Ambiguity in matter sector for modified gravity involving $\delta^2 \mathcal{L}_{m}/\delta g^{\mu\nu}\delta g^{\alpha\beta}$ and its implications to astrophysics and cosmology

A. Sulaksono; B. N. Jayawiguna

REVIEW 3 major objections 5 minor 102 references

Ambiguity in matter sector for modified gravity involving $\delta^2 \mathcal{L}_{m}/\delta g^{\mu\nu}\delta g^{\alpha\beta}$ and its implications to astrophysics and cosmology

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single normalization rule makes modified-gravity equations independent of the matter-Lagrangian choice.

desk verdict The paper's resolution of the L_m ambiguity rests on an invalid derivation of Eq. (25), so the central claim is unsupported despite a clear presentation and coherent applications. read the letter →

arxiv 2608.04867 v1 pith:QIXHNVQE submitted 2026-08-05 gr-qc astro-ph.HE

B. N. Jayawiguna , A. Sulaksono This is my paper

classification gr-qcastro-ph.HE MSC 83D0583C5583F05 PACS 04.50.Kd04.40.Dg98.80.-k

keywords matterLagrangianambiguitymodifiedgravityfour-velocitynormalizationf(RT)energy-momentumsquaredneutronstarsquarkcosmology

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

Modified gravity theories that couple curvature directly to matter require a second derivative of the matter Lagrangian with respect to the metric. In general relativity, choosing $L_m=p$ or $L_m=-\rho$ is interchangeable, but in these theories the two choices produce different effective Einstein equations, so predictions for stars and cosmology become ambiguous. The paper claims this ambiguity disappears if one stops treating the four-velocity and the metric as independent variables and instead enforces the four-velocity normalization inside the variation. With that step, the paper derives an identity for the variation of $u^{\alpha}u^{\beta}$ and shows that the troublesome tensors $\Theta$, $\theta$, $\Gamma$, and $\Xi$ are identical for $p$ and $-\rho$ in $f(R,T)$, $f(R,\tau)$, $f(R,T,P)$, and $f(R,T_G,T_{GD})$ gravity. The authors then demonstrate the consequences on neutron- and quark-star mass-radius relations and on radiation-dominated cosmology, where the resulting dynamics stay close to general relativity.

What carries the argument

The load-bearing object is the identity $\delta(u^{\alpha}u^{\beta})/\delta g^{\mu\nu}=u^{\alpha}u^{\beta}u^{\mu}u^{\nu}$, which the paper derives in Appendix A from the normalization condition $g_{\alpha\beta}u^{\alpha}u^{\beta}=-1$. This identity makes the four-velocity part of $\delta\rho/\delta g^{\mu\nu}$ contribute correctly to second derivatives, turning $\delta^2 p$ and $\delta^2\rho$ into expressions that cancel the $L_m$-dependent terms in $\Theta$, $\theta$, $\Gamma$, and $\Xi$. The paper deliberately drops the adiabatic sound-speed terms that appear in a fuller fluid treatment, so the resulting second derivatives depend only on $\rho+p$ and the metric and velocity structure, giving a matter-model-independent result.

What would settle it

Compute the second derivatives of $p$ and $-\rho$ with respect to the metric using a perfect-fluid variation that retains adiabatic sound-speed terms, and evaluate $\Theta_{\mu\nu}(L_m=p)-\Theta_{\mu\nu}(L_m=-\rho)$; any nonzero result would show the universality claim depends on dropping those terms. A simpler check is to derive the same field equations from an explicit action with $u^{\alpha}$ as an independent constrained field and compare the transverse part of $\delta u^{\alpha}$ that the normalization condition leaves free.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the equality $\delta(u^{\alpha}u^{\beta})/\delta g^{\mu\nu}=u^{\alpha}u^{\beta}u^{\mu}u^{\nu}$, obtained from the normalization condition $g_{\alpha\beta}u^{\alpha}u^{\beta}=-1$, removes the $L_m$ ambiguity. Using this identity, the second derivatives of $p$ and $-\rho$ combine with the definitions of $\Theta_{\mu\nu}$, $\theta_{\mu\nu}$, $\Gamma_{\mu\nu}$, and $\Xi_{\mu\nu}$ to give Eqs. (29)-(32): each tensor evaluates identically for $L_m=p$ and $L_m=-\rho$. Consequently, for $f(R,T)$, $f(R,\tau)$, $f(R,T,P)$, and $f(R,T_G,T_{GD})$, the effective Einstein equations no longer depend on which matter Lagrangian is chosen. Applied to $f(R,T)=R+2\chi T$ with $\chi=1$, the new prescription produces quark-star mass-radius curves close to general relativity and neutron-star curves with higher maximum masses; applied to energy-momentum squared gravity, it yields simpler Friedmann equations whose radiation-era evolution has a finite maximum density.

Load-bearing premise

The load-bearing premise is that the normalization condition $g_{\alpha\beta}u^{\alpha}u^{\beta}=-1$ fully fixes the metric variation of $u^{\alpha}u^{\beta}$ to the paper's Eq. (25); that condition fixes only the contracted part, and the paper also sets aside the sound-speed terms that a fuller fluid variation produces, so if either step fails the equalities (29)-(32) and the applications built on them do not follow.

Editorial extensions

If this is right

For $f(R,T)$, $f(R,\tau)$, $f(R,T,P)$, and $f(R,T_G,T_{GD})$, the effective Einstein equations become the same whether $L_m=p$ or $L_m=-\rho$, so prior results that depend on that choice need revisiting.
In $f(R,T)=R+2\chi T$ with $\chi=1$, the universal prescription yields quark-star mass-radius relations almost identical to general relativity and neutron-star maximum masses above general relativity for the MPA1 equation of state.
In energy-momentum squared gravity, universality makes $\theta_{\mu\nu}=0$, so the modified Friedmann equations reduce to general relativity plus quadratic density corrections.
In the radiation-dominated early universe, the new prescription gives a finite maximum radiation density for $\alpha>0$ around $10^{-5}\text{ s}\lesssim t\lesssim 10^{-4}\text{ s}$, rather than the unbounded early-time behavior of the old prescriptions.
The framework supplies a single matter Lagrangian that can be used consistently in both ultraviolet (compact star) and infrared (cosmological) applications of matter-coupled modified gravity.

Reading between the lines

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

Beyond the paper: the identity in Eq. (25) is one possible completion of the four-velocity variation, because the normalization condition fixes only the contracted part; the universality claim should be checked against a fully covariant fluid action before treating the ambiguity as settled.
If the claim holds, it implies that many published results in $f(R,T)$ and energy-momentum squared gravity based on $L_m=p$ or $L_m=-\rho$ are artifacts of an arbitrary variational choice and may need recalculation.
The same normalization-based prescription could plausibly extend to anisotropic fluids, charged fluids, or scalar-field matter, where second derivatives of the matter Lagrangian would similarly need a metric-consistent variation.
A testable extension is to compare the new radiation-era density bound, $\rho_{r,\max}=3/(4\alpha)$, with big-bang nucleosynthesis or early-universe bounds, which would constrain $\alpha$ independently of late-time cosmology.

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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

Punchline: the central claim doesn't hold. Equation (25) is not derived from the normalization condition, and the equalities (29)-(32) fall with it. The rest of the paper is a coherent scan over models and applications, but it's conditional on an unsupported step.

What is worthwhile: the paper tackles a real ambiguity that has nagged the f(R,T), EMSG, and related literatures for years. The presentation is clear. The TOV equations for the old prescriptions reproduce standard f(R,T) results, and the Appendix B cancellations are correct if you accept Eq. (25). The astrophysical and cosmological applications are honestly explored, and the plots are readable.

Where it goes wrong: Appendix A. Varying gαβ uα uβ = -1 with respect to gμν gives exactly gαβ δ(uαuβ)/δgμν = -uμ uν. That equation fixes one trace-like combination of the four-index object. It does not determine the whole object. The authors multiply by uαuβ and then treat the result as if it had solved for the full tensor. Adding Wαβ_{μν} = (gαβ+4uαuβ)uμ uν to any solution preserves the trace condition, so the normalization condition does not select Eq. (25). The step in (A2) is an algebraic non sequitur. Maybe there is a physical argument for that particular completion, but the paper does not give one. It also drops the sound-speed terms that appear in its own cited Ref. [73], saying only that it wants to be general.

Because Eq. (25) is the load-bearing input for every subsequent equality and for the "new" TOV and Friedmann equations, the main result is not established. The equalities (29)-(32) and the applications are internally consistent, but they are consequences of an ansatz, not theorems.

For a reader: this is a cautionary example, not a resolution. The intended audience is researchers working on curvature-matter couplings, and they should read the appendix carefully before adopting the formalism. I would not cite it as a resolution. If it landed on a desk, I would send it to a referee—the question is important and the error is worth documenting clearly—but I would expect the referee to recommend rejection unless the authors can justify Eq. (25) on independent grounds. My guess is they can't.

Referee Report

3 major / 5 minor

Summary. The paper addresses the well-known ambiguity in curvature-matter-coupled modified gravity theories (f(R,T), f(R,τ), f(R,T,P), f(R,T_G,T_GD)) concerning whether the matter Lagrangian L_m should be chosen as p or −ρ. The authors claim that enforcing the four-velocity normalization condition during the metric variation removes the ambiguity, making the effective field equations independent of the L_m choice. The key step is Eq. (25), δ(u^α u^β)/δg^{μν} = u^α u^β u^μ u^ν, derived in Appendix A. From it the authors obtain new second-derivative formulas (27)–(28), prove the equalities (29)–(32) in Appendix B, and then apply the resulting field equations to quark and neutron stars and to FLRW cosmology with dust, radiation, and stiff equations of state.

Significance. If correct, the paper would resolve a long-standing ambiguity in a broad class of modified gravity models and would have direct astrophysical and cosmological consequences. The paper is useful in clearly identifying where the ambiguity enters—the second variational derivative of L_m—and in presenting explicit old-prescription TOV equations for f(R,T). However, the central derivation is flawed: Eq. (25) does not follow from the normalization condition, and the second-derivative formulas (27)–(28) drop sound-speed terms that appear in the cited Ref. [73] without physical justification. The claimed L_m independence is therefore not established, so the astrophysical and cosmological predictions in Sections IV and V are not consequences of a well-defined variational principle. The manuscript's positive contribution is limited to a clear formulation of the known ambiguity, not its resolution.

major comments (3)

[Appendix A, Eq. (A2); Sec. III, Eq. (25)] The derivation of Eq. (25) is invalid. Varying the normalization condition g_{αβ}u^α u^β = −1 yields only the trace condition g_{αβ} δ(u^α u^β)/δg^{μν} = −u_μ u_ν. This scalar-trace equation does not determine the four-index tensor δ(u^α u^β)/δg^{μν}. For example, adding W^{αβ}_{μν} = (g^{αβ} + 4u^α u^β) u_μ u_ν leaves the trace condition unchanged because g_{αβ} W^{αβ}_{μν} = 0, yet W is nonzero and satisfies all index symmetries. The paper's step of multiplying the trace equation by u^α u^β and using g_{αβ}u^α u^β = −1 to solve for the full tensor is an algebraic non sequitur. Since Eq. (25) is the foundation for the second derivatives (27)–(28) and hence for the equalities (29)–(32), the paper's central claim is unsupported.
[Sec. III, Eqs. (27)–(28)] Even setting aside Eq. (25), the derivation of the second derivatives (27)–(28) is incomplete. The paper differentiates the explicit u_μ u_ν factors in the first derivatives (26) but treats ρ+p as independent of the metric. However, for a perfect fluid ρ and p are determined by thermodynamic variables whose metric variation does not vanish, and a consistent second variation contains adiabatic sound-speed terms. The paper itself acknowledges that Ref. [73] obtains such terms in its eqs. (48) and (51); the paper drops them 'to make it as general as possible,' but this is not a derivation from the fluid action. Consequently (27)–(28) constitute an additional ansatz, not a consequence of the normalization condition, and they are not valid for a general equation of state.
[Sec. III, Eqs. (29)–(32); Secs. IV–V] Because the two preceding points invalidate the derivation of Eqs. (27)–(28), the equalities (29)–(32) and the field equations (35), (36)–(38), and (86)–(88) used in the astrophysical and cosmological applications do not follow. The TOV equations (36)–(38) and the Friedmann equations (86)–(88) are presented as new predictions, but they are consequences of an unsupported prescription. The old-prescription results (39)–(46) and (52)–(84) are standard and not at issue; however, they do not support the paper's central claim of L_m independence.

minor comments (5)

[Sec. V.B, Eq. (65); Sec. V.C, Eq. (81)] The acceleration equation for the w = −1 case is written with −2 Ḣ − 3H^3, which should presumably be −2 Ḣ − 3H^2; this typo should be corrected.
[Sec. II.A and Sec. II.C] In the sentences after Eqs. (4) and (10), 'left hand side' should be 'right hand side,' since the ambiguous second-derivative terms appear on the right-hand side of the displayed definitions of Θ_{μν} and Γ_{μν}.
[Appendix B] The word 'Threfore' in the opening sentence should be 'Therefore.'
[Sec. V.D] The phrase 'the trio equations' is awkward; it should be 'the three equations.'
[Throughout] The abbreviation 'FLR W' is inconsistent; it should be 'FLRW' (Friedmann–Lemaître–Robertson–Walker).

Circularity Check

1 steps flagged · score 7.0 of 10

The central L_m-independence equalities (29)-(32) are forced by the velocity-variation ansatz in Eq. (25)/(A2), which the normalization condition does not determine.

self definitional [Section III, Eq. (25); Appendix A, Eq. (A2); applied in Eqs. (27)-(28) to obtain (29)-(32)]
"The form can be written as δ(u^α u^β)/δg^{μν} = u^α u^β u^μ u^ν, where we attach the detailed derivation in the APPENDIX (A). ... g_{αβ} δ(u^αu^β)/δg^{μν} = −u_μu_ν, ... g_{αβ}u^αu^β |−1 δ(u^αu^β)/δg^{μν} = −u^αu^βu_μu_ν, δ(u^αu^β)/δg^{μν} = u^αu^βu^μu^ν. (A2)"

The constraint g_{αβ}u^αu^β = −1 fixes only the g-trace of the four-index object, g_{αβ} δ(u^αu^β)/δg^{μν} = −u_μu_ν; it leaves the transverse part completely free. In the displayed derivation the paper replaces g_{αβ} by u^αu^β via g_{αβ}u^αu^β = −1, which is legitimate only if δ(u^αu^β)/δg^{μν} is already proportional to u^αu^β. That is exactly the conclusion the paper then uses. Adding any W^{αβ}_{μν} with g_{αβ}W^{αβ}_{μν}=0 preserves the normalization yet changes the second derivatives; the paper supplies no principle excluding such terms. Equations (27)-(28) and the equalities (29)-(32) therefore follow from the chosen ansatz, not from the normalization condition, so the advertised resolution of the L_m ambiguity is built into the input rather than derived.

full rationale

The paper's central claim is that imposing u^μu_μ = −1 removes the L_m ambiguity. That claim rests entirely on Eq. (25)/(A2). The derivation of (A2) in Appendix A is an algebraic non sequitur: the normalization equation fixes one scalar-trace combination of δ(u^αu^β)/δg^{μν}, and the paper treats this as if it had solved for the full tensor, thereby assuming the tensor is u^αu^βu^μu^ν. All subsequent equalities (29)-(32) and the astrophysical/cosmological applications inherit this input. This is not a case of fitted data renamed as prediction or of a self-citation chain; Ref. [73] is external and would in fact introduce sound-speed terms the paper drops. The circularity is that the conclusion — L_m-independence — is installed by the particular velocity variation rather than derived from the normalization, so the central result reduces by construction to the ansatz. Score 7 reflects that the main derivation is forced by an assumed form, while the numerical applications themselves are honestly computed from the resulting equations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central argument depends entirely on an unproved ansatz for the metric variation of the four-velocity, plus the customary perfect-fluid and FLRW/TOV assumptions. The coupling constants chi and alpha are illustrative values taken from the literature, not fitted. No new entities are introduced.

free parameters (3)

chi (f(R,T) coupling) = 1 (illustrative)
Coupling constant in f(R,T)=R+2*chi*T; set to 1 for the mass-radius plots, not fitted to data.
alpha (EMSG coupling) = 10^-37 cm^3/erg
Coupling constant in f(R,tau) = f(R) + 16*pi*f(tau); value taken from Ref. [80] for the cosmological plots, not fitted here.
B (MIT bag constant) = 60 MeV/fm^3
Input parameter for the quark-star equation of state, taken from the bag model literature; not fitted in this paper.

assumptions (5)

ad hoc to paper The normalization condition g_alpha beta u^alpha u^beta = -1 uniquely determines delta(u^alpha u^beta)/delta g^mu nu.
Section III and Appendix A use this to claim Eq. (25), but the normalization condition only fixes one scalar combination; the full velocity variation requires fluid equations or an equation of state.
ad hoc to paper The second derivative of L_m can be computed without specifying the equation of state, so no sound-speed term appears.
Section III drops the c_s^2 dependence present in Ref. [73] with no physical justification.
domain assumption The matter is a perfect fluid with T_mu nu = (rho + p) u_mu u_nu + p g_mu nu.
Standard matter model used throughout the paper.
domain assumption Cosmology uses zero cosmological constant, spatially flat FLRW metric, and barotropic equation of state p = w rho.
Assumptions stated in Section V for the cosmological applications.
domain assumption Astrophysics uses a spherically symmetric static metric and the TOV equations.
Assumptions stated in Section IV for the compact-star applications.

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Cite this review

Pith. "Pith review of Ambiguity in matter sector for modified gravity involving $\delta^2 \mathcal{L}_{m}/\delta g^{\mu\nu}\delta g^{\alpha\beta}$ and its implications to astrophysics and cosmology." pith.science (2026). https://pith.science/paper/QIXHNVQE

@misc{pith2026260804867,
  author       = {Pith},
  title        = {Pith review of: Ambiguity in matter sector for modified gravity involving $\delta^2 \mathcalL_m/\delta g^\mu\nu\delta g^\alpha\beta$ and its implications to astrophysics and cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIXHNVQE}},
  note         = {Machine review of arXiv:2608.04867}
}

abstract

Matter density ($\rho$) and radial pressure ($p$) are often used as the matter Lagrangian density ($\mathcal{L}_{m}$) because both are thermodynamically consistent and produce the same Einstein field equation (EFE) in general relativity (GR). New gravity models with explicit links between matter and geometry instead involve second-order derivatives of $\mathcal{L}_{m}$ relative to the metric tensor. So, picking either $p$ or $-\rho$ for $\mathcal{L}_{m}$ gives different effective EFEs. This confusion appears because one usually treats the four-velocity ($u_\mu$) and the metric tensor ($g_{\mu \nu}$) as independent. Here, we revisit the basics and offer a consistent framework by relaxing that assumption, thereby making the modified gravity theory independent of the choice of $\mathcal{L}_{m}$. Finally, we test this approach on neutron and quark stars (ultraviolet region) and on cosmological situations with radiation-dominated ($p=\rho/3$) equations of state (infrared region), showing how it clarifies the ambiguity in picking $\mathcal{L}_{m}$ for gravity models.

Figures

Figures reproduced from arXiv: 2608.04867 by the authors.

**Figure 1.** FIG. 1: The mass and radius relation (top) and the mass and central density (bottom) for quark [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** FIG. 2: [Top] Hubble time parameter (H) against the energy density ( [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

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Works this paper leans on

102 extracted references · 39 canonical work pages

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The density expression after using the first line reads ρ= 1 2α −1 + r 1 + 3αH 2 2π ! ,(58) 14 where we choose the physical root since the latter equation will reduce to GR whenα→0

Dust (w= 0) For the dust case in EMSG, the field equations are given by 3H 2 = 8π(ρ+αρ 2),(55) −2 ˙H−3H 2 = 8παρ 2,(56) (1 + 2αρ)( ˙ρ+ 3Hρ) = 0.(57) In evaluating the Hubble time and the density, we can only use the first two equations. The density expression after using the first line reads ρ= 1 2α −1 + r 1 + 3αH 2 2π ! ,(58) 14 where we choose the physi...
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Radiation (w= 1/3) In the reference [80] (L m =p), the field equations read 3H 2 = 8πρ r(1 + 4αρr),(60) −2 ˙H−3H 2 = 8π ρr 3 (1 + 4αρr),(61) (1 + 8αρ) ˙ρ+ 4Hρ(1 + 4αρ) = 0.(62) In [80], the solution for the Hubble parameter is given byH(t) = 1/(2t+C), whereCis an integration constant, and the authors chooseC= 0. Therefore, we later follow the same convent...
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Dust (w= 0) In the dust case, the field equations are given by 3H 2 = 8π(ρ+ 3αρ 2),(70) 2 ˙H+ 3H 2 = 8παρ 2,(71) 3Hρ(1 + 2αρ) + (1 + 6αρ) ˙ρ= 0.(72) The solution for Hubble time and density, after some algebra, can be expressed into t= √ 18αH 2 + 4π−2 √ 6√αHtanh −1 2 √ 3√αH√ 9αH 2+2π + 2 √ 6√αHtanh −1 q 3 2π √αH + 2√π 6√πH (73) ρ= 1 6α r 1 + 9αH 2 2π −1 !...
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Radiation (w= 1/3) The field equations can be written as follows 3H 2 = 8πρ r 1 + 20 3 αρr ,(75) −2 ˙H−3H 2 = 8π ρr 3 (1 + 4αρr),(76) 12Hρ(1 + 6αρ) + (3 + 40αρ) ˙ρ= 0.(77) The solutions are as follows ρr = 3 40α r 1 + 10αH 2 π −1 ! ,(78) t= 2 √ 10αH 2 +π+ √π −11 √αHtanh −1 11√αH 2 √ 10αH 2+π + 11√αHtanh −1 9√αH 2√π 8√παH .(79)
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Dust (w= 0) For dust case, the expression reads 3H 2 = 8πρ(1−αρ),(89) −2 ˙H−3H 2 = 8παρ 2,(90) 3Hρ+ (1−2αρ) ˙ρ= 0.(91) 18 ρ= 1 2α 1− r 1− 3αH 2 2π ! (92) t= q 4− 6αH 2 π + q 6 π √αHsin −1 q 3 2π √αH + 2 6H (93)

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Radiation (w= 1/3) For a radiation-dominated universe, the corresponding equations become 3H 2 = 8πρ r 1− 4 3 αρr ,(94) −2 ˙H−3H 2 = 8π ρr 3 (1 + 4αρr),(95) 12Hρ+ (3−8αρ) ˙ρ= 0 (96) Compared with the dust case, the quadratic correction is enhanced by the radiation equation of ...
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Reviewed August 6, 2026 · model on record in the stance chip above.

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The density expression after using the first line reads ρ= 1 2α −1 + r 1 + 3αH 2 2π ! ,(58) 14 where we choose the physical root since the latter equation will reduce to GR whenα→0

Dust (w= 0) For the dust case in EMSG, the field equations are given by 3H 2 = 8π(ρ+αρ 2),(55) −2 ˙H−3H 2 = 8παρ 2,(56) (1 + 2αρ)( ˙ρ+ 3Hρ) = 0.(57) In evaluating the Hubble time and the density, we can only use the first two equations. The density expression after using the first line reads ρ= 1 2α −1 + r 1 + 3αH 2 2π ! ,(58) 14 where we choose the physi...

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Therefore, we later follow the same convention in both the new formalism and the old prescription withL m =−ρ

Radiation (w= 1/3) In the reference [80] (L m =p), the field equations read 3H 2 = 8πρ r(1 + 4αρr),(60) −2 ˙H−3H 2 = 8π ρr 3 (1 + 4αρr),(61) (1 + 8αρ) ˙ρ+ 4Hρ(1 + 4αρ) = 0.(62) In [80], the solution for the Hubble parameter is given byH(t) = 1/(2t+C), whereCis an integration constant, and the authors chooseC= 0. Therefore, we later follow the same convent...

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Stiff (w=−1) This choices lead us to the vacuum energy case 3H 2 = 8π(ρ−4αρ 2),(64) −2 ˙H−3H 3 = 8π(−ρ+ 4αρ 2),(65) ˙ρ= 0.(66) From the above expressions, it is shown that ˙H= 0 and ˙ρ= 0. 15 C. Old Prescription (Lm =−ρ) In this subsection, we emphasize the field equations forL m =−ρfor generalwas well as the specific case. For the general case, the equat...

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Dust (w= 0) In the dust case, the field equations are given by 3H 2 = 8π(ρ+ 3αρ 2),(70) 2 ˙H+ 3H 2 = 8παρ 2,(71) 3Hρ(1 + 2αρ) + (1 + 6αρ) ˙ρ= 0.(72) The solution for Hubble time and density, after some algebra, can be expressed into t= √ 18αH 2 + 4π−2 √ 6√αHtanh −1 2 √ 3√αH√ 9αH 2+2π + 2 √ 6√αHtanh −1 q 3 2π √αH + 2√π 6√πH (73) ρ= 1 6α r 1 + 9αH 2 2π −1 !...

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Radiation (w= 1/3) The field equations can be written as follows 3H 2 = 8πρ r 1 + 20 3 αρr ,(75) −2 ˙H−3H 2 = 8π ρr 3 (1 + 4αρr),(76) 12Hρ(1 + 6αρ) + (3 + 40αρ) ˙ρ= 0.(77) The solutions are as follows ρr = 3 40α r 1 + 10αH 2 π −1 ! ,(78) t= 2 √ 10αH 2 +π+ √π −11 √αHtanh −1 11√αH 2 √ 10αH 2+π + 11√αHtanh −1 9√αH 2√π 8√παH .(79)

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Stiff (w=−1) In the vacuum energy case, the final results are 3H 2 = 8π(ρ−20αρ 2),(80) −2 ˙H−3H 2 = 8π(−ρ+ 4αρ 2),(81) 48αHρ 2 + (40αρ+ 1) ˙ρ= 0.(82) In this case, the Hubble time and the density are not constant. ρ= 1 40α 1− r 1− 30αH 2 π ! (83) t=− 15αH 2 −2 q 1− 30αH 2 π −3 + √π √ π−30αH 2 +π 54αH 3 (84) In the limitα→0, the density readsρ= 3H 2/8πand ...

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Dust (w= 0) For dust case, the expression reads 3H 2 = 8πρ(1−αρ),(89) −2 ˙H−3H 2 = 8παρ 2,(90) 3Hρ+ (1−2αρ) ˙ρ= 0.(91) 18 ρ= 1 2α 1− r 1− 3αH 2 2π ! (92) t= q 4− 6αH 2 π + q 6 π √αHsin −1 q 3 2π √αH + 2 6H (93)