[Paper Review] Cosmological models with spinor and scalar fields by Noether symmetry approach

Gilberto M. Kremer, Rudinei C de Souza|arXiv (Cornell University)|Jan 22, 2013

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper uses the Noether symmetry approach to constrain potentials and couplings in cosmological models with spinor and scalar fields, enabling analytical integration of the field equations. It finds that spinor fields can mimic dark matter or inflaton behavior, while scalar fields induce oscillatory expansion with alternating accelerated and decelerated phases, consistent with observational cosmology.

ABSTRACT

General cosmological models with spinor and scalar fields playing the role of gravitational sources are analyzed. The Noether symmetry approach is taken as a criterion to constrain the undefined potentials and couplings of the generic actions. For all the found Noether symmetries the corresponding dynamical systems can be analytically integrated. The obtained cosmological solutions describe the early and late Universe as expected by basing on the known eras of the Universe.

Motivation & Objective

• To constrain generic potentials and couplings in cosmological models with spinor and scalar fields using the Noether symmetry criterion.
• To identify physically meaningful models without ad hoc assumptions by requiring invariance under Noether symmetries.
• To analytically integrate the resulting dynamical systems and derive cosmological solutions for early and late-time Universe.
• To explore whether spinor fields can account for dark matter and scalar fields for dark energy, or vice versa.
• To determine the dynamical behavior of the scale factor and scalar field under Noether symmetry constraints.

Proposed method

• Apply the Noether symmetry approach to a general action involving spinor and scalar fields with unspecified potentials and couplings.
• Identify infinitesimal generators of symmetry via the Lie derivative condition $ L_{\mathbf{X}}\mathcal{L} = 0 $, ensuring conservation of a quantity $ M_0 $.
• Use the cyclic variable condition $ \sum_k \alpha_k \partial z / \partial q_k = 1 $ to reduce the system to fewer variables.
• Transform the original dynamical system into a reduced system involving only the scale factor $ a $ and scalar field $ \phi $, using canonical transformations.
• Integrate the reduced system analytically by expressing solutions in terms of $ z(t) = z_1 t + z_2 $ and $ u(t) = u_0 \sin(\omega t + b_0) $, with $ \omega^2 = 3ABU_0 $.
• Reconstruct explicit time-dependent solutions for $ a(t) $, $ \phi(t) $, and $ \psi(t) $ using the transformation relations and conserved quantities.

Experimental results

Research questions

• RQ1 Which forms of potentials and couplings for spinor and scalar fields lead to Noether symmetries in cosmological models?
• RQ2 Can the Noether symmetry approach yield analytically integrable cosmological solutions for spinor-scalar field systems?
• RQ3 How do the resulting solutions describe the early (inflationary) and late (accelerated) Universe?
• RQ4 Can the spinor field exhibit properties of dark matter or inflaton depending on coupling type?
• RQ5 What is the time evolution of the scale factor and scalar field when the system admits a cyclic variable and conserved quantity?

Key findings

• The Noether symmetry approach successfully constrains the potentials and couplings, leading to analytically integrable dynamical systems.
• For the non-minimally coupled spinor field, the equation of state approaches that of a cosmological constant, mimicking inflation.
• In the minimally coupled case, the spinor field behaves as standard matter, with energy density $ \rho_\psi = m\Psi_0 / a^3 $, supporting its identification as dark matter.
• The scalar field exhibits an oscillating equation of state, alternating between matter-like and dark energy-like behavior.
• The scale factor $ a(t) $ shows oscillatory expansion with alternating accelerated and decelerated phases, described by $ a(t) = \left( \frac{\omega^2}{2U_0} \right)^{1/3} \left\{ z_1^2 t^2 + \cdots \right\}^{1/3} $.
• The spinor field solution is $ \psi(t) = \frac{\sqrt{2U_0}}{\omega} \begin{pmatrix} \psi_1^0 e^{-i m t} \\ \psi_2^0 e^{-i m t} \\ \psi_3^0 e^{i m t} \\ \psi_4^0 e^{i m t} \end{pmatrix} \Theta(t) $, with $ \Theta(t) \propto \{ \cdots \}^{-1/2} $, indicating time-dependent normalization.