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Effective description of generalized disformal theories

This paper extends the effective field theory of cosmological perturbations to encompass generalized disformal Horndeski and U-DHOST theories by incorporating operators with higher spatial derivatives of the lapse function.

Original authors: Kazufumi Takahashi, Masato Minamitsuji, Hayato Motohashi

Published 2026-09-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kazufumi Takahashi, Masato Minamitsuji, Hayato Motohashi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity, as we understand it from Einstein's general relativity, is a beautiful and precise description of how matter and energy curve the fabric of space and time. For over a century, this theory has passed every test we have thrown at it, from the bending of starlight to the ripples of gravitational waves detected from colliding black holes. Yet, cosmologists and physicists have long suspected that Einstein's equations might be only part of a larger story. To explain the mysterious forces driving the accelerating expansion of the universe or the invisible mass holding galaxies together, scientists have proposed thousands of alternative theories. These "modified gravity" models often add a new, invisible field to the mix, interacting with the geometry of space in ways that could subtly change how gravity behaves on cosmic scales. The challenge has always been to find a version of these theories that is mathematically consistent, avoiding hidden instabilities that would cause the universe to collapse or behave in impossible ways, while still remaining flexible enough to match the observations we see today.

In this context, a team of researchers has taken a significant step forward by mapping out the most general class of these stable, modified gravity theories known to date. They focused on a specific mathematical tool called a "disformal transformation," which is a way of reshaping the geometry of space-time using a scalar field. Imagine the geometry of space as a flexible sheet; a standard transformation might stretch or shrink this sheet uniformly, but a disformal transformation can stretch it differently depending on how fast the field is changing at that specific point. The researchers had previously shown that if you apply a specific, reversible version of this transformation to the most famous class of modified gravity theories, you get a new, larger family of theories that remain stable. In their latest work, they pushed this idea further. They developed a more complex version of the transformation that includes higher-order changes, effectively allowing the geometry to react to the acceleration of the field, not just its speed. By applying this advanced transformation to existing stable theories, they constructed what they call the "generalized disformal Horndeski" class. This new framework represents the broadest possible collection of stable, modified gravity theories that can be built without introducing mathematical ghosts—unphysical particles that would break the laws of physics.

To make these abstract theories useful for real-world astronomy, the authors extended a powerful tool known as the effective field theory. This framework allows scientists to study the tiny ripples and fluctuations in the early universe without needing to know the exact details of every single theory. Think of it as a universal language that describes how gravity and matter interact on a cosmic background, using a set of adjustable knobs and dials. The researchers showed how to turn these dials to accommodate their new, more complex theories. A key discovery in this process was that these generalized theories introduce new types of interactions that involve the rate of change of time itself, specifically affecting how the "lapse function"—a measure of how time flows relative to space—varies across the universe. These new interactions manifest as operators with higher spatial derivatives, meaning the theory becomes sensitive to how rapidly the geometry changes from one point to another in space. This is a distinct feature that sets these theories apart from previous models, which were limited to simpler, lower-order changes.

The team then used this expanded framework to calculate how these theories would behave in the real universe, specifically looking at how gravitational waves and density fluctuations would propagate. They found that for gravitational waves to travel at the speed of light, a strict condition must be met regarding how the transformation reshapes the geometry. This condition is crucial because it aligns perfectly with the observation of a neutron star merger detected in 2017, where gravitational waves and light arrived at Earth almost simultaneously, ruling out many other modified gravity models that predicted a different speed. When they turned their attention to the scalar perturbations—the clumps of matter that eventually form galaxies—they discovered that the behavior of these structures depends heavily on whether the universe is empty or filled with matter. In a vacuum, the new theories behave much like their simpler predecessors, with standard wave patterns. However, when matter is present, the new, higher-order spatial derivatives kick in, creating a more complex dynamic. This complexity is linked to a "shadowy mode," a type of instantaneous, non-propagating disturbance that exists in the theory but does not travel through space like a wave. While this mode does not cause the theory to collapse, it suggests that the presence of matter fundamentally alters how these generalized theories operate, adding a layer of nuance that was previously unexplored.

Ultimately, this work provides a comprehensive map for the landscape of stable modified gravity. By extending the effective field theory to include these generalized transformations, the researchers have given cosmologists a robust toolkit to test the most extreme versions of gravity against observational data. They have identified exactly which parameters control the speed of gravitational waves and how matter influences the stability of the universe in these models. While the theories remain mathematical constructs, the framework allows scientists to systematically check them against the cosmic microwave background and the large-scale structure of the universe. The authors suggest that future studies could use this framework to see if the unique signatures of these generalized theories leave any trace in the temperature fluctuations of the early universe or in the way gravitational waves decay over time. For now, the work stands as a rigorous definition of the boundaries of what is possible in a universe governed by a scalar field and a modified geometry, ensuring that any future theory of gravity must fit within this newly drawn, and far more expansive, circle.

Problem
Scalar-tensor theories represent a primary avenue for exploring modifications to General Relativity, particularly those involving additional degrees of freedom (DOFs). While the Horndeski theory constitutes the most general class of ghost-free scalar-tensor theories with second-order Euler-Lagrange equations, the space of viable theories has been expanded by the Degenerate Higher-Order Scalar-Tensor (DHOST) class, which allows higher derivatives subject to degeneracy conditions. Further extensions include the Unitary-DHOST (U-DHOST) class. However, most DHOST subclasses suffer from phenomenological issues such as ghost/gradient instabilities or non-dynamical tensor modes. The only phenomenologically viable DHOST subclass currently known is the Disformal Horndeski (DH) class, generated by invertible disformal transformations of Horndeski theories.

Recent developments have introduced "generalized disformal transformations" that include higher derivatives of the scalar field (specifically terms involving $\phi_{(\mu}X_{\nu)}$ ) while maintaining invertibility. Applying these transformations to Horndeski theories yields the Generalized Disformal Horndeski (GDH) class, and applying them to U-DHOST theories yields the Generalized Disformal Unitary-Degenerate (GDU) class. These classes represent the most general known ghost-free scalar-tensor theories to date. However, the existing Effective Field Theory (EFT) framework for cosmological perturbations, which accommodates Horndeski and standard DHOST theories, does not yet incorporate GDH and GDU theories. A systematic EFT description is required to study the cosmological perturbations of these generalized theories in a model-independent manner.

Methodology
The authors extend the standard EFT of inflation/dark energy to accommodate GDH and GDU theories. The methodology proceeds as follows:

Definition of Transformations: The paper focuses on a specific subclass of generalized disformal transformations defined by $g_{\mu\nu} \to \bar{g}_{\mu\nu} = f_0 g_{\mu\nu} + f_1 \phi_\mu \phi_\nu + 2f_2 \phi_{(\mu}X_{\nu)}$ , where $X_\mu$ involves derivatives of the kinetic term $X$ . This transformation is invertible under specific conditions on the coefficient functions $f_i$ .
Seed Action: The authors start with the quadratic action for cosmological perturbations of the "seed" theory (Horndeski or U-DHOST) expressed in the unitary gauge. This seed action is formulated using ADM variables (lapse $N$ , shift $N^i$ , spatial metric $\gamma_{ij}$ ) and geometric quantities like the extrinsic curvature $K_{ij}$ and spatial curvature ${}^{(3)}R$ .
Transformation of Building Blocks: The authors derive the transformation laws for the EFT building blocks (ADM variables, extrinsic curvature, and spatial curvature) under the generalized disformal transformation. Crucially, they account for the fact that the transformation introduces higher spatial derivatives of the lapse function perturbation ( $\delta N$ ) due to the $f_2$ term.
Construction of Extended EFT: By applying these transformation rules to the seed quadratic action, the authors derive a new quadratic action for the generalized theories. This action retains the structure of the standard EFT but introduces new time-dependent coefficients.
Perturbation Analysis: Using the derived extended EFT action, the authors analyze tensor and scalar cosmological perturbations. They derive the conditions for the absence of ghost and gradient instabilities and determine the propagation speeds of gravitational waves (GWs) and scalar modes. They also consider the coupling to matter fields (specifically a $k$ -essence scalar) to investigate the emergence of "shadowy modes."

Key Contributions

Extended EFT Framework: The paper constructs an extended EFT for cosmological perturbations that explicitly incorporates GDH and GDU theories.
New Operators: The extended EFT introduces three new parameters ( $\gamma_1, \gamma_2, \gamma_3$ ) that multiply operators involving higher spatial derivatives of the lapse function perturbation (e.g., $(\partial_i \delta N)^2$ , $\Delta \delta N$ , $(\Delta \delta N)^2$ ). These operators arise specifically from the $f_2$ term in the generalized disformal transformation.
Parameter Mapping: The authors provide explicit formulas mapping the EFT parameters of the seed theory (Horndeski or U-DHOST) to the parameters of the transformed GDH/GDU theories. This includes the transformation of the standard parameters ( $\tilde{M}, \tilde{\alpha}_L, \tilde{\alpha}_T, \tilde{\alpha}_K, \tilde{\alpha}_B, \tilde{\alpha}_H$ ) and the derivation of the new $\beta$ and $\gamma$ parameters.
Stability and Propagation Conditions: The paper derives the conditions for the absence of Ostrogradsky ghosts and gradient instabilities for both tensor and scalar perturbations within this extended framework.

Results

Tensor Perturbations: The squared sound speed for tensor perturbations is found to be $c_T^2 = 1 + \alpha_T$ . The authors derive the specific condition on the transformation function $f_0$ required to ensure $c_T^2 = 1$ , ensuring consistency with the simultaneous detection of GW170817 and GRB 170817A.
Scalar Perturbations (Vacuum): In the vacuum case, the dispersion relation for scalar perturbations depends on the parameter $\alpha_L$ $α_{L}$ .
- For GDH theories (generated from Horndeski), $\alpha_L = 0$ , resulting in an ordinary quadratic dispersion relation.
- For GDU theories (generated from U-DHOST), $\alpha_L$ can be non-zero. This leads to a dispersion relation with fourth-order spatial derivative terms, a phenomenon known as the "scordatura effect," which is characteristic of generic U-DHOST theories.
Scalar Perturbations (Matter Coupled): When matter fields are coupled, the analysis reveals that higher-order spatial derivatives appear in the equations of motion even for GDH theories (where $\alpha_L=0$ in vacuum). This is due to the non-vanishing of parameters like $\gamma_1$ and $f_{0Z}$ in the presence of matter, implying the existence of a "shadowy mode" (an instantaneous, non-propagating mode) in matter-coupled GDH theories.
Distinction from DH Class: The paper notes that if the transformation is restricted to $f_2 = 0$ (the conventional disformal transformation), the new parameters $\gamma_i$ vanish, and the theory reduces to the DH class. Thus, deviations from the DH class at the level of linear cosmological perturbations only manifest when $f_2 \neq 0$ .

Significance
The paper claims to provide the most general effective description of ghost-free scalar-tensor theories to date by extending the EFT framework to include GDH and GDU classes. By incorporating operators with higher spatial derivatives of the lapse function, the authors enable a model-independent study of cosmological perturbations for these theories. This framework allows for the systematic investigation of stability conditions and observational signatures, such as the speed of gravitational waves and the behavior of scalar modes, which are crucial for constraining these theories against cosmological data (e.g., CMB observations). The work establishes that while GDH theories offer a broader theoretical landscape than DH theories, they share similar linear perturbation properties in the absence of matter, with distinct features (shadowy modes and higher-derivative terms) emerging primarily through matter coupling or when generated from U-DHOST seeds.

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