New Agegraphic Dark Energy Model in Modified Symmetric Teleparallel Theory

Madiha Ajmal; M. Sharif

REVIEW 4 major objections 4 minor 40 references

New Agegraphic Dark Energy Model in Modified Symmetric Teleparallel Theory

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that the interacting new agegraphic dark energy model, reconstructed in f(Q) gravity with a power-law scale factor, behaves as quintessence, lies in the freezing region of the $\omega_D$-$\omega'_D$ plane, corresponds to…

desk verdict The factor-of-two error in the central reconstruction step invalidates every quantitative result in Section 4; this is a desk reject. read the letter →

arxiv 2501.00721 v1 pith:MX7CHFS6 submitted 2025-01-01 gr-qc

Madiha Ajmal , M. Sharif This is my paper · ORCID

classification gr-qc MSC 83F0583D05 PACS 95.36.+x04.50.Kd64.30.+t

keywords newagegraphicdarkenergyf(Q)gravitysymmetricteleparallelnon-metricitycosmologicalreconstructionquintessenceequationofstatesquaredspeedsoundcosmiccoincidenceproblem

topics Dark Energy Dark Matter

open problems Dark Matter 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 tries to show that the new agegraphic dark energy (NADE) model, whose energy density is fixed by conformal time, can be rebuilt as a specific function $f(Q)$ of the non-metricity scalar within modified symmetric teleparallel gravity, with dark energy and dark matter allowed to interact. The reconstruction starts from equation (19), $f/2 - 6H^2 f_Q = 3n^2/\eta^2$, and yields $f(Q) = c\sqrt{Q} + 12n^2/\eta^2$; using a power-law scale factor $a(t) = t^{1/(1+q)}$ with $q = -0.832$ converts all quantities into redshift functions. The authors find that the equation of state lies in the quintessence band $-1 < \omega_D < -1/3$, the $(\omega_D, \omega'_D)$ plane falls in the freezing region, and the $(r,s)$ plane corresponds to the Chaplygin gas. The squared speed of sound $\nu_s^2$ is negative throughout, so the model is unstable, and the slowly evolving density ratio $\rho_m/\rho_D$ is said to relieve the cosmic coincidence problem. If the central claim is correct, this gives an explicit, though unstable, interacting dark-energy realization in $f(Q)$ gravity.

What carries the argument

The load-bearing object is the identification of the two dark-energy densities, written as equation (19), $f/2 - 6H^2 f_Q = 3n^2/\eta^2$. Solving this linear first-order equation in $Q$ gives $f(Q) = c\sqrt{Q} + 12n^2/\eta^2$, and combining $Q = 6H^2$ with the power-law scale factor $a(t) = t^{1/(1+q)}$ (with $q = -0.832$ fixed by an observational estimate) converts the model into explicit functions of redshift. This machinery turns the abstract NADE density into a concrete $f(Q)$ Lagrangian and drives every later plot of $\omega_D$, $\omega'_D$, $r$, $s$, and $\nu_s^2$.

What would settle it

Compare the predicted expansion history $H(z) = H_0(1+z)^{1+q}$ with model-independent cosmic-chronometer or supernova data over a range of redshifts; any significant deviation from the single power law invalidates the reconstructed $f(Q)$. Alternatively, solve equation (19) numerically without treating the conformal time $\eta$ as independent of $Q$; if the resulting equation of state leaves the band $-1<\omega_D<-1/3$ or the squared speed of sound $\nu_s^2$ becomes positive, the paper's central claim is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the correspondence between the NADE energy density $\rho_D = 3n^2/\eta^2$ and the $f(Q)$ dark-energy density $\rho_D = f/2 - 6H^2 f_Q$ closes to a first-order ODE whose solution is $f(Q) = c\sqrt{Q} + 12n^2/\eta^2$. With $Q = 6H^2$ and the power-law scale factor, the resulting interacting model has a quintessence-like equation of state for the chosen values $n = 11, 11.4, 11.8$ and small negative couplings, a freezing-region trajectory in the $\omega_D$-$\omega'_D$ plane, a Chaplygin-gas signature in the $(r,s)$ plane, and a negative squared speed of sound at all redshifts. The paper further claims that the interaction makes the dark-energy-to-dark-matter ratio evolve slowly enough to address the coincidence problem.

Load-bearing premise

The reconstruction stands on the assumption that the entire expansion history is a single power law $a(t)=t^{1/(1+q)}$ with $q=-0.832$ fixed from one observational estimate, and that the NADE density can be set equal to the $f(Q)$ dark-energy density in equation (19); if either gives way, the derived function $f(Q)$ and all Section 4 conclusions do not follow.

Editorial extensions

If this is right

If the model is right, NADE admits an explicit $f(Q)$ realization whose equation of state stays in the quintessence band, making it a candidate for the late-time acceleration.
The freezing-region trajectory implies the model's dark energy is decelerating in $\omega_D$ space, corresponding to faster-than-thawing expansion.
The Chaplygin-gas correspondence in the $(r,s)$ plane means the model can be reinterpreted as a unified dark-sector fluid, not just a geometric modification.
The negative squared speed of sound implies the model is unstable to perturbations, so it can only serve as a background cosmology unless an additional stabilization mechanism is introduced.
Because the density ratio $\rho_m/\rho_D$ evolves slowly, the interaction term may ease the coincidence problem without fine-tuning the initial densities.

Reading between the lines

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

A straightforward test would be to repeat the reconstruction with a model-independent $H(z)$ dataset rather than the single power-law $a(t)=t^{1/(1+q)}$; deviations in the expansion history would likely move $\omega_D$ outside the quintessence band.
The same correspondence scheme, applied to holographic or pilgrim dark energy models in $f(Q)$ gravity, may show that a negative $\nu_s^2$ is a generic feature of such reconstructions rather than a peculiarity of NADE.
If the model is only a background cosmology, the negative sound speed implies growing modes under perturbations, so a full linear perturbation analysis would decide whether the instability is fatal or merely a signal that the reconstruction is an effective description.

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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

Short take: the reconstruction step that everything else rests on is algebraically wrong by a factor of two, and the paper also treats conformal time as independent of Q when it isn't. So Section 4 is describing a model that doesn't satisfy the assumed NADE/f(Q) correspondence. I'd reject.

On the plus side, the paper is a recognizable reconstruction study: it uses the standard correspondence idea, works through the interacting FRW equations, and presents the usual diagnostics (EoS, omega_D-omega'_D, r-s, sound speed). The exposition is clear enough and the references to earlier NADE and f(Q) work are adequate. The specific combination—NADE with a coupling in f(Q)—is not something I've seen before, though it's a modest extension of the same authors' previous f(Q) reconstructions with other dark energy models.

Now the soft spots. The central derivation: Eq. (19) is f/2 − 6H^2 f_Q = 3n^2/eta^2. With Q = 6H^2 that's f/2 − Q f_Q = 3n^2/eta^2. Substituting their solution f = c sqrt(Q) + 12n^2/eta^2 gives f/2 − Q f_Q = 6n^2/eta^2, not 3n^2/eta^2. The correct constant particular solution is +6n^2/eta^2. That factor of two propagates to Eq. (25) and to every plotted quantity in Section 4. This is not a typo: the EoS, phase-plane trajectories, statefinder planes, and stability curves are all evaluated with the wrong f(Q), so the results are not properties of the model they claim to study.

Second, the solution treats eta as independent of Q. For the power-law a(t) = t^j, eta ∝ t^{1−j}, and Q ∝ t^{−2}, so eta^2 ∝ Q^{j−1} = Q^{−q/(1+q)}. The right-hand side of the reconstruction ODE is not constant in Q, so the closed-form solution in (20) is not correct even with the factor fixed. That is a second independent reason the derived results don't follow.

Also, the coincidence-problem claim in the conclusion is asserted without calculation. Saying the density ratio "evolves slowly" is not the same as showing it does, and given that n, c, psi, q are fixed from a single value of q and otherwise chosen post hoc, the claim isn't quantitatively supported.

Who is this for? A reader wanting to see another example of the NADE reconstruction recipe applied in f(Q) might skim it, but the central flaw makes the quantitative results untrustworthy. The qualitative instability is consistent with earlier NADE studies, but that's already known.

Recommendation: don't send to review in current form. If the authors fix the factor and re-derive with Q-dependent eta, and actually compute the coincidence ratio, it could become a marginal but publishable reconstruction paper. As is, I'd desk reject.

Referee Report

4 major / 4 minor

Summary. This paper proposes an interacting new agegraphic dark energy (NADE) model in f(Q) gravity. The authors equate the NADE energy density with the geometric dark-energy density of f(Q), assume a power-law scale factor a(t)=t^{1/(1+q)} with q=-0.832, solve the resulting first-order differential equation for f(Q), and then compute the equation-of-state parameter, the (ω_D, ω'_D) phase plane, the (r, s) statefinder, and the squared speed of sound. They report quintessence behavior, a freezing-region trajectory, Chaplygin-gas-like statefinder behavior, and instability, and they claim that the interaction alleviates the cosmic coincidence problem.

Significance. If the derivation were correct, the paper would provide a workmanlike example of a NADE/f(Q) reconstruction, with explicit analytic expressions and parametric plots. The qualitative outcomes are, however, strongly shaped by the reconstruction input: the geometric dark-energy density is forced to equal the assumed NADE density, so the diagnostics mostly recirculate known NADE phenomenology. The paper does not perform a comparison with observational data, and no machine-checkable derivation or reproducible code is supplied. The usefulness of the manuscript therefore rests entirely on the correctness of the algebraic reconstruction, and that correctness is the main problem.

major comments (4)

[§3, Eqs. (19)–(20)] Substituting the claimed solution f(Q)=c√Q+12n²/η² into the left-hand side of Eq. (19) gives f/2 − Q f_Q = 6n²/η², not 3n²/η². The correct particular solution is f(Q)=c√Q+6n²/η². Every quantity derived from Eq. (20) — ρ_D, p_D, ω_D, ω'_D, r, s, and ν_s² — inherits this factor of two, so the Section 4 conclusions describe a model that does not satisfy the assumed NADE/f(Q) density correspondence.
[§3, Eqs. (25)–(26)] The redshift conversion leading to Eq. (25) is not dimensionally consistent. For a(t)=(t/t0)^{1/(1+q)} with σ=q+1, the conformal time is η=(σ/q)t0 Ψ^{-q}, so 1/η²=q²H0²Ψ^{2q}. The second term in Eq. (25) should therefore be 12n²q²H0²Ψ^{2q}, not 12n²q²Ψ^{2q}/σ². As written, the two terms in Eq. (25) carry inconsistent powers of H0. Relatedly, the c-dependent part of Eq. (26) for ρ_D vanishes identically because √(H0²Ψ^{2q+2})=H0Ψ^σ; thus the plotted ρ_D is independent of c, and the explicit c terms in later formulas are not physically interpretable.
[§3, Eq. (19)] Solving Eq. (19) also treats η as independent of Q. With the power-law background, η²=(q²H0²)^{-1}Ψ^{-2q} and Q=6H0²Ψ^{2σ}, so η²∝Q^{-q/σ}. Equation (19) is therefore f/2 − Q f_Q = K Q^{q/σ} rather than a constant-source equation, and its solution is c√Q plus a term proportional to Q^{q/(q+1)}, not c√Q plus a constant. This Q-dependence enters before any of the Section 4 diagnostics, so this is a second, independent reason why the reconstructed f(Q) and all derived results are not the ones claimed.
[§3, after Eq. (15)] The interaction term is misstated. From the definitions, ρ_m+p_D=ρ_D(χ+ω_D), not ρ_D(1+χ); the equality Γ=3ψH(ρ_m+p_D)=3ψHρ_D(1+χ) would require ω_D=1, which contradicts the quintessence result obtained later. In addition, Eq. (17) for ω_D is asserted without derivation and does not follow from the NADE density and the continuity equations. Since Eq. (17) is the basis for the subsequent ω_D, ω'_D, r, and s calculations, this is a load-bearing gap.

minor comments (4)

[§3, Eq. (13)] Equation (13) contains the term 2H f_QQ, which is dimensionally inconsistent with the other terms; it should presumably be 2 \dot{H} f_Q or the notation should be corrected.
[§3, Eqs. (21)–(25)] The paper sets H0=70 km/s/Mpc while also setting a0=1 and effectively t0=1 in Eq. (25); the normalization of the scale factor and the time unit should be stated explicitly, since they are needed to verify Eq. (25).
[Throughout] There are several typos, including "disfomation" before Eq. (5), "constsnt" after Eq. (25), and "chossen" in Section 5; the figure labels such as "/ScriptZ" should be replaced by z.
[§4.2, Fig. 4] Figure 4 axes show ω_D and ω'_D over the range −2.4 to −1.2, while the text defines the freezing region only by ω_D<0 and ω'_D<0; the caption should explain the plotted range so the reader can relate it to the standard (ω_D, ω'_D) plane.

Circularity Check

2 steps flagged · score 6.0 of 10

EoS conclusions reduce to a self-cited NADE ansatz and an engineered f(Q); the reconstruction is definitional, not an independent prediction.

ansatz smuggled in via citation [Section 3, Eq. (17) and Section 4.1, Eq. (28)]
"We can represent ω D using the parameters that have been established previously [29] ω D = − 1/2 − Ω D ( 1 + 2ψ/Ω D ) . (17) ... Referring to Eq.(17), we can derive [Eq. (28)]."

Eq. (17) is an explicit formula for the EoS ωD, exactly the quantity Section 4.1 claims to determine, and it is imported from the authors' own prior work [29] rather than derived in this paper. Eq. (28) is then presented as the derived EoS, but it is a rewrite of Eq. (17) in Q- and z-variables using the reconstructed f(Q). All later outputs—the quintessence interval, the freezing-region plot, the r−s Chaplygin classification, and the ν_s^2<0 instability—are computed from Eq. (28)/Eq. (17). The central qualitative results are therefore the self-cited input ansatz, made to look like predictions by selecting n=11,11.4,11.8 and ψ=−0.001,−0.004,−0.006; the text even says these values produce 'favorable' phase-plane graphs. The reconstruction does not independently generate the EoS behavior.
self definitional [Section 3, Eqs. (19)-(20) and Section 4]
"Taking the equivalent densities equal to each other, we demonstrate the connection between NADE and the f(Q) gravity [33]. From Eqs.(12) and (18), it is clear that f/2 − 6H²fQ = 3n²/η². This is the first-order linear differential equation in Q and its solution is f(Q) = c√Q + 12n²/η², (20)"

The f(Q) model is constructed by requiring its dark-energy density to equal the assumed NADE density (Eq. 18). Every Section 4 quantity—ωD, ω′D, r, s, and ν_s²—is then evaluated with this same f(Q). Consequently the EoS and phase-plane conclusions are not independent tests of a first-principles f(Q) theory; they are consequences of the NADE input by construction. The paper reports these consequences as findings, but the model has been engineered to reproduce NADE, so the claimed behavior is built into the definition rather than predicted by the geometry.

full rationale

The paper is a reconstruction exercise: f(Q) is solved from Eq. (19) so that the f(Q) dark-energy density equals the NADE density, and Section 4 then rediscovers the EoS behavior already encoded in the self-cited Eq. (17). The central qualitative claims (quintessence-like EoS, freezing region, Chaplygin-gas statefinder, instability) are thus inherited from the assumed NADE ansatz and hand-chosen parameters, not independently derived from f(Q) gravity. This is partial circularity rather than total: the explicit f(Q) form and its graphical behavior are new content, and the reconstruction procedure is transparent. I also note, separately, that the algebraic factor in Eq. (20) does not satisfy Eq. (19) (the constant should be 6n²/η² rather than 12n²/η²), and that the derivative in the ωD−ω′D plane is taken with respect to Q rather than ln a; these are correctness defects, not circularity, and I have not scored them as circular steps.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the standard NADE density, the f(Q) field equations, a power-law background, and the correspondence identification. The constants n, c and psi are chosen by hand; q is taken from a single cited measurement. No new entities are introduced.

free parameters (5)

n (NADE parameter) = 11, 11.4, 11.8
Arbitrary choices for the NADE parameter; all figures and conclusions depend on these values.
c (integration constant) = 2
Set arbitrarily in Section 3; the text says it 'negligibly impacts' plots but it enters every expression.
psi (interaction coupling) = -0.001, -0.004, -0.006
Negative values chosen because they 'provided consistent and meaningful results'; not determined by data.
q (deceleration parameter) = -0.832
Fixed from a single cited observational estimate [34] and used to set the power-law exponent; results are conditional on this value.
H0 (Hubble constant) = 70 km/s/Mpc
Input from observations used in redshift expressions; not fitted here.

assumptions (6)

domain assumption Flat FRW metric with scale factor a(t)
Used throughout; Eq. (10).
domain assumption Power-law scale factor a(t) = t^(1/(1+q))
Adopted after Eq. (20); all redshift expressions rely on this form.
domain assumption NADE energy density rho_D = 3n^2/eta^2
Standard NADE definition from [4], invoked in Eq. (18); not derived here.
domain assumption Correspondence principle: NADE density equals f(Q) dark-energy density
Eq. (19); the whole reconstruction rests on this identification.
domain assumption Interacting fluids with Gamma = 3 psi H (rho_m + p_D)
Eq. (15) and surrounding text; constant coupling psi.
standard math f(Q) field equations and energy density definitions
Taken from [18]; Eqs. (9), (12), (13).

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Cite this review

Pith. "Pith review of New Agegraphic Dark Energy Model in Modified Symmetric Teleparallel Theory." pith.science (2026). https://pith.science/paper/MX7CHFS6

@misc{pith2026250100721,
  author       = {Pith},
  title        = {Pith review of: New Agegraphic Dark Energy Model in Modified Symmetric Teleparallel Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MX7CHFS6}},
  note         = {Machine review of arXiv:2501.00721}
}

abstract

In this manuscript, we examine the cosmological significance of the new agegraphic dark energy model by investigating different cosmological parameters such as the equation of state parameter, $\omega_{D}-\omega^{\prime}_{D}$ and the $r-s$ planes in the framework of $f(\mathcal{Q})$ theory. We consider flat Friedmann-Robertson-Walker universe model under interacting conditions between dark energy and dark matter. The equation of state parameter indicates a quintessence-like characteristic of the universe. The stability of the model is analyzed using the squared speed of sound parameter which demonstrates the unstable behavior of the new agegraphic dark energy model throughout the cosmic evolution. The freezing region is represented by the $\omega_{D}-\omega^{\prime}_{D}$ plane, while the Chaplygin gas model corresponds to the $r-s$ plane. It is worthwhile to mention here that the interacting new agegraphic dark energy model addresses the cosmic coincidence problem by allowing the energy density ratio between dark energy and dark matter to evolve slowly over cosmic time.

Figures

Figures reproduced from arXiv: 2501.00721 by the authors.

**Figure 1.** Graph of f(Q) against z and Q. Applying the value of H, we obtain Q = 6H 2 0Ψ 2+2q . (24) When we substitute this value in Eq.(20), We can express the solution in terms of z as follows f(Q) = √ 6c q H2 0Ψ2q+2 + 12n 2 q 2Ψ2q (q + 1)2 . (25) For the purpose of analysis, we use three fixed values of n = 11, 11.4 and 11.8 to explore the graphical behavior in the f(Q) theory. If we change the value of n, it has a distinc… view at source ↗

**Figure 2.** Graphs of ρD and pD against z. also examine the characteristics of ρD and pD in the context of NADE reconstructed f(Q) gravity model. Applying Eq.(20) to (12) and (13), we derive ρD = 6n 2 η 2 − 1 2 c √ 6H − √ Q , pD = cη2 Q(2H˙ − Q) + 6H2Q − H − 12n 2Q3/2 2η 2Q3/2 , where σ = q + 1 for further simplification. In terms of redshift parameter, these equations take the following form ρD = r 3 2 c q H2 0Ψ2q+2 −… view at source ↗

**Figure 3.** Plots of ωD versus z. × cσ2 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Graphs of ω ′ D versus ωD. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Graphs of s versus r. × 6H 3 0Ψ 3q+3 + 144c 2H 5 0Ψ 5q+5 − H 2 0Ψ 2q+25 2 ψ − 24√ 6c 2H 2 0Ψ 2q+2 × 3(6ψ + 3) H 2 0Ψ 2q+25/2 − H 2 0Ψ 2q+2ψ − 3H0Ψ σ 6H 4 0Ψ 4q+4(2ψ + 1) − q H2 0Ψ2q+2ψ + H 2 0Ψ 2q+2 3 H 2 0Ψ 2q+23 2 (6ψ + 3) − 2ψ Ψ −6q × q 6 …

**Figure 6.** Figure 6: Graphs of ν 2 s versus z. − H0Ψ σ √ 6cΨ −2q q 2 −1 + 12n 2 2−1 . (30) Several studies have investigated this aspect for different DE models. For instance, Setare [38] examined the interacting HDE model with the Chaplygin gas and found that both models exhibi…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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