Cosmological Coupling: Mechanisms and Implications

• Cosmological coupling is a framework where local systems acquire scale-factor dependence in an expanding universe, modifying mass laws and local observables.
• Different approaches include phenomenological, geometric, and non-minimal interactions, each altering continuity equations and implying unique gravitational behavior.
• Observational studies, especially of black holes and dark sectors, constrain coupling strengths, offering insights into dark energy and the universe’s expansion history.

Cosmological coupling is a non-unique term in contemporary theoretical cosmology. In current usage it denotes, in different subliteratures, the possibility that a local gravitating system acquires scale-factor dependence when embedded in an expanding universe; an interaction between dark matter and dark energy or quintessence; a non-minimal coupling between matter, scalar fields, and curvature; or, in quantum-field settings, an explicit dependence of particle production or detector response on cosmological expansion. Across these usages, the common theme is that large-scale cosmological dynamics do not remain cleanly separated from local masses, local fields, or local observables. The literature therefore treats “cosmological coupling” less as a single theory than as a family of coupling mechanisms defined by the sector under consideration and by the mass, stress-energy, or interaction concept used to quantify the effect (Farrah et al., 2023).

1. Terminological scope and conceptual structure

Several distinct research programs use the same expression for different mechanisms. In the black-hole and compact-object literature, cosmological coupling usually means a scale-factor dependence of an effective mass law,

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$

or equivalently

$M(z=0)=M(z_i)(1+z_i)^k,$

with $k$ the coupling strength; the case $k=3$ is singled out because it yields a constant black-hole energy density and has therefore been proposed as a possible dark-energy source (Andrae et al., 2023). In dark-sector phenomenology, the term refers to an exchange current $Q$ between dark matter and dark energy, modifying the continuity equations of the two components and affecting both background evolution and perturbations (Pettorino et al., 2012). In modified-gravity work, cosmological coupling may instead mean a non-minimal coupling between curvature and matter, or field-dependent gravitational couplings $G(\xi)$ and $\Lambda(\xi)$, so that the effective strength of gravity evolves with cosmic time (Bertolami et al., 2017); (Sengupta, 25 Feb 2025).

The black-hole program further distinguishes between phenomenological and geometric notions of coupling. Phenomenological analyses treat $k$ as an empirical parameter inferred from mass-redshift data, while geometric analyses derive coupling from exact or quasi-local GR constructions, typically using the Misner–Sharp mass in spherically symmetric cosmological embeddings (Cadoni et al., 2023); (Cadoni et al., 2023). A related but distinct extension studies cosmological coupling of local gravitational systems such as galaxies, again through the Misner–Sharp mass and its scale-factor dependence (Cadoni et al., 2024).

The phrase also appears in quantum-field settings. One direction studies cosmological particle production in a strongly coupled plasma placed in an expanding Friedmann–Robertson–Walker (FRW) background, where particle production and entropy production become inseparable (Rangamani et al., 2015). Another introduces an “expansion-sensitive” detector coupling in de Sitter space by coupling an Unruh–DeWitt detector to a conformally rescaled field rather than to the original field, thereby building background dependence directly into the interaction Hamiltonian (Louko et al., 7 Sep 2025).

This multiplicity of meanings is a recurrent source of confusion. A common misconception is that all claims about “cosmological coupling” refer to black-hole mass growth. The literature does not support that identification. Instead, the term labels a broader class of proposals in which cosmological expansion modifies a local quantity, local interaction, or effective conservation law, but the mathematical implementation varies substantially across subfields.

2. Black holes and local gravitating systems

The most visible recent use of the term concerns black-hole mass growth in an expanding universe. A phenomenological form widely discussed in this context is

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$

with $k=0$ corresponding to decoupled Kerr-like evolution and $M(z=0)=M(z_i)(1+z_i)^k,$0 associated with vacuum-energy-like interiors in the phenomenological framework of supermassive black holes (SMBHs) in elliptical galaxies (Farrah et al., 2023). The same work reports

$M(z=0)=M(z_i)(1+z_i)^k,$1

with $M(z=0)=M(z_i)(1+z_i)^k,$2 excluded at $M(z=0)=M(z_i)(1+z_i)^k,$3 confidence in their SMBH sample over $M(z=0)=M(z_i)(1+z_i)^k,$4, and interprets $M(z=0)=M(z_i)(1+z_i)^k,$5 as implying an effectively constant cosmological energy density for the black-hole population (Farrah et al., 2023).

That interpretation is directly tied to the continuity equation. If individual masses scale as $M(z=0)=M(z_i)(1+z_i)^k,$6 while number density dilutes as $M(z=0)=M(z_i)(1+z_i)^k,$7, then the total density remains approximately constant, and stress-energy conservation implies $M(z=0)=M(z_i)(1+z_i)^k,$8, i.e. vacuum-energy behavior (Farrah et al., 2023). On this basis, the same analysis proposes stellar-remnant black holes as an astrophysical origin of dark energy and connects the onset of accelerated expansion to black-hole production from the cosmic star-formation history (Farrah et al., 2023).

This proposal is controversial in the observational literature. A direct test using the field binaries Gaia BH1 and Gaia BH2 assumes the same $M(z=0)=M(z_i)(1+z_i)^k,$9 law and infers the masses these $k$0 black holes would have had at formation from the companion-star ages (Andrae et al., 2023). For Gaia BH2, the paper gives a median age of $k$1 Gyr and a median inferred formation mass of $k$2; the probability that the formation mass was below $k$3 is $k$4. For Gaia BH1, the corresponding probability is $k$5. Assuming statistical independence, the probability that both original masses were above $k$6 is only $k$7 (Andrae et al., 2023). The paper therefore concludes that strong cosmological coupling with $k$8 is disfavored by these systems, while values $k$9 are not decisively excluded.

A separate GR analysis of nonsingular black holes reaches a different theoretical scaling. In that framework, singularity-free compact objects do cosmologically couple, but the leading contribution from the curvature term yields

$k=3$0

and the linear law is stated to be universal for spherically symmetric regular black holes (Cadoni et al., 2023). For horizonless regular compact objects there is an additional subleading model-dependent term, but the dominant scaling remains linear in $k=3$1 (Cadoni et al., 2023). Because $k=3$2 corresponds to an effective density scaling like $k=3$3, this framework concludes that cosmologically coupled nonsingular black holes in GR are unlikely to explain dark energy (Cadoni et al., 2023).

The program has also been extended to black holes immersed in dark halos. An exact analytical solution for a black hole in an anisotropic dark-sector background embedded in FLRW derives a radius-dependent coupling exponent,

$k=3$4

with explicit form

$k=3$5

for the specific seed metric used there (Wu et al., 23 Mar 2026). In that construction, mass growth is attributed not to modifying the black-hole interior equation of state, but to the anisotropic external halo profile and its response to the Hubble flow.

The same general question can be posed for noncompact local systems. For spherical astrophysical systems embedded in FLRW, a general Misner–Sharp mass formula is derived as

$k=3$6

with $k=3$7 the local coupling exponent at the object’s radius (Cadoni et al., 2024). Applying this to galaxies modeled with Navarro–Frenk–White and Einasto profiles, the paper concludes that galactic mass can be cosmologically coupled, but for realistic halos the growth is mild, monotonic, and typically sublinear in $k=3$8 (Cadoni et al., 2024).

3. Quasi-local mass, Misner–Sharp versus ADM, and the role of regularity

A central theoretical dispute concerns which notion of mass is appropriate in cosmological backgrounds. One line of argument states that cosmological coupling is controlled by whether the relevant energy is quasi-local or asymptotic. In spherically symmetric spacetimes the Misner–Sharp mass is

$k=3$9

or, in areal-radius gauge,

$Q$0

and is treated as the natural mass concept for non-asymptotically flat cosmological settings (Cadoni et al., 2023). By contrast, the ADM mass is a boundary quantity defined at infinity and is appropriate only when asymptotic flatness and scale separation are meaningful (Cadoni et al., 2023).

This distinction underlies a sharp criterion. Cosmological coupling occurs when the relevant mass is genuinely quasi-local and described by the Misner–Sharp mass; cosmological decoupling occurs when the Misner–Sharp mass is fully equivalent to the ADM mass outside the object (Cadoni et al., 2023). On this basis, singular black holes embedded in cosmological backgrounds are argued not to couple.

Two standard examples are emphasized. For Schwarzschild–de Sitter,

$Q$1

the Misner–Sharp mass is

$Q$2

The first term is the ADM mass $Q$3; the second is the cosmological background mass enclosed within radius $Q$4. There is no term in which $Q$5 itself grows with the scale factor (Cadoni et al., 2023). The same conclusion is reached for McVittie spacetime, where

$Q$6

and

$Q$7

Again, the central mass remains separate from the cosmological contribution (Cadoni et al., 2023).

The situation changes for regular or matter-supported compact objects. In the Sultana–Dyer solution, the Misner–Sharp mass is

$Q$8

so the first term grows linearly with the scale factor (Cadoni et al., 2023). For a broad anisotropic-fluid class with metric

$Q$9

the mass integrated over a sphere of radius $G(\xi)$0 becomes

$G(\xi)$1

with

$G(\xi)$2

and the paper interprets the leading coupling as a universal linear scaling with $G(\xi)$3 (Cadoni et al., 2023).

Weak-field systems introduce an additional condition. In the weak-field limit, cosmological coupling is found to be allowed only if there are pressure anisotropies (Cadoni et al., 2024). If the Newtonian Poisson equation holds, then one obtains

$G(\xi)$4

so that

$G(\xi)$5

The isotropic case $G(\xi)$6 then forces $G(\xi)$7, i.e. the trivial Newtonian limit with no coupling (Cadoni et al., 2024). This is an important restriction: within this framework, cosmological coupling is not generic for weak-field isotropic objects.

A plausible implication is that future observational support for nonzero cosmological coupling would discriminate not only between coupling models but also between underlying mass concepts and internal structures. One paper makes this explicit, concluding that observational evidence of cosmological coupling of astrophysical black holes would be the “smoking gun” of their nonsingular nature (Cadoni et al., 2023).

4. Dark-sector interactions and coupled cosmologies

In dark-energy phenomenology, cosmological coupling usually denotes energy-momentum exchange within the dark sector. A representative coupled dark energy model begins from

$G(\xi)$8

with interaction current

$G(\xi)$9

and, for a constant coupling,

$\Lambda(\xi)$0

The background continuity equations become

$\Lambda(\xi)$1

so the coupling directly alters how the two dark components dilute with expansion (Pettorino et al., 2012).

This class of models affects the cosmic microwave background through two main mechanisms: a peak shift to larger multipoles $\Lambda(\xi)$2 because the comoving distance to last scattering is changed, and a modified baryon-to-CDM ratio at decoupling because coupled dark matter dilutes faster than uncoupled CDM (Pettorino et al., 2012). Using WMAP7 and SPT data, the coupling strength is constrained to

$\Lambda(\xi)$3

and with HST, BAO, and SN Ia priors the bound strengthens to

$\Lambda(\xi)$4

A mild likelihood peak at $\Lambda(\xi)$5 is reported in the latter case, but remains compatible with zero at $\Lambda(\xi)$6 (Pettorino et al., 2012).

A related phenomenological program constrains the interaction directly from cosmography. For the modified continuity equations

$\Lambda(\xi)$7

three linear choices are analyzed: $\Lambda(\xi)$8 Using cosmographic parameters $\Lambda(\xi)$9, explicit expressions for $k$0 are derived, and the flat-universe estimates are found to be small in magnitude; for $k$1,

$k$2

(Bolotin et al., 2015). The same analysis finds common roots where the three interaction models become observationally degenerate (Bolotin et al., 2015).

Not all dark-sector couplings operate through energy exchange. A distinct model couples only a fraction of the cold dark matter to a massless scalar field through

$k$3

so that the coupled species feels an additional long-range force while the background dark energy remains a cosmological constant (Morris et al., 2013). The coupled density obeys a modified Einstein-frame scaling, but the quantity

$k$4

still scales as $k$5 (Morris et al., 2013). The paper concludes that for coupling of order

$k$6

more than half of the dark matter can be coupled without producing significant deviations from $k$7CDM in the CMB spectrum (Morris et al., 2013).

A more specialized case is momentum coupling between dark matter and quintessence. In the Type-3 model explored with $k$8-body simulations, the dark-matter continuity equation remains standard, but the Euler equation is modified. For the choice

$k$9

the subhorizon Euler equation becomes

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$0

with explicit coupling-dependent coefficients $M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$1 (Palma et al., 2023). Implemented in RAMSES, this coupling suppresses small-scale structure and enhances large-scale structure for sufficiently strong positive coupling, while also reducing halo inner densities and increasing velocity dispersions (Palma et al., 2023). The strongest case highlighted has, at $M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$2,

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$3

so the usual Hubble drag becomes a “cosmological push” (Palma et al., 2023).

Another dark-energy interaction program is motivated by cosmic age problems. With OHD and BAO constraints, a constant coupling between dark energy and dust matter,

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$4

is reported as the preferred model among those considered, with OHD+BAO best fit

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$5

and is described as giving a noticeably larger age of the universe than $M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$6CDM while avoiding the nonphysical density oscillations found in an alternative oscillatory-coupling model (Cao et al., 2021).

5. Geometry–matter couplings and varying gravitational couplings

Another major use of cosmological coupling refers to non-minimal interactions between matter, scalar fields, and curvature. One class of models introduces a scalar field with non-minimal kinetic coupling through

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$7

For a spatially flat FRW universe with constant potential $M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$8, the early-time limit is dominated by the coupling term and yields

$M(a)=M(a_i)\left(\frac{a}{a_i}\right)^k,$9

with the paper estimating

$k=0$0

for the end of the initial inflationary stage (Sushkov, 2012). The same model then passes to a matter-dominated epoch and later to $k=0$1-driven acceleration, with estimated present acceleration parameter $k=0$2 (Sushkov, 2012).

A different route to cosmological coupling employs a scalar-Euler form interaction in first-order gravity,

$k=0$3

In this framework the torsion equation,

$k=0$4

shows that a nonconstant scalar sources torsion, and the usual equivalence between first-order and second-order formalisms fails (Toloza et al., 2013). In FRW symmetry the torsion has two functions $k=0$5 and $k=0$6, and the modified Friedmann system can be rewritten in terms of effective torsion density and pressure,

$k=0$7

so torsion acts as an effective cosmological fluid (Toloza et al., 2013).

Non-minimal matter–curvature couplings have also been proposed as a sequestering mechanism for vacuum energy. In the action

$k=0$8

the matter stress tensor is no longer covariantly conserved,

$k=0$9

and in the asymptotic “relaxed regime”

$M(z=0)=M(z_i)(1+z_i)^k,$00

the curvature and matter trace become algebraically tied (Bertolami et al., 2017). The proposal is explicitly compared with unimodular gravity and Kaloper–Padilla sequestering, but is presented as a local matter–curvature mechanism rather than a global nonlocal one (Bertolami et al., 2017).

Generalized Rastall theory offers yet another coupling language. There the non-conservation law is

$M(z=0)=M(z_i)(1+z_i)^k,$01

and the field equations read

$M(z=0)=M(z_i)(1+z_i)^k,$02

Through the Newtonian-limit relation

$M(z=0)=M(z_i)(1+z_i)^k,$03

a time-dependent $M(z=0)=M(z_i)(1+z_i)^k,$04 induces a time-dependent $M(z=0)=M(z_i)(1+z_i)^k,$05 (Moradpour et al., 2021). The paper emphasizes reconstruction in both directions: a chosen $M(z=0)=M(z_i)(1+z_i)^k,$06 determines the coupling, and a chosen coupling determines $M(z=0)=M(z_i)(1+z_i)^k,$07. One model with

$M(z=0)=M(z_i)(1+z_i)^k,$08

is stated to allow both late-time acceleration and a Dirac-compatible decreasing $M(z=0)=M(z_i)(1+z_i)^k,$09, with transition redshift $M(z=0)=M(z_i)(1+z_i)^k,$10 (Moradpour et al., 2021).

Field-dependent couplings can also be incorporated directly into the gravity action via a scalar $M(z=0)=M(z_i)(1+z_i)^k,$11, with

$M(z=0)=M(z_i)(1+z_i)^k,$12

For FLRW power-law solutions

$M(z=0)=M(z_i)(1+z_i)^k,$13

the model gives

$M(z=0)=M(z_i)(1+z_i)^k,$14

with $M(z=0)=M(z_i)(1+z_i)^k,$15 for $M(z=0)=M(z_i)(1+z_i)^k,$16, implying $M(z=0)=M(z_i)(1+z_i)^k,$17 in the solution class considered (Sengupta, 25 Feb 2025). Using supernova cosmology, the paper states that present data prefer a slowly growing Newton coupling and rule out Dirac’s large-number hypothesis, with bounds

$M(z=0)=M(z_i)(1+z_i)^k,$18

and

$M(z=0)=M(z_i)(1+z_i)^k,$19

as quoted there (Sengupta, 25 Feb 2025).

A further action-based construction starts from Brown’s variational formulation of a relativistic fluid and couples a perfect fluid, a scalar field, and the boundary pseudovector $M(z=0)=M(z_i)(1+z_i)^k,$20. Among the interaction terms considered, the derivative coupling

$M(z=0)=M(z_i)(1+z_i)^k,$21

is identified as the cosmologically relevant case (Boehmer et al., 2024). In flat FLRW it leads to

$M(z=0)=M(z_i)(1+z_i)^k,$22

$M(z=0)=M(z_i)(1+z_i)^k,$23

while still preserving the standard fluid continuity equation $M(z=0)=M(z_i)(1+z_i)^k,$24 (Boehmer et al., 2024). The constant-coupling version mimics interacting dark-energy models, whereas non-constant couplings produce a richer two-dimensional phase space with scaling and accelerated attractors (Boehmer et al., 2024).

6. Quantum and strong-coupling realizations

In quantum-field theory on cosmological backgrounds, cosmological coupling need not refer to mass growth or dark-sector exchange. One example is the dynamics of a strongly coupled confining gauge theory placed on

$M(z=0)=M(z_i)(1+z_i)^k,$25

with

$M(z=0)=M(z_i)(1+z_i)^k,$26

so that the system undergoes a homogeneous isotropic expansion over timescale $M(z=0)=M(z_i)(1+z_i)^k,$27 (Rangamani et al., 2015). Using holography, the theory is dual to a classical gravitational problem in asymptotically $M(z=0)=M(z_i)(1+z_i)^k,$28, and the renormalized boundary stress tensor yields the energy density, pressure, and entropy (Rangamani et al., 2015).

A central result is that at strong coupling particle production cannot be separated from entropy production. In the hydrodynamic regime,

$M(z=0)=M(z_i)(1+z_i)^k,$29

the stress tensor is well described by first-order fluid/gravity hydrodynamics, and temperature obeys

$M(z=0)=M(z_i)(1+z_i)^k,$30

The right-hand side is a positive viscous correction to adiabatic cooling, so expansion produces entropy even in the hydrodynamic regime (Rangamani et al., 2015). At large expansion amplitude the final energy density becomes nearly independent of the initial temperature, which the paper interprets as rapid loss of memory of the initial state; this contrasts sharply with weak coupling, where Bogoliubov-produced particles retain more direct information about the initial occupation numbers (Rangamani et al., 2015).

A different quantum realization modifies the detector–field interaction itself. In de Sitter FLRW with compact spatial sections, a comoving Unruh–DeWitt detector is compared under conventional coupling to $M(z=0)=M(z_i)(1+z_i)^k,$31 and a novel coupling to the conformal field

$M(z=0)=M(z_i)(1+z_i)^k,$32

The novel interaction Hamiltonian is

$M(z=0)=M(z_i)(1+z_i)^k,$33

so the coupling explicitly incorporates the cosmological expansion (Louko et al., 7 Sep 2025). For detectors coupled to a single mode in the long-interaction limit, the responses are

$M(z=0)=M(z_i)(1+z_i)^k,$34

$M(z=0)=M(z_i)(1+z_i)^k,$35

both satisfying detailed balance at the Gibbons–Hawking temperature

$M(z=0)=M(z_i)(1+z_i)^k,$36

(Louko et al., 7 Sep 2025). The paper finds that the expansion-sensitive coupling tends to enhance de-excitation peaks, particularly for single-mode detectors (Louko et al., 7 Sep 2025).

These quantum examples broaden the meaning of cosmological coupling. In one case, the background couples irreversibly to a strongly interacting plasma through entropy-generating dynamics; in the other, the detector coupling itself is modified to track the expansion. This suggests that “cosmological coupling” can denote either dynamical response to expansion or a deliberate insertion of expansion dependence into a local interaction law.

7. Observational status, controversies, and open structure

The observational situation is mixed and heavily model-dependent. The strongest empirical claim in favor of black-hole cosmological coupling is the SMBH analysis of elliptical galaxies, which reports evidence against $M(z=0)=M(z_i)(1+z_i)^k,$37 and a preferred value near $M(z=0)=M(z_i)(1+z_i)^k,$38 (Farrah et al., 2023). However, analyses based on stellar-mass black holes in Gaia BH1 and BH2 argue that the same strong-coupling law implies sub-$M(z=0)=M(z_i)(1+z_i)^k,$39 formation masses with high probability and therefore substantially tensions standard stellar-collapse expectations (Andrae et al., 2023). Within GR-based nonsingular-black-hole models, the theoretically preferred scaling is instead $M(z=0)=M(z_i)(1+z_i)^k,$40, not $M(z=0)=M(z_i)(1+z_i)^k,$41, and the authors explicitly conclude that such objects are unlikely to source dark energy (Cadoni et al., 2023).

The theoretical controversy is therefore not merely empirical. It concerns what mass concept is being fit, what class of interior or exterior solutions is admissible, and whether singular or nonsingular compact objects are under discussion. One side treats $M(z=0)=M(z_i)(1+z_i)^k,$42 as a largely phenomenological index inferred from mass-redshift data (Farrah et al., 2023); another argues that, within general relativity, singular black holes generically decouple while regular compact objects couple quasi-locally through the Misner–Sharp mass (Cadoni et al., 2023). Yet another extension locates the coupling in the anisotropic dark halo surrounding a black hole, making the exponent radius-dependent rather than constant (Wu et al., 23 Mar 2026).

Dark-sector couplings are less controversial conceptually, because they are framed directly as interacting-fluid or scalar-field models, but they remain tightly constrained. Constant coupled dark energy is bounded at $M(z=0)=M(z_i)(1+z_i)^k,$43 when HST+BAO+SN Ia are added to WMAP7+SPT (Pettorino et al., 2012). Momentum coupling between dark matter and quintessence produces sizable nonlinear effects in simulations, but the strongest regimes approach the singular limit

$M(z=0)=M(z_i)(1+z_i)^k,$44

which the simulation paper flags as a potential viability issue (Palma et al., 2023).

Modified-gravity and varying-coupling models face a different challenge: many are cosmologically viable only in restricted regimes or under specific ansätze. The non-minimal matter–curvature sequestering proposal is explicitly stated to apply to a late-time relaxed attractor, not to local astrophysical systems (Bertolami et al., 2017). The varying-coupling gravity-action model derives its bounds within a power-law ansatz for $M(z=0)=M(z_i)(1+z_i)^k,$45 and $M(z=0)=M(z_i)(1+z_i)^k,$46 (Sengupta, 25 Feb 2025). The generalized Rastall analysis likewise presents reconstruction rather than a unique underlying microphysical mechanism (Moradpour et al., 2021).

A plausible synthesis is that cosmological coupling is best understood as a diagnostic umbrella rather than a settled theory class. It diagnoses the failure of strict decoupling between local and cosmological sectors, but the underlying cause may be quasi-local mass structure, stress anisotropy, dark-sector exchange, non-minimal geometric interaction, or explicitly expansion-dependent quantum coupling. The present literature supports no single universal form. Instead, it supports a structured taxonomy in which the meaning of coupling depends on the sector, the observables, and the mass or conservation concept chosen.