SVT Rep. in Cosmology

• Scalar-Vector-Tensor representation systematically decomposes gravitational perturbations into scalar, vector, and tensor modes, clarifying their distinct roles in modified gravity theories.
• It provides a robust framework to parameterize field content and impose gauge symmetries, resulting in consistent quadratic actions and precise degree-of-freedom counting.
• SVT models underpin effective field theories in cosmology, enabling direct comparisons of dark energy, inflation, and modified gravity scenarios.

The scalar-vector-tensor (SVT) representation refers to the systematic decomposition of perturbations or field content in gravitational and cosmological theories into components characterized by their transformation properties: scalars (invariant under spatial rotations), vectors (transforming as spatial 3-vectors), and tensors (symmetric, traceless rank-2 objects under rotations). This framework is central in modern cosmology, gravitational theory, and high-energy physics for classifying degrees of freedom, constructing gauge-invariant actions, and diagnosing possible new gravitational phenomenology beyond general relativity. SVT representations play a critical role in parametrizing general modifications of gravity, analyzing cosmological perturbations, and constructing effective field theories for inflation and dark energy.

1. Parametrization Framework for SVT Theories

The systematic SVT parametrization for linear cosmological perturbations is based on several steps:

• Field content and symmetry selection: The gravitational action is specified by a metric $g_{\mu\nu}$, possibly supplemented by a scalar field $\chi$ and/or a vector field $A^\mu$, along with matter content (e.g., perfect fluid or scalar).
• Background and perturbation expansion: All fields are split into homogeneous background and linear perturbations:

$$g_{\mu\nu} = \bar{g}_{\mu\nu} + \delta g_{\mu\nu}, \quad \chi = \chi_0(t) + \delta\chi, \quad A^\mu = (A(t), \vec{0}) + \alpha^\mu$$

The vector perturbation is itself decomposed into irreducible scalar and solenoidal vector parts:

$$\alpha^i = \bar{h}^{ij}\partial_j\alpha + \alpha^{Ti}, \quad \partial_i\alpha^{Ti} = 0.$$

• General quadratic action: The gravitational and matter actions are Taylor-expanded to second order in perturbations using the 3+1 ADM decomposition. All possible quadratic terms respecting desired symmetries (e.g., diffeomorphism invariance, Lorentz invariance) are included, parameterizing the most general quadratic action in terms of the perturbed "building blocks" (e.g., lapse, shift, spatial metric, extrinsic curvature, field perturbations).
• Imposing gauge symmetry (Noether identities and constraints): Enforcing invariance under infinitesimal coordinate transformations (diffeomorphisms) and any internal symmetries yields algebraic constraints among the free functions in the quadratic action. The number of propagating degrees of freedom and relevant free parameters is thereby fixed.

This construction yields a set of quadratic actions whose form and free parameters are directly linked to the field content and gauge structure of the underlying theory. The approach is systematic and sufficiently general to recover the standard scalar-tensor (Horndeski), vector-tensor (generalized Proca, Einstein-Aether), and higher-derivative (Beyond Horndeski, higher-order vector-tensor) models (Lagos et al., 2016).

2. Scalar-Tensor Sector: Horndeski and Beyond

For scalar-tensor theories, the SVT formalism recovers and extends the effective field theory of linear perturbations:

• The quadratic action (up to three derivatives) is written in terms of time-dependent free functions:

$$S^{(2)}_G = \int d^4x\,a^3 M^2\left\{ \frac{1}{2}H^2(\alpha_K-12\alpha_B-6)\Phi^2 - 6H(1+\alpha_B)\Phi\dot\Psi + 2(1+\alpha_H)\Psi\partial^2\Phi -3\dot\Psi^2 - (1+\alpha_T)\Psi\,\partial^2\Psi + \cdots \right\}$$

where $M^2$ (effective Planck mass), $\alpha_K$ (kineticity), $\alpha_B$ (braiding), $\alpha_T$ (tensor speed excess), and $\alpha_H$ (beyond-Horndeski) are free functions of time.

• Higher spatial-derivative corrections (fourth-order operators) introduce additional functions (e.g., $\alpha_{Q1}, \alpha_{Q2}, \dots, \alpha_P$), capturing systematic deviations beyond Horndeski, but always constructed to avoid Ostrogradsky instabilities (at most two time derivatives).
• The full scalar sector is thus parameterized by a finite set of background functions, directly tied to theoretical microphysics or specified phenomenologically in data analysis (Lagos et al., 2016).

3. Vector-Tensor Sector: General and Einstein–Aether Models

For theories with an explicit vector field $A^\mu$, the general quadratic action for perturbations involves:

• The vector perturbation yields two scalar modes ($\alpha, \alpha^0$), which couple to the metric perturbations. The quadratic action can be organized as

$$S^{(2)}_G = \int d^4x\,\left[{\mathcal L}^{i}_T + {\mathcal L}^{i}_{\alpha^0} + {\mathcal L}^{i}_{\alpha} + {\mathcal L}^{i}_{\alpha^0\alpha}\right], \quad i=0,1,2$$

with each Lagrangian piece capturing zero, one, or two derivatives.

• Applying gauge invariance imposes constraints on the $L_*$ and regrouped coefficients $T_*$. Generically, up to ten free functions remain for the most general quadratic action with at most two derivatives for vector-tensor models.
• In the absence of further restrictions, two scalar degrees of freedom are propagated—one associated with the usual metric scalar perturbation, the other associated with the longitudinal mode of the vector field.
• Specializing to a time-like vector (Einstein–Aether theory), a Lagrange multiplier enforces $A^\mu A_\mu = -1$, eliminating one combination of the time-component vector and the metric lapse perturbation. In this case, only one extra scalar degree of freedom survives, and the action contains just four free functions (plus the background) (Lagos et al., 2016).

4. Physical Parameter Counting and Model Classification

The SVT parametrization enables a clean classification of modified gravity models by field content, highest number of allowed time derivatives per field, and number of free background functions. For linear cosmological perturbations:

Theory Class	Max Time Derivatives	Free Functions	Propagating Scalars
General Relativity (GR)	2	1 ($M^2$)	0
Horndeski	2	4 ($M^2, \alpha_K, \alpha_B, \alpha_T$)	1
Beyond Horndeski	3	5 ($+\alpha_H$)	1
Fourth-order Scalar-Tensor	4 spatial/2 time	9+ ($+\alpha_{Qi},\alpha_P$)	1
General Vector-Tensor	2	10 ($$M^2, \alpha_T, \alpha_H, \alpha_{D1-3}, \alpha_{V0-3}$$)	2
Einstein–Aether	2	4	1

This transparent mapping allows for direct interpretation and implementation of constraints from cosmological observables and gravitational wave data (Lagos et al., 2016).

5. Gauge Structure and Degree-of-Freedom Counting

Gauge invariance, and possible enhancements (e.g., $U(1)$ symmetry for the vector sector), further control the propagating degrees of freedom:

• A $U(1)$ gauge invariance in the vector sector removes the longitudinal mode, reducing the number of propagating scalar modes from two to one (as in Einstein–Maxwell theory).
• Field redefinitions (such as $\tilde{\alpha} = \alpha + A B$, $\tilde{\alpha}^0 = \alpha^0 + A \Phi$) can further clarify which combinations of the perturbations remain dynamical after imposing all constraints.
• Noether identities from the imposed symmetries strictly dictate the allowed form and interrelations of coefficients in the quadratic action, and hence the possible phenomenological behaviors of SVT models (Lagos et al., 2016).

6. Cosmological Implications and Applications

The SVT representation has direct and wide-ranging applications:

• Constraining modified gravity: Cosmological perturbation observables (growth of structure, CMB anisotropies, lensing) constrain the allowed ranges of the free functions (e.g., $\alpha_K, \alpha_T$), thereby probing the underpinning SVT theory.
• Phenomenological mapping: The explicit parameterization allows translation between microphysical models (e.g., particular scalar-tensor Lagrangians) and their phenomenology at observed cosmological scales, enabling a high degree of model independence.
• Model-building: The framework is employed in constructing consistent theories with multiple degrees of freedom, including those matched to Horndeski, Beyond Horndeski, generalized Proca, and Einstein–Aether theories, and provides the backbone for effective field theory studies of dark energy, inflation, and other phenomena.

7. Summary and Outlook

The scalar-vector-tensor representation, as implemented in a parametrized quadratic action for linear cosmological perturbations, enables a systematic and comprehensive approach to exploring gravitational theories with extra degrees of freedom. By organizing perturbations according to their scalar, vector, and tensor character, and imposing symmetries via Noether constraints, the formalism yields well-defined predictions with a transparent mapping between field content, propagating modes, and phenomenological parameters. The resulting framework encompasses and generalizes Horndeski, Beyond Horndeski, vector-tensor, and Einstein–Aether models, and is ideally suited to the confrontation with current and future cosmological observations (Lagos et al., 2016).