Dynamical Dark Sector: A Joint Two-Scalar-Field Model for Dark Matter and Quintessence

1. Introduction

The standard ΛCDM model, while remarkably successful in describing the large-scale universe, is built upon two components of unknown origin: a cosmological constant (Λ), responsible for the observed cosmic acceleration [2,3], and cold dark matter (CDM). The cosmological constant faces a severe fine-tuning problem [4], and the fundamental nature of dark matter remains elusive despite decades of experimental searches. These foundational challenges have motivated a rich exploration of alternative theories. Scalar fields are compelling candidates, with precedents in cosmic inflation and the Standard Model’s Higgs mechanism. Quintessence models, where a slowly rolling scalar field generates negative pressure [5,11,13], provide a dynamic alternative to Λ. On the dark matter front, models featuring an ultra-light scalar field—often termed Fuzzy Dark Matter (FDM)—have gained traction for their ability to resolve potential small-scale structure anomalies [6,7,8].

While the conceptual framework combining an axion-like quintessence [12] and FDM is well-established, this work presents a dedicated, end-to-end quantitative analysis connecting the model’s fundamental parameters to key observables. Our central contribution is to demonstrate that this unified model can simultaneously reproduce the standard cosmic expansion history and possess the inherent flexibility to address the S₈ tension. By solving the full background and linear perturbation equations using a modified version of the CLASS code [15], we explicitly map the dependence of the linear matter power spectrum, the resulting S₈ parameter, and the dark energy equation of state on the model’s parameters. This phenomenological exploration aims to provide sharp, testable predictions and motivate a full-scale statistical analysis against cosmological data.

2. Theoretical Framework

We consider a system governed by the action

(1)

where R is the Ricci scalar and M_pl = (8πG)^−1/2 is the reduced Planck mass. We work in natural units (c = ℏ = 1). _matter represents perfect fluids for baryons and radiation. The Lagrangians for the scalar fields Ψ (dark matter) and ϕ (quintessence) are:

(2)

(3)

The fields are minimally coupled to gravity and are assumed not to interact directly. We adopt this non-interacting scenario as the most predictive and parsimonious baseline, allowing for the isolation of the distinct phenomenological consequences of each field. It is important to note that this is an idealization. In a more complete field theory framework, coupling terms (e.g., βϕ²Ψ²) could exist. Such interactions would introduce new free parameters and could lead to a richer phenomenology, such as energy transfer between the dark components. However, without a strong theoretical motivation for a specific interaction form, the minimally coupled model serves as the most robust and falsifiable starting point. Together, they constitute a self-contained “dark sector.”

3. Cosmological Field Equations

Varying the action in a spatially flat (k = 0) Friedmann-Lemaître-Robertson-Walker (FLRW) background yields the Friedmann equations and the Klein-Gordon equations for the homoge-neous fields:

(4)

(5)

(6)

(7)

where H = a˙/a is the Hubble parameter and the total energy density ρ_total and pressure p_total are the sums of the individual components.

4. Background Cosmological Dynamics

4.1. Numerical Setup and Parameter Space

The coupled system of equations is solved numerically from an initial scale factor a_initial = 10⁻⁸ to the present day at a = 1.

4.1.1. Quintessence Potential

We adopt the well-motivated axion-like potential

(8)

This form is theoretically appealing as its underlying shift symmetry can protect the potential from large quantum corrections, making a small mass scale technically natural [12].

4.1.2. Parameter Selection and Benchmark

The model’s key new parameters are the FDM mass m_Ψ, and the quintessence scales M and f . To demonstrate the model’s viability and explore its predictions, we define a benchmark scenario calibrated to match Planck 2018 data [1] (H₀ ≈ 67.4 km/s/Mpc, Ω_b,0 ≈ 0.05, Ω_Ψ,0 ≈ 0.26, Ω_ϕ,0 ≈ 0.69):

• FDM Mass (m_Ψ): We select m_Ψ = 10⁻²² eV for our benchmark. This value is specifically chosen because it lies in a range known to suppress small-scale structure [8].
• Quintessence Scales (M , f): We set M = 2.5 × 10⁻³ eV and f = M_pl. The energy scale M is tuned to yield the correct dark energy density today, while f = M_pl is motivated by high-energy physics contexts.

4.1.3. Initial Conditions and Robustness

We use a numerical shooting method to find initial conditions ϕ_initial and ϕ˙initial that yield the target densities. For the field to be slow-rolling today, it must start near its potential maximum with negligible initial velocity. This requires a small initial displacement of δϕ ≈ 10⁻⁵f . This sensitivity to initial conditions is an intrinsic feature and a significant challenge for “thawing” models [13].

4.2. Results: Cosmic Expansion History

The numerical solution for the benchmark scenario successfully replicates the canonical cosmic history, transitioning from radiation- to matter- to dark energy-domination. The FDM field behaves as pressureless matter (ρ_Ψ ∝ a⁻³) after its oscillations commence, while the quintessence field begins to dominate at late times (a ≳ 0.5), driving cosmic acceleration.

Figure 1. Cosmic expansion history showing transition from radiation to matter to dark energy domination.

Figure 1. Cosmic expansion history showing transition from radiation to matter to dark energy domination.

Figure 2. Axion-like potential V (ϕ) = M ⁴[1 + cos(ϕ/f )]. The field starts near the maximum.

Figure 2. Axion-like potential V (ϕ) = M ⁴[1 + cos(ϕ/f )]. The field starts near the maximum.

5. Observational Signatures and Discussion

5.1. A Dynamic Equation of State for Dark Energy

The model predicts a dynamic equation of state, w_ϕ(a). The field is frozen by Hubble friction (w_ϕ ≈ −1) for most of cosmic history before “thawing” at late times. The present-day value is w_ϕ,0 ≈ −0.92. This value is primarily sensitive to the axion decay constant, f . Larger values of f make the potential flatter, pushing w_ϕ,0 closer to −1, while smaller values would lead to more significant deviations. The benchmark prediction of w_ϕ,0 ≈ −0.92 represents a potentially detectable deviation from ΛCDM with future surveys like EUCLID [14].

Figure 3. “Thawing” behavior of w_ϕ(a). Present-day: w_ϕ,0 ≈ −0.92.

Figure 3. “Thawing” behavior of w_ϕ(a). Present-day: w_ϕ,0 ≈ −0.92.

5.2. Suppression of Small-Scale Structure and the S₈ Tension

The scalar nature of FDM introduces an effective “quantum pressure” that counteracts gravity and suppresses structure formation below a characteristic Jeans scale [6]. To quantify this, we solved the complete set of linearized Einstein-Boltzmann equations by modifying the public code CLASS [15]. The resulting linear matter power spectrum, P (k), exhibits a distinct suppression at high wavenumbers compared to ΛCDM.

This power suppression directly impacts the S₈ = σ₈(Ω_m,0/0.3)^0.5 parameter. For our benchmark, with a self-consistent total matter density Ω_m,0 ≈ 0.31, we calculate S₈ ≈ 0.79. This value significantly reduces the ~3–5σ tension between Planck’s ΛCDM inference (S₈ ≈ 0.83) [1] and measurements from large-scale structure (LSS) surveys [9,10].

A Testable Observational Trade-Off: The S₈ tension can be addressed by choosing a mass in the range 10⁻²²–10⁻²¹ eV. This choice of mass, however, is in noteworthy tension with some of the most stringent lower bounds from the Lyman-alpha forest (e.g., m_Ψ ≳ 2 × 10⁻²¹ eV) [16]. If we were to enforce the Lyman-alpha bound and choose a mass of m_Ψ = 3 × 10⁻²¹ eV, the suppression of power would be minimal, yielding an S₈ value of approximately 0.82, offering little relief.

Figure 4. Suppression of P (k) at small scales due to FDM quantum pressure.

Figure 4. Suppression of P (k) at small scales due to FDM quantum pressure.

Figure 5. S₈ decreases with lighter m_Ψ. Lyman-alpha bound: m_Ψ ≳ 2 × 10⁻²¹ eV [16].

Figure 5. S₈ decreases with lighter m_Ψ. Lyman-alpha bound: m_Ψ ≳ 2 × 10⁻²¹ eV [16].

5.3. Parameter Degeneracies and Model Robustness

Our analysis focuses on a fixed benchmark cosmology. In a full statistical analysis, the new parameters (m_Ψ, M , f ) would exhibit degeneracies with standard ΛCDM parameters. A full statistical analysis using MCMC methods is necessary to properly explore the multi-dimensional parameter space [18].

6. Conclusion

We have analyzed a two-scalar-field model that provides a unified, dynamical origin for the cosmic dark sector. By solving the coupled background and perturbation equations for a well-motivated benchmark scenario, we have demonstrated that the model not only provides a viable cosmic history but also possesses the inherent capability to address the S₈ tension. The key results are:

• A Mechanism for S₈ Tension Relief: The FDM component can suppress the matter power spectrum on small scales. We showed that a benchmark mass of m_Ψ = 10⁻²² eV can yield S₈ ≈ 0.79.
• Testable Dark Energy Dynamics: The quintessence component leads to a dynamic equation of state, with w_ϕ,0 ≈ −0.92.
• A Clear Observational Trade-Off: The model creates a testable tension between the FDM mass required to lower S₈ and existing constraints from the Lyman-alpha forest.

The crucial next step is a full Bayesian statistical analysis using MCMC methods to rigorously test the model’s entire parameter space against a combination of CMB, LSS, and supernova datasets.

Figure 6. Schematic of perturbation evolution in CLASS (stiff system handled robustly).

Figure 6. Schematic of perturbation evolution in CLASS (stiff system handled robustly).