Sign-Changeable Interaction Models in Cosmology

• Sign-changeable interaction models are phenomenological frameworks where the interaction term Q dynamically reverses sign, enabling energy flow direction to change during cosmic evolution.
• They employ diverse ansätze—such as q-weighted, running, and density-difference couplings—to model transitions from cosmic deceleration to acceleration in an FRW universe.
• Observational constraints and dynamical systems analyses indicate that, while these models can alleviate the coincidence problem, their stability and viability are highly model-dependent.

Sign-changeable interaction models are phenomenological frameworks in which the interaction term $Q$ between two sectors is allowed to reverse sign during evolution, so the direction of energy transfer is not fixed a priori. In cosmology this idea is used primarily for interacting dark matter–dark energy systems, often by multiplying $Q$ by the deceleration parameter $q$ or by promoting the coupling itself to a time-dependent function. The program was motivated in part by the result, emphasized in subsequent work, that the sign of the dark-sector interaction may have changed in the approximate redshift interval $0.45\lesssim z\lesssim 0.9$; it has since been developed in holographic, ghost, agegraphic, Chaplygin-gas, vacuum-energy, Tsallis-holographic, and Brans–Dicke settings (Wei, 2010, Zadeh et al., 2016).

1. Conceptual basis and covariant background structure

The common background is an FRW or FLRW universe in which dark matter and dark energy do not conserve separately but satisfy split continuity equations of the form

$$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$

or equivalent conventions with $Q$ entering with the opposite sign in the dark-energy equation. In the flat case this is combined with the Friedmann constraint $H^2\propto \rho_m+\rho_{de}$, while nonflat, anisotropic, and scalar–tensor extensions modify the constraint but retain the same basic interaction logic (Zadeh et al., 2016, Zadeh et al., 2017, Zadeh et al., 2018).

The defining device is that the sign of $Q$ is made dynamical. In the canonical $q$-weighted construction,

$q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$

and one takes $Q$0. Since $Q$1 in a decelerating era and $Q$2 in an accelerating era, $Q$3 changes sign when the universe passes through the deceleration-to-acceleration transition (Wei, 2010, Zadeh et al., 2017). This links the interaction directly to background kinematics rather than to a fixed-sign ansatz such as $Q$4.

A technical complication is that the sign convention for $Q$5 is not universal. Several papers adopt $Q$6 as energy flow from dark energy to dark matter, while vacuum-energy studies often define $Q$7 as energy flow from dark matter to dark energy (Zadeh et al., 2016, Guo et al., 2017). Accordingly, “sign change” is invariant, but the physical interpretation of a positive $Q$8 is paper-dependent.

2. Principal interaction ansätze

Several distinct constructions have been used to realize sign reversals. The most common are summarized below.

Class	Representative form	Sign-change trigger
$Q$9-weighted coupling	$q$0, $q$1	$q$2
Running coupling	$q$3 or $q$4, $q$5	$q$6
Density-difference coupling	$q$7, $q$8	$q$9 or $0.45\lesssim z\lesssim 0.9$0
Extended sign-variable coupling	$0.45\lesssim z\lesssim 0.9$1 with $0.45\lesssim z\lesssim 0.9$2 factors	Controlled by $0.45\lesssim z\lesssim 0.9$3, not locked to $0.45\lesssim z\lesssim 0.9$4
Nonlinear factorized coupling	$0.45\lesssim z\lesssim 0.9$5	Polynomial in $0.45\lesssim z\lesssim 0.9$6

The $0.45\lesssim z\lesssim 0.9$7-weighted form was proposed precisely because the familiar couplings $0.45\lesssim z\lesssim 0.9$8, $0.45\lesssim z\lesssim 0.9$9, and $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$0 have fixed sign once their parameters are chosen and therefore cannot accommodate a reversal during the full cosmic history (Wei, 2010). In this family, examples include $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$1 for holographic and ghost dark energy, $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$2 for agegraphic dark energy, and $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$3 in anisotropic Tsallis holographic models (Zadeh et al., 2016, Xu, 2015, Zadeh et al., 2019).

Running-coupling models shift the time dependence from $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$4 into the coupling itself,

$$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$5

and then use $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$6 or $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$7. These constructions are explicitly free of the assumption that $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$8 must be proportional to $$\dot{\rho}_m+3H\rho_m=Q,\qquad \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q,$$9 and the instantaneous dark-sector densities in the standard way, and they allow the interaction to cross zero even when the sign change is not synchronized with the acceleration transition (Li et al., 2011, Guo et al., 2017, Li et al., 13 Jan 2025).

A separate branch uses the relative dark-sector densities themselves. In

$Q$0

the coupling changes sign when $Q$1; in

$Q$2

the reversal occurs when $Q$3 (Pan et al., 2019). Extended sign-changeable interactions go further by liberating the reversal from both $Q$4 and the density ratio, introducing additional parameters such as $Q$5 and allowing the sign change and the acceleration transition to occur at different redshifts (Forte, 2013).

3. Realizations in dark-energy model building

In holographic dark energy, the interaction $Q$6 has been studied with the future event horizon, Hubble, and Granda–Oliveros cutoffs. The future-event-horizon and Granda–Oliveros versions were found to be in good agreement with observational data and to predict the transition from deceleration to acceleration around $Q$7. The Hubble-cutoff case is notable because the noninteracting model gives $Q$8 and cannot drive acceleration, whereas the sign-changeable interaction makes acceleration possible. Phantom-divide crossing is cutoff- and parameter-dependent: for the future event horizon it can occur for $Q$9, and for the Granda–Oliveros cutoff it is possible when the relevant parameter is below unity (Zadeh et al., 2016).

Ghost dark energy and generalized ghost dark energy exhibit a different pattern. With

$H^2\propto \rho_m+\rho_{de}$0

the late-time flat-universe behavior satisfies $H^2\propto \rho_m+\rho_{de}$1 while the equation-of-state parameter does not cross the phantom line, $H^2\propto \rho_m+\rho_{de}$2. The same non-phantom behavior persists in generalized ghost dark energy for both flat and nonflat universes, even though the models still admit a deceleration-to-acceleration transition around $H^2\propto \rho_m+\rho_{de}$3 (Zadeh et al., 2017).

Agegraphic dark energy with

$H^2\propto \rho_m+\rho_{de}$4

in a non-flat universe yields an accelerated scaling attractor. For the physically relevant attractor one needs $H^2\propto \rho_m+\rho_{de}$5, in which case the interaction reverses from dark matter $H^2\propto \rho_m+\rho_{de}$6 agegraphic dark energy at early times to agegraphic dark energy $H^2\propto \rho_m+\rho_{de}$7 dark matter at late times. The model was reported to alleviate the coincidence problem and to be consistent with the Union2.1 supernova sample (Xu, 2015).

Chaplygin-gas constructions show that sign-changeability is not uniformly benign. In phase-space analyses of interacting Chaplygin gas, fixed-sign linear and nonlinear interactions admit de Sitter and scaling late-time attractors, whereas sign-changeable linear interactions are more restrictive and the nonlinear sign-changeable examples studied there do not possess late-time attractors (Khurshudyan et al., 2015). In the reexamined generalized Chaplygin gas system, only

$H^2\propto \rho_m+\rho_{de}$8

was found to be physically acceptable for the full cosmological sequence; it possesses a matter-like point $H^2\propto \rho_m+\rho_{de}$9, a de Sitter attractor $Q$0, and an oscillatory interaction $Q$1 that tends to zero at late times (Xi et al., 2015).

Later extensions include sign-changeable interacting Tsallis holographic dark energy in Bianchi type I spacetime, where some cutoffs can be classically stable today but all studied models become classically unstable as $Q$2, and recent ghost-dark-energy models in Brans–Dicke cosmology with a logarithmic scalar field, where quintessence-like present and future behavior is typical, phantom-like behavior is possible for suitable parameters, and a second future transition back to deceleration can occur (Zadeh et al., 2019, Mehta et al., 2 Jan 2026).

4. Dynamical-systems structure, attractors, and stability

A major use of sign-changeable interactions is dynamical-systems analysis. One goal is to recover the standard cosmic sequence

$Q$3

while also producing a stable accelerated critical point. For a class of models $Q$4 and $Q$5, all studied cases admit a stable critical point corresponding to an accelerated phase. In the observationally fitted $Q$6 model, the best-fit coupling was reported as $Q$7, consistent with the existence conditions obtained from the phase-space analysis (Arevalo et al., 2022).

Scaling attractors are especially relevant because they keep the dark-matter and dark-energy densities comparable for longer and therefore alleviate the coincidence problem. This mechanism is explicit in non-flat agegraphic dark energy and in the viable generalized Chaplygin-gas model, where a heteroclinic orbit connects the matter-like regime to the de Sitter attractor (Xu, 2015, Xi et al., 2015).

Stability beyond background dynamics is model-dependent. In Brans–Dicke holographic dark energy, the squared sound speed

$Q$8

shows that the future-event-horizon cutoff can be stable for suitable parameters, the Granda–Oliveros cutoff is stable only in special parameter ranges, and the Ricci cutoff remains classically unstable for the studied ranges (Zadeh et al., 2018). In the anisotropic Tsallis holographic case, the general conclusion is that all cutoffs become classically unstable in the far future (Zadeh et al., 2019). By contrast, observational studies of density-difference interactions concluded that the corresponding sign-changeable models can produce stable perturbations at the linear level (Pan et al., 2019).

Thermodynamic consistency has also been analyzed. For sign-changeable models constrained against large-scale data, the generalized second law is always satisfied and the second derivative of the total entropy becomes negative at late times, implying approach to thermodynamic equilibrium (Pan et al., 2019). A Brans–Dicke ghost-dark-energy realization with $Q$9 likewise satisfies the generalized second law in its present interacting formulation (Mehta et al., 2 Jan 2026).

5. Observational constraints and phenomenology

The first observational motivation came from a model-independent analysis reporting that the sign of the interaction $q$0 changed in the approximate redshift range $q$1 (Wei, 2010). This motivated explicit fits of $q$2-weighted couplings to Union2 Type Ia supernovae, the WMAP7 CMB shift parameter, and the BAO distance parameter. In that framework the constraints on the coupling were found to be fairly tight. For example, the matter-coupled model $q$3 gave the best fit

$q$4

while the vacuum-coupled model $q$5 gave

$q$6

The sign-changeable models fit nearly as well as $q$7CDM, but $q$8CDM remained the best overall fit under $q$9, BIC, and AIC (Wei, 2010).

Running-coupling analyses using Union2, BAO, WMAP7, $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$0, and X-ray gas mass fraction data found $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$1 and $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$2, with the coupling crossing the noninteracting line around $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$3 at about $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$4 confidence level (Li et al., 2011). A later six-model I$q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$5CDM analysis with JLA supernovae, Planck 2015 distance priors, BAO, and a direct $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$6 measurement found $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$7 and $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$8 at around the $q=-\frac{\ddot a}{aH^2}=-1-\frac{\dot H}{H^2},$9 level for all six models, together with an extremely strong anti-correlation between $Q$00 and $Q$01; most reconstructions crossed sign around $Q$02, while one model crossed closer to $Q$03 (Guo et al., 2017).

More recent constraints with Planck+ACT CMB, first-year DESI BAO, and DES Year 5 supernovae studied

$Q$04

with $Q$05. In that analysis, $Q$06 crossed zero at the $Q$07 confidence level for the $Q$08, $Q$09, and $Q$10 cases, but not for the $Q$11 case. Bayesian evidence moderately preferred the $Q$12 and $Q$13 models over $Q$14CDM, with the latter identified as the best-performing model among those studied (Li et al., 13 Jan 2025).

Other observationally constrained sign-changeable models use density-difference couplings rather than running couplings. In these cases the data slightly favored non-zero interaction, but within $Q$15 confidence level the scenarios could not be distinguished from noninteracting cosmologies; the best-fit dark-energy equation of state lay in the phantom regime, the $Q$16 tension could be reconciled, and the $Q$17 tension persisted. Raw CMB TT and matter power spectra remained close to $Q$18CDM, but residual plots made the interaction traceable (Pan et al., 2019).

6. Interpretation, limitations, and broader generalizations

Several recurrent conclusions emerge. First, sign change does not imply a unique physical outcome. Some realizations suppress phantom behavior, as in ghost and generalized ghost dark energy where $Q$19 at late times; others permit phantom crossing only in restricted parameter ranges; still others, such as some event-horizon or Granda–Oliveros holographic models, cross the phantom divide more readily (Zadeh et al., 2017, Zadeh et al., 2016, Zadeh et al., 2018).

Second, the sign reversal need not be tied rigidly to the deceleration-to-acceleration transition. In the original $Q$20-weighted models $Q$21 at the acceleration transition by construction, but extended sign-changeable couplings were designed precisely to decouple the sign change from both $Q$22 and from a fixed density ratio. In fitted examples of these extended models, the sign change occurred either well before or after the onset of acceleration (Forte, 2013).

Third, dynamical viability is highly ansatz-dependent. Some sign-changeable models furnish accelerated scaling attractors, stable critical points, or perturbatively stable backgrounds, whereas others lose attractors or become classically unstable in the far future (Arevalo et al., 2022, Khurshudyan et al., 2015, Zadeh et al., 2019). This suggests that “sign-changeable interaction” is a structural property of the coupling, not a guarantee of phenomenological success.

Finally, the idea has analogues outside cosmology. In the signed simplicial contagion model, contagion proceeds through positive pairwise edges and balanced triangles, with the negative-edge fraction $Q$23 suppressing pairwise transmission linearly and group transmission nonlinearly through

$Q$24

The model exhibits discontinuous transitions, bistability, and hysteresis for strong group effects, but increasing the negative-edge fraction drives a shift from discontinuous to continuous phase transitions (Ma et al., 2024). This is not a dark-sector model, but it shows that sign-sensitive interactions can alter both the existence and the type of collective transitions in a mathematically controlled way.

Taken together, the literature treats sign-changeable interaction models as a flexible phenomenological language for systems in which the interaction direction is itself dynamical. In cosmology, the central question is whether the dark-sector energy flow changes sign near, before, or after the epoch when acceleration begins; current data permit such behavior in several constructions and, for some recent running-coupling models, even give moderate preference over $Q$25CDM, but the result remains strongly model-dependent (Li et al., 13 Jan 2025).