Mimetic Embedding Gravity

• Mimetic embedding gravity is defined as a generally covariant construction where the physical metric is reparametrized via an auxiliary metric and scalar field, isolating the conformal degree of freedom to mimic dark matter.
• The framework extends to theories like k-essence, modified Gauss–Bonnet, and massive gravity, offering unified insights into dark energy, UV behaviors, and embedding matter.
• Applications range from cosmological model reconstruction and compact object dynamics to stability analyses, emphasizing the role of a positive mimetic energy density in maintaining healthy gravitational dynamics.

Searching arXiv for the cited mimetic gravity and embedding-gravity papers to ground the article and confirm identifiers. Mimetic embedding gravity denotes a class of generally covariant constructions in which the physical metric is written in terms of auxiliary variables—most commonly an auxiliary metric and a scalar field—so that the conformal degree of freedom of gravity is isolated and reappears as an effective matter sector. In the minimal formulation introduced by Chamseddine and Mukhanov, this sector behaves as pressureless dust and can mimic cold dark matter; in later developments the same constrained metric-scalar mechanism was used to reformulate covariant renormalizable gravity, $k$-essence, massive gravity, modified Gauss–Bonnet gravity, and Regge–Teitelboim embedding theory (Chamseddine et al., 2013, Myrzakulov et al., 2015, Jiroušek et al., 2022, Paston et al., 2018).

1. Foundational mimetic construction

The foundational construction defines the physical metric $g_{\mu\nu}$ in terms of an auxiliary metric $\tilde g_{\mu\nu}$ and a scalar field $\phi$ through

$$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$

This parametrization is invariant under Weyl rescalings of $\tilde g_{\mu\nu}$, so the conformal factor of the auxiliary metric is encoded covariantly in $\phi$ rather than being an independent metric variable (Chamseddine et al., 2013).

A direct consequence is the mimetic constraint

$g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi = -1$

or $+1$ in alternate sign conventions. In the original formulation this follows from the metric definition itself; in equivalent formulations it is imposed with a Lagrange multiplier (Chamseddine et al., 2013, Golovnev, 2013).

Varying the Einstein–Hilbert action with respect to the auxiliary variables yields modified Einstein equations of the form

$$G_{\mu\nu}=T_{\mu\nu}+(G-T)\,\partial_\mu\phi\,\partial_\nu\phi,$$

together with

$g_{\mu\nu}$0

The trace of the Einstein equations is therefore not fixed algebraically as in ordinary GR; instead, the trace sector is governed dynamically by the scalar equation. This is the sense in which the conformal mode becomes dynamical even in vacuum (Chamseddine et al., 2013).

2. Effective fluid interpretation and constrained formulations

The additional contribution

$g_{\mu\nu}$1

has the form of a pressureless perfect fluid once one identifies $g_{\mu\nu}$2 and $g_{\mu\nu}$3. In a spatially flat FLRW background with $g_{\mu\nu}$4, the conservation equation gives

$g_{\mu\nu}$5

so the mimetic component reproduces cold-dark-matter scaling at the background level (Chamseddine et al., 2013).

Golovnev gave a transparent equivalent action without auxiliary metric,

$g_{\mu\nu}$6

showing explicitly that the theory can be viewed as GR plus a constrained scalar whose Lagrange multiplier $g_{\mu\nu}$7 plays the role of the effective dust density (Golovnev, 2013). This reformulation also clarifies the variational mechanism: because the original metric parametrization is derivative-dependent, the allowed variations are restricted, and the missing trace equation reappears as the scalar conservation law rather than as an algebraic Einstein equation (Golovnev, 2013).

Across the literature, this constrained-scalar representation functions as the basic template for “mimetic embedding” constructions. The common feature is not a single unique action, but a non-invertible conformal or disformal map that embeds the physical metric into a larger field space while producing an effective fluid sector.

3. Representative embeddings and extensions

Several later works generalized the same template to distinct gravitational frameworks.

Realization	Defining ingredient	Effective interpretation
Minimal mimetic gravity (Chamseddine et al., 2013)	Non-invertible conformal metric map	GR plus dust-like mimetic matter
Covariant renormalizable gravity (Myrzakulov et al., 2015)	Constrained scalar $g_{\mu\nu}$8 in Hořava-like operators	Mimetic scalar as Lorentz-breaking fluid
Mimetic $g_{\mu\nu}$9-essence (Jiroušek et al., 2022)	Weyl-invariant $\tilde g_{\mu\nu}$0 built from scalar and gauge invariants	On-shell GR plus $\tilde g_{\mu\nu}$1-essence
Embedding theory (Paston et al., 2018)	Induced metric $\tilde g_{\mu\nu}$2	GR plus embedding matter
Mimetic massive gravity (Golovnev, 2018)	Disformal extension of dRGT	dRGT plus constrained scalar
Mimetic $\tilde g_{\mu\nu}$3 (Astashenok et al., 2015)	Lagrange-multiplier mimetic sector in Gauss–Bonnet gravity	Dark-matter-like mode plus modified curvature dynamics

In covariant renormalizable gravity, Nojiri and Odintsov’s “unknown fluid” is identified with the mimetic scalar and therefore with the conformal mode of gravity. In that theory the constrained scalar supplies a unit timelike vector $\tilde g_{\mu\nu}$4, dynamically breaks Lorentz invariance, and enters Hořava-like higher-derivative operators that produce UV scaling $\tilde g_{\mu\nu}$5; in the limit $\tilde g_{\mu\nu}$6 and $\tilde g_{\mu\nu}$7, the action reduces to minimal mimetic gravity (Myrzakulov et al., 2015).

“Mimetic K-essence” generalizes the construction further by taking

$\tilde g_{\mu\nu}$8

with $\tilde g_{\mu\nu}$9 and $\phi$0, and imposing the Weyl-homogeneity condition $\phi$1. The resulting theory is on-shell equivalent to GR plus a $\phi$2-essence scalar with effective Lagrangian $\phi$3, where the overall scale $\phi$4 appears as an integration constant rather than a coupling in the action (Jiroušek et al., 2022).

In the massive-gravity context, the disformal extension studied in “Beyond dRGT as Mimetic Massive” is not a new ghost-free potential beyond dRGT, but equivalent either to ordinary dRGT if the transformation is invertible or to dRGT plus a constrained mimetic scalar if it is non-invertible. The pure gravity sector is therefore equivalent to the usual dRGT theory coupled to a pressureless ideal-fluid-like scalar sector (Golovnev, 2018).

4. Cosmological uses

The cosmological literature exploits the fact that the mimetic sector is fixed by a constraint rather than by a standard kinetic term. With the action

$\phi$5

the mimetic constraint forces $\phi$6 in spatially flat FLRW, and the background evolution reduces to

$\phi$7

Writing $\phi$8, this becomes the linear equation

$\phi$9

This permits reconstruction of essentially arbitrary expansion histories from a suitable choice of $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$0, including inflationary, quintessence-like, and bouncing cosmologies (Chamseddine et al., 2014).

The same work shows that minimal mimetic inflation produces scalar perturbations with zero sound speed, so all modes behave like long-wavelength dust perturbations. Adding the higher-derivative term $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$1 yields

$$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$2

restores a genuine wave equation for scalar perturbations, and allows simple mimetic inflation to generate red-tilted scalar perturbations largely enhanced over gravity waves (Chamseddine et al., 2014). This became a standard motivation for higher-derivative completions of the minimal model.

Modified-curvature extensions broaden the range of cosmological applications. Mimetic $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$3 gravity with a Lagrange multiplier can realize accelerated expansion, unified inflation with dark energy, and bouncing cosmologies, while still producing a mimetic dark-matter component whose density scales as

$$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$4

The same framework also admits reconstruction schemes for $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$5, $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$6, and $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$7 from a prescribed Hubble history (Astashenok et al., 2015).

“Mimetic K-essence” adds a further restriction: the mimetic realization is obstructed when the effective fluid reaches the ultra-relativistic equation of state $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$8. The paper shows that the breakdown of the mimetic description coincides with tangency of the $$g_{\mu\nu} = -(\tilde g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi)\,\tilde g_{\mu\nu}.$$9-essence curve to a parabola $\tilde g_{\mu\nu}$0, so a single-branch mimetic realization cannot smoothly pass through radiation (Jiroušek et al., 2022). This is a substantive limitation on cosmological model building.

A different line of work studies interacting dark-energy–dark-matter–radiation systems in mimetic gravity through autonomous dynamical systems. “Mimetic Attractors” identifies critical points, stable attractors, and stable invariant submanifolds for several phenomenological interaction terms $\tilde g_{\mu\nu}$1, showing that late-time accelerating attractors can arise within the mimetic background dynamics (Raza et al., 2015).

5. Geometrical embedding, strong-field sectors, and exact solutions

A geometrically distinct realization appears in Regge–Teitelboim embedding theory, where spacetime is represented as a four-dimensional surface embedded in flat higher-dimensional space,

$\tilde g_{\mu\nu}$2

“Embedding theory as new geometrical mimetic gravity” rewrites this theory as GR plus an additional matter sector described by conserved currents $\tilde g_{\mu\nu}$3 and an action containing $\tilde g_{\mu\nu}$4. In the single-current limit it reduces to the standard perfect-fluid action used to represent mimetic gravity, while in general it behaves as a multi-current generalization of a perfect fluid (Paston et al., 2018).

The non-relativistic limit of this embedding matter makes the analogy with dark matter explicit. In that limit the embedding matter behaves as cold dust sourced by a density $\tilde g_{\mu\nu}$5, but also acquires a self-interaction determined by the embedding variables. The authors argue that this self-interaction could be useful in addressing the core–cusp problem in $\tilde g_{\mu\nu}$6CDM (Paston, 2020). This suggests a geometrical route from embedding variables to a self-interacting dark sector without introducing new particles.

Strong-field applications show that the mimetic sector modifies compact objects and exact solutions even in the minimal theory. “Mimetic Compact Stars” derives modified Tolman–Oppenheimer–Volkoff equations for quark stars and neutron stars, finding viable quark-star configurations for empirical equations of state and more exotic behavior for simple neutron-star polytropes (Momeni et al., 2015). “Cylindrical solutions in Mimetic gravity” finds exact non-Kasner cylindrical metrics, shows that quasi-Kasner solutions do not exist in mimetic gravity, and proves a no-go theorem for Linet–Tian-type solutions with cosmological constant in the non-GR mimetic regime (Momeni et al., 2015). In $\tilde g_{\mu\nu}$7 dimensions, mimetic gravity admits BTZ-like black-hole solutions; static charged and uncharged cases develop a curvature singularity at $\tilde g_{\mu\nu}$8, whereas the rotating solutions have finite curvature invariants at the origin, so angular momentum regularizes the central singularity (Sheykhi, 2020).

6. Hamiltonian structure, stability, and open issues

The most systematic Hamiltonian analysis in the supplied literature establishes a sharp criterion for healthy mimetic scalar sectors: for healthy mimetic scalar-tensor theories, the mimetic contribution to the Hamiltonian is bounded from below when the mimetic field energy density satisfies $\tilde g_{\mu\nu}$9 (Ganz et al., 2018). In the shift-symmetric mimetic dark-matter model, $\phi$0 is conserved, so positivity of $\phi$1 is preserved in time if it holds initially and suitable boundary conditions are imposed (Ganz et al., 2018).

When shift symmetry is absent, positive energy density can be enforced structurally by replacing $\phi$2, which guarantees $\phi$3 identically (Ganz et al., 2018). The same analysis extends to mimetic $\phi$4 gravity: the theory is healthy if the usual $\phi$5 stability conditions are imposed and $\phi$6 (Ganz et al., 2018).

A contrary result is equally important: adding a mimetic sector to an unhealthy seed action does not in general remove the original instability. The explicit example with a seed higher-derivative scalar-tensor theory containing $\phi$7 shows that Ostrogradski ghosts remain even after imposing the mimetic constraint (Ganz et al., 2018). This sharply limits the idea that non-invertible metric redefinitions by themselves provide a universal cure for higher-derivative pathologies.

The perturbative picture is correspondingly mixed. In FLRW, adding ordinary scalar matter does not revive dynamical UV ghost modes; the problematic modes already present in mimetic dark matter are non-propagating, have zero sound speed, and are associated with the mimetic component itself rather than with the conventional matter sector (Ganz et al., 2018). Other papers identify related structural limitations: the minimal perturbation sector has zero sound speed (Chamseddine et al., 2014), covariant renormalizable gravity requires careful arrangement of higher-derivative operators to avoid Ostrogradsky ghosts and strong coupling (Myrzakulov et al., 2015), mimetic $\phi$8-essence cannot cross $\phi$9 in a regular single-branch realization (Jiroušek et al., 2022), and cylindrical solutions with cosmological constant exhibit a no-go result in the genuinely mimetic sector (Momeni et al., 2015).

Taken together, these results indicate that mimetic embedding gravity is best understood not as a single theory but as a family of constrained, non-invertibly parametrized gravitational models. Their unifying idea is that an auxiliary description of the metric turns part of the gravitational sector into an effective matter component—dust-like in the minimal case, but capable in extended models of reproducing $g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi = -1$0-essence, dark-energy-like behavior, Hořava-like UV structure, embedding matter, or modified strong-field geometries (Chamseddine et al., 2013, Myrzakulov et al., 2015, Jiroušek et al., 2022, Paston et al., 2018).