023 arXiv

  28. [42]

    and Poulin, V

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

    2023 arXiv

  29. [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 =

    2025 doi

  30. [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 =

    2013 arXiv

  31. [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 =

    2017 arXiv

  32. [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 =

    2019 arXiv

  33. [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 =

    2016 arXiv

  34. [48]

    and Zumalac

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

    2017 arXiv

  35. [49]

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

    2016 arXiv

  36. [50]

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

    2024 arXiv

  37. [51]

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

    2017 arXiv

  38. [53]

    and Komatsu, Eiichiro , title =

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

    2018 arXiv

  39. [55]

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

    2001 arXiv

  40. [56]

    , title =

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

    2003 arXiv

  41. [58]

    and others , collaboration =

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

    2020 arXiv

  42. [60]

    and others , collaboration =

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

    2025 arXiv

  43. [61]

    and others , title =

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

    2013 arXiv

  44. [62]

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

    2020 arXiv

  45. [63]

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

    2020 arXiv

  46. [64]

    Astrophys

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

    2023 arXiv

  47. [65]

    and others , collaboration =

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

    2023 arXiv

  48. [66]

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

    2023 arXiv

  49. [67]

    and others , title =

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

    2015 arXiv

  50. [68]

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

    2021 arXiv

  51. [71]

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

    2008 arXiv

  52. [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 =

    2010 arXiv

  53. [73]

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

    2018 arXiv

  54. [74]

    and others , collaboration =

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

    2020 arXiv

  55. [75]

    and Mirmelstein, M

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

    2022 arXiv

  56. [76]

    and Lewis, A

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

    2022 arXiv

  57. [77]

    and others , title =

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

    2017 arXiv

  58. [78]

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

    2025

  59. [79]

    and others , title =

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

    2023 arXiv

  60. [80]

    and others , collaboration =

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

    2023 arXiv

  61. [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 =

    2013 arXiv

  62. [82]

    and Lesgourgues, J

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

    2019 arXiv

  63. [83]

    and others , collaboration =

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

    2025 arXiv

  64. [84]

    Astrophys

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

    2025

  65. [85]

    2012 , eprint=

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

    2012

  66. [86]

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

    2017 arXiv

  67. [87]

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

    2026

  68. [88]

    and others , collaboration =

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

    2020 arXiv

  69. [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 =

    2010 arXiv

  70. [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 =

    2017 arXiv

  71. [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 =

    2020 arXiv

  72. [92]

    and Jimenez, R

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

    2015 arXiv

  73. [93]

    and Sawicki, I

    Bellini, E. and Sawicki, I. , title =. J. Cosmol. Astropart. Phys. , volume =. 2014 , doi =. 1404.3713 , archivePrefix =

    2014 arXiv

  74. [94]

    and Sawicki, I

    Bellini, E. and Sawicki, I. and Zumalac\'arregui, M. , title =. J. Cosmol. Astropart. Phys. , volume =. 2020 , doi =. 1909.01828 , archivePrefix =

    2020 arXiv

  75. [95]

    and others , title =

    Bellini, E. and others , title =. Phys. Rev. D , volume =. 2018 , doi =. 1709.09135 , archivePrefix =

    2018 arXiv

  76. [96]

    Bloomfield, J. K. and Flanagan, \'E. \'E. and Park, M. and Watson, S. , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , doi =. 1211.7054 , archivePrefix =

    2013 arXiv

  77. [97]

    and Tsedrik, M

    Bose, B. and Tsedrik, M. and Kennedy, J. and Lombriser, L. and Pourtsidou, A. and Taylor, A. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2023 , doi =. 2210.01094 , archivePrefix =

    2023 arXiv

  78. [98]

    and Koyama, K

    Brando, G. and Koyama, K. and Winther, H. A. , title =. J. Cosmol. Astropart. Phys. , volume =. 2023 , doi =. 2303.09549 , archivePrefix =

    2023 arXiv

  79. [99]

    Breton, M. -A. , title =. Astron. Astrophys. , volume =. 2025 , doi =. 2407.13636 , archivePrefix =

    2025 arXiv

  80. [100]

    Breton, M. -A. and Rasera, Y. and Taruya, A. and Lacombe, O. and Saga, S. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2019 , doi =. 1803.04294 , archivePrefix =

    2019 arXiv

  81. [101]

    and Jaffe, A

    Colombi, S. and Jaffe, A. and Novikov, D. and Pichon, C. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2009 , doi =. 0811.0313 , archivePrefix =

    2009 arXiv

  82. [102]

    and Vernizzi, F

    Creminelli, P. and Vernizzi, F. , title =. Phys. Rev. Lett. , volume =. 2017 , doi =. 1710.05877 , archivePrefix =

    2017 arXiv

  83. [103]

    and Lewandowski, M

    Cusin, G. and Lewandowski, M. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =

  84. [104]

    and Pantiri, M

    de Boe, D. and Pantiri, M. and Ye, G. and Silvestri, A. , title =. 2026 , eprint =

    2026

  85. [105]

    and Gao, X

    Deffayet, C. and Gao, X. and Steer, D. A. and Zahariade, G. , title =. Phys. Rev. D , volume =

  86. [106]

    and Papadomanolakis, G

    Frusciante, N. and Papadomanolakis, G. , title =. J. Cosmol. Astropart. Phys. , volume =

  87. [107]

    and Perenon, L

    Frusciante, N. and Perenon, L. , title =. Phys. Rep. , volume =

  88. [108]

    and Langlois, D

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

  89. [109]

    and Langlois, D

    Gleyzes, J. and Langlois, D. and Vernizzi, F. , title =. Int. J. Mod. Phys. D , volume =

  90. [110]

    and Piazza, F

    Gubitosi, G. and Piazza, F. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =

  91. [111]

    and Teyssier, R

    Guillet, T. and Teyssier, R. , title =. J. Comput. Phys. , volume =

  92. [112]

    Gupta, A. S. and Fiorini, B. and Baker, T. , title =. J. Cosmol. Astropart. Phys. , volume =

  93. [113]

    and Adamek, J

    Hassani, F. and Adamek, J. and Kunz, M. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =

  94. [114]

    and Lombriser, L

    Hassani, F. and Lombriser, L. , title =. Mon. Not. Roy. Astron. Soc. , volume =

  95. [115]

    Horndeski, G. W. , title =. Int. J. Theor. Phys. , volume =

  96. [116]

    Effective field theory of cosmic acceleration: An implementation in CAMB , author =. Phys. Rev. D , volume =. 2014 , month =. doi:10.1103/PhysRevD.89.103530 , url =

    2014 doi

  97. [117]

    , title =

    Langlois, D. , title =. Int. J. Mod. Phys. D , volume =

  98. [118]

    and Mancarella, M

    Langlois, D. and Mancarella, M. and Noui, K. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =

  99. [119]

    and Noui, K

    Langlois, D. and Noui, K. , title =. J. Cosmol. Astropart. Phys. , volume =

  100. [120]

    and Allen, L

    Levi, M. and Allen, L. E. and Raichoor, A. and others , title =. Bull. Am. Astron. Soc. , volume =

  101. [121]

    and Challinor, A

    Lewis, A. and Challinor, A. and Lasenby, A. , title =. Astrophys. J. , volume =

  102. [122]

    and Zhao, G.-B

    Li, B. and Zhao, G.-B. and Koyama, K. , title =. J. Cosmol. Astropart. Phys. , volume =

  103. [123]

    , title =

    Llinares, C. , title =. Int. J. Mod. Phys. D , volume =

  104. [124]

    and Simon, T

    Lu, Z. and Simon, T. and Zhang, P. , title =. 2025 , eprint =

    2025

  105. [125]

    and Shapiro, P

    Martel, H. and Shapiro, P. R. , title =. Mon. Not. Roy. Astron. Soc. , volume =

  106. [126]

    and Frusciante, N

    Moretti, T. and Frusciante, N. and Verza, G. and Pace, F. , title =. 2026 , eprint =

    2026

  107. [127]

    and Hassani, F

    Nouri-Zonoz, A. and Hassani, F. and Bellini, E. and Kunz, M. , title =. 2025 , eprint =

    2025

  108. [128]

    and Pichon, C

    Prunet, S. and Pichon, C. and Aubert, D. and others , title =. Astrophys. J. Suppl. Ser. , volume =

  109. [129]

    and Breton, M.-A

    Rasera, Y. and Breton, M.-A. and Corasaniti, P.-S. and others , title =. Astron. Astrophys. , volume =

  110. [130]

    and Corasaniti, P.-S

    Rasera, Y. and Corasaniti, P.-S. and Alimi, J.-M. and others , title =. Mon. Not. Roy. Astron. Soc. , volume =

  111. [131]

    and Klypin, Anatoly and Prada, Francisco

    Ruan, Cheng-Zong and Hern \'a ndez-Aguayo, C \'e sar and Li, Baojiu and Arnold, Christian and Baugh, Carlton M. and Klypin, Anatoly and Prada, Francisco. Fast full N-body simulations of generic modified gravity: conformal coupling models. J. Cosmol. Astropart. Phys. 2022. doi:...

    2022

  112. [132]

    and Klypin, Anatoly and Prada, Francisco

    Hern \'a ndez-Aguayo, C \'e sar and Ruan, Cheng-Zong and Li, Baojiu and Arnold, Christian and Baugh, Carlton M. and Klypin, Anatoly and Prada, Francisco. Fast full N-body simulations of generic modified gravity: derivative coupling models. J. Cosmol. Astropart. Phys. 2022. doi...

    2022

  113. [133]

    Realistic simulations of galaxy formation in f(R) modified gravity

    Arnold, Christian and Leo, Matteo and Li, Baojiu. Realistic simulations of galaxy formation in f(R) modified gravity. Nature Astron. 2019. doi:10.1038/s41550-019-0823-y. arXiv:1907.02977

    2019 arXiv

  114. [134]

    Galaxy formation in the brane world I: overview and first results

    Hern \'a ndez-Aguayo, C \'e sar and Arnold, Christian and Li, Baojiu and Baugh, Carlton M. Galaxy formation in the brane world I: overview and first results. Mon. Not. Roy. Astron. Soc. 2021. doi:10.1093/mnras/stab694. arXiv:2006.15467

    2021 arXiv

  115. [135]

    Modified Gravity-GADGET: A new code for cosmological hydrodynamical simulations of modified gravity models

    Puchwein, Ewald and Baldi, Marco and Springel, Volker. Modified Gravity-GADGET: A new code for cosmological hydrodynamical simulations of modified gravity models. Mon. Not. Roy. Astron. Soc. 2013. doi:10.1093/mnras/stt1575. arXiv:1305.2418

    2013 arXiv

  116. [136]

    and Winther, Hans A

    Llinares, Claudio and Mota, David F. and Winther, Hans A. ISIS: a new N-body cosmological code with scalar fields based on RAMSES. Code presentation and application to the shapes of clusters. Astron. Astrophys. 2014. doi:10.1051/0004-6361/201322412. arXiv:1307.6748

    2014 arXiv

  117. [137]

    Systematic simulations of modified gravity: symmetron and dilaton models. J. Cosmol. Astropart. Phys. , keywords =. doi:10.1088/1475-7516/2012/10/002 , archivePrefix =. 1206.3568 , primaryClass =

    2012 arXiv

  118. [138]

    and Li, Baojiu and Winther, Hans A

    Brax, Philippe and Davis, Anne-C. and Li, Baojiu and Winther, Hans A. and Zhao, Gong-Bo. Systematic simulations of modified gravity: chameleon models. J. Cosmol. Astropart. Phys. 2013. doi:10.1088/1475-7516/2013/04/029. arXiv:1303.0007

    2013 arXiv

  119. [141]

    PySCo-EFT and ECOSMOG-EFT: a tandem of N-body simulation codes for the Effective Field Theory of Dark Energy , journal =

    Ganjoo, Himanish and Rasera, Yann and Bellini, Emilio and Breton, Michel-Andr. PySCo-EFT and ECOSMOG-EFT: a tandem of N-body simulation codes for the Effective Field Theory of Dark Energy , journal =. 2026 , eprint =

    2026

  120. [142]

    Matter power spectra in modified gravity: a comparative study of approximations and N-body simulations

    Bose, Benjamin and others. Matter power spectra in modified gravity: a comparative study of approximations and N-body simulations. Mon. Not. Roy. Astron. Soc. 2024. doi:10.1093/mnras/stae2562. arXiv:2406.13667

    2024 arXiv

  121. [143]

    and Pontzen, Andrew

    Angulo, Raul E. and Pontzen, Andrew. Cosmological N -body simulations with suppressed variance. Mon. Not. Roy. Astron. Soc. 2016. doi:10.1093/mnrasl/slw098. arXiv:1603.05253

    2016 arXiv

  122. [144]

    and Sen Gupta, Ashim and Baker, Tessa and Valogiannis, Georgios and Fiorini, Bartolomeo , year=

    Wright, Bill S. and Sen Gupta, Ashim and Baker, Tessa and Valogiannis, Georgios and Fiorini, Bartolomeo , year=. Hi-COLA: fast, approximate simulations of structure formation in Horndeski gravity , volume=. J. Cosmol. Astropart. Phys. , publisher=. doi:10.1088/1475-7516/2023/0...

    2023 doi

  123. [145]

    S. Mon. Not. Roy. Astron. Soc. , volume =

  124. [146]

    and Jain, B

    Sakstein, J. and Jain, B. , title =. Phys. Rev. Lett. , volume =

  125. [147]

    and Crocce, M

    Sefusatti, E. and Crocce, M. and Scoccimarro, R. and Couchman, H. , title =. Mon. Not. Roy. Astron. Soc. , volume =

  126. [148]

    10.1051/0004-6361:20011817

    Cosmological hydrodynamics with adaptive mesh refinement - A new high resolution code called RAMSES , DOI= "10.1051/0004-6361:20011817", url= "https://doi.org/10.1051/0004-6361:20011817", journal =

    doi

  127. [149]

    , title =

    Villaescusa-Navarro, F. , title =

  128. [150]

    Will, Clifford M. , year=. The Confrontation between General Relativity and Experiment , volume=. Living Reviews in Relativity , publisher=. doi:10.12942/lrr-2014-4 , number=

    2014 doi

  129. [151]

    Winther, H. A. and Ferreira, P. G. , title =. Phys. Rev. D , volume =

  130. [152]

    Winther, H. A. and others , title =. Mon. Not. Roy. Astron. Soc. , volume =

  131. [153]

    Wright, B. S. and Sen Gupta, A. and Baker, T. and Valogiannis, G. and Fiorini, B. , title =. J. Cosmol. Astropart. Phys. , volume =

  132. [154]

    and Yokoyama, S

    Yamauchi, D. and Yokoyama, S. and Tashiro, H. , title =. Phys. Rev. D , volume =

  133. [155]

    Zumalac. J. Cosmol. Astropart. Phys. , volume =

  134. [156]

    Zumalac. Phys. Rev. D , volume =

  135. [157]

    Riess, A. G. and others , title =. Astron. J. , volume =. 1998 , eprint =

    1998

  136. [158]

    , keywords =

    Measurements of and from 42 High-Redshift Supernovae. , keywords =. doi:10.1086/307221 , archivePrefix =. astro-ph/9812133 , primaryClass =

    arXiv

  137. [159]

    and Peebles, P

    Ratra, B. and Peebles, P. J. E. , title =. Phys. Rev. D , volume =

  138. [160]

    Ferreira, P. G. and Joyce, M. , title =. Phys. Rev. Lett. , volume =. 1997 , eprint =

    1997

  139. [161]

    Copeland, E. J. and Liddle, A. R. and Wands, D. , title =. Phys. Rev. D , volume =. 1998 , eprint =

    1998

  140. [162]

    Caldwell, R. R. and Dave, R. and Steinhardt, P. J. , title =. Phys. Rev. Lett. , volume =. 1998 , eprint =

    1998

  141. [163]

    and Wang, L

    Zlatev, I. and Wang, L. and Steinhardt, P. J. , title =. Phys. Rev. Lett. , volume =. 1999 , eprint =

    1999

  142. [164]

    and Odintsov, S

    Nojiri, S. and Odintsov, S. D. , title =. Int. J. Geom. Meth. Mod. Phys. , volume =. 2007 , eprint =

    2007

  143. [165]

    Sotiriou, T. P. and Faraoni, V. , title =. Rev. Mod. Phys. , volume =. 2010 , eprint =

    2010

  144. [166]

    Armend. Phys. Lett. B , volume =. 1999 , eprint =

    1999

  145. [167]

    and Okabe, T

    Chiba, T. and Okabe, T. and Yamaguchi, M. , title =. Phys. Rev. D , volume =. 2000 , eprint =

    2000

  146. [168]

    Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress tensors , author =. Phys. Rev. D , volume =. 2009 , month =. doi:10.1103/PhysRevD.80.064015 , url =

    2009 doi

  147. [169]

    Physics Letters B , keywords =

    4D gravity on a brane in 5D Minkowski space. Physics Letters B , keywords =. doi:10.1016/S0370-2693(00)00669-9 , archivePrefix =. hep-th/0005016 , primaryClass =

    arXiv

  148. [170]

    and Shafer, D

    Huterer, D. and Shafer, D. L. , title =. Rep. Prog. Phys. , volume =. 2018 , eprint =

    2018

  149. [171]

    and others , title =

    Moresco, M. and others , title =. Living Rev. Rel. , volume =. 2022 , eprint =

    2022

  150. [172]

    ACCELERATING UNIVERSES WITH SCALING DARK MATTER , volume=

    Chevallier, Michel and Polarski, David , year=. ACCELERATING UNIVERSES WITH SCALING DARK MATTER , volume=. International Journal of Modern Physics D , publisher=. doi:10.1142/s0218271801000822 , number=

    doi

  151. [173]

    Linder, E. V. , title =. Phys. Rev. Lett. , volume =. 2003 , eprint =

    2003

  152. [174]

    Constraints on a new post-general relativity cosmological parameter , volume=

    Caldwell, Robert and Cooray, Asantha and Melchiorri, Alessandro , year=. Constraints on a new post-general relativity cosmological parameter , volume=. Physical Review D , publisher=. doi:10.1103/physrevd.76.023507 , number=

    doi

  153. [175]

    and Kunz, M

    Amendola, L. and Kunz, M. and Sapone, D. , title =. J. Cosmol. Astropart. Phys. , volume =. 2008 , eprint =

    2008

  154. [176]

    and Piazza, F

    Gubitosi, G. and Piazza, F. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , eprint =

    2013

  155. [177]

    and Flanagan,

    Bloomfield, J. and Flanagan,. J. Cosmol. Astropart. Phys. , volume =. 2013 , eprint =

    2013

  156. [178]

    and Langlois, D

    Gleyzes, J. and Langlois, D. and Piazza, F. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , eprint =

    2013

  157. [179]

    and Papadomanolakis, G

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

    2016

  158. [180]

    and Papadomanolakis, G

    Frusciante, N. and Papadomanolakis, G. , title =. J. Cosmol. Astropart. Phys. , volume =. 2017 , eprint =

    2017

  159. [181]

    and Lewandowski, M

    Cusin, G. and Lewandowski, M. and Vernizzi, F. , title =. J. Cosmol. Astropart. Phys. , volume =. 2018 , eprint =

    2018

  160. [182]

    and Nakamura, T

    Chiba, T. and Nakamura, T. , title =. 20th Texas Symposium on relativistic astrophysics , volume =. 2001 , publisher =

    2001

  161. [183]

    and Barrow, J

    Rubano, C. and Barrow, J. D. , title =. Phys. Rev. D , volume =. 2001 , eprint =

    2001

  162. [184]

    Gerke, B. F. and Efstathiou, G. , title =. Mon. Not. Roy. Astron. Soc. , volume =. 2002 , eprint =

    2002

  163. [185]

    , title =

    Zhang, X. , title =. Phys. Lett. B , volume =. 2007 , eprint =

    2007

  164. [186]

    and Ma, D.-Z

    Wu, J.-P. and Ma, D.-Z. and Ling, Y. , title =. Phys. Lett. B , volume =. 2008 , eprint =

    2008

  165. [187]

    and Mukherjee, A

    Sangwan, A. and Mukherjee, A. and Jassal, H. K. , title =. J. Cosmol. Astropart. Phys. , volume =. 2018 , eprint =

    2018

  166. [188]

    and Raveri, M

    Park, M. and Raveri, M. and Jain, B. , title =. Phys. Rev. D , volume =. 2021 , eprint =

    2021

  167. [189]

    and Odintsov, S

    Nojiri, S. and Odintsov, S. D. , title =. Gen. Rel. Grav. , volume =. 2006 , eprint =

    2006

  168. [190]

    and Nojiri, S

    Capozziello, S. and Nojiri, S. and Odintsov, S. D. , title =. Phys. Lett. B , volume =. 2006 , eprint =

    2006

  169. [191]

    and Esposito-Far

    Boisseau, B. and Esposito-Far. Phys. Rev. Lett. , volume =. 2000 , eprint =

    2000

  170. [192]

    and Odintsov, S

    Nojiri, S. and Odintsov, S. D. and Sami, M. , title =. Phys. Rev. D , volume =. 2006 , eprint =

    2006

  171. [193]

    and Elizalde, E

    Cognola, G. and Elizalde, E. and Nojiri, S. and Odintsov, S. D. and Zerbini, S. , title =. Phys. Rev. D , volume =. 2007 , eprint =

    2007

  172. [194]

    and Odintsov, S

    Nojiri, S. and Odintsov, S. D. , title =. Journal of Physics Conference Series , volume =. 2007 , publisher =. hep-th/0611071 , archivePrefix =

    2007 arXiv

  173. [195]

    and Nesseris, S

    Capozziello, S. and Nesseris, S. and Perivolaropoulos, L. , title =. J. Cosmol. Astropart. Phys. , volume =. 2007 , eprint =

    2007

  174. [196]

    and Davis, A.-C

    Brax, P. and Davis, A.-C. and Li, B. , title =. Phys. Lett. B , volume =. 2012 , eprint =

    2012

  175. [197]

    and Odintsov, S

    Nojiri, S. and Odintsov, S. D. , title =. Phys. Rev. D , volume =. 2006 , eprint =

    2006

  176. [198]

    Elizalde, E. and S. Phys. Rev. D , volume =. 2009 , eprint =

    2009

  177. [199]

    and Khaledian, M

    Karami, K. and Khaledian, M. S. , title =. J. High Energy Phys. , volume =. 2011 , eprint =

    2011

  178. [200]

    and Goswami, R

    Carloni, S. and Goswami, R. and Dunsby, P. K. S. , title =. Class. Quant. Grav. , volume =. 2012 , eprint =

    2012

  179. [201]

    Houndjo, M. J. S. , title =. Int. J. Mod. Phys. D , volume =. 2012 , eprint =

    2012

  180. [202]

    and Momeni, D

    Jamil, M. and Momeni, D. and Raza, M. and Myrzakulov, R. , title =. Eur. Phys. J. C , volume =. 2012 , eprint =

    2012

  181. [203]

    Singh, C. P. and Singh, V. , title =. Gen. Rel. Grav. , volume =

  182. [204]

    and Myrzakulov, R

    Elizalde, E. and Myrzakulov, R. and Obukhov, V. V. and S. Class. Quant. Grav. , volume =. 2010 , eprint =

    2010

  183. [205]

    and Rodrigues, M

    Hamani Daouda, M. and Rodrigues, M. E. and Houndjo, M. J. S. , title =. Eur. Phys. J. C , volume =. 2012 , eprint =

    2012

  184. [206]

    Elkhateeb, E. A. , title =. Found. Phys. , volume =. 2024 , eprint =

    2024

  185. [207]

    , title =

    Tsujikawa, S. , title =. Phys. Rev. D , volume =. 2005 , eprint =

    2005

  186. [208]

    Sen, A. A. , title =. J. Cosmol. Astropart. Phys. , volume =. 2006 , eprint =

    2006

  187. [209]

    and Nojiri, S

    Matsumoto, J. and Nojiri, S. , title =. Phys. Lett. B , volume =. 2010 , eprint =

    2010

  188. [210]

    Perkovi. Class. Quant. Grav. , volume =

  189. [211]

    , title =

    Zhang, X. , title =. Phys. Rev. D , volume =. 2006 , eprint =

    2006

  190. [212]

    and Zhang, X

    Zhang, J. and Zhang, X. and Liu, H. , title =. Mod. Phys. Lett. A , volume =. 2008 , eprint =

    2008

  191. [213]

    and Steinhardt, P

    Ijjas, A. and Steinhardt, P. J. , title =. Phys. Rev. Lett. , volume =. 2016 , eprint =

    2016

  192. [214]

    Dobre, D. A. and Frolov, A. V. and G. J. Cosmol. Astropart. Phys. , volume =. 2018 , eprint =

    2018

  193. [215]

    and Cardona, W

    Arjona, R. and Cardona, W. and Nesseris, S. , title =. Phys. Rev. D , volume =. 2019 , eprint =

    2019

  194. [216]

    , title =

    Arjona, R. , title =. 2020 , eprint =

    2020

  195. [217]

    Bernardo, R. C. and Levi Said, J. , title =. J. Cosmol. Astropart. Phys. , volume =. 2021 , eprint =

    2021

  196. [218]

    , title =

    Nesseris, S. , title =. Universe , volume =. 2022 , eprint =

    2022

  197. [219]

    Bernardo, R. C. and Vega, I. , title =. J. Cosmol. Astropart. Phys. , volume =. 2019 , eprint =

    2019

  198. [220]

    , title =

    Herrera, R. , title =. Phys. Rev. D , volume =. 2020 , eprint =

    2020

  199. [221]

    and Myrzakul, S

    Sebastiani, L. and Myrzakul, S. and Myrzakulov, R. , title =. Eur. Phys. J. Plus , volume =. 2017 , eprint =

    2017

  200. [222]

    and Housset, M

    Herrera, R. and Housset, M. and Osses, C. and Videla, N. , title =. Phys. Dark Univ. , volume =. 2024 , eprint =

    2024

  201. [223]

    , title =

    Herrera, R. , title =. Phys. Rev. D , volume =. 2018 , eprint =

    2018

  202. [224]

    and Lombriser, L

    Kennedy, J. and Lombriser, L. and Taylor, A. , title =. Phys. Rev. D , volume =

  203. [225]

    and Lombriser, L

    Kennedy, J. and Lombriser, L. and Taylor, A. , title =. Phys. Rev. D , volume =. 2018 , eprint =

    2018

  204. [226]

    and Lombriser, L

    Kennedy, J. and Lombriser, L. and Taylor, A. , title =. Phys. Rev. D , volume =. 2019 , eprint =

    2019

  205. [227]

    and Kennedy, J

    Renevey, C. and Kennedy, J. and Lombriser, L. , title =. J. Cosmol. Astropart. Phys. , volume =. 2020 , eprint =

    2020

  206. [228]

    , title =

    Kobayashi, T. , title =. Rep. Prog. Phys. , volume =. 2019 , eprint =

    2019

  207. [229]

    Phys. Rev. Lett. , volume =. 2017 , eprint =

    2017

  208. [230]

    and Jain, B

    Sakstein, J. and Jain, B. , title =. Phys. Rev. Lett. , volume =. 2017 , eprint =

    2017

  209. [231]

    Dark Energy After GW170817: Dead Ends and the Road Ahead , author =. Phys. Rev. Lett. , volume =. 2017 , month =. doi:10.1103/PhysRevLett.119.251304 , url =

    2017 doi

  210. [232]

    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 , eprint =

    2017

  211. [233]

    and Sawicki, I

    Bellini, E. and Sawicki, I. , title =. J. Cosmol. Astropart. Phys. , volume =. 2014 , eprint =

    2014

  212. [234]

    and Raveri, M

    Hu, B. and Raveri, M. and Frusciante, N. and Silvestri, A. , title =. Phys. Rev. D , volume =. 2014 , eprint =

    2014

  213. [235]

    Zumalac. J. Cosmol. Astropart. Phys. , volume =. 2017 , eprint =

    2017

  214. [236]

    and Sawicki, I

    Bellini, E. and Sawicki, I. and Zumalac. J. Cosmol. Astropart. Phys. , volume =. 2020 , eprint =

    2020

  215. [237]

    and Odintsov, S

    Nojiri, S. and Odintsov, S. D. and Tsujikawa, S. , title =. Phys. Rev. D , volume =. 2005 , eprint =

    2005

  216. [238]

    and Koyama, K

    Clemson, T. and Koyama, K. and Zhao, G.-B. and Maartens, R. and V. Phys. Rev. D , volume =. 2012 , eprint =

    2012

  217. [239]

    , title =

    Vikman, A. , title =. Phys. Rev. D , volume =. 2005 , eprint =

    2005

  218. [240]

    Caldwell, R. R. and Doran, M. , title =. Phys. Rev. D , volume =. 2005 , eprint =

    2005

  219. [241]

    and Cai, Y.-F

    Xia, J.-Q. and Cai, Y.-F. and Qiu, T.-T. and Zhao, G.-B. and Zhang, X. , title =. Int. J. Mod. Phys. D , volume =. 2008 , eprint =

    2008

  220. [242]

    and Saridakis, E

    Cai, Y.-F. and Saridakis, E. N. and Setare, M. R. and Xia, J.-Q. , title =. Phys. Rep. , volume =. 2010 , eprint =

    2010

  221. [243]

    Abramo, L. R. and Pinto-Neto, N. , title =. Phys. Rev. D , volume =. 2006 , eprint =

    2006

  222. [244]

    and Rattazzi, R

    Nicolis, A. and Rattazzi, R. and Trincherini, E. , title =. Phys. Rev. D , volume =. 2009 , eprint =

    2009

  223. [245]

    and Tsujikawa, S

    de Felice, A. and Tsujikawa, S. , title =. Phys. Rev. Lett. , volume =. 2010 , eprint =

    2010

  224. [246]

    and Tsujikawa, S

    De Felice, A. and Tsujikawa, S. , title =. J. Cosmol. Astropart. Phys. , volume =. 2012 , eprint =

    2012

  225. [247]

    and De Felice, A

    Giacomello, F. and De Felice, A. and Ansoldi, S. , title =. J. Cosmol. Astropart. Phys. , volume =. 2019 , eprint =

    2019

  226. [248]

    Phenomenology of the generalized cubic covariant Galileon model and cosmological bounds , author =. Phys. Rev. D , volume =. 2020 , month =. doi:10.1103/PhysRevD.101.064001 , url =

    2020 doi

  227. [249]

    and Tsujikawa, S

    Kase, R. and Tsujikawa, S. , title =. Phys. Rev. D , volume =. 2018 , eprint =

    2018

  228. [250]

    and Benevento, G

    Peirone, S. and Benevento, G. and Frusciante, N. and Tsujikawa, S. , title =. Phys. Rev. D , volume =. 2019 , eprint =

    2019

  229. [251]

    and Pujol

    Deffayet, C. and Pujol. J. Cosmol. Astropart. Phys. , volume =. 2010 , eprint =

    2010

  230. [252]

    and Pascoli, Silvia , title =

    Barreira, Alexandre and Li, Baojiu and Baugh, Carlton M. and Pascoli, Silvia , title =. Journal of Cosmology and Astroparticle Physics , abstract =. 2014 , month =. doi:10.1088/1475-7516/2014/08/059 , url =

    2014 doi

  231. [253]

    and Davis, A.-C

    Brax, P. and Davis, A.-C. and Li, B. and Winther, H. A. , title =. Phys. Rev. D , volume =. 2012 , eprint =

    2012

  232. [254]

    and Davis, A.-C

    Brax, P. and Davis, A.-C. and Li, B. and Winther, H. A. and Zhao, G.-B. , title =. J. Cosmol. Astropart. Phys. , volume =. 2012 , eprint =

    2012

  233. [255]

    and Davis, A.-C

    Brax, P. and Davis, A.-C. and Li, B. and Winther, H. A. and Zhao, G.-B. , title =. J. Cosmol. Astropart. Phys. , volume =. 2013 , eprint =

    2013

  234. [256]

    and Raveri, M

    Hu, B. and Raveri, M. and Frusciante, N. and Silvestri, A. , title =. 2014 , journal =. 1405.3590 , archivePrefix =

    2014 arXiv

  235. [257]

    Covariant Galileon , author =. Phys. Rev. D , volume =. 2009 , month =. doi:10.1103/PhysRevD.79.084003 , url =

    2009 doi

  236. [258]

    Atayde, Lu. Phys. Rev. D , volume =. 2024 , eprint =

    2024

  237. [259]

    N-body simulations of large-scale structure in the generalized cubic covariant Galileon model , journal =

    Atayde, Lu. N-body simulations of large-scale structure in the generalized cubic covariant Galileon model , journal =. 2026 , eprint =

    2026

  238. [260]

    and Pascoli, Silvia

    Barreira, Alexandre and Li, Baojiu and Baugh, Carlton M. and Pascoli, Silvia. Spherical collapse in Galileon gravity: fifth force solutions, halo mass function and halo bias. J. Cosmol. Astropart. Phys. 2013. doi:10.1088/1475-7516/2013/11/056. arXiv:1308.3699

    2013 arXiv

  239. [261]

    and Ferreira, Pedro G

    Winther, Hans A. and Ferreira, Pedro G. Vainshtein mechanism beyond the quasistatic approximation. Phys. Rev. D. 2015. doi:10.1103/PhysRevD.92.064005. arXiv:1505.03539

    2015 arXiv

  240. [262]

    How deep can a cosmic void be? Voids-informed theoretical bounds in Galileon gravity

    Moretti, Tommaso and Frusciante, Noemi and Verza, Giovanni and Pace, Francesco. How deep can a cosmic void be? Voids-informed theoretical bounds in Galileon gravity. Phys. Rev. D. 2026. doi:10.1103/nk9f-dw1z. arXiv:2601.05145

    2026 arXiv

  241. [263]

    and Pichon, C

    Prunet, S. and Pichon, C. and Aubert, D. and Pogosyan, D. and Teyssier, R. and Gottloeber, S. Initial Conditions for Large Cosmological Simulations. Astrophys. J. Suppl. 2008. doi:10.1086/590370. arXiv:0804.3536

    2008 arXiv

  242. [264]

    and Pueblas, S

    Crocce, M. and Pueblas, S. and Scoccimarro, R. Transients from Initial Conditions in Cosmological Simulations. Mon. Not. Roy. Astron. Soc. 2006. doi:10.1111/j.1365-2966.2006.11040.x. arXiv:astro-ph/0606505

    2006

  243. [265]

    and Arnold, C

    Becker, C. and Arnold, C. and Li, B. and Heisenberg, L. , title =. J. Cosmol. Astropart. Phys. , volume =. 2020 , doi =. 2007.03042 , archivePrefix =

    2020 arXiv

  244. [266]

    Barreira, Alexandre and Bose, Sownak and Li, Baojiu , title =. J. Cosmol. Astropart. Phys. , volume =. 2015 , doi =. 1511.08200 , archivePrefix =

    2015 arXiv

  245. [267]

    Journal of Cosmology and Astroparticle Physics , issn =

    Li, Baojiu and Zhao, Gong-Bo and Koyama, Kazuya and Teyssier, Romain , title =. Journal of Cosmology and Astroparticle Physics , issn =. 2012 , month =. doi:10.1088/1475-7516/2012/01/051 , url =

    2012 doi

  246. [268]

    Barreira, Alexandre and Li, Baojiu and Baugh, Carlton and Pascoli, Silvia , title =. Phys. Rev. D , volume =. 2012 , doi =. 1208.0600 , archivePrefix =

    2012 arXiv

  247. [269]

    De Felice, Antonio and Tsujikawa, Shinji , title =. Phys. Rev. D , volume =. 2011 , doi =. 1011.6132 , archivePrefix =

    2011 arXiv

  248. [270]

    Oyaizu, Hiroaki and Lima, Marcos and Hu, Wayne , title =. Phys. Rev. D , volume =. 2009 , doi =. 0906.4548 , archivePrefix =

    2009 arXiv

  249. [271]

    Zhao, Gong-Bo and Li, Baojiu and Koyama, Kazuya , title =. Phys. Rev. D , volume =. 2013 , doi =. 1303.0008 , archivePrefix =

    2013 arXiv

  250. [272]

    and Baugh, Carlton M

    Barreira, Alexandre and Li, Baojiu and Hellwing, Wojciech A. and Baugh, Carlton M. and Pascoli, Silvia , title =. Phys. Rev. D , volume =. 2013 , doi =. 1308.3491 , archivePrefix =

    2013 arXiv

  251. [273]

    Khoury, Justin and Wyman, Mark , title =. Phys. Rev. D , volume =. 2009 , doi =. 0903.1292 , archivePrefix =

    2009 arXiv

  252. [274]

    Generalization of the Proca Action. J. Cosmol. Astropart. Phys. , year = 2014, month = may, volume =. doi:10.1088/1475-7516/2014/05/015 , archivePrefix =. 1402.7026 , primaryClass =

    2014 arXiv

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.