Emergent Universe in f(Q) gravity theories

Avik De; Hamid Shabani; Tee-How Loo

REVIEW 3 major objections 5 minor 91 references

Emergent Universe in f(Q) gravity theories

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Quadratic f(Q) gravity can host a stable Einstein-static universe and then exit into an inflationary expansion, replacing the Big Bang singularity with a past-eternal phase.

desk verdict Exact conserved ES result is solid, but the advertised emergent-universe exit rests on non-conserved approximate orbits that are not full f(Q) solutions. read the letter →

arxiv 2412.13242 v2 pith:GBQDUAML submitted 2024-12-17 gr-qc

Hamid Shabani , Avik De , Tee-How Loo This is my paper · ORCID

classification gr-qc MSC 83D0583F05 PACS 04.50.Kd98.80.-k

keywords f(Q)gravitysymmetricteleparallelEinsteinstaticuniverseemergentnon-metricitycosmologicalsingularityinflationcurvedFRWcosmology

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 establish that the emergent-universe scenario—a long-lived Einstein-static phase that replaces the Big Bang singularity and later gives way to inflation—can be realized inside $f(Q)$ gravity, a modified theory built on spacetime non-metricity rather than curvature. The concrete model is $f(Q)=\alpha Q+\beta Q^2$ on a curved Friedmann universe filled with a perfect fluid. The authors derive an Einstein-static radius, identify equation-of-state windows in which small perturbations oscillate instead of growing, and show that a time-dependent equation of state can push the universe out of the static phase into expansion. If correct, this offers a classical, singularity-free entrance to inflation without requiring a scalar inflaton field.

What carries the argument

The load-bearing object is the quadratic symmetric-teleparallel model $f(Q)=\alpha Q+\beta Q^2$ on a spatially curved FRW background, with a free function $\gamma(t)$ in the torsion-free, curvature-free affine connection. For a perfect fluid with equation of state $p=w\rho$, setting all time derivatives to zero gives the Einstein-static radius $a_{\mathrm{ES}}=\sqrt{6k(1-3w)/(1+3w)\,\beta/\alpha}$, and stability is decided by the eigenvalues of the dynamical system $\dot x=y$, $\dot y=$ a rational function of $x,y,w,k,\gamma$, for several choices of $\gamma(t)$: $\gamma=\pm a$, $\gamma=\gamma_0$, $\gamma=\gamma_0 a^n$, and a $\gamma(t)$ fixed by solving a second-order differential equation. The exit mechanism is a time-dependent $w(t)$ that moves the system out of the stable windows, converting the oscillatory Einstein-static state into an expanding universe.

What would settle it

Impose the exact conservation condition $\Sigma=0$ while numerically integrating the $\gamma(t)=\gamma_0$ system; if the only bounded solution is the static point and no oscillatory orbit around $a_{\mathrm{ES}}=1$ survives, the claimed Einstein-static phase is not an exact solution of $f(Q)$ gravity.

Watch

Extended reading notes

Core claim

The paper claims that $f(Q)$ gravity can realize the emergent scenario: for $f(Q)=\alpha Q+\beta Q^2$ with a curved FRW metric and a perfect fluid, there are stable Einstein-static solutions around which the scale factor oscillates, and a smoothly running equation-of-state parameter can break that stability and send the universe into an expanding phase. Stable oscillations are found for $\gamma(t)=\pm a(t)$ with $k=-1$ when $-7/15<w<-1/3$ or $w>1/3$; for constant $\gamma(t)=\gamma_0$ in closed and open geometries under specific parameter windows; and for $\gamma(t)=\gamma_0 a^n$ with $n=3w$ in both curvatures. In the cases where the continuity equation is not identically conserved, the residual $\Sigma$ oscillates with tiny amplitude around zero. In the open case the post-static phase is a decelerated expansion with $a\propto t$, while in the closed $\gamma_0 a^n$ case it is an accelerated, inflationary expansion; a decreasing $w(t)$ can also produce a sequence from a past-eternal static state through accelerated expansion to a decelerated phase.

Load-bearing premise

The construction assumes that a small, oscillating violation of energy conservation still counts as a valid $f(Q)$ solution, and that the hand-picked time-dependent equation-of-state parameter driving the exit is physically realizable.

Editorial extensions

If this is right

A stable Einstein-static phase exists in $f(Q)$ gravity for both open and closed spatial curvature, so the Big Bang singularity can be replaced by a past-eternal oscillating state.
The static radius is fixed by the model and the matter content through $a_{\mathrm{ES}}=\sqrt{6k(1-3w)/(1+3w)\,\beta/\alpha}$, which selects the allowed signs of $\beta/\alpha$ and the admissible matter equation of state.
In the open case with $\gamma=\pm a$, stability holds for $-7/15<w<-1/3$ or $w>1/3$, and leaving that window produces a decelerated expansion with $a\propto t$.
In the closed case with $\gamma=\gamma_0 a^n$ and $n=3w$, a running equation-of-state parameter can trigger an inflationary accelerated expansion after the static phase.
A decreasing $w(t)$ can produce the full sequence of a past-eternal Einstein-static state, an accelerated expansion, and finally a decelerated phase that matches the entrance to the standard hot big bang.

Reading between the lines

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

Because the $\gamma=\gamma_0$ and $\gamma=\gamma_0 a^n$ cases only minimize the continuity residual $\Sigma$ rather than setting it to zero, a stricter reading of $f(Q)$ gravity would require searching for connections or matter couplings that make $\Sigma=0$ exactly; without that, the oscillatory states are approximate solutions rather than exact ones.
The hand-picked linear functions $w(t)$ could be translated into a minimally coupled scalar-field potential; deriving that potential explicitly would turn the exit mechanism into a physical model with its own dynamics rather than a kinematic choice.
A testable extension is the primordial spectrum: an Einstein-static phase with a running equation of state generically imprints a low-multipole suppression in the CMB temperature spectrum, so the parameter windows found here could be constrained by Planck low-$\ell$ data.

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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 version: the paper has one clean exact result, and the rest of the constructed emergent scenario does not hold up as a solution of the full f(Q) theory.

What is actually new: for f(Q)=αQ+βQ² with curved FRW, the ES radius (23), the conserved γ=±a, k=-1 case with pure imaginary eigenvalues (27), and the general-γ differential equation (39) are all derived, not assumed. The conserved stability windows are a genuine extension of the ES program. I give credit for that.

The soft spot is load-bearing. Sections VI and VII use γ=γ0 and γ=γ0 a^n with k=+1. There the continuity relation (20) is not satisfied: the paper solves (14)-(15) for ˙y and then checks numerically that Σ is small, with amplitude around 10^-4. But Σ=0 is the content of the connection field equation. A trajectory with Σ≠0 is not a solution of the full f(Q) system. The paper's own wording—"the continuity equation ... is not conserved except at the coordinate ... minimal variations"—concedes exactly this. And this matters because the only route to an inflationary exit (lower panels of Fig. 5, Fig. 6) uses the k=+1, γ0 a^n case, which is non-conserved. The conserved γ=±a case exits to a~t, i.e. decelerated expansion, not inflation. So the advertised claim that f(Q) "can accurately depict the emergence of the universe" is unsupported.

Secondary points: the exit mechanism uses w(t) chosen by hand as linear functions such as w=-0.323-0.0000129t, with no scalar-field or microphysical derivation, so it is a tuned toy rather than a mechanism. The normalization α=6k(1-3w)/(1+3w)β also sets the ES radius to unity and removes the model parameters from the stability analysis, so the stability windows are partly engineered. And pure imaginary eigenvalues give only linear neutral stability; the paper does not address nonlinear stability. These are not the main problem; the main problem is the non-conserved approximation being used for the central claim.

Who this is for: someone working in f(Q) cosmology who wants the specific eigenvalue algebra and the connection-gauge effects. The paper is honest about what it does, and the exact conserved case is a useful reference.

Recommendation: a serious editor should send this to peer review rather than desk reject, because the conserved-case result and the general-γ derivation are real work and a referee can pinpoint the above flaw. My own verdict would be reject in current form, with the fix being either an exact conserved exit to inflation or a reframed, explicitly approximate claim with controlled errors.

Referee Report

3 major / 5 minor

Summary. The paper studies Einstein static (ES) solutions in f(Q)=αQ+βQ^2 gravity with spatial curvature k=±1, using a non-coincident symmetric teleparallel connection with a free temporal function γ(t). It derives the ES scale factor a_ES (Eq. 23), fixes the model parameters by setting a_ES=1 (Eq. 24), and then performs a linear stability analysis for several choices of γ(t): the exactly conserved case γ=±a with k=−1 (Section V), γ=γ0 (Section VI), γ=γ0 a^n with n=3w (Section VII), and a general γ(t) constrained by a differential equation (Section VIII). It claims that for certain equation-of-state windows the ES solution is stable in the sense of small oscillations, and that a time-dependent EoS w(t) can produce an exit to inflation, especially for k=+1. The paper concludes that f(Q) gravity can 'accurately depict the emergence of the universe.'

Significance. If valid, the paper would be a useful contribution to the ES/emergent-universe literature in modified gravity, since the f(Q) setup with a nontrivial connection is technically involved and the paper derives an explicit ES equilibrium, Eq. (23), with transparent dependence on model parameters and curvature. The exactly conserved case γ=±a, k=−1 is a clean result, and the algebraic derivation of the ES radius and of the stability conditions is internally consistent. However, the advertised emergent-universe scenario is not established: the k=+1 exit to inflation relies on approximate orbits that violate the connection field equations, and the time-dependent EoS is inserted ad hoc into equations derived for constant w. The central claim therefore outruns what the calculations actually show.

major comments (3)

[Sections VI–VII and Figs. 2–4] The oscillatory solutions around the ES fixed point are not exact solutions of the full f(Q) system. The authors state in Section VI that for γ=γ0 (and similarly for γ=γ0 a^n) the continuity equation (20) is not conserved except at (x=1,y=0), and that they 'choose appropriate values of the constants to have minimal variations' of Σ rather than enforcing Σ=0. In f(Q) gravity, Eq. (10) and its cosmological reduction Eq. (16) show that Σ=0 is exactly the content of the connection field equation (8); a trajectory with Σ≠0 violates that equation and is not a solution of the theory. The small numerical values of Σ (10^-4 to 10^-5, Figs. 2–4) do not change this. Since the k=+1 stability windows and the inflationary exit in Section IX (lower panels of Figs. 5–6) are obtained from these non-conserved cases, the central claim that f(Q) can depict the emergent universe is unsupported.
[Section IX, Figs. 5–6] The exit mechanism is built on hand-picked linear functions w(t), such as w=−0.323−0.0000129t in Fig. 6 and w=0.35−0.000021t in Fig. 5, but the dynamical systems (25)–(26) and (28)–(33) were derived under the assumption p=wρ with constant w. Substituting a time-dependent w(t) into these equations without re-deriving the field equations introduces additional ˙w terms and changes the continuity relation, so the numerical evolution is not a solution of the original system. The reference to a minimally coupled scalar field in Section IX is only a motivation, not a derivation of the specific w(t); thus the claimed transition from ES to inflation is a numerical experiment under an unverified assumption rather than a demonstrated exit.
[Sections V–VII, eigenvalues (27), (30), (31), (34)] The stability analysis uses purely imaginary eigenvalues, which establishes linear neutral stability only. No Lyapunov function, conserved quantity, or higher-order analysis is provided, so the claim that the ES solution is 'stable' for finite perturbations, and hence the 'past-eternal' character of the oscillatory phase, is not fully established even in the exact conserved case of Section V. This is a standard caveat in ES studies, but it is load-bearing here because the paper explicitly invokes long-lasting oscillatory behavior before the exit.

minor comments (5)

[Section IV, first paragraph] Typo: 'ﬁst' should be 'first'.
[References [20] and [21]] References [20] and [21] are identical (Chanda et al., Eur. Phys. J. C 84, 658 (2024)); the duplication should be removed and the intended citations checked.
[Eq. (23) and stability windows] For k=+1, the stability windows w>1/3 and w<−1/3 require β/α<0 for a real positive a_ES; the paper does not explicitly discuss this sign requirement, which is important for the physical viability of the model.
[Section VI, Eq. (30)] The text states 'approximately −4.18 < γ0 < −0.71 ∧ 0.71 < γ0 < 4.18' without showing how these endpoints follow from the radicand in Eq. (30); the derivation should be stated explicitly.
[Fig. 5] The upper right panel caption and the text (Section IX) both describe a linearly decreasing w(t), but the panel uses w=0.35−0.000021t while the adjacent text mentions w=−0.35+...; the notation should be made consistent so the reader can identify which curve corresponds to which parameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ES solution and stability analysis are derived from the f(Q) field equations, and the hand-picked running EoS is an explicit model assumption rather than a disguised prediction.

full rationale

The central ES construction is not circular. Equations (21)-(23) solve the f(Q) Friedmann equations at equilibrium, and the stability eigenvalues are computed from the Jacobians of the derived dynamical systems (25)-(26), (28)-(29), and (32)-(33), rather than being imposed to match the desired emergent-universe behavior. The normalization a_ES=1, which gives the parameter relation (24), is a legitimate choice of units/parameters and does not inject the target conclusion. The exit to inflation in Sect. IX is obtained by explicitly choosing a running EoS, e.g. 'w=-0.323-0.0000129t' in Fig. 6, and the paper openly states 'we have used a linear time-varying forms of EoS with suitable coefficients so that they can evade intervals of w mentioned within the list.' Because the running EoS is presented as an assumed mechanism rather than as a prediction uniquely forced by f(Q), the resulting inflationary behavior is an existence argument, not a circular reduction. The admitted non-conservation of the continuity equation in Sects. VI and VII ('the continuity equation (20) is not conserved except at the coordinate (x=1,y=0) ... minimal variations of the expression (20)') is a correctness or validity limitation of the approximate oscillatory orbits, but it is not circularity: the ES equilibrium itself satisfies Sigma=0, and the approximate orbits are not used to define the equilibrium. The self-citations, such as [81] for the conservation property of the gamma=±a case and [88] for the energy-conservation identity, support auxiliary mathematical facts that are directly checkable from the paper's own Eq. (20) and are not load-bearing for the main conclusion; hence they do not raise the circularity score. Overall, the paper's claim is a fragile existence demonstration with many free choices, but the derivation chain is not equivalent to its inputs by construction.

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

The central claim relies heavily on model and parameter choices: the f(Q) form, the connection function γ(t), the normalization a_ES=1, the ranges for w, and the hand-tuned time-dependent EoS. No new entities are invented, but the freedom in γ(t) and w(t) supplies the main degrees of freedom.

free parameters (6)

β (Q^2 coupling in f(Q)=αQ+βQ^2)
Model coefficient chosen by hand; examples β=1,-1,0.1,-0.5 in figures; affects the continuity violation Σ amplitude.
α (linear Q coupling) = α=6k(1-3w)/(1+3w)β
Imposed by setting a_ES=1 (Eq. 24); represents a fine-tuning constraint between model parameters and the matter equation of state.
w (EoS parameter)
Free matter parameter; stability windows are expressed as ranges of w, e.g., -7/15<w<-1/3 or w>1/3.
γ0 (constant connection function)
Free parameter in the connection (12); stability ranges, e.g., 0.71<|γ0|<4.18 for k=+1.
n (power in γ=γ0 a^n) = n=3w
Chosen because only n=3w gives pure imaginary eigenvalues (Eq. 34).
time-dependent EoS coefficients = examples: w=-0.35+0.000021t, w=0.323-0.0000122t, w=-0.323-0.0000129t
Hand-picked linear functions used to force the ES-to-inflation transition in Section IX.

assumptions (5)

domain assumption The symmetric teleparallel connection (12) with free function γ(t) is the appropriate non-coincident connection for k≠0 FRW.
Standard f(Q) framework from Ref. [69]; not re-derived in this paper.
domain assumption Matter is a perfect fluid with barotropic EoS p=wρ and vanishing hypermomentum.
Assumed in Eqs. (7) and (17); w is later taken to be time-dependent by hand.
ad hoc to paper f(Q)=αQ+βQ^2 is sufficient to capture the relevant early-universe physics.
Motivated by Starobinsky model and small deviation from GR in Section IV, but not derived from a deeper principle.
ad hoc to paper The dynamics near ES is captured by the 2D system (x,y), and the continuity violation Σ can be made small by parameter choice.
Sections VI-VIII solve for ẏ from Eqs. (18)-(19) while only minimizing Σ in Eq. (20), not setting it to zero.
ad hoc to paper A hand-picked time-dependent w(t) models the physical exit from ES.
Section IX uses linear w(t) without deriving it from a scalar field or other microphysical mechanism.

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Cite this review

Pith. "Pith review of Emergent Universe in f(Q) gravity theories." pith.science (2026). https://pith.science/paper/GBQDUAML

@misc{pith2026241213242,
  author       = {Pith},
  title        = {Pith review of: Emergent Universe in f(Q) gravity theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBQDUAML}},
  note         = {Machine review of arXiv:2412.13242}
}

abstract

One resolution of the ancient cosmic singularity, i.e., the Big Bang Singularity (BBS), is to assume an inflationary stage preceded by a long enough static state in which the universe and its physical properties would oscillate around certain equilibrium points. The early period is referred to as the Einstein Static (ES) Universe phase, which characterizes a static phase with positive spatial curvature. A stable Einstein static state can serve as a substitute for BBS, followed by an inflationary period known as the Emergent Scenario. The initial need has not been fulfilled within the context of General Relativity, prompting the investigation of modified theories of gravity. The current research aims to find such a solution within the framework of symmetric teleparallel gravity, specifically in the trendy $f(Q)$ theories. An analysis has been conducted to investigate stable solutions for both positively and negatively curved spatial FRW universes, in the presence of a perfect fluid, by utilizing various torsion-free and curvature-free affine connections. Additionally, we propose a method to facilitate an exit from a stable ES to a subsequent inflationary phase. We demonstrate that $f(Q)$ gravity theories have the ability to accurately depict the emergence of the universe.

Figures

Figures reproduced from arXiv: 2412.13242 by the authors.

**Figure 2.** FIG. 2. Cosmological quantities for models with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. Properties of the oscillatory dynamics when [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Important quantities numerically obtained for the m [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5. Transition from an oscillatory to a decelerated expa [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. A decreasing form of the EoS parameter under consider [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

Discussion (0). Continue with ORCID to comment.

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