Interacting Dark Sector Models

• Interacting dark sector frameworks are cosmological models incorporating non-gravitational couplings between dark matter and dark energy to address cosmic acceleration and the coincidence problem.
• They employ a gauge-invariant perturbative methodology, ensuring that observable predictions remain free from spurious coordinate artifacts while accurately capturing energy and momentum exchanges.
• Observational effects include modifications to the matter power spectrum and ISW effect, with stability constraints guiding the selection of viable interaction models for future experiments.

Interacting dark sector frameworks describe cosmological models in which dark matter and dark energy exchange energy and momentum through non-gravitational couplings. These frameworks generalize the ΛCDM paradigm by introducing dynamical dark energy (typically a scalar field) and allowing for explicit interaction terms that modify the conservation equations at both the background and perturbative levels. The general aim is to capture a broader array of cosmic behaviors, address the "coincidence problem," test the robustness of cosmic acceleration mechanisms, and generate distinct observational signatures in cosmic structure, the cosmic microwave background, and large-scale gravitational potentials.

1. Gauge-Invariant Perturbative Methodology

The analysis of interacting dark sector models is formulated within the gauge-invariant framework for cosmological perturbations, ensuring that physically observable quantities are free from spurious coordinate (gauge) artifacts. Scalar perturbations to the Friedmann–Robertson–Walker (FRW) metric are typically analyzed in the Newtonian gauge:

$$ds^2 = -(1 + 2\Phi) dt^2 + a(t)^2 (1 - 2\Phi) dx_i dx^i,$$

where $\Phi$ is the gravitational potential. Gauge-invariant density perturbations for any species $x$ are constructed as

$$\Delta_x \equiv \delta_x + 3 \frac{aH}{k}(1-q_x)v_x,$$

$q_x = \frac{Q_x}{3H\rho_x(1+w_x)},$

where $Q_x$ denotes the source/sink term for the background energy-conservation equation of component $x$, $w_x$ its equation of state, $\delta_x$ the fractional density perturbation, $v_x$ its peculiar velocity, and $H$ the Hubble parameter. For the dark energy scalar field (quintessence) $\phi$, the corresponding gauge-invariant perturbation involves the field and its velocity perturbation as well as a weighted total energy density perturbation.

These definitions are essential for computing robust predictions for observables such as matter power spectra and the integrated Sachs–Wolfe (ISW) effect, eliminating ambiguities due to coordinate choices (Potter et al., 2011).

2. Covariant Description of Energy and Momentum Exchange

Interactions are implemented by modifying the energy–momentum conservation equations:

$\dot{\rho}_x + 3 H \rho_x (1 + w_x) = Q_x,$

subject to $\sum_x Q_x = 0$ to ensure total energy–momentum conservation. The interaction terms can represent dark matter decay ($Q_m = -A\rho_m$), dark energy decay, or more general scalar-tensor couplings (e.g., $Q_{(\phi)}^\mu = A \rho_m \nabla^\mu\phi$). The interaction is described covariantly by

$$\nabla_\nu T_x^{\mu\nu} = Q_x^\mu, \quad Q_{(x)\mu} = Q_x u_{(a)\mu},$$

where $u_{(a)\mu}$ is the reference four-velocity. For perturbations, the energy and momentum transfer terms are captured by gauge-invariant variables $E_x$ and $F_x$ (energy and momentum exchange, respectively):

$$E_x = \epsilon_x - \frac{a\dot{Q}_x}{k Q_x} v_x, \quad F_x = f_x - \frac{Q_x (v_x - \bar{v})}{H \rho_x (1 + w_x)},$$

where $f_x$ is the momentum flux and $\bar{v}$ is the mean velocity.

These additional terms allow a consistent treatment of energy–momentum exchange at the perturbative level, ensuring both the correct evolution of inhomogeneities and compliance with general covariance (Potter et al., 2011).

3. Physical Realizations and Model Taxonomy

Several physically distinct models for dark sector interactions are analyzed:

1. Dark Matter Decay Model: Here, $Q_m = -A\rho_m$ leads to $\epsilon_m = \delta_m$ and $E_m = \Delta_m$, with momentum exchange computed for the dark energy field.
2. Dark Energy Decay Model: For $Q_\phi = -A\rho_\phi$, the structure is similar but with component roles reversed. However, this model is generally unstable as the effective kinetic term (parameterized by $x \propto \dot{\phi}$) can vanish, leading to divergent perturbations.
3. Scalar–Tensor (Derivative) Interaction: The coupling $Q_{(\phi)}^\mu = A \rho_m \nabla^\mu \phi$ introduces energy transfer terms entangling the dark matter and dark energy fluids. Stability requires careful sign choices for the coupling constant, ensuring energy flow from dark matter to dark energy.

The general class of interacting models can generate energy exchange through coupling functions dependent on the fields (scalar–tensor), simple decay terms, or even more complex functions including kinetic dependence (Potter et al., 2011).

4. Impact on Observables: Power Spectra and the ISW Effect

Interactions affect key cosmological observables:

• Matter Power Spectrum: Interactions can enhance the matter power spectrum on large scales ($k \lesssim 0.001~\text{Mpc}^{-1}$). In scalar–tensor models, tuning the coupling $A$ boosts power at these scales, while corrections at $k > 0.01~\text{Mpc}^{-1}$ remain at the percent level, much smaller than previous estimates using phenomenological constant-$w$ models.
• Dark Energy Power Spectrum: This spectrum remains subdominant to matter by several orders of magnitude, turns over at horizon scales ($$k \sim~ \text{a~few~}\times 10^{-4}~\text{Mpc}^{-1}$$), and decays at small scales.
• Integrated Sachs–Wolfe (ISW) Effect: The ISW effect is sensitive to the time evolution of the gravitational potential $\Phi$, itself determined by Poisson's equation:

$\Phi = -\frac{3}{2} \frac{(a H)^2}{k^2} \bar{Y},$

where $\bar{Y}$ is the weighted total energy density perturbation. In these frameworks, interactions modify the evolution of $\Delta_x$, and thereby $d\Phi/dz$, leading to an enhanced ISW effect in the lowest CMB multipoles. However, the gauge-invariant approach demonstrates that these enhancements are modest and only significant near horizon scales, where cosmic variance limits their detectability. This contrasts with more pronounced enhancements found in treatments neglecting energy–momentum exchange or dark energy perturbations (Potter et al., 2011).

5. Stability Analysis and Role of Momentum Exchange

A nontrivial interplay exists between the form of the interaction and the stability of cosmological perturbations:

• Instabilities: In dark energy decay models, as $x \rightarrow 0$, terms in the evolution equation (e.g., $dx/dN$) diverge, rendering the perturbation theory ill-defined. This issue is absent in the properly sign-chosen dark matter decay or scalar–tensor cases.
• Momentum Exchange: Without a corresponding momentum transfer (i.e., if $F_x = 0$), energy exchange alone fails to conserve total 4-momentum and results in inconsistent evolution of velocity perturbations and the gravitational potential. The inclusion of momentum transfer parameters is thus essential for theoretical consistency and for predicting correct modifications to the large-scale structure (Potter et al., 2011).

6. Comparison with Phenomenological and Previous Treatments

Earlier phenomenological models often adopted a simple constant-$w$ equation of state for dark energy and ignored both clustering of the scalar field and momentum transfer. These approximations:

• Overstate enhancements in the matter power spectrum and the ISW effect.
• Omit the possibility of instabilities in certain interaction schemes.
• Fail to capture sub-horizon scale corrections and subtle, field-induced clustering effects.

The gauge-invariant covariant formulation incorporates both energy–momentum exchange, clustering of the scalar field, and velocity perturbations, yielding a more accurate and subtle phenomenology, more controlled enhancements, and a clear diagnosis of possible pathologies (Potter et al., 2011).

7. Implications for Experimental Detection and Future Directions

The predictions of interacting dark sector frameworks carry implications for observational cosmology:

• Large-Scale Effects: Modest enhancements in the matter and ISW signals would be most noticeable at horizon and super-horizon scales—challenging to probe given cosmic variance.
• Small-Scale Robustness: Effects at $k > 0.01~\text{Mpc}^{-1}$ are small ($\lesssim$ a few percent), making model discrimination through galaxy surveys or weak lensing subtle without precision measurements.
• Stability Constraints: The necessity for consistent treatment of energy and momentum exchange and the avoidance of instability-prone interactions place strong theoretical constraints on viable models.
• Model Selection: Observational preference will depend on a combination of CMB low-$\ell$ power, galaxy clustering, lensing, and the growth of cosmic structure, requiring detailed comparison with high-precision surveys.

The rigorous approach presented sets a benchmark for future assessments of interacting dark sector models, relying on both theoretical consistency and careful confrontation with a spectrum of cosmological data (Potter et al., 2011).