DBI Galileons: Unified Scalar Theories

• DBI Galileons are scalar field theories that derive second-order equations from Lovelock invariants, ensuring ghost-free interactions.
• They are constructed via a higher-dimensional probe-brane approach that unifies the relativistic DBI and nonrelativistic Galileon models through systematic derivative expansion.
• The framework naturally incorporates nonminimal gravitational couplings, supporting cosmological applications like inflation and modified gravity.

The Dirac–Born–Infeld (DBI) Galileons are scalar field theories that unify and generalize Galileon and DBI scalar effective field theories through a higher-dimensional probe-brane construction. The central idea is that all interactions leading to second-order equations of motion (thus evading the Ostrogradsky instability) can be derived from geometric invariants—specifically, Lovelock bulk and boundary terms—associated with the position modulus of a relativistic probe brane embedded in a five-dimensional (5D) bulk. The DBI Galileon framework not only reproduces the well-known Galileon and DBI theories as different limits but also naturally incorporates gravitational couplings and enables systematic construction of higher-derivative, yet healthy, interactions. This structure provides a robust and geometrically motivated unification of brane-world inflation, infrared modifications of gravity, and related cosmological models.

1. Higher-dimensional Brane Construction and Relativistic Galileons

DBI Galileons originate from effective field theories describing the motion of a probe brane in a higher-dimensional ambient space. The dynamics of the brane are characterized by its position modulus, $T(x)$, encoding the brane's transverse displacements. The worldvolume action is constructed from geometric quantities induced on the brane:

$$g_{\mu\nu} = q_{\mu\nu} + \partial_\mu T \partial_\nu T$$

where $q_{\mu\nu}$ is the metric on constant-$y$ slices in the bulk.

The full relativistic (DBI) structure is retained in this description, with the brane tension providing a canonical square-root kinetic term. Expansions in derivatives produce higher-order corrections, fully determined by induced curvature and extrinsic geometric invariants. In the non-relativistic (Galilean contraction) limit, $(\partial T)^2 \ll 1$, the non-linear structure reduces to the classic Galileon Lagrangian, and the Galileon field emerges as the brane-bending mode. This connection systematically extends the decoupling-limit theory of the Dvali–Gabadadze–Porrati (DGP) model and its covariant/conformal extensions.

2. Systematic Generation of Higher-Order Interactions

All higher-derivative, healthy interactions in the DBI Galileon construction arise from Lovelock invariants and their Gibbons–Hawking–York (GHY) boundary terms. The first few Lagrangians are constructed as follows:

$$S_\text{brane} = \int d^4 x \sqrt{-g} \left[-V(g) + A R - M_3 K - \frac{B}{m^2} K_\mathrm{GB}\right] + A(T)$$

Here:

• $R$ is the induced Ricci scalar,
• $K$ the extrinsic curvature scalar,
• $K_\mathrm{GB}$ represents the GHY term associated with the Gauss–Bonnet combination,
• $V(g)$ and $A(T)$ are brane tension and possible interaction potentials.

The explicit form for the GHY term in the context of the 5D Gauss–Bonnet invariant is:

$$K_\mathrm{GB} = -K + K^2 - K^3 - 2 G_{\mu\nu} K^{\mu\nu}$$

where $G_{\mu\nu}$ is the brane-indeced Einstein tensor. The structure of these terms ensures recursive relationships between variations:

$$\delta(\sqrt{-g}) = K,\quad \delta(\sqrt{-g} K) = R, \quad \delta(\sqrt{-g} R) = K_\mathrm{GB}$$

Such recursive identities guarantee that although the Lagrangian contains higher derivatives, the equations of motion remain second-order.

3. Covariant Coupling to Gravity and Nonminimal Interactions

To couple the DBI Galileon scalar consistently to four-dimensional gravity, one promotes $q_{\mu\nu}$ to a dynamical metric. The induced metric naturally generates nonminimal gravitational couplings, with the scalar sector and gravitational curvature intertwined:

$$g_{\mu\nu} = q_{\mu\nu}(x) + \partial_\mu T \partial_\nu T$$

The induced curvature terms match those required for covariant Galileon models. Field redefinitions and integrations by parts confirm the equivalence between the covariance structure derived from the brane and nonminimal terms previously found by consistency arguments in four-dimensional effective theories. This approach sidesteps the need for ad hoc gravitational couplings and ensures that the correct form emerges automatically from geometric construction.

4. Lovelock Invariants and the Well-Posed Cauchy Problem

Lovelock invariants serve as the unique set of geometric terms in higher-dimensional gravity whose associated equations of motion do not introduce derivatives higher than second order. The DBI Galileon theory inherits this property because all allowed interaction terms, including those in four and five dimensions, are constructed from these invariants and the corresponding boundary terms.

The recursive relations relating variations of the action (e.g., $\delta(\sqrt{-g}) = K$ and subsequent steps) demonstrate concretely that higher derivatives in the Lagrangian conspire to yield strictly second-order field equations, distinguishing DBI Galileons from generic higher-derivative scalar-tensor theories. This property ensures that the Cauchy problem for the system is well posed—solutions with given initial data evolve predictably without the emergence of extra ghostlike degrees of freedom.

5. Unified Effective Theory: The DBI and Galileon Limits

DBI Galileons provide a unified effective field theory framework interpolating between the Dirac–Born–Infeld action (relativistic regime) and the classic Galileon action (non-relativistic, small derivatives). In the purely relativistic limit, large brane velocities and tensions are correctly described, incorporating all geometric corrections. In the $(\partial T)^2 \ll 1$ (Galilean contraction) limit, the action automatically reduces to the ghost-free Galileon generalization familiar from the DGP decoupling limit. Further, in anti-de Sitter (AdS) bulk backgrounds, the same construction yields the conformal Galileon effective theory.

The unification is formalized through recursive formulas linking all allowed terms, permitting algorithmic construction of the full class of second-order scalar-tensor interactions in both regimes. This incorporates brane-world motivation (DBI models suited to high-energy inflation) and IR modifications of gravity (Galileons as effective theories of large-scale modification).

6. Observational and Phenomenological Consequences

The DBI Galileon theory facilitates model building in several phenomenological arenas:

• Cosmology and Inflation: The framework supports consistent generation of large primordial non-Gaussianities in models of inflation, as well as predicts possible late-time acceleration scenarios through non-trivial scalar field dynamics.
• Modified Gravity: The construction produces models with screening mechanisms and IR modifications of gravity; extra nonlinear structure can modulate modifications to gravitational potentials in screened (e.g., solar system) environments.
• Consistency with Gravity: The approach ensures that all nonminimal gravitational couplings required for ghost-freedom and cosmological viability are automatically included, reducing parameter tuning and ambiguities in model construction.

The embedding into higher-dimensional geometry and the explicit link to Lovelock invariants provide systematic control over theoretical consistency and predictive power.

7. Limitations and Theoretical Consistency

The DBI Galileon framework, while systematically unifying previously distinct classes of scalar theories, rests on the assumption of a higher-dimensional brane embedding and mathematical consistency of the probe brane expansion. It relies crucially on Lovelock invariance for the ghost-freedom and well-posedness of the resulting effective theory. The central claims—second-order equations of motion, natural inclusion of all nonminimal gravitational couplings, and explicit correspondence to both DBI and Galileon models—are validated within the context of four- and five-dimensional analysis and via the recursive relationships among induced terms.

A notable theoretical consequence is that all ghost-free scalar–tensor interactions derivable from higher-dimensional brane physics are classified within the DBI Galileon scheme, with the inclusion of all necessary nonminimal couplings and boundary terms regulated by the geometric (Lovelock) invariants. This positions DBI Galileons as a central class of effective field theories for modifications of gravity and cosmological model building based directly on fundamental geometric principles (1003.5917).