Constraining An Exact Brans-Dicke gravity theory with Recent Observations

Anil Kumar Yadav; Hassan Amirhashchi

REVIEW 4 major objections 5 minor 1 cited by

Constraining An Exact Brans-Dicke gravity theory with Recent Observations

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper claims that an exact Brans-Dicke cosmological solution with a cosmological constant, fitted to Hubble, BAO, and Planck CMB data, matches general relativity, with the Brans-Dicke coupling exceeding 1560 at 95% confidence.

desk verdict A routine MCMC fit with a load-bearing sign error: the paper's headline ω>1560 bound is contradicted by its own Eq. (20). read the letter →

arxiv 1908.04735 v3 pith:4K7PS7M7 submitted 2019-08-12 gr-qc

Hassan Amirhashchi , Anil Kumar Yadav This is my paper · ORCID

classification gr-qc PACS 98.80.-k04.20.Jb04.50.kd

keywords Brans-Dickegravityscalar-tensortheorycosmologicalconstantdarkenergyexactsolutiongravitationalvariationparameterconstraints

topics 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

This paper derives an exact cosmological solution in Brans-Dicke gravity with a cosmological constant and uses it to ask how far current data allow gravity to depart from general relativity. The authors re-express the Brans-Dicke coupling constant $\omega$ as a scalar-field density parameter $\Omega_\phi$, fit the model to a joint sample of cosmic-chronometer Hubble data, baryon acoustic oscillation distance ratios, and Planck CMB distance priors, and find no significant deviation from general relativity. Their headline result is $\Omega_\phi = 0.010^{+0.021}_{-0.012}$, which they translate into a lower bound $\omega > 1560$ at 95% confidence and a best-fit $\omega = 308.452$. The same fit gives a present-day drift of the gravitational constant between $1.150\times10^{-13}$ and $1.198\times10^{-13}\,\mathrm{yr}^{-1}$ at $1\sigma$, and a total change since recombination between $-0.0084$ and $-0.0082$ at 68% confidence. If the result is right, cosmological data already push Brans-Dicke gravity very close to Einstein's theory, with a sub-percent variation in $G$ since recombination.

What carries the argument

The load-bearing object is the scalar-field density parameter $\Omega_\phi$ defined in Eq. (20), a reparametrization of the Brans-Dicke coupling that measures the fraction of the cosmic energy budget carried by the scalar field. It enters the model through the Hubble rate $H_{\rm BD}=H_0(1-\Omega_\phi)^{1/2}[\Omega_m(1+z)^{3+2.5(\sqrt{1+0.96\Omega_\phi}-1)}+\Omega_\Lambda]^{1/2}$, obtained from the exact solution $\phi=(a/a_0)^{1/(\omega+1)}$ after setting the integration constant in Eq. (16) to zero. The MCMC fit constrains $\Omega_\phi$ directly, and $\omega$, $\dot G/G$, and $\delta G/G$ are then derived from it. The near-zero fitted value of $\Omega_\phi$ is what produces the large lower bound on $\omega$.

What would settle it

Recompute the posterior for $\omega$ by translating the fitted $\Omega_\phi$ interval through Eq. (20) without imposing the $\Omega_\phi\to0$ branch; at $\Omega_\phi=0.010$ the formula gives $\omega\simeq-1.2$, which would show that the quoted $\omega>1560$ depends on the branch choice. A second check is to re-fit the model with the integration constant $\kappa$ in Eq. (16) left free; if the data accept $\kappa\neq0$ just as well, the reported constraints are specific to that special solution.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the exact spatially flat Brans-Dicke solution with a cosmological constant is statistically indistinguishable from $\Lambda$CDM when fitted to OHD, BAO, and CMB distance data. The paper defines the scalar-field density parameter $\Omega_\phi = -(5\omega+6)/(6(\omega+1)^2)$ and treats $\Omega_\phi\to0$ as equivalent to $\omega\to\infty$, so the fitted value of $\Omega_\phi$ measures how far the theory sits from general relativity. Marginalizing over $\{H_0,\Omega_m,\Omega_\phi\}$ yields $\Omega_\phi = 0.010^{+0.021}_{-0.012}$, which the authors convert to $\omega>1560$ at 95% CL and use to derive $\dot G/G = 1.17\times10^{-13}\,\mathrm{yr}^{-1}$ and $\delta G/G = -0.00825$ since recombination. They conclude that no significant deviation from general relativity is present and that the additional scalar degree of freedom does not shift the best-fit values of $H_0$, $\Omega_m$, or $\Omega_\Lambda$.

Load-bearing premise

The quoted bound on the Brans-Dicke coupling hangs on the assumption that a fitted scalar-field density of about $0.01$ corresponds to an almost infinite coupling rather than to the other branch of the same relation, where the coupling would be near $-1.2$; the exact-solution derivation also assumes that an integration constant in the first integral is zero because the solution is singular at $a=0$ and $\phi=0$.

Editorial extensions

If this is right

If the result holds, Brans-Dicke gravity with a cosmological constant is observationally indistinguishable from $\Lambda$CDM over the fitted redshift range, with $H_0$, $\Omega_m$, and $\Omega_\Lambda$ statistically consistent between the two models.
The lower bound $\omega>1560$ at 95% confidence means any cosmological deviation from general relativity is smaller than the joint OHD+BAO+CMB data can currently resolve.
The gravitational constant was slightly larger in the past, with a total change since recombination of about $-0.8\%$ at 68% confidence.
The deceleration-to-acceleration transition happens at $z_t\simeq0.60$ in both models, so the scalar field does not shift the epoch of cosmic acceleration.
The paper's bound on $\omega$ is stronger than earlier cosmological bounds but remains far below the solar-system bound of roughly $40{,}000$ from Cassini ranging.

Reading between the lines

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

Beyond the paper: the other branch of Eq. (20) would read a fitted $\Omega_\phi=0.010$ as $\omega\simeq-1.2$, so the physical meaning of the fit depends on the branch choice.
Beyond the paper: allowing a nonzero integration constant in Eq. (16) would generate a different family of Hubble laws; fitting those to the same data would show whether the reported bounds are special to the $\kappa=0$ solution.
Beyond the paper: the $\Omega_\phi$ parametrization could be applied to other scalar-tensor theories, turning a fitted density parameter near zero into a direct measure of proximity to general relativity.
Beyond the paper: the sub-percent integrated change in $G$ implies that standard analyses treating $G$ as constant are safe at the percent level over this redshift range, a check the paper does not perform.

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.

Desk Editor's Note

read the letter

Colleague,

This one is straightforward: the central result is wrong. The headline bound ω>1560 is built on an inconsistent mapping between Ωφ and ω. Eq. (20) gives Ωφ = −(5ω+6)/(6(ω+1)^2), which is negative for every ω > −6/5 and tends to 0 from below as ω→∞. So a positive fitted Ωφ like 0.010 corresponds to ω around −1.2, not to large positive ω. The value ω=308.452 they quote for the best fit comes from the opposite sign; that is the convention implicit in Eq. (22), while Eq. (20) has a minus sign. The paper runs two incompatible mappings and all derived constraints on ω, Gdot/G, and δG/G flow through the error. The positive Gdot/G bound (1.15–1.20×10^-13 yr^-1) is itself a sign red flag: with positive ω, φ grows and G_eff∝1/φ falls, so Gdot/G should be negative.

The rest is thin. The 'new exact solution' is the standard power-law φ∝a^{1/(ω+1)}. The κ=0 step at Eq. (16)–(17) is unproven; setting the integration constant to zero because a=0 and φ=0 is singular excludes the general solution without justification. Eq. (23) also fails to normalize at z=0: if Eq. (21) holds, H(0)=H0(1−Ωφ), not H0.

What the paper does do competently is the data analysis: the MCMC implementation is straightforward, the datasets are public, and the comparison with ΛCDM yields H0 and Ωm consistent with Planck. That is not enough to carry the paper.

Who is this for? Someone cataloguing BD constraints should skip it. The pipeline could be instructive after the sign error is fixed and the fit redone, but as published the numbers are not usable.

Recommendation: desk reject, with a note that the Ωφ–ω sign convention must be corrected and the analysis rerun before resubmission. If a referee is used, that is the first check.

Referee Report

4 major / 5 minor

Summary. The paper derives an exact Brans-Dicke solution with a cosmological constant in a spatially flat Robertson-Walker metric, introduces a scalar-field density parameter Omega_phi in Eq. (20), and fits the resulting LambdaBD model to 31 observational Hubble data points, Planck CMB distance priors, and BAO data using an MCMC. It also fits flat LambdaCDM to the same data as a baseline. The paper reports Omega_phi = 0.010^{+0.021}_{-0.012}, translates this into omega > 1560 at 95% CL, and gives constraints on Gdot/G and delta G/G, concluding that the data show no significant deviation from general relativity.

Significance. If the Omega_phi--omega mapping and the Hubble evolution formula were correct, the reported limits would be competitive cosmological constraints on scalar-tensor gravity and would strongly support general relativity. The manuscript uses standard public data, describes MCMC convergence checks, and compares against a LambdaCDM baseline. However, the conversion from the fitted Omega_phi to the headline omega and Gdot/G constraints is internally inconsistent, and the Hubble formula is mis-normalized; as written, the central results are not supported by the model's own equations.

major comments (4)

[Eq. (20), Eq. (22), Table II] The mapping between Omega_phi and omega changes sign between equations, and this invalidates the central claim of the paper. Eq. (20) defines Omega_phi = -(5 omega + 6)/(6(omega+1)^2), which is negative for every omega > -6/5 and tends to zero from below as omega tends to infinity. Under this definition, the reported positive value Omega_phi = 0.010 would give omega approximately -1.20 or -84, not omega > 1560. The opposite sign, Omega_phi = +(5 omega + 6)/(6(omega+1)^2), is what reproduces omega = 308.452 for the best-fit Omega_phi = 0.0027 and is also the relation used in Eq. (22), where solving for 1/(1+omega) yields the quoted exponent 2.5(sqrt(1+0.96 Omega_phi) - 1). Eq. (22) itself gives omega approximately 83 for Omega_phi = 0.010, and omega > 1560 would require Omega_phi < 5.4 x 10^-4. The paper therefore works with two incompatible definitions of Omega_phi, and all derived bounds on omega, Gdot/G, and delta G/G inherit the inconsistency.
[Eq. (23)] The Hubble rate in Eq. (23) does not satisfy H(0) = H0 at z = 0 for the model's own closure relation. Combining Eqs. (19)-(21) gives Omega_m + Omega_Lambda = 1 - Omega_phi. Substituting z = 0 into Eq. (23) then yields H_BD(0) = H0(1 - Omega_phi), not H0. The required factor is (1 - Omega_phi)^(-1/2) if Omega_m and Omega_Lambda are the present-day density parameters. Since all three likelihoods are evaluated with Eq. (23), the fitted H0 and the associated parameter uncertainties are biased; for Omega_phi = 0.010 the normalization mismatch is about 1%, comparable to the quoted uncertainty on H0.
[Eqs. (16)-(17)] The passage from the general first integral to the claimed exact solution sets the integration constant kappa to zero without justification. The text says this follows because the solution has a singularity at a = 0 and phi = 0, but the presence of a standard big-bang-type singularity does not imply that the integration constant must vanish. Setting kappa = 0 restricts the analysis to a special case of Eq. (16). The subsequent derivation of phi(a), psi = 1/(omega+1), the density-parameter sum rule, and Eq. (23) all rely on this restriction, so the claimed exactness of the solution is unsupported.
[Eq. (43), Table IV] The quoted error bars on Gdot/G are inconsistent by a factor of 10. Table II gives Gdot/G = (0.1147 +/- 0.0024) x 10^-12 yr^-1 at 68%, which equals (1.147 +/- 0.024) x 10^-13 yr^-1 and agrees with the interval in Eq. (43). Table IV, whose header is in units of 10^-13 yr^-1, lists 1.174 +/- 0.0024 at 1 sigma and +/- 0.0046 at 2 sigma, i.e., errors ten times smaller. The abstract and Eq. (43) therefore advertise a precision for Gdot/G that Table IV does not support.

minor comments (5)

[Eq. (33)] The integral for the sound horizon r_s(z*) is written with limits that appear as 'from infinity to z'; it should presumably run from 0 to z*.
[Table II] The table header says 'at 2 sigma & 2 sigma confidence levels' but the table reports 68% and 95% confidence levels; please correct the header.
[Table IV] The table caption says 'Constraints on the BD coupling constant', but the table lists bounds on Gdot/G rather than on omega; the caption should be corrected.
[Appendix A, Eq. (A2)] The expression for q(z) appears to be missing the -4 Omega_Lambda term in the numerator. As written it equals (1+z) H'/H rather than q = (1+z) H'/H - 1, and at z = 0 it does not reduce to Eq. (26) unless Omega_Lambda = 0. The transition-redshift formula in Eq. (A3) inherits this issue.
[Throughout] There are numerous typos and grammatical slips, including 'Rece nt Observations' in the title, 'Brans-Dike' in the keywords, 'Before solving above questions' in Section II, and 'we also obtained constraint the rate of change' in the abstract; the manuscript needs a careful proofreading pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ω and G-dot/G constraints are derived from the fitted Ωφ via the model's own defining relation, with external data and a ΛCDM baseline; the serious issues are sign and branch-choice errors, not circularity.

full rationale

The derivation is self-contained against external data. The paper solves the Brans-Dicke field equations, defines Ωφ by Eq. (20) as a function of ω, fits Ωφ to OHD+BAO+CMB data by MCMC, and then reports ω, G-dot/G, and δG/G as derived parameters. Because the paper explicitly labels Ωφ as the constrained parameter and ω as derived from the same defining relation, the reported ω constraints are a deterministic reparametrization of the fitted posterior rather than an independent prediction smuggled in as new evidence. The comparison with ΛCDM fitted to the same external data and with previous independent solar-system and cosmological bounds provides independent anchoring. There is no load-bearing self-citation: the cited prior works by the same authors appear in the introduction and in transition-redshift comparisons and do not carry the derivation. No uniqueness theorem or ansatz is imported from the authors' prior work. The two substantive problems are correctness/fragility issues, not circularity: (i) Eq. (20) gives Ωφ = -(5ω+6)/(6(ω+1)^2), which is negative for ω > -6/5, so the positive fitted Ωφ = 0.0027 should not map to ω = 308.452; Table II and Eq. (22) evidently use the opposite sign, making the headline ω > 1560 internally inconsistent. (ii) The step at Eqs. (16)-(17) setting κ = 0 because the general solution has singularities at a = 0 and φ = 0 is an unproven restriction to a special branch. Neither of these is a circular reduction of a prediction to an input.

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

The central result rests on three fitted cosmological parameters, plus one additional parameter (Omega_b h^2) whose status is ambiguous. The most serious entry is the ad hoc setting of kappa=0, which discards the general solution, and the branch-choice error in the Omega_phi-to-omega mapping, which is not captured as a parameter but as a misidentification of the physical branch.

free parameters (4)

H0 (present Hubble parameter) = 69.334 (best fit, Lambda BD)
Sampled with uniform prior U(60,80) as a free parameter in both models.
Omega_m (matter density parameter) = 0.318 (best fit, Lambda BD)
Sampled with uniform prior U(0,0.5); constrained by OHD+BAO+CMB.
Omega_phi (scalar field density parameter) = 0.0027 (best fit), quoted 0.0101+0.0026-0.010 at 68%
Treats the Brans-Dicke coupling as a new density parameter; however the mapping to omega is sign-inconsistent, so the fitted value does not support the quoted omega bounds.
Omega_b h^2 (baryon density parameter) = 0.0227 (best fit, Lambda BD)
Listed in the parameter space but claimed to be derived; the CMB and BAO likelihoods depend on it, so it must enter the fit either as a varied parameter or a fixed input, which is not clarified.

assumptions (4)

domain assumption Spatial flatness (k=0) and pressureless matter (p=0) for the universe.
Used throughout Section II to derive the exact solution and the Hubble parameter (Eqs. 5-23).
ad hoc to paper The first-integral constant kappa in Eq. (16) is set to zero.
The paper states the solution has a singularity at a=0 and phi=0 'which in turn gives kappa=0'; no mathematical proof is provided, and this removes the general solution.
domain assumption Planck 2015 CMB distance priors with fixed decoupling redshift z*=1090 are valid for the Brans-Dicke model.
The CMB likelihood (Eqs. 31-36) uses shift parameters and acoustic scale computed for a standard recombination history; applying them to a nonstandard expansion history is an approximation.
standard math The Brans-Dicke field equations with cosmological constant as written in Eqs. (1)-(2) are the correct starting point.
Standard textbook equations, but sign conventions for the Lambda term in the wave equation are not checked.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Constraining An Exact Brans-Dicke gravity theory with Recent Observations." pith.science (2026). https://pith.science/paper/4K7PS7M7

@misc{pith2026190804735,
  author       = {Pith},
  title        = {Pith review of: Constraining An Exact Brans-Dicke gravity theory with Recent Observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4K7PS7M7}},
  note         = {Machine review of arXiv:1908.04735}
}

abstract

In this paper first we study Brans-Dicke equations with the cosmological constant to find an exact solution in the spatially flat Robertson-Walker metric. Then we use Observational Hubble data, the baryon acoustic oscillation distance ratio data as well as cosmic microwave background data from Planck to constrain parameters of the obtained Brans-Dicke model. To compare our results and find out the amount of deviation from general relativity, we also constrain concordance cosmological model using the same data. In our theoretical model the Brans-Dicke coupling constant is replaced by a , say new, parameter namely \textgravedbl scalar field density $\Omega_{\phi}$\textacutedbl. Therefore, as $\Omega_{\phi}\to 0$ which is equivalent to $\omega\to \infty$ general relativity is recovered. In general, we found no significant deviation from general relativity. Our estimations show $\Omega_{\phi}=0.010^{+0.021}_{-0.012}$ which is equivalent to $\omega>1560$ at $95\%$ confidence level. We also obtained constraint the rate of change of gravitational constant, $\dot{G}/G$, at present time as $1.150\times 10^{-13}yr^{-1}<\dot{G}/G<1.198\times 10^{-13}yr^{-1}$ (at $1\sigma$ error). The total variation of gravitational constant, since the epoch of recombination, is also constrained as $-0.0084<\delta G/G<-0.0082$ at $68\%$ confidence level.

Figures

Figures reproduced from arXiv: 1908.04735 by the authors.

**Figure 1.** FIG. 1: One-dimensional marginalized distribution, and [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 4.** FIG. 4: Constraints in the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5: Constraints in the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7: The plot of deceleration parameter versus the redshif [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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

Lyra's cosmology of homogeneous and isotropic universe in Brans-Dicke theory
gr-qc 2019-09 reject novelty 3.0 of 10

A Brans-Dicke-Lyra model is fitted to H(z) and supernova data, but the acceleration claim conflicts with the model's own field equations and the fit reduces to ΛCDM.

Reference graph

Works this paper leans on

114 extracted references · 71 canonical work pages · cited by 1 Pith paper

[1]

198 × 10−13, − 0

150 × 10−13< ˙G/G< 1. 198 × 10−13, − 0. 0084<δG/G< −0. 0082, (43) with the best-ﬁt values ˙G/G = 1. 17 × 10−13, δG/G = −0. 00825. (44) We have summarized constraints on ˙G/G obtained from diﬀerent methods in Table. IV which is an update of Ta- ble I of Ref [30]. From this table we observe that our constrain is much tighter with respect to other listed pre...
[2]

galaxy diﬀerential age

(24) We also note that a de-Sitter solution arises whenever q = −1 ⇒ ˙H = 0 ⇒H =H0 ⇒a(t) ∝eH0t. (25) Since for our ΛBD model we have qz=0 = Ω m ( √ 24Ω φ + 25 − 3 ) − 4Ω Λ 4 (Ω Λ + Ω m) , (26) it is clear to conclude that this condition can not be achieved in our ΛBD model (see V for derivation of DP an transition redshift). In the next section we use joi...
[3]

We use the CMB distance priors based and Planck 2015 collaboration [3] to constrain our models

7 ) −4 , (34) whereh is the normalized Hubble constant and z∗ = 1090 is decoupling redshift. We use the CMB distance priors based and Planck 2015 collaboration [3] to constrain our models. According to Planck 2015 data, the mean values of shift and acoustic scale parameters are ⟨R⟩ = 1. 7488 and ⟨la⟩ = 301. 76 with the deviations σ (R) = 0. 0074 and σ (la...

2015
[4]

106) − 0

0152 [ rs(zd) DV (z = 0. 106) − 0. 336 ] 2 + 1 ( 25
[5]

32) rs(zd) − 1264

28 )2 [ DV (z = 0. 32) rs(zd) − 1264
[6]

Padmanabhan, Phys

T. Padmanabhan, Phys. Rept. 380, 234 (2003)

2003
[7]

00712 [ rs(zd) DV (z = 0. 44) − 0. 0916 ] 2 + 1 ( 19
[8]

51) rs(zd) − 1877

78 )2 [ DV (z = 0. 51) rs(zd) − 1877

Show all 114 references

[9]

57) rs(zd) − 2056

28 )2 [ DV (z = 0. 57) rs(zd) − 2056
[10]

00342 [ rs(zd) DV (z = 0. 6) − 0. 0726 ] 2 + 1 ( 22
[11]

61) rs(zd) − 2140

78 )2 [ DV (z = 0. 61) rs(zd) − 2140
[12]

00322 [ rs(zd) DV (z = 0. 73) − 0. 0592 ] 2 (41) Finally, since these three datasets are independent, the total chisqure could be given by χ 2 tot =χ 2 OHD +χ 2 CM B + χ 2 BAO . Therefore, we evaluated the following total likeli- hood for statistical analysis. Ltot ∝ exp ( − 1...
[13]

G. K. Goswami, Research in Astron. Astrophys. 17, 27 (2017); arXiv: 1710.0728 [gr-qc] (2017)

2017
[14]

4 67 . 2 68 . 0 68 . 8 69 . 6 70 . 4 71 . 2 H0 − 0. 0084 − 0. 00836 − 0. 00832 − 0. 00828 − 0. 00824 − 0. 0082 − 0. 00816 δG/G FIG. 5: Constraints in the (δG/G, H 0) plane with 1σ -3σ conﬁdent level. tively. The 68% marginalized limits on these two param- eters are as bellow
[15]

Copeland, A.R

E.J. Copeland, A.R. Liddle and D. Wands, Phys. Rev. D 57, 4686 (1988)

1988
[16]

174+0. 0024+0. 0046 − 0. 0024− 0. 0046 (1σ 2σ ) CC+CMB+BAO This paper estimated Hubble constant H0 for both ΛCDM and ΛBD models are in excellent agreement with those of Chen & Ratra (68 ± 2. 8) [83], Aubourg et al (BAO:
[17]

Zlatev, L

I. Zlatev, L. Wang and P.J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999)

1999
[18]

321+0. 012 −0. 0096, Ω Λ = 0. 669 ± 0. 0089) for these models re- spectively. Obviously, these results are in excellent agree- ment with those obtained by Planck 2018 collaboration[4] (Ω m = 0. 3103 ± 0. 0057, Ω Λ = 0. 6897 ± 0. 0057). The FIG. 6: The plot of Hubble rate versu...

2018
[19]

525 0 . 550 0 . 575 0 . 600 0 . 625 0 . 650 0 . 675 zt
[20]

R. R. Caldwell and M. Kamionkowski, Ann. Rev. Nucl. Part. Sci. 59, 397 (2009)

2009
[21]

Akarsu et al

O. Akarsu et al. arXiv: 1903.06679v1 [gr-qc] (2019)

2019 arXiv
[22]

Komastu et al

E. Komastu et al. Astrophys. J. Suppl. Ser. 192, 18 (2011)

2011
[23]

P. A. R. Ade et al. Astron. Astrophys. 594, A13 (2016)

2016
[24]

Aghamin et al

N. Aghamin et al. [Plank Collaboration], arXiv: 1807.06209 [astro-ph.CO] (2018)

2018 arXiv
[25]

P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003)

2003
[26]

C. M. Will, Living Rev. Rel. 9, 3 (2006)

2006
[27]

Weinberg, Rev

S. Weinberg, Rev. Mod. Phys. 61, 1 (1989)

1989
[28]

Peebles, B

P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003)

2003
[29]

Sahni, A

V. Sahni, A. Shaﬁeloo and A. A. Starobinsky, Astrophys. J. 793, 2 (2014)

2014
[30]

Shaﬁeloo, B

A. Shaﬁeloo, B. L. Huillier and A. A. Starobinsky, Phys. Rev. D 98, 083526 (2018)

2018
[31]

A. K. Yadav, Astrophys. Space Sc. 335,565 (2011)

2011
[32]

A. K. Yadav, Astrophys. Space Sc. 361,276 (2016)

2016
[33]

Sen and A

S. Sen and A. A. Sen, Phys. Rev. D 63, 124006 (2001)

2001
[34]

E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2007)

2007
[35]

Garc ´ ıa-Salcedo, T

R. Garc ´ ıa-Salcedo, T. Gonz´ alez, and I. Quiros, Phys. Rev. D 92, 124056 (2015)

2015
[36]

Peebles and B

P.J.E. Peebles and B. Ratra, Astrophys. J. Lett. 325, L17 (1988)

1988
[37]

H. W. Lee, K. Y. Kim, Y. S. Myung, Eur. Phys. J. C 71, 1585 (2011)

2011
[38]

Bartolo and M

N. Bartolo and M. Pietroni, Phys. Rev. D 61, 023518 (1999)

1999
[39]

E. J. Copeland, M. Sami and Shinnji Tsujikava, Int. J. Mod. Phys. D 15, 1753 (2006)

2006
[40]

J. M. Server´ o and P. G. Est´ evez, Gen. Rel. Gray.15, 351 (1983)

1983
[41]

Brans and R

C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961)

1961
[42]

R. H. Dicke, ibid 125, 2163 (1962)

1962
[43]

Fujii, K.-I

Y. Fujii, K.-I. Maeda, The Scalar-Tensor Theory of Grav - itation (Cambridge University Press, UK, 2003)

2003
[44]

Faraoni, Cosmology in Scalar-Tensor Gravity (Kluwe r Academic Publishers, The Netherlands, 2004)

V. Faraoni, Cosmology in Scalar-Tensor Gravity (Kluwe r Academic Publishers, The Netherlands, 2004)

2004
[45]

Iess,and P

B.Bertotti, L. Iess,and P. Tortora, Nature 425, 374 (2003)

2003
[46]

Singh, T

T. Singh, T. Singh, J. Math Phys. 25, 9 (1984)

1984
[47]

Perivolaropoulos, Phys

L. Perivolaropoulos, Phys. Rev. D 81, 047501 (2010)

2010
[48]

Tegmark et al

M. Tegmark et al. (SDSS Collaboration), Phys. Rev. D 74, 123507 (2006)

2006
[49]

Acquaviva, C

V. Acquaviva, C. Baccigalupi, S. M. Leach, A. R. Liddle, and F. Perrotta, Phys. Rev. D 71, 104025 (2005)

2005
[50]

Li, F.-Q

Y.-C. Li, F.-Q. Wu, and X. Chen, Phys. Rev. D 88, 084053 (2013)

2013
[51]

Hrycyna, M

O. Hrycyna, M. Szydlowski and M. Kamionka, Phys. Rev. D 90, 124040 (2014)

2014
[52]

Avilez and C

A. Avilez and C. Skordis, Phys. Rev. Lett. 113, 011101 (2014)

2014
[53]

A. T. Ali, A. K. Yadav S.R. Mahmoud, Astrophys Space Sci 349, 539 (2014)

2014
[54]

M. Arik, M. Calik, M. B. Sheftel, Int. J. Mod. Phys. D 17, 225 (2008)

2008
[55]

J. Ryan, S. Doshi, B. Ratra, MNRAS 480, 759 (2018)

2018
[56]

Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)

S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)

1972
[57]

Zhang, H

C. Zhang, H. Zhang, S. Yuan, S. Liu, T.-J. Zhang, Y.-C. Sun, Research in Astronomy and Astrophysics, 14, 1221 (2014)

2014
[58]

Romero, A

C. Romero, A. Barros, Astrophys. Space Sci. 192, 263 (1992)

1992
[59]

Uehara and C

K. Uehara and C. W. Kim, Phys. Rev. D 26, 2575 (1982)

1982
[60]

Moresco, et al,J

M. Moresco, et al,J. Cosmology Astropart. Phys 5, 014 (2016)

2016
[61]

Lorenz-Petzold, Phys

D. Lorenz-Petzold, Phys. Rev. D 29, 2399 (1984)

1984
[62]

Pimentel, Astrophys

L.O. Pimentel, Astrophys. Space Sci. 112, 175 (1984)

1984
[63]

T. Etoh, M. Hashimoto, K. Arai, S. Fujimoto, Astron. Astrophys. 325, 893 (1997)

1997
[64]

L. E. Gurevich, A. M. Finkelstein, V. A. Ruban, Astro- phys. Space Sci. 22, 231 (1973)

1973
[65]

D. A. Tretyakova, A. A. Shatskiy, I. D.Novikov and S. O.Alexeyev, Phys. Rev. D 85, 124059 (2012)

2012
[66]

4 67 . 2 68 . 0 68 . 8 69 . 6 70 . 4 71 . 2 H0 0 20000 40000 60000 80000 100000ω FIG. 3: Constraints in the (ω, H 0) plane with 1σ -3σ conﬁdent level in the ﬂat Λ BD model ﬁtted over CC+ABO+CMB data. combination era to the current time [73]. Therefor, one can estimate the vari...
[67]

1) [84], Chen et al (68

3 ± 1. 1) [84], Chen et al (68 . 4+2. 9 −3. 3) [85], Aghanim et al (Planck 2018: 67 . 66 ± 0. 42) [4], and 9-years WMAP mission (68 . 92+0. 94 −0. 95) [86]. Figure. 6 depicts the robustness of our ﬁts for H(z). When we ﬁt both (Θ Λ CDM, Θ Λ BD ) to CC+CMB+BAO data we obtain (Ω...

2018
[68]

A. K. Azad, J. N. Islam, Pramana 60, 2127 (2003)

2003
[69]

Qiang, G

Li. Qiang, G. M. Yong, H. Muxin, Y. Dan, Phys. Rev D 71, 061501 (2005)

2005
[70]

8: H0 − zt plane with 1σ − 3σ conﬁdent level

5 H0 FIG. 8: H0 − zt plane with 1σ − 3σ conﬁdent level. Solid contours stand for Λ BD model V. CONCLUDING REMARKS In this paper ﬁrst we tried to obtain a new exact so- lution for Brans-Dicke equations with the cosmological constant in the spatially ﬂat Robertson-Walker metric....
[71]

M. A. Smolyakov, arXiv: 0711.3811 [gr-qc] (2007)

2007 arXiv
[72]

Das and N

S. Das and N. Banerjee, Phys. Rev. D. 78, 043512 (2008)

2008
[73]

M. R. Setare & M. Jamil, Physics Letters B 690, 1 (2010)

2010
[74]

A. K. Yadav. Research in Astron. Astrophys. 13, 772 (2013)

2013
[75]

Gaztanaga, E

E. Gaztanaga, E. Garcia-berro , J. Isern, E. Bravo, and I. Dominguez, Phys. Rev. D 65, 023506 (2002)

2002
[76]

Moresco, M., et al.: JCAP 05, 014 (2016)

2016
[77]

Bambi, M

C. Bambi, M. Giannotti, and F. L. Villante, Phys. Rev. D 71, 123524 (2005)

2005
[78]

Simon J, L

J. Simon J, L. Verde, R. Jimenez, Phys. Rev. D 71, 123001 (2005)

2005
[79]

Moresco M, et al, J

M. Moresco M, et al, J. Cosmology Astropart. Phys 8, 006 (2012)

2012
[80]

R. W. Hellings, et al, Phys. Rev. Lett. 51, 1609 (1983)

1983
[81]

A. L. Ratsimbazafy, et al, MNRAS, 467, 3239 (2017)

2017
[82]

Stern D, R

D. Stern D, R. Jimenez, L. Verde, M. Kamionkowski, S. A. Stanford, J. Cosmology Astropart. Phys 2, 008 (2010)

2010
[83]

Moresco, MNRAS, 450, L16 (2015)

M. Moresco, MNRAS, 450, L16 (2015)

2015
[84]

E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)

2006
[85]

Hu and N

W. Hu and N. Sugiyama, Astrophys. J. 444, 489 (1995)

1995
[86]

Hinshaw et al

G. Hinshaw et al. [WMAP Collaboration], Astrophys. J. Suppl. 208, 19 (2013)

2013
[87]

Beutler et al., Mon

F. Beutler et al., Mon. Not. Roy. Astron. Soc. 423, 3430 (2012)

2012
[88]

A. J. Ross, et al, Mon. Not. Roy. Astron. Soc. 449, 835 (2015)

2015
[89]

Alam et al

S. Alam et al. [BOSS Collaboration], arXiv:1607.03155 [astro-ph.CO]

arXiv
[90]

Anderson et al

L. Anderson et al. [BOSS Collaboration], Mon. Not. Roy. Astron. Soc. 441, 24 (2014)

2014
[91]

E. A. Kazin et al., Mon. Not. Roy. Astron. Soc. 441, 3524 (2014)

2014
[92]

D. J. Eisenstein et al. [SDSS Collaboration], Astrophy s. J. 633, 560 (2005). 10

2005
[93]

D. J. Eisenstein and W. Hu, Astrophys. J. 496, 605 (1998)

1998
[94]

F. Q. Wu and X. Chen, Phys. Rev. D 82, 083003 (2010)

2010
[95]

Muller and L

J. Muller and L. Biskupek, Classical Quantum Gravity 24, 4533 (2007)

2007
[96]

C. J. Copi, A. N. Davis, and L. M. Krauss, Phys. Rev. Lett. 92, 171301 (2004)

2004
[98]

D. B. Guenther, L. M. Krauss, and P. Demarque, Astro- phys. J. 498, 871 (1998)

1998
[99]

S. E. Thorsett, Phys. Rev. Lett. 77, 1432 (1996)

1996
[101]

V. M. Kaspi, J. H. Taylor, and M. F. Ryba, Astrophys. J. 428, 713 (1994)

1994
[102]

Chang and M

K.-C. Chang and M. C. Chu, Phys. Rev. D 75, 083521 (2007)

2007
[103]

G. Chen, B. Ratra, B, PASP 123, 1127 (2011)

2011
[104]

Aubourg, et al, Phys

E. Aubourg, et al, Phys. Rev D 92, 123516 (2015)

2015
[105]

G. Chen, S. Kumar, B. Ratra, B, Astrophys. J. 835, 86 (2017)

2017
[106]

Hinshaw, et al, Astrophys.J.Suppl.Ser 208, 25 (2013)

G. Hinshaw, et al, Astrophys.J.Suppl.Ser 208, 25 (2013)

2013
[107]

Amirhashchi, Phys

H. Amirhashchi, Phys. Rev D 97, 063515 (2018)

2018
[108]

12, 045 (2015)

Nisha Rani et al, JCAP. 12, 045 (2015)

2015
[109]

Vargas dos Santosa, R

M. Vargas dos Santosa, R. R. R. Reisa and I. Wagaa, JCAP. 02, 066 (2016)

2016
[110]

Amirhashchi, arXiv: 1811.05400 [astro-ph.CO] (201 8)

H. Amirhashchi, arXiv: 1811.05400 [astro-ph.CO] (201 8)

arXiv
[111]

H. Yu, B. Ratra and Fa-Yin, Wang, Astrophys. J. 856, 3 (2018)

2018
[112]

Goistri, et al, JCAP 03, 027 (2012)

R. Goistri, et al, JCAP 03, 027 (2012)

2012
[113]

Capozziello, O

S. Capozziello, O. Luongo, E.N. Saridakis, Phys. Rev. D 91, 124037 (2015)

2015
[114]

Amirhashchi, S

H. Amirhashchi, S. Amirhashchi, Phys. Rev. D 99, 023516 (2019)

2019
[147]

38) rs(zd) − 1477

78 )2 [ DV (z = 0. 38) rs(zd) − 1477
[149]

15) rs(zd) − 664

69 )2 [ DV (z = 0. 15) rs(zd) − 664

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

198 × 10−13, − 0

150 × 10−13< ˙G/G< 1. 198 × 10−13, − 0. 0084<δG/G< −0. 0082, (43) with the best-ﬁt values ˙G/G = 1. 17 × 10−13, δG/G = −0. 00825. (44) We have summarized constraints on ˙G/G obtained from diﬀerent methods in Table. IV which is an update of Ta- ble I of Ref [30]. From this table we observe that our constrain is much tighter with respect to other listed pre...

[2] [2]

galaxy diﬀerential age

(24) We also note that a de-Sitter solution arises whenever q = −1 ⇒ ˙H = 0 ⇒H =H0 ⇒a(t) ∝eH0t. (25) Since for our ΛBD model we have qz=0 = Ω m ( √ 24Ω φ + 25 − 3 ) − 4Ω Λ 4 (Ω Λ + Ω m) , (26) it is clear to conclude that this condition can not be achieved in our ΛBD model (see V for derivation of DP an transition redshift). In the next section we use joi...

[3] [3]

We use the CMB distance priors based and Planck 2015 collaboration [3] to constrain our models

7 ) −4 , (34) whereh is the normalized Hubble constant and z∗ = 1090 is decoupling redshift. We use the CMB distance priors based and Planck 2015 collaboration [3] to constrain our models. According to Planck 2015 data, the mean values of shift and acoustic scale parameters are ⟨R⟩ = 1. 7488 and ⟨la⟩ = 301. 76 with the deviations σ (R) = 0. 0074 and σ (la...

2015

[4] [4]

106) − 0

0152 [ rs(zd) DV (z = 0. 106) − 0. 336 ] 2 + 1 ( 25

[5] [5]

32) rs(zd) − 1264

28 )2 [ DV (z = 0. 32) rs(zd) − 1264

[6] [6]

Padmanabhan, Phys

T. Padmanabhan, Phys. Rept. 380, 234 (2003)

2003

[7] [7]

00712 [ rs(zd) DV (z = 0. 44) − 0. 0916 ] 2 + 1 ( 19

[8] [8]

51) rs(zd) − 1877

78 )2 [ DV (z = 0. 51) rs(zd) − 1877