Cosmological solutions in $f(Q)$ gravity via Noether symmetry approach

Technical Summary: Cosmological Solutions in f(Q) Gravity via Noether Symmetry Approach

Problem Statement
The accelerated expansion of the universe remains a central challenge in modern cosmology, typically attributed to dark energy or modifications to General Relativity (GR). While various modified gravity theories, such as $f(R)$ and $f(T)$, have been explored, they often rely on metric-compatible manifolds. This paper investigates $f(Q)$ gravity, an alternative theory based on the non-metricity scalar $Q$ within a non-metric compatible manifold. A primary challenge in this field is identifying specific functional forms of $f(Q)$ that yield exact cosmological solutions without arbitrary assumptions. Furthermore, while recent studies have applied Noether symmetry to $f(Q)$ gravity, this work aims to provide a distinct derivation of exact solutions and analyze their dynamical stability and consistency with observational data.

Methodology
The authors employ the Noether symmetry approach as a selection criterion for viable $f(Q)$ models. The methodology proceeds through the following steps:

1. Point-like Lagrangian Formulation: The action for $f(Q)$ gravity is reduced to a point-like Lagrangian $L(a, \dot{a}, Q, \dot{Q})$ within a spatially flat Friedmann-Robertson-Walker (FRW) universe. A Lagrange multiplier is introduced to enforce the definition of the non-metricity scalar $Q$ in terms of the scale factor $a(t)$.
2. Noether Symmetry Analysis: The authors define a vector field generator $X$ in the configuration space $(a, Q)$. By imposing the condition that the Lie derivative of the Lagrangian vanishes ($L_X L = 0$), they derive a system of partial differential equations. Solving these equations yields the specific functional form of $f(Q)$ and the associated Noether charge (conserved quantity).
3. Exact Solution Derivation: Using the derived $f(Q)$ function and the conserved Noether charge, the authors solve the Euler-Lagrange equations to obtain an exact analytical solution for the scale factor $a(t)$.
4. Dynamical System Analysis: To study the evolution and stability of the model, the field equations are transformed into an autonomous dynamical system using dimensionless variables ($x$ and $y$) representing dark energy and radiation density parameters, respectively. The authors identify critical points, construct the Jacobian matrix, and analyze eigenvalues to determine the stability of these points.
5. Scalar-Tensor Extension: The approach is extended to $f(Q)$ scalar-tensor cosmology, where a scalar field $\phi$ is non-minimally coupled to $f(Q)$, to find consistent cosmological models under Noether symmetry.
6. Observational Constraints: The derived power-law solutions are tested against inflationary observational data, specifically the scalar spectral index ($n_s$) and the tensor-to-scalar ratio ($r$), using the slow-roll approximation.

Key Contributions and Results

• Functional Form of f(Q): The application of Noether symmetry uniquely determines the functional form of the gravity theory as:
  $$f(Q) = c(Q - nQ)^{\frac{3}{2-2n}}$$
  (Note: The paper presents this as $f(Q) = c(Q - nQ)^{\frac{3}{2-2n}}$, derived from the differential equation $2Q(n-1)F_Q + Q(2n+1) + 3F = 0$).

• Power-Law Expansion: The exact solution for the scale factor is found to follow a power law:
  $$a(t) \sim t^{\frac{1}{1-n}}$$
  The authors demonstrate that for $n < 1$ (and $n \neq 0$), the universe undergoes accelerated expansion. Specifically, the condition $n < 1$ is necessary to guarantee expansion, and specific values of $n$ correspond to different cosmological eras.

• Dynamical System Stability: The dynamical analysis reveals several critical points corresponding to different cosmological epochs:

  • Case I ($n=0.5$): Corresponds to a $\Lambda$CDM-like model with an effective equation of state $\omega_{eff} = -2/3$ and $a(t) \sim t^2$.
  • Case II ($n=-0.5$): Represents a matter-dominated universe with $\omega_{eff} = 0$ and $a(t) \sim t^{2/3}$.
  • Case III ($n=-1$): Represents a radiation-dominated universe with $\omega_{eff} = 1/3$ and $a(t) \sim t^{1/2}$.
  • Case IV: Indicates a future singularity where $\omega_{eff} \to \infty$.
    The stability analysis classifies these points as saddle points, unstable nodes, or oscillatory instabilities depending on the parameter $n$.

• Inflationary Consistency: By applying the slow-roll parameters to the power-law solution, the authors derive constraints on the parameter $n$. To satisfy the Planck 2018 bound on the tensor-to-scalar ratio ($r < 0.064$), the parameter must satisfy $n < 0.9959$. Under these conditions, the predicted scalar spectral index is $n_s \approx 0.9920$, which is consistent with observational data.

• Scalar-Tensor Extension: In the scalar-tensor framework, the Noether symmetry approach yields oscillating or exponential solutions for the Brans-Dicke parameter $\omega(\phi)$, depending on the sign of a derived constant $k^2$.

Significance and Claims
The paper claims that the Noether symmetry approach provides a robust, assumption-free method for selecting viable $f(Q)$ models and deriving exact cosmological solutions. By deriving the specific functional form $f(Q) = c(Q - nQ)^{\frac{3}{2-2n}}$, the authors establish a direct link between the symmetry of the Lagrangian and the physical evolution of the universe.

The study highlights that $f(Q)$ gravity, being a second-order theory free from the pathologies often associated with fourth-order $f(R)$ theories, offers a promising alternative for explaining cosmic acceleration. The results demonstrate that the model naturally accommodates the transition from radiation and matter domination to an accelerated expansion phase (resembling $\Lambda$CDM) without the need for an explicit cosmological constant. Furthermore, the model's ability to satisfy current inflationary constraints on $n_s$ and $r$ suggests its viability as a candidate for early-universe cosmology. The authors note that while a recent study [37] also addressed Noether symmetry in $f(Q)$, their work differs in both content and results, offering a distinct derivation and dynamical analysis.