SVT Decomposition in Cosmological Perturbations

• SVT decomposition is a method that categorizes cosmological perturbations into scalar, vector, and tensor components based on their transformation properties under spatial rotations.
• It employs both non-local harmonic techniques and local differential operators to extract gauge-invariant variables essential for CMB analysis, gravitational wave physics, and modified gravity studies.
• The framework underpins the formulation of decoupled evolution equations and facilitates degree-of-freedom counting in cosmological models and alternative theories of gravity.

The scalar–vector–tensor (SVT) decomposition provides a unique, gauge-invariant organization of cosmological perturbations by exploiting the background spatial symmetries of homogeneous, isotropic spacetimes. Metric and matter field perturbations are separated into modes according to how they transform under three-dimensional (spatial) rotations—or, in alternative approaches, under full four-dimensional general coordinate transformations. This procedure plays a central role in cosmological perturbation theory, the theory of linearized gravity in modified gravity scenarios, and in constructing gauge-invariant observables for cosmic microwave background (CMB) and large-scale structure (LSS) analyses, as well as in formulating well-posed evolution equations for gravitational wave physics.

1. SVT Decomposition on Homogeneous, Isotropic Backgrounds

The SVT “split” classifies perturbations through their transformation properties under the 3-dimensional rotation group. Around a spatially flat Friedmann–Robertson–Walker (FRW) metric,

$ds^2 = -dt^2 + a^2(t)d\vec{x}^2,$

any symmetric rank-2 perturbation $h_{ij}$ (the spatial part of $\delta g_{\mu\nu}$) decomposes uniquely as follows (Lagos et al., 2016, Phelps et al., 2019):

• Scalars: $\phi, \psi, B, E$ (4 functions)
• Transverse vectors: $S_i, F_i$ with $\partial^i S_i = \partial^i F_i = 0$ (2+2 functions)
• Transverse–traceless tensors: $h_{ij}$ with $\partial^i h_{ij} = h^i{}_i = 0$ (2 functions)

The explicit form for the full metric perturbation is

$$\begin{aligned} \delta g_{00} &= -2\phi, \ \delta g_{0i} &= \partial_i B + S_i, \ \delta g_{ij} &= a^2\left[ -2\psi\delta_{ij} + 2\partial_i \partial_j E + \partial_i F_j + \partial_j F_i + h_{ij} \right], \end{aligned}$$

with the constraints detailed above. Under spatial diffeomorphisms, these variables transform such that $h_{ij}$ remains gauge-invariant, while combinations like $\hat{\alpha}^0 = \alpha^0 + A\phi$ and $\hat{\alpha} = \alpha + A B$ may be constructed for vector fields to remain invariant (Lagos et al., 2016).

The decomposition organizes degrees of freedom for both gravity and matter fields, providing a clean basis for writing, analyzing, and counting propagating modes.

2. Gauge-Invariant Variables and the SVT Extraction Operators

Gauge-invariant combinations are essential for physical interpretation and for the decoupling of equations at linear order. For the metric perturbation, under an infinitesimal coordinate shift, six independent gauge-invariant combinations may be constructed (Phelps et al., 2019): $$\Phi = \phi + \dot{B} - \tfrac{1}{2} \ddot{E}, \quad \Psi = \psi - H(B - \dot{E}), \quad Q_i = B_i - \dot{E}_i, \quad E_{ij}$$ with $H = \dot{a}/a$.

The extraction of SVT modes from arbitrary rank-2 (projected, symmetric, trace-free, PSTF) tensors can be non-local if performed via harmonic expansion or elliptic operator inversion, requiring boundary data. Clarkson and Osano (Clarkson et al., 2011) defined fully local extraction operators:

• Scalar extractor: $\mathcal{S}[X] = \nabla^a\nabla^b X_{ab}$
• Vector extractor: $$\mathcal{V}_a[X] = \varepsilon_a{}^{cd}\nabla_c\nabla^b X_{bd}$$
• Tensor extractor: $$\mathcal{T}_{ab}[X] = [-\Delta + 2K + 2\,\text{Dis\,Div}]\,[\text{curl\,}X]_{ab}$$ These operators annihilate all but their target mode and commute with the Laplacian and with linear wave operators, enabling local, gauge-invariant extraction of SVT content at arbitrary order (Clarkson et al., 2011).

3. The Decomposition Theorem: Decoupling and its Limitations

The decomposition theorem states that, in linearized gravity on a homogeneous background, the equations for scalars, vectors, and tensors decouple, permitting independent solutions in each sector. This holds generically only at the level of higher-derivative (typically fourth-order) combinations. For decoupling at the level of the linearized field equations themselves, boundary conditions are crucial:

• For flat or de Sitter ($K = 0$) backgrounds, vanishing of gauge-invariants at spatial infinity is sufficient.
• For $K \neq 0$ (curved) Robertson–Walker backgrounds, regularity at the origin in addition to asymptotic boundary conditions is required (Phelps et al., 2019).
• In more general backgrounds, full separation at second order for each SVT sector is obstructed unless additional initial data or source terms are imposed.

In the absence of boundary conditions, the gauge-invariant combinations isolating pure SVT modes solve fourth-order spatial equations: $$(\nabla^2 + 3K)(\Phi - \Psi) = 0, \quad (\nabla^2 - 2K)Q_i = 0, \quad (\nabla^2 - 2K)(\nabla^2 - 3K)E_{ij} = 0$$ with direct decoupling requiring imposition of spatial infinity and/or regularity constraints (Phelps et al., 2019).

4. Alternative SVT Bases: Four-Dimensional Covariance and TAM Formalism

The standard SVT decomposition respects only 3D rotational invariance. A manifestly four-covariant decomposition expresses the metric perturbation as

$$h_{\mu\nu} = -2g_{\mu\nu} X + 2\nabla_{(\mu} F_{\nu)} + 2F_{\mu\nu}$$

where $X$ is a scalar, $F_\mu$ a transverse four-vector ($\nabla^\mu F_\mu = 0$), and $F_{\mu\nu}$ a transverse-traceless tensor (Phelps et al., 2019). In this SVT₄ basis, the linearized Einstein equations become compact, but decoupling at the second-order field equation level does not occur in general backgrounds without further data.

The total angular momentum (TAM) formalism (Dai et al., 2012) constructs SVT modes as eigenfunctions of both the Helmholtz operator and total angular momentum, which is advantageous for analyses involving the spherical sky (e.g., the CMB). Scalar, vector, and tensor fields are written as expansions over TAM waves, each generated via application of differential operators on scalar TAM eigenfunctions:

• Vectors: Decomposed into longitudinal (L), divergence-free electric (E), and magnetic (B) parity modes.
• Tensors: Further into longitudinal (L), two vector-type (VE, VB), and two tensor (TE, TB) modes.

This basis guarantees orthonormality, completeness, and straightforward connections to observable angular power spectra.

5. SVT Decomposition in Modified Gravity and Degree-of-Freedom Counting

In theories beyond general relativity—such as Horndeski, beyond-Horndeski, and vector-tensor scenarios—SVT decomposition remains central for classifying propagating and constrained modes and for constructing quadratic actions for linear perturbations (Lagos et al., 2016). Explicitly:

• Scalar–tensor and vector–tensor actions organize into individual SVT sectors at linear order, each sector parameterized by time-dependent effective masses and “α-functions”.
• Degree-of-freedom counting follows: for a theory with $N_s$ scalar, $N_v$ vector, and $N_t$ tensor variables, the physical DoFs are given by the variables minus the number of constraints (auxiliary fields) and gauge redundancies. For instance, scalar–tensor Horndeski theories with $\{ \phi, B, \psi, E, \delta\chi \}$ have φ and B auxiliary, two diffeomorphism gauge freedoms, and propagate one scalar DoF.
• Vector–tensor theories propagate additional vector and/or scalar DoFs depending on constraints, e.g., in Einstein-Aether, an additional Lagrange multiplier removes the α^0 variable in favor of φ, leaving a single dynamical scalar.

These features allow a scheme to parameterize linear perturbations in a wide class of modified gravity models via a small set of time-dependent functions: mass scales and α-functions specify the physical content of each SVT sector (Lagos et al., 2016).

6. Local vs Non-Local SVT Splits and Applications

Traditional SVT decomposition relies on harmonic (non-local) methods and integration over spatial hypersurfaces with fixed spatial curvature, making mode extraction inherently non-local and dependent on boundary conditions. The purely local operators constructed by Clarkson and Osano enable extraction of scalar, vector, and tensor modes directly from the tensor fields via differentiation, without recourse to non-local inversions or boundary data. These operators preserve gauge invariance and commute with all relevant evolution operators, making them particularly powerful for constructing gauge-invariant observables at second and higher perturbative order, with direct relevance for the study of non-linear cosmological perturbations (Clarkson et al., 2011).

In observational cosmology, SVT decomposition underpins the classification and extraction of CMB anisotropies (E-/B-modes), gravitational lensing effects, and stochastic gravitational wave backgrounds by relating observables to irreducible SVT modes of underlying perturbation fields (Dai et al., 2012).

7. Summary and Theoretical Significance

The scalar–vector–tensor decomposition is foundational for the analysis of cosmological fluctuations, gravitational wave physics, and modified gravity. It organizes perturbations into irreducible representations tied to background symmetries, enabling gauge-invariant mode separation. While true second-order decoupling across SVT sectors—the decomposition theorem—depends on background spacetime and the enforcement of appropriate boundary or initial conditions, the methodology is robust. Local extraction techniques and alternate bases such as the manifestly covariant SVT₄ decomposition and the TAM wave formalism augment the versatility of the SVT approach. The parametrization of quadratic actions for linearized theories via SVT variables and the associated counting of physical degrees of freedom enables systematic exploration of the theoretical landscape of gravity, both in general relativity and in its generalizations (Phelps et al., 2019, Lagos et al., 2016, Clarkson et al., 2011, Dai et al., 2012).