Quintessence Dark Energy Model

• Quintessence dark energy is a theoretical framework characterized by a dynamic scalar field with a redshift-dependent equation-of-state (−1 < w < −1/3) that drives cosmic acceleration.
• The model integrates standard matter with quintessence fields in Einstein’s equations, providing insights into anisotropic stresses and their impact on compact star configurations.
• It employs matching conditions with the Schwarzschild metric and rigorous stability analyses to ensure fully regular and physically viable equilibrium systems.

The quintessence dark energy model is a theoretical framework in which the accelerated expansion of the Universe is driven by a dynamical, minimally coupled scalar field with a suitable potential, as opposed to a strict cosmological constant. Quintessence models admit a broad range of cosmological and astrophysical implications, encompassing both the large-scale cosmological background and local structures such as anisotropic stars. The core property that distinguishes quintessence from a cosmological constant is a redshift-dependent equation-of-state parameter, typically obeying $-1 < w_q < -1/3$ for cosmic acceleration.

1. Quintessence Equation of State and Model Specification

Quintessence is formalized through a scalar field $q$ (or, in some contexts, $\phi$) whose energy–momentum tensor supplements standard matter contributions in Einstein’s equations. A defining parameter is the dark energy equation of state $w_q$, restricted to $-1 < w_q < -1/3$, ensuring violation of the strong energy condition compatible with an accelerating universe, but remaining distinct from phantom regimes ($w < -1$).

The total energy–momentum tensor is constructed as

$$T^{(\mu)}_{\,\,\nu} = (\rho+p_t)u^\mu u_\nu - p_t\delta^\mu_\nu + (p_r-p_t)\eta^\mu\eta_\nu\,,\quad \tau^t_t = \tau^r_r = -\rho_q\,,\quad \tau^\theta_\theta = \tau^\phi_\phi = \frac{1}{2}(3w_q+1)\rho_q$$

where $u^\mu$ is the fluid 4-velocity, $\eta^\mu$ a radial spacelike unit vector, $\rho$ the standard matter density, and $p_r$, $p_t$ the radial and transverse pressures, with $\rho_q$ the quintessence energy density. The degree of anisotropy is captured by $\Delta(r) = (2/r)\,[p_t-p_r]$.

For equilibrium compact objects, one often introduces a matter equation of state, $p_r = m\rho$, with $m > 0$, and prescribes a functional form for $\rho_q$, e.g., $\rho_q = A(2b-r)\exp(r)$, with $A$ a model parameter and $b$ the boundary radius.

2. Interior Geometry: Finch–Skea Metric and Matching to Schwarzschild Exterior

The interior geometry is realized via the Finch–Skea metric,

$$ds^2 = -e^{\nu(r)}dt^2 + \left(1+\frac{r^2}{R^2}\right)dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)$$

where $R$ sets the geometric scale and $\nu(r)$ is a metric function determined by integrating the generalized field equations. The choice of the Finch–Skea paraboloidal geometry is motivated by its application in describing the structure of compact stars.

To ensure physical viability, the interior metric is smoothly matched with the Schwarzschild solution at the stellar boundary $r = b$, enforcing both metric coefficient and first derivative continuity: $$\left(1+\frac{b^2}{R^2}\right)^{-1} = 1-\frac{2M}{b},\qquad \nu(b) = \ln(1-2M/b)$$ where $M$ is the total mass enclosed within radius $b$.

3. Field Equations, Effective Energetics, and Anisotropy

The Einstein equations incorporating both matter and quintessence yield a coupled system: $$8\pi(\rho+\rho_q) = \frac{1}{R^2} \frac{3 + (r^2/R^2)}{(1 + r^2/R^2)^2}$$

$$8\pi(p_r-\rho_q) = \frac{1}{1 + r^2/R^2}\left(\frac{\nu'}{r} + \frac{1}{r^2}\right) - \frac{1}{r^2}$$

$$8\pi\left[p_t + \frac{(3w_q + 1)}{2}\rho_q\right] = \frac{1}{1 + r^2/R^2}\left(\frac{\nu''}{4} + \frac{\nu'}{2r} + \frac{(\nu')^2}{4}\right) - \frac{1}{R^2}(1 + r^2/R^2)^{-2}[1 + (r\nu'/2)]$$

The function $\nu(r)$, determined by integrating these equations with the prescribed $p_r = m\rho$ and $\rho_q$, encapsulates the gravitational potential including both matter and quintessence effects: $$\nu(r) = \frac{m}{2}\left[2\ln\left(1 + \frac{r^2}{R^2}\right) + \frac{r^2}{R^2}\right] + \ln\frac{1}{R} + \frac{r^2}{2R^2} - 8\pi(1+m)\int r(1 + r^2/R^2)\rho_q dr$$ Effective energy density and pressures, central for physical regularity and matching, are defined by including both components: $\rho_\text{eff} = \rho + \rho_q$ with analogous definitions for $p_{r,\text{eff}}$, $p_{t,\text{eff}}$.

Anisotropy is governed by

$\Delta(r) = \frac{2}{r}(p_t - p_r)$

with both standard and quintessence contributions, the latter featuring directly in the balance between radial and tangential stresses depending on $w_q$ and $\rho_q$.

4. Regularity, Boundary Conditions, and Physical Constraints

Physically admissible configurations require regular (finite and positive) $\rho_\text{eff}$ and $p_{r,\text{eff}}$ at the center ($r=0$), with extremal (maximum) central values and correct derivatives: $d\rho_\text{eff}/dr=0$, $d^2\rho_\text{eff}/dr^2<0$. The chosen forms for $\rho_q$ and $p_r$ ensure that the matter dominates at the center and the quintessence component becomes increasingly important near the boundary.

Matching to the Schwarzschild exterior solution via (14) and (15) ensures global regularity and continuity of the geometry across $r = b$.

5. Equilibrium and Stability Analysis: Tolman–Oppenheimer–Volkoff Equation

For equilibrium, the generalized Tolman–Oppenheimer–Volkoff (TOV) equation for an anisotropic fluid including quintessence reads

$$\frac{dp_{r,\text{eff}}}{dr} + \frac{\nu'}{2}(\rho_\text{eff} + p_{r,\text{eff}}) + \frac{2}{r}(p_{r,\text{eff}} - p_{t,\text{eff}}) = 0$$

This can be reformulated as a force balance: $F_g + F_h + F_a = 0$ with gravitational ($F_g$), hydrostatic ($F_h$), and anisotropic ($F_a$) forces defined as \begin{align*} F_g &= -\frac{1}{2}(\rho_\text{eff} + p_{r,\text{eff}})\nu' \ F_h &= -\frac{dp_{r,\text{eff}}}{dr} \ F_a &= \frac{2}{r}(p_{t,\text{eff}} - p_{r,\text{eff}}) \end{align*} The mutual balance of these forces is verified throughout the interior, ensuring static stability.

Additional stability is established via the sound speed constraints

$$0 \leq v_{sr}^2 = \frac{dp_{r,\text{eff}}}{d\rho_\text{eff}} \leq 1,\qquad 0 \leq v_{st}^2 = \frac{dp_{t,\text{eff}}}{d\rho_\text{eff}} \leq 1$$

and the “cracking” stability condition $|v_{st}^2 - v_{sr}^2| \leq 1$, precluding any local instability in the pressure configuration.

6. Implications for Astrophysics and Dark Energy Phenomenology

The integration of a quintessence field into the structure of a relativistic star introduces a physically motivated scenario in which exotic energy components can coexist with ordinary matter in strongly gravitating systems. Near the center, matter dominates and ensures regularity, while toward the edge the repulsive nature of the quintessence field influences the total pressure and density, subject to the constraints on $w_q$ and $\rho_q$.

This approach implies that observed astrophysical compact objects, such as neutron or strange stars, could be partially supported by an internal dark energy component, bridging the conceptual gap between cosmological observations of acceleration and high-density stellar phenomena.

7. Mathematical Summary and Core Relationships

The main mathematical structure underlying the anisotropic quintessence star model can be summarized as follows:

Relation	Formula	Role
Metric	$$ds^2 = -e^{\nu(r)}dt^2 + (1 + r^2/R^2)dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$	Interior geometry
Anisotropic stress	$\Delta(r) = (2/r)[p_t - p_r]$	Measures pressure anisotropy
Effective energy density	$\rho_\text{eff} = \rho + \rho_q$	Total density
Matching condition	$(1 + b^2/R^2)^{-1} = 1 - 2M/b$	Boundary to Schwarzschild metric
TOV equilibrium	$$\frac{dp_{r,\text{eff}}}{dr} + \frac{\nu'}{2}(\rho_\text{eff} + p_{r,\text{eff}}) + \frac{2}{r}(p_{r,\text{eff}} - p_{t,\text{eff}}) = 0$$	Hydrostatic equilibrium
Sound speed stability	$0 \leq v_{sr}^2, v_{st}^2 \leq 1$	Causality and micro-stability

This framework demonstrates that, with the stipulated forms for $w_q$, $\rho_q$ and matter EoS, one can construct fully regular, stable, and observationally relevant equilibrium configurations for compact stars—including both ordinary matter and quintessence-type dark energy contributions—thereby making a direct connection between local compact astrophysical structures and the global cosmic acceleration problem (Kalam et al., 2013).