Extended Mimetic Gravity Models

• Extended mimetic gravity models are scalar–tensor theories that enforce non-invertible conformal transformations to isolate a mimetic degree of freedom.
• They incorporate Lagrange multipliers and higher-derivative operators to control degrees of freedom, enabling modifications of dark matter and cosmological behavior.
• Hamiltonian analyses confirm these models avoid Ostrogradsky ghosts while unifying dark matter, dark energy, and inflationary dynamics.

Extended mimetic gravity models constitute a broad class of scalar–tensor and higher-derivative covariant gravitational theories that generalize the original mimetic gravity construction. They are based on enforcing a non-invertible, typically Weyl-invariant (conformal) transformation between an auxiliary (“seed”) metric and the physical metric, often through the introduction of Lagrange multipliers and higher-derivative operators. Such models are motivated by the need to capture a variety of cosmological and phenomenological features—including the emergence of pressureless (dark matter–like) components, nonzero sound speed, modified structure formation, and potential unification of dark matter, dark energy, and early-universe inflation—while maintaining strong constraints on degrees of freedom (avoiding Ostrogradsky ghosts).

1. Lagrangian Framework and Generalization of Mimetic Gravity

The prototypical extended mimetic gravity action arises from a singular Weyl-symmetric redefinition of the Einstein–Hilbert action. Explicitly, the physical metric is related to an auxiliary metric $h_{\mu\nu}$ and scalar $\phi$ via the disformal transformation $$g_{\mu\nu} = (h^{\alpha \beta} \partial_\alpha \phi \partial_\beta \phi) h_{\mu\nu}$$, imposing the constraint $g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi = 1$ on-shell. The action

$$S_0[h, \phi] = -\tfrac12 \int d^4x \sqrt{-g[g(h, \phi)]} R[g(h, \phi)]$$

is manifestly invariant under Weyl rescalings of $h_{\mu\nu}$ and can be recast in a scalar-tensor form equivalent to a singular Brans–Dicke theory with $\omega = -3/2$: $$S_0 = -\int d^4x \sqrt{-h} \left[ X R(h) + \frac{3}{2X} h^{\alpha\beta} \partial_\alpha X \partial_\beta X \right], \quad X = \tfrac12 h^{\alpha\beta} \partial_\alpha \phi \partial_\beta \phi.$$ This structure admits systematic higher-derivative generalizations. The canonical extension is the inclusion of a Weyl-invariant $(\Box_g \phi)^2$ term, yielding

$$S_\gamma[g, \phi, \rho] = \int d^4x \sqrt{-g} \left[ -\tfrac12 R(g) + \tfrac12 \rho (g^{\mu\nu}\partial_\mu\phi \partial_\nu\phi - 1) + \tfrac12 \gamma (\Box_g \phi)^2 \right],$$

where $\rho$ is a Lagrange multiplier field. In the auxiliary $h_{\mu\nu}$ frame, the action contains explicit terms beyond the Horndeski structure: $$S_\gamma[h, \phi, X] = -\int d^4x \sqrt{-h} \left[ X R(h) + \frac{3}{2X} h^{\alpha\beta}\partial_\alpha X\partial_\beta X - \frac{\gamma}{2} \left( \Box_h \phi + \frac{h^{\alpha\beta}\phi_{,\alpha} X_{,\beta}}{X} \right)^2 \right].$$ These extensions generalize to formulations involving multiple mimetic scalars, curved field-space metrics, explicit non-minimal curvature couplings, and covariant higher-derivative terms. Notably, such generalized actions remain constrained to propagate only a single (or $M$ for $M$-field versions) scalar degree of freedom in addition to the two standard tensor modes (Hammer et al., 2015, Mansoori et al., 2021, Zheng, 2018).

2. Weyl Symmetry and Frame Representations

Weyl invariance is structured so that physical quantities are independent of the normalization of the seed metric. In the "dust frame," the variables $(g_{\mu\nu}, \phi, \rho)$ lead directly to the Lagrangian for irrotational pressureless dust, with $\phi$ interpreted as velocity potential and $\rho$ as energy density. In the "singular Brans–Dicke frame," alternative gauge-invariant combinations (e.g., $(\hat h_{\mu\nu}, \phi, \chi)$ with $h_{\mu\nu} = \lambda^{-1} \hat h_{\mu\nu}$, $X = \lambda \chi$) establish equivalence with $\omega = -3/2$ Brans–Dicke theory with the dust sector coupled to $2\chi\,\hat h_{\mu\nu}$. Both frames encapsulate all physical content of the theory (Hammer et al., 2015).

In the context of generalized or multi-field mimetic models, the non-invertible map can be extended to several scalars with targets in a curved, Riemannian field space, with the mimetic constraint imposing a unit "kinetic norm" for the field-space trajectory: $$g^{\mu\nu}G_{IJ}(\phi)\partial_\mu\phi^I\partial_\nu\phi^J = -1$$. The resulting phase-space and constraint algebra remain regular and no additional Ostrogradsky degrees of freedom arise (Mansoori et al., 2021).

3. Hamiltonian Analysis and Absence of Ostrogradsky Ghosts

In both single-field and generalized extensions, the mimetic constraint (plus residual gauge symmetry) removes all would-be higher-derivative Ostrogradsky ghosts. The generalized $(\Box\phi)^2$ term yields higher-order derivatives in the action but, due to the algebraic constraint, the gauge-invariant equations are strictly second order in time in the physical sector (Hammer et al., 2015, Zheng, 2018, Takahashi et al., 2017).

Extended versions generated by non-invertible conformal transformations on arbitrary scalar-tensor "seed" theories generically propagate at most three physical degrees of freedom: two tensor gravitons plus a single scalar. A full Dirac–Bergmann analysis shows that, depending on gauge choice (e.g., unitary gauge $\phi=t$), the Dirac matrix of secondary constraints may become singular, reducing the DOF count further (e.g., from four to three in certain higher-derivative cases (Zheng, 2018)). For multi-field generalizations with $M$ fields, the total propagating DOF is $M+2$ (Mansoori et al., 2021).

4. Cosmological Solutions and Phenomenological Implications

Exact cosmological solutions can be constructed in synchronous gauge or the comoving (velocity-potential) frame. For the standard case $\phi(\tau, x) = \tau$ and metric $ds^2 = d\tau^2 - \ell_{ij}(\tau, x) dx^i dx^j$, the energy density $\rho(\tau,x)$ solves an explicit ODE: $$\rho(\tau,x) = \frac{\rho_0(x) + \gamma \sqrt{\ell}\;\partial_\tau^2 \ln\sqrt{\ell} - \gamma \partial_i \int_0^\tau d\tau' \sqrt{\ell}\; \ell^{ik} \partial_k \partial_{\tau'} \ln\sqrt{\ell}}{\sqrt{\ell}},$$ where $\ell \equiv \det\ell_{ij}$. In shear-free or test-field limits, the solution reduces to

$$\rho(\tau,x) = \frac{\rho_0(x)}{\sqrt{\ell}(\tau,x)} + \frac{\gamma}{\sqrt{\ell}}\;\partial_\tau^2\ln\sqrt{\ell}.$$

This approach allows the mimetic field to source exact pressureless dust cosmologies and their modifications due to the $\gamma$-term (Hammer et al., 2015).

Fluid decomposition in the local rest frame yields the energy density, pressure, heat flow, and a shift-charge density: \begin{align*} & \varepsilon = \rho - \gamma(\partial_\tau \theta - \tfrac{1}{2}\theta^2), \ & p = -\gamma(\partial_\tau \theta + \tfrac{1}{2}\theta^2), \ & q_\mu = -\gamma \perp_\mu{}^\nu \partial_\nu\theta, \ & n = \rho - \gamma \partial_\tau\theta, \end{align*} where $\theta$ is the expansion scalar. The effective sound speed $c_s^2 \simeq \gamma$ shows that for small $\gamma$, the model mimics cold dark matter to leading order, with nonzero pressure and heat flux allowing for imperfect fluid behavior (Hammer et al., 2015).

Phenomenological constraints arise from large scale structure (LSS) and cosmic microwave background (CMB): e.g., $|\gamma| \lesssim 10^{-7}$–$10^{-9}$, to avoid excessive suppression of structure on sub-sound-horizon scales and to prevent excessive diffusion damping in the CMB. The model can also generate the correct dark matter abundance during radiation domination if $\gamma(\phi)$ is allowed to be time-dependent. Anisotropic stress and heat flow may be relevant for galaxy rotation curves and lensing (Hammer et al., 2015).

5. Beyond the Minimal Model: Extensions and Limitations

Systematic procedures exist for constructing more general (including multi-field and higher-derivative) mimetic gravities. Non-invertible conformal maps acting on generic scalar-tensor seed actions generate degenerate higher-order scalar-tensor theories (DHOST–like) with at most three propagating DOF (Takahashi et al., 2017). However, the extended mimetic models generically encounter gradient or ghost instabilities at the level of cosmological perturbations, depending on the precise sign structure of the tensor and scalar kinetic coefficients. Simultaneously achieving stability of both tensor and scalar sectors is generally impossible except in degenerate cases, which may render the scalar strongly coupled or revive ghosts in the presence of additional matter (Takahashi et al., 2017).

For multi-field mimetic models realized via a singular kinetic–type conformal transformation, field-space geometry becomes crucial: entropy perturbations associated with directions in field space normal to the background trajectory propagate with unit sound speed if the field-space metric is positive-definite; otherwise, ghost or gradient instabilities can occur (Mansoori et al., 2021). The full non-linear Hamiltonian analysis confirms that these models avoid Ostrogradsky instabilities and propagate strictly $M+2$ DOF for $M$ mimetic fields.

A series of “extended mimetic” models have also been developed to address cosmological structure and phenomenology via explicit non-minimal curvature couplings, higher-order spatial derivatives, and matter couplings. These generalizations preserve the core mechanism for mimetically induced dust, but require model-specific analyses to establish stability and compatibility with local gravity constraints and cosmological data.

6. Physical Interpretation and Geometric Structure

Extended mimetic gravity provides a unified geometric interpretation for dark-matter–like degrees of freedom as a consequence of the isolated conformal mode, enforced as a constraint on the metric via a non-invertible transformation. The higher-derivative extensions parameterized by $\gamma$ or analogous variables encode imperfect and non-trivial fluid properties while strictly controlling propagating DOF and avoiding Ostrogradsky instabilities (Hammer et al., 2015, Zheng, 2018).

Exact solutions, including synchronous gauge dust cosmologies, are available, and the models make concrete predictions for pressureless dust, small sound speed, and explicit higher-derivative corrections. The close connection to singular limits of Brans–Dicke theory, DHOST and beyond-Horndeski models underscores the geometrical versatility and unifying capabilities of the mimetic paradigm.

Nonetheless, the extended mimetic sector remains subject to stability, causality, and phenomenological constraints. Gradient or ghost instabilities are endemic to the most generic higher-derivative extensions, though carefully constrained models (with judicious choices of higher-derivative terms and field-space signatures) can maintain physical viability. The Weyl invariance and gauge artifact structure of the formulation ensure that only physically meaningful (observable) quantities are propagated, a feature central to the theoretical consistency of these models (Hammer et al., 2015, Mansoori et al., 2021, Takahashi et al., 2017).