Nonlinear reconstruction of general dark energy theories

Baojiu Li; Jie Wang; Yunhao Gao

REVIEW 2 major objections 4 minor 106 references

Nonlinear reconstruction of general dark energy theories

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper claims that the time evolution of the first five effective-field-theory functions of dark energy determines the full nonlinear Lagrangians of quintessence, scalar-tensor, k-essence, and shift-symmetric cubic Galileon models…

desk verdict Solid new reconstruction algebra for k-essence and cubic Galileon, but the screening/simulation promise is unsupported and needs a fiducial test. read the letter →

arxiv 2507.01442 v2 pith:CL5UJXOU submitted 2025-07-02 astro-ph.CO gr-qc

Yunhao Gao , Baojiu Li , Jie Wang This is my paper

classification astro-ph.COgr-qc

keywords darkenergyeffectivefieldtheoryofHorndeskiLagrangianreconstructionquintessencek-essenceshift-symmetriccubicGalileonscalar-tensor

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

The paper argues that the background and linear-perturbation parametrization of dark energy contains enough information to reconstruct the fully nonlinear Lagrangian for four broad classes of single-scalar-field theories. The key step is to read the effective-field-theory dictionary backwards: the five time functions $f(t)$, $\Lambda(t)$, $c(t)$, $M_2^4(t)$, $\bar M_1^3(t)$ are treated as ordinary differential equations for $\phi(t)$, $X(t)$, and for the Horndeski functions $K$, $G_3$, $G_4$ along the background trajectory. Numerical examples show the reconstruction works for quintessence, scalar-tensor theory, $k$-essence, and shift-symmetric cubic Galileon models, and the paper identifies when it fails: non-monotonic scalar motion, sign changes of $X(t)$, and phantom-crossing instabilities. If the method is right, measured expansion histories and linear perturbation data can be converted directly into Lagrangian inputs for nonlinear cosmological simulations, allowing whole theory classes to be tested rather than one model at a time.

What carries the argument

The central object is the dictionary in Eq. (2.5) between the EFT time functions $f(t)$, $\Lambda(t)$, $c(t)$, $M_2^4(t)$, $\bar M_1^3(t)$ and the Horndeski functions $K(\phi,X)$, $G_3(\phi,X)$, $G_4(\phi)$. Treating this dictionary as five ordinary differential equations along the background trajectory yields the reconstructed functions; field-redefinition freedom lets one choose a convenient monotonic $\phi(t)$. For the shift-symmetric cubic Galileon the tracker identity $E L(X)=1/3$, with $L=H_0\sqrt{-X}\,G_{3X}/K_X$, converts part of the differential system into algebraic equations.

What would settle it

Take a known Horndeski model from one of the four classes with an explicit Lagrangian, compute its first five EFT functions from Eq. (2.5), feed only those functions back through the reconstruction, and compare the recovered $K$, $G_3$, $G_4$ along the trajectory with the original. A mismatch, or a multi-valued recovered potential for a model whose background $\phi(t)$ is monotonic, would falsify the claimed invertibility; likewise, a cubic Galileon whose equation of state crosses $-1$ but whose reconstructed $L(X)$ remains single-valued past the crossing would contradict the paper's instability argument.

Watch

Extended reading notes

Core claim

The paper's discovery is an inversion: the same five time-dependent EFT functions that describe background and linear perturbations of a Horndeski dark-energy model also fix, along the background trajectory, the functional forms of the Lagrangian. Reading Eq. (2.5) as differential equations for $\phi(t)$, $X(t)$, $K(t)$, $G_3(t)$, and $G_4(t)$, the authors reconstruct the potential of quintessence and scalar-tensor theories, the functions $V(\phi)$, $f(\phi)$, $q(\phi)$, $h(X)$ in $k$-essence models, and $K(X)$, $L(X)$ of shift-symmetric cubic Galileons. The reconstruction is demonstrated numerically and its validity conditions identified: monotonic scalar evolution and, for the Galileon, convergence to the tracker $E L(X)=1/3$. In this sense the nonlinear Lagrangian is not an independent input but a consequence of the measured EFT time functions, provided the theory lies in one of the considered classes and satisfies the stated monotonicity conditions.

Load-bearing premise

The reconstruction yields a single-valued Lagrangian only if the background scalar field moves monotonically in time (no turning points in $\phi(t)$, and no sign change in $X(t)$ for $k$-essence and cubic Galileon), and for the cubic Galileon only if the universe sits on or has converged to the tracker solution.

Editorial extensions

If this is right

For quintessence and scalar-tensor theories, $c(t)$ fixes the scalar velocity and $\Lambda(t)$ fixes the potential along the trajectory, so a field shift removes the integration constant and no initial condition is needed.
For $k$-essence of the form $K=V(\phi)+f(\phi)X+q(\phi)X^2$, choosing a convenient $\phi(t)$ via field redefinition lets the three EFT functions recover $V$, $f$, and $q$; for factorizable $K=q(\phi)h(X)$ the same data recover both factors up to a physically irrelevant overall scaling.
For the shift-symmetric cubic Galileon, the tracker condition $E L(X)=1/3$ turns the reconstruction into algebraic relations, with $K(t)=-\rho_{\rm DE}(t)$, so $K(X)$ and $L(X)$ follow from $c$, $\Lambda$, and $\bar M_1^3$ alone.
The method is limited to monotonic scalar trajectories; a phantom crossing ($w_{\rm DE}$ crossing $-1$) makes factorizable $k$-essence multi-valued and destabilizes the Galileon tracker, so such observed histories lie outside the reconstructed classes.
The reconstructed Lagrangians can be fed into cosmological simulations, so nonlinear structure-formation data can test entire theory classes instead of a single hand-picked model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's own 'just enough' remark implies a pipeline: fit EFT functions to expansion and linear-growth data, reconstruct the Lagrangian, then use nonlinear clustering or screening signatures as a consistency test; the authors announce follow-up simulation work but do not run it here.
The same inversion could be extended to more EFT parameters to reconstruct $X$-dependent $G_4$ or higher-order Horndeski terms, since each added EFT function supplies one more equation along the trajectory; the paper notes this extension is straightforward but does not carry it out.
A testable consequence is that a future data set with a phantom-crossing signal would push the allowed theory space away from the four classes studied here, because their reconstructions become multi-valued or unstable in exactly that regime.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Briefly: the genuinely new pieces are the ODE-based reconstruction of factorizable k-essence (Sec. 4) and the simultaneous reconstruction of both K(X) and G3(X) for shift-symmetric cubic Galileon (Sec. 5), including the tracker simplification. The algebra is self-consistent, the overlap with Refs. [72-75] and earlier quintessence/scalar-tensor work is acknowledged honestly, and the paper is unusually explicit about where the method breaks down: non-monotonic fields, phantom crossing, underdetermination, and the fact that only a 1D curve in the (phi,X) plane is recovered.

What it does well: the initial-condition analysis is careful. For factorizable k-essence, field redefinition and scaling arguments show X_i and h_i drop out, leaving a genuinely parameter-free reconstruction of q(phi) and h(X) along the background path. The Galileon section is more involved: the J0 dependence is mapped, the tracker solution turns differential equations into algebraic ones, and the stability analysis around the tracker is a real contribution. The numerics support the algebra, but they are consistency checks, not recovery tests—the background parts equal the input EFT functions by construction, and the non-trivial X-dependence is derived from the ODEs, not compared against a known Lagrangian.

The main soft spot is the simulation promise. The abstract says the reconstruction 'will enable cosmological simulations' for theories with screening; Section 6 concedes that the reconstruction cannot produce the full functional forms and that the method relies on perturbations of X and phi being small, which fails precisely for thin-shell screening (chameleon, symmetron, Vainshtein). In a screened environment the field sits at a density-dependent value far from the cosmological background trajectory. The reconstructed V(phi) over the background interval simply does not constrain the potential there, and extrapolation is unconstrained—giving the wrong chameleon mass and fifth force. So the paper oversells itself on the screening front. A fiducial recovery test with a chameleon or f(R) model, where the true Lagrangian is known, would settle it and should be required.

Also minor: no code or data, which makes reproducibility harder.

Who it's for: people working on EFTofDE reconstruction, designer Horndeski models, and anyone wanting concrete Lagrangians to feed into simulations for weakly-screened models. It deserves a serious referee and a conditional accept, with the recovery test and a tempered abstract as conditions. I would not cite the simulation claim, but I would cite the reconstruction method for the Galileon and factorizable k-essence.

Referee Report

2 major / 4 minor

Summary. The paper presents a method to reconstruct the nonlinear Lagrangians of several single-scalar-field dark-energy theories from the time-dependent EFTofDE functions f, Λ, c, M2^4, and Mbar1^3. The central idea is to regard Eq. (2.5) as a system of ordinary differential equations for the background φ(t), X(t) and for K, G3, and G4 evaluated along the background trajectory. Separate sections treat quintessence and scalar-tensor theories (§3), k-essence with ansätze K=V+fX+qX^2 and factorizable K=q(φ)h(X) (§4), and shift-symmetric cubic Galileon with tracker and off-tracker cases (§5). The paper honestly identifies conditions under which the inversion fails, including non-monotonic φ(t) or X(t), phantom crossing, and loss of tracker stability. Numerical examples for CPL-type background evolutions and parametrized α_B or c_s^2 show that the relevant differential equations can be solved and that the reconstructed functions are single-valued in the chosen cases.

Significance. If fully established, this would be a useful systematic framework for connecting background and linear EFT measurements to the nonlinear Lagrangian of restricted Horndeski subclasses, potentially valuable for simulation studies. The paper's strengths are its transparent algebraic derivations, its explicit treatment of initial-condition degeneracies (e.g., the φ-shift invariance of quintessence and the scaling degeneracies in k-essence and cubic Galileon), and its candid analysis of singular limits such as phantom crossing and multi-valued inversions. The treatment of cubic Galileon tracker stability, including Eq. (5.25), is a useful contribution. However, two issues bound the significance: the reconstructed functions are known only on the one-dimensional background trajectory, so the abstract's promise of enabling simulations of screening theories is not supported; and the numerical examples are not recovery tests against known fiducial Lagrangians. The paper is therefore a solid methodological construction whose scope and validation need to be tightened before the advertised conclusions are accepted.

major comments (2)

[§6 (also Abstract; §4)] The abstract and §6 claim that the reconstructed Lagrangians 'will enable cosmological simulations' to study theories with screening mechanisms, but the construction provides the functions K(φ,X), G3(φ,X), and G4(φ) only along the background trajectory (φ(t),X(t)). For non-separable K(φ,X) this is a one-dimensional curve, as the paper itself states in §4; the full two-dimensional functional form is not determined. In chameleon/symmetron and Vainshtein screening, the relevant field values in screened environments are density-dependent and generically lie far outside the cosmological background interval, so the reconstructed interval does not constrain the screening behaviour. The paper concedes this in §6 ('Our reconstruction framework cannot produce the full functional forms...', and the reassurance that follows is restricted to models 'that do not involve thin-shell screening'). Since screening is the stated primary motivation, this is a load-bearing gap: either the claims must be restricted to non-screening applications, or a concrete screening-theory example (e.g., an f(R)/chameleon fiducial) must be shown to be recoverable.
[§4 and §5, Eqs. (4.13)–(4.16) and (5.15)–(5.17)] The numerical demonstrations are not recovery tests. The examples specify the EFT functions directly (via w0, wa, c_s^2 or c_B) and then solve the reconstruction equations; they never start from a known K(φ,X), G3(X), or G4(φ), generate the corresponding EFT functions, and check that the procedure recovers the input functions. This matters because at least in the tracker case the reconstructed K is partly imposed by construction: Eq. (5.16) sets K(t) = -ρ_DE(t) from the tracker condition J0=0, so the resulting K(X) largely reproduces the input background rather than providing an independent test. In addition, no numerical demonstration is given for the scalar-tensor case advertised in §3. Adding fiducial recovery tests for, e.g., a covariant cubic Galileon and a factorizable k-essence model with known q(φ)h(X) would directly test the inversion and would also probe the sensitive X(t)-monotonicity and phantom-crossing limits discussed in the text.

minor comments (4)

[§5, after Eq. (5.28)] The phrase 'the sound speed will become imaginative' should read 'imaginary'.
[§1, first paragraph] The abbreviation 'SNIe' is a typo; it should be 'SNIa'.
[Reference list, Ref. [57]] Reference [57] contains an unresolved LaTeX artifact, '[ image ]', in the title; the citation should be completed and the artifact removed.
[Figures 10 and 11] The last-column panels would be easier to interpret if the tracker value E(a)L(a)=1/3 and the phantom-crossing line X_c were labeled consistently in both figures; currently the horizontal gray line is only described in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reconstruction is an explicit inversion of the stated EFT-to-Lagrangian mapping, and the numerical examples are self-consistency demonstrations rather than independent predictions.

full rationale

The paper proposes a reconstruction method rather than a first-principles prediction. Equations such as (3.2) (Λ(t)=V(t) for quintessence) and (5.16) (K(t)=-ρ_DE(t) on the tracker) are algebraic consequences of the EFT mapping (2.5) and the assumed theory subclass; they are used as the reconstruction equations themselves, not as derived outputs that secretly re-enter the inputs. The numerical examples specify the EFT functions (e.g., via CPL parameters and α_B parametrization) and then reconstruct the corresponding Lagrangian, which is a consistency check of the inversion, not a fitted parameter renamed as a prediction. The paper also explicitly acknowledges the main limitation: only the values along the background trajectory (φ(t), X(t)) are reconstructed, and extending to the full functional forms needed for simulations, especially for thin-shell screening models, is an extrapolation. This is a scope/validity limitation, not circularity. Self-citations (e.g., refs. [43,103-106]) are contextual and not load-bearing to the derivation chain. No circular step meeting the required 'exhibit the specific reduction' standard was identified.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The reconstruction is an inverse problem: outputs are determined by inputs plus structural assumptions. Free parameters are the hand-chosen inputs of the toy examples, including CPL parameters, density fractions, sound speed, alpha_B amplitude, Galileon integration constant, and the turning-point scale factor. Axioms are the standard EFT mapping, the GW170817 restriction, monotonicity of the background path, and the tracker condition. No new physical entities are introduced.

free parameters (7)

w0 (dark energy EoS at a=1, CPL) = -1.0, -0.8, etc. per model (Tables 1-4)
Chosen by hand to define the background expansion history; all reconstructed functions inherit this input.
wa (CPL slope) = 0.2, -0.2, -0.4 per model
Chosen by hand together with w0 to specify c(t) and Lambda(t) in the toy models.
Omega_m0 and Omega_DE0 = 0.3, 0.7
Fixed matter and dark energy density fractions at z=0 in all numerical examples.
c_s^2 (k-essence sound speed squared) = 0.1 or 0.01
Used to fix the EFT parameter M2^4(t); reconstruction of f(phi) and q(phi) depends on it.
c_B (amplitude of alpha_B parametrization) = 0.5 or -0.5
Sets the evolution of Mbar1^3(t) in cubic Galileon examples.
J0 (Galileon equation-of-motion constant, tied to K_i) = 0 (tracker) or +/- 4e-8 M* H0
Specifies the initial condition for K; off-tracker values change reconstructed K(X) and L(X).
a_t (turning-point scale factor for QIII/QIV) = 0.7
Chosen to create a non-monotonic phi(t) example that yields a multi-valued V(phi).

assumptions (4)

domain assumption The EFT-to-Horndeski mapping of Eq. (2.5), taken from Refs. [23,24], is correct and complete for the considered subclasses.
All reconstruction equations are built on this mapping; if it omits relevant couplings or higher-order EFT operators, the reconstructed Lagrangian will not reproduce the true non-linear dynamics.
domain assumption GW170817 constraints restrict the action to G5=0 and G4 independent of X (Eq. 2.3).
This excludes beyond-Horndeski and G5-type theories from the reconstruction space.
ad hoc to paper The background scalar field phi(t) is monotonic over the reconstruction interval, and X(t) is monotonic when inversion to X is needed.
The paper requires this for single-valued V(phi), q(phi), h(X), and K(X); it is not derived from the EFT inputs but imposed as a condition for the method to apply (Sections 3-5).
domain assumption For shift-symmetric cubic Galileon, the solution sits on (or has converged to) the tracker solution with J0=0.
Turns the differential equation for K into the algebraic K(t)=-rho_DE(t) (Eq. 5.16); the paper tests convergence but does not derive J0=0 from observations.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Nonlinear reconstruction of general dark energy theories." pith.science (2026). https://pith.science/paper/CL5UJXOU

@misc{pith2026250701442,
  author       = {Pith},
  title        = {Pith review of: Nonlinear reconstruction of general dark energy theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CL5UJXOU}},
  note         = {Machine review of arXiv:2507.01442}
}

abstract

The large variety and number of dark energy (DE) theories make it impractical to perform detailed analyses on a case-by-case basis, which has motivated proposals to ``parameterize" theories to reduce the size of theory space. The leading approach to do this is the effective field theory of dark energy (EFTofDE), which can describe general Horndeski-type theories with a small number of observationally accessible time-dependent functions. However, the EFTofDE primarily works for linear perturbations, and extending it to obtain a fully non-linear description of DE theories, which is critical for theories with screening mechanisms, is challenging. In this paper, we present a general method for reconstructing the non-linear DE Lagrangian from the background expansion history and certain linear-perturbation quantities, building upon the EFTofDE framework. Using numerical examples, we demonstrate that this method is applicable to a wide range of single-scalar-field dark energy and modified gravity theories, including quintessence, scalar-tensor theory, $k$-essence, and generalized cubic Galileon with shift symmetry. For each of these theories, we discuss the validity of the method and factors affecting its results. While this method involves solving differential equations, we find that the initial conditions are not important for quintessence, scalar-tensor theory and $k$-essence, while for shift-symmetric cubic Galileon, the generic tracker solution can help transform differential equations into algebraic equations. This offers a useful framework to connect cosmological observations at the background and linear-perturbation levels to the underlying non-linear dynamics of dark energy, and will enable cosmological simulations to analyze and examine DE theories systematically and in much greater detail.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

106 extracted references · 27 canonical work pages

[1]

Riess, A.V

A.G. Riess, A.V . Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J.116(1998) 1009 [astro-ph/9805201]

arXiv 1998
[2]

J.517(1999) 565 [astro-ph/9812133]

SUPERNOVACOSMOLOGYPROJECTcollaboration,Measurements ofΩandΛfrom 42 High Redshift Supernovae,Astrophys. J.517(1999) 565 [astro-ph/9812133]. – 30 –

arXiv 1999
[3]

Ratra and P.J.E

B. Ratra and P.J.E. Peebles,Cosmological consequences of a rolling homogeneous scalar field,Phys. Rev. D37(1988) 3406

1988
[4]

Ferreira and M

P.G. Ferreira and M. Joyce,Structure Formation with a Self-Tuning Scalar Field,Phys. Rev. Lett.79 (1997) 4740 [astro-ph/9707286]

arXiv 1997
[5]

Copeland, A.R

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

arXiv 1998
[6]

Caldwell, R

R.R. Caldwell, R. Dave and P.J. Steinhardt,Cosmological Imprint of an Energy Component with General Equation of State,Phys. Rev. Lett.80(1998) 1582 [astro-ph/9708069]

arXiv 1998
[7]

Zlatev, L

I. Zlatev, L. Wang and P.J. Steinhardt,Quintessence, Cosmic Coincidence, and the Cosmological Constant,Phys. Rev. Lett.82(1999) 896 [astro-ph/9807002]

arXiv 1999
[8]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Introduction to Modified Gravity and Gravitational Alternative for Dark Energy,International Journal of Geometric Methods in Modern Physics04(2007) 115 [hep-th/0601213]

arXiv 2007

Show all 106 references

[9]

Sotiriou and V

T.P. Sotiriou and V . Faraoni,f(R) theories of gravity,Reviews of Modern Physics82(2010) 451 [0805.1726]

2010 arXiv
[10]

Armend ´ariz-Pic´on, T

C. Armend ´ariz-Pic´on, T. Damour and V . Mukhanov,k-Inflation,Phys. Lett. B458(1999) 209 [hep-th/9904075]

1999 arXiv
[11]

Chiba, T

T. Chiba, T. Okabe and M. Yamaguchi,Kinetically driven quintessence,Phys. Rev. D62(2000) 023511 [astro-ph/9912463]

2000 arXiv
[12]

Horndeski,Second-order scalar-tensor field equations in a four-dimensional space, International Journal of Theoretical Physics10(1974) 363

G.W. Horndeski,Second-order scalar-tensor field equations in a four-dimensional space, International Journal of Theoretical Physics10(1974) 363

1974
[13]

Deffayet, S

C. Deffayet, S. Deser and G. Esposito-Far `ese,Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress tensors,Phys. Rev. D80 (2009) 064015 [0906.1967]

2009 arXiv
[14]

Dvali, G

G. Dvali, G. Gabadadze and M. Porrati,4D gravity on a brane in 5D Minkowski space,Phys. Lett. B 485(2000) 208 [hep-th/0005016]

2000 arXiv
[15]

Huterer and D.L

D. Huterer and D.L. Shafer,Dark energy two decades after: observables, probes, consistency tests, Reports on Progress in Physics81(2018) 016901 [1709.01091]

2018 arXiv
[16]

Moresco, L

M. Moresco, L. Amati, L. Amendola, S. Birrer, J.P. Blakeslee, M. Cantiello et al.,Unveiling the Universe with emerging cosmological probes,Living Reviews in Relativity25(2022) 6 [2201.07241]

2022 arXiv
[17]

Chevallier and D

M. Chevallier and D. Polarski,Accelerating Universes with Scaling Dark Matter,International Journal of Modern Physics D10(2001) 213 [gr-qc/0009008]

2001 arXiv
[18]

Linder,Exploring the Expansion History of the Universe,Phys

E.V . Linder,Exploring the Expansion History of the Universe,Phys. Rev. Lett.90(2003) 091301 [astro-ph/0208512]

2003 arXiv
[19]

Caldwell, A

R. Caldwell, A. Cooray and A. Melchiorri,Constraints on a new post-general relativity cosmological parameter,Phys. Rev. D76(2007) 023507 [astro-ph/0703375]

2007 arXiv
[20]

Amendola, M

L. Amendola, M. Kunz and D. Sapone,Measuring the dark side (with weak lensing),JCAP2008 (2008) 013 [0704.2421]

2008 arXiv
[21]

Gubitosi, F

G. Gubitosi, F. Piazza and F. Vernizzi,The effective field theory of dark energy,JCAP2013(2013) 032 [1210.0201]

2013 arXiv
[22]

Bloomfield, ´E

J. Bloomfield, ´E. ´E. Flanagan, M. Park and S. Watson,Dark energy or modified gravity? An effective field theory approach,JCAP2013(2013) 010 [1211.7054]

2013 arXiv
[23]

Gleyzes, D

J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi,Essential building blocks of dark energy,JCAP 2013(2013) 025 [1304.4840]. – 31 –

2013 arXiv
[24]

Frusciante, G

N. Frusciante, G. Papadomanolakis and A. Silvestri,An extended action for the effective field theory of dark energy: a stability analysis and a complete guide to the mapping at the basis of EFTCAMB, JCAP2016(2016) 018 [1601.04064]

2016 arXiv
[25]

Frusciante and L

N. Frusciante and L. Perenon,Effective field theory of dark energy: A review,Phys. Rep.857(2020) 1 [1907.03150]

2020 arXiv
[26]

Frusciante and G

N. Frusciante and G. Papadomanolakis,Tackling non-linearities with the effective field theory of dark energy and modified gravity,JCAP2017(2017) 014 [1706.02719]

2017 arXiv
[27]

Cusin, M

G. Cusin, M. Lewandowski and F. Vernizzi,Nonlinear effective theory of dark energy,JCAP2018 (2018) 061 [1712.02782]

2018 arXiv
[28]

Chiba and T

T. Chiba and T. Nakamura,Feasibility of reconstructing the quintessential potential using SNIa data, in20th Texas Symposium on relativistic astrophysics, vol. 586 ofAmerican Institute of Physics Conference Series, pp. 319–321, AIP, 2001, DOI

2001
[29]

Rubano and J.D

C. Rubano and J.D. Barrow,Scaling solutions and reconstruction of scalar field potentials,Phys. Rev. D64(2001) 127301 [gr-qc/0105037]

2001 arXiv
[30]

Gerke and G

B.F. Gerke and G. Efstathiou,Probing quintessence: reconstruction and parameter estimation from supernovae,Mon. Not. Roy. Astron. Soc.335(2002) 33 [astro-ph/0201336]

2002 arXiv
[31]

Zhang,Reconstructing holographic quintessence,Phys

X. Zhang,Reconstructing holographic quintessence,Phys. Lett. B648(2007) 1 [astro-ph/0604484]

2007 arXiv
[32]

Wu, D.-Z

J.-P. Wu, D.-Z. Ma and Y . Ling,Quintessence reconstruction of the new agegraphic dark energy model,Phys. Lett. B663(2008) 152 [0805.0546]

2008 arXiv
[33]

Sangwan, A

A. Sangwan, A. Mukherjee and H.K. Jassal,Reconstructing the dark energy potential,JCAP2018 (2018) 018 [1712.05143]

2018 arXiv
[34]

M. Park, M. Raveri and B. Jain,Reconstructing quintessence,Phys. Rev. D103(2021) 103530 [2101.04666]

2021 arXiv
[35]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Unifying phantom inflation with late-time acceleration: scalar phantom-non-phantom transition model and generalized holographic dark energy,General Relativity and Gravitation38(2006) 1285 [hep-th/0506212]

2006 arXiv
[36]

Capozziello, S

S. Capozziello, S. Nojiri and S.D. Odintsov,Unified phantom cosmology: Inflation, dark energy and dark matter under the same standard,Phys. Lett.s B632(2006) 597 [hep-th/0507182]

2006 arXiv
[37]

Capozziello, S

S. Capozziello, S. Nojiri and S.D. Odintsov,Dark energy: the equation of state description versus scalar-tensor or modified gravity,Phys. Lett. B634(2006) 93 [hep-th/0512118]

2006 arXiv
[38]

Boisseau, G

B. Boisseau, G. Esposito-Far `ese, D. Polarski and A.A. Starobinsky,Reconstruction of a Scalar-Tensor Theory of Gravity in an Accelerating Universe,Phys. Rev. Lett.85(2000) 2236 [gr-qc/0001066]

2000 arXiv
[39]

Nojiri, S.D

S. Nojiri, S.D. Odintsov and M. Sami,Dark energy cosmology from higher-order, string-inspired gravity, and its reconstruction,Phys. Rev. D74(2006) 046004 [hep-th/0605039]

2006 arXiv
[40]

Cognola, E

G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov and S. Zerbini,String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy,Phys. Rev. D75(2007) 086002 [hep-th/0611198]

2007 arXiv
[41]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Modified gravity and its reconstruction from the universe expansion history, inJournal of Physics Conference Series, vol. 66 ofJournal of Physics Conference Series, p. 012005, IOP, 2007, DOI [hep-th/0611071]

2007 arXiv
[42]

Capozziello, S

S. Capozziello, S. Nesseris and L. Perivolaropoulos,Reconstruction of the scalar tensor Lagrangian from aΛCDM background and Noether symmetry,JCAP2007(2007) 009 [0705.3586]

2007 arXiv
[43]

Brax, A.-C

P. Brax, A.-C. Davis and B. Li,Modified gravity tomography,Phys. Lett. B715(2012) 38 [1111.6613]. – 32 –

2012 arXiv
[44]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Modified f(R) gravity consistent with realistic cosmology: From a matter dominated epoch to a dark energy universe,Phys. Rev. D74(2006) 086005 [hep-th/0608008]

2006 arXiv
[45]

Elizalde and D

E. Elizalde and D. S ´aez-G´omez,F(R) cosmology in the presence of a phantom fluid and its scalar-tensor counterpart: Towards a unified precision model of the evolution of the Universe,Phys. Rev. D80(2009) 044030 [0903.2732]

2009 arXiv
[46]

Karami and M.S

K. Karami and M.S. Khaledian,Reconstructing f( R) modified gravity from ordinary and entropy-corrected versions of the holographic and new agegraphic dark energy models,Journal of High Energy Physics2011(2011) 86 [1004.1805]

2011 arXiv
[47]

Carloni, R

S. Carloni, R. Goswami and P.K.S. Dunsby,A new approach to reconstruction methods in f(R) gravity, Classical and Quantum Gravity29(2012) 135012 [1005.1840]

2012 arXiv
[48]

Houndjo,Reconstruction of f(R, t) Gravity Describing Matter Dominated and Accelerated Phases,International Journal of Modern Physics D21(2012) 1250003 [1107.3887]

M.J.S. Houndjo,Reconstruction of f(R, t) Gravity Describing Matter Dominated and Accelerated Phases,International Journal of Modern Physics D21(2012) 1250003 [1107.3887]

2012 arXiv
[49]

Jamil, D

M. Jamil, D. Momeni, M. Raza and R. Myrzakulov,Reconstruction of some cosmological models in f( R, T) cosmology,European Physical Journal C72(2012) 1999 [1107.5807]

2012 arXiv
[50]

Singh and V

C.P. Singh and V . Singh,Reconstruction of modified gravity with perfect fluid cosmological models, General Relativity and Gravitation46(2014) 1696

2014
[51]

Elizalde, R

E. Elizalde, R. Myrzakulov, V .V . Obukhov and D. S´aez-G´omez,ΛCDM epoch reconstruction from F(R, G) and modified Gauss-Bonnet gravities,Classical and Quantum Gravity27(2010) 095007 [1001.3636]

2010 arXiv
[52]

Hamani Daouda, M.E

M. Hamani Daouda, M.E. Rodrigues and M.J.S. Houndjo,Reconstruction of f( T) gravity according to holographic dark energy,European Physical Journal C72(2012) 1893 [1111.6575]

2012 arXiv
[53]

Elkhateeb,Reconstruction of f(R) Gravity from Cosmological Unified Dark Fluid Model, Foundations of Physics54(2024) 18 [2301.13858]

E.A. Elkhateeb,Reconstruction of f(R) Gravity from Cosmological Unified Dark Fluid Model, Foundations of Physics54(2024) 18 [2301.13858]

2024 arXiv
[54]

Tsujikawa,Reconstruction of general scalar-field dark energy models,Phys

S. Tsujikawa,Reconstruction of general scalar-field dark energy models,Phys. Rev. D72(2005) 083512 [astro-ph/0508542]

2005 arXiv
[55]

Sen,Reconstructing k-essence,JCAP2006(2006) 010 [astro-ph/0512406]

A.A. Sen,Reconstructing k-essence,JCAP2006(2006) 010 [astro-ph/0512406]

2006 arXiv
[56]

Matsumoto and S

J. Matsumoto and S. Nojiri,Reconstruction of k-essence model,Phys. Lett. B687(2010) 236 [1001.0220]

2010 arXiv
[57]

Perkovi ´c and H

D. Perkovi ´c and H. ˇStefanˇci´c,Analytical reconstruction of equivalent purely kinetic k-essence description for [ image ] barotropic fluid models,Classical and Quantum Gravity42(2025) 075017

2025
[58]

Zhang,Dynamical vacuum energy, holographic quintom, and the reconstruction of scalar-field dark energy,Phys

X. Zhang,Dynamical vacuum energy, holographic quintom, and the reconstruction of scalar-field dark energy,Phys. Rev. D74(2006) 103505 [astro-ph/0609699]

2006 arXiv
[59]

Zhang, X

J. Zhang, X. Zhang and H. Liu,Reconstructing Generalized Ghost Condensate Model with Dynamical Dark Energy Parametrizations and Observational Datasets,Modern Physics Letters A23(2008) 139 [astro-ph/0612642]

2008 arXiv
[60]

Ijjas and P.J

A. Ijjas and P.J. Steinhardt,Classically Stable Nonsingular Cosmological Bounces,Phys. Rev. Lett. 117(2016) 121304 [1606.08880]

2016 arXiv
[61]

Dobre, A.V

D.A. Dobre, A.V . Frolov, J.T. G´alvez Ghersi, S. Ramazanov and A. Vikman,Unbraiding the bounce: superluminality around the corner,JCAP2018(2018) 020 [1712.10272]

2018 arXiv
[62]

Arjona, W

R. Arjona, W. Cardona and S. Nesseris,Designing Horndeski and the effective fluid approach,Phys. Rev. D100(2019) 063526 [1904.06294]

2019 arXiv
[63]

Arjona,The effective fluid approach for modified gravity,arXiv e-prints(2020) arXiv:2010.04764 [2010.04764]

R. Arjona,The effective fluid approach for modified gravity,arXiv e-prints(2020) arXiv:2010.04764 [2010.04764]

2020 arXiv
[64]

Bernardo and J

R.C. Bernardo and J. Levi Said,A data-driven reconstruction of Horndeski gravity via the Gaussian processes,JCAP2021(2021) 014 [2105.12970]. – 33 –

2021 arXiv
[65]

Nesseris,The Effective Fluid Approach for Modified Gravity and Its Applications,Universe9 (2022) 13 [2212.12768]

S. Nesseris,The Effective Fluid Approach for Modified Gravity and Its Applications,Universe9 (2022) 13 [2212.12768]

2022 arXiv
[66]

Bernardo and I

R.C. Bernardo and I. Vega,Tailoring cosmologies in cubic shift-symmetric Horndeski gravity,JCAP 2019(2019) 058 [1903.12578]

2019 arXiv
[67]

Herrera,Reconstructing k -essence: Unifying the attractor n S(N ) and the swampland criteria, Phys

R. Herrera,Reconstructing k -essence: Unifying the attractor n S(N ) and the swampland criteria, Phys. Rev. D102(2020) 123508 [2009.01355]

2020 arXiv
[68]

Sebastiani, S

L. Sebastiani, S. Myrzakul and R. Myrzakulov,Reconstruction of k-essence inflation in Horndeski gravity,European Physical Journal Plus132(2017) 433 [1702.00064]

2017 arXiv
[69]

Herrera, M

R. Herrera, M. Housset, C. Osses and N. Videla,Reconstructing k-inflation from n s(N) and reheating constraints,Physics of the Dark Universe43(2024) 101386 [2305.05042]

2024 arXiv
[70]

Herrera,Reconstructing G inflation: From the attractors n S(N ) and r (N ),Phys

R. Herrera,Reconstructing G inflation: From the attractors n S(N ) and r (N ),Phys. Rev. D98(2018) 023542 [1805.01007]

2018 arXiv
[71]

DESI collaboration,DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints,2503.14738

arXiv
[72]

Kennedy, L

J. Kennedy, L. Lombriser and A. Taylor,Reconstructing Horndeski models from the effective field theory of dark energy,Phys. Rev. D96(2017) 084051

2017
[73]

Kennedy, L

J. Kennedy, L. Lombriser and A. Taylor,Reconstructing Horndeski theories from phenomenological modified gravity and dark energy models on cosmological scales,Phys. Rev. D98(2018) 044051 [1804.04582]

2018 arXiv
[74]

Kennedy, L

J. Kennedy, L. Lombriser and A. Taylor,Screening and degenerate kinetic self-acceleration from the nonlinear freedom of reconstructed Horndeski theories,Phys. Rev. D100(2019) 044034 [1902.09853]

2019 arXiv
[75]

Renevey, J

C. Renevey, J. Kennedy and L. Lombriser,Parameterised post-Newtonian formalism for the effective field theory of dark energy via screened reconstructed Horndeski theories,JCAP2020(2020) 032 [2006.09910]

2020 arXiv
[76]

Kobayashi,Horndeski theory and beyond: a review,Reports on Progress in Physics82(2019) 086901 [1901.07183]

T. Kobayashi,Horndeski theory and beyond: a review,Reports on Progress in Physics82(2019) 086901 [1901.07183]

2019 arXiv
[77]

LIGO SCIENTIFIC, VIRGOcollaboration,GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,Phys. Rev. Lett.119(2017) 161101 [1710.05832]

2017 arXiv
[78]

Creminelli and F

P. Creminelli and F. Vernizzi,Dark Energy after GW170817 and GRB170817A,Phys. Rev. Lett.119 (2017) 251302

2017
[79]

Sakstein and B

J. Sakstein and B. Jain,Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories,Phys. Rev. Lett.119(2017) 251303 [1710.05893]

2017 arXiv
[80]

Ezquiaga and M

J.M. Ezquiaga and M. Zumalac ´arregui,Dark Energy After GW170817: Dead Ends and the Road Ahead,Phys. Rev. Lett.119(2017) 251304 [1710.05901]

2017 arXiv
[81]

Baker, E

T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller and I. Sawicki,Strong Constraints on Cosmological Gravity from GW170817 and GRB 170817A,Phys. Rev. Lett.119(2017) 251301 [1710.06394]

2017 arXiv
[82]

Bellini and I

E. Bellini and I. Sawicki,Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity,JCAP2014(2014) 050 [1404.3713]

2014 arXiv
[83]

B. Hu, M. Raveri, N. Frusciante and A. Silvestri,Effective field theory of cosmic acceleration: An implementation in CAMB,Phys. Rev. D89(2014) 103530 [1312.5742]

2014 arXiv
[84]

B. Hu, M. Raveri, N. Frusciante and A. Silvestri,EFTCAMB/EFTCosmoMC: Numerical Notes v3.0, arXiv e-prints(2014) arXiv:1405.3590 [1405.3590]. – 34 –

2014 arXiv
[85]

Zumalac ´arregui, E

M. Zumalac ´arregui, E. Bellini, I. Sawicki, J. Lesgourgues and P.G. Ferreira,hi class: Horndeski in the Cosmic Linear Anisotropy Solving System,JCAP2017(2017) 019 [1605.06102]

2017 arXiv
[86]

Bellini, I

E. Bellini, I. Sawicki and M. Zumalac ´arregui,hi class background evolution, initial conditions and approximation schemes,JCAP2020(2020) 008 [1909.01828]

2020 arXiv
[87]

Nojiri, S.D

S. Nojiri, S.D. Odintsov and S. Tsujikawa,Properties of singularities in the (phantom) dark energy universe,Phys. Rev. D71(2005) 063004 [hep-th/0501025]

2005 arXiv
[88]

Clemson, K

T. Clemson, K. Koyama, G.-B. Zhao, R. Maartens and J. V ¨aliviita,Interacting dark energy: Constraints and degeneracies,Phys. Rev. D85(2012) 043007 [1109.6234]

2012 arXiv
[89]

Vikman,Can dark energy evolve to the phantom?,Phys

A. Vikman,Can dark energy evolve to the phantom?,Phys. Rev. D71(2005) 023515 [astro-ph/0407107]

2005 arXiv
[90]

Caldwell and M

R.R. Caldwell and M. Doran,Dark-energy evolution across the cosmological-constant boundary, Phys. Rev. D72(2005) 043527 [astro-ph/0501104]

2005 arXiv
[91]

Xia, Y .-F

J.-Q. Xia, Y .-F. Cai, T.-T. Qiu, G.-B. Zhao and X. Zhang,Constraints on the Sound Speed of Dynamical Dark Energy,International Journal of Modern Physics D17(2008) 1229 [astro-ph/0703202]

2008 arXiv
[92]

Cai, E.N

Y .-F. Cai, E.N. Saridakis, M.R. Setare and J.-Q. Xia,Quintom cosmology: Theoretical implications and observations,Phys. Rep.493(2010) 1 [0909.2776]

2010 arXiv
[93]

Abramo and N

L.R. Abramo and N. Pinto-Neto,Stability of phantom k-essence theories,Phys. Rev. D73(2006) 063522 [astro-ph/0511562]

2006 arXiv
[94]

Nicolis, R

A. Nicolis, R. Rattazzi and E. Trincherini,Galileon as a local modification of gravity,Phys. Rev. D79 (2009) 064036 [0811.2197]

2009 arXiv
[95]

Deffayet, G

C. Deffayet, G. Esposito-Far `ese and A. Vikman,Covariant Galileon,Phys. Rev. D79(2009) 084003 [0901.1314]

2009 arXiv
[96]

de Felice and S

A. de Felice and S. Tsujikawa,Cosmology of a Covariant Galileon Field,Phys. Rev. Lett.105(2010) 111301 [1007.2700]

2010 arXiv
[97]

De Felice and S

A. De Felice and S. Tsujikawa,Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models,JCAP2012 (2012) 007 [1110.3878]

2012 arXiv
[98]

Giacomello, A

F. Giacomello, A. De Felice and S. Ansoldi,Bounds from ISW-galaxy cross-correlations on generalized covariant Galileon models,JCAP2019(2019) 038 [1811.10885]

2019 arXiv
[99]

Frusciante, S

N. Frusciante, S. Peirone, L. Atayde and A. De Felice,Phenomenology of the generalized cubic covariant Galileon model and cosmological bounds,Phys. Rev. D101(2020) 064001 [1912.07586]

2020 arXiv
[100]

Kase and S

R. Kase and S. Tsujikawa,Dark energy scenario consistent with GW170817 in theories beyond Horndeski gravity,Phys. Rev. D97(2018) 103501 [1802.02728]

2018 arXiv
[101]

Peirone, G

S. Peirone, G. Benevento, N. Frusciante and S. Tsujikawa,Cosmological data favor Galileon ghost condensate overΛCDM,Phys. Rev. D100(2019) 063540 [1905.05166]

2019 arXiv
[102]

Deffayet, O

C. Deffayet, O. Pujol `as, I. Sawicki and A. Vikman,Imperfect dark energy from kinetic gravity braiding,JCAP2010(2010) 026 [1008.0048]

2010 arXiv
[103]

Barreira, B

A. Barreira, B. Li, C.M. Baugh and S. Pascoli,The observational status of Galileon gravity after Planck,JCAP2014(2014) 059 [1406.0485]

2014 arXiv
[104]

Brax, A.-C

P. Brax, A.-C. Davis, B. Li and H.A. Winther,A Unified Description of Screened Modified Gravity, Phys. Rev. D86(2012) 044015 [1203.4812]

2012 arXiv
[105]

Brax, A.-C

P. Brax, A.-C. Davis, B. Li, H.A. Winther and G.-B. Zhao,Systematic simulations of modified gravity: symmetron and dilaton models,JCAP2012(2012) 002 [1206.3568]. – 35 –

2012 arXiv
[106]

Brax, A.-C

P. Brax, A.-C. Davis, B. Li, H.A. Winther and G.-B. Zhao,Systematic simulations of modified gravity: chameleon models,JCAP04(2013) 029 [1303.0007]. – 36 –

2013 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Riess, A.V

A.G. Riess, A.V . Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J.116(1998) 1009 [astro-ph/9805201]

arXiv 1998

[2] [2]

J.517(1999) 565 [astro-ph/9812133]

SUPERNOVACOSMOLOGYPROJECTcollaboration,Measurements ofΩandΛfrom 42 High Redshift Supernovae,Astrophys. J.517(1999) 565 [astro-ph/9812133]. – 30 –

arXiv 1999

[3] [3]

Ratra and P.J.E

B. Ratra and P.J.E. Peebles,Cosmological consequences of a rolling homogeneous scalar field,Phys. Rev. D37(1988) 3406

1988

[4] [4]

Ferreira and M

P.G. Ferreira and M. Joyce,Structure Formation with a Self-Tuning Scalar Field,Phys. Rev. Lett.79 (1997) 4740 [astro-ph/9707286]

arXiv 1997

[5] [5]

Copeland, A.R

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

arXiv 1998

[6] [6]

Caldwell, R

R.R. Caldwell, R. Dave and P.J. Steinhardt,Cosmological Imprint of an Energy Component with General Equation of State,Phys. Rev. Lett.80(1998) 1582 [astro-ph/9708069]

arXiv 1998

[7] [7]

Zlatev, L

I. Zlatev, L. Wang and P.J. Steinhardt,Quintessence, Cosmic Coincidence, and the Cosmological Constant,Phys. Rev. Lett.82(1999) 896 [astro-ph/9807002]

arXiv 1999

[8] [8]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Introduction to Modified Gravity and Gravitational Alternative for Dark Energy,International Journal of Geometric Methods in Modern Physics04(2007) 115 [hep-th/0601213]

arXiv 2007