arxiv: 2604.25412 · v1 · submitted 2026-04-28 · 🌀 gr-qc

Recognition: unknown

Dynamical analysis of the covariant f(Q) gravity models

S. A. Narawade , S. A. Kadam

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:26 UTC · model grok-4.3

classification 🌀 gr-qc

keywords covariant f(Q) gravitydynamical systemscritical pointscosmological evolutionstability analysislate-time accelerationphase space

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The pith

Covariant f(Q) gravity with Hubble-proportional coupling reproduces the full sequence of radiation, matter, and dark energy eras.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies two covariant f(Q) gravity models, one with power-law and one with logarithmic form, in which a coupling function changes in direct proportion to the Hubble parameter. The authors convert the field equations into an autonomous dynamical system, locate the critical points that mark each cosmological era, and determine their stability including non-hyperbolic cases via center manifold theory. Both models are shown to support stable transitions through radiation-dominated, matter-dominated, and late-time accelerating phases, with the expected values of the equation-of-state parameter, density parameters, and deceleration parameter at each stage. This dynamical consistency indicates that the chosen f(Q) forms can account for the observed expansion history within a single modified-gravity framework.

Core claim

In the covariant formulation of f(Q) gravity, the power-law and logarithmic models with Hubble-proportional coupling admit a sequence of critical points whose stability properties allow the universe to evolve from a radiation-dominated era through a matter-dominated era into a dark-energy-dominated attractor, as verified by phase-space trajectories, density-parameter evolution, and the total equation-of-state parameter.

What carries the argument

The autonomous dynamical system constructed from the cosmological field equations, whose critical points are classified by stability and whose trajectories are shown to connect the successive eras.

If this is right

Both models produce stable critical points that correspond to radiation, matter, and dark energy domination with the correct equation-of-state values.
Phase-space trajectories connect these eras in the expected order for each functional form.
The deceleration parameter changes sign at the appropriate transitions, confirming late-time acceleration.
The total equation-of-state parameter approaches -1 in the final attractor for both models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the Hubble-proportional coupling holds, similar dynamical-system techniques could be applied to other covariant modified-gravity theories to check whether they also generate the observed era sequence.
The results imply that late-time acceleration can arise from the geometric sector alone, potentially reducing reliance on separate dark-energy fields in model building.
Extension of the analysis to include linear perturbations around the critical points would test whether the models remain viable when structure formation is considered.
Comparison of the predicted density-parameter evolution curves with current supernova and CMB data could directly constrain the free parameters of the power-law and logarithmic forms.

Load-bearing premise

The coupling function is taken to evolve in direct proportion to the Hubble parameter, and the two chosen functional forms are assumed to be representative of the covariant f(Q) framework.

What would settle it

A set of cosmological observations that showed no stable radiation-dominated epoch or that lacked the predicted transitions in the density parameters between radiation, matter, and dark energy would falsify the models' ability to reproduce the cosmic sequence.

Figures

Figures reproduced from arXiv: 2604.25412 by S. A. Kadam, S. A. Narawade.

**Figure 1.** Figure 1: Phase portraits of the system illustrating the critical points view at source ↗

**Figure 2.** Figure 2: Cosmological evolution with respect to redshift: (a) density parameter view at source ↗

**Figure 3.** Figure 3: Phase portraits of the system illustrating the critical points view at source ↗

**Figure 4.** Figure 4: Unstable region for the critical point C2 utilizing CMT. Ωr Ωm ΩDE -2 -1 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 log₁₀(1 + z) Density Parameters (a) Evolution of the density parameters. q ωtot -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 log₁₀(1 + z) EoS, Deceleration Parameter (b) Evolution of q, ωtot. ωDE -10 -5 0 5 10 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 log₁₀(1 + z) EoS for Dark Energy (c) EoS parameter ωDE view at source ↗

**Figure 5.** Figure 5: Cosmological evolution with respect to redshift: (a) density parameter view at source ↗

read the original abstract

In this study, we explore the cosmological evolution of the Universe in the framework of covariant $f(Q)$ gravity, with a coupling function that evolves dynamically in proportion to the Hubble parameter. Two specific forms of the function are examined: a power-law model and a logarithmic model. By rewriting the cosmological field equations as an autonomous dynamical system, we determine and classify the corresponding critical points and analyze their stability. Our results show that both models are able to reproduce the sequence of cosmic evolution, including radiation, matter, and dark energy-dominated eras, along with the transitions between them. The physical properties at each critical point are described using key cosmological quantities such as the total EoS parameter, density parameters, and the deceleration parameter. The stability of the non-hyperbolic critical point is analyzed through center manifold theory. In addition, we present phase space trajectories along with the stability behavior of each critical point. The evolution plots for the density parameters of radiation, matter, and dark energy, along with the EoS parameter for the total, are illustrated for further analysis. Overall, the analysis suggests that the $f(Q)$ models considered here, within the context of covariant formulation, provide a consistent description of cosmic evolution and offer a promising approach to explaining the late-time acceleration of the Universe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper runs two new Hubble-proportional coupling forms through a standard dynamical-systems analysis in covariant f(Q) gravity and locates the expected critical points, but local stability alone does not confirm the claimed global sequence of eras.

read the letter

The authors pick a power-law and a logarithmic form for the coupling function that scales with the Hubble parameter, rewrite the field equations as an autonomous system, find the critical points, classify their stability, and apply center-manifold theory to the non-hyperbolic case. They also report phase-space trajectories and plots of the density parameters and total equation of state. That is the concrete content. The specific functional choices are new in this setting, and the local analysis follows the usual playbook for modified-gravity cosmologies without obvious algebraic errors in the setup described in the abstract. The center-manifold step is a small step beyond the most basic eigenvalue checks. The paper therefore supplies two additional working examples that people already studying f(Q) or similar theories can compare against existing results. The coupling is introduced as a modeling choice rather than derived, which is common and acceptable for this kind of exploration. The soft spot is the gap between local stability and the global history. The abstract states that both models reproduce the radiation-to-matter-to-late-acceleration sequence, yet the available description gives no numerical integration of sample trajectories, no explicit heteroclinic orbits, and no confirmation that orbits starting near the radiation fixed point actually pass through a matter-dominated saddle before reaching the accelerating attractor. Local classification is necessary but not sufficient for that claim. If the full paper contains only qualitative sketches around the fixed points, the central viability statement rests on an assumption rather than a demonstrated flow. This is the sort of incremental paper that belongs in the modified-gravity dynamical-systems literature. Specialists who track f(Q) models or autonomous-system techniques in cosmology will find the specific critical-point data useful for reference. It is coherent on its own terms and formally grounded enough to merit referee time, though any review should focus on whether the global trajectories are actually shown or merely inferred from the fixed points. I would send it for peer review with that request to the referees.

Referee Report

1 major / 2 minor

Summary. The manuscript performs a dynamical systems analysis of covariant f(Q) gravity with a Hubble-proportional coupling function, considering power-law and logarithmic forms. The cosmological field equations are rewritten as an autonomous system; critical points are identified, classified by stability (including eigenvalues and center-manifold reduction for a non-hyperbolic point), and their cosmological properties (EoS, density parameters, deceleration) are computed. Phase-space trajectories and evolution plots for density parameters and total EoS are presented. The authors conclude that both models reproduce the observed cosmic sequence (radiation → matter → dark-energy domination) and provide a consistent description of late-time acceleration.

Significance. If the global flow is shown to connect the fixed points in the required order, the results would strengthen the case for covariant f(Q) gravity as a viable alternative to ΛCDM that naturally produces the full cosmic history from a single modified-gravity action. The explicit use of center-manifold theory for the non-hyperbolic point is a technical strength that goes beyond standard linear stability analysis.

major comments (1)

[Results / abstract] The central claim that both models reproduce the radiation-matter-dark-energy sequence rests only on local stability classifications and center-manifold analysis. No numerical integration of the autonomous system, explicit heteroclinic orbits, or phase-portrait trajectories starting near the radiation fixed point and passing through a matter-dominated saddle before reaching the accelerating attractor are provided (see abstract and results sections). Local stability alone does not establish the global evolutionary history asserted in the conclusions.

minor comments (2)

The explicit autonomous equations and the concrete numerical values chosen for the power-law exponent and logarithmic coefficient are not stated in the provided text; these should be displayed prominently so that the stability results can be reproduced.
Figure captions and labels should explicitly indicate which model (power-law or logarithmic) and which initial conditions correspond to each plotted trajectory and evolution curve.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major point below.

read point-by-point responses

Referee: [Results / abstract] The central claim that both models reproduce the radiation-matter-dark-energy sequence rests only on local stability classifications and center-manifold analysis. No numerical integration of the autonomous system, explicit heteroclinic orbits, or phase-portrait trajectories starting near the radiation fixed point and passing through a matter-dominated saddle before reaching the accelerating attractor are provided (see abstract and results sections). Local stability alone does not establish the global evolutionary history asserted in the conclusions.

Authors: We agree that local stability classifications, even when supplemented by center-manifold analysis, do not by themselves establish the global flow connecting the fixed points in the observed cosmic sequence. The manuscript does contain phase-space trajectories and evolution plots of the density parameters and total equation-of-state parameter that are obtained by numerical integration of the autonomous system; these plots illustrate the overall behavior and the transitions between eras. However, the referee is correct that the existing figures do not explicitly display trajectories initiated near the radiation-dominated critical point that pass through the matter-dominated saddle before reaching the accelerating attractor. To strengthen the claim, we will add dedicated numerical integrations with initial conditions close to the radiation fixed point, together with the corresponding heteroclinic orbits and additional phase-portrait insets, in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation follows directly from modeling choices and standard dynamical systems methods

full rationale

The paper introduces the coupling function's proportionality to the Hubble parameter and selects power-law and logarithmic forms as explicit modeling assumptions. It then rewrites the covariant f(Q) field equations as an autonomous system, locates critical points, classifies them via eigenvalues, and applies center-manifold analysis to the non-hyperbolic point. These steps are standard and self-contained; no reported outcome is obtained by fitting a parameter to data and relabeling it a prediction, nor does any load-bearing step reduce to a self-citation whose content is itself unverified or tautological. The claimed reproduction of radiation-matter-dark-energy sequence is asserted from the stability classification of the fixed points, which is a direct consequence of the chosen equations rather than a circular redefinition of the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the chosen power-law and logarithmic coupling functions are physically reasonable and that the standard FLRW reduction of covariant f(Q) gravity holds; no new physical entities are introduced.

free parameters (2)

exponent or coefficient in power-law coupling
Specific numerical choice required to define the model and obtain the reported critical points.
coefficient in logarithmic coupling
Specific numerical choice required to define the model and obtain the reported critical points.

axioms (1)

domain assumption Covariant formulation of f(Q) gravity with matter coupling on an FLRW background
Standard background assumption invoked when rewriting the field equations as an autonomous system.

pith-pipeline@v0.9.0 · 5529 in / 1440 out tokens · 74159 ms · 2026-05-07T15:26:57.328847+00:00 · methodology

discussion (0)

Reference graph

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