Two-Field Chiral Cosmological Model

• The two-field chiral cosmological model is a framework where gravity couples to two interacting scalar fields with a nontrivial target space metric, enabling mixed kinetic and potential effects.
• It incorporates phantom and quintom behaviors that allow the effective equation-of-state parameter to cross the phantom divide, linking early inflation and late-time dark energy.
• The model supports a range of scenarios—including emergent, non-singular universes and unified dark sector dynamics—with exact solutions and detailed perturbative analyses.

The two-field chiral cosmological model is a cosmological framework in which gravity is coupled to two scalar fields whose dynamics are governed by a nontrivial metric structure on field (“target”) space. The model generalizes single-field inflationary or dark energy scenarios, allowing for kinetic and potential interactions between the fields. The field-space metric can encode negative curvature, non-canonical signatures, and phantom-like behavior, leading to rich phenomenology bridging early- and late-universe physics, unification of dark sector components, and the possibility of non-singular cosmological evolution.

1. Mathematical Formulation and Geometry of the Model

The general two-field chiral cosmological model has an action of the form

$$S = \int d^4x \sqrt{-g} \left[ R - \frac{1}{2} h_{AB}(\Phi) g^{\mu\nu} \nabla_\mu \Phi^A \nabla_\nu \Phi^B - V(\Phi) \right],$$

where $\Phi^A=(\phi,\psi)$ are the two scalar fields and $h_{AB}(\Phi)$ is the field-space metric, often chosen with non-trivial structure, e.g., negative constant curvature for hyperbolic models or Lorentzian signature for “quintom” models (Paliathanasis et al., 2021, Paliathanasis, 2023).

Typical forms include:

• Canonical/phantom mix: $h_{AB} = \mathrm{diag}(1, -e^{\kappa\phi})$, yielding a maximally symmetric target space of Lorentzian signature (Paliathanasis, 2023).
• Hyperbolic geometry: $h_{AB} = \mathrm{diag}(1, e^{\kappa\phi})$, corresponding to a field-space of constant negative curvature (Paliathanasis et al., 2021).

The equations of motion comprise the Einstein–Friedmann equations,

$$3 H^2 = \frac{1}{2} h_{AB} \dot{\Phi}^A \dot{\Phi}^B + V(\Phi),$$

and the coupled chiral field equations,

$$D_t(h_{AB} \dot{\Phi}^B) + 3 H h_{AB} \dot{\Phi}^B + \partial_A V = 0$$

with $D_t$ the covariant derivative on field-space.

A key feature is that mixed kinetic terms—controlled by $h_{12}$ or $h_{22}$ as functions of either field—can couple the dynamics between fields nontrivially. This induces effective “kinetic interactions” essential for features such as crossing the phantom divide, emergent universe scenarios, and unified dark sector models (Beesham et al., 2013, Chervon, 2014, Abbyazov et al., 2014, Paliathanasis et al., 2021).

2. Phantom, Quintom, and Dark Sector Interactions

Promoting one field to have negative kinetic energy (a “phantom” field) is crucial in these models for accessing a broader range of cosmological evolutions. For instance,

• Chiral-Phantom (quintom) models: Implemented via the $h_{AB}$ signature or phantom kinetic terms, such models allow the effective equation-of-state parameter

$w_{\mathrm{eff}} = \frac{K - V}{K + V}$

to cross the “phantom divide” at $w = -1$ (where $K$ is total kinetic energy) (Paliathanasis, 2023, Paliathanasis et al., 2020, Chervon, 2014). This is forbidden in single-field, canonical-kineic energy models with monotonic potentials.

A chiral metric of constant negative curvature introduces target-space-induced coupling. For a representative action,

$$S = \int \sqrt{-g} \left[ R - \frac{1}{2} (\nabla\phi)^2 + \frac{1}{2} e^{\kappa\phi} (\nabla\psi)^2 - V(\phi) \right] d^4x,$$

the $\phi$ field is canonical and $\psi$ is phantom; their interaction via the $e^{\kappa\phi}$ factor embodies the chiral (sigma model) structure (Paliathanasis et al., 2021).

3. Dynamical Systems, Asymptotics, and Attractor Structure

Phase space analysis in Hubble-normalized variables is a principal tool for understanding the global dynamics:

Define,

$$x = \frac{\dot\phi}{\sqrt{6} H},\quad z = \frac{e^{\kappa\phi/2} \dot\psi}{\sqrt{6} H},\quad y^2 = \frac{V(\phi)}{3 H^2},$$

with a constraint relating the variables (for canonical–phantom mix, e.g., $1 - x^2 + z^2 - y^2 = 0$).

Critical points of the resulting autonomous system correspond to distinct cosmological epochs:

• Kinetic-dominated solutions: $w_{\mathrm{eff}} = 1$.
• Scaling attractors (where kinetic and potential energies are proportional): $w_{\mathrm{eff}}> -1$, depending on χ of potential and kinetic couplings.
• Phantom-dominated or de Sitter attractors: $w_{\mathrm{eff}}\leq -1$.

Analysis shows that:

• Depending on the signature and curvature of the chiral metric, the model supports regimes with transitions between matter, quintessence, and phantom dominance (Paliathanasis et al., 2020, Paliathanasis et al., 2020).
• For $N>2$ chiral fields, the dynamical system generically reduces to the two-field case: additional fields do not yield new stable physical attractors (only mathematical generalizations) (Paliathanasis et al., 2020).

4. Emergent Universe and Non-singular Cosmological Evolution

The two-field chiral cosmological model provides a framework for emergent universe (EmU) scenarios, characterized by a past-eternal, non-singular, quasi-static phase transitioning smoothly into inflation. Key features (Beesham et al., 2013, Chervon et al., 2014):

• The scale factor is typically chosen as

$a(t) = A (\beta + e^{\alpha t})^m,$

which is regular as $t\rightarrow -\infty$.

• The chiral metric is engineered so that the kinetic term of the phantom field (with $h_{11} < 0$) supplies the negative energy required for $\dot{H}>0$ at early times, while the canonical field (with positive $h_{22}$) modulates contributions from spatial curvature and transitions through inflation.

Notably, exact solutions can capture the entire cosmic evolution from static past to accelerating future, with the equation-of-state parameter evolving $w(-\infty) = -1/3 \to w(+\infty) = -1$, realizing a non-singular departure from conventional big bang cosmologies (Beesham et al., 2013).

Generalizations to Einstein–Gauss–Bonnet gravity show that similarly constructed two-field chiral models (sometimes with three fields if curvature is included) permit explicit solutions with both fields often possessing "phantom" character (Chervon et al., 2014, Maharaj et al., 2017).

5. Unification of Dark Energy and Dark Matter Components

In several models, dark energy and dark matter are unified within the chiral framework by prescribing the field kinetic and potential terms to mimic observational scalings:

• By setting, e.g., $\tfrac{1}{2} h_{11} \dot\phi^2 = B = \text{const}$, and $\tfrac{1}{2} h_{22} \dot\psi^2 = C a^{-3}$, dark matter-like ($a^{-3}$) and dark energy–like (constant or slowly varying) behaviors are realized in the chiral sector (Abbyazov et al., 2014).
• The reconstruction procedure consists of determining the chiral metric component $h_{22}$ and the potential $V(\phi,\psi)$ as functions of $\phi$ (or scale factor) to match observational expansion (via $H(a)$), yielding e.g.,

$V(a) = -6 B \ln a + C a^{-3} + V^*$

corresponding to a dynamically unified dark sector (Abbyazov et al., 2014).

This chiral unification provides an alternative to standard $\Lambda$CDM, replacing phenomenological fluids with interacting scalar fields that can interpolate between matter and dark energy dominance, offering new handles on cosmic coincidence and fine-tuning problems.

6. Cosmological Perturbations and Structure Formation

Perturbation theory in chiral cosmological models extends the usual analysis of inflationary cosmologies by showing how both canonical and dark sector (phantom) fields contribute to primordial fluctuations:

• Perturbing both metric and chiral fields, the energy–momentum tensor sources for scalar perturbations (e.g., in the longitudinal gauge) can be explicitly decomposed into inflaton and dark sector contributions (Chervon, 2014).
• In the short-wavelength regime, dark sector field perturbations are temporally damped and decay faster than adiabatic (inflaton) modes, consistent with negligible impact on observable large-scale power.
• In the long-wavelength regime, power-law solutions for perturbations differ from canonical single-field results, with the possibility for additional isocurvature or entropic modes depending on the couplings and initial conditions.
• The stability and decay properties of these additional scalar perturbations help clarify their role in the formation of cosmic structures and inform constraints on multi-field inflationary dynamics (Chervon, 2014, Paliathanasis et al., 2021).

7. Extensions, Exact Solvability, and Connections to Modified Gravity

Many studies focus on integrable subcases, explicit solution construction, and links to generalized gravity theories:

• By performing conformal transformations on modified gravity actions (e.g., $f(R)$ or $F(R,\xi)$ models), a systematic map to two-field chiral cosmological models in the Einstein frame is established (Ivanov et al., 2021, Pozdeeva et al., 22 Jan 2024).
• This correspondence allows general analytic solutions (often involving hyperbolic functions or Jacobi elliptic functions in conformal time) for the scale factor and scalar field evolution, including exact results in spatially flat, open, and closed universes (Ivanov et al., 17 Jul 2024).
• Hybrid inflationary models and primordial black hole formation are unified by specifying special forms of the function $F(R,\xi)$ so that the effective two-field potential supports both inflation and PBH-enhancing features (e.g., saddle points in the potential landscape) (Pozdeeva et al., 22 Jan 2024).
• Analytical control allows exploration of cosmologically significant phenomena without slow-roll or numerical approximations, and can accommodate cyclic, bouncing, or asymptotically de Sitter solutions, and the emergence of antigravity regimes (Ivanov et al., 17 Jul 2024).

Summary Table: Roles of Scalar Fields in Canonical Two-Field Chiral Cosmological Models

Field / Term	Kinetic Character	Function
$\phi$	canonical or phantom	Drives inflation / emergent universe / dark energy
$\psi$	canonical or phantom	Modulates curvature, matter-like, or dark sector roles
$V(\phi,\psi)$	—	Encodes self and cross-interaction effects
$h_{AB}$	metric on field space	Controls kinetic mixing and target-space geometry

The two-field chiral cosmological model encompasses a flexible and mathematically rich framework for realizing scenarios including unification of dark matter and dark energy, smooth phantom divide crossing, quasi-static emergent universes, exact inflation and anisotropic solutions, with robust links to modified gravity, all while capturing essential phenomenology for both background and perturbative cosmological evolution (Beesham et al., 2013, Chervon, 2014, Abbyazov et al., 2014, Paliathanasis et al., 2020, Paliathanasis, 2023, Pozdeeva et al., 22 Jan 2024, Ivanov et al., 17 Jul 2024).