Dark Matter–Dark Energy Interactions

• Dark matter–dark energy interactions are hypothesized non-gravitational couplings that affect cosmic expansion and address issues like the coincidence problem.
• Models incorporate an energy–momentum exchange term into standard cosmological equations using various functional forms and stability criteria.
• Observational constraints from CMB, BAO, and SNe Ia refine coupling parameters, influencing measures such as H₀ and structure growth rates.

Dark matter–dark energy (DM–DE) interactions are hypothesized non-gravitational couplings between the two dominant components of the cosmic energy budget. Such models have been developed to address theoretical puzzles like the coincidence problem and to alleviate empirical tensions in cosmology, such as those related to the Hubble constant ($H_0$) and the amplitude of structure growth ($S_8$ and $\sigma_8$). The literature has established a landscape of phenomenological, field-theoretical, and simulation-calibrated frameworks to study these interactions, with increasing observational constraints.

1. Theoretical Frameworks and Interaction Forms

Models of DM–DE interactions are constructed by modifying the standard cosmological conservation equations to include a direct energy–momentum exchange term $Q$: $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$ where $\rho_c$ and $\rho_{\text{de}}$ are the dark matter and dark energy densities, $w$ is the DE equation of state, and $Q$ encodes the interaction rate (Zimdahl, 2012, Wang et al., 2024).

Canonical functional forms for $Q$ found in the literature include:

• $S_8$0 (proportional to DM density)
• $S_8$1 (proportional to DE density)
• $S_8$2 (linear in both densities)
• $S_8$3 (bilinear, generalized) Field-theoretic models often generate $S_8$4 via a coupling between a quintessence field and DM, leading to terms like $S_8$5 or more complex structures involving both conformal and disformal couplings (Bruck et al., 2017, Wang et al., 2024).

Stability considerations dictate the sign and allowed magnitude of $S_8$6 for a given $S_8$7 (typically $S_8$8 for accelerated expansion). For example, stability of early-time perturbations for $S_8$9 with $\sigma_8$0 requires $\sigma_8$1 (Lucca et al., 2020).

2. Dynamical Implications and Motivations

Non-gravitational coupling between DM and DE profoundly modifies cosmic expansion dynamics. A key motivation is the “coincidence problem”—the unexplained near-equality of DM and DE densities today despite their different redshift scalings in $\sigma_8$2CDM. Many interaction models produce a scaling solution $\sigma_8$3, with $\sigma_8$4 slowing the decay of $\sigma_8$5 and potentially yielding $\sigma_8$6 over cosmological timescales (Zimdahl, 2012, Wang et al., 2016).

In models like $\sigma_8$7, the energy flow at late times can raise the Hubble rate inferred from CMB, bringing it closer to local $\sigma_8$8 measurements and thus partially alleviating the $\sigma_8$9 tension (Lucca et al., 2020, Wang et al., 2024). Conversely, $Q$0 tends to suppress the amplitude of structure growth at late times, addressing the $Q$1 or $Q$2 tension between CMB and weak-lensing/galaxy-clustering data (Lucca, 2021, Wang et al., 2024).

For specific interaction forms:

• Bilinear models $Q$3 can be motivated by analogy with decay and annihilation rates in particle physics, and their parameter space can interpolate between various phenomenological scenarios (Cheng et al., 2019).
• Field-theoretic constructions, including Chameleon-type and axion-monodromy models, can naturally produce late-time DM–DE coupling through the dependence of DM mass or couplings on scalar fields (Bruck et al., 2017, D'Amico et al., 2016, Chakraborty et al., 2024).

3. Linear Perturbations, Stability, and Structure Growth

DM–DE coupling modifies not only background dynamics but also the evolution of cosmic perturbations. In synchronous or Newtonian gauge, additional source and friction terms proportional to $Q$4 appear in the equations for DM and DE density contrasts ($Q$5, $Q$6) and velocity divergences ($Q$7, $Q$8) (Lucca et al., 2020, Cheng et al., 2019, Lucca, 2021): $Q$9 Correctly specifying the perturbative $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$0 is crucial for avoiding non-adiabatic and large-scale instabilities, especially at early times (Tamanini, 2015, Cheng et al., 2019).

Phenomenologically, $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$1 (DE $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$2 DM) can suppress the late-time growth rate $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$3, aligning theory with the observed low clustering amplitude; $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$4 (DM $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$5 DE) may instead enhance growth unless tightly constrained. Best-fit values of parameters like $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$6 in interacting models typically show improved compatibility with redshift-space distortion and weak-lensing datasets (Cheng et al., 2019, Lucca, 2021, Rezaei, 2020).

4. Observational Constraints and Model Selection

Cosmological constraints on DM–DE interactions are derived from multi-probe analyses employing:

• CMB anisotropies (Planck 2015/2018, WMAP9)
• BAO (6dFGS, SDSS/BOSS/CMASS/DR14, DESI DR2)
• SNe Ia (Pantheon, Pantheon+, JLA, Union)
• Growth/RSD ($$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$7)
• Local $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$8 direct measurements (SH0ES/R19)
• Cosmic chronometers, lookback time, lensing, and cluster counts

Representative model constraints include:

• For $$\dot\rho_c + 3H\rho_c = +Q, \qquad \dot\rho_{\text{de}} + 3H(1+w)\rho_{\text{de}} = -Q,$$9, joint CMB+R19+BAO+Pantheon fits favor $\rho_c$0 with $\rho_c$1 km/s/Mpc, reducing the Hubble tension to $\rho_c$2 (Lucca et al., 2020).
• For $\rho_c$3, $\rho_c$4 is obtained from CMB+BAO+SNIa, with the 1D PDF of cosmic void ellipticity shifting at the $\rho_c$5 level (Rezaei, 2020).
• Power-law interaction forms $\rho_c$6 yield best-fit $\rho_c$7, $\rho_c$8 for $\rho_c$9 (Cheng et al., 2019).
• Direct field-theoretic analyses (Chameleon, conformal, and disformal couplings) report best-fit $\rho_{\text{de}}$0 at $\rho_{\text{de}}$1 when DESI DR2 and SH0ES are included, with preference driven primarily by the late ISW effect (Chakraborty et al., 2024).
• Current constraints generally require $\rho_{\text{de}}$2–$\rho_{\text{de}}$3 at 95% C.L. for most phenomenological forms (Wang et al., 2024, Nunes et al., 2014, Bruck et al., 2017).

Statistical evidence for or against interaction is mixed. While some datasets or combinations can provide $\rho_{\text{de}}$4–$\rho_{\text{de}}$5 preference for nonzero coupling when including local $\rho_{\text{de}}$6 or specific weak-lensing datasets, combined global analyses using the full array of data typically yield inconclusive or weak (statistically insignificant) evidence, with Bayesian model comparison failing to favor interacting models over $\rho_{\text{de}}$7CDM (Lucca et al., 2020, Lucca, 2021, Wang et al., 2024).

Model/Class	Key Coupling(s)	Best-fit/Constraint	Tension Addressed
$\rho_{\text{de}}$8	$\rho_{\text{de}}$9	$w$0 (all probes)	$w$1
$w$2	$w$3	$w$4–$w$5, $w$6–$w$7 (phantom)	$w$8, $w$9
Chameleon/Conformal	$Q$0	$Q$1	ISW anomaly, $Q$2
Power-law ($Q$3)	$Q$4	$Q$5, $Q$6	Early/late $Q$7

5. Nonlinear Regime, Simulations, and Physical Novelty

Large-scale N-body simulations for interacting models require careful modifications:

• Time-dependent DM mass evolution
• Drag forces from energy–momentum exchange
• Modified Poisson equation (fifth-force terms)
• Clustering signatures: changes in halo mass function, power spectrum, and void ellipticity

These simulations have shown:

• Modest changes in the high-mass end of the halo mass function for $Q$8 (Wang et al., 2024)
• Matter power spectrum modifications with non-trivial scale dependence, requiring customized Halofit-like prescriptions or full N-body to match nonlinear observables

More speculative work has proposed that mutual DM–DE interactions may exhibit chaotic attractor behavior under general thermodynamic open-system principles, associated with the emergence of novel macroscopic order parameters and possibly observable fractal-like properties in cosmological structure (Aydiner, 2023).

6. Advanced Models and Fundamental Constraints

Field-theory-based models allow for a theoretically consistent embedding of DM–DE couplings:

• Chameleon models use conformal coupling of DM to a scalar field $Q$9, with action $Q$0, motivated by string compactifications. Current $Q$1 constraints are compatible with a fifth force exclusively in the dark sector (Chakraborty et al., 2024).
• Monodromy, axion, and multi-scalar constructions provide radiatively stable mechanisms for mixing DE and ultra-light axion DM, with associated signatures in $Q$2 oscillations and structure growth (D'Amico et al., 2016).
• “Holographic” and Ricci-scalar-cutoff DE models exploit connections between IR physics and the vacuum energy, introducing further covariant interaction possibilities (Forte et al., 2012, Micheletti, 2010, 0901.1215).

First-principles kinetic theory, e.g., via Boltzmann equations for a Yukawa coupling between DM and DE scalars, predicts physically allowed microphysical $Q$3 to be vastly smaller than phenomenological fits ($Q$4–$Q$5 eV$Q$6) (Ludwick et al., 2019), suggesting that empirically meaningful interactions are likely emergent or effective rather than fundamental.

7. Current Status, Limitations, and Future Prospects

Current cosmological data provide model-dependent upper bounds on DM–DE couplings at the $Q$7–$Q$8 level, with no robust global detection across all datasets; some specific extensions or data combinations suggest mild preference but remain statistically inconclusive (Lucca et al., 2020, Bruck et al., 2017, Wang et al., 2024, Chakraborty et al., 2024). Constraints degrade rapidly if only background expansion is used—growth and lensing data are crucial.

Major open difficulties include:

• Maintaining stability and avoiding non-adiabatic instabilities at the perturbative level (Tamanini, 2015)
• Ambiguity in defining the energy–momentum transfer at both background and perturbation order, unless a Lagrangian approach is specified
• Non-trivial degeneracies with other cosmological parameters and the broad phenomenological flexibility permitted by current precision

Forthcoming probes will decisively test much of the allowed parameter space:

• ESA/Euclid, LSST, DESI for weak lensing, large-scale structure, and BAO
• CMB Stage-4 (lensing, ISW)
• 21cm intensity mapping, gravitational wave standard sirens, and fast radio bursts for geometry–growth combination

The minimum coupling viable with $Q$9CDM is expected to be probed at the $S_8$00 level, with nonlinear clustering statistics and cosmic void shapes providing independent constraints (Rezaei, 2020). Laboratory and astrophysical searches for fifth forces in the dark sector may become relevant if evidence continues to mount for nonzero DM–DE interaction (Chakraborty et al., 2024). The ongoing search for theoretical consistency, data-driven phenomenology, and robust observable signatures remains central in the field.

---

Key references:

(Lucca et al., 2020, Wang et al., 2024, Cheng et al., 2019, Nunes et al., 2014, Rezaei, 2020, Zimdahl, 2012, Richarte et al., 2015, Lucca, 2021, Bruck et al., 2017, Micheletti, 2010, 0901.1215, Chakraborty et al., 2024, Aydiner, 2023, Wang et al., 2016, Tamanini, 2015, Forte et al., 2012, D'Amico et al., 2016, Ludwick et al., 2019, Ashmita et al., 2024).