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REVIEW 3 major objections 4 minor 75 references

Lyra's cosmology of homogeneous and isotropic universe in Brans-Dicke theory

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A Brans-Dicke scalar field in Lyra's geometry drives cosmic acceleration without a cosmological constant.

desk verdict The fitted H(z) is not a solution of the model's own field equations, so the central claim fails; the paper is mostly a routine data fit of a known form with no new physics. read the letter →

arxiv 1909.01998 v4 pith:X5CFV2GW submitted 2019-09-04 gr-qc

Rajendra Prasad , Lalit Kumar Gupta , Anil Kumar Yadav This is my paper · ORCID
classification gr-qc PACS 04.50.kd98.80.-k98.80.JK
keywords Brans-DicketheoryLyrageometryscalarfieldcosmologylate-timeaccelerationdecelerationparameterobservationalHubbledataTypeIasupernovaecosmologicalparameters
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 claims that a Brans-Dicke scalar field living in Lyra's geometry can reproduce the observed expansion history of a flat, homogeneous, isotropic universe without a cosmological constant or a separate dark-energy fluid. The authors solve the field equations for pressureless matter using a power-law scalar field, obtain a closed-form Hubble function, and fit it to 46 cosmic-chronometer $H(z)$ points plus a Type Ia supernova sample, finding $H_0 \approx 67.44$ km/s/Mpc and $\Omega_{m0}\approx 0.28$ from the Hubble data. The same solution gives a deceleration-to-acceleration transition at $z_t\approx 0.52$, a present deceleration $q_0\approx -0.58$, an age of about 14.29 Gyr, and a jerk parameter near 0.80. If correct, the model would show that the observed late-time acceleration can be a dynamical scalar-field effect rather than the cosmological constant.

What carries the argument

The carrying object is the power-law scalar field ansatz $\varphi = \varphi_0 (a/a_0)^{1/(\omega+1)}$, taken together with the Lyra displacement vector $\beta$ inside the effective density and pressure. This ansatz converts the two Friedmann-type equations into a single algebraic constraint, $\Omega_m + \Omega_\beta = 1 + (5\omega+6)/(6(\omega+1)^2)$, which fixes the $\beta$-energy fraction whenever the matter density and $\omega$ are specified. Feeding that constraint into the Friedmann equation produces the closed-form Hubble function in Eq. (20); the rest of the paper is the cosmographic series derived from that function: $q = -1 + (1+z)H'(z)/H(z)$ for deceleration, the age integral $t_0 = \int_0^\infty dz / [(1+z)H(z)]$, and the jerk expression obtained from derivatives of $H(z)^2$.

What would settle it

Substitute Eq. (20) and the power-law $\varphi$ back into the Raychaudhuri equation (Eq. 6) and compute the deceleration parameter $q$ directly from the field equations; if it does not reproduce Eq. (23) for the fitted $\Omega_{m0}\approx 0.28$ and $\omega = 40000$, the fitted expansion history is not a solution of the model as written.

Watch

Extended reading notes

Core claim

The paper's central claim is that the expansion of a pressureless universe obeying the Brans-Dicke field equations in Lyra's manifold is governed by $H(z) = H_0\,[1+C]^{-1/2}\,\bigl[1+\Omega_{m0}(1+z)^{(3\omega+4)/(\omega+1)} + C - \Omega_{m0}\bigr]^{1/2}$, where $C = (5\omega+6)/(6(\omega+1)^2)$, and that this history fits both the 46-point $H(z)$ sample and the Type Ia supernova sample used in the paper. Fitting yields $H_0=67.44$ km/s/Mpc with $\Omega_{m0}=0.28$ from the Hubble data and $H_0=70.02$ km/s/Mpc with $\Omega_{m0}=0.272$ from the supernova data. The same $H(z)$ is then differentiated to obtain a deceleration parameter that changes sign at $z_t\approx 0.52$ and equals $q_0\approx -0.58$ today, integrated to give an age of about 14.29 Gyr, and differentiated twice to give a present jerk $j_0\approx 0.80$. The authors read these numbers as showing that the model describes a universe that decelerates early, accelerates now, and needs no cosmological constant, with the acceleration supplied by the dynamics of the Brans-Dicke scalar field rather than by Lyra's displacement vector.

Load-bearing premise

The load-bearing premise is that the assumed power-law form of the scalar field really solves the full field equations, so the fitted $H(z)$ is a true solution of the theory and not just a convenient curve; if that premise gives way, the derived expansion history and its parameters no longer describe the model.

Editorial extensions

If this is right

  • If Eq. (20) is the correct expansion history, the universe's transition from deceleration to acceleration occurs at $z_t\approx 0.52$, so independent high-redshift probes of the expansion rate should see deceleration above that redshift.
  • The model yields a present age of about 14.29 Gyr, which is consistent with independent CMB-based age estimates and removes the need for a cosmological constant to stretch the age.
  • The fitted matter density $\Omega_{m0}\approx 0.28$ and Hubble constant $H_0\approx 67.44$ km/s/Mpc from the Hubble data fall inside the ranges reported by CMB experiments, so the model is not immediately excluded by these basic expansion probes.
  • Because the predicted present jerk $j_0\approx 0.80$ differs from the $\Lambda$CDM value $j=1$, future cosmographic measurements of higher-order expansion can distinguish this scalar-field model from vacuum-energy cosmology.

Reading between the lines

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

  • Not stated in the paper: at $\omega \approx 40000$ the correction $C = (5\omega+6)/(6(\omega+1)^2)$ is tiny, so the fitted $H(z)$ closely tracks flat $\Lambda$CDM with a rescaled matter density; the successful fit may be largely a test of how close the ansatz is to $\Lambda$CDM, not of the Lyra-specific terms.
  • Not stated in the paper: since the displacement vector has effective equation of state $+1$ in the matter-free limit, the acceleration must be carried by the scalar field; a high-precision measurement of the late-time effective equation of state would separate these two contributions.
  • Not stated in the paper: applying the same power-law scalar-field ansatz to anisotropic Bianchi models would give the Lyra displacement field a testable shear or anisotropy signature that the isotropic fit cannot reveal.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a flat FLRW Brans-Dicke cosmological model formulated in Lyra's geometry with a time-like displacement vector. After writing the field equations (5)-(7), the authors adopt a power-law scalar field φ ∝ a^{1/(ω+1)}, derive the Hubble function (20), and fit its two free parameters H0 and Ωm0 to 46 H(z) points and an SN Ia Gold sample. From the best fit they compute a present deceleration q0=-0.58, a transition redshift zt=0.52, an age of 14.29 Gyr, and j0=0.80, concluding that the model yields late-time cosmic acceleration without a cosmological constant.

Significance. If the model were correct, it would provide an interesting example of an accelerating dust-filled universe without dark energy, with the Brans-Dicke scalar field playing the role usually assigned to a cosmological constant in Lyra geometry. The paper reports a standard χ² likelihood analysis with contour plots, which is a useful route to falsifiability. However, the central expansion history is not a solution of the field equations: the best-fit H(z) violates the scalar-wave equation. Because every subsequent observable (q0, age, jerk, transition redshift) is derived from this H(z), the claimed cosmological conclusions do not follow. The paper also correctly notes that the Lyra displacement vector β cannot act as Λ in Brans-Dicke theory, but this insight is not developed into a workable model.

major comments (3)
  1. [§III, Eqs. (20) and (23); §II, Eq. (7)] The best-fit Hubble function is not a solution of the model's field equations. For the ansatz (16), one has φ˙/φ = nH with n = 1/(ω+1). With dust (p=0), the scalar-wave equation (7) becomes n(\dot H/H²) + n(n+3) = 3Ωm/(2ω+3). Evaluating this at z=0 with ω=40000 and Ωm0=0.28 gives q0 = -1 - \dot H/H² ≈ +1.58, whereas Eq. (23), obtained from the fitted H(z), gives q0 = -0.58. Both expressions describe the same model at the same point, so Eq. (20) cannot satisfy Eqs. (5)-(7) simultaneously; the fitted H(z) solves only the Friedmann equation used to construct it.
  2. [§III, Eqs. (15) and (16)] The statement that Eq. (16) is 'the general solution' of Eq. (15) is incorrect. Substituting φ ∝ a^{1/(ω+1)} makes Eq. (15) an identity for arbitrary a(t); it imposes no restriction on H(z). Consequently, the power-law ansatz does not select an expansion history, and the derivation of Eq. (20) uses only Eq. (5) together with the density-parameter decomposition. This is the structural reason why the scalar-wave constraint is lost and why the fitted H(z) is not a solution of the full system.
  3. [§III, Table I and Eq. (21); §IV, Eqs. (23)-(27)] The observational validation is partly circular and the data handling is inconsistent. The deceleration parameter, transition redshift, age, and jerk are all deterministic functions of H0 and Ωm0 obtained from the same OHD/SNIa fits; their agreement with WMAP/Planck values is therefore a restatement of the fit rather than an independent prediction. In addition, Table I is described in the text as cosmic-chronometer data, but it includes BAO measurements (for example, entries cited to refs. [49], [51], [53], [55], [57]-[59]). Cosmic-chronometer and BAO data have different systematics and should not be combined in a single 'OHD' likelihood without modeling those systematics.
minor comments (4)
  1. [Throughout] There are numerous typographical errors ('obtian', 'Brans-Dike', 'Plank collaboration', 'co-moving co-ordinate', 'generak', 'red-shift' as one word) that should be corrected in a thorough proofread.
  2. [§III, Eq. (16) and Eq. (18)] The equation cross-references are unreliable: the sentence before Eq. (16) refers to 'equation (18)' when it should refer to Eq. (15), and the sentence before Eq. (18) says 'Using equation (23) in equation (6)' when it should refer to Eq. (16).
  3. [§IV.C] The text says the jerk parameter is 'graphed in Figure 5', but Figure 5 is the age plot H0(t0−t) versus z; the jerk plot is Figure 6.
  4. [§III, SN Ia data] The SN Ia sample is described only as 'Gold Sample and New Gold Sample' from ref. [60]; this is an old compilation, and the paper should state why more recent samples (e.g., Pantheon or DES) were not used.

Circularity Check

3 steps flagged · score 6.0 of 10

q0, age and jerk are re-expressions of fitted H0, Ωm0 from a ΛCDM-form H(z) built on a self-definitional φ ansatz.

  1. self definitional [Section III, Eqs. (15), (16), (18), (20)]
    "The general solution of equation (18) is given by φ =φ0 ( a a0 ) 1 ω+1 ... Using equations (16), (23), (18) and (19) in equation (5), we obtain the expression for Hubble's function in terms of red-shift as [Eq. (20)]"

    Substituting φ = φ0(a/a0)^n with n = 1/(ω+1) into Eq. (15) makes the left and right sides identical for arbitrary H: ä/a + 2H² = (ω+1)(nḢ+n²H²)+(3ω+2)nH² = Ḣ + 3H². Thus Eq. (15) imposes no restriction on a(t); the 'general solution' is a self-definitional ansatz. Eq. (20) follows by inserting that φ-law plus the closure Ωm+Ωβ = 1+C into Eq. (5) alone, leaving H0 and Ωm0 as free fit parameters. The expansion history used for all later predictions is therefore constructed, not solved from the full BD-Lyra system (5)-(7).

  2. fitted input called prediction [Section III, Eq. (21) and Table II; Section IV.A Eq. (23); Section IV.B Eqs. (26)-(27)]
    "For statistical analysis, we define χ2 ... The estimated values of H0 = 67 .44 km s−1 Mpc−1 and Ω m0 = 0 .28 with χ2 min = 26.36. ... From equation (23), one can obtain the present value of deceleration parameter as q0 = −0.58 by putting z = 0, Ω m0 = 0 .28 and ω = 40000. ... H0t0 = 0.983019 ... t0 = 0.983019 H−1 0 = 14.288 Gyrs."

    q0, age, and jerk are computed by inserting the best-fit Ωm0 = 0.28 and H0 = 67.44 into kinematically defined formulas (22), (24)-(26), (28). For ω = 40000, Eq. (23) gives q0 ≈ −1 + 1.5Ωm0, so the quoted q0 = −0.58 is just the fitted matter density rewritten; similarly t0 = 0.983/H0 is the fitted Hubble constant rewritten as an age. These quantities therefore contain no new information beyond the fit and cannot serve as independent confirmation of the model, even though the paper presents them as 'nice match' with observations.

1 more flagged steps
  1. renaming known result [Eq. (20) with ω = 40000 used throughout Section IV]
    "H(z) = H0 [1 + (5ω+6)/(6(ω+1)^2)]^{-1/2} [1 + Ωm0(1+z)^{(3ω+4)/(ω+1)} + (5ω+6)/(6(ω+1)^2) − Ωm0]^{1/2}"

    With ω = 40000, C = (5ω+6)/(6(ω+1)^2) ≈ 2×10^-5 and (3ω+4)/(ω+1) ≈ 3, so Eq. (20) reduces to H² = H0²[Ωm0(1+z)^3 + 1 − Ωm0] — the flat ΛCDM Hubble rate. All of the paper's 'derived' results (q0 = −0.58, zt = 0.521, t0 = 14.29 Gyr, j0 = 0.8002) are the standard ΛCDM values for the same fitted Ωm0 and H0. Presenting them as predictions of Brans-Dicke/Lyra theory is a renaming of the known ΛCDM expansion rather than an independent derivation.

full rationale

The paper's derivation chain is not circular because of self-citation: its references to the authors' own prior Lyra/Brans-Dicke papers are background, not load-bearing, and the WMAP/Planck comparisons are external. The circularity sits in the construction of H(z) and in the use of fitted parameters. Eq. (15) is identically satisfied by φ ∝ a^{1/(ω+1)} for any H(t), so calling this the 'general solution' and then deriving Eq. (20) from Eq. (5) makes the expansion history an input ansatz rather than a consequence of the field equations. The subsequent q0 = −0.58, t0 = 14.29 Gyr, and j0 = 0.8002 are obtained by inserting the best-fit H0 and Ωm0 into formulas that are algebraic rearrangements of those same fitted parameters; hence they are forced by the fit, not independent predictions. For ω = 40000, Eq. (20) collapses to the flat ΛCDM H(z) = H0[Ωm0(1+z)^3 + 1 − Ωm0]^{1/2}, so the 'derived' cosmography is the standard ΛCDM result presented as a new BD-Lyra model. The quoted acceleration epoch and age are therefore not novel predictions of the BD-Lyra field equations, but consequences of the ansatz and fit. Score 6 reflects that the central phenomenology reduces by construction, while stopping short of 8 because the fitted parameters are compared to some external data (WMAP/Planck) and the paper does contain genuine field-equation formalism.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The fitted H(z) is essentially flat ΛCDM with an exponent (3ω+4)/(ω+1) ≈ 3 and a constant C that acts like ΩΛ. All numerical results assume ω=40000, so the model is never tested away from that value. The scalar field ansatz and the sign convention for the β terms are load-bearing for the acceleration claim.

free parameters (4)
  • H0 = OHD: 67.44, SNIa: 70.02 km/s/Mpc
    Hubble constant in Eq. (20), determined by chi-square minimization against OHD and SNIa data.
  • Ωm0 = OHD: 0.28, SNIa: 0.272
    Present matter density parameter in Eq. (20), determined by chi-square minimization.
  • ω (Brans-Dicke coupling) = 40000
    Fixed at 40000 from solar system bounds, not fitted; all numerical results assume this value and no sensitivity study is given.
  • β (Lyra displacement field) = implied by Ωβ0 ≈ 0.72 at present
    Not directly fitted; its value is set by the relation Ωβ = 1 + C - Ωm0 after fitting Ωm0, and it supplies the constant term in H(z) that acts like dark energy.
assumptions (4)
  • domain assumption Brans-Dicke field equations in Lyra's manifold, Eqs. (1)-(2), with the adopted sign convention for the β^2 terms.
    These equations are the starting point; the sign of the Lyra contribution is load-bearing for the acceleration result.
  • domain assumption Flat, homogeneous, isotropic FLRW metric and dust matter (p=0) at late times.
    Assumed in Section II and III before deriving H(z); the model is not validated outside this regime.
  • ad hoc to paper The scalar field ansatz φ = φ0(a/a0)^{1/(ω+1)} is an exact solution of the full system.
    Labeled 'the general solution' of Eq. (15) in Section III, but it is a particular ansatz and its consistency with Eq. (9) is not established.
  • domain assumption The 46 OHD points are model-independent and uncorrelated cosmic-chronometer data.
    The chi-square analysis assumes this, but Table I includes BAO measurements (refs [57],[58],[59]) that are correlated and not cosmic chronometers.
invented entities (1)
  • Lyra displacement field β (with associated Ωβ energy)
    purpose: Postulated geometric vector field whose constant time component adds a constant term to the Friedmann equation, mimicking dark energy in the fit.
    No independent observation isolates β; its value is set by the same fit that claims agreement. The paper asserts β cannot act as a cosmological constant, yet the fitted H(z) requires a constant term, and the field equations give conflicting signs for its dynamical effect.

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Cite this review

Pith. "Pith review of Lyra's cosmology of homogeneous and isotropic universe in Brans-Dicke theory." pith.science (2026). https://pith.science/paper/X5CFV2GW

@misc{pith2026190901998,
  author       = {Pith},
  title        = {Pith review of: Lyra's cosmology of homogeneous and isotropic universe in Brans-Dicke theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5CFV2GW}},
  note         = {Machine review of arXiv:1909.01998}
}
abstract

In this paper, we investigate a scalar field Brans-Dicke cosmological model in Lyra's geometry which is based on the modifications in geometrical term as well as energy term of Einstein's field equations. We have examined the validity of proposed cosmological model on observational scale by performing statistical analysis from latest $H(z)$ and SN Ia observational data. We find that the estimated values of Hubble's constant and matter energy density parameter are in agreement with their corresponding values, obtained from recent observations of WMAP and Plank collaboration. We also derived deceleration parameter, age of the universe and jerk parameter in terms of red-shift and computed its present values. The dynamics of deceleration parameter in derived model of the universe shows a signature flipping from positive to negative value and also indicates that the present universe is in accelerating phase.

Figures

Figures reproduced from arXiv: 1909.01998 by the authors.

Figure 1
Figure 1. FIG. 1: One-dimensional marginalized distributions and two [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The Hubble rate versus red-shift error bar plot with 46 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: One-dimensional marginalized distributions and two [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The variation of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The plot of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The plot of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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