Pure Disformal Coupling in Cosmology

• Pure disformal coupling is defined by a fixed conformal factor with nontrivial derivative-dependent modifications of the metric.
• It plays a crucial role in dark-sector cosmology, influencing dark energy dynamics, dark matter interactions, and the effective sound speed in fluctuations.
• The mechanism appears in various frameworks—from inflation and reheating to collider EFTs and CMB analyses—yielding distinctive observable signatures.

Searching arXiv for recent and foundational papers on pure disformal coupling. arxiv_search(query="\"pure disformal coupling\" disformal dark sector cosmology", max_results=10, sort_by="relevance") Pure disformal coupling denotes a class of metric-mediated interactions in which the physically coupled metric differs from a reference metric only through derivative-dependent terms aligned with scalar gradients, while the conformal factor is trivial, constant, or otherwise dynamically inessential. In the standard Bekenstein form,

$$\tilde g_{\mu\nu}=C(\phi,X)\,g_{\mu\nu}+D(\phi,X)\,\partial_\mu\phi\,\partial_\nu\phi,$$

the pure disformal limit places the nontrivial interaction entirely in $D$, with no scalar-dependent conformal rescaling. Across recent literature this notion appears in dark-sector EFTs, CMB/radiation couplings, inflation and preheating, collider EFTs, and consistency analyses with fermions. The common core is that the leading observable effects arise from derivative deformation of the effective metric and hence of causal structure, kinetic mixing, and matter propagation, rather than from ordinary conformal rescaling (Bansal et al., 23 Aug 2025, Bruck et al., 2013).

1. Formal definition and frame structure

A common definition of pure disformal coupling sets the conformal factor to a constant and retains a nontrivial disformal factor. In the dark-sector construction of "Disformal interactions in the Dark Sector: From driving Early Dark Energy to confronting cosmological tensions" (Bansal et al., 23 Aug 2025), the defining subclasses are

$C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$

with the simplest representative

$C=c_0,\qquad D=d_0=\text{const}.$

Because $C$ is constant, all interaction effects are genuinely disformal and there is no scalar-dependent metric rescaling. The same logic appears in earlier radiation-sector work, where pure disformal coupling is implemented by setting $C_i(\phi)=1$ and $D_i(\phi)=M_i^{-4}$ for each species $i$, so that photons and matter can couple to different derivative-deformed metrics (Bruck et al., 2013).

The definition is not fully uniform across subfields. In collider EFT treatments, the pure disformal limit is written directly as

$$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$

with no conformal factor at all, leading at lowest order to the universal dimension-8 operator

$$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$

for SM matter (Brax et al., 2015). In multi-field fermionic analyses, the phrase instead refers to matter coupled minimally to a metric whose nontrivial structure is entirely encoded in derivative-dependent tetrad deformations, with any conformal factor either trivial or absorbable (Domènech et al., 8 Oct 2025). This suggests that “pure disformal” is best understood as a structural criterion—derivative metric deformation as the sole source of nontrivial coupling—rather than a single universal parametrization.

The frame map is subject to nontrivial consistency conditions. In the dark-sector scalar-tensor construction, invertibility and Lorentzian signature require

$D$0

These conditions ensure that the disformal transformation is well defined and that the transformed metric preserves causal structure (Bansal et al., 23 Aug 2025).

2. Field-theoretic realizations and matter sectors

In dark-sector EFT realizations, pure disformal coupling is obtained by starting from a two-scalar action in a tilde frame and recasting it by a disformal transformation into Einstein frame. The resulting theory has standard Einstein gravity, a DE scalar $D$1, and a DM scalar $D$2 whose kinetic sector is non-minimally coupled to $D$3 through the transformed scalar

$D$4

In this representation the interaction is not an explicit potential term but a non-gravitational DE–DM coupling induced by the metric seen by the DM sector (Bansal et al., 23 Aug 2025).

In brane constructions, pure disformal coupling arises geometrically from the induced metric on a fluctuating brane. Expanding the induced metric in branon fields $D$5 produces

$D$6

and the leading interaction with SM matter becomes

$D$7

In that setting the derivative structure is protected by shift symmetry and brane parity, and odd-branon couplings are absent, so branons are produced and annihilate in pairs and can act as stable DM candidates (Cembranos et al., 2016).

For fermions, the situation is more restrictive. A generic disformal tetrad transformation maps the minimally coupled Dirac action into an action with a modified kinetic term, a conformal mass rescaling, and a universal axial-current coupling,

$D$8

Because $D$9 generically contains higher derivatives of the disformal fields, ghost-freedom requires strong degeneracy conditions. For two-scalar disformal transformations, the consistent branch forces the field-space metric in the disformal sector to be degenerate, so that the metric deformation is effectively along a single direction $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$0 (Domènech et al., 8 Oct 2025). A plausible implication is that bosonic pure disformal constructions do not automatically extend to consistent fermionic matter sectors without additional degeneracy structure.

3. Dark-sector cosmology: interacting Early Dark Sector and late-time acceleration

In the pure disformal dark-sector model with

$C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$1

the scalar equation simplifies to

$C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$2

At early times, when $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$3 is large, suitable $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$4 gives $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$5. The inertial prefactor becomes large, $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$6 is suppressed, and the field enters a regime with approximately constant $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$7, hence approximately constant kinetic energy $C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$8. The DE density then behaves as

$C(\phi,X)=c_0=\text{const},\qquad D(\phi,X)\neq 0,$9

yielding a kinetic-driven cosmological-constant-like plateau that mimics EDE without requiring a tuned freezing potential. As DM dilutes, $C=c_0,\qquad D=d_0=\text{const}.$0 falls, the equation tends back to Klein–Gordon form, and if the potential is still subdominant the kinetic energy redshifts as

$C=c_0,\qquad D=d_0=\text{const}.$1

At later times the potential $C=c_0,\qquad D=d_0=\text{const}.$2 can dominate and drive $C=c_0,\qquad D=d_0=\text{const}.$3. The resulting scaling pattern is

$C=c_0,\qquad D=d_0=\text{const}.$4

which unifies an early interacting dark component and late-time DE within one scalar sector (Bansal et al., 23 Aug 2025).

This mechanism differs sharply from conventional scalar-field EDE. In standard EDE, the early plateau is usually engineered by the shape of the potential and by Hubble freezing; in the pure disformal case, the plateau is a consequence of the DE–DM interaction itself, and the exit is triggered by the natural dilution of DM rather than by a specially designed potential. The paper identifies this as a “more fundamental and less ad hoc” route to EDE-like behavior (Bansal et al., 23 Aug 2025).

The background effect on pre-recombination expansion depends on the transition redshift $C=c_0,\qquad D=d_0=\text{const}.$5. For $C=c_0,\qquad D=d_0=\text{const}.$6, the early plateau increases $C=c_0,\qquad D=d_0=\text{const}.$7 before recombination, reduces the sound horizon $C=c_0,\qquad D=d_0=\text{const}.$8, and allows a larger inferred $C=c_0,\qquad D=d_0=\text{const}.$9. For $C$0, the model instead lowers $C$1. The same setup modifies growth: for smaller $C$2, matter perturbations are suppressed and $C$3 decreases, while larger $C$4 can enhance small-scale growth. In the CMB TT spectrum, Model M3 produces strong suppression at low multipoles, significant for $C$5, with only modest changes near the acoustic peaks; the paper notes that Planck 2018 shows a low-$C$6 anomaly in which the best-fit amplitude of the TT spectrum at $C$7 is about $C$8 lower than at high $C$9, and the pure disformal model yields a suppression of similar type (Bansal et al., 23 Aug 2025).

4. Radiation, CMB observables, and the fine-structure constant

For radiation, pure disformal coupling is most often studied in a species-dependent form,

$C_i(\phi)=1$0

with distinct scales $C_i(\phi)=1$1 and $C_i(\phi)=1$2 for photons and matter. In an FRW background, the Jordan-frame and Einstein-frame radiation descriptions no longer coincide. For a single disformally coupled fluid,

$C_i(\phi)=1$3

so the equation-of-state parameter becomes frame dependent: $C_i(\phi)=1$4 The photon distribution function is also not frame invariant; instead of $C_i(\phi)=1$5, one finds

$C_i(\phi)=1$6

This yields an effective collision term in the Einstein-frame Boltzmann equation and implies that the observed CMB need not remain a perfect blackbody in that frame (Bruck et al., 2013).

Observable CMB effects depend crucially on whether photons and baryons couple identically. The measured speed of light in the matter frame is

$C_i(\phi)=1$7

and the effective chemical potential in the observed photon spectrum is

$C_i(\phi)=1$8

Hence $C_i(\phi)=1$9-distortions arise only if $D_i(\phi)=M_i^{-4}$0. The same paper reports that current bounds from COBE/FIRAS,

$D_i(\phi)=M_i^{-4}$1

exclude regions of the $D_i(\phi)=M_i^{-4}$2 plane, and that these bounds are generally stronger than those from the temperature–redshift relation $D_i(\phi)=M_i^{-4}$3 (Bruck et al., 2013).

A related but distinct line of work studies disformal modifications of electromagnetism and $D_i(\phi)=M_i^{-4}$4. When matter and radiation couple to different disformal metrics and the electromagnetic sector also has a coupling $D_i(\phi)=M_i^{-4}$5, the effective fine-structure constant scales as

$D_i(\phi)=M_i^{-4}$6

This makes clear that a varying $D_i(\phi)=M_i^{-4}$7 can arise from disformal couplings alone if $D_i(\phi)=M_i^{-4}$8 but $D_i(\phi)=M_i^{-4}$9, whereas if matter and photons couple in the same way, $i$0 and disformal terms by themselves do not vary $i$1. The same framework modifies distance duality,

$i$2

and therefore connects $i$3-variation, varying photon speed, and reciprocity violation in a single formalism (Bruck et al., 2015, Brax et al., 2013).

5. Inflation, reheating, and derivative transfer of energy

In two-field inflation with a disformally coupled spectator or matter field, the Einstein-frame action contains a derivative mixing term

$i$4

in addition to the modified $i$5 kinetic prefactor $i$6, where

$i$7

The pure disformal limit corresponds to $i$8 and nontrivial $i$9. In the inflationary model of "Disformally coupled inflation" (Bruck et al., 2015), the canonical perturbation sector is reorganized into two propagating modes with sound speeds

$$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$0

For the parameter regimes studied there are no superluminal or unstable scalar fluctuation modes, and a generic prediction is low tensor-to-scalar ratio. The constant-coupling regime $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$1, $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$2 is especially close to a pure disformal limit, with very small sound speeds and extremely low $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$3 in the numerical examples (Bruck et al., 2015).

Pure disformal derivative couplings also modify preheating. In "Preheating in an inflationary model with disformal coupling" (Karwan et al., 2017), neglecting backreaction reduces the inflaton background to the standard oscillatory solution, but the matter-field mode equation inherits disformal kinetic structure and can be brought to Mathieu form. In the pure disformal case ($$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$4, meaning constant $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$5 and nontrivial $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$6), the resonance parameter is

$$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$7

Unlike in the mixed conformal–disformal case, this $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$8 grows with time. The paper finds broad resonance for heavy $$\tilde g_{\mu\nu}=g_{\mu\nu}+\frac{1}{M^4}\partial_\mu\phi\,\partial_\nu\phi,$$9 with moderate $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$0, and also for light $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$1 with sufficiently large $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$2, including instability bands with $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$3. The result is that efficient preheating can occur without an enormous disformal scale (Karwan et al., 2017).

A third inflationary usage appears in "Potential-driven Inflation with Disformal Coupling to Gravity" (Qiu et al., 2020). There the conformal piece is chosen to remove explicit nonminimal curvature coupling, leaving a one-parameter derivative modification controlled by $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$4. In that sense the remaining new physics is purely disformal. The theory gives

$$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$5

so both tensor and scalar sectors acquire the same nontrivial sound speed. The paper finds modest shifts in $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$6, $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$7, and $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$8, with monodromy inflation moving toward the center of Planck contours as $$\mathcal L_{\text{int}}=\frac{1}{M^4}\,\partial_\mu\phi\,\partial_\nu\phi\,T^{\mu\nu}$$9 increases (Qiu et al., 2020). A common misconception is that every “pure disformal” model must have literally no conformal factor in every frame; these inflationary examples show that some authors instead use the phrase for cases where the conformal part is fixed so as not to introduce independent dynamics, while the phenomenologically relevant modification is derivative.

6. Consistency conditions, pathologies, and observational status

Pure disformal coupling is not automatically healthy. In fermionic sectors, generic disformal tetrad transformations induce the axial-current coupling $D$00, and removal of higher-time-derivative ghosts forces strong degeneracy conditions. In the two-scalar case, the allowed branch has a rank-1 degenerate field-space metric,

$D$01

so the consistent disformal transformation is effectively along one field-space direction. The inverse transformation of the Einstein–Hilbert action then yields two-field Horndeski or two-field DHOST, and analogous constructions extend to generalized U-DHOST and certain degenerate beyond generalized Proca classes (Domènech et al., 8 Oct 2025).

Other sectors display explicit pathologies. In vector–tensor theories with matter coupled through

$D$02

the pure vector disformal coupling causes ghost or gradient instability of GR stellar backgrounds and indicates breakdown of hyperbolicity, whereas the pure conformal coupling produces a tachyonic instability and can support spontaneous vectorization (Minamitsuji, 2020). In cosmological scalarization models with

$D$03

a late-time attractor to GR requires $D$04, while the scale relevant for spontaneous scalarization of neutron stars is $D$05. The required large $D$06 suppresses scalarization and induces ghost instabilities on scalar perturbations, which the authors trace to the dimensionful nature of the disformal scale (Silva et al., 2019).

Observationally, pure disformal EFTs are constrained very differently across sectors. In collider language, the operator

$D$07

evades fifth-force searches because static non-relativistic matter does not source it efficiently, but high-energy processes are sensitive. Recasting LHC Run 1 gives the strongest bound from CMS monojet searches,

$D$08

with a projected improvement to roughly $D$09 at 13 TeV with $D$10 (Brax et al., 2015). In gravitational two-body dynamics, the disformal coupling gives negligible Shapiro delay and therefore no useful Cassini bound, but it contributes to perihelion advance with a term scaling quadratically with the heavy mass, yielding a weak solar-system bound $D$11; the paper notes that Eöt-Wash provides the stronger bound $D$12 (Brax et al., 2018).

Early-universe and DM phenomenology yield another class of signatures. In a D-brane-inspired pure disformal model with $D$13 and $D$14, the matter-frame Hubble rate satisfies $D$15, so the expansion rate is always enhanced before BBN. Freeze-out then occurs earlier, and reproducing the observed relic abundance requires larger annihilation cross sections than in standard thermal cosmology (Dutta et al., 2017). In the branon realization, the same derivative coupling structure produces stable, weakly coupled scalars with relic-density, direct-detection, indirect-detection, collider, BBN, and supernova constraints distributed across a broad $D$16 parameter space (Cembranos et al., 2016).

Taken together, these results show that pure disformal coupling is not a single model but a family of derivative metric interactions with sharply context-dependent consequences. Its most distinctive signatures are suppression or enhancement of kinetic terms, modified light cones, species-dependent frame effects, and couplings that are weak in static low-energy environments but competitive in cosmology, strong gravity, and high-energy scattering. The literature therefore treats pure disformal coupling simultaneously as a phenomenological mechanism, an EFT organizing principle, and a nontrivial consistency problem.