In this work, we analyze the dynamical system in f(Q) gravity coupled with a scalar field, considering the specific model <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(Q) = -Q + \frac{\alpha }{Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mi>Q</mi> <mo>+</mo> <mfrac> <mi>α</mi> <mi>Q</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is a free parameter. We investigate the dynamical behavior of the system for two different choices of scalar field potentials: the exponential potential <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V(\phi ) = V_0 e^{-\beta \phi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>β</mi> <mi>ϕ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and the power-law potential <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$V(\phi ) = V_0 \phi ^{-k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <msup> <mi>ϕ</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. By examining the equilibrium points and their stability properties, we determine the qualitative evolution of the system through phase-space analysis. For the exponential potential, we identify five equilibrium points and classify them according to their stability. Our results show that the de Sitter-like phase acts as a stable attractor, indicating a late-time accelerated expansion of the Universe, whereas the stiff matter-dominated phase are generally unstable. The remaining equilibrium points exhibit stability only for specific ranges of the model parameters, demonstrating the flexibility of the framework in describing different cosmological phases. Similarly, for the power-law potential, we also obtain five equilibrium points, three of which are stable and ensure the existence of a viable late-time attractor solution. The presence of unstable and saddle points enables transitions between different cosmological epochs, further supporting the relevance of f(Q) gravity in cosmological dynamics. Our results suggest that f(Q) gravity with a scalar field can successfully describe key phases of cosmic evolution, including a transition from a stiff matter-dominated era to an accelerated expansion phase. The stability properties of the equilibrium points and the corresponding phase-space trajectories further reinforce the viability of this model as an alternative to standard dark energy scenarios. Our model exhibits excellent consistency with observational datasets, such as Hubble, DESI DR 2 BAO and Pantheon+ datasets which affirm its capability to offer fresh perspectives on the accelerating Universe.

Phase space analysis of scalar field evolution in f(Q) gravity

期刊论文
发表日期：

2026年1月10日

查看全文

Shaibal Mani,
S. Surendra Singh,
Amit Samaddar

摘要

In this work, we analyze the dynamical system in f(Q) gravity coupled with a scalar field, considering the specific model $f(Q) = -Q + \frac{\alpha }{Q}$ $f (Q) = - Q + \frac{α}{Q}$ , where $\alpha $ $α$ is a free parameter. We investigate the dynamical behavior of the system for two different choices of scalar field potentials: the exponential potential $V(\phi ) = V_0 e^{-\beta \phi }$ $V (ϕ) = V_{0} e^{- β ϕ}$ and the power-law potential $V(\phi ) = V_0 \phi ^{-k}$ $V (ϕ) = V_{0} ϕ^{- k}$ . By examining the equilibrium points and their stability properties, we determine the qualitative evolution of the system through phase-space analysis. For the exponential potential, we identify five equilibrium points and classify them according to their stability. Our results show that the de Sitter-like phase acts as a stable attractor, indicating a late-time accelerated expansion of the Universe, whereas the stiff matter-dominated phase are generally unstable. The remaining equilibrium points exhibit stability only for specific ranges of the model parameters, demonstrating the flexibility of the framework in describing different cosmological phases. Similarly, for the power-law potential, we also obtain five equilibrium points, three of which are stable and ensure the existence of a viable late-time attractor solution. The presence of unstable and saddle points enables transitions between different cosmological epochs, further supporting the relevance of f(Q) gravity in cosmological dynamics. Our results suggest that f(Q) gravity with a scalar field can successfully describe key phases of cosmic evolution, including a transition from a stiff matter-dominated era to an accelerated expansion phase. The stability properties of the equilibrium points and the corresponding phase-space trajectories further reinforce the viability of this model as an alternative to standard dark energy scenarios. Our model exhibits excellent consistency with observational datasets, such as Hubble, DESI DR 2 BAO and Pantheon+ datasets which affirm its capability to offer fresh perspectives on the accelerating Universe.