Scalar-tensor gravity in an accelerating universe

D. Polarski; G. Esposito-Farese

REVIEW

Scalar-tensor gravity in an accelerating universe

G. Esposito-Farese, D. Polarski

General Relativity and Quantum Cosmology · Submitted 11 Sep 2000 · 47 references

Pith's verdict: A solid, honest reconstruction paper: the general (F,Z,U) machinery and the z_max bounds are real, but the numbers are conditional on an assumed GR+Lambda Hubble diagram, which the authors say plainly.

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arXiv IDgr-qc/0009034 (version 1)

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CitationD. Polarski, G. Esposito-Farese. "Scalar-tensor gravity in an accelerating universe." arXiv:gr-qc/0009034 (2000). https://pith.science/paper/XRF4EXTC

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Editor's note

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The thing worth your time here is the machinery, not the headline number. Esposito-Farese and Polarski set up the background and perturbation equations for a completely general scalar-tensor action, then run the reconstruction backwards: given H(z), Eq. (5.3) fixes F(z) up to initial conditions, and the requirement that both graviton and scalar carry positive energy turns into a hard upper bound on the redshift over which any given subclass can survive. For a potential-free model that bound is z_max ~ 0.66; for a massless scalar with a cosmological constant it stays under ~1.5 unless Omega_V,0 nearly saturates the observed Omega_Lambda,0; for scaling solutions F = (1+z)^-p it is ~1.43 even as p -> 0. Those are clean, checkable results, and the algebra in Eqs. (6.4)-(6.7) is correct as far as I traced it. The observation in Sec. V.A that the Z=1 parametrization can go pathological (Phi'^2 < 0) while the Einstein-frame theory is perfectly healthy is a genuinely useful caution, and I suspect it is the part people will cite. Fig. 4, where they deform H(z) by up to 30% and still never get past z_max ~ 2, is the right robustness test to have run. One correction to the record: the classifier reports no stated limitations, but the paper does flag the key one itself, in Sec. VI.C.2, where they concede that a slightly different H(z) could allow positive-energy solutions to much higher redshift. So the soft spot is acknowledged, not hidden. What is not really derived is the abstract's closing claim (s4) that the most natural models come out marginally closed. That rests on the visual smoothness of the reconstructed V(phi) in Fig. 6, which is a judgment call, not a criterion. It is hedged in the text, but it is doing more work in the abstract than the evidence supports. The perturbation section (IV.B, V.B) is complete and never used in the numerical results; fine for completeness, but it makes the paper longer than its argument. None of this is load-bearing. The central claim holds: luminosity-distance data to z ~ 2 will constrain these theories through positivity of energy, and more sharply than solar-system tests, because it probes the whole shape of A(phi) rather than two derivatives. Send it out. The referee should press on the closed-universe claim and on whether Omega_m,0 was varied independently of the H(z) shape, but I expect it to come back with minor revisions.

Referee report

Major objections (2)

[Sec. VI.A, Eq. (6.4) and paragraph following Eq. (6.1)] The paper identifies the fit parameter Ω_m,0 in the assumed H(z) of Eq. (6.1) with the true present dust density entering Eq. (5.3). This is stated explicitly (the text says the value is assumed to be the same), but the two are not varied independently: Sec. VI.A.3 varies Ω_Λ,0 with Ω_m,0=1-Ω_Λ,0, which moves both together. Since z_max is sensitive to the true Ω_m,0, a short robustness check (vary the true Ω_m,0 at fixed fit value, or vice versa) would strengthen the quantitative bounds. I note this does not affect the qualitative conclusion, which is robust for any realistic Ω_m,0 in 0.2-0.4.
[Sec. VI.C.2, Eq. (6.15)] I attempted to reproduce Eq. (6.15) by substituting F=(1+z)^{-p} into Eq. (5.6b) at z=0 and using α_0²=(F'/F)²/(4φ'²). This gives φ'²_0=p²/4+p(2+3Ω_m,0+2Ω_κ,0)/4, and hence the linear relation p(1-α_0²)=α_0²(2+3Ω_m,0+2Ω_κ,0), not the printed quadratic. The p→0 limit Eq. (6.16) is consistent with this direct evaluation, which makes the discrepancy with Eq. (6.15) more puzzling. Please check the derivation and the numerical coefficient in the '12α_0²' bound that follows. The qualitative conclusion (p small) is unaffected, but the curvature dependence and the open-universe statement depend on this equation.

Minor objections (6)

[Sec. III, first paragraph] The word 'desintegrate' should be 'disintegrate'.
[Fig. 1] The two ordinates (F/F_0 and φ-φ_0) share a single axis; a secondary axis or separate panels would improve readability.
[Sec. VI.A.3, footnote 5] The nucleosynthesis bound F_nuc > 0.86F_0 from Ref. [34] is contradicted by Ref. [27]; the text notes this but the discussion could be clearer about which bound is adopted and why.
[Abstract] The phrase 'the most natural ones are obtained if the universe is (marginally) closed' is a qualitative smoothness judgment; the abstract could say 'in the sense of smooth reconstructed potentials' to avoid over-reading.
[Sec. VI.C.2] The statement that 'a slightly different function may of course allow a positive-energy scalar field up to much higher redshifts' is a good acknowledgment; it should be echoed in the abstract or conclusion so that the conditional nature of the z_max numbers is clear to a casual reader.
[References and nomenclature] The collaboration names should be rendered consistently as 'BOOMERANG' and 'MAXIMA' (or consistently in title case) throughout the text and references.

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The paper

The paper claims

If the Hubble diagram keeps its general-relativity shape out to redshift 1–2, most scalar-tensor gravity models die there, because their scalar partner would have to carry negative energy.

The reading

The paper asks a reverse question: instead of picking a scalar-tensor theory and predicting the expansion history, suppose future supernova surveys hand us the luminosity distance $D_L(z)$ out to $z$ of order 1–2, and ask which scalar-tensor theories could possibly have produced it. The authors write the background and perturbation equations for the fully general action with arbitrary $F(Φ)$, $Z(Φ)$ and $U(Φ)$, and then run the reconstruction chain backwards: $D_L(z)$ gives $H(z)$, which gives a differential equation for the coupling function $F(z)$, which in turn gives the Einstein-frame scalar $φ(z)$ and the potential $V(φ)$. The constraint that does the work is not a fit to data but a consistency requirement: the graviton needs $F>0$ and the helicity-0 field needs $(dφ/dz)^2>0$, and in a wide class of models both fail at a modest redshift. For a potential-free theory the failure comes at $z_{max}≈0.66$; for a massless scalar it stays below about 1.5 unless a genuine cosmological constant already supplies nearly all of the observed $Ω_Λ$; for power-law scaling solutions it comes at $z_{max}≈1.43$ even as the exponent goes to zero. The authors conclude that a Hubble diagram of the general-relativity-plus-$Λ$ form measured out to $z∼2$ would be more constraining on these theories than solar-system and binary-pulsar tests, and that the models that survive with a natural, smooth potential are the ones in a marginally closed universe.

What carries the argument

The load-bearing object is the pair of positivity conditions applied along the reconstruction chain. In the Jordan frame the coefficient $F(Φ)$ multiplies the curvature scalar, so $F$ must stay strictly positive for the graviton to carry positive energy; in the Einstein frame the true helicity-0 field is $φ$, defined by $(dφ/dΦ)^2 = (3/4)(dlnF/dΦ)^2 + Z/(2F)$, and its kinetic term must have the standard sign. The authors stress that these two conditions are not the same statement: in the popular $Z=1$ parametrization $Φ'^2$ can go negative in perfectly regular situations, so an apparent singularity there is an artifact of the variables, whereas $F=0$ is a real breakdown. Because Eq. (5.6) expresses $(dφ/dz)^2$ purely in terms of the observable $H(z)$ and the reconstructed $F(z)$, one can scan over initial conditions and find, for each model class, the redshift at which the theory stops being definable. That redshift, not a goodness-of-fit statistic, is the quantity the paper reports.

Extended reading notes

Core claim

The central claim is that low-redshift cosmology, read as a reconstruction problem rather than a model-fitting problem, is enough to eliminate large subclasses of scalar-tensor gravity. The argument is a chain of algebraic and differential steps laid out in Section V: Eq. (5.1) turns a measured $D_L(z)$ into $H(z)$; Eq. (5.3) is a second-order linear equation for $F(z)$ sourced by the unknown potential $U$; and Eq. (5.6) gives $(dφ/dz)^2$ in terms of $F$, $H$ and their derivatives. Two positivity conditions must hold everywhere the theory is defined — $F>0$ so gravitons carry positive energy, and $(dφ/dz)^2>0$ so the Einstein-frame scalar does — and the authors show that for potential-free models, for massless scalars with a separate cosmological constant, and for scaling solutions $F=(1+z)^{-p}$, one or the other condition fails at a computable redshift. Assuming the observed $H(z)$ has exactly the flat $Λ$CDM form with $Ω_{Λ,0}=0.7$ and $Ω_{m,0}=0.3$, they obtain $z_{max}≈0.66$ analytically for $U=0$ in a flat universe, pushed only slightly higher by spatial curvature or by the largest matter-scalar coupling still allowed by solar-system tests. They also report the converse branch: when $F$ is prescribed and the potential is reconstructed instead, smooth and stable $V(φ)$ emerges only if the universe is closed, with $|Ω_{κ,0}|$ large enough (their example is $-0.1$) that the observed $Λ$ is the present value of the potential rather than a parameter inserted by hand.

Load-bearing premise

The quantitative redshift limits all assume the observed expansion history has exactly the flat general-relativity-plus-cosmological-constant form with $Ω_{Λ,0}=0.7$ and $Ω_{m,0}=0.3$ out to $z∼2$; the authors themselves note in Section VI.C.2 that a slightly different function could allow a positive-energy scalar field up to much higher redshifts.

Editorial extensions

If this is right

A supernova Hubble diagram of the $Λ$CDM form measured to $z∼2$ would exclude potential-free scalar-tensor gravity, and would exclude a massless scalar explanation of the acceleration unless a separate cosmological constant already accounts for nearly all of the observed $Ω_Λ$.
Power-law scaling solutions $F=(1+z)^{-p}$ are excluded beyond $z≈1.43$ for any nonzero $p$, however small, so the limit is not a small-coupling effect that better data could soften.
Cosmological distance measurements can probe the full shape of the matter-scalar coupling function $A(φ)$, whereas solar-system and binary-pulsar tests only constrain its first two derivatives at the present epoch.
If a smooth, stable reconstructed potential is the criterion for naturalness, then a marginally closed universe with $|Ω_{κ,0}|$ of order 0.1 is favoured over a flat one, since in the flat case the reconstructed $V(φ)$ has a sudden kink near the present epoch.
Reconstructed coupling functions $\ln A(φ)$ come out convex, so the present tight solar-system bound is a consequence of the cosmological attractor mechanism rather than a fine-tuned coincidence.

Reading between the lines

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

The logic inverts the usual direction of cosmological model testing: the paper treats a null result (no deviation from $Λ$CDM in $H(z)$) as the most informative outcome, because the deviations it forbids are not in the fit but in the sign of a kinetic term.
Because the excluded models fail by losing positivity rather than by fitting badly, the bounds are one-sided in a useful way: a future measurement that does deviate from $Λ$CDM would not automatically rescue scalar-tensor theories, it would just move the reconstruction onto a different branch that must be re-checked for positivity.
The same positivity scan could be run against any parametrized dark-energy equation of state, not only the exact $Λ$CDM form used here, and the resulting $z_{max}$ map would be a compact way to state which regions of dark-energy parameter space admit a healthy scalar partner at all.
The authors' remark that a slightly different $H(z)$ could allow a positive-energy scalar to much higher redshift means the quantitative numbers in this paper are hostage to the assumed functional form; the durable content is the method, and the method's verdicts should be quoted as conditional on the fitted shape of $H(z)$.

Original abstract

We consider scalar-tensor theories of gravity in an accelerating universe. The equations for the background evolution and the perturbations are given in full generality for any parametrization of the Lagrangian, and we stress that apparent singularities are sometimes artifacts of a pathological choice of variables. Adopting a phenomenological viewpoint, i.e., from the observations back to the theory, we show that the knowledge of the luminosity distance as a function of redshift up to z ~ (1-2), which is expected in the near future, severely constrains the viable subclasses of scalar-tensor theories. This is due to the requirement of positive energy for both the graviton and the scalar partner. Assuming a particular form for the Hubble diagram, consistent with present experimental data, we reconstruct the microscopic Lagrangian for various scalar-tensor models, and find that the most natural ones are obtained if the universe is (marginally) closed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,413 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.58 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.67 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We consider scalar-tensor theories of gravity in an accelerating universe. The equations for the background evolution and the perturbations are given in full generality for any parametrization of the Lagrangian, and we stress that apparent singularities are sometimes artifacts of a pathological choice of variables. Adopting a phenomenological viewpoint, i.e., from the observations back to the theory, we show that the knowledge of the luminosity distance as a function of redshift up to z ~ (1-2), which is expected in the near future, severely constrains the viable subclasses of scalar-tensor theories. This is due to the requirement of positive energy for both the graviton and the scalar partner. Assuming a particular form for the Hubble diagram, consistent with present experimental data, we reconstruct the microscopic Lagrangian for various scalar-tensor models, and find that the most natural ones are obtained if the universe is (marginally) closed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider scalar-tensor theories of gravity in an accelerating universe.	background 0.64	0.70	0.03	asserted 0.65
s1	The equations for the background evolution and the perturbations are given in full generality for any parametrization of the Lagrangian, and we stress that apparent singularities are sometimes artifacts of a pathological choice of variables.	contribution 0.65	0.90	0.06	asserted 0.94
s2	Adopting a phenomenological viewpoint, i.e., from the observations back to the theory, we show that the knowledge of the luminosity distance as a function of redshift up to z ~ (1-2), which is expected in the near future, severely constrains the viable subclasses of scalar-tensor theories.	result 0.75	0.90	0.64	asserted 0.99
s3	This is due to the requirement of positive energy for both the graviton and the scalar partner.	interpretation 0.95	0.59	0.12	asserted 1.00
s4	Assuming a particular form for the Hubble diagram, consistent with present experimental data, we reconstruct the microscopic Lagrangian for various scalar-tensor models, and find that the most natural ones are obtained if the universe is (marginally) closed.	result 0.91	0.95	0.33	asserted 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 3.** FIG. 3. Two versions of the reconstructed coupling function

**Figure 6.** On the contrary, if |Ωκ,0| is large enough, the same analytical expression (6.14) gives nice regular potentials, like the one displayed in the right panel of

Showing 4 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/0009034 by the authors.

References

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astro-ph 2006-12 conditional novelty 6.0 of 10

Viable f(R) gravity models that pass solar system tests cannot produce dark energy distinguishable from a cosmological constant, whether they hide the scalar via large mass or via the chameleon effect.
Assessing cosmological evidence for non-minimal coupling
astro-ph.CO 2025-04 conditional novelty 5.0 of 10

Bayesian model comparison of current cosmological data favors a scalar-field dark energy that couples non-minimally to gravity over both ΛCDM and minimally coupled quintessence.
Late-transition vs smooth $H(z)$ deformation models for the resolution of the Hubble crisis
astro-ph.CO 2021-10 conditional novelty 5.0 of 10

A late, sharp transition in supernova absolute magnitude at z about 0.01 fits CMB, BAO, and supernova data better than smooth dark-energy models and can reconcile local and CMB Hubble constant estimates.
Brans-Dicke cosmology with a $\Lambda$- term: a possible solution to $\Lambda$CDM tensions
astro-ph.CO 2020-06 conditional novelty 5.0 of 10

A Brans-Dicke plus Lambda model with two extra parameters fits the joint cosmological data better than GR-Lambda-CDM and pulls both the H0 and sigma8 tensions down to about 1.5 sigma.
Evolution of the $f\sigma_8$ tension with the Planck15/$\Lambda$CDM determination and implications for modified gravity theories
astro-ph.CO 2018-03 conditional novelty 5.0 of 10

The fσ8 tension with Planck15/ΛCDM is 5σ for the full 63-point RSD compilation but below 1σ for the 20 most recent points, a shift the authors attribute partly to higher redshifts and larger error bars.
Tension and constraints on modified gravity parametrizations of $G_{\textrm{eff}}(z)$ from growth rate and Planck data
astro-ph.CO 2017-03 conditional novelty 5.0 of 10

Growth-rate data prefer a weakening effective Newton constant at low redshift, but that preference conflicts with scalar-tensor theory and with the low-multipole CMB.
Solutions in the scalar-tensor theory with nonminimal derivative coupling
gr-qc 2013-12 accept novelty 5.0 of 10

Exact static black hole solutions are found for a scalar-tensor theory with derivative coupling to the Einstein tensor, showing that the asymptotic geometry is set by the coupling constant, not the cosmological constant.
Cosmic history of viable exponential gravity: Equation of state oscillations and growth index from inflation to dark energy era
gr-qc 2012-07 conditional novelty 5.0 of 10

Adding a cube-root correction term to viable F(R) gravity stabilizes dark-energy oscillations at constant frequency, and two exponential-gravity models with an R^α term produce inflation that exits into a ΛCDM-like ph...
Nonminimal Coleman--Weinberg Inflation with an $R^2$ term
gr-qc 2018-10 conditional novelty 4.0 of 10

Adding an R² term to a nonminimally coupled Coleman-Weinberg inflation model reduces the observable predictions to one mixing angle and brings the tensor-to-scalar ratio into agreement with Planck 2018.
Frame-dependence of higher-order inflationary observables in scalar-tensor theories
gr-qc 2017-07 conditional novelty 4.0 of 10

Third-order slow-roll corrections leave Einstein-frame and Jordan-frame spectral indices identical, but the two frames disagree on how many e-folds a given field excursion produces, shifting predictions by amounts com...
Dark energy in Horndeski theories after GW170817: A review
gr-qc 2018-09 accept novelty 3.0 of 10

After GW170817, dark energy in Horndeski theories is restricted to four classes of models with distinct growth and lensing signatures.
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gr-qc 2026-09 conditional novelty 2.0 of 10

A doctoral dissertation synthesizing scalar-tensor, inflationary EFT, quintessence, and ekpyrotic and cyclic cosmology into a single scalar-field framing of the dark sector, built largely from prior published work.
Modified Gravity and Cosmology: An Update by the CANTATA Network
gr-qc 2021-05 conditional novelty 2.0 of 10

A large multi-author review catalogues modified gravity theories, their theoretical consistency conditions, and the observational constraints on each, without producing new derivations or data.
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astro-ph.CO 2010-09 accept novelty 2.0 of 10

The review argues that testing the constancy of the fundamental constants is a direct test of local position invariance and the universality of free fall, and catalogues every current observational constraint on such drift.
Infrared-modified gravities and massive gravitons
hep-th 2008-02 accept novelty 2.0 of 10

Lorentz-violating massive gravity can avoid the ghost and strong-coupling problems of the Fierz-Pauli theory, at the price of giving up Lorentz invariance and relying on residual gauge symmetries.

How later papers used it

86 later papers cite it

Evidence: substantial. 84 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 144 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 6 papers with this label

Wrote its action in a canonical form by defining a redefinition of the scalar field from this paper.

Spontaneous symmetry breaking in the late Universe and glimpses of early Universe phase transitions \`a la baryogenesis (2021) · 0.80 · paraphrased: the licence does not allow quoting
Followed the approach discussed in this paper to constrain the underlying parameters of a theory by requiring agreement with the observed expansion history derived from ΛCDM.

A Unified Approach to Cosmic Acceleration (2010) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

73 arXiv papers cite this in 125 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 76 sentences without a confident label

Contests it 1 paper

However, these models are predicting a blue spectrum [ref] i.e. $n_s > 1$, and are thus ruled out by present CMB observations[While the interest in these models to describe inflation was damped by this evidence, these models were extensively studied as possible models for quintessence [ref] .].

Classification of inflationary models and constraints on fundamental physics · Generalized models. · contest 0.82

Uses its method or tool 5 papers

The action in this frame becomes (see [ref] for further details) S_E= 12∫d^4x √(-g̅)[M^2 R̅-K(ϕ)(∂̅ϕ)^2- 2 L̅_SM ] , K(ϕ)≡2f(ϕ)+3 M^2 f^'2(ϕ)2f^2(ϕ) .

The Higgs mass range from Standard Model false vacuum Inflation in scalar-tensor gravity · The inflationary model · 0.82
This scalar-tensor theory is defined by the functional forms of the scalar field potential $U(ϕ)$ and nonminimal coupling $F(ϕ)$ that are reconstructed uniquely using the method of Refs [ref] .

Tension and constraints on modified gravity parametrizations of $G_{\textrm{eff}}(z)$ from growth rate and Planck data · Introduction · 0.98
To this end, following the approach in [ref] , we will use a subhorizon approximation under which we assume that time derivative of any perturbation variable is much less than its spatial derivative or equivalently, in terms of Fourier space, a term multiplied by $k$.

Linear growth of matter density perturbations in f(R,G) theories · Effective gravitational constant · 1.00

Compares with it 2 papers

This is quite similar to the result in Ref. [ref] where $F(n)/F_0$ is defined as $e^-2n f(n)$ with $f(n)$ is a general function of $n$.

Constraints on scalar-tensor theories of gravity from observations · Comparison with Dark Energy models · 1.00
By using the approximate expression for the metric perturbation in ([ref]), we find the equation describing the evolution of the density contrast δ̈_m+2Hδ̇_m-4 πG_eff ρ̅ δ_m ≈0, which again is the result obtained in [ref] .

A Unified Approach to Cosmic Acceleration · Evolution of Density Perturbations · 0.94

Cites it as support 1 paper

Such scalar-tensor models have no ghost [ref] .

Possible evolution of a bouncing universe in cosmological models with non-minimally coupled scalar fields · Introduction · 0.96

Cites it as background 34 papers

We consider the following ST action in the physical Jordan frame [ref] S=116πG ∫d^4x √(-g) (F(Φ) R - Z(Φ) g^μν ∂_μΦ∂_νΦ- 2U(Φ) ) + S_m[ψ_m; g_μν] .

Evolution of Dark Energy Perturbations in Scalar-Tensor Cosmologies · Perturbations in Scalar-Tensor Cosmologies · 0.91
A more generic scalar-tensor gravity (STG) action is characterized by two arbitrary functions, the coupling function $ω(Ψ)$ in the kinetic term of the scalar field and the potential $V(Ψ)$ [ref] .

Post-Newtonian parameters gamma and beta of scalar-tensor gravity with a general potential · Introduction · 0.98
with $κ=8πG_∗$, where $G_∗$ represents the “bare” gravitational constant ( [ref] ).

The galaxy cluster concentration-mass relation in dark energy cosmologies · Dark energy models · 0.92

Discussion

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